Fermi seas from Bose condensates in Chern-Simons matter theories and a bosonic exclusion principle

We generalize previously obtained results for the (all orders in the 't Hooft coupling) thermal free energy of bosonic and fermionic large N Chern-Simons theories with fundamental matter, to values of the chemical potential larger than quasiparticle thermal masses. Building on an analysis by Geracie, Goykhman and Son, we present a simple explicit formula for the occupation number for a quasiparticle state of any given energy and charge as a function of the temperature and chemical potential. This formula is a generalization to finite 't Hooft coupling of the famous occupation number formula of Bose-Einstein statistics, and implies an exclusion principle for Chern-Simons coupled bosons: the total number of bosons occupying any particular state cannot exceed the Chern-Simons level. Specializing our results to zero temperature we construct the phase diagrams of these theories as a function of chemical potential and the UV parameters. At large enough chemical potential, all the bosonic theories we study transit into a compressible Bose condensed phase in which the runaway instability of free Bose condensates is stabilized by the bosonic exclusion principle. This novel Bose condensate is dual to - and reproduces the thermodynamics of - the fermionic Fermi sea.


Introduction
There is now considerable evidence to support the conjectured Bose-Fermi duality between families of 2 + 1 dimensional Chern-Simons gauge theories coupled to scalars and (roughly level-rank) dual Chern-Simons gauge theories coupled to fermions . In the simplest large N examples with the matter fields in the fundamental of the gauge group, the 'elementary' bosonic fields on the LHS of the duality create stable bosonic particles that map, under duality, to the fermionic particles created by the 'elementary' fermionic fields on the RHS of the duality 1 .
That bosons and fermions can be dual to each other in this elementary fashion may seem surprising. In most familiar contexts, fermions have half-integer spin while bosons have integer spin 2 . While this distinction is completely sharp in D ≥ 4 dimensions, it, 1 In this sense the 2 + 1 dimensional dualities at large N are qualitatively different from famous examples of bosonization in 1 + 1 dimensions, in which particles created by the elementary fermionic field ψ map to particles created by a complicated composite (sometimes called solitonic) bosonic operator e φ . This 1+1 dimensional duality is, in some respects, more similar to the N = 1 case of the 2+1 dimensional dualities [36][37][38] in which elementary fermions map under duality to vortices bound to a single boson. The role of the vortex is analogous to the role of the Jordan-Wigner 'string' in the 1+1 dimensional context. We thank D. Radicevic for a discussion on this point. 2 Bosonisation is less surprising in 1 + 1 dimensions as there is insufficient room to rotate in a one dimensional space (more formally the massive little group, SO (1), is trivial) and so massive excitations do not carry a spin quantum number.
however, blurs out in D = 3 where particle spins are allowed to take arbitrary value 3 . Indeed, it turns out that the coupling to a Chern-Simons gauge field renormalizes the integer (resp. half-integer) spins of the bosons (resp. fermions) to general real values. For various specific conjectured Bose-Fermi dualities one can check that the renormalized boson spin does indeed match the renormalized fermion spin across the duality [81]. The matching of single particle spins does not address all paradoxes. Given the differences between Bose and Fermi statistics, we must still understand how the spectrum of multi-particle states can match across the duality. This question has already been investigated in the context of two particle scattering states in [20]. At least in large N limit, it turns out that the key to answer this lies in the colour quantum numbers of the bosonic and fermionic states 4 .
Let us recall that the gauge groups are different on the bosonic and the fermionic sides of the duality 5 . As the elementary bosons and fermions each transform in the fundamental representations of their respective gauge groups, it is clear that all single particle quantum numbers cannot match across the duality. While the 'physical' spacetime quantum numbers (mass and spin) match precisely, the 'unphysical' gauge quantum numbers lie in different spaces and so cannot be identified.
Of course, this 'mismatch' does not necessarily falsify duality. Suppose we work on three dimensional spacetime which is S 2 × R. 6 Gauss' law then forces the physical states to be colour singlets. Charged single-particle (or multi-particle) states are not by themselves physical, but are instead the building blocks for physical states. It was demonstrated in [20] that, under duality, two-particle states that are symmetric (antisymmetric) in colour indices map to two-particle states that are antisymmetric (symmetric) in colour. Thus, while dual pairs of bosonic and fermionic two-particle states have opposite statistics under the interchange of all quantum numbers, they have identical statistics under the interchange of all physical -i.e. non-gauge -quantum numbers 7 . We see that, as promised, the 'hidden' 3 Massive particles in D dimensions transform in representations of the little group SO(D − 1) -or more precisely Spin(D − 1), the covering group of SO(D − 1). For D ≥ 3, this group is non-abelian and its unitary representations are discrete. Fermions transform in spinorial representations (those in which a 2π rotation is implemented by the operator −1) while bosons transform in discretely distinct faithful representations (those in which a 2π rotation is implemented by the operator 1). In D = 3, this little group is R (the covering group of SO (2)). The unitary representations of R are labelled by one continuous parameter (the spin, an arbitrary real number) rather than a collection of discrete parameters as in higher dimensions. 4 We have ignored the fact that the coupling to the Chern-Simons gauge field modifies the statistics of both bosonic and fermionic excitations. This is consistent in the large N limit as the effective anyonic phase in the scattering of two fundamental fermions or bosons is of order 1/N [20]. At finite values of N (in particular when N = 1) the modification of statistics plays a key role in reconciling the difference between bosonic and fermionic statistics and cannot be ignored. 5 SU (NB) or U (NB) for the bosons, and SU (NF ) or U (NF ) for the fermions in the situation studied by [33,40]. In general NF = NB. 6 This is one way of regulating the spatial R 2 in the IR. Another way would be to replace R 2 with a disk of large radius. This is a more complicated regulator as it introduces boundary degrees of freedom -roughly the boundary WZW modes. We will not study such a regulator in this paper. 7 So, for example, a two-particle fermion state that is antisymmetric in colour indices is forced, by Fermi statistics, to be symmetric under the interchange of all physical (non-gauge) quantum numbers like momenta. Its dual two-particle boson state is symmetric in colour indices and so is also forced by Bose statistics to be colour indices are the key to reconciling duality with the difference between Bose and Fermi statistics.
In this paper we study four families of quantum field theories: the quasi-fermionic theories which are the regular fermion and the critical boson, and the quasi-bosonic theories which are the regular boson and the critical fermion 8 . In our study of the quasi-fermionic theories, we adopt the conventions and regularization scheme of [64] (see Section 1.1 of that paper including footnotes [5][6][7][8]. Similarly, in our study of quasi-bosonic theories, we adopt the conventions and regularization scheme of [70] (see Section 2.1 of that paper). The Euclidean actions for these theories are listed in Section 2.1.
We will address three related but distinct questions. We first revisit the question of how Bose-Fermi duality can be reconciled with the difference between bosonic and fermionic statistics, but this time working with thermodynamic systems with macroscopic numbers of particles rather than few-particle scattering states. Specialising this discussion to zero temperature, we then investigate how Bose condensates can replicate the behaviour of Fermi seas. Finally, we perform a detailed quantitative study of the zero temperature, finite chemical potential phase diagrams of the critical boson and regular boson theories (and hence, by duality, of the regular fermion and critical fermion respectively). This study is related to the previous question because Bose condensates (and their dual Fermi seas) play a key role in these phase diagrams. In the rest of this introduction we briefly describe our results on each of these counts in turn.

Reconciling Statistics with Duality
How do bosonic and fermionic thermodynamic ensembles 9 manage to exhibit the same physics despite their apparent difference in statistics? A study of this question was initiated in the pioneering article of Geracie, Goykhman and Son [32]. The authors of [32] studied the analytically tractable theory of regular fermions coupled to SU (N ) Chern-Simons theories in the large N limit 10 . The effective excitations in these grand canonical ensembles at finite temperature and finite chemical potential had previously been demonstrated to be symmetric in all physical (non-gauge) quantum numbers. 8 The names quasi-fermionic and quasi-bosonic [7,8] indicate only the following: by definition the large N single-trace operator spectrum of quasi-fermionic theories is identical to the same spectrum of operators in a free, gauged, large N fermion theory. Similarly, the large N single-trace operator spectrum of quasi-bosonic theories is identical to the same spectrum of operators in a free, gauged, large N bosonic theory. The names carry no other significance. In particular, there is no sense in which all quasi-fermionic theories are 'almost fermionic'. As an example, the ungauged large N bosonic Wilson-Fisher theory -certainly a bosonic theory from every other point of view -is quasi-fermionic with our definitions. We hope this terminology will not prove confusing. We thank A. Karch for a discussion on this point. 9 Our systems are defined on an S 2 with effective volume V , where the volume is large (V = V2N ). While, in general, V2 is held fixed in the large N limit, in some parts of our analysis we also study the limit V2 → ∞. 10 In the large N limit that we study throughout this paper, the thermodynamics of SU (N ) and U (N ) theories are effectively identical, even though the nature of the U (1) global symmetries of SU (N ) theories and U (N ) theories, at first sight, appear to be qualitatively different. We explain how this works in Appendix B.
parametrically long-lived fermionic excitations with a dynamically determined mass 11 . The authors of [32] (see Section 4 below for a review and generalisations) demonstrated that the large N exact charge of the ensemble with temperature β −1 and chemical potential µ in the infinite volume limit is that of a non-interacting gas of these fermionic quasiparticles, with the average occupation number of each quasiparticle state with energy and charge q given bȳ (λ F is the 't Hooft coupling defined in (2.7); the first line of (1.1) was presented in [32]; the second line of (1.1) is derived in Section 5.1 of this paper). The equation ( and (1.1) reduces to (1.2) in the limit λ F → 0. What is the bosonic analogue of (1.1)? In addressing this question 12 , we face the following obstacle. The free energy of the bosonic theories (see (2.4) and (2.5) for the Lagrangians) and the fermionic theories (see (2.1) and (2.3) for the Lagrangians) have previously been computed only for a range of temperatures and chemical potentials: those values of these parameters for which the quasiparticle thermal masses turn out to be larger than the chemical potential 13 . As the first step in the analysis of this paper we remedy this technical defect in (see Section 3 below) by generalising previously obtained results to all values of the chemical potential and temperature.
The method we employ is to analytically continue previously computed results for the 'off-shell free energy' of matter Chern-Simons theories in the chemical potential. This procedure makes sense because off-shell free energy depends on the chemical potential only through a finite temperature determinant; as this determinant is effectively two dimensional we do not expect it to undergo a sharp phase transition as µ exceeds the thermal mass, see Section 3.1. 14 This analytic continuation is conveniently accomplished by modifying a contour of integration in the space of holonomy eigenvalues. The final results of this exercise are listed in detail in (3.35) and (3.36) for the bosonic theories in the infinite volume limit. For general finite volume phases, the formulas are in (3.46), (3.47) for the bosonic theories, and (3.44) and (3.45) for the fermionic theories.
Armed with exact results for the relevant thermodynamical formulae, it is not difficult to imitate the analysis of [32] for bosonic Chern-Simons-matter theories. Once again we find that the large N exact charge of the ensemble of temperature β −1 and chemical potential µ in the infinite volume limit 15 is that of a non-interacting gas of these quasiparticles, with the average occupation number of each quasiparticle state of energy and charge q given bȳ n B ( , µ) where Θ(x) is the Heaviside step-function. As above, (5.21) may be regarded as the generalisation to finite 't Hooft coupling λ B of the standard formula of Bose-Einstein statisticsn For quasiparticle energies > qµ, (1.3) reduces to (1.4) in the limit λ B → 0. Unlike (1.4), however,n B ( , µ) listed in (1.3) is positive at all values of (even when < qµ). In the limit λ B → 0,n B ( , µ) given in (1.3) diverges; this is the physically correct answer for a free Bose ensemble at qµ > (see below for more discussion). The modified occupation numbers (1.1) and (1.3) are one-parameter generalisations of the standard formulae (1.2) and (1.4) for fermions and bosons respectively, with the parameter being the 't Hooft coupling λ F or λ B respectively. As remarked in Footnote 4, the anyonic phase obtained by exchanging two particles with each other or two antiparticles with each other is proportional to 1/N and hence is negligible in the large N limit. 16 However the ensembles we study in this paper contain a thermodynamic number of particles since we are working in the large N and large volume limit of [15]. Consequently, the modified occupation number formulae (1.1) and (1.3) are presumably a consequence of these anyonic phases, whose net effect builds up to an O(1) number. It would be very interesting to understand this better, perhaps in the context of a Schrodinger type description of the non relativistic limit of this system, see [81].
(1.5) 15 See Section 3.4.3 for the results that apply in more general phases that dominate away from the infinite volume limit. 16 In contrast the anyonic phase obtained in taking a particle around an antiparticle in the so called singlet channel [20] is given π|λF | or π|λB| (for the fermions or bosons respectively), and so is of order unity in the large N limit. 17 One version of Bose Fermi duality relates SU (NF ) theories to U (NB) theories. While the chemical potential on the fermion side is standard, the chemical potential on the bosonic side couples to the topological U(1) symmetry of this theory. As we explain in Appendix B, at finite NB there is a sense in which out the dynamical U (1) gauge field of the U (NB) theory turns this deformation into a standard chemical potential plus a standard background magnetic field for the left over SU (NB) theory. As we explain in Appendix B, The fact thatn F ( , µ) andn B ( , µ) are not simply equal to each other may, at first, seem to contradict Bose-Fermi duality. In fact, the opposite is true. Equation (1.5) is precisely what is needed in order to ensure that the net occupation number of a particular fermionic state matches the net occupation number of bosonic state once we sum over the 'invisible' colour quantum numbers of these states. Effective occupation numbers as a function of physical quantum numbers agree exactly across the duality once we sum over the unphysical gauge quantum numbers. As in the case of two-particle scattering states reviewed earlier in this introduction, the 'invisible' colour quantum numbers play a key role in reconciling statistics with duality.

Fermi seas from Bose condensates
It is useful to focus attention on the ensemble in which the dichotomy between Bose and Fermi statistics intuitively appears most pronounced viz. finite chemical potential and zero temperature. In this limit we expect the fermions to arrange themselves into a Fermi sea while the bosons are expected to form a Bose condensate. We now explain how these two rather different phases manage to be dual to each other.
In the zero temperature limit 18 the formula for the Fermionic occupation number (1.1) simplifies ton F ( , µ) = Θ(qµ − ) , (1.6) demonstrating that the fermionic phase is a vanilla Fermi sea at every value of λ F (this was already noted in [32]). On the other hand the formula for the bosonic occupation number (1.3) simplifies in the zero temperature limit tō (1.7) In fact, the state described by (1.7) is a regularized Bose condensate. Recall that in a free bosonic theory with λ B = 0, the partition function Tr e −β( −qµ) defines an ensemble which infinitely populates 19 every single-particle state with energy < qµ 20 . As a consequence, the ensemble with qµ strictly larger than the thermal mass c B is simply ill-defined in the free however, this mixing goes away in the large NB limit. In this limit there is no difference between the U (NB) and SU (NB) theory and the duality maps a 'standard' fermionic chemical potential to a 'standard' bosonic chemical potential. 18 All statements here are accurate when we first take N to infinity and then take the temperature to zero.
It is possible that the phase obtained by first taking T to zero and then taking N to infinity has additional complications. See Section 8 for additional remarks. 19 To forestall confusion we recall that the standard discussion of Bose condensates in free theories involves taking the limit µq → mB (mB is the boson mass) from below. This limit populates the boson zero mode in a macroscopic manner. qµ is never taken above mB, precisely because the free ensemble with qµ > mB infinitely populates some states -including the zero mode -and so yields an ill defined ensemble. We thank G. Mandal for a related discussion. 20 Analytic continuation of this formula from values of qµ < to values of qµ > gets rid of the divergence, but evaluates to an unphysical negative number (see (1.4)).
theory. In agreement with this fact,n B ( , µ) diverges 21 when < µq in the limit λ B → 0. This divergence is regulated at nonzero λ B . Quite remarkably, in the theories studied in this paper the resultant 'non-perturbative' phase is remarkably simple; it can be described by a collection of non-interacting quasi particles with occupation numbers given by (1.7). We see that the regulated Bose condensate (1.7) is very similar to a Fermi sea with one key difference; every state with energy less than qµ in this phase is occupied not once (as in the case of a Fermi sea) butn times (see (1.7)). As in the previous subsection, the difference in occupation numbers between the Fermi sea and the Bose condensate is perfectly consistent with duality. It is easily verified (using the definition of the 't Hooft couplings (2.7) and the duality map (2.8)) that and hence the occupation numbers agree once we sum on both sides over 'invisible' colour quantum numbers.

Bosonic Exclusion Principle
The discussion above suggests that the large N bosonic matter Chern-Simons theory appears to obey an effective exclusion principle. As the upper limit on the occupation number of single particle states 22 ,n (see around (2.6) for definitions) is not an integer, this exclusion principle does not operate at the level of single particle states with given colour quantum numbers. Instead, it asserts the total number of bosons one can add in any given single particle state after summing over all colour indices is the integer N Bn = |k B |. We pause to explore this result a little further. Consider a collection of M bosons each of which occupies the same (not counting colour quantum numbers) single particle state. As the net wave function of these particles is symmetric under interchange, and as all non-colour quantum numbers are identical for each of the particles, the wave function of these M particles must be symmetric in colour. In other words, these M particles transform in the SU (N B ) representation with a single row and M columns in the Young Tableaux as below The exclusion principle above asserts that the length of the Young Tableaux row cannot exceed k B .
The restriction on representations described in the previous paragraph is, of course, familiar. Recall that all integrable representations of SU (N B ) k B WZW theory are described 21 The formula (1.7) applies both to the regular boson theory and the critical boson theory. It follows that the critical boson self-interactions are insufficient to cure the µ > cB divergence of the free boson. 22 We thank J. McGreevy for a very useful discussion on the topic of this subsection.
with Young Tableaux have with no more than k B columns. It seems very likely to us that this well-known property of WZW theory will turn out to be the underlying explanation for the bosonic exclusion principle. The discussion of the previous paragraph fits very neatly with the ideas of level-rank duality. Recall that M fermions in a given single particle state (ignoring colour quantum numbers) transform in an SU (N F ) representation with a single column with M rows as below The fact that M ≤ N F is just a statement of the Fermi exclusion principle. However, the representation with a single column and M rows maps, under level-rank duality, to a representation with a single row and M columns. The restriction M ≤ N F is dual to M ≤ |k B |, i.e. the restriction described in the previous paragraph. We hope to return to this important and interesting point in future work.

