Condensates of interacting non-Abelian $SO(5)_{N_f}$ anyons

Starting from a one-dimensional model of relativistic fermions with $SO(5)$ spin and $U(N_f)$ flavor degrees of freedom we study the condensation of $SO(5)_{N_f}$ anyons. In the low-energy limit the quasi-particles in the spin sector of this model are found to be massive solitons forming multiplets in the $SO(5)$ vector or spinor representations. The solitons carry internal degrees of freedom which are identified as $SO(5)_{N_f}$ anyons. By controlling the external magnetic fields the transitions from a dilute gas of free anyons to various collective states of interacting ones are observed. We identify the generalized parafermionic cosets describing these collective states and propose a low temperature phase diagram for the anyonic modes.


I. INTRODUCTION
The fractionalized quasi-particle excitations of topological states of matter have attracted a lot of attention in the recent years. A particulary interesting class of these quasi-particles are so-called non-Abelian anyons. Their remarkable exchange statistics makes them a resource for decoherence-free quantum computing [1] which has further driven the search for physical realizations. Candidate systems are the topologically ordered phases of two-dimensional quantum matter such as the fractional quantum Hall states or p + ip superconductors where non-Abelian anyons may appear as zero-energy degrees of freedom of gapped excitations in the bulk [2][3][4].
Mathematically, anyons are objects in a braided tensor category. In this description they are characterized by their braiding and fusion properties. These completely determine the physics of a dilute anyon gas and quantum computing operations can be realized based on the braiding of the anyonic quasi-particles [5]. The fusion rules on the other hand determine the Hilbert space of a many-anyon system as well as the possible local interactions between pairs of anyons. The presence of the latter lifts the degeneracy of the zero-energy modes and leads to the anomalous collective behaviour of systems with a finite density of anyons, e.g. when they condense at the boundaries between phases of different topological order. This can be exploited to stabilize topological quantum memories [6].
The properties of interacting anyons forming a high density condensate on the edge of the arXiv:1906.09929v1 [cond-mat.str-el] 21 Jun 2019 topologically ordered phase of a two dimensionsal quantum system have been studied in various effective lattice models [7][8][9][10][11][12][13]. Combining numerical methods with insights from exactly solvable models and conformal field theory important insights into the collective behaviour of different types of non-Abelian anyons have been obtained. Unfortunately, these lattice models do not allow to tune the anyon density. To study the transition between the low density phase of 'bare' anyons and the collective state realized at high anyon densities one can follow the approach of [14][15][16] In the present paper we extend this approach to fermions with an SO(5) spin degree of freedom. 1 Again we find that the excitations in the spin sector are massive solitons. They form multiplets in the SO(5) vector and spinor representations and, in addition, carry an internal degree of freedom which we identify as non-Abelian SO(5) N f anyons. The density of solitons can be controlled by external fields coupling to the SO(5) Cartan generators. For sufficient large magnetic fields solitons in form a condensate described by a U (1) Gaussian model. The condensation of the solitons is accompanied by the formation of collective states for the anyon degrees of freedom which are found to be described by generalized parafermionic conformal field theories. We note that this complements previous results obtained for the high density collective states of interacting SO(5) 2 anyons in lattice models [10,13].
In [19] the magnetic properties of relativistic chiral fermions with SO(5)×SO(5) symmetry have been studied. Here we consider a generalization of this model given by a Hamiltonian densities describing fermions in external magnetic fields h i (i = 1, 2) with anisotropic spin-spin interaction where ψ f α are Dirac spinors with U (N f ) 'flavor' indices f = 1, . . . , N f and SO(5) 'spin' indices a = 1, . . . , 5 (the latter are suppressed in (2.1)). H i c (i = 1, 2) are the generators of the so(5) Cartan subalgebra while the so(5) ladder operator for a root α in the Cartan-Weyl basis is denoted by E α . Moreover, the γ µ (µ = 1, 2) are Dirac matrices andψ f = γ 0 ψ † f . The low-energy excitations of (2.1) carry SO(5) spin and U (N f ) charge degrees of freedom (cf. [20] for the isotropic case). Here we are interested solely in the SO(5) spin degrees of freedom.
By placing N fermions into a box of length L with periodic boundary conditions the energy contribution of the spin excitations specified by quantum numbers N 1 , N 2 is where g, p 0 are functions of the coupling constants λ and λ ⊥ in (2.1). H 1 and H 2 are linear combinations of the magnetic fields introduced above, i.e. H 1 ≡ α 1 · h, H 2 ≡ α 2 · h with the simple roots α 1 , α 2 of so (5). Also notice that the relativistic invariance of the fermion model is broken by the choice of boundary conditions but will be restored later by considering observables in the scaling limit L, N → ∞ and g 1 such that the mass of the elementary excitations is small compared to the particle density N /L. The complex parameters λ Refs. [19,21] for the isotropic case) τ =±1 where e k (x) = sinh π 2p 0 (x + ik) / sinh π 2p 0 (x − ik) . Based on equations (2.3), (2.2) the thermodynamics of the model can be studied provided that the solutions to the Bethe equations describing the eigenstates in the limit N → ∞ are known. Here we argue that the root configurations corresponding to the ground state and excitations relevant for the low-temperature behavior of (2.1) can be built based on a generalized string hypothesis, see e.g. Refs. [22,23]: in the thermodynamic limit the Bethe roots λ (m) α are grouped into j-strings of length n j and with parity v n j ∈ {±1} The allowed lengths and parities depend on the parameter p 0 . To simplify the the discussion below we assume that p 0 = N f + 1/ν with integer ν > 2 where only a few string configurations are relevant for the low-temperature thermodynamics n (1) together with the string configuration for even ν Within the root density approach the Bethe equations are rewritten as coupled integral equations for the densities of these strings [24]. For vanishing external fields one finds that the Bethe root see Appendix A. As mentioned above relativistic invariance is restored in the scaling limit g 1 where the solitons are massive particles with bare densities ρ (m) 0,j 0,m and bare energies where z = (1 + N f ν). The prefactors 2 √ 3M 0 and 2M 0 with M 0 ≡ e −π/3g are the masses of the [1, 0]-and [1, 1]-solitons, respectively. Furthermore, the corresponding charges can be read off from (2.7): for a general excitation with mass M and bare energy 0 (λ) its charges (q 1 , q 2 ) are defined by where ω 1 , ω 2 are the components of a weight in a so ( Similarly, the bare densities and energies of thej 0,m -strings are 0,j 2 (λ). The energy density of a macro-state with densities given by (2.5) is Furthermore, it is convenient to define the masses M To derive the physical properties of the different quasi-particles appearing in the Bethe ansatz solution of the model (2.1) its low-temperature thermodynamics is studied. The equilibrium state at finite temperature is obtained by minimizing the free energy, F/N = E −T S, with the combinatorial The resulting thermodynamic Bethe ansatz (TBA) equations read where the dressed energies For studying the properties of free and interacting solitons it is convenient to rewrite the integral equations of the auxiliary modes: the auxiliary modes become independent of λ for (m) j 0,m (0)/T → ∞ as well as for finite λ with (m) j 0,m (0)/T → ±∞. In these cases the effective equations describing the auxiliary modes are In terms of the dressed energies the free energy per particle is Similarly, for sufficiently small  Figure 3 shows

B. Non-interacting solitons
For fields zH 2 < min 2 √ 3M 0 − zH 1 , 2M 0 − zH 1 /2 temperatures below the gaps of the solitons are considered, i.e. T min 0,j 0,2 (0) . Analogously to [14][15][16] the nonlinear integral equations (2.13) can be solved iteratively in this regime: the energies (m) k of solitons are well described by their first order approximation while those of the auxiliary modes can be replaced by the asymptotic solution for |λ| → ∞, see Table I for is obtained for m = 1, j ∈ {j 0,1 ,j 0,1 } and m = 2, j ∈ {j 0,2 ,j 0,2 } resulting in the free energy  j0,m (0)) close and the degeneracy of the auxiliary modes is lifted. where Q (m) k (k = j 2,m ) depends on the asymptotic solution of the auxiliary modes .

(2.17)
See j0,2 ) derived from (2.17) using the asymptotic solutions of the auxiliary modes (see Table I where θ is the highest root. In terms of the Dynkin labels (m 1 , m 2 ), the condition (2.19) results in  [2,2] .
The corresponding fusion rules can be found in [10] using the identification Notice that these fusion rules are consistent with the tensor product reductions of so (5) irreducible representations with reasonable modifications due to the level N f = 2 2 . The quantum dimensions extracted from the fusion rules for N f = 2 are given by Therefore, the appearance of the internal degrees of freedom, Q (1) and Q (2) , can be interpreted as [1,1] the dominant contribution to F is that of the [1, 0]-solitons with [1,1] anyons being bound to them.
In the remaining region of non-interacting solitons the [

21)
2 An elegant graphical method for deriving tensor product reductions for Lie algebras with rank r ≤ 2 can be found in [27].

