Dual QED3 At" NF= 1/2" Is an Interacting CFT in the Infrared

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

Dietrich Roscher,∗ Emilio Torres,† and Philipp Strack‡ Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany (Dated: July 26, 2021) Abstract

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 β-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. arXiv:1605.05347v2 [cond-mat.str-el] 19 Jul 2016

∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected]; URL: http://www.thp.uni-koeln.de/~strack/

1 Contents

I. Introduction 2 A. Key results and outline of paper4

(4) II. Conformal scaling and its breakdown for QED3 with Nf 4-component fermions 5 A. Fierz-complete action5 B. β-functions5 (4) (4) C. Recap of interacting conformal ﬁxed point for Nf > Nf,c 7 (4) D. Breakdown criterion of conformal scaling at Nf,c 8

(2) III. Dual QED3 with a single 2-component fermion Nf = 19 A. Interacting conformal ﬁxed point 10 B. Discussion 11

IV. Implications of a possible “strong” Dirac fermion duality 12 A. Constraining the topological current correlator 12 B. Scaling dimension of composite mass operator at one-loop 13

V. Conclusions 16

Acknowledgments 16

References 16

A. Derivation of β-function ∂tλ for four-fermi coupling λ 18

2 B. Cancellation of λ term from ﬂavor trace in ∂tλ 20

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 [1–3]:

Lc f = ψ¯ i∂/ + e¯a/ ψ + ... . (1)

2 Here ψ¯ are not the fundamental electrons but two-component composite fermions. Ande ¯ is not

the physical, electric charge but the charge with respect to the emergent gauge ﬁeld aµ. There is no

Chern-Simons term for aµ 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 (4) “NF ≡ Nf = 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 in ψψ¯ channels, thereby breaking time-reversal and space- reflection symmetry [4]. But their proof only holds for a number of two-component fermions (2) Nf ≥ 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 QED3 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 QED3, Eq. (1), remains conformal and by which mechanism interactions can generate a mass for the (composite) fermions at low energies. To (4) achieve this, we will adapt the symmetry-breaking analysis of Nf four-component fermions cou- (2) pled to a U(1) gauge field of Ref.6 to a single two-component Nf = 1 fermion and compute the low-energy fixed-point structure for an initially weakly coupled Eq. (1). The question of (chiral) symmetry-breaking of QED3 has been tackled intensely and we refer to the Introduction and Bibliography of Refs.6–8 for an overview.

3 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 QED3 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.IVA.

In light of strong (chiral and vector) symmetry-breaking tendencies for QED3 at single-digit (4) (2) flavor number Nf [6], this result may seem surprising. On the other hand, the Nf = 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 symmetry- breaking sufficiently strongly; instead, the flow is always attracted toward an infrared stable fixed- point for the four-fermi coupling, which preserves the scaling/conformal symmetry of the gauge sector. Surprisingly, this is due to the absence of a λ2 term in the β-function for the four-fermi (2) 2 (2) coupling: the flavor trace carries a ∼ (Nf − 1)λ . This is what stabilizes the CFT at Nf = 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 [9–11]. 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 di fferent physical origin have been (4) detected in Ref.6. We present the Fierz-complete action of Nf 4-component Dirac fermions from (2) which we project out the β-functions for a single two-component fermion, Nf = 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.IVB 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 four- fermi coupling λ are relegated to two AppendicesA,B.

4 (4) II. CONFORMAL SCALING AND ITS BREAKDOWN FOR QED3 WITH Nf 4-COMPONENT FERMIONS

In Ref.6, β-functions for QED3 were calculated using a 4D-reducible representation of the fermionic spinor ﬁelds. The main scope of this work was an investigation of the symmetry break-

ing patterns of QED3, including chiral channels, by detection of runaway ﬂows for fermionic

couplings caused by ﬁxed point annihilation. To set the stage for dual QED3, we now recapitulate the key elements of this analysis.