Zero temperature phase diagrams at finite chemical potential
The exact (to all orders in 't Hooft coupling) large N results for the thermal free energy at finite chemical potential described above become completely explicit and fairly simple in the zero temperature limit 23 . We now describe the detailed and quantitative phase diagrams for the bosonic and fermionic theories (see (2.4), (2.1), (2.5) and (2.3) for the Lagrangians) at zero temperature but arbitrary chemical potential (see Sections 6 and 7 below) that follow from these results.
Before presenting our quantitative phase diagrams, it is useful to first address a preliminary qualitative question. What is the precise nature of the charged bosonic excitations that condense to form the Bose condensate? This question is more subtle than it might first seem, as we now explain.
At zero chemical potential and temperature, a SU (N B ) k B Chern-Simons gauge theory coupled to fundamental bosons has two distinct phases as a function of the UV parameters. The critical boson theory (2.4) has one UV parameter: the mass deformation m cri B . At zero temperature and chemical potential, this theory lies in the unHiggsed phase m cri B > 0 but in the Higgsed phase for m cri B < 0 (m cri B is the mass parameter defined in (2.4)). There is a sharp second order phase transition between these two phases at m cri B = 0. The situation is qualitatively similar but quantitatively more complicated in the regular boson theory and will be addressed below.
The zero temperature, zero chemical potential phase diagram as a function of m cri the spin one W -bosons. The transmutation of degrees of freedom from scalars to vectors is a consequence of the Higgs mechanism. Given that the charged excitations are qualitatively distinct in the two phases depicted in Figure 1, it might thus appear that there are two distinct Bose condensed phases; the condensate of scalar excitations about the unHiggsed vacuum and the condensate of spin one W -bosons around the Higgsed vacuum. At the level of calculations this is true at intermediate steps; the computations that determine the thermodynamics of these two phases appear completely different. Remarkably, it turns out that the final results for the thermodynamics in these two phases -at least at leading order in large N -turn out to be analytic continuations of each other. This result holds in both the critical boson theory (2.4) and the regular boson theory (2.5). It follows that these two apparently distinct Bose condensates are simply different but equivalent descriptions of the same physical state 24 .
The phase diagram of the critical boson theory as a function of its mass deformation parameter m cri B at zero temperature but fixed nonzero µ, is depicted in Figure 2. As and denotes a change in description from that a phase of condensed scalars to one of phase of condensed W -bosons though the condensed phase is itself unique. discussed in the previous subsection, the fermionic dual of the Bose condensed phase is a Fermi sea phase. The absence of phase transitions within the Fermi sea phase as the effective fermion mass passes through zero is hardly a surprise. Thus, Bose-Fermi duality can be thought of as a sort of 'explanation' for the lack of an invariant distinction between the Bose condensate of scalars and W bosons.
In the rest of this introduction we describe the more intricate phase diagram of the regular boson theory (2.5) (and so, of its dual, the critical fermion theory (2.3).) Let us first recall that in the large N limit, in addition to λ B , the regular theory is labelled by the dimensionless U V parameter x B 6 , the (mass-)dimension one parameter b 4 and the dimension two parameter m 2 B . These correspond to the sextic, quartic and mass couplings respectively. At zero temperature and zero chemical potential, and at fixed λ B and x B 6 , the phase diagram of this theory is a function of b 4 and m 2 B subject to the scaling equivalence One way to incorporate the above equivalence while also keeping track of the signs of m 2 B and b 4 is to consider the following section of the (m 2 B , λ B b 4 ) plane: The phase diagram is then the above ellipse-like curve. We present this phase diagram (which was worked out in great detail in [70], [71]) in Figure 3. When the chemical potential . Phase diagram for the regular boson theory at zero temperature and chemical potential for the stable range φ h < x B 6 < φ u of the marginal parameter x B 6 (refer to (7.14) for the definitions of φ h and φ u ). The letter u stands for the unHiggsed phase while the letter h stands for the Higgsed phase. The blue dots are second order transitions while the green dots are first order transitions. The blue lines are is non-zero, the phase diagram of the regular boson theory is two dimensional rather than one dimensional. It is efficiently parametrized by the two dimensionless variables In the stable range of parameters (φ u < x 6 < φ h , see equation (7.14) for the definition of φ u and φ h ) the phase diagram of our theory takes one of three qualitatively distinct forms depending on the precise range in which x B 6 lies. These three phase diagrams have been sketched in Figure 4 below.
The phase diagrams in Figure 4 allow one to determine the thermodynamical behaviour of any given regular boson theory (i.e. a theory at given fixed values of UV parameters) as a function of the chemical potential (see Section 7.6 for details). When the chemical potential is small, the theory lies in the uncondensed phase 25 -unHiggsed or Higgsed -as 25 Note that the quantity plotted on the y axis scales like 1 |µ| while the quantity plotted on the x axis scales like 1 |µ| 2 . Going to small µ at fixed values of UV parameters therefore corresponds to moving along a parabola of the schematic form y = α √ x. Every such parabola moves into an uncondensed phase at infinity (corresponding to small |µ|). See Section 7.6 for details.
Refer to (7.14) for the definitions of φ u and φ h . The blue lines are second order transitions while the green lines are first order transitions. Actual numerical plots of the above phase diagrams can be found in Section 7.7 in Figures 20,21 and 22. depicted in Figure 3 (b). As the (modulus of the) chemical potential is increased, for every value of UV parameters, the theory undergoes a single phase transition into the condensed phase and then stays in this phase at every higher value of the chemical potential. The phase transition is sometimes of second order and sometimes of first order depending on the precise value of UV parameters.

The Regular Fermion
The Regular Fermion theory is defined by the Lagrangian (see e.g. [64]) where the Chern-Simons levelκ F is given bỹ Note that the second term is subleading in the 't Hooft limit; hence, we shall ignore this O(1) shift in all subsequent calculations. The same comments apply for the critical fermion action below.

The Critical Fermion
The critical fermion theory is defined by the Lagrangian (see e.g. [70]) where ζ F is an auxiliary field.

The Critical Boson
The critical boson theory is defined by the Lagrangian [64] where σ is an auxiliary field.

The Regular boson
The regular boson theory is defined by the Lagrangian [70] The dimensionless parameters x F 6 and x B 6 run under the renormalization group. However the beta functions of these parameters are of order 1 N (see [69] for a detailed study) and so vanish in the strict large N limit. In the strict large N limit it is thus consistent to treat x F 6 and x B 6 as freely tunable parameters -which one can later set equal to any desired values (e.g. values at the fixed points of beta functions). This is the attitude we adopt in this paper. See [69] and [70] for further discussion.

Symmetries and Operators
Each of the theories described above enjoys invariance under a U (1) global symmetry. The precise nature of this symmetry depends on whether the gauge group of the theory in question is SU (N ) or U (N ). In the case that the gauge group is SU (N ), the symmetry simply corresponds to a uniform global phase rotation of all fundamental fields and a simultaneous uniform global rotation of all antifundamental fields by the inverse phase. In the case that the gauge group is U (N ), the global symmetry is the 'topological U (1) symmetry' corresponding to the U (1) part of the U (N ) gauge group. In this paper we will turn on chemical potentials corresponding to these global symmetries. See Appendix B for more details about these symmetries.
At first sight the global symmetries of the SU (N ) and U (N ) theories appear to be rather different from each other. In Appendix B we argue, however, that this distinction goes away in the large N limit. For this reason, the thermodynamical formulae presented in this paper apply equally well to both the SU (N ) and the U (N ) theories.
At large N the gauge invariant operators for each of the theories listed above falls into two classes 26 . The first class consists of products of single trace (or more accurately single sum) operators. A single sum operator is given by the colour contraction of (derivatives acting on) a fundamental field and (derivatives acting on) an antifundamental field. Such operators generically have dimensions of order unity and are all uncharged under the U (1) global symmetries that each of these theories enjoys.
In the case of the SU (N ) theories, the second class of operators is given by products of 'baryons' (operators with, for instance, (derivatives of) N fundamental fields whose colour indices are contracted with an SU (N ) Levi Civita tensor). In the case of U (N ) theories the second class of operators consists of monopoles. In the simple case of a conformal theory these monopole operators are best understood via the state operator map. These operators correspond to states with one unit of flux, in, say, one of the diagonal entries of the U (N ) matrix. The Chern-Simons equation of motion forces this configuration to be dressed by k quanta of the fundamental field. Here k is the Chern-Simons level, see the next subsection. See [31,33] for more details of these operators.
This second class of operators are 'heavy'; their scaling dimensions are of order N in the large N limit. These operators -unlike those of the first class -are charged under the global U (1) symmetries. Through this paper we work with global U (1) symmetries normalised so that (roughly speaking) fundamental fields carry unit charge under this symmetry. This means, for instance, that the simplest baryonic operators carry charge N under this symmetry. Note that the U (1) charges of these operators, like their scaling dimensions, are of order N in the large N limit.

A note on the Chern-Simons levels
We pause here to elaborate on the notation used in the actions (2.1), (2.3), (2.4) and (2.5) above.
When the fermions or bosons in the theories above are all massive, the low energy effective dynamics of our system is governed by a pure Chern-Simons theory. In the case of the bosonic (resp. fermionic) theories we use the symbols k B (resp. k F ) to denote the level of the SU (N ) part of the WZW theory dual to the pure Chern-Simons theory obtained after integrating out the matter fields with a positive mass. The symbols κ B and κ F denote the 'renormalized levels' defined by (2.6) 26 We thank T. Senthil and A. Vishwanath for discussions on this point.
In this paper we will study the theories above in the large N limit. The true coupling constant of these theories in this limit is the 't Hooft couplings λ B (resp. λ F ) defined by See [64] and [70] for further details of notations and conventions.

The conjectured Bose-Fermi duality map
The quasi-fermionic theories i.e. regular fermion (2.1) and critical boson (2.4) theories have been conjectured to be dual to each other. Similarly, the quasi-bosonic theories i.e. the critical fermion (2.3) and regular boson (2.5) theories also have been conjectured to be dual to each other. For each of these conjectured dualities, the map between the renormalized levels and ranks of the CS gauge groups is given by The above relationship (2.8) is conjectured to be exact. In particular, they are precisely the rules for the well-established level-rank duality of WZW theories that are dual to the pure Chern-Simons theories that we obtain after integrating out the massive matter fields. From (2.7) it follows that the 't Hooft couplings of the dual pairs of theories are related by In addition to the Chern-Simons levels and ranks, the quasi-fermionic theories are each characterised by an additional mass parameter, m F and m cri B respectively for the regular fermion and critical boson theories. In the large N limit these mass parameters are conjectured to be related via the duality map (2.10) The quasi-bosonic theories are similarly characterised by three UV parameters apart from the Chern-Simons ranks and levels. These correspond to the sextic, quartic and mass terms in the Lagrangian (see (2.5), (2.3)). In the large N limit these are conjectured to be related via the duality maps As mentioned above, while the relations (2.8) and (2.9) are expected to hold even at finite N , the relations (2.10) and (2.11) are conjectured to hold exactly at only infinite N and may receive finite-N corrections. See e.g. [64,70] for more details of the conjectured duality maps, including a detailed discussion of the levels of U (1) factors, as well as the regulation scheme in which the duality map above is conjectured to hold.

Large N thermal partition functions
All four theories described in Section 2.1 are effectively solvable in the large N limit. In particular, the free energy of each of these theories at finite temperature and finite chemical potential (in a range of chemical potentials, see below) has previously been computed at every value of the 't Hooft coupling in the large N limit. In this section we first review previously obtained results and then extend them to the range of chemical potentials of interest to this paper.
The partition function Z on S 2 × S 1 of each of these theories is given by an expression of the sort where [dU ] CS is the Chern-Simons modified Haar measure over U (N ) [14,15]. At the physical level, the matrix U is the zero mode of the gauge field holonomy around the thermal circle S 1 . V 2 is the volume of the two dimensional space S 2 and T is the temperature which is related to the radius β of the thermal circle S 1 as T = β −1 . The quantity ρ is the holonomy eigenvalue distribution function. As explained in e.g. Section 1.2.1 in [64], the zero mode of the holonomy U of the gauge field around S 1 is completely specified (upto a permutation) by its eigenvalues, e iα j where, j = 1, · · · , N and α j ∈ (−π, π]. In the large N limit, the eigenvalue locations on the circle are effectively specified by a continuous distribution function ρ(α) defined by The quantity v[ρ] that appears in (3.1) is simply the result of path integral over all nonholonomy modes including the gauge field and matter fields. Indeed, where [dφ] in (3.3) is the path integral measure over all matter and gauge degrees of freedom except the holonomy of the gauge field around the thermal circle. The subscript R 2 × S 1 indicates that the path integral is actually computed for the theory on R 2 × S 1 . The information that we are in fact on S 2 × S 1 goes into the Chern-Simons modified Haar measure [dU ] CS in (3.1). Partition functions at finite temperature and chemical potential are physical observables, and so should be the same for two dual theories. From the definition of the partition function in (3.1) it might at first seem that the requirement that thermal partition functions match across a duality requires that the quantities v[ρ] match across the duality. However that is not quite the case; recall that unitary matrices are of different ranks on the bosonic and fermionic sides of the duality, and on each side one also has to take into account the contribution from the Chern-Simons modified Haar measure over the unitary matrices in (3.1), which depends on the 't Hooft couplings λ B and λ F in a non-trivial way. Nonetheless the bosonic and fermionic v[ρ] functions must clearly be related in some way in order for the partition functions to match. It was demonstrated in [15] that the thermal partition functions given by (3.1) of the bosonic (regular or critical) and fermionic (critical or regular) theories will in fact agree with each other in the large N limit provided the functions v[ρ] obey the more subtle relationship where the holonomy distribution functions ρ B and ρ F are not equal to each other but are instead related by the equation

The off-shell free energy
In the large N limit, the path integral (3.3) is very effectively computed by the saddle point approximation. The final results for v [ρ], obtained this way in earlier work, are most conveniently presented in terms of a so-called 'off-shell free energy' [70]. The off-shell free energy is a quantity that depends on some auxiliary variables in addition to the UV parameters of the theory, the temperature, chemical potential and eigenvalue distribution ρ. It turns out that once we perform the path integral (3.3) via saddle point approximation and obtain an answer for v[ρ] at each of the various large N saddle points, the whole procedure can be replaced by a simpler one: that of performing an ordinary integral over a few auxiliary variables, denoted collectively as ϕ aux , with the integrand being a function of these variables rather than being a functional of fields: where, the object F (ϕ aux ; ρ] is the off-shell free energy per unit volume. The funny brackets (· · · ] indicate that F is still a functional of the holonomy distribution ρ while being an ordinary function of the auxiliary variables ϕ aux . The off-shell free energy F has an explicit factor of N in its expression and hence, in the large N limit, the integral above may be evaluated by saddle point approximation. The quantity v[ρ] is thus obtained from the off-shell free energy by extremizing the latter w.r.t. its auxiliary variables [70]. The existing results in the literature for the free energies v[ρ] [5, 10, 12, 14-18, 32, 64, 69-71, 77] and F (ϕ aux ; ρ] in the presence of chemical potential were carefully computed only for values of the chemical potential smaller than the thermal masses of the bosonic or fermionic excitations in the respective theories. We present these expressions for the free energy in the next subsection. It turns out that some interesting modifications are required in the expressions for the free energies in order to extend their validity for chemical potentials larger than the thermal masses. We describe these modifications in the subsection after next.

Previously known results
Note: Throughout this paper, we use the convention that, given a quantity x of mass dimension a, the corresponding dimensionless hatted quantityx is defined to be the quantity x in the units of temperature, i.e.,x = β a x. For example, the mass parameter m cri B of the critical boson theory has its dimensionless counterpartm cri B = βm cri B .

The Regular Fermion
The off-shell free energy for the RF theory, presented as a function of the two auxiliary variablesC and c F , is [70,71] F RF (c F ,C) Extremization of (3.7) w.r.t. c F andC gives, respectively, where C is a particular moment of the holonomy distribution given by The extremum value of the variable c F gives the thermal mass of the fermionic excitations which is also denoted by the same symbol c F . We do not know of a physical interpretation for the variableC except that it becomes the function C(c F , µ) on-shell 27 .
paper -is defined to have a branch cut for negative real arguments, so that the imaginary part of ln(x + iy) is +iπ just above the negative x axis, but is −iπ just below the negative x axis. In deriving the equation of motion (3.8) we have used that when x > 0 π −π dα ρF (α) log 2 cosh We use this identity once with x =ĉF +μ and once with x =ĉF −μ. The condition x > 0 is met in both cases because we have assumedĉF > |μ|. In going from the first to the second line of (3.10) we use the fact that when the log-function is defined to be real on the the positive real axis and with branch cut along the negative real axis The conditions on the arguments of z1 and z2 in (3.11) are met in going from the first to the second line of (3.10), because we have assumed x > 0. In going from the second to the third line of (3.10) we have used the fact that ρF (α) is an even function that integrates to unity.

The Critical Fermion
The off-shell free energy for the CF theory was presented in [70] (see equation (A.3) of [70]) and is given by (3.12) In (3.12), c F , ζ F andC are auxiliary quantities (w.r.t. which we have to extremize F CF in order to obtain v CF [ρ]). The gap equations that follow from this extremization process are derived and described in detail in the Section A.1 of [70]. We present the equations here for completeness. The extremization of (3.12) w.r.t.C, c F and ζ F respectively gives the following equationŝ As for the regular fermion, the extremum value of the variable c F gives the thermal mass of the fermionic excitations and the extremum value of the variableC is C(c F , µ). The variable ζ F in the critical fermion free energy (3.12) is precisely the (constant) expectation value of the auxiliary field ζ F (x) that appears in the critical fermion Lagrangian (2.3). If we (classically) integrate out the variables c F andC from the free energy (3.12) and perform the integral over the holonomy, the resultant function of ζ F is the (large N exact) quantum effective potential of the theory as a function of ζ F .

The Critical Boson
The off-shell free energy for the critical boson theory, presented in terms of the two auxiliary variables c B andS, is given by [70] F CB (c B ,S) (3.14) Extremizing the free energy (3.14) w.r.t.S and c B respectively gives where the function S is defined by (the logarithm in (3.16) is defined in exactly the same way as the logarithm in (3.9) in Footnote 27). The variable c B has a direct physical interpretation: its extremum value is the thermal mass of the bosonic excitations. We do not have a similar physical interpretation for the variableS except that it becomes the function S(c B , µ) above.

The Regular Boson
As demonstrated in [70], the off-shell thermal free energy (per unit volume βV 2 ) of the RB theory at a finite chemical potential µ is given by (3.17) In (3.17), c B , σ B andS are auxiliary quantities (w.r.t. which we have to extremize F RB in order to obtain v RB [ρ] ). The gap equations that follow from this extremization process are derived in detail in equation (4.3) of [70], and for completeness, we list these here. Extremization of (3.17) w.r.t.S,ĉ B andσ B respectively gives the following equations The quantity σ B in the regular boson free energy (3.17) also has a simple physical interpretation. It is related to the expectation value of the lightest gauge-invariant operator,φφ, of the regular theory as As for the critical boson, the variable c B has the physical interpretation that, on shell, it is the thermal mass of the bosonic excitations. The variable σ B also has an important physical origin. In fact, if we (classically) integrate out the variables c B andS in the regular boson free energy (3.17), and then substitute (3.19) for σ B and finally perform the integral over the holonomy, the resultant function of φ φ is simply the large-N -exact quantum effective potential of the theory as a function of its 'lightest' gauge invariant observable φ φ . The objectsS and S(c B , µ) are different off-shell: one is a variable while the other is a function of a different variable. However, on shell, we havẽ S = S(c B , µ) , (3.20) which is easy to see by comparing the first two equations in (3.18).

Duality of off-shell variables and off-shell free energies
Under duality, the off-shell variables have to be appropriately mapped to each other in addition to the duality identifications between the coupling constants given in Section 2.4. The duality map between the off-shell variables is as follows: It can be shown that the off-shell free energies of the regular fermion and critical boson map to each other under the Bose-Fermi duality map. Similarly, it can be shown that the off-shell free energies of the critical fermion and regular boson map to each other under duality. We relegate these calculations to Appendix C.

Conjecture for the off-shell free energy for |µ| greater than thermal mass
In Section 3.2, we presented the expressions for the off-shell thermal free energies of the four classes of theories of interest to us in this paper. The results of Section 3.2 apply whenever the thermal mass of the bosonic or fermionic excitations is larger than the modulus of the chemical potential. In this subsection, we will present a conjecture for the extension of these results to the situation in which the chemical potential is larger than the quasiparticle thermal masses. In order to motivate our conjecture, consider the case of the bosonic theories. The effective squared mass of the boson (c 2 B − µ 2 ), vanishes when µ = c B and becomes negative when µ 2 > c 2 B , encouraging the boson to 'condense'. As we will explain in detail in the next subsection, the off-shell free energies of these theories depend on the chemical potential only through a bosonic determinant. As this finite temperature determinant is two dimensional at long distances, however, we expect this 'condensation' to occur without a sharp phase transition (recall that spontaneous symmetry breaking of continuous symmetries is forbidden in two dimensions). Note this expectation continues to hold even in the large N limit, as the logarithm of a large N determinant depends on N only through an overall multiplicative factor of N which does not affect its analyticity properties. 28 The expectation on the fermionic side is similar: Fermi sea formation, while sharp at zero temperature, is a crossover at finite temperature.
At finite temperature, we then expect the off-shell free energies of both the bosonic and fermionic theories to be analytic functions of µ. We thus conjecture that the correct formulae for the off-shell free energies for |µ| ≥ c B are simply the analytic continuation (in the variable µ) of the relevant formulae for |µ| < c B (presented in The conjecture presented in this section matches the expectations of Bose-Fermi duality more or less trivially. Since the fermionic and bosonic free energies have already been shown to match when the chemical potential is smaller than the quasiparticle masses, the analytic continuations of these two expressions necessarily agree at all values of the chemical potential.

Non-analyticity of the naive expressions
The off-shell free energies of the previous section were computed in the previous literature by evaluating the field theory partition functions (3.6) via the following (schematic) procedure. One first integrates out the gauge field (this is possible because it appears in the action in a quadratic manner). One then introduces various singlet Lagrange multiplier and Hubbard-Stratanovich fields chosen so as to ensure that the action is quadratic in the matter fields. The matter fields are then integrated out, yielding a determinant (which exponentiates to a log-determinant). The resultant action is a complicated non-local functional of the auxiliary singlet fields.
One then derives the large N saddle point equations for these singlet fields. Remarkably, it turns out to be possible to exactly solve the resultant non-linear integral equations. What is more remarkable is that, these complicated non-linear integral equations for the singlet fields can be recast into much simpler algebraic or transcendental equations for a few c-number variables rather than fields. These c-number variables are the 'auxiliary' variables that appear in the off-shell free energies (3.14), (3.17), (3.7) and (3.12), and the algebraic or transcendental equations involving these variables are the ones obtained by extremizing these simple off-shell free energies listed in these four equations w.r.t. their auxiliary variables as discussed in detail in Section 3.2.
What is important for the current discussion is that the log-determinant described in the first paragraph above is the only part of the effective action that had its origin in a term in the Lagrangian involving derivatives acting on the matter fields. Therefore, it is the only part of this effective action that is sensitive to the chemical potential. This fact is also visible in the explicit expressions (3.14), (3.17), (3.7) and (3.12) (the log-determinant is the last line in each of those equations).
In summary, the general structure of the off-shell free energies (3.14), (3.17), (3.7) and (3.12) of the previous is where F det is the log-determinant term and F int (which is the rest of the free energy) is independent of µ. As reflected in the final answers (3.14), (3.17), (3.7) and (3.12), the determinants F det take a simple universal form, correspond to the free energy of a single massive boson (of a mass c B that remains to be determined) or a single massive fermion (of a mass c F that remains to be determined) modified by its coupling to the holonomy distribution ρ 29 . Let us first focus on the bosonic determinant. At small enough values of µ, F det is, of course, an analytic function of µ. At |µ| = c B something special happens. At this value the effective mass of the bosonic field -which is proportional to c 2 B − µ 2 -vanishes. After Kaluza-Klein reduction on the thermal circle, this effectively massless scalar field reduces to an infinite collection of two dimensional fields of squared masses where the Kaluza-Klein momentum n runs from −∞ to ∞, α is the holonomy and β is the inverse temperature. When α = 0 the Kaluza-Klein mode with n = 0 is a massless two dimensional scalar field. The determinant over such a field has a logarithmic divergence. As a consequence we should expect our determinant to have a potential non-analyticity of the schematic form in the neighbourhood of α = 0. Indeed the exact expression for the Bosonic determinant does have precisely such a singularity. Similar remarks apply to fermionic fields. Once again the field is effectively massless at µ = c F . Once again a Kaluza-Klein reduction at this value of µ gives an infinite number of two dimensional fields whose squared masses, this time, are (the shift of n by 1 2 is a consequence of the antiperiodic boundary conditions of fermions around the thermal circle). This time the mode with n = 0 is effectively massless at α = π 29 Recall that this holonomy distribution function ρB or ρF which appears in the determinant is itself determined at the saddle-point by extremizing its effective action obtained by integrating out the matter fields. and so we should expect the fermionic determinant to have a potential non-analyticity of the form in the neighbourhood of α = π. Indeed, the exact expression for the fermionic determinant does have precisely such a singularity 30 .
In summary, the bosonic determinant (3.23) is non-analytic at |µ| = c B if ρ B (α) is non-vanishing in a neighbourhood of α = 0. In a similar way the fermionic determinant (3.24) is potentially non-analytic at |µ| = c F if ρ F (α) is non-vanishing in a neighbourhood of α = π. In the rest of this subsection we will explore the nature of these non-analyticities and explain how we can analytically continue around them.

Analytic continuation in bosonic upper cap phases and fermionic lower gap phases
In this subsection, we consider phases of the fermionic theory [15] in which the eigenvalue distribution ρ F (α) vanishes in an interval around α = π. Correspondingly, the dual bosonic eigenvalue distribution ρ B (α) saturates the upper cutoff 1/2π|λ B | in an interval around α = 0. Concretely, we focus on phases in which From the duality map relating ρ B and ρ F (3.5), it follows that the dual eigenvalue distribution obeys When the large N saddle-point eigenvalue distributions ρ B and ρ F satisfy (3.25) and (3.26), the fermionic theory is said to be in a lower gap phase while the bosonic theory is said to be in an upper cap phase [15]. We return to the study of more general phases in the next subsection.

Fermions in the lower gap phase
Let us first examine the fermionic determinant (3.24). The term that depends on µ is the integral The second equality in both (3.23) and (3.24) holds in the dimensional regulation scheme.
We recast the integral over α into a contour integral over the unit circle in the complex z-plane with z = e iα , see Figure 5. Let us study the branch cut structure of the logarithms Figure 5. The contour of integration in the z-plane for the integral (3.27). The angle α is related to z as z = e iα .
(3.27). When c F > |µ|, the branch cut that appears in both logarithms does not intersect the unit circle in the z-plane (see Figure 6(a)). As a consequence, the expression (3.27) is analytic in µ for |µ| < c F . On the other hand, when |µ| > c F , the unit circle intersects the branch cut of the second logarithm in (3.27) when lies in the range c F < < |µ|. For a general distribution ρ F (α), this intersection implies a breakdown of the analyticity of one of the logarithms in (3.27) as a function of µ at µ = c F or at µ = −c F , depending on the sign of µ (recall that c F is a positive quantity by definition). However, when the fermion is in the lower gap phase (3.25) the integration contour is not the full unit circle but the open arc corresponding to (3.28) The branch cut of the logarithms in (3.27) lies at α = π and so does not intersect the open contour specified by (3.28) (see Figure 6(b)). It follows that when ρ F (α) obeys (3.25), the expression (3.27) is analytic as a function of µ at all values of µ including |µ| ≥ c F . Recall that (3.27) is the only term in the fermionic free energies that depends on µ. Now, since this term is analytic for all µ, we conclude that whenever the fermion is in a lower gap phase (3.25), the formulae for the fermionic off-shell free energies are fully analytic in µ and apply without modification for all values of µ including the range |µ| > c F .

Bosons in an upper cap phase
Let us now consider the bosonic determinant (3.23). The term that depends on µ is When the boson is in an upper cap phase (3.26), the integral over α in (3.29) can be split up into the the integral over the range α ∈ (−b, b) on the unit circle and the integral over the rest of the unit circle. The integral over the rest of the unit circle is manifestly analytic in µ for all values of µ. The part of (3.29) that is potentially non-analytic in µ is The potentially non-analytic part is one of the logarithms of (3.29) depending on the sign of µ. In the above expression (3.30), we treat the two cases together by noticing that the non-analyticity in either case depends only on the absolute value |µ|. As written, (3.30) is indeed non-analytic as a function of µ at |µ| = . However, it is easy to find a modification of (3.30) that agrees with (3.30) for |µ| < but is analytic for all µ. We first note that (3.30) can be rewritten as an open contour integral where the contour C starts at e −ib and runs counterclockwise along the unit circle to e ib (see Figure 7). The modification of (3.31) that makes it analytic everywhere replaces the contour C with a contour C that begins and ends at the same points, but no longer runs along the unit circle all the way. It lies on the unit circle except in the neighbourhood of the real axis where it makes a hairpin bend around the branch cut so that it cuts the positive real axis at a point x 0 < eˆ −|μ| (see Figure 7(b)). This condition on the contour C is chosen to ensure that this contour never passes through the branch cut of the logarithm in (3.31) . With this choice of contour, (3.31) defines a function that manifestly agrees with (3.30) when |µ| < c B and is analytic everywhere. Note that when z is slightly below (resp. above) the branch cut, the argument of the logarithm 1 − z e −ˆ +|μ| , has a small positive (resp. negative) imaginary part. It follows that the integral in (3.31) evaluates to The first term in (3.32) is the integral over the arc on the unit circle; the second term is the discontinuity across the branch cut of the logarithm. Performing the integral over x and then the integral over in the second term, we find that (3.32) reduces to It follows from this analysis that whenever the boson is in the upper cap phase (i.e. when (3.26) holds), the analytic continuation of the log-determinant (3.23) is The log-determinants in the last line of the critical boson free energy (3.14) and the regular boson free energy (3.17) have to be modified to include the additional term in (3.34) when c B < |µ|. We conjecture that the above procedure gives the correct expressions for the off-shell free energies of the critical boson and regular boson theories for all values ofμ in upper cap bosonic phases.
Note that the new term in (3.34) is proportional to 1 λ B and hence is singular in the weak coupling limit λ B → 0. As we will see below, this term will play a key role in curing the runaway singularity of Bose condensation observed in the free bosonic theories.

Analytic continuation and off-shell free energies in general phases
Though the formulae of Section 3.4.2 are all we will need later in this paper, in this subsection we take the opportunity to present the (conjecturally) correct expressions for the off-shell free energy for both the bosonic and the fermionic theories with |µ| > c B and |µ| > c F in more general phases than the bosonic upper cap phases and dual fermionic theory lower gap phases dealt with in the previous subsubsection.
When the boson is in an upper cap phase, the eigenvalue distribution ρ B was constant in a finite interval around α = 0 (see equation (3.26); its analytic continuation to the interior of the unit circle where the hairpin part of the contour was present (see Figure 7(b)) was trivially the same constant. However, when the bosonic theory is not in a upper cap phase, this analytic continuation has to be done more carefully. For a general eigenvalue distribution ρ B , it is still true [15] that there is a sufficiently small interval, say |α| < c, in which ρ B (α) is an analytic function. Similarly, the dual fermionic eigenvalue distribution ρ F is also analytic in the interval |α| ≥ π − c. Within these intervals, these eigenvalue distributions may be expanded into Fourier series as We emphasize that the right hand sides of (3.37) -which are well-defined and analytic everywhere on the unit circle -are guaranteed to agree with the actual eigenvalue distributions only in the restricted intervals described above, and not necessarily on the whole unit circle. We expect that the coefficients a n and b n in (3.37) decay with n at least as fast as ε n for some number ε < 1 since the series are convergent on the unit circle z = e iα . It is then easy to see that the right hand sides can be analytically continued at least in the annulus These functions clearly agree with ρ B (α) and ρ F (α) in the restricted intervals shown in (3.37). Hence,ρ B (z) andρ F (z) are the correct analytic continuations of the eigenvalue distributions ρ B and ρ F , at least in the strips Of course, using the Bose-Fermi duality map of the holonomy distributions (3.5) we can relate the a n and b n coefficients in (3.37) which can then be plugged back into the analytic continuationsρ B andρ F to get the duality relation With all these definitions in place, we can now imitate the analysis of the previous subsection to get expressions for the log-determinants that are analytic for all µ by deforming the contours of integration to go around the branch cuts as shown in Figures 8 and 9.
Re(z) Figure 8. Bosons in non-upper cap phases: The branch point is at z = +eˆ −|μ| and is denoted by the black dot on the +ve real axis. The orange curve is the contour (counterclockwise) over which the holonomy is integrated over in this phase. The red line is the branch cut for the logarithm appearing in the bosonic free energies. Figure 9. Fermions in non-lower gap phases: The branch point is at z = −eˆ −|μ| and is denoted by the black dot on the -ve real axis. The purple curve is the contour (counterclockwise) over which the holonomy is integrated over in this phase. The green line is the branch cut for the logarithm appearing in the fermionic free energies.
Finally, incorporating the modifications to the log-determinants into the off-shell free energies given in Section 3.2, we get

Gap Equations
For chemical potentials less than the gap c B or c F , the equations that followed from varying the off-shell free energy were presented for the regular fermions in (3.8), for the critical fermions in (3.13), for the critical bosons in (3.15) and for the regular bosons in (3.18). In general, these equations of motion are modified when the chemical potential is greater than c B or c F since the off-shell free energies are themselves modified in this case. The modified equations of motion can be obtained by varying the modified off-shell free energies presented in the previous subsection. Alternatively, note that the only explicit occurrences of µ in the equations are in the quantities S(c B , µ) and C(c F , µ) which are the same factors which appear in the log-determinants (3.23) and (3.24) under the -integral. Thus, the analytic continuation procedure for the equations is identical to the one used for the off-shell free energies. The final result is that the equations continue to hold for |µ| > c B and |µ| > c F provided we make the replacements x , where, recall thatρ F (x) andρ B (x) are the analytic continuation of the holonomy distributions ρ F (α) and ρ B (α) to a small neighbourhood of the unit circle. In an upper cap phase for the bosons (3.26), we can use the second equation in (3.48) after settingρ B (x) = 1/2π|λ B |, i.e. make the replacement The equation (3.49) was already guessed in [64] (see equation A.5 of [64]).

Quasiparticle Occupation Numbers
The usual thermodynamical formulae inform us that the charge of an ensemble is obtained by varying the free energy F w.r.t. the chemical potential µ while keeping β and the spatial volume V 2 fixed. In equations, The quantity F that appears in (4.1) is the thermodynamic free energy related to the partition function Z as The free energy F is typically a rather complicated function. As we have discussed in the previous section, in the large N limit, it is obtained by extremizing the effective action functional s[ρ] w.r.t. the holonomy distribution ρ. This action functional consists of two pieces. The first is the contribution v[ρ] that we discuss in this paper due to integrating out all modes except the gauge field holonomy ρ and the second is the contribution due to the Chern-Simons modified path integral measure (discussed extensively in [15]). Note, of course, that this measure is independent of µ.
We have seen in Section 3.1, the final (usually complicated) answer for v[ρ] can be obtained by extremizing a rather simple 'off-shell' free energy function F (ϕ aux ; ρ] with respect to the auxiliary variables denoted collectively as ϕ aux . Thus the full free energy F is obtained by extremizing the effective action functional S(ϕ aux ; ρ] which consists of the 'off-shell' free energy F (ϕ aux ; ρ] and the contribution due to the measure which depends only on ρ but not the other auxiliary variables (and is also independent of µ).
Let the full set of variables (auxiliary variables as well as holonomies) be denoted by {ϕ i }. The on-shell values ϕ * i of the variables ϕ i are determined by extremizing the effective action S(ϕ aux ; ρ] w.r.t. each of the ϕ i : The final free energy is obtained by evaluating S(ϕ aux ; ρ] at these extremum values. In general, the extremum values of ϕ i are functions of µ. Hence, the final free energy F has two sources of dependence on µ: (a) the explicit dependence on µ coming from the off-shell free energy F (b) the implicit dependence coming from the fact that the extremum values ϕ * i are functions of µ. In other words, is the effective action evaluated at one of its extrema. It follows that Now, we use the equations of motion (4.3) to set the second term above to zero. Note that the only explicit dependence on µ in the effective action S is through the off-shell free energy F . Thus, the formula for the charge (4.5) simplifies to Moreover the quantity F in (4.6) can further be replaced by (the corrected form of) F det as this is the only part of the off-shell free energy that depends on µ.

Chemical Potential smaller than Quasiparticle Thermal Mass
When the chemical potential is smaller than the thermal masses of the excitations, the offshell free energy formulae are given in Section 3.2. Recall that the only explicit dependence on µ of these formulae is in the log-determinants in the last line of these expressions. Evaluating the derivative w.r.t. µ (see (4.6)) we find that for both the regular (3.7) and critical fermionic (3.12) theories 31 Now recall that the number dn of single-particle states of any fixed colour is given by (where we have used the usual relativistic dispersion relation 2 = c 2 F + k 2 ). It follows from (4.8) that (4.7) can be rewritten as The expression (4.9) has an obvious interpretation. At least as far as the charge goes, the system under study consists of a collection of effectively non-interacting particles where the occupation number of a single-particle state at energy and charge q is given byn F ( , µ) wheren where q = 1 for fundamental excitations and q = −1 for antifundamental excitations. The two terms under the integral in (4.9) then represent the occupation numbers of the fundamental quasiparticle (charge +1) and antifundamental (charge −1) states respectively. Note: The formula (4.10) and the interpretation proposed here, first appeared in the paper [32] in the form where the notation n F ( p) stands for the sum of the charges of a fundamental quasiparticle state and its antifundamental counterpart with momentum p and energy p . It is easy to rewrite the formula (4.7) for the charge above to match the one in [32] using the trigonometric identity The analysis of the bosonic theories proceeds along completely identical lines. Differentiating the free energies for the critical boson (3.14) and the regular boson (3.17) w.r.t µ, we find a formula for the net charge that admits an interpretation in terms of effectively non-interacting particle states of energy and charge q with average occupation number given by the formulan The charge of the bosonic ensemble is then given by 14) or, in terms of the occupation numbern( , µ),

Chemical potential Larger than Quasiparticle Mass
The generalisation of the fermionic (4.10) and bosonic (4.13) occupation numbers to |µ| ≥ c F and |µ| ≥ c B may be obtained in one of two ways. First, we could recompute the charge of the ensemble by differentiating the corrected off-shell free energy formulae (3.44), (3.45) for the fermions and (3.46), (3.47) for the bosons w.r.t. µ and then reinterpret the result in terms of a modified occupation number for each single-particle state. Alternatively (and more simply as well as more informatively), we could simply analytically continue (4.10) and (4.13) to values of µ larger than c F and c B . We briefly outline this second method (which proceeds along the lines of Section 3.4.3).
The formula (4.10) for the fermionic occupation number is analytic at all values of µ such that sgn(qµ) = −1. On the other hand this expression has a potential non-analyticity 32 at qµ = (recall q = ±1) . In order see this non-analyticity, we rewrite the integral over holonomies in (4.10) as a contour integral over z = e iα : The contour C in (4.16) is initially the unit circle, but with this definition (4.16) is not analytic in µ at qµ = . We obtain the analytic continuation of (4.16) by deforming the contour C to the contour C (see Figure 10(a)). C is chosen so that it continues to circle the origin, but cuts the negative x axis at a point closer to the origin than −e β( −qµ) . The formula (4.16), with C replaced by C , is thus the correct analytic continuation of (4.16) to values ofμ s.t. qµ > . The contour C can be deformed to the sum of two contours without changing the value of the contour integral: the first contour is the original contour C while the second is a second smaller loop that runs clockwise around the pole at z = −e β( −qµ) (see Figure 10(b)). The contribution of the small loop encircling the pole is where the Heaviside Θ-function is to indicate that this additional contribution is present only when qµ > . It follows that the analytic continuation of (4.10) -and hence the true fermionic occupation number of a quasiparticle state of energy -is obtained by subtracting the contribution due to the pole: In the special case of a lower gap fermionic phase,ρ F vanishes, and so, (4.18) reduces to (4.10). The analysis of the bosonic case proceeds along similar lines. The occupation number (4.13) can be rewritten as a contour integral as Again, the analytic continuation of (4.13) (and hence the correct formula for the occupa- Figure 11. Bosons: the pole is at z = +e β( −qµ) and is denoted by the black dot on the +ve real axis. Radius of the big circle is unity. The orange curve is the contour C (counterclockwise) over which the holonomy is integrated over.
tion number of a bosonic quasiparticle state at energy ) is obtained by subtracting the contribution due to the pole at z = +e β( −qµ) . The calculation can be followed pictorially in Figure 11. The modified occupation number is then In the particular case of a bosonic upper cap phase, (4.20) simplifies tō (4.21)

The thermodynamic limit
In the rest of the paper we specialise to the study of the quasi-fermionic theories and quasi-bosonic theories in the thermodynamic limit defined by taking the spatial volume V 2 to infinity V 2 → ∞. In this limit, the dominant phase of the holonomy eigenvalues is one in which the bosonic and fermionic eigenvalue distributions take the 'tabletop' form which we reproduce below: We will further be interested in the so-called 'thermodynamic, zero-temperature limit' obtained by first taking the limit V 2 → ∞ and only then taking the limit T → 0. The importance of the above order of limits is the following. When we take the limit V 2 → ∞ the eigenvalue distribution is frozen to (5.1). The zero-temperature limit is then taken with the eigenvalue distribution frozen at (5.1).
In this section, we first give expressions for the occupation numbers in terms of explicit well-known functions in the thermodynamic limit. We then consider the 'thermodynamic, zero-temperature limit' of those expressions. We also provide the expressions for the off-shell free energies in the limit since they will be used extensively in the subsequent sections.

Occupation Numbers in the infinite volume limit
In this subsection we specialise the formulae for occupation numbers presented in the previous section to the infinite volume limit V 2 → ∞ where the eigenvalue distributions take the universal 'tabletop' profile (5.1).
It turns out to be convenient to phrase the discussion in terms of the variable x defined as We shall encounter the multivalued function tan −1 ζ in the formulas of this section. This is defined unambiguously by choosing the following branch: Note that with this definition of tan −1 (ζ), we have tan −1 (−ζ) = − tan −1 (ζ).

Fermionic Occupation Number
The fermionic occupation number is given by equation (4.10) where ρ F (α) takes the universal tabletop form In terms of the variable x defined in (5.2), the formula for the occupation number of a quasiparticle state at energy and of charge q = ±1 simplifies tō (an equivalent formula first appeared in [32]). In going from the first to the second line of (5.5) we have used the fact that the integration limit is symmetric about α = 0. The integral on the RHS of the second line of (5.5) can be performed explicitly and we find the relatively simple analytic expression where x = e β( −qµ) is defined in (5.2). As indicated in (5.3), the multivalued function tan −1 (ζ) that appears in (5.6) is unambiguously defined by choosing the branch − π 2 < tan −1 (ζ) < π 2 . Substituting (5.2) in (5.6) it follows that (5.6) and (5.7) are the main results of this subsubsection. In the rest of this subsubsection we will study various limits of (5.7).

Zero-temperature limit
In the zero temperature limit (i.e., β → ∞), we have It follows immediately from (5.7), that in this limit, Note in particular that all states with energy smaller than qµ are occupied with unit occupancy. In the zero temperature limit, in other words, the fermions form a free Fermi sea at every value of λ F (this fact was already noted in [32]).

Small-λ F Expansion
Working at arbitrary temperature and µ, (5.7) is easily expanded in a power series expansion in λ F about zero; we find (as the expression (5.7) is an even function of λ F , the power series on the RHS of (5.10) has only even powers in λ F ). The expansion (5.10) applies at all values of the energy . Note that the first term on the RHS of (5.10) is, of course, the familiar formula of Fermi Statistics for the occupation number of a single particle state. Subsequent terms in this expansion represents universal 'anyonic' corrections to this familiar formula.

Small-x expansion
When the energy of our quasiparticle state lies well below qµ in units of the temperature, then x = e β( −qµ) 1. Equations (5.6) or (5.7) takes the following form The first term on the RHS of (5.11) is unity indicating that at leading order in this limit, all states are occupied with unit occupancy. The first correction to unity on the RHS of (5.11), − sin πλ F πλ F x, gives the exponentially suppressed probability of this state becoming unfilled (equivalently, of a 'quasihole' being created). The intriguing factor of sin πλ F πλ F (which is unity in the free limit λ F → 0 ) indicates that the formation at a hole is less likely at nonzero λ F than in the free limit. It would be very interesting to understand why this is the case directly from the Schrodinger equation in an appropriate non-relativistic limit.

Large-x expansion
On the other hand when the energy of our quasiparticle state lies well above qµ in units of the temperature, then x = e β( −qµ) 1. (5.6) or (5.7) takes the form (5.7) takes the following formn (5.12) (5.12) tells us that the occupation number of a state with such energies is exponentially suppressed. Once again note the interesting suppression of this probability by the factor of sin πλ F πλ F , which once again calls out for a non-relativistic explanation.

Bosonic Occupation Number
Recall the formula for the bosonic occupation number in an upper cap phase (4.21): Using the tabletop form of the bosonic eigenvalue distribution (5.1), we find the following formula for the bosonic occupation number in the thermodynamic limit: As in the case of fermions, the integral on the RHS of the second line of (5.14) can be exactly evaluated and we find 33 Using the fact that x > 1 for > qµ , the Θ-function appearing in (5.15) can also be rewritten as Θ(1 − x). So, (5.15) can be rewritten as The tan −1 and the Θ terms on the RHS above discontinuous at x = 1. However these discontinuities cancel each other and the final result is perfectly smooth at x = 1. In order to see this we proceed as follows. Using 18) and the identity we see that (5.17) can be rewritten as (5.20) and (5.21) are the main results of this subsection. In the rest of this subsection we study the various limits of these formulae.

Zero-temperature limit
Using (5.8), it follows that in the limit β → ∞.
It follows from (5.22), in particular, that all states with energies less than qµ (states that would have been infinitely occupied by a Bose condensate in the free limit) have an occupation number given by Note that the additional piece in the occupation number of the boson (the second term in the formula forn B in (5.13)) -plays a key role in the formula (5.22). Without this term we would have foundn B ( , µ) = −1; this is the nonsensical 'divergent' occupation number of a theory of free bosons. The second term in (5.13) 'regulates' this free result into a physically sensible answer, albeit one that does diverge in the limit λ B → 0.

Small-λ B expansion
Working at arbitrary values of x, (5.20) is easily expanded in a power series in λ B . We find The above equation can be rewritten more explicitly as (5.24) The first term on the first line of (5.24) is, of course, the familiar expression for the occupation number of a single particle state with > qµ using the formulae of Bose statistics; the remaining terms on this line are the universal 34 'anyonic' corrections to this formula.
The second line of (5.24) deals with the more interesting case of a state whose energy is less than qµ. Any such state would be infinitely occupied in a free Bose ensemble. The first terms on the RHS of (5.24) tell us that finite λ B effects regulate this infinite occupation to a finite occupation number given by 1 |λ B | . The remaining subleading terms in this expansion give us the (temperature dependent) corrections to this result.
The corrections to the leading order results in (5.24) are of order λ B β( −qµ) ; at generic values of , these corrections are small if λ B is small. No matter how small λ B is, however, these corrections are non-negligible when At small λ B , it follows thatn B ( , µ) takes the value 1 |λ B | (for < qµ) and 1 e β( −qµ) −1 for > qµ. The transition between these two values takes place in the range of energies 34 These corrections are universal in the following sense. They apply both to the critical and the regular boson theory, and all values of the UV parameters of these theories. Note, however, that these formulae only apply in the infinite volume limit where the holonomy eigenvalue distribution takes the tabletop form. (5.25). In fact at leading order in small λ B ,n B ( , µ) is everywhere (i.e. at all energies) well approximated byn Note that the RHS of tends to zero when β( − qµ) |λ B | but tends to 1 |λ B | for β( − qµ) −|λ B |.

Small-x expansion
As in the case of fermions, x is small for states whose energies lie much below qµ in units of the temperature. The occupation number for such states is well approximated by the small-x expansion of (5.20), given bȳ We see that at leading order in this expansion, all states are occupied with occupation number 1−|λ B | |λ B | . The probability of a quasihole appearing in this condensate is, at leading order, proportional to sin πλ B πλ B x. As in the case of fermions it would be interesting to find a physical interpretation of the prefactor sin πλ B πλ B .

Large-x expansion
x is large for states whose energies lie much above qµ in units of the temperature. The occupation number of such states is well approximated expanding (5.20) in a power series in 1/x. This expansion is given bȳ We see that the probability of occupation for any of these states is given by sin πλ B πλ B x −1 at leading order. The factor of x −1 is the familiar Boltzmann suppression of a high energy state. The universal prefactor is more mysterious and requires an explanation.

Duality of explicit forms of occupation number
In this brief subsection we directly verify that the explicit expressions (5.7) and (5.21) are dual to each other. Using (5.7) and (5.21) we see that The duality map of the coupling constants 35 . It then follows that the right hand side of (5.29) vanishes. We conclude that as required by duality.

Free energies at zero temperature
The expressions for the free energies in Section 3.4.2 simplify in the limit β → ∞ in that that the holonomy integrals become almost trivial. Consider the holonomy integral in the bosonic case: In the limit β → ∞ with and µ fixed, the quantitiesˆ andμ go to ∞. It follows that the second logarithm above in (5.31) simply vanishes. The first logarithm also vanishes when is larger than |µ|. On the other hand, when < µ, we have where θ B (α) = α − π when α > 0 and θ B (α) = α + π when α < 0 . (5.33) Using the fact that ρ B (α) is an even function of α, it follows that The holonomy integral (5.31) thus simplifies to (5.34) 35 Throughout this section and in other sections, (unless mentioned explicitly) we have implicitly used the fact that cF = cB, so that the single particle energy spectra, = p 2 + c 2 F = p 2 + c 2 B , are identical for both fermions and bosons.
The same calculation works for the fermionic case except for one simplification: there is no additional choice of branch for the logarithm in (5.33) since there is no additional minus sign in the logarithm log(1 + e −ˆ +|μ|+iα ) for the fermions.
Thus, in the zero-temperature limit, the off-shell free energies for the bosonic and fermionic theories become (5.38)

Quasi-fermionic theories
In this section, we analyse in detail the thermodynamical zero-temperature phase structure of the quasi-fermionic theories i.e. the regular fermion and the critical boson. We mainly work in the bosonic theory and appeal to Bose-Fermi duality in order obtain the corresponding results for the fermions. The off-shell free energy of the critical boson theory in the zero-temperature limit was obtained in (5.37) and is reproduced here for convenience: The equation of motion that follows by varying (6.1) w.r.t.S is 2) and the equation of motion from varying c B is Comparing (6.2) and (6.3), we see thatS is given on-shell by Plugging this back into the off-shell free energy (6.1), we get, for c B < |µ|, and for c B > |µ|, Plugging the on-shell value ofS into the equation of motion for c B (6.3), we get These phases occur for the following ranges of m cri B that we obtain from the restriction c B > |µ|: On the contrary, in the condensed phase (corresponding to c B < |µ|), there seems to be a unique solution: This phase exists for the following range of m cri B obtained from the restriction c B < |µ|: There are two possibilities of reaching the condensed phase from an uncondensed phase by raising the chemical potential above c B : either from the uncondensed unHiggsed side or from the uncondensed Higgsed side. Recall that the excitations in the unHiggsed phase are scalars while those in the Higgsed phase are W -bosons. Depending on the side we approach from, the description of the condensed phase is either in terms of condensed W bosons or condensed scalars. This can be seen in the formula for c B (6.10) which is non-analytic in the coupling constant m cri B due to the absolute value on the right hand side. The sign of the quantity inside the modulus on the right hand side distinguishes between these two cases: Though the description of the condensed branch in terms of condensed scalars or condensed W bosons seems manifestly different, it turns out that there is no real distinction between these two descriptions. This can be seen from the free energy in the condensed phase (6.5) which does not depend on the variable c B at all! Thus, the non-analyticity in the equation of motion for c B does not affect the free energy in the condensed phase. Plugging the solutions for each of the uncondensed phases into (6.6), we find the free energy in the unHiggsed and Higgsed phases to be

Phase Diagram
Note that the conditions (6.9) and (6.11) on m cri B are mutually non-overlapping. Together they give a single cover of the range of masses At any given value of |µ| and m cri B , there exists exactly one solution of the gap equations. At every value of parameters, therefore, the system has a unique phase. It follows that the phase diagram of the system is as depicted in Figure 12. As depicted in Figure 12 our system undergoes a sharp second order phase transition at the edges of the range (6.11). Within the range (6.11), our system is in a compressible phase, i.e. the charge is a non-trivial function of the chemical potential. Outside the range (6.11), the system is in an incompressible phase: its charge is independent of µ (and vanishes in particular).
Within the compressible phase, the charge (per unit volume) as a function of µ is given by the formula Note that, the charge vanishes at both the edges of the condensed range for m cri B (6.11), but that its first derivative is non-vanishing at these boundary points. This discontinuity in the second derivative of the free energy demonstrates that our system undergoes a second order phase transition at the edges of the range (6.11). The number of single particle states (per unit volume) in the condensed phase, upto the energy |µ| is given by Thus, we see that (6.14) can be thought of as the product of the number of single particle states (6.15) and the occupation numbern listed in (1.7). Plugging in the expression for c B as a function of µ into this expression ((6.10)) we find an explicit expression for the charge as a function of UV parameters and the chemical potential: µ-dependence, the charge can be written as It is clear from (6.16) that the charge Q is a quadratic function of µ. The compressibility of this phase is given by Note that ∂Q ∂µ is always positive in the condensed phase and vanishes at the edges.