(2.24)
For the remaining term, f (φ (m) j 2,m (∞)), an analytical expression is not known. However, it can be computed numerically using the results for the asymptotic behavior of the auxiliary modes from Table I. From (2.24) one can further conclude that the densities for |λ| < λ δ are given by L sin 2 (π/n) sin 2 (πk/n) = π 2 6 2(n − 3) n (2. 25) it is found that In general Rogers dilogarithm identities giving the relationship between Lie algebras and central charges of parafermion conformal field theories have only been proven for the simply laced case [29].
However, for the non-simply laced Lie algebra so(5) similar relations can be verified numerically Hence, we obtain the following low-temperature behavior of the entropy which is consistent with a conformal field theory describing the collective modes given by the coset Using the conformal embedding where Z G denotes generalized parafermions given as the quotient G/U (1) rank(G) involving the group G [30], the collective modes can equivalently be described by a product of a free U (1) boson and a parafermion coset Following [15] the entropy S = − d dT F N is computed numerically to study the transition from free anyons to a condensate of anyons. In the region 2 √ 3M 0 −zH 2 zH 1 the entropy deviates from the asymptotic expression (2.27): in this range of H 1 the auxiliary modes of the first level propagate with a velocity (independent of j 2,1 ) differing from that of the [1, 0]-solitons, v [1,0] , namely where Λ 1 denotes the Fermi point of [1, 0]-solitons defined by (1) j 0,1 (±Λ 1 ) = 0. Also notice that Fermi velocities of the second level do not exist in this regime. As a consequence the bosonic (spinon) and parafermionic degrees of freedom in the first level separate and the low-temperature entropy is Actually, this behavior can only be seen for temperatures T < 0.02 M0, which was not accessible by available numerical methods. To overcome this problem the entropy for T = 0.02 M0 was computed, while already neglecting the contribution of (2) j 0,2 in the integral equations (2.13). together with the numerical expressions for f (φ (m) j 2,m (∞)) using the asymptotic behavior of the auxiliary modes from Table I. The densities for |λ| < λ δ following from (2.30) are where e − (1) j 2,1 /T = const. for |λ| < λ δ . Since the integral equations (2.5) for ρ h(1) j 2,1 simplify in this regime to ρ h(1) (λ δ ) = 0 for all j, m is obtained. Using the Rogers dilogarithm identity (2.25) the relation for Z SU (2) N f parafermions is found: which is consistent with a conformal field theory describing the collective modes given by the coset Using the conformal embedding [30], the collective modes can equivalently be described by a product of a free U (1) boson and a parafermion coset Notice that for N f = 2 the central charge of the coset Z SO(5) N f /Z SU (2) N f is c = 3/2, which is consistent with the results for interacting chains of [1, 0] SO(5) N f anyons [13].
Analogously to the regime discussed in Section II C, the entropy deviates from the asymptotic expression in the region 2M 0 − zH 1 zH 2 , since the auxiliary modes of the second level propagate with a velocity differing from that of the [1, 1]-solitons, v [1,1] , namely where Λ 2 denotes the Fermi point of [1,1]-solitons defined by (2) j 0,2 (±Λ 2 ) = 0. Also notice that Fermi velocities of the first level do not exist in this regime. As a consequence the bosonic (spinon) and parafermionic degrees of freedom in the first level separate and the low-temperature entropy such that Λ 2 (H 2,δ ) > λ δ . In Figure 5 (b) the computed entropy is shown as a function of the field H 2 together with the T → 0 behavior expected from conformal field theory.  [1,1] and v (2) pf , respectively (full red line). For magnetic fields zH 2 < 2M 0 and temperature T 2M 0 the entropy is that of a dilute gas of non-interacting quasi-particles with degenerate internal degree of freedom due to the anyons. v [1,0] , v [1,1] , v to the central charge, respectively.
For fields H 1 , H 2 such that v [1,0] pf < 1 the degeneracy between the solitons and the parafermions is lifted resulting in the low-temperature behavior of the entropy given by Additionally, the fields can be chosen such that the remaining degeneracies are lifted, i.e. v [1,0] < v [1,1] and v (1) pf < v (2) pf . In this case the entropy becomes which is consistent with the conformal embedding Figure 6 (a) for the Fermi velocities and Figure 6 (b) for the entropy in this regime.
At last, for Fermi velocities v [1,1] < v [1,0] and v pf the entropy results in which is consistent with the conformal embedding [1,0] v [1,1] (a) In summary we can conclude that the effective model describing the SO(5) spin excitations is the SO(5) N f WZNW model with an anisotropic current-current perturbation. In contrast to the previous application of this approach to SU (k) N f anyons in [15,16]  For small fields (regions Q (1) , Q (2) ) a dilute gas of non-interacting quasi-particles with an internal anyonic (zero-energy) degree of freedom with quantum dimension Q (1) or Q (2) is realized. In the shaded region The degeneracy of the zero modes is lifted by the presence of thermally activated solitons with a small but finite density. All the other phases are labelled by the CFT describing the collective state formed by the condensed degrees of freedom.
and U (N f ) flavour degrees of freedom are missing.

ACKNOWLEDGMENTS
Funding for this work has been provided by the School for Contacts in Nanosystems. Additional support by the research unit Correlations in Integrable Quantum Many-Body Systems (FOR2316) is gratefully acknowledged.
The bare densitiesρ (1) 0,j (λ) and the kernels A (A4) Using (2.2) and the solutions ρ (m) k of (A3) the energy density E = E/N is rewritten as where the bare energies (1) 0,j (λ) = τ =±1 τ 2 t j,N f (λ + τ /g) + n j H 1 ,˜ 0,j (λ) = n j H 2 were introduced. It turns out that the energy (A5) is minimized by a configuration, where only the strings of length N f on the first level and strings of length 2N f on the second level have a finite density (cf. Ref. [19] for the isotropic case). After inverting the kernels A j 0,1 j 0,2 (ω) .