A. Fierz-complete action

(4) Based on the bare action of a Maxwell term for the photons coupled to a set of Nf ﬂavors of 4-component Dirac fermions (ﬂavor index implicit) ( ) 1 S = d3 x ψ¯ i∂/ + e¯a/ ψ + F Fµν , (2) ˆ 4 µν

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    Za Za g¯˜ g¯  Γ [ψ,¯ ψ, a] = d3 x ψ¯ iZ ∂/ + e¯a/ ψ + F F + (∂ a )2 + (ψγ¯ ψ)2 + (ψγ¯ ψ)2 . k ˆ  ψ 4 µν µν 2ξ µ µ (4) 45 (4) µ   2Nf 2Nf  (3) Here the last two terms g¯˜,g ¯ 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 (Zψ,e ¯, Za, g¯˜ andg ¯), 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

5 gauge coupling e2 and the two four-fermi couplingsg ˜, g of Eq. (3) are:

2 2 ∂te = (ηa − 1)e (4a)   2N(4) − 1 3 2  ∂ g˜ = g˜(1 + 2η ) − l1  f g˜2 − gg˜ − g2 − l1,1 2˜ge2 + 4ge2 + l2,1 2N(4)e4 (4b) t ψ ψ  (4) (4) (4)  a,ψ a,ψ f Nf Nf Nf  (4)  1,1  1 2N + 1  la,ψ ∂ g = g(1 + 2η ) + l1  gg˜ + f g2 − 4˜ge2 − 2ge2 . (4c) t ψ ψ  (4) (4)  3 Nf 3Nf

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. ηψ for the electrons e2 h i η = − 2m ˜ 1,1 − 2m2,1 (5) ψ 3 a,ψ a,ψ turns out to be negative in the regimes of interest and the threshold coeﬃcients here take the form 3 1 1 m˜ 1,1 = − η − η a,ψ 2 6 ψ 4 a 1 m2,1 = 1 − η . (6) a,ψ 4 a

The photon anomalous dimension ηa is physically caused by decay and recombination into electron-positron pairs. It takes the form

∞ ( 1 √ 2 1 2 ∂trψ(y) − ηψrψ(y) 1 yx − ζx η =N(4)e2 dy − dx √ (7) a f 2 √ h i3 2 ζ ˆ0 3 2 ˆ−1 y − 2ζx y + ζ y 1 + rψ(y) " √ 2 √ 2 #) [∂ r ](y) − η r (y) [∂trψ](y − 2ζx y + ζ ) − ηψrψ(y − 2ζx y + ζ ) t ψ ψ ψ , 2 √ 2 + √ 2 2 [1 + rψ(y)] [1 + rψ(y − 2ζx y + ζ )] [1 + rψ(y)][1 + rψ(y − 2ζx y + ζ )] which has a ﬁnite ζ → 0 limit. We also absorbed explicit k-dependences into renormalized gauge and induced four-fermion couplings gk¯˜ gk¯ e¯2 ≡ ≡ 2 ≡ g˜ 2 , g 2 , e 2 . (8) Zψ Zψ ZψZak The threshold functions appearing in Eq. (4) are (for the linear Litim regulator) 2 1 l1 = − η (9) ψ 3 6 ψ 4 1 2 l1,1 = − η − η (10) a,ψ 3 6 ψ 15 a 1 4 l2,1 = 2 − η − η (11) a,ψ 6 ψ 15 a 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.

6 0 N(4) η∗ e2 0.1 f ψ ∗ − 0.2 5 -0.066 0.195 − 0.3 6 -0.055 0.163 ∗ ψ − η 0.4 − 7 -0.048 0.140 0.5 − 8 -0.042 0.123 0.6 − 0.7 9 -0.037 0.109 − 12345678910 10 -0.033 0.099 (4) Nf Figure 1: Fermion anomalous dimension and value of the gauge coupling in the conformal phase as a (4) (4) function of the ﬂavor number Nf . The dashed line indicates the regime below Nf,c , where the conformal ﬁxed point becomes unstable and likely spontaneous symmetry breaking sets in.

The 5 β-functions Eq. (4) and Eqs. (5,7) have scale-invariant, real-valued solutions for large (4) (4) enough Nf > Nf,c ; these signify a conformal phase. We now ﬁrst describe the nature of these (4) (4) conformal ﬁxed-points and subsequently explain how the scaling breaks down at Nf = Nf,c .

(4) (4) C. Recap of interacting conformal ﬁxed point for Nf > Nf,c

Due to charge conservation, the photon anomalous dimension is exactly equal to one for any (4) (4) value of N f > Nf,c :

a η∗ = 1 , (12) that is, along the line of interacting conformal fixed points corresponding to the conformal phase. a 2 This follows from Eq. (4a). Since η∗ depends on e∗ itself, this fixes the numerical value of the (4) ∗ gauge coupling, given in Fig.1, as a function of Nf . The values of ηψ 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/Nf expansion, which offers perturbative control for sufficiently large Nf (e.g.: [7, 13–19]), 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 2 universal fixed-point values e∗(Nf) shown in Fig.1 do not depend on ˜g or g. At the level of the

℄ g˜

∗ ℄ g˜

∗ g ℄

∗

(4) Nf

Figure 2: Fixed point values of the fermionic couplingsg ˜ and g at the scale-invariant/conformal ﬁxed point O (4) (4) 2 (4) for a range of Nf > Nf,c . The gauge coupling is set to the respective ﬁxed point value e∗(Nf ). Finite (4) imaginary parts indicate spontaneous symmetry-breaking. Here this happens below Nf,c = 4.7.

perturbative Ward identity, this is diagrammatically due to Furry’s theorem, i.e., the vanishing of graphs with an odd number of external gauge field insertions. This decoupling of the gauge flow from the fermion sector permits a simplified analysis of symmetry breaking patterns. e2 may be viewed as an external parameter for the fermionic flow equations (4b) and (4c). Here we have

taken the fermionic couplings to be not fundamental in the UV,g ˜k=Λ = gk=Λ = 0. They first need to be generated by gauge field fluctuations. We now recapitulate how to detect symmetry breaking from the flows of Eqs. (4,5,7).

(4) D. Breakdown criterion of conformal scaling at Nf,c

(4) (4) For sufficiently large Nf > Nf,c , the initial values, gk=Λ = g˜k=Λ = 0 lie in the basin of attraction of a conformal fixed point. Note that in general there are four fixed point solutions, only one of 2 2 which is infrared attractive. This is the conformal fixed-point at finite ek→0 = e∗ (labelled as O). (4) (4) For Nf ≤ Nf,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. (4) In Fig.2, we plot the fixed-point values of g andg ˜ in the complex plane for varying Nf . (4) (4) We observe that at Nf ≤ Nf,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

8 (4) (4) (4) (4)

(a): Nf = 6 > Nf,c (b): Nf = 4 < Nf,c

¼º¾ ¼º5

¼º½

¹¼º5

¹¼º½

¹½

¹¼º¾

¹¼º¿ ¹½º5 g˜(k) ¹¼º4 g(k) 2 e (k) ¹¾ g˜(k) ¹¼º5 g(k)

e2(k)

¹¾º5

¹¼º6

¹½¾ ¹½¼ ¹8 ¹6 ¹4 ¹¾ ¼ ¹½4 ¹½¾ ¹½¼ ¹8 ¹6 ¹4 ¹¾ ¼ t = ln(k/Λ) t = ln(k/Λ)

Figure 3: Explicit solutions of the ﬂow equations in the conformal phase (a) and in the phase with putatively

broken symmetry (b). In the second case, the fermionic ﬂow diverges at tsb = −12.9.

(4) (4) Nf,c ≈ 4.4 [8] and another recent estimate from the -expansion at Nf,c ≈ 4.5 [12]. An explicit solution of the 5 coupled flow equations as a function of k confirms this picture: for (4) Nf ≤ 4.7, the four-fermi couplings diverge at some finite scale ksb. 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.

(2) III. DUAL QED3 WITH A SINGLE 2-COMPONENT FERMION Nf = 1

For 2D surfaces of certain 3D topological insulators and possibly other topological phases, we

are interested in QED3 with a single two-component (composite) fermion coupled to an emergent U(1) gauge ﬁeld. Therefore, the 4D representation for the fermionic spinors is not applicable and (2) we need to move to the 2D-representation and set Nf = 1. To that end, consider the two linearly independent fermionic interactions in the 2D representation deﬁned in the appendix of Ref.6:

a a2 i i2 a a2 i i2 ¯ (4) → ¯ → ψ γ45ψ a=1...N χ¯ χ (2) , ψ γµψ (4) χ¯ σµχ (2) (13) f i=1...Nf a=1...Nf i=1...Nf

9 (2) In the special case of Nf = 1, it is

2 2 − 3(χχ ¯ ) = (χσ ¯ µχ) . (14)