Quasi-bosonic theories
In this section, we consider the detailed phase structure of the quasi-bosonic theories viz. the critical fermion and the regular boson. We explicitly work in the bosonic setting and use the Bose-Fermi duality map discussed previously to obtain the analogous results for the critical fermion. The off-shell free energy for the regular boson at zero temperature is given by (5.38) and is reproduced below: The variation w.r.t. c B andS give the equations Consistency of the two equations of motion gives the following on-shell condition forS in the zero temperature limit: We substitute the above expression forS back into the off-shell free energy (7.1) and the c B equation of motion in the second line of (7.2) respectively. The off-shell free energy becomes : Each of the equations in (7.5) is quadratic in c B . It may thus seem that the function c B (σ) obtained from the solution of (7.5) is multivalued with upto four branches. In actuality this is not the case as we now explain. Solutions to the equation in the second of (7.5) are valid only when 0 ≤ c B ≤ |µ| . (7.6) It follows that the second equation in (7.5) has at most one valid solution -the so called condensed phase solution -given by This solution is valid if and only if σ B satisfies The expression for c B in (7.7) as a function of σ B is not analytic at σ B = − |µ| 2 1 − 1 |λ B | . Intuitively, this non-analyticity is a consequence of the fact that the solution migrates from the condensed unHiggsed to the condensed Higgsed phase at this value of σ B . Remarkably enough, however, the free energy in the second line of (7.4) is independent of c B , and so the non-analyticity of c B as a function of σ B does not translate into a non-analyticity of the free energy as a function of σ B .
Notice that the above non-analytic point of c B is at σ B = 0 when the chemical potential is zero. This was the point at which the free energy was non-analytic and there was a phase transition between the unHiggsed and Higgsed phases. The point σ B = − |µ| 2 1 − 1 |λ B | is the analog of such a transition when we turn on a chemical potential. The discussion in the previous paragraph indicates that, at least in the large N limit, there is no distinction between the condensed unHiggsed and the condensed Higgsed phases; we just have one single condensed phase. The contribution of this phase to the free energy as a function of σ B is simply the second line of (7.4). This expression for the free energy is valid when (7.8) is obeyed.
There are clearly two different branches of solutions to the first line of (7.5); these are the solutions These solutions are only valid when c B > |µ| which translates to the following conditions on σ B : Plugging the solutions (7.9) into the first line of (7.4) yields the free energy as a function of σ B in the ranges (7.10).
Notice that the ranges of σ B in (7.10) are non-overlapping, and are also exactly complementary to the range (7.8). It follows that the three ranges listed in (7.8) and (7.10) together cover the σ B line in a non-overlapping manner. The transitions between the three ranges take place at the points We may set |µ| = 1 in the above expressions since it just corresponds to a choice of scale and we will do so in the intermediate steps of the analysis below. However, it will be useful in the end to restore the |µ| dependence; for this we just rescale all dimensionful quantities with appropriate powers of |µ|: In summary, the off-shell free energy as a function of the single variable σ B is given by piecewise analytic expressions in the three ranges (7.8) and (7.10): The quantities φ h and φ u are functions of |λ B | given by and satisfy, for all |λ B | ≤ 1, φ φ turns the free energy into the quantum effective potential for the lightest gauge invariant operatorφφ of our theory (see [70]). We frequently refer to U eff (σ B ) as the effective potential while keeping in mind the above fact. Notice that U eff (σ B ) presented in (7.13) is a piecewise cubic function of σ B . Independently in each of the three intervals in (7.13), the equation U eff (σ B ) = 0 is quadratic in σ B . Real solutions to the quadratic equations exist only when their discriminants are positive. They first start existing on the curves along which the discriminants vanish i.e. the parabolas The real solutions that are created are double zeroes of U eff (σ B ) 36 and are located on the σ B line respectively at Crossing any of the curves listed in (7.16) signals the (dis)appearance of new extrema in the corresponding branch of the effective potential U eff (σ B ). Of course the discriminant curves (7.16) are physically relevant only when the solutions (7.17) lie within their respective domains of validity (7.8), (7.10). In other words, only certain parts of the discriminant curves (7.16) are relevant for the phase diagram.

Strategy to obtain the phase diagram
In the rest of this paper, we work out the phase diagram of the theory that follows from (7.13). We split the analysis into four cases depending on the range of the marginal parameter x B 6 : Looking at the potential (7.13), it is clear that the potential is unbounded below for large and positive σ B when x B 6 < φ h (Case A) and is unbounded below for large and negative σ B when x B 6 > φ u (Case D). When x B 6 is in the range φ h < x B 6 < φ u corresponding to Cases B and C, the potential is bounded below for both positive and negative σ B . Thus, the potential is stable for cases B and C while it is unstable for cases A and D. We treat each of 36 For a generic potential U (σ) = Aσ 3 + Bσ 2 + Cσ + D, the extrema of this function are given by U (σ) = 3Aσ 2 + 2Bσ + C = 0. The solutions of this quadratic equation are When the discriminant B 2 − 3AC vanishes, this simplifies to σ± the cases in (7.18) separately below. The stable cases are addressed in Section 7.4 while the analysis for the unstable cases is pushed to the appendix and is presented in Appendix E. The phase diagram of the theory is two-dimensional and can be conveniently described by the two dimensionless ratios We have one phase diagram for every range of x B 6 in (7.18). In drawing these diagrams, we place the variable λ B b 4 |µ| along the Y -axis and We obtain the detailed structure of the potential for each range of x B 6 at each point of the phase diagram. Our strategy is as follows. First, recall that there are two values of σ B in the neighbourhood of which the potential (7.13) is non-analytic. These points, which we designate and r, are The point is the junction between the unHiggsed and the condensed branches of the potential, while the point r is the junction between the condensed and the Higgsed branches of the potential.
In order to work out the full phase diagram we study the behaviour of the potential near each of the non-analytic points above. To study the potential near a non-analytic point, say σ B = , we shift our origin in σ B to that point. Then, we study the behaviour of the effective potential (7.13) as a function of m 2 B and λ B b 4 by pretending that the effective potential has only the unHiggsed and the condensed branches which meet at σ B = i.e. ignoring the existence of the Higgsed branch that, in reality, cuts off the condensed phase at σ B = r. This description is not expected to hold for all values of m 2 B and λ B b 4 but only for a small region in the phase diagram where the σ B = r cutoff is not important. Similarly, we study the behaviour of the potential near σ B = r by ignoring the existence of the unHiggsed branch of the potential which cuts off the condensed branch at σ B = .
The two exercises above furnish two 'local patches' of the behaviour of the effective potential on the phase diagram. We then appeal to the simplicity of the form of the effective potential to interpolate between these two patches and obtain a global picture of the behaviour of the effective potential. It is then a simple exercise to determine the dominant minimum of the potential and thus the phase structure of the theory. We now provide the expressions for the potential in the neighbourhood of the two non-analytic points in σ B described above in (7.20).

Near : σ
In the local patch around σ B = 0, we have There is an additional common constant to both the expressions above which is unimportant for understanding the behaviour near σ B = 0: The potential (7.21) matches the general form of the potential D.1 considered in Appendix D which we reproduce here for convenience: The quantities A, a, B, b and C in the present case are given by As we have mentioned above, we use λ B b 4 = 0 and m 2 B = 0 as the X-axis and Y -axis for the full phase diagram. However, the local coordinate axes for the potential (7.21) are different are obtained by setting the quantities B and C in (7.24) to zero: The origin of the local coordinate axes above lies at B = C = 0, given by There is another point on this phase diagram in which the effective potential (7.21) becomes particularly simple: the point L in the phase diagram at which B = −b and C = 0 in (7.24): L : Borrowing results from the general analysis of Appendix D, we see that this 'local patch' of the phase diagram has the following special curves.
1. The Y -axis (7.25) is divided into three segments by the local origin O (7.26) and the point L (7.27). These segments correspond to B > −b, 0 < B < −b and B < 0 in (7.24). These translate to the following conditions on λ B b 4 : Please refer to Section D.1 for a detailed description of these segments and the role they play in understanding the structure of the potential.
2. We also have the discriminant parabolas (7.16) for the unHiggsed and condensed branches of the potential: Note that the curve D c passes through the point L at the junction of S and N ; in fact, this parabola is tangent to the Y -axis (7.25) at L. In a similar manner the curve D u is tangent to the Y -axis at the local origin O (7.26) which is at the junction of N and M .
3. In addition to the curves we have drawn above, there may be first order transition lines which have to be determined numerically.

Near r: σ
We look at the effective potential (7.13) in the neighbourhood of σ B = 2−|λ B | 2|λ B | by setting σ B = σ B − 2−|λ B | 2|λ B | and expanding the relevant expressions for U eff . Again, note that σ B = 0 when σ B = r. In a neighbourhood around σ B = 0, we have Again, there is an additional constant in both expressions which is unimportant while looking at the behaviour near σ B = 0: Again, the potential in (7.30) is of the form of the general potential (D.1) in Appendix D (also reproduced in (7.23) above) with the following identification of the various parameters: The local coordinate axes for the above potential in the (m 2 B , λ B b 4 ) plane are given by setting the quantities B and C to zero: The origin of the above local coordinate axes is at the point R specified by B = C = 0 in (7.32): R : There is another point on this phase diagram in which the effective potential (7.30) becomes particularly simple. This is the point P r at which B = −b and C = 0 in (7.32): Appealing to the results of Appendix D, we see that this 'local patch' of the phase diagram has the following special curves.
1. The Y -axis (7.33) is divided into three segments S , N and M by the local origin R (7.34) and the point P r (7.35). The importance of these segments are elucidated in Section D.1 in the Appendix. The segments are again specified by the following inequalities on λ B b 4 : 2. The parabolas D c and D h were discussed around equation (7.16) and correspond to the discriminants in the condensed and the Higgsed branches.
Note that the curve D c passes through the local origin R at the junction of S and N ; in fact, this parabola is tangent to the Y -axis (7.33) at the point R. In a similar manner the curve D h is tangent to the Y -axis at the point P r which is at junction of N and M .
3. In addition to the curves we have described above, there may be first order transition lines which have to be determined numerically.

Some broad features of the phase diagram
As we have mentioned above, we will present the phase diagram of our theory on a plane whose Y -axis is m 2 B = 0 and whose X-axis is λ B b 4 = 0. With these conventions we note the following general properties of the curves we have defined above.

Relative locations of the local patches
The local patches of the phase diagram corresponding to the behaviour of the potential near σ B = and σ B = r are described separately in the previous subsection. Now, we try to understand their relative locations in the full phase diagram.
1. The Y -axis in (7.25) has slope 1 2 . Substituting the coordinates of the point R (7.34) into the expression for Y in (7.25), we get Thus, the point R lies to the right of the Y -axis when x B 6 is positive, and to its left when x B 6 is negative.
2. The slope of the Y -axis in (7.33) is − |λ B | 2(2−|λ B |) . Similar to the previous case, substituting the coordinates of the point L (7.27) into the expression for Y , we get Thus, the point L lies to the right of Y when x B 6 > 0 and to its left when x B 6 < 0.
3. The points L and R are coincident when x B 6 = 0 and are located at

The phase near the origin of the phase diagram
Let us focus on the origin of the phase diagram (m 2 B , λ B b 4 ) = (0, 0). The effective potential (7.13) becomes The potential is monotonic in the unHiggsed and Higgsed branches since it contains only the cubic term. The only extrema possible are then in the condensed branch. Depending on the choice of x B 6 , there may be extrema or not in the range of validity of σ B of the condensed branch. Thus, the origin of the phase diagram is either in the condensed phase when an extremum exists or in a 'no phase' region if the potential is monotonic in the condensed branch as well. The same logic applies if m 2 B and lambda B b 4 are non-zero but are small when compared to the chemical potential. Thus, a neighbourhood of the origin lies in the condensed phase or in a 'no phase' region.

The behaviour for |m
When the parameters m 2 B and λ B b 4 are much larger in modulus than |µ| 2 and |µ| respectively, the effective potential (7.13) takes the simple form Since the extrema of the potential scale like λ B b 4 or m 2 B , they will not lie in the condensed branch of the potential whose range of validity is O(|µ|) around σ B = 0. Thus, when m 2 B and λ B b 4 are both large, we can ignore the presence of the condensed branch. In this case the analysis reduces to that of the regular boson without chemical potential [70] whose effective potential has only two branches viz. the unHiggsed and the Higgsed.
Then the above analysis tells us that, after a certain point on any parabola of the type given above, the system is in one of the two uncondensed phases i.e. either the unHiggsed or the Higgsed phase (or in a no-phase region if x B 6 is in one of the unstable ranges). For the regular boson without chemical potential with x B 6 in the stable range φ h < x B 6 < φ u , there is a first order curve for λ B b 4 < 0 along the following parabola: where ν(x B 6 , |λ B |) is a numerically determined function. The relevant information about ν that is needed is that it is monotonically decreasing as a function of x B 6 from φ h to φ u and is zero when . It can be easily seen from the expressions for φ h and φ u in (7.14) that φ h + φ u is always positive for any value of |λ B | ≤ 1. Thus, the above parabola is in the fourth quadrant when φ h < x B 6 < 1 2 (φ u + φ h ) and is in the third quadrant when From the discussion above, it is clear that we should see the above first order curve even in phase diagrams of the current paper for sufficiently large λ B b 4 and m 2 B .

The phase diagram
In this subsection, we focus on the ranges of x B 6 in (7.18) that correspond to the potential being stable i.e. it is bounded below for large and positive σ B as well as for large and negative σ B . This corresponds to the cases B and C in (7.18). The unstable ranges of x B 6 are addressed in Appendix E.

Case B:
Case III of Appendix D is pertinent near the non-analytic point since the quantity A in (7.24) satisfies A < 0 and A + a < 0, while Case II of Section D is pertinent near the point r since the quantity A in (7.32) satisfies A < 0 and A + a > 0. This implies that the global potential is bounded for both positive and negative σ B . The unboundedness of the local potential near σ B = 0 for large and positive σ B is not very serious since it is cutoff at The following conditions specify the segments of D u , D c and D h that appear in this phase diagram: We display the detailed structure of the potential near each local patch in Figure 13(a) and put together these two local behaviours of the potential in Figure 13(b). The final phase structure is displayed in Figure 14. The first order transition lines shown in solid green in Figures 13 and 14 are determined numerically. The part of the first order line that separates the unHiggsed and Higgsed phases is determined by comparing the value of the potential at the respective minima in each branch. As was discussed in Section 7.3.3 (also see the discussion in [70] and [69]), this line goes to infinity in the fourth quadrant when x B 6 < 1 2 (φ u + φ h ) and in the third quadrant when x B 6 > 1 2 (φ u + φ h ). Since φ h + φ u is always positive, it follows that the first order line always curves towards infinity in the fourth quadrant for all values of x B 6 such that φ h < x B 6 < 0. Figure 15. Structure of the potential for 0 < x B 6 < φ u . We have restored the |µ| dependence of the phase diagram by rescaling m 2 B and λ B b 4 by appropriate powers of |µ|. The green curve is a first order transition while the blue curves are second order transitions. The above plot is for the subrange 0 < x B 6 < 1 2 (φ h + φ u ) where the first order curve goes to infinity in the fourth quadrant. The analysis proceeds similarly when the first order curve goes to infinity in the third quadrant for

Case
For this range of x B 6 , Case II of Section D applies for the potential near σ B = and Case I applies for the potential near σ B = r. The conditions specifying which segments of the discriminant parabolas appear in this phase diagram are The detailed structure of the potential is displayed in Figure 15. In Figure 15(a), we provide a small inset plot of the local potential near σ B = and r in each region demarcated by the special lines of the potentials (7.21) and (7.30). In Figure 15 where the first order line goes to infinity in the fourth quadrant. The figure on the right corresponds to 1 2  potential in each of the above regions. Finally, we display the phase structure of the theory in Figure 16.
The part of the first order line that separates the unHiggsed and Higgsed phases goes to infinity in the fourth quadrant when x B 6 < 1 2 (φ u + φ h ) as explained in Section 7.3.3. As indicated there, φ u + φ h is always positive for all values of |λ B |. It follows that the first order line curves towards infinity in the fourth quadrant for x B 6 such that 0 < x B 6 < 1 2 (φ u + φ h ) and goes to infinity in the third quadrant when 1 2 (φ u + φ h ) < x B 6 < φ u . These two cases are displayed in Figure 16.

A closer view at the triple point
In the stable cases, Case B: φ h < x B 6 < 0 and Case C: 0 < x B 6 < φ u , there is an interesting point labelled T on the corresponding phase diagrams Figures 14 and 16. At this point, all three phases -the condensed, the (uncondensed) unHiggsed and the (uncondensed) Higgsed -of the system coexist. Equivalently, this 'triple point' is the intersection of two first order lines and one second order line. In this subsection, we study the relative orientations of the two first order lines that meet at the 'triple point'.
The first order curve F separating the unHiggsed and the Higgsed phases takes the shape of a parabola as discussed in Section 7.3.3 which starts at the triple point T and goes off to infinity. The first order curve F which separates the condensed phase and one of the uncondensed phases (Higgsed for Case B while unHiggsed for Case C) has to be determined numerically and is generally more complicated than a parabola. However, there are some features which can be understood analytically.
Suppose we continue the first order curve F that separates the unHiggsed and Higgsed phases slightly further past the triple point and closer to the origin. Then, the question is as follows: does the other first order curve F go (1) to the left of, (2) along, or (3) to the right of the extended curve F ? We have illustrated the possibilities in Figure 17. It is in fact Case C Figure 17. Different possibilities for the first order curve F , labelled as (1), (2) and (3), in relation to the first order curve F . The two subfigures differ in the relative placements of the second order line (blue) and the first order line.
quite simple to figure out which of the above possibilities actually occurs in our situation given the forms of the effective potential (7.21) and (7.30) near the two non-analytic points σ B = and σ B = r respectively. We study Case B for concreteness -the results for Case C can be obtained by swapping the unHiggsed and Higgsed phases. Recall that F is the curve that separates the condensed phase and the Higgsed phase. On the extended line F , the Higgsed and unHiggsed phases coexist with equal free energy: (7.49) where σ 1 is the minimum in the Higgsed branch of the potential while σ 2 is the minimum in the unHiggsed branch. On this line and very close to the triple point, we compare the free energy at the condensed phase minimum and at the Higgsed phase minimum (which is the same as the free energy unHiggsed phase minimum). If these free energies are the same, then F coincides with the extended F . On the other hand, if the condensed phase minimum has lower free energy, the the extended curve F is in fact in the condensed phase, implying that the first order curve F is to the left of the extended F . Analogously, if the condensed phase minimum has the higher free energy, then the first order curve F is to the right of the extended F . The triple point T is on the second order line separating the condensed and the unHiggsed phases. From the general analysis of Section D.1 in the Appendix and the expression for the potential (7.21) in Section 7.2.1, we know that the minimum in the condensed phase and the minimum in the unHiggsed phase are both situated at the junction between the branches σ B = 0. When we vary the parameters to go slightly into the condensed phase along the extended F , both the condensed and unHiggsed minima go to the right of σ B = 0, 37 and if we are really close to the triple point, then we can take these minima to be at the same value of σ B = σ * ≈ 0. In this case, the free energies of the unHiggsed and condensed minima are given by (7.50) where the parameters A, a, B, b and C are given in equation (7.24). The difference between the condensed and unHiggsed phase minimal free energies is then given by where we have discarded the (σ * ) 3 term since σ * is very close to 0. Since the parameter b = −(1 − |λ B |) is negative for the potential (7.21), it follows that the condensed phase has lower free energy than the unHiggsed phase and hence, also lower free energy than the Higgsed phase since we are on the extended F line. Thus, possibility (1) in Figure 17 is realised for Case B.
In the same way, for Case C, one must compare the free energies at the condensed and Higgsed phase minima (7.52) where the parameters A, a, B, b and C are given in (7.32) corresponding to the potential (7.30). Again, the sign of the parameter b decides the phase with lower free energy. It is easy to see from (7.32) that the parameter b is positive, and hence the condensed phase has lower free energy than the Higgsed phase, and consequently, than the unHiggsed phase since we are on the extended F line. Thus, possibility (3) in Figure 17 is realised for Case C. 37 Note that the unHiggsed minimum is not inside the range of validity σ B < 0 of the unHiggsed branch of the potential and is not useful for determining the phase. However, it is indeed useful for the ongoing analysis.

Varying the chemical potentials for fixed UV parameters
In the analysis above we have studied the phase diagram of our system as a function of the three dimensionful parameters, λ B b 4 , m 2 B and |µ|. While the first two of these parameters appear in the Lagrangian of the theory (and so define its behaviour in the deep ultraviolet), the third parameter (the chemical potential) specifies the ensemble of the theory, and so the 'vacuum' about which we work.
A question of clear physical interest is the following: what is the phase structure of any particular theory as a function of the modulus of the chemical potential? What phases, in other words, do we encounter if we start at |µ| = 0 and then steadily raise the chemical potential to ∞, all the while keeping the UV parameters of the theory fixed?
The phase diagrams in Figures 14 and 16 allow us to answer the questions posed in the previous paragraph rather easily. Recall that that we have plotted |µ| on the X-axis and Y -axis respectively of the phase diagrams referred to above. Now consider the following two sets of parabolas The important point here is that α and γ depend only on (and are fixed by) the UV parameters of the theory but are independent of |µ|. In other words, as we move along the foliating parabolas (7.53), we are working in the same UV theory but at different values of |µ|. Each of the parabolas run from the origin of the phase diagram to infinity. When |µ| = 0 we start out at infinity. As |µ| is increased we move in towards the origin, reaching the origin only at |µ| = ∞. Tracing this path in the phase diagrams in Figures 34, 14, 16 and 36 gives the phase diagram as a function of µ in any particular theory given by the fixed UV parameters. In the rest of this subsection we explain in more detail how this works for values of x B 6 that lie in the 'stable range'.
In this case the phase diagram of our system is depicted in Figure 14 and reproduced in Figure 18. There are two interesting parabolas of the form (7.53) associated with this phase Figure 18. Foliating parabolas in gray for the phase diagram with x B 6 in the range φ h < x B 6 < 0. Increasing the chemical potential at fixed values of UV parameters corresponds to moving along one of these parabolas (the values of the UV parameters determine which one). The parabolas from T to F (green line) and the parabola that pass through R are distinguished. They separate parabolas across which the phase transition to a compressible phase is of second order from those along which the same transition is of first order.
diagram. The first of these passes through the triple point T in the fourth quadrant and goes along the green first order phase transition line F that starts at T in Figure 18. This curve is precisely a parabola of the form of the first of (7.53) with a particular value of α =α, as explained in Section 7.3.3. Note thatα is a negative number 38 .
A second interesting parabola associated with the same phase diagram is a parabola that passes through the point R in the second quadrant in Figure 18. This is of the the form of the second curve in (7.53) with a specific positive value for γ given by γ = γ * . The value of γ * can be determined by plugging in the coordinates of the point R (7.34) in the formula for γ in (7.55): A glance at Figure 18 will convince the reader that if our UV parameters are such that we are on the foliating parabolas with α * < α < ∞, our theory starts out in the unHiggsed phase at small |µ|. As |µ| is increased, the theory undergoes a single order second order phase transition into a condensed phase and then stays in this phase for all larger values of |µ|.
If the UV parameters are such that we are on the foliating parabolas with γ * < γ < ∞, then our theory starts out in the Higgsed phase at small |µ|. As |µ| is increased the theory undergoes a single order second order phase transition into a condensed phase and then stays in this phase for all larger values of |µ|.
Finally, if the UV parameters are such that we are on the parabolas with −∞ < α < α * or quadrants with −∞ < γ < γ * , then our theory starts out in the Higgsed phase at small |µ|, and then undergoes a first order phase transition to the condensed phase at a particular value of |µ|. The theory then stays in this phase for all larger values of |µ|. Figure 19. Foliating parabolas for the range 0 < x B 6 < φ u . The significance of these parabolas was recounted in the caption to the previous figure. The distinguished parabolas (see caption to the previous figure) are the curves TF (green line) and the parabola that passes through L.
In this case the phase diagram of our system is depicted in Figure 16 and reproduced in Figure 19. Once again, there are two interesting foliating parabolas (7.53) associated with this phase diagram. The first of these passes through the triple point T and continues along the first order curve F that separates the unHiggsed and Higgsed phases. As above, this curve is precisely a parabola. For 0 < x B 6 < 1 2 (φ h + φ u ), this curve is an α-parabola with a particular value of α =α. For the complementary range 1 2 (φ h + φ u ) < x B 6 < φ u , the curve is a γ parabola with a particular value of γ =γ. Note that bothα andγ are negative are determined numerically in general.
A second interesting parabola associated with the phase diagram is an α-parabola which passes through the point L in the first quadrant of Figure 19. This corresponds to a positive value of α given by α = α * . We have an analytic expression for α * that we obtain by plugging in the coordinates of the point L (7.27) in the formula for α in (7.55): For the range 0 < x B 6 < 1 2 (φ h + φ u ), suppose our UV parameters are such that we are on a γ-parabola with any value of γ, or an α-parabola with −∞ < α <α. Then, our theory starts out in the Higgsed phase at small |µ|. As |µ| is increased the theory undergoes a single second order phase transition into the condensed phase and then stays in this phase for all larger values of |µ|.
On the other hand if the UV parameters are such that we are on α-parabolas with α * < α < ∞ then our theory starts out in the unHiggsed phase at small |µ| and as |µ| is increased it undergoes a single order second order phase transition into a condensed phase and then stays in this phase for all larger values of |µ|. Whenα < α < α * , the theory starts out in the unHiggsed phase at small |µ|, undergoes a first order phase transition at an intermediate value of |µ| into the condensed phase and stays in that phase for all higher values of |µ|.
A similar analysis can be performed in a straightforward way for the case 1 6 < φ u depicted on the right in Figure 19.

Numerical plots
In this section we have so far presented only schematic phase diagrams for the quasi bosonic theories. However the analysis that led up to these phase diagrams was completely quantitative. Consequently, it is not difficult, at any given value of x 6 λ B to plot fully quantitative phase diagrams (the precise shapes of the first order phase transitions in these diagrams can be determined only numerically).
To illustrate this fact, in this subsection we present three sample numerical plots, in Figures 20, 21 and 22 for the three subranges in the stable range of x B 6 for the choice of 't Hooft coupling |λ B | = 1/4. The schematic plots corresponding to these are in Figures 14,  16(a) and 16(b).

Discussion and Conclusions
Following the analysis of [32], in this paper we have presented completely explicit formulaē for the occupation numbers of fermionic and bosonic quasiparticle states in large N matter Chern-Simons theories at infinite volume as a function of temperature and chemical potentials. These formulae reduce to the famous formulae (1.2) and (1.4) in the weak coupling limit, and may be regarded as 'anyonic' generalisations of these (undergraduate) textbook formulae. In our opinion the formulae (8.1) are very interesting, and deserve to be understood and rederived from many points of view. Specialising (8.1) to zero temperature we have demonstrated that the runaway instability in the Bose condensate phase of free (or large N Wilson-Fisher) bosons at values of the chemical potential larger than the thermal mass is cured by the coupling to Chern-Simons gauge theory in large N matter Chern-Simons theories. The final stabilised Bose condensate is extremely simple: it can be viewed as a theory of free bosonic quasiparticles, with each quasiparticle state of energy less than |µ| occupied with an occupation number given by (1.7). This regulated Bose condensate is the dual of an effectively free Fermi sea of fermions coupled to an analogous Chern-Simons gauge theory.
We find it fascinating that bosonic quasiparticle states appear to obey a sort of modified exclusion principle, as a consequence of which the Bose condensed phase is conceptually very similar to a Fermi Sea and should display many of its characteristic features. As in a Fermi sea, in this phase every single particle state with energy less than |µ| is fully occupied (i.e. has the maximum allowed occupation numbern). As a consequence, excitations about this Bose condensate are qualitatively similar to those about a Fermi Sea. In particular this Bose condensate hosts charged hole type excitations. It also hosts uncharged particle hole pair excitations. As in the case of the Fermi Sea, when the particle carries momentum k p with | k p | just larger than k F (k F is the modulus of the momentum of the last occupied state) and the hole carries momentum k h with | k h | just smaller than k F then this particle-hole state carries nonzero momentum but arbitrarily small energy. As in the case of a Fermi Sea, the modulus of the momentum of these particle hole excitations is always smaller than 2k F . As a consequence, we expect to see 2k F singularities -similar to those in Fermi liquid theory -in correlation functions of operators like the stress tensor that are built out of bilinears of bosonic operators. The bosonic theory should also host collective excitations of its condensate analogous to the zero sound of a Fermi surface 39 .
It would be very interesting to understand whether and how the modified 'bosonic' exclusion principle manifests itself in dynamical processes. Let us recall that for usual bosons a † |n = √ n + 1|n + 1 (8.2) (here |n and |n + 1 are the normalised states corresponding to n particles occupying a given single particle energy level). The factor of √ n + 1 in this formula tells us that bosons are gregarious; they preferentially transit to states that are highly occupied. It would be fascinating to investigate what the analogue of (8.2) (if it exists) is for Chern-Simons coupled bosons. At small λ B one would intuitively expect (8.2) to continue to apply for n n (see (1.7) for a definition: recalln is large when λ B is small). However (8.2) will certainly be significantly modified for values of n of ordern. The computation of scattering processes at finite chemical potential and temperature (which may be technically feasible at large N along the lines of [20]) could shed light on this question, and could also make connection with Haldane's characterisation of statistics in terms of occupation numbers [82] 40 . The discussion of Section 1.3 suggests that these S-matrices could also make contact with SU (N B ) k B representation theory.
The results of this paper have been obtained starting from expression for the leading order in large N (but exact to all orders in the 't Hooft coupling) partition function of the system at finite temperature and chemical potential. Explicit all-orders expressions for this partition function had previously been derived at values of the (modulus of the) chemical potential smaller than the quasiparticle masses. In the current paper we have extended these known formulae to values of the chemical potential larger than quasiparticle masses (see Section 3.3). We obtained the new free energy formulae of this paper by analytically continuing previously obtained formulae in µ. It would be interesting to confirm these results from a direct analysis of the path integral at values of the modulus of the chemical potential larger than the thermal mass.
Our extended free energy formulae involved the addition of extra terms to the free energies. These new terms qualitatively modify the dynamics of the Bose condensate, stabilising it. Equivalent new terms in the free energy expression for fermions are absent in the infinite volume limit that we have focused on in this paper; as a consequence the dynamics of the fermionic compressible phase in this limit is essentially identical to that of a free Fermi sea. Away from the infinite volume limit, however, the modification to the free energy does not vanish in other phases [15] of the fermionic theory (see subsection 3.4.2). This observation strongly suggests that the thermodynamics of fermions in such phases differs from that of a non-interacting Fermi sea; plausibly each single particle fermion state has an occupation number less than unity in such phases. This phenomenon deserves analysis, both from a direct study of the relevant free energy formulae, as well as from a study of the Schrodinger equation in the appropriate non-relativistic limits.
The fermionic and bosonic compressible phases described in this paper can both be thought of as collections of quasiparticles which occupy all states with energy smaller than the chemical potential with a given occupation number (unity for the fermions andn (1.7) for the bosons at zero temperature). The fact that each of these quasiparticles each carry a definite spin 41 suggests that each of these phases have extensive angular momenta (i.e. have definite angular momenta per unit area). In Appendix F, we have verified that this is indeed true of the free Fermi sea. It would be interesting to compute the angular momentum per unit volume of all the compressible phases studied in this paper as a function of the chemical potential and the 't Hooft coupling.
Since the results of this paper apply at every value of the chemical potential µ and the quasiparticle mass c B , they apply, in particular, in the limit |µ|−c B c B 1. In this limit, the particles that make up the Bose condensate are all non-relativistic and the Bose condensate described in this paper should admit a description as a solution to the multi-particle Schrodinger equation [81]. It would be very interesting to reproduce the 'bosonic exclusion principle' formula of Section 1.3 (also the more general formula (8.1)) from this point of view. Such an analysis would likely furnish qualitatively new understanding.
The finite chemical potential phase diagram for the regular boson theory, presented e.g. in Figures 14 and 16, is rather intricate and has many interesting features. At fixed values of the UV parameters, upon increasing the chemical potential we always encounter phase transition indicating the formation of a Bose condensate. For a range of values of the chemical potential, this transition turns out to be of first order rather than second order. It would be interesting to better understand the physics of this switch of phase transitions between first and second order.
Finally, we have studied compressible phases by first taking N → ∞ and then taking T → 0. In other words, the results of this paper apply at temperatures that are much smaller than the chemical potential but much larger than (for instance) the chemical potential divided by any given positive power of N . It is possible that, at still lower temperatures (temperatures of order of the chemical potential divided by a positive power of N ), new infrared divergences invalidate the naive 1/N expansion (so that diagrams at apparently different orders in 1/N all end up actually contributing at the same order, see e.g. [83]) leading to very interesting low temperature dynamics. It would be very interesting to investigate this further. U (N ) and SU (N ) gauge groups is lost. At finite N and κ, it is important to distinguish between U (N ) and SU (N ) groups and Bose-Fermi duality works correctly only when the groups as well as the corresponding levels are correctly chosen.
Below, we give our conventions for the Chern-Simons action for gauge fields. Then, we list the various dualities between fundamental bosons and fundamental fermions coupled to Chern-Simons gauge fields with SU (N ) or U (N ) gauge groups.

