Dual QED3 at"NF = 1/2"is an interacting CFT in the infrared

We study the fate of weakly coupled dual QED3 in the infrared, that is, a single two-component Dirac fermion coupled to an emergent U(1) gauge field, but without Chern-Simons term. This theory has recently been proposed as a dual description of 2D surfaces of certain topological insulators. Using the renormalization group, we find that the interplay of gauge fluctuations with generated interactions in the four-fermi sector stabilizes an interacting conformal field theory (CFT) with finite four-fermi coupling in the infrared. The emergence of this CFT is due to cancellations in the $\beta$-function of the four-fermi coupling special to"NF = 1/2". We also quantify how a possible"strong"Dirac fermion duality between a free Dirac cone and dual QED3 would constrain the universal constants of the topological current correlator of the latter.


I. INTRODUCTION
There are several proposals claiming that more than one strongly interacting topological phase of matter might be describable by a unifying dual theory: a single two-component (composite) Dirac fermion coupled to an emergent U(1) gauge field [1][2][3]: Hereψ are not the fundamental electrons but two-component composite fermions. Andē is not the physical, electric charge but the charge with respect to the emergent gauge field a µ . There is no Chern-Simons term for a µ and this is due to a restricted set of large gauge transformations. That is, only certain types of electric fluxes are allowed on the surface of the topological insulator [2].
The dots stand for terms that are allowed by symmetries to be generated under a renormalization group (RG) flow, such as four-fermi interactions and radiative corrections to the propagators.
In high-energy physics, Dirac fermions are typically collected in a 4D-representation of the fermion spinor algebra; the case of interest in the present paper therefore laxly corresponds to "N F ≡ N (4) f = 1/2". It has not received much attention so far due to its novelty and only very recent association to topological condensed matter systems. Moreover, this case falls outside the region of validity of the Vafa-Witten theorem, which states that vector-like gauge theories cannot dynamically generate fermion masses inψψ channels, thereby breaking time-reversal and spacereflection symmetry [4]. But their proof only holds for a number of two-component fermions N (2) f ≥ 4 and even. More broadly, a Dirac fermion analog to the particle-vortex duality of bosons has been proposed [2] and an explicit construction of a possible "strong" duality of Eq. (1) to a free Dirac cone of electrically charged electrons on the level of partition functions has been put forward in Ref. 5.
It is fair to say that whether, and in what form, and on what energy scales this Dirac fermion duality holds is an open question with fundamental implications across a variety of physical systems. In particular, not much is known about the low-energy fate and ground state phase diagram of Eq. (1), which we will refer to as dual QED 3 in what follows.
In this paper, we want to start filling this knowledge gap and we ask if/under what conditions the low-energy dynamics of weakly coupled dual QED 3 , Eq. (1), remains conformal and by which mechanism interactions can generate a mass for the (composite) fermions at low energies. To achieve this, we will adapt the symmetry-breaking analysis of N (4) f four-component fermions coupled to a U(1) gauge field of Ref. 6 to a single two-component N (2) f = 1 fermion and compute the low-energy fixed-point structure for an initially weakly coupled Eq. (1). The question of (chiral) symmetry-breaking of QED 3 has been tackled intensely and we refer to the Introduction and Bibliography of Refs. 6-8 for an overview.