There is only one independent fermionic interaction term left. The ansatz for the eﬀective action (3) then reduces to ( ) Z Z Γ [ψ,¯ ψ, a] = d3 x χ¯ iZ ∂/ + e¯a/ χ + a F F + a (∂ a )2 + λ¯(χχ ¯ )2 (15) k ˆ ψ 4 µν µν 2ξ µ µ with λ¯ = g¯˜k − 3¯gk. Making use of the ﬂow equations (4), the β-function for λ¯ can be obtained from

∂tλ¯ = ∂tg¯˜k − 3∂tg¯k. Consequently, the ﬂow equation for the dimensionless renormalized coupling ¯ −2 λ = λkZψ is given by 1,1 2 2,1 4 ∂tλ = λ(1 + 2ηψ) + 2la,ψλe + la,ψe . (16)

The key feature of this equation is the absence of a λ2 term. This is due to cancellations in the (2) β-function special to the Nf = 1 case as we explain in AppendixB. This structural specialty is (4) 2 already visible upon setting Nf = 1/2 in Eq. (4b); then theg ˜ in that equation disappears and the ﬁxed-point structure qualitatively changes. Further implementing the symmetry Eq. (14), then yields Eq. (16) for a single four-fermi coupling. Of course, this β-function Eq. (16) may also be derived directly from applying the Polchinski-Wetterich equation [20, 21] to the ansatz (15). This is performed explicitly in AppendixA.

A. Interacting conformal ﬁxed point

We now adapt the symmetry-breaking analysis explained above to a reduced number of equations, feeding the gauge sector and anomalous dimensions into Eq. (16). Given the IR ﬁxed point 2 2 ∗ ∗ of the gauge sector (e = e∗ = 1.59, ηa = 1, ηψ = −0.64) as an input, and because ∂tλ is linear in λ, there can be only one ﬁxed point solution to Eq. (16):

l2,1 e4 A,ψ ∗ ∂(∂tλ) 1,1 2 λ∗ = − = −1.20, = 1 + 2ηψ + l e∗ = 3.87 (17) 1,1 2 ∂λ A,ψ 1 + 2ηψ + 2lA,ψe∗ (λ∗) The numerical values are provided for the linear regulator at ζ = 0 and in Landau gauge as before.

Given the positive slope of ∂tλ, the fixed point λ∗ is found to be infrared attractive. Therefore, 2 no runaway flow occurs for arbitrary initial values ek=Λ and the fixed-point structure and explicit flows (see Fig.4) preserve scaling /conformal invariance as k → 0.

10 (a) (b)

1.5

1 λ(k) e2(k) 0.5

2 2 eΛ/e = 0.1 0 ∗ 2 2 eΛ/e = 0.5 ∗ 2 2 0.5 eΛ/e = 1.2 − ∗ 1 − 1.5 − 7 6 5 4 3 2 1 0 − − − − − − − t = ln(k/Λ)

Figure 4: (a): Universal location of the interacting conformal ﬁxed point in the plane of interaction couplings (2) for Nf = 1. Arrows point towards the infrared. (b): Explicit numerical solutions for a range of initial values 2 2 eΛ showing convergence of both, the gauge coupling e and the four-fermi coupling λ, toward ﬁxed-point plateaus in the infrared as k → 0.

B. Discussion

In principle, by virtue of Eqs. (13,14) a runaway flow in the four-fermi sector with gauge- invariant regularization can lead to quadratic mass terms ∼ mχχ¯ and spontaneous background currents ∼ h jµiχσ¯ µχ 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 (2) and Chern-Simons terms are induced [22–25]. But Eq. (2) with Nf = 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 (2) Nf = 1 to the partition function of a single, non-interacting Dirac cone [5]. In this scenario, it can be asserted that if me = 0 for the free Dirac cone, than m = 0 for the composite fermions of dual QED3. Our computation is in line with this reasoning.

11 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 fermions ψ¯ e, ψe [5]. For a given physical electromagnetic ﬁeld Aµ coupling the electric charge, we have that

h µ i L f ree = ψ¯ e iγ (∂µ − iephysAµ) ψe , (18)

is “dual” to the theory

h i gc f L = ψ¯ iγµ(∂ − ig a ) ψ + µνλA ∂ a . (19) c f µ c f µ 4π µ ν λ

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)

Aµ 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.IVA and operator dimensions in Subsec.IVB.

A. Constraining the topological current correlator

The current-current correlator on the free Dirac cone side of the duality Eq. (18) is ! pµ pν h j (−p) j (p)i = −N(2)C f ree|p| δ − (20) µ ν f j µν p2

where jµ = ψ¯ eγµψe is the physical current dual to the physical electromagnetic gauge ﬁeld Aµ, (2) Nf = 1 for one single two-component fermion, and

2 ephys C f ree = . (21) j 16

This equation is exact; there are no interaction corrections. On the composite fermion (cf) side, the physical electromagnetic gauge ﬁeld couples (via a top Aµ Jµ term in the Lagrangian) to the topological current via

gc f Jtop = νκ∂ a . (22) µ 4π µ ν κ

12 Since both sides of the duality should describe the same physical reality, we should have

top top h jµ(−p) jν(p)i f ree = hJµ (−p)Jν (p)ic f (23)

top Now, for zero doping and in a conformal phase with conserved topological current ∂µ Jµ = 0, the correlator must also have the form of Eq. (20) but with an independent universal constant ! pµ pν hJtop(−p)Jtop(p)i = −N(2)Ctop|p| δ − (24) µ ν c f J µν p2

(2) top with Nc f = 1. However, the number CJ is not known, since the composite fermion theory is top interacting. We can now use Eq. (23) to determine CJ exactly thus constraining perturbative top top computations for CJ . The best estimate for CJ is in Eq. (4.3.) of Ref. [7], abbreviated as GTK: ! ! ! 8|p| 1 736 p p hJtop,GTK −p Jtop,GTK p i − − O /N2 δ − µ ν µ ( ) ν ( ) = 2 1 + 8 2 + (1 GTK) µν 2 (25) π NGTK NGTK 9π p " !! # ! 8 1 736 p p ≡ −|p| − Xtop N δ − µ ν , 2 1 + 8 2 + ∆ ( GTK) µν 2 π NGTK NGTK 9π p where in the second line we have the subsumed the unknown interaction corrections to all orders top in 1/NGTK into the variable ∆X (NGTK). After matching conventions

(2) NGTK = 2Nf top,GTK top Jµ = 2Jµ (26)

(2) and invoking Eqs. (21,23), we obtain for Nf = 1 (in units where coupling constants are unity)

top ∆X (NGTK = 2) = −0.0973652 . (27)

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 QED3 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:

d3 x m Z Z ψψ¯ (28) ˆ m ψ

13 q

k k

Figure 5: One-loop fermion self energy for Zψ and the corresponding anomalous dimension ηψ.

Deﬁning the anomalous exponents (Λ is the running cutoﬀ scale),

∂ log Zψ ∂ log Z η = − , η = − m , (29) ψ ∂ log Λ m ∂ log Λ the total correction to the scaling dimension of the mass operator is then

ηmass = ηm + ηψ . (30)

We first compute the fermionic field renormalization Zψ and ηψ from the one-loop self-energy shown in Fig.5. The photon anomalous dimension is ηa = 1 and we can use the standard over- (2) damped one-loop form for Nf = 1: ! 16 q q D q η − − ξ µ ν , µν( ) = 2 µν (1 ) 2 (31) gcf|q| q where gcf 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: d3 p δΣ (q) = g2 (−γµ)G (k + q)(−γν)D (q) ψ cf ˆ (2π)3 ψ µν ! 8 Λ = − k/ log . (32) 3π2 |k| Invoking the usual RG improvement, the anomalous dimension for the electrons obtains as 8 η = , (33) ψ 3π2 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 ! 24 Λ 24 δm(k) = 1 log , η = − . (34) π2 |k| m π2 Adding the exponents as per Eq. (30) we get the final result 64 η = − , (35) mass 3π2

14 (a) (b)

k 1

1 q k k

Figure 6: One-loop graphs for insertions of the mass operator m = ψψ¯ . (a) is computed in the text. (b) arises from the four-fermi coupling λ and does not produce singular corrections, when approximating λ as momentum-independent. Upon keeping the momentum-dependence of the four-fermi coupling, for example via σ-meson exchange [27], (b) can also generate singular corrections to operator dimensions. 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 ηmass = 0 [5]. Implementing the momentum- dependent 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. 3 ¯ µ We checked that for the insertion of the (conserved) current operator, d x jµ Z jµ Zψ ψcfγ ψcf, ´ with jµ = ψ¯ cfγµψcf, these cancellations appear explicitly. 8 8 η = η + η = − + = 0 . (36) current jµ ψ 3π2 3π2 Evaluating Fig.7 results in ! ! 8 Λ kµ δ jµ(k) = γµ log − k/ , (37) 3π2 |k| k2

and produces a singular correction with opposite sign η jµ = −ηψ ensuring the cancellation Eq. (36).

jµ q

Figure 7: One-loop correction to insertions of the current operator jµ = ψ¯ cfγµψcf.

15 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 (2) Eq. (1) by generalizing the Vafa-Witten theorem to smaller flavor numbers down to N f = 1.

A strong-form duality between dual QED3 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 QED3 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

QED3 such as the kinematics, velocities, additional interactions, and energy scales below which the continuum ﬁeld theories for the eﬀective degrees of freedoms emerge.

Acknowledgments

We thank A. Altland, J. Braun, S. Diehl and A. Rosch for fruitful discussions. We also thank H. Gies, I. Klebanov and M. Metlitski for insightful correspondences. This work was supported by the Leibniz Prize of A. Rosch and by the German Research Foundation (DFG) through the Institutional Strategy of the University of Cologne within the German Excellence Initiative (ZUK 81).

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Appendix A: Derivation of β-function ∂tλ for four-fermi coupling λ

The beta function for the single fermionic coupling λ can be derived without making reference to the ﬂow equations (4). Here we employ the Polchinski-Wetterich equation [21]

   ∞  1  ∂ R  1 h i 1  X (−1)n−1  ∂ Γ = STr  t k  = STr ∂˜ P + STr ∂˜ (P−1F )n (A1) t k 2  (2)  2 t k 2  t n k k  Γk + Rk n=1

18 to extract ∂tλ from the ansatz (15). The expansion of this equation on terms of propagator and (2) −1 fluctuation matrices Γk + Rk = Pk + Fk facilitates a projection onto the respective operator structures. The propagator matrix is given by    µν  Pa 0 0    P−1 δ(3) p−q   , P q· Z r −1, P δ q2 −q q · Z q2 r −1. k = ( )  0 0 −Pχ χ = / [ ψ(1+ ψ)] a = [ µν µ ν] [ a (1+ a)]    T  0 −Pχ 0 (A2) Here, only terms ∼ (χχ ¯ ) are of interest. Therefore, any explicit dependence of the fluctuation matrix on the gauge field aµ can safely be dropped from the outset and we get    0e ¯χ¯ σ −e¯χT σT   q−p µ p−q µ   h i F =  T T ¯ T ¯ T T 1  (A3) k −e¯σν χ¯ q−p −2λ χ¯ p χ¯ p−q+p1 2λ χ¯ p χp−q+p − (χ ¯ p1 χp−q+p1 )   p1 1 p1 1 1   h ´ i ´ T   e¯σνχp−q 2λ¯ (χ ¯ p χp−q p )1 + χp χ¯ p−q p −2λ¯ χp χ  p1 1 + 1 1 + 1 p1 1 p−q−p1 ´ ´ (3) Projecting onto spatially constant fermion fields χp := χδ (p), the basic building block of the expansion (A1) can be expressed as    µν µν T T   0 Pa e¯χσ¯ ν −Pa e¯χ σν  h i   −1 (3)  T  P Fk = δ (p − q) −P e¯σ χ −P 2λ¯ (χχ ¯ )1 + χχ¯ P 2λχχ¯  . (A4) k µν  χ µ χ χ   h i  T T T T ¯ T T ¯ T T 1  Pχ e¯σν χ¯ Pχ 2λχ¯ χ¯ −Pχ 2λ χ¯ χ − (χχ ¯ ) There are three contributions to the flow of the bare coupling λ¯ which can be depicted diagrammatically as in fig.8 below. Here, double lines symbolize the (full) renormalized propagators in (A2). a. Fermionic self interaction This contribution does not involve the gauge vertex. Conse- quently, only the lower right submatrix of eq. (A4) is needed. Projecting onto (χχ ¯ )2 gives 2 1 1 −1 −1 2 ∂tλ¯|¯ 2 (χχ ¯ ) = STr P Fk = 0 (A5) λ Ω 2 2 k λ¯ 2 where Ω is the three-dimensional spacetime volume. Thus, tracelessness of the quadratic term

in the expansion enforces linearity of the β-function. This ﬁnding is crucial for QED3 being (2) conformal for Nf = 1 and it only occurs for this ﬂavor number. Some details of why this is the case are given in App.B below. b. Triangle diagram Both gauge and fermionic sectors are involved in the computation of this diagram. It is given by l1,1 2 1 1 1 −1 3 a,ψ 2 2 ∂tλ¯|¯ 2 (χχ ¯ ) = STr P Fk = (χχ ¯ ) (A6) λe¯ k λ¯e¯2 2 2 Ω 2 3 π kZψZa

19 Figure 8: Diagrammatic representation of the contributions to ∂tλ¯. It is a special feature here that diagram (a) vanishes after the trace.

Here, the usual compact notation in terms of a threshold function has been introduced, " # n ,n ∂tra − ηara 1 + rψ −n a ψ 2nA−2nψ−3 4 − −na ψ la,ψ = k dqq na + 2nψ (∂trψ ηψrψ) Pra Prψ , (A7) ˆ Pra Prψ with 2 2 2 Pra = q [1 + ra] , Prψ = q [1 + rψ] . (A8)

c. Box diagram The last diagram is particularly important as it is responsible to generate

the fermionic interaction when starting from the QED3 action in the UV, where λ¯ Λ = 0. When computing its value, the lower right submatrix of Eq.(A4) may be ignored as only the gauge vertex contributes. l2,1 2 1 1 −1 −1 4 a,ψ 1 4 2 ∂tλ¯|e4 (χχ ¯ ) = STr P Fk = e¯ (χχ ¯ ) (A9) ¯ k e¯4 2 3 2 2 Ω 2 4 π k ZψZa d. Renormalization and rescaling In a last step, a renormalization of the couplings as in p 2 2 eq. (8) and the ﬁelds as χ → χ/ Zψ as well as a rescaling of e and λ with π provides the ﬁnal result for the β-function: k ¯ 1,1 2 2,1 4 ∂tλ = λ(1 + 2ηψ) + 2 ∂tλ = λ(1 + 2ηψ) + 2lA,ψλe + lA,ψe . (A10) Zψ This reproduces Eq. (16) of the main text.

2 Appendix B: Cancellation of λ term from ﬂavor trace in ∂tλ

In order to understand the origin of the cancellation enforcing linearity of the β-function (A10) with respect to λ, it is necessary to revisit the corresponding contributions for general ﬂavor num- (2) (2) ber Nf . Although there are two independent quartic fermion terms for Nf > 1 (see Eq. (13)), it is suﬃcient to consider the generalized interaction

2 i i2 (χχ ¯ ) → χ¯ χ (2) . (B1) i=1...Nf

20 The fermion sector of the analog to eq. (A4) is then given by

 h i T  i j i ¯j i j i j ¯ −Pχ (χχ ¯ )1δ + χ χ Pχχ χ  h −1 i (3) 2λ   P Fk = δ (p − q)  T T T  , (B2) k χχ¯ (2)  T i j T i j 1 i j  Nf  Pχ χ¯ χ¯ −Pχ χ¯ χ − (χχ ¯ ) δ  where the ﬂavor dependence of the fermion propagator has already been absorbed. It is then

(2) ! l1 i i 2 Nf 1 −1 −1 2 ψ h a a i j a a 2 i ji ∂tλ¯|λ¯ 2 (χ ¯ χ ) = STr P Fk = 2 Trf (χ¯ χ ) χ¯ χ − (χ¯ χ ) δ Ω 2 2 k λ¯ 2 (2) 2 Nf π 1 (B3) lψ 2 = 2 1 − N(2) λ¯ 2 χ¯ iχi (2) 2 f Nf π where in the ﬁrst line all traces but the ﬂavor one have been performed and

1 + rψ ln = 2nk2n−3 dqq4(∂ r − η r ) . (B4) ψ ˆ t ψ ψ ψ n+1 Prψ

(2) We emphasize the prefactor in this expression vanishes for the case of interest Nf = 1 as an- nounced in the main text. This is due to the absence of flavor off-diagonal terms in Eq. (B3) for (2) this case. For Nf = 1, the flavor indices a = i = j must fall onto each other and the bracket [...] in the first line of Eq. (B3) vanishes. Related vanishing of β-functions in the single-flavor case also appears in the Gross-Neveu model and in particular also holds at higher loop orders [9–11]. Complete cancellations of individual contractions in a β-function for a four-fermion vertex also appear in Luttinger liquids [29], although these are specific to spatial dimension D = 1.