A.1 Pure Chern-Simons theories
In the Lagrangians presented in this paper and the subsequent calculations that were performed, the level that appears in the SU (N ) Chern-Simons gauge action is the so-called 'renormalized level' κ (see Section 2.3). This is valid in the dimensional regularization scheme that is adopted throughout this paper (see [64,70] for more details). This is related to the 'bare level' k via the equation (κ, k and N are generically used to denote either the bosonic quantities κ B , k B and N B or the fermionic quantities κ F , k F and N F ). The bare Chern-Simons level k appears in the current algebra of the chiral WZW model that is dual to the pure Chern-Simons theory obtained after integrating out the massive matter fields. In this appendix, we use bare Chern-Simons levels in the notation for the Chern-Simons theories. So, for example, an SU (N ) k Chern-Simons gauge field A has the following action in the dimensional regularization scheme: Incidentally, in another regularization scheme -the Yang-Mills scheme -the level that appears in the Chern-Simons action is the bare level k (see e.g. [33] for a detailed discussion regarding both the dimensional regularization and the Yang-Mills regularization schemes). The group U (N ) has a semisimple factor SU (N ) and an abelian U (1) factor, with the centre Z N of SU (N ) identified with a corresponding Z N subgroup of the U (1) factor: (we frequently suppress writing the quotient by the diagonal Z N subgroup). Thus, each factor has its own corresponding Chern-Simons level. We use the following notation U (N ) k,k to denote the levels of a U (N ) Chern-Simons gauge group: There are two kinds of U (N ) Chern-Simons theories that are of interest for us -the Type I and the Type II U (N ) theories (see Section 2 of [69]). The two theories differ in the levels of the U (1) factor of the gauge group. Type I theories have the level N κ for the U (1) factor whereas Type II theories have level N k for the U (1) factor (κ and k are the renormalized and bare levels of the SU (N ) factor (A.1)).
(A.5) Suppose the U (N ) gauge field is X µ . It can be split into a traceless SU (N ) part and a pure-trace U (1) part: where 1 N is the N × N identity matrix. The actions for the Type I and Type II theories are then given below: where the SU (N ) trace is chosen such that Tr (T a T b ) = 1 2 δ ab when the SU (N ) generators T a are in the fundamental representation. The action for the Type II U (N ) theories is given by Note: The bare level k of a pure Chern-Simons theory is quantized to be an integer due to topological considerations. Consequently, the renormalized level κ is also an integer.

A.2 Coupling fundamental matter to Chern-Simons gauge fields
In this paper, a Chern-Simons gauge theory is minimally coupled to scalar or spinor matter fields in the fundamental representation of the gauge group. When these matter fields are massive, one can integrate them out of the path integral to obtain a pure Chern-Simons theory at low energies. This integrating out of massive matter sometimes induces shifts in the level of the low-energy Chern-Simons theory compared to the level in the ultraviolet. Integrating out massive bosons does not alter the level of the Chern-Simons action while integrating out massive fermions gives rise to a shift in the level. Let us consider the case of massive fundamental fermions coupled to an SU (N F ) Chern-Simons gauge field with bare levelk F (similar remarks apply to U (1) gauge fields). When one integrates out massive fundamental fermions, there is a shift in the Chern-Simons level which depends on the sign of the fermion mass m F : The above shift is correct in the dimensional regularization scheme that is employed throughout this paper. Hence, we havẽ Since the level of pure Chern-Simons theory is constrained to be an integer, we are forced to impose the condition that the ultraviolet levelk F is a half-integer. In order to exhibit this clearly, we define so that the quantity k F is integer. The level of the low-energy Chern-Simons theory in the two cases in (A.12) is then given bỹ Thus, the level of the low-energy pure Chern-Simons theory after integrating out massive fundamental fermions is either k F or k F − sgn(k F ) depending on whether the fermion mass has the same or opposite sign as k F . Convention: When we describe the Chern-Simons theory that a fundamental fermion is coupled to, we write G k where G is the gauge group and k is the bare level of the pure Chern-Simons theory obtained by integrating out the fermion with a mass of the same sign as the level. Note that the ultraviolet bare level is not k butk = k − 1 2 sgn(k). Our convention avoids cluttering of notation due to the additional shifts.  Table 1. The levels and ranks map to each other under duality according to (2.8) which we reproduce here in terms of bare Chern-Simons levels: It must be noted that the actions written in (2.1), (2.3), (2.4) and (2.5) are a priori only for SU (N ) gauge groups. One must include extra Chern-Simons terms in the action for the U (1) factors with the (bare) levels listed in Table 1.
For the sake of completeness, we list the low-energy Chern-Simons theories that one obtains by integrating out the massive matter fields. Under duality, the massive phase of 42 For example, Type I U (N ) Chern-Simons theory coupled to fermions is denoted by the notation 2 )sgn(k F ) though the ultraviolet levels are indeed kF − 1 2 sgn(kF ) and NF (kF + (NF − 1 2 )sgn(kF )) for the SU (NF ) and U (1) factors respectively.

Bosons
Fermions the fermion with m F k F > 0 maps to the massive boson in the unHiggsed phase where the gauge group is not Higgsed and the bosonic excitations are scalars. This is presented in Table 2.
The massive phase of the fermion with m F k F < 0 loses one unit of Chern-Simons level compared to the phase with positive m F k F . This phase with negative m F k F maps to the Higgsed phase of the massive boson where the SU (N B ) part of the gauge group loses one unit of rank due to the Higgs mechanism. The duality of low-energy Chern-Simons theories in this situation is presented in Table 3.
Note that the pure Chern-Simons theories in the bosonic and fermionic theories map to each other under the level-rank duality map (A.13) in both Tables 2 and 3.

Bosons
Fermions Table 3. Low energy pure Chern-Simons theories obtained after integrating out the bosonic excitations in the Higgsed phase and correspondingly, the fermionic excitations in the phase with m F k F < 0. Note that the bosonic gauge group has its rank reduced by one unit compared to the unHiggsed gauge group while the fermionic level has reduced by one unit compared to the case with For the U (N ) theories, the global U (1) symmetry is the topological U (1) symmetry with the following topological current: where a ρ is the dynamical gauge field for the U (1) factor of the gauge group, κ is the renormalized level of the SU (N ) factor and k is the bare level of the SU (N ) factor. The coupling of a background gauge field A µ for the global U (1) symmetry to both the baryonic current and the topological current is of the standard form: Consider the so-called Type I U (N F ) Chern-Simons theory coupled to fundamental fermions discussed in Appendix A. The system is governed by the Euclidean action where X µ is the U (N F ) gauge field and the '· · · ' denotes other terms in the action (e.g. the fermion mass) that will play no role in the analysis of this section. In this appendix, we are interested in the U (1) part of the gauge field. Let where TrX µ = 0 and 1 N F is the N F × N F unit matrix. Working in the gauge in which the X 3 term vanishes and ignoring all terms that are independent of a µ , the action (B.3) reduces to The coupling of the action (B.5) to a background gauge field for its global U (1) symmetry is achieved by modifying (B.5) to As the field a µ enters quadratically in the action (B.6), it can be integrated out exactly by replacing a µ by the solution to its classical equation of motion That is, by setting Turning on a chemical potential µ amounts to setting A 0 = iµ. We see from (B.8) that this corresponds to setting a ν = iµ δ ν0 +ã ν . (B.10) If the second term in (B.10) was zero then (B.10) would assert that turning on a chemical potential for the topological U(1) symmetry of the Type 1 theory has exactly the same effect -after integrating out the U (1) part of the gauge field -as turning on a chemical potential for the resultant effective SU (N F ) fermionic theory. Whileã ν in (B.10) is not zero, it is self consistently small in the large N limit. To see this let us first supposeã ν = 0. a µ is then equal to just a chemical potential which results in turning on a matter charge density so thatψ It follows from (B.9) thatf Therefore, at least roughly speaking, the gauge field a ν (B.10) is a sum of two terms, the first of which corresponds to chemical potential of order unity in the time direction, and the second of which is the gauge potential for a uniform magnetic field of order 1 κ F . If κ F were of order unity, the second term would be very important and would greatly affect the physics of the situation. However, in the 't Hooft large N limit, κ F ∼ O(N F ) → ∞ and the magnetic fieldf 12 is effectively zero. In this limit, therefore, a turning on a chemical potential for the global 'topological' U (1) symmetry has exactly the same effect as turning on a more standard chemical potential for the effective SU (N F ) theory obtained after integrating out the dynamical U (1) gauge field. 43 In order to get (B.9), we have used the fact that, in Euclidean space, αβρ µνρ = δ µ α δ ν β − δ µ β δ ν α with the choice 123 = 123 = 1.
In the discussion above, we have worked with a Chern-Simons theory coupled to fermions; however the fermions played no real role in the discussion above, which therefore goes through also for bosons (we need to replace the LHS of (B.11) with the appropriate expression for the U (1) current for the bosonic theory).
Also, in the discussion above we worked with a U (N F ) Chern-Simons gauge theory of Type I. However, the final result followed just from the fact that the effective level of the dynamical U (1) gauge field a µ is of order N 2 F in the 't Hooft large N F limit. Thus, the above result holds in U (N F ) theories of Type II where the Chern-Simons level for the U (1) part of the gauge field is given by N F k F . In this case as well, turning on a chemical potential for the global 'topological' U (1) symmetry is equivalent, at leading order in the large N limit, to turning on a chemical potential for the effective SU (N F ) theory obtained after integrating out the U (1) gauge factor.
Recall from Appendix A that Bose-Fermi dualities map e.g. SU (N B ) theories to Type II U (N F ) theories while the Type I U (N ) theories are mapped to each other. Turning on a chemical potential on the SU (N ) side for the global baryonic U (1) symmetry maps, under duality, to turning on a chemical potential for the global topological U (1) symmetry on the U (N ) side of the duality. It follows from the discussion of this appendix that, in the U (N ) theory, this is effectively the same as turning on both a chemical potential and a magnetic field for an effective SU (N ) theory that we obtain after integrating out the dynamical U (1) gauge field. In the 't Hooft large N limit, however, the strength of the magnetic field is very weak, and this mixing between chemical potentials and magnetic fields goes away. The duality effectively relates an SU (N B ) theory to an SU (N F ) theory, where both theories are coupled only a chemical potential and the value of the chemical potential is equal on the two sides of the duality.

C Duality of off-shell free energies and occupation numbers
We show that the off-shell free energies of the appropriate fermionic and bosonic theories map to each other under the duality map. This requires the mapping of levels and ranks (2.8) the identification of the UV parameters (2.10) for the quasi-fermionic theories and (2.11) for the quasi-bosonic theories the map between the holonomy distributions (3.5) and the following identification of the off-shell variables (3.21): We divide the checking of duality into two parts. Note that the holonomy distributions appear only in one term in each of the free energies of the two bosonic and two fermionic theories. We first check that the terms independent of the holonomy distribution in the free energies map to each other under duality. We then separately check that the terms involving the holonomy distributions map to each other as well.
It is easy to check that the terms independent of the holonomy in the free energies of the the critical boson and regular fermion map to each other provided we make the duality identifications (C.1), (C.2) and (C.5) (see e.g. around equations A.6 in [64]). Similarly, it is easy to check (see e.g. [70]) that that the regular boson and critical fermion free energies (3.17) and (3.12)) map to each other once we make the duality identifications (C.1), (C.3) and (C.5). Next, we check the duality of the holonomy-dependent term in the free energies.

C.1 Duality of the off-shell free energy when c B > |µ|
It is useful to introduce the following notation for the holonomy integral in the last line of the each of the bosonic (see equations (3.14) and (3.17)) and fermionic (see equations (3.7) and (3.12)) free energies.
where, C 1 is the holonomy integral in the last line of the free energies for both the regular (in (3.12)) and critical fermion (in (3.7)) while S 1 is the holonomy integral for both the regular (in (3.17)) and critical bosonic (in (3.14)) theories. It is then easy to verify using the map (C.4) between the holonomy distributions that To verify (C.7), we start with the expression for S 1 ( , |µ|) presented in the second line of (C.6), substitute ρ B in terms of ρ F using (C.4) and make the change of variables α = π −α , when 0 < α < π , and α = −π +α , when −π < α < 0 . (C.8) We find (C.9) When > |µ|, the integral in the second term on the RHS vanishes. One can see this by power-series-expanding each of the logarithms in e −ˆ ±(|μ|+iα) (which is less than 1 sincê > |μ|) and integrating term by term. Alternatively, we may convert the integral over α into a contour integral which nowhere intersects the branch cut of the log because > µ, and use Cauchy's theorem. We have thus established (C.7).
There is a further integral of C 1 or S 1 over the energy in each of the bosonic and fermionic free energies (3.14), (3.17), (3.12) and (3.7). The range of the integral is from c B to infinity for the bosons and fromĉ F to infinity for the the fermions. Since we work with values of µ such that c B > |µ| and c F > |µ|, it follows that all the values of that appear in these integrals obey the condition > |µ| and hence (C.7) applies. Thus, the terms with the holonomy integrals also map to each other under Bose-Fermi duality.
In summary, it follows that the off-shell free energies of the regular fermion and critical boson theories map to each other under duality and similarly for the critical fermion and regular boson theories. It follows that the expressions for the full thermal free energies map to each other under duality, at least when |µ| ≤ c B and |µ| ≤ c F . In the next subsection, we check the duality of the modified off-shell free energies in the situation when |µ| > c B or |µ| > c F .

C.2 Duality of the modified off-shell free energies for c B < |µ|
The holonomy dependent parts of the modified bosonic free energies (3.47) and (3.46) x , (C. 10) and those of the modified fermionic free energies (3.44) and (3.45) are Similar to the discussion in the previous subsection, it is clearly sufficient to show that in order to establish that (C.10) and (C.11) map to each other under duality. We will now proceed to verify (C.12). We need the duality relations between the holonomies ρ B , ρ F (C.4) and the analytically continued holonomiesρ B ,ρ F (3.41) which we reproduce below: It is then not difficult to verify that the LHS of (C.12) simplifies to (C.14) The above equation can be rewritten using the variable change z = e iα as where the contour C runs over the unit circle. The second logarithm in the first term of (C.15) is analytic inside the unit circle (its branch cut starts outside the unit circle) and so, by Cauchy's theorem, the contour integral involving this logarithm vanishes. It follows that (C.15) simplifies to Now the contour integral involving the logarithm in (C.16) does not vanish, but receives contributions only from the discontinuity across the cut of this logarithm. Evaluating this discontinuity we find that (C.16) becomes establishing (C.12).

C.3 Duality of the occupation numbers
We will now examine how the occupation numbersn B ( , µ) (as computed in (4.20)) and n F ( , µ) (as computed in (4.18)) map to each other under duality. We have 19) to argue that the third line of (C.18) evaluates to the second term on the fourth line of the same equation. We have also broken up the integral in the second term in the second line (the term involving ρ F ) into an integral from −π to 0 and 0 to π, and then made the change of variables (C.8) and used e ±iπq = −1 and (C.13) to demonstrate that the second line of (C.18) reduces to the first term on the fourth line of the same equation. In going from fourth line to the last line of (C.18), we have used

D Curve sketching
In this appendix we study a toy model which is governed the following quantum effective potential Note that U eff (σ) is analytic (and polynomial) everywhere away from σ = 0. At σ = 0, U eff (σ) is continuous and once-differentiable, but its second and third derivatives are discontinuous.
We designate the σ < 0 branch in (D.1) as the '−' branch and the σ > 0 branch as the '+' branch. Whenever a local minimum exists in the − or + branch and we choose our theory to be in that vacuum, the theory is said to be in the − or + phase.
The potential U eff (σ) has 5 parameters, A, B, C, a and b. We will also impose the condition a > 0 44 but allow all the other four parameters to have either sign 45 . In this section we will study the phase diagram of the model (D.1) at fixed values of A, a, b, but as functions of the two parameters B and C. Thus, the phase diagram will be two dimensional, with C parametrizing the x-axis and B parametrizing the y-axis.
It turns out that the phase diagrams in question are qualitatively different depending on the signs of A, A + a and b. As a consequence we will present 2 3 − 2 different phase diagrams, one for each of the allowed signs for these three 'parameters' (since we have a > 0, the case A > 0, A + a < 0 with either sign of b is excluded from the analysis).
The sign of A (resp. A + a) decides the boundedness of the potential for large negative (resp. positive) σ. We thus consider the three cases separately 46 : Case I: A > 0 , A + a > 0 , Case II: A < 0 , A + a > 0 , Case III: A < 0 , A + a < 0 .
(D.2) Cases I and III are unbounded below (for negative σ and positive σ respectively) while Case II is stable. Within each case, we consider two subcases corresponding to b > 0 and b < 0.

D.1 Some analytic features of the phase diagram
The extremization equation of the effective potential, U eff (σ) = 0, is separately quadratic both when σ < 0 and σ > 0. The discriminants of these two quadratic equations are given 44 Simply because this condition always turns out to be true in the case of physical interest to us, see Section 7.2. 45 Under the change of variables σ → ασ (with α > 0), these 5 variables scale as so in actuality our toy model (D.1) has 4 continuous parameters. 46 When one of the σ < 0 or σ > 0 branches in (D.1) corresponds to the condensed branch of the regular boson effective potential (7.13) which exists only for a finite range of σ, the question of boundedness is not very relevant since σ is not allowed to grow too large.
The extremum at σ = 0 is always a legitimate one for both branches. However, the other extremum must be discarded if it occurs outside the range of validity σ ≶ 0 for the − or + branches. The second derivative of the potential in either branch at the extremum σ = 0 is given by Based on the above facts, we deduce three different kinds of transitions in the behaviour of the potential (D.1) depending on the range of B. We give the details separately for b > 0 and b < 0. The potential is sketched in Figure 23 for each of the different cases. (c) B < 0: Both branches of the potential have a maximum at σ = 0, thus giving rise to a maximum for the full potential at σ = 0.
So far, we have looked at the behaviour on the potential by setting C = 0 and varying B. Suppose we now broaden our focus to a small neighbourhood of the C = 0 line. To begin with, note that the slope of the potential U eff (σ) at σ = 0 is given by C in both branches of the potential. Let us study the potential as we take C through zero from positive to negative values by choosing B to be in one of the three ranges discussed previously. When B is such that we are on the blue line in Figure 23, a local minimum crosses from the − branch to the + branch. When we choose B to be on the black line in Figure 23, a local maximum crosses from the + branch to the − branch.
Finally, when we choose B such that we are on the red line segment, two kinds of behaviour are possible depending on the sign of b. First, consider the case b > 0. The potential is monotonic near the origin when C > 0, has a point of inflection at C = 0 and develops a local maximum in the − branch and a local minimum in the + branch for C < 0. Second, when b < 0, the potential starts with a local minimum in the − branch and a local maximum in the + branch for C > 0, has a point of inflection at C = 0 and is monotonic near σ = 0 for C < 0.
We stress that the above description applies to only the local behaviour of the potential near σ = 0, the junction of the two branches. In particular, there may be other local extrema of the potential as one goes deeper into either of its branches. A full analysis of the potential is required in order to understand its global structure, which is what we turn to next.

D.2 Case I:
The potential is bounded below for σ > 0 and unbounded below for σ < 0. We have the following cases: Similarly, the potential is monotonic in the + branch when B + b > 0 or (B + b < 0 with D + < 0) and has extrema when (B + b < 0 with D + > 0). Then, we have the following possibilities.

D.3 Case II:
The potential is bounded below for both σ < 0 and σ > 0.  Figure 27(b). The minimum in the + branch could also be lower than the minimum in the − branch though it is not depicted in Figure 27(b).  Figure 27(d). The minimum in the − branch could also be lower than the minimum in the + branch though it is not depicted in Figure 27(d).

D.4 Case III:
In this case, the effective potential is bounded below for σ < 0 and unbounded below for σ > 0. Figure 27. Effective potential for Case II: The minimum in one of the branches is typically lower than the other. There will be a first order phase transition when the dominant minimum switches from one branch to the other. We have shown the two possibilities in Figure 30(d).
2. C > 0: There is always a minimum in − branch and a maximum in the + branch, irrespective of the value of the parameter B. Figure 30. Effective potential for Case III: Figure 31. Detailed structure of the potential for Case III: A < 0, A + a < 0. Refer to the caption of Figure 24 for details on the various marked lines.

E The phase diagram of the regular boson theory with an unstable potential
In this section, we present the analysis of the effective potential (7.13) for the unstable ranges of x B 6 viz. Case A and Case D in (7.18): As we have remarked earlier, since the potential is unbounded below either for large and positive σ B (Case A) or for large and negative σ B (Case D), the phases that may exist are metastable at best. However, we present the calculations simply because it is informative and for the sake of completeness. The analysis of the current section may also be useful in the study of the N = 2 supersymmetric Chern-Simons theory with one fundamental chiral multiplet in the presence of chemical potential 47 .
Recall that we work only at leading order in the large N limit in which case the parameter x B 6 is exactly marginal. However, this is no longer the case when when the subleading corrections in 1/N for the β-function of x B 6 are taken into account [69]. The current analysis may be useful in this situation since the fixed points of x B 6 may well end up being in one of the above unstable ranges (see [69] for more details).

E.1 Case
For this range of x B 6 , both local expressions of the potential (7.21) and (7.30) are unbounded below for large and positive σ B and σ B respectively since the quantity A satisfies A < 0 and A + a < 0 for both (7.21) and (7.30). Thus, Case III of Appendix D applies to the potential near both non-analytic points. We must remark that the unboundedness of (7.21) is not very serious since it occurs in the condensed branch which is eventually cut off at We plot the special lines corresponding to the potential near both non-analytic points in Figure 33. The segments of D u , D c and D h which are present in this phase diagram are 47 The phase diagram of this theory with zero chemical potential was computed in [71]. The main strategy in that paper was to use the results for the effective potential for the regular boson to understand the local behaviour of the effective potential for the N = 2 theory near its non-analytic points (analogous to what we do in this paper). The local behaviour of the potential was sometimes decided by the unstable cases of the regular boson, though the final global potential was stable. The global stability of the potential essentially meant that the unboundedness of the local potential was replaced by a very deep minimum in the global potential which was situated outside the range of applicability of the local potential. specified by the following conditions: For each region demarcated by the special lines in Figure 33(a), we display the shape of the potential in the neighbourhood of the points σ B = and σ B = r as a small inset figure (with the plot on the left (resp. right) corresponding to (resp. r)). In Figure 33(b), we combine the information from the plots near and r to give a global shape of the potential in each region demarcated by the special lines. Note that one is able to obtain this information just by comparing the common region of the two plots and without any detailed quantitative analysis of the potential. Since the potential is unbounded from below for σ B 0, all the phases for this range of x B 6 are metastable at best. We show this "phase" diagram in Figure 34. Figure 33. Case I of Appendix D applies to the behaviour of the potential near both non-analytic points. The conditions on λ B b 4 which specify the segments of the discriminant parabolas that appear in the phase diagram are Higgsed Figure 34. Phase diagram for x B 6 < φ h . The blue lines are second order transitions, the green line is a first order transition while the red lines demarcate regions of runaway potential from the regions where metastable phases exist. Recall that the potential is unstable and all the phases displayed above are metastable at best. We have restored the |µ| dependence of the phase diagram by rescaling m 2 B and λ B b 4 by appropriate powers of |µ|.
The detailed structure of the potential is given as small inset plots in Figure 35(a) in each region demarcated by the special lines of the two local potentials (7.21) and (7.30). The global structure of the potential is displayed in each of these regions in Figure 35(b). Since the potential is unbounded below in this range of x B 6 , the phases in this case are metastable and are displayed in Figure 36. Figure 35. Structure of the potential for φ u < x B 6 . Recall that the effective potential is unbounded from below in this case so that the 'phases' displayed here are dominant metastable phases when they exist. To the right of the red curve the effective potential is monotonic so no metastable phases exist. The green curve is a first order transition (between dominant metastable phases) while the blue curves are second order transitions between such phases. We have restored the |µ| dependence of the phase diagram by rescaling m 2 B and λ B b 4 by appropriate powers of |µ|.  Recall that the effective potential is unbounded from below in this case so that the 'phases' displayed here are dominant metastable phases when they exist. To the right of the red curve the effective potential is monotonic so no meta-table phases exist. The blue lines are second order transitions and the green curve is a first order transition between metastable phases. To the 'right' of the red curve no metastable phases exist. We have restored the |µ| dependence of the phase diagram by rescaling m 2 B and λ B b 4 by appropriate powers of |µ|.

F Preliminary discussions on angular momentum
The 2 + 1 dimensional bosons and fermions that we study in the main text of this paper do not appear in spin multiplets but instead each carry the same spin in the rest frame. For this reason we expect a macroscopic collection of these particles to carry angular momentum that is extensive (i.e. scales like the volume).
In this appendix, we will demonstrate that -at least with one reasonable definition of the angular momentum -this expectation is indeed correct for the Fermi sea of a collection of free fermions of unit charge in 2 + 1 dimensions. We leave the definition and computation of the angular momentum of thermodynamical ensembles of Chern-Simons-matter theories to future work 48 .

F.1 The free Dirac equation and its symmetries
The free Dirac equation follows from the action We make the following choice of gamma matrices Note that γ 0 γ i = − ij γ j , (F.4) where 12 = − 21 = 1.

F.1.1 Symmetries of the Lagrangian
Parity: It is easy to check that the field redefinition χ(x, y, t) = γ y Ψ(x, −y, t) ≡ (P y · Ψ)(x, y, t) , (F.5) turns the Dirac action (F.2) into In other words the parity operation (F.5) (corresponding to y → −y) flips the sign of the mass term. Equivalently, the parity operation combined with a flip in the sign of the fermion mass is a symmetry of the system. Charge conjugation: In a similar vein, it is easy to check that the field redefinition χ(x, y, t) = γ x Ψ * (x, y, t) ≡ (C · Ψ)(x, y, t) , (F.7) 48 One additional complication that the Chern-Simons gauge field would introduce is the presence of WZW boundary degrees of freedom, which, by themselves, could carry extensive angular momentum.
turns the Dirac action (F.2) into In other words, complex conjugation (F.7) preserves the Dirac action and is a symmetry of the system. Time reversal: Finally, note that the field redefinition χ(x, y, t) = γ 0 Ψ(x, y, −t) ≡ (T · Ψ)(x, y, t) , (F.9) turns the Dirac Lagrangian (F.2) into The 'time reversal transformation' T (F.9) 49 and a flip in the sign of the fermion mass together flips the overall sign of the Lagrangian and so preserves the Dirac equation.

F.1.2 The Hamiltonian, angular momentum and symmetries
The (single-particle) Dirac Hamiltonian The single-particle angular momentum operator is given by As expected, x i and γ i rotate the same way under rotations (both are vectors in the spatial plane). It follows that the operators x i γ i and γ 0 both commute with J. We now study the interplay between solutions of the Dirac equation and the discrete symmetries discussed in the previous subsubsection.
First, let us look at parity. The analysis around (F.6) tells us that if Ψ(x, y, t) is a solution of the Dirac equation then γ y Ψ(x, −y, t) is also a solution of the same equation with flipped mass. Applying this result to solutions of the form ψ(x, y)e −iEt , we conclude that if ψ(x, y) is an eigenstate of the Dirac Hamiltonian H(m F ) with energy E, then γ y ψ(x, −y) is an eigenstate of the Dirac Hamiltonian H(−m F ) with the same energy. This is easy to check directly. Let ψ(x, y) be an eigenstate of H(m F ) with energy E i.e. iγ 0 γ i ∂ i + im F γ 0 ψ(x, y) = Eψ(x, y) .
Then it is easily verified that iγ 0 γ i ∂ i − im F γ 0 γ y ψ(x, −y) = Eγ y ψ(x, −y) . (F. 16) Then, the above discussion can be phrased in terms of the parity operator P y as It follows that parity operator flips angular momentum; it maps an angular momentum eigenstate with eigenvalue j to an eigenstate of eigenvalue −j. Consider a simultaneous eigenstate of the Dirac Hamiltonian H(m F ) and the angular momentum operator J with eigenvalues E and j respectively. Then the parity operator P y maps this eigenstate to an eigenstate of H(−m F ) and J with eigenvalues E and −j respectively. We can repeat this discussion for the charge conjugation operation. It follows from the discussion around (F.8) that if ψ(x, y)e −iEt is a solution to the Dirac equation with mass m F , then γ x ψ * (x, y)e iEt is also a solution to the Dirac equation with mass m F . In other words, if ψ(x, y) is an eigenstate of H(m F ) with energy E then γ x ψ * (x, y) is another eigenstate of H(m F ) but with energy −E. Again this is easy to check explicitly. Given that (F.15) holds, it follows that iγ 0 γ i ∂ i + im F γ 0 γ x ψ * (x, y) = −Eγ x ψ * (x, y) , i.e. CH(m F ) = −H(m F )C . (F. 22) Similarly, it is easy to verify that 51 J (γ x ψ * (x, y)) = −γ x (Jψ(x, y)) * , i.e. JC = −CJ .
(F. 24) In summary, if the state ψ has energy E and angular momentum J, then the state γ x ψ * has energy −E and angular momentum −J. 50 If Ψ(x, y, t) is an eigenstate of the angular momentum operator (F.12) with eigenvalue j i.e.
JΨ(x, y, t) = jΨ(x, y, t) , (F. 18) then, J γ y Ψ(x, −y, t) = −j γ y Ψ(x, −y, t) (F.19) 51 More explicitly, if Ψ(x, y, t) is an eigenstate of the angular momentum operator (F.12) with eigenvalue j, then, it is easy to convince oneself that J γ x Ψ * (x, y, t) = −j γ x Ψ * (x, y, t) (F. 23) Finally, the discussion around (F.10) tells us that if ψ(x, y)e −iEt is a solution to the Dirac equation with mass m F , then γ 0 ψ(x, y)e iEt is also a solution to the Dirac equation with mass −m F . It follows that if ψ(x, y) is an eigenstate of the Dirac Hamiltonian H(m F ) with energy E, then γ 0 ψ(x, y) is another eigenstate of H(−m F ) with energy −E. Again this is easy to check explicitly: iγ 0 γ i ∂ i + im F γ 0 ψ(x, y, t) = i∂ t ψ(x, y, t) .
( In summary, if a simultaneous eigenstate ψ of H(m F ) and J has energy E and angular momentum j respectively, then the state γ 0 ψ is a simultaneous eigenstate of H(−m F ) and J with energy −E and angular momentum j respectively.

F.2 Boundary conditions at infinity
Since the angular momentum of a Fermi sea on a truly infinite space will turn out to be infinite, we need an IR regulator to get a sensible answer. We adopt the following strategy. Let us define the radial spatial variable r and the unit position vector by We study the Dirac on the cut off plane r ≤ R, subject to one of the two the boundary conditions γ ixi ψ = ±ψ at r = R . (F.30) The boundary conditions with the ± signs in (F.30) define two distinct Hilbert Spaces H + and H − respectively. We will now discuss the interplay between the boundary conditions (F.30) and the symmetry operations (F.5), (F.7) and (F.9). Consider solutions ψ ± ∈ H ± that obeys the boundary conditions (F.30). It is easy to check that γ ixi γ y ψ ± (x, −y) = ∓γ y ψ ± (x, −y) γ ixi γ x ψ * ± (x, y) = ±γ x ψ * ± (x, y) γ ixi γ 0 ψ ± (x, y) = ∓γ 0 ψ ± (x, y) (F.31) Thus, parity and time reversal map H + to H − and vice versa (recall that these two operations also flip the sign of the mass parameter m F ). On the other hand the charge conjugation operation acts separately within H + and H − .

F.3 A convenient definition of the angular momentum of the Fermi sea
Let J ± (µ, m F ) be the angular momentum of the Fermi sea with chemical potential µ of fermions with mass m F in the Hilbert Space H ± . The fact that the charge conjugation operation maps every state of energy E and angular momentum j in H + (resp. H − ) to another state of energy −E and angular momentum −j in H + (resp. H − ) (and the fact that a antiparticle is the removal of a particle in a state of negative energy 53 and therefore has minus the charges of the negative energy state in question) tells us that the spectrum of particle and antiparticle states are exactly isomorphic. For every particle state of charge 1, energy E and angular momentum j, there is a corresponding antiparticle state of charge −1, energy E and angular momentum j. In other words  53 In the first quantized formalism in which we work in this Appendix, the fermion has both positive and negative energy states. The vacuum of the theory at µ = 0 is the Dirac Sea in which all negative energy states are filled and all positive energy states are unfilled. The positive µ Fermi Sea is the state in which all negative energy states continue to be filled, but in addition, positive energy states with energies less than µ are also filled. At negative µ, on the other hand, no positive energy states are filled and, in addition, some negative energy states -those with > µ -are also now empty (the other negative energy states continue to be filled). In contrast to this Appendix, in the main text we have adopted the second quantized viewpoint in which all fermions states have positive energy, but there are two kinds of fermion states; those corresponding to positive charge 'particles' (these map to the positive energy states of the Dirac Sea picture) and those corresponding to negative charge 'antiparticles' (these map to the removal of negative energy states in the Dirac Sea picture).

F.4 Solutions of the Dirac equation
With the choice of γ matrices (F.3), it is not difficult to verify that a basis of regular solutions to the Dirac equation in polar coordinates is given -upto normalisation -by where n is an integer, positive or negative and the energy is given by E = η|E| = η p 2 + m 2 F . η = ±1 for positive/negative energy solutions, respectively 54  Let ψ E,j,m F denote the eigenfunction of the Dirac Hamiltonian H(m F ), energy E and angular momentum j (E and j both range over positive and negative values in our notation). The explicit form of ψ E,j,m F is easily read off from (F.37), (F.42) and (F.43). As a check on our algebra we have verified that, these explicit solutions obey γ y ψ E,j,m F (r, −φ) ∝ ψ E,−j,−m F (r, φ) , (F. 44) as predicted on general grounds in Section F.1.2.

F.5 Boundary conditions on the solutions
In writing (F.37) we have not yet imposed any boundary conditions at r = R. Turning to the boundary conditions (F.30) now, it is easy to check that the solution (F.37) belongs to H ± provided χ ± n,η (p) ≡ J |n|+sgn(n) (pr) ± sgn(n)η |E| − ηm F |E| + ηm F J |n| (pr) = 0 .

F.6 Computation of the Angular Momentum
We wish to enumerate the states in both H + and H − graded by their angular momenta that contribute to the Fermi sea of chemical potential µ. In the computations presented in the rest of this appendix we will assume that µ > 0 so that we need to deal with only particles with positive energy states rather than 'antiparticles' with positive energy (i.e. states in which particles of negative energy are removed or rather, absent). As we know that J ± (µ, m F ) is an even function of µ (see (F.32)), this restriction results in no loss of generality.
We proceed with our computation as follows. Let N ± n (µ, m F ) denote the number of eigenstates of the Dirac Hamiltonian H(m F ) in the Hilbert Space H ± , with discrete label n and with energy less than µ. These are states such that i.e with p < p F where p F is defined by the equation It follows that be reorganised in three useful ways, each of which we now list: (F. 54) In particular note that the difference between N values in the first line in (F.54) is of order unity rather than order R (it is easy to check that this quantity vanishes when m F = 0 using the boundary condition equations (F.45)). This establishes that the angular momentum is actually of order R 2 rather than R 3 .

F.7 Angular momentum in the non-relativistic limit
The utility of the second line of (F.54) is that it allows us to see that the angular momentum of the Fermi sea is simply 1 2 times the number of occupied single particle states in the non-relativistic limit at positive mass m F > 0. To see this, we note that, for large enough n, it follows from (F.53) that N ± n = N ± −n−2 . And so, we get from the second line of (F. and that, in this limit, the angular momentum is exactly half of the charge of the Fermi sea. To see this explicitly in formulas, we note that at large n, we can replace this summation by an integral and use (F.53) to get where, we have defined α = n/(p F R) for brevity. Performing the integral, the final result is This result is exactly the 1/2 times number of occupied single-particle states 56 in the non-relativistic limit at positive mass m F > 0. 56 This can be easily obtained from the phase-space counting, p≤p F , where, V2 = πR 2 is the volume of the two dimensional space with a radial cutoff at R.
In the same way, the third line allows us to see that the angular momentum of the Fermi sea is simply − 1 2 times the number of occupied single-particle states in the non-relativistic limit at negative mass m F < 0. The argument is similar to the positive mass case. The final answer in this case is given by Combining (F.58) and (F.59), we see that in the non-relativistic limit Note, in particular, that this angular momentum is extensive.

F.8 General form of angular momentum
Away from the non-relativistic limit the angular momentum of the Fermi sea, defined in this appendix, takes the form is a yet to be carefully computed function which has the properties We leave the verification or improvement of (F.63) (a relatively easy exercise) and its generalisation to nonzero 't Hooft coupling (a more interesting exercise) to future work.