A. Key results and outline of paper
Our main result is that Eq. (1), if initially weak-to-moderately coupled, flows toward an interacting conformal field theory (CFT) in the infrared in which generated four-fermi couplings attain finite values. In the condensed matter context of dual QED 3 as a surface description for topological insulators, this implies that interactions preserve the gapless responses and the intrinsic topological order also at lowest energies. This result is derived and presented in Subsec. IV A.
In light of strong (chiral and vector) symmetry-breaking tendencies for QED 3 at single-digit flavor number N (4) f [6], this result may seem surprising. On the other hand, the N (2) f = 1 case does not have the full chiral symmetry to begin with, as it may be viewed to operate within one chiral sector. The number of symmetries that can be broken is now reduced and there remains essentially only one independent four-fermi coupling (λ) of the associated Fierz algebra (see Sec. III). With this, we find that gauge fluctuations never destabilize the four-fermi sector toward symmetrybreaking sufficiently strongly; instead, the flow is always attracted toward an infrared stable fixedpoint for the four-fermi coupling, which preserves the scaling/conformal symmetry of the gauge sector. Surprisingly, this is due to the absence of a λ 2 term in the β-function for the four-fermi coupling: the flavor trace carries a ∼ (N (2) f − 1)λ 2 . This is what stabilizes the CFT at N (2) f = 1. Related cancellations of β-functions in the single-flavor case also appear in the Gross-Neveu model; in particular these also hold at higher loop orders [9][10][11].
We believe our results are not limited to the RG technique used. An -expansion [12] around the weakly interacting Gaussian fixed-point in D = 4 should recover the same physics.
In Sec. II, we recapitulate how related runaway flows of different physical origin have been detected in Ref. 6. We present the Fierz-complete action of N (4) f 4-component Dirac fermions from which we project out the β-functions for a single two-component fermion, N (2) f = 1 in Sec. III. In Sec. IV, we explore consequences of a possible strong Dirac fermion duality. We determine exactly the universal constant of the topological current correlator of Eq. (1), an interacting theory, by relating it to the electromagnetic response of a free Dirac cone. Finally, in Subsec. IV B we point out the need to include the generated four-fermi coupling (and possibly other ingredients) in order to establish exponent identities for operator dimensions in compliance with the duality.
In Sec. V, we conclude the paper. Details of a direct derivation of the β-function for the fourfermi coupling λ are relegated to two Appendices A, B.
the following Fierz-complete ansatz for the euclidean, scale (k-) dependent effective action is sufficient to study symmetry-breaking into the complete set of all possible fermionic channels Here the last two termsg,ḡ are two four-fermi couplings from which all possible interaction channels, which can lead to condensation of fermion bilinears, can be constructed. ξ is a gauge fixing parameter which will be set to ξ = 0 in the following (Landau gauge). In total, Eq. (3) has 5 running couplings (Z ψ ,ē, Z a ,g andḡ), which depend on the cutoff scale k. We are interested in their evolution in the infrared as we take k → 0.

B. β-functions
In the simplest, point-like truncation for the couplings, projected onto the most singular point in frequency-and momentum space (the origin at q = 0), the leading order β-functions for the gauge coupling e 2 and the two four-fermi couplingsg, g of Eq.
(3) are: ∂ t e 2 = (η a − 1)e 2 (4a) Here we have abbreviated the scale-derivative ∂ t = k ∂ k . The set of β-functions Eq. (4) is closed by two anomalous dimensions making it 5 equations and 5 couplings to solve. η ψ for the electrons turns out to be negative in the regimes of interest and the threshold coefficients here take the form The photon anomalous dimension η a is physically caused by decay and recombination into electron-positron pairs. It takes the form which has a finite ζ → 0 limit. We also absorbed explicit k-dependences into renormalized gauge and induced four-fermion couplings The threshold functions appearing in Eq. (4) are (for the linear Litim regulator) and are positive in the regimes of interest, that is, the "RG-corrections" by the anomalous dimensions are subdominant when compared to the leading term.
f,c ; these signify a conformal phase. We now first describe the nature of these conformal fixed-points and subsequently explain how the scaling breaks down at N (4) Due to charge conservation, the photon anomalous dimension is exactly equal to one for any that is, along the line of interacting conformal fixed points corresponding to the conformal phase.
This follows from Eq. (4a). Since η a * depends on e 2 * itself, this fixes the numerical value of the gauge coupling, given in Fig. 1, as a function of N (4) f . The values of η * ψ depend on the number of fermion flavors. This follows from a solution of the coupled equations for the anomalous dimensions Eqs. (5,7). Its values are given in Fig. 1 for the linear regulator and ζ → 0. Alternative techniques to access the conformal phase and its exponents and operator dimensions are the 1/N f expansion, which offers perturbative control for sufficiently large N f (e.g.: [7,[13][14][15][16][17][18][19]), and the -expansion around d = 4 in the limit → 1 (e.g.: [7,8,12]).
It is a feature of the β-functions that the flow of the gauge coupling (4a) and consequently the universal fixed-point values e 2 * (N f ) shown in Fig. 1 do not depend ong or g. At the level of the . For N (4) f ≤ N (4) f,c , however, the four-fermi couplings at O start developing imaginary parts, which is indicative of spontaneous symmetry breaking and the phase boundary between the conformal phase and a phase with spontaneously broken symmetry.
In Fig. 2, we plot the fixed-point values of g andg in the complex plane for varying N (4) f . We observe that at N (4) f ≤ N (4) f,c = 4.7 the couplings develop imaginary parts. This estimate is coincidentally close to a recent computation from the F-theorem and a resummed -expansion  N (4) f,c ≈ 4.4 [8] and another recent estimate from the -expansion at N (4) f,c ≈ 4.5 [12]. An explicit solution of the 5 coupled flow equations as a function of k confirms this picture: for N (4) f ≤ 4.7, the four-fermi couplings diverge at some finite scale k sb . These runaway flows indicate that fluctuations in one, or several, fermion bilinear channels become so strong that one, or a combination, of bilinears are likely to condense and spontaneously break the conformal symmetry.
In the special case of There is only one independent fermionic interaction term left. The ansatz for the effective action (3) then reduces to withλ =g k − 3ḡ k . Making use of the flow equations (4), the β-function forλ can be obtained from ∂ tλ = ∂ tgk − 3∂ tḡk . Consequently, the flow equation for the dimensionless renormalized coupling The key feature of this equation is the absence of a λ 2 term. This is due to cancellations in the The numerical values are provided for the linear regulator at ζ = 0 and in Landau gauge as before.
Given the positive slope of ∂ t λ, the fixed point λ * is found to be infrared attractive. Therefore, no runaway flow occurs for arbitrary initial values e 2 k=Λ and the fixed-point structure and explicit flows (see Fig. 4) preserve scaling/conformal invariance as k → 0.

B. Discussion
In principle, by virtue of Eqs. (13,14) a runaway flow in the four-fermi sector with gaugeinvariant regularization can lead to quadratic mass terms ∼ mχχ and spontaneous background currents ∼ j µ χσ µ χ thereby breaking time-reversal and space-reflection symmetry. In fact, this parity anomaly appears generically if the entire large group of gauge transformations is allowed and Chern-Simons terms are induced [22][23][24][25]. But Eq. (2) with N (2) f = 1 is an effective dual theory for the 2D surface of certain topological insulators. Then χ is actually a composite fermion field, which is electrically neutral and the standard large gauge transformations need to be modified such that no Chern-Simons term from a parity anomaly is allowed [2].
We note that a possible "strong form" of a Dirac fermion duality would relate Eq. (2) with N (2) f = 1 to the partition function of a single, non-interacting Dirac cone [5]. In this scenario, it can be asserted that if m e = 0 for the free Dirac cone, than m = 0 for the composite fermions of dual QED 3 . Our computation is in line with this reasoning.

IV. IMPLICATIONS OF A POSSIBLE "STRONG" DIRAC FERMION DUALITY
In this section, and based on the considerations above, we want to take for granted this strong duality on the level of the path integral between Eq. (1) and a free Dirac cone of electrically charged fermionsψ e , ψ e [5]. For a given physical electromagnetic field A µ coupling the electric charge, we have that is "dual" to the theory The duality then implies that both Eq. (1) and Eq. (18) describe the same underlying physical system and the response to actual physical electromagnetic fields must be the same. In Eq. (18), the physical electromagnetic field couples, as usual, to the electronic current, whereas in Eq. (19) A µ couples to the gauge flux of the emergent photon.
We now survey two aspects of this duality: constraints for the topological current correlator in Subsec. IV A and operator dimensions in Subsec. IV B.

A. Constraining the topological current correlator
The current-current correlator on the free Dirac cone side of the duality Eq. (18) is where j µ =ψ e γ µ ψ e is the physical current dual to the physical electromagnetic gauge field A µ , N (2) f = 1 for one single two-component fermion, and This equation is exact; there are no interaction corrections.
On the composite fermion (cf) side, the physical electromagnetic gauge field couples (via a A µ J top µ term in the Lagrangian) to the topological current via Since both sides of the duality should describe the same physical reality, we should have Now, for zero doping and in a conformal phase with conserved topological current ∂ µ J top µ = 0, the correlator must also have the form of Eq. (20) but with an independent universal constant with N (2) c f = 1. However, the number C top J is not known, since the composite fermion theory is interacting. We can now use Eq. (23) to determine C top J exactly thus constraining perturbative computations for C top J . The best estimate for C top J is in Eq. (4.3.) of Ref. [7], abbreviated as GTK: where in the second line we have the subsumed the unknown interaction corrections to all orders in 1/N GT K into the variable ∆X top (N GT K ). After matching conventions and invoking Eqs. (21,23), we obtain for N (2) f = 1 (in units where coupling constants are unity) Knowledge of the exact value for this and other universal constants may help to constrain perturbative computations of thermodynamics, entanglement, and response functions of interacting, dual QED 3 and possibly extensions thereof.

B. Scaling dimension of composite mass operator at one-loop
Here we compute the singular corrections to the scaling dimension of the composite mass operator m =ψψ for the fermion fields at the fixed-point of Subsec. III A: Defining the anomalous exponents (Λ is the running cutoff scale), the total correction to the scaling dimension of the mass operator is then We first compute the fermionic field renormalization Z ψ and η ψ from the one-loop self-energy shown in Fig. 5. The photon anomalous dimension is η a = 1 and we can use the standard overdamped one-loop form for N (2) f = 1: where g cf is the photon-fermion coupling as denoted in Eq. (19). We may use Feynman gauge ξ = 1 in what follows. With this, the singular fermion self-energy correction is: Invoking the usual RG improvement, the anomalous dimension for the electrons obtains as in agreement with a previous calculation upon setting their N to 1/2 and gauge fixing ξ = 1 [26].
The one-loop graphs for the correction to the mass operator δm(k) are shown in Fig. 6. Graph (a) is non-vanishing and yields the divergent correction Adding the exponents as per Eq. (30) we get the final result which agrees with the leading term in a computation of the same quantity by Gracey upon setting his N f = 1/2 [28].
A strong form Dirac fermion duality would require η mass = 0 [5]. Implementing the momentumdependent four-fermi coupling via σ-meson exchange [27] will produce corrections to Eq. (35), as may other additional terms to the truncation Eq. (15). It is an interesting project to establish explicitly the exponent identities conjectured by the duality.
We checked that for the insertion of the (conserved) current operator,´d 3 x j µ Z j µ Z ψψcf γ µ ψ cf , with j µ =ψ cf γ µ ψ cf , these cancellations appear explicitly.
Evaluating Fig. 7 results in and produces a singular correction with opposite sign η j µ = −η ψ ensuring the cancellation Eq. (36).

V. CONCLUSIONS
This paper explored aspects of a particle-vortex duality for Dirac fermions in two space dimensions. We raised the question whether a single two-component Dirac fermion coupled to U(1) gauge field can have a conformally invariant ground state and found indications that this may the case, at least for the initially only weakly coupled model. The CFT we found is not free but has non-trivial anomalous dimensions and finite four-fermi couplings. It would desirable to have a (non-perturbative) proof of the absence of spontaneous symmetry-breaking in the ground state of Eq. (1) by generalizing the Vafa-Witten theorem to smaller flavor numbers down to N (2) f = 1. A strong-form duality between dual QED 3 and a free Dirac cone would constrain operator dimensions and, as we tried to show, also universal constants of electromagnetic response functions.
In the future, it will be interesting to establish exponent identities and possibly emergent conservation laws of dual QED 3 more completely and to higher loop order. To strengthen the link to a specific condensed matter situation, a further topic of interest are the "non-universalities" of dual QED 3 such as the kinematics, velocities, additional interactions, and energy scales below which the continuum field theories for the effective degrees of freedoms emerge.
to extract ∂ t λ from the ansatz (15). The expansion of this equation on terms of propagator and fluctuation matrices Γ (2) k + R k = P −1 k + F k facilitates a projection onto the respective operator structures. The propagator matrix is given by Here, only terms ∼ (χχ) are of interest. Therefore, any explicit dependence of the fluctuation matrix on the gauge field a µ can safely be dropped from the outset and we get Projecting onto spatially constant fermion fields χ p := χδ (3) (p), the basic building block of the expansion (A1) can be expressed as There are three contributions to the flow of the bare couplingλ which can be depicted diagrammatically as in fig. 8 below.
a. Fermionic self interaction This contribution does not involve the gauge vertex. Consequently, only the lower right submatrix of eq. (A4) is needed. Projecting onto (χχ) 2 gives where Ω is the three-dimensional spacetime volume. Thus, tracelessness of the quadratic term Here, the usual compact notation in terms of a threshold function has been introduced, l n a ,n ψ a,ψ = k 2n A −2n ψ −3ˆd qq 4 n a ∂ t r a − η a r a P r a + 2n ψ 1 + r ψ P r ψ (∂ t r ψ − η ψ r ψ ) P −n a r a P −n ψ r ψ , with P r a = q 2 [1 + r a ] , P r ψ = q 2 [1 + r ψ ] 2 . (A8) c. Box diagram The last diagram is particularly important as it is responsible to generate the fermionic interaction when starting from the QED 3 action in the UV, whereλ Λ = 0. When computing its value, the lower right submatrix of Eq.(A4) may be ignored as only the gauge vertex contributes.