QED s in 2+1 dimensions: complex fixed points and dualities

Transcription

1 Prepared for submission to JHEP SISSA 53/2018/FISI arxiv: v1 [hep-th] 4 Dec 2018 QED s in 2+1 dimensions: complex fixed points and dualities Sergio Benvenuti, 1,2 Hrachya Khachatryan 1 1 International School of Advanced Studies (SISSA), Via Bonomea 265, Trieste, Italy 2 INFN, Sezione di Trieste, Via Valerio 2, Trieste, Italy benve79@gmail.com, hrachya.khachatryan@sissa.it Abstract: We consider Quantum Electrodynamics with an even number N f of bosonic or fermionic flavors, allowing for interactions respecting at least U(N f /2) 2 global symmetry. Both in the bosonic and in the fermionic case, we find four interacting fixed points: two with U(N f /2) 2 symmetry, two with U(N f ) symmetry. Large N f arguments suggest that, lowering N f, all these fixed points merge pairwise and become complex CFT s. In the bosonic QED s the merging happens around N f 8 10 and does not break the global symmetry. In the fermionic QED s the merging happens around N f 3 7 and breaks U(N f ) to U(N f /2) 2. When N f = 2, we show that all four bosonic fixed points are one-to-one dual to the fermionic fixed points. The merging pattern suggested at large N f is consistent with the four N f = 2 boson fermion dualities, providing support to the validity of the scenario.

2 Contents 1 Introduction and summary 1 2 Fixed points of easy-plane -QED s, U(N f /2) 2 symmetry Bosonic QED Fermionic QED 12 3 Boson fermion dualities for QED 3 with 2 flavors QED-GN + bosonic QED (CP 1 model) fermionic QED easy plane CP 1 model QED-GN bosonic QED QED-NJL tricritical bosonic QED 20 1 Introduction and summary Quantum Electro Dynamics (QED) in 2+1 dimensions, with either fermionic or bosonic flavors, is a paradigmatic example of a Quantum Field Theory with a strongly coupled infrared behaviour. Both in the bosonic and in the fermionic case, if the number of flavors N f is large enough, the Renormalization Group (RG) flows to a unitary interacting Conformal Field Theory (CFT). For small N f, however, other possibilities remain open. One option is that lowering N f the RG fixed point becomes complex : the real RG flow slows down ( walking ) close to the complex fixed point, and the phase transition is weakly first order instead of second order, see [1] for a modern perspective. Starting from a ultraviolet unitary gauge theory, the mechanism is that two real fixed points, varying continuously some parameter like space-time dimension or N f, annihilate into each other and become a pair of complex conjugate fixed points. The case of QED 3 with N f =2, two bosonic or two fermionic flavors, is particularly interesting: it describes the Nèel Valence Bond Solid quantum phase transition [2, 3], moreover non-trivial boson fermion dualities are expected to hold [4, 5]. Among other things, the dualities imply symmetry enhancements to O(4) or SO(5) at the fixed point, depending on the model. The dualities [4, 5] are part of recent interesting progress in 3d dualities, see for instance [6 24], and [25] for a review. 1

3 In this paper we consider QED s with an even number N f of bosonic or fermionic flavors. We allow for quartic interactions respecting U(N f /2) 2 global symmetry 1. In both cases we argue that there are four interacting fixed points, two with U(N f /2) 2 global symmetry, two with U(N f ) global symmetry. Bosonic QED with N f /2 flavors φ i plus N f /2 flavors φ i, and U(N f /2) 2 global symmetry, has four fixed points, which we denote as bqed (tricritical), U(N f ) global symmetry. Both the mass term and the quartic scalar interactions are tuned to zero, so the potential vanishes: V (φ i, φ j ) = 0. bqed + (standard, or CP N f 1 model), U(N f ) symmetry. V ( i φ i 2 + φ i 2 ) 2. ep-bqed (easy plane QED), U(N f /2) 2 symmetry. V ( i φ i 2 ) 2 +( j φ j 2 ) 2. bqed. U(N f /2) 2 symmetry. V ( i φ i 2 φ i 2 ) 2. The fermionic QED s have N f /2 flavors ψ i plus N f /2 flavors ψ i. We introduce one or two real scalars, interacting with the fermions via cubic Yukawa couplings, such models are called QED-Gross-Neveu or QED-Nambu-Jona-Lasinio. 2 Allowing for U(N f /2) 2 global symmetry gives four fixed points, which we denote as fqed (standard), U(N f ) global symmetry. The non-gauge interacting part of the Lagrangian, L int, vanishes. QED-GN + (QED-Gross-Neveu), U(N f ) symmetry. L int = ρ + i ( ψ i ψ i + ψi ψi ). QED-NJL (gauged Nambu-Jona-Lasinio), U(N f /2) 2 symmetry. There are two real scalars: L int = ± ρ ±( i ψ i ψ i ± ψi ψi ). QED-GN. U(N f /2) 2 symmetry. L int = ρ i ( ψ i ψ i ψi ψi ). In section 2 we discuss the fixed points and report the the scaling dimensions of quadratic and quartic scalar operators, computed in the large-n f expansion in [26]. For N f large enough the fixed points are real CFT s, but what can we say about the smallest possible N f, namely N f =2? 1 We are more precise about the global structure of the symmetry group in eqs. 2.3, 2.7, 2.20, In the literature there are two different models called QED-Gross-Neveu: one with 3d global symmetry U(N f ) (which we name QED-GN + ) and one with 3d global symmetry U(N f /2) 2 (which we name QED-GN ). For us each ψ, ψ is a complex two-component 3d fermion. See also footnote 9. 2

4 Considering all the different models together allows for a useful unified perspective. First, dualities can help us. If N f =2, the four bqed s are dual to the four fqed s, in the following fashion ( stands for dual to ) ep-bqed, U(1) 2 fqed, U(2) bqed, U(1) 2 bqed +, U(2) QED-GN, U(1) 2 QED-GN +, U(2) bqed, U(2) QED-NJL, U(1) 2 (1.1) The first and third dualities were discovered in [4, 5]. We obtain the second and fourth dualities (which are new) in section 3. The dualities do not tell us if the fixed points are real or complex, but suggest the all 8 fixed points share a similar fate at small N f, and restrict the possible scenarios. Numerical simulations in N f =2 fqed, bqed + and ep-bqed suggest second order or weakly first order transitions with certain critical exponents [27 31]. However the numerical bootstrap [32 35] shows that there are no 3d unitary CFT s with those critical exponents and O(4)/SO(5) symmetry. Lowering N f as a continuos variable, on the fermionic side it has long been suspected that fqed dynamically develops quartic interactions that break the global symmetry U(N f ) U(N f /2) 2 [36 40]. On the bosonic side, [41, 42] proposed that the CP N f 1 model merges with tricritical QED and become a pair of complex CFT s, with SO(5) global symmetry at N f = 2. Keeping track of the N f = 2 dualities 1.1, we can propose a scenario which is consistent with all the above observations/proposals: all the 8 fixed points merge pairwise at some Nf > 2, below which the fixed points are complex CFT s. The fixed point ep-bqed merges with bqed, becoming a pair of complex O(4)- CFT s. On the dual fermionic side, fqed merges with QED-GN and become the same complex O(4)-CFT s. The fixed point bqed + merges with bqed, becoming a pair of complex SO(5)-CFT s. On the dual fermionic side, QED-GN + merges with QED-NJL and become the same complex SO(5)-CFT s. In a cartoon, we draw the path of the fixed points as an analytically continued function of N f (continuos line for real fixed points at N f >Nf, dashed line for complex 3

5 fixed points at N f <N f, arrows go in the direction of decreasing N f): ep-bqed, U(N f /2) 2 bqed, U(N f /2) 2 CFT O(4) CFT O(4) fqed, U(N f ) QED-GN, U(N f /2) 2 bqed +, U(N f ) bqed, U(N f ) CFT SO(5) CFT SO(5) QED-GN +, U(N f ) QED-NJL, U(N f /2) 2 N f = N f =2 N f = Estimation of merging points from large N f. (1.2) Let us consider the scaling dimensions of the quadratic operators in tricritical bqed and in fqed, at 1 st order in 1/N f : { [φ φ adjoint ] = 1 64 [φ φ singlet ] = { [ ψψadjoint ] = 2 64 [ ψψ singlet ] = (1.3) Decreasing N f, in bqed the singlet approaches from below = 3, while in fqed it 2 is the adjoint that approaches = 3, from above. When this happens, in the large 2 approximation, the quartic SU(N f )-invariant operators in bqed hit = 3, so N f tricritical QED merges with the CP N f 1 model. In the fermionic QED instead quartic interactions (SU(N f ) symmetry breaking) become relevant. A simple estimate of the merging points is then possible: 3 NbQED π 8.6 N 2 fqed (1.4) 3π2 In section 2 we give various less crude estimates of Nf in all 8 models, studying the actual quartic operators that hit = 3 at the merging, and using scaling dimensions at 2 nd order in 1/N f when available. We consistently find that in the bosonic QEDs the merging point sits around Nf 8 10, while in fermionic QEDs the merging (with symmetry breaking) happens around N f 3 7. Especially for bosonic QEDs, we 3 Both in the bosonic and fermionic QED s, it is a one-loop Feynman diagram, describing the decay into two photons, that gives a big contribution to the singlet operators,

6 expect these estimates to be pretty robust, basically hoping that at N f 9 perturbation theory in 1/N f works pretty well. Precisely beacause the coefficient 128 is quite large, 3π 2 the operators run toward = 3 pretty fast, and bqed s arrive at their merging points at quite large values of N f. Of course, if the 2 nd order correction is larger and with the opposite sign, the merging might disappear. In fqed and QED-GN, it turns out that 2 nd order results actually increase the value of N fqed.4 One of the main points of the paper is that the merging pattern suggested by large N f arguments (no symmetry breaking in bqeds and symmetry breaking in fqed) is in agreement with the pattern dictated by the N f = 2 dualities. Let us close this discussion comparing with other large N f models. In O(N) models or O(N)-Gross-Neveu models, the 1 st order corrections to the singlets operators are smaller, 32, and there is a unitary CFT for all N 1. 3π 2 N Yukawa and quartic interactions are weaker than gauge interactions. In the minimally supersymmetric QED with N f flavors, that we study in [43], 5 the 1 st order correction to the SU(N f )-singlet quadratic operator, instead of being large as in non supersymmetric QED s, is zero. Moreover, setting N f =2, the 1 st order scaling dimensions of all quadratic and quartic operators agree pretty well with the scaling dimensions of a dual N =1 Gross-Neveu-Yukawa model [44, 45], computed in the D=4 ε expansion [45]. Since bosonic theories merge without symmetry breaking, and fermionic theories merge with symmetry breaking, a supersymmetric theory must live the single life. On the other hand, it is natural to expect that non supersymmetric gauge theories with non-abelian gauge groups, and possibly Chern-Simons interactions, display a qualitative behavior similar to QED. The large-n f expansion might be useful for instance to improve our understanding of the quantum phase scenarios of [19, 20, 24]. 2 Fixed points of easy-plane -QED s, U(N f /2) 2 symmetry In this section we describe our interacting bosonic and fermionic QED s with U(N f /2) 2 - invariant quartic couplings, and we report the scaling dimensions of simple scalar op- 4 In any case we cannot say anything about the fate of fermionic QED at N f = 4, which has many interesting physical applications in condensed matter [46 48]. 5 The theory is N =1 QED with zero superpotential. The matter content consists of N f fermionic flavors, N f bosonic flavors and a gauge invariant Majorana fermion. The matter interactions are cubic SU(N f )-invariant Yukawa couplings. 5

7 erators, computed at 1 st order in the large-n f expansion [26] (some of the results were already computed in [49 55]). Recent advances in the large-n f limit of QED 3 s include [56 61]. For recent investigations of QED 3 s in the context of N f =2 dualities or quantum critical points see [46, 55, 62 65]. We first discuss the RG fixed points in the ungauged models, where the existence of four unitary fixed points can be established rigorously for any even N f, large or small. Upon gauging the U(1) symmetry, the RG flow structure is certainly the same for large enough N f, but for small N f the fate of the gauged fixed points can be different. We estimate in each case the Nf where the real fixed points merge into a pair complex fixed points. Let us emphasize that our QED s do not contain monopoles in the action, that would break the U(1) top topological symmetry. Let us observe that, at 1 st order in 1/N f, the anomalous (not the total) dimensions of the fermionic fixed points 2.24, 2.28 are equal to the anomalous dimensions of the bosonic fixed points 2.8, This is true for the Hubbard-Stratonovich fields and for quadratic operators in the charged fields. This coincidence is partially explained studying QED s with N =1 and N =2 supersymmetry [43]. 2.1 Bosonic QED We start with bosonic QED, with with N f /2 flavors φ i plus N f /2 flavors φ i, all of gauge charge 1, organizing all the fixed points in the plane of quartic couplings of the potential V = λ 1 (( φ i 2 ) 2 + ( φ i 2 ) 2 ) + 2λ 2 ( φ i 2 )( φ j 2 ) (2.1) This potential preserve U(N f /2) 2 Z e 2 symmetry, where φ i, φ j are in the fundamental of the two SU(N f /2), and Z e 2 exchanges φ i φ i. These symmetries prevent other couplings to be generated. On the locus λ 1 = λ 2 the global symmetry is enhanced to U(N f ). O(2N f ) model Let us first consider the ungauged model, with 2N f real scalars and global symmetry is (O(N f ) O(N f )) Z e 2, becoming O(2N f ) on the locus λ 1 = λ 2. There are four fixed points: 1. Free fixed point, with λ 1 = λ 2 = 0. Both quartic couplings are relevant, obviously. 2. Decoupled fixed point, with λ 1 > 0, λ 2 = 0. It describes two decoupled O(N f ) models. We know from the numerical bootstrap [66] that [ φ 2 singlet ] O(N f ) > 3 2 6

8 (if N f > 1 6 ), so [ φ i 2 φ j 2 ] decoupled = 2 [ φ 2 singlet ] O(N f ) > 3. This proves rigorously that, for any N f > 1, this fixed point is attractive. 3. O(2N f ) model, with λ 1 = λ 2 > 0. O(2N f ) global symmetry. A relevant symmetry breaking quartic deformation, ( φ i 2 φ i 2 ) 2, drives the theory to the decoupled fixed point. 4. Model-3 with λ 1 > 0, λ 2 < 0. Global symmetry is (O(N f ) O(N f )) Z e 2. A relevant quartic deformation triggers an RG flow to the decoupled fixed point. The RG flows looks as follows Model-3 O(N f ) 2 Z 2 λ 1 Two decoupled O(N f )-models O(2N f )-model 2N f Free scalars λ 2 (2.2) Let us emphasize that this is an exact result valid for any N f > 1. The pattern agrees with the findings of [67]. Gauging the U(1) symmetry. points flow to four interacting QED fixed points. 7 When we gauge the global symmetry the four fixed The global symmetry includes a topological (or magnetic) U(1) top, under which only monopole operators are charged. The full UV global symmetry is ( SU(Nf /2) SU(N f /2) Z Nf ) U(1) b Z e 2 U(1) top Z C 2 (2.3) 6 In the O(n) vector model, the rigorous scaling dimensions of the quadratic singlet operator is (10) if n = 1 (Ising model), (25) for the O(2)-model, (55) for the O(3)-model, [66] and goes up to π 2 n at large n. Notice the different qualitative structure at N f = 1. 7 In the 4 ε expansion, tricritical QED is described by a small λ 1 = λ 2 1/Nf 2 fixed point, the ep-bqed has λ 1 1/N f, λ 2 1/Nf 2, while the other two fixed points have λ 1 1/N f, λ 2 1/N f. 7

9 Z Nf acts as {φ i, φ i } e 2πi/N f {φi, φ i }, which is a gauge transformation, so we need to quotient by this factor (it does not act on U(1) top because in bosonic QED the bare monopoles are gauge invariant, so the monopoles are not dressed). U(1) b acts as {φ i, φ i } {e iα φ i, e iα φi }. Z e 2 acts as φ i φ i, so it does not commute with the symmetries appearing on its left. Z C 2 is the charge-conjugation symmetry φ i φ i, A µ A µ. There is also time-reversal (or parity) symmetry Z T 2. It is convenient to rewrite the quartic potential using two Hubbard-Stratonovich real scalars σ ± : L = 1 N f /2 4e F µνf µν + ( D 2 µ φ i 2 + D µ φi 2 )+ ± i=1 σ ± N f /2 i=1 ( φ i 2 ± φ i 2 ) η 1 2 (σ2 ++σ 2 ) η 2 (σ 2 + σ 2 ). (2.4) Integrating out σ ±, one recovers the potential 2.1, with {λ 1, λ 2 } expressed in terms of {η 1, η 2 }. At large enough N f the first and last term are irrelevant (their N f = scaling dimension in 4), so it is enough to work with L = N f /2 ( D µ φ i 2 + D µ φi 2 ) + ± i=1 σ ± N f /2 ( φ i 2 ± φ i 2 ). (2.5) i=1 where the photon and the Hubbard-Stratonovich fields σ ± have an effective propagator obtained geometrically resumming an infinite number of Feynman graphs. If N f is large enough, the qualitative features of the RG flows are not changing when turning on the U(1) gauge coupling, which triggers an RG flow from 2.9 to four interacting bqed s: ep-bqed ( easy plane QED) U(N f /2) 2 bqed U(N f /2) 2 bqed + (CP N f 1 -model) U(N f ) Tricritical bqed U(N f ) (2.6) 8

10 Assuming that below a certain Nf two or four fixed points become complex, the picture of the RG flows below Nf is different for the RG flows between complex conjugates CFT s, but there still are RG flows from the complex conjugated pair coming from bqed bqed + to the complex conjugated pair coming from bqed ep-bqed. At the fixed points bqed and bqed + the global symmetry is enhanced to ( ) SU(Nf ) U(1) top Z C 2 (2.7) Z Nf where Z Nf is the center of SU(N f ). All gauge invariant local operators, including the monopoles, transform in SU(N f ) representations with zero N f -ality. The scaling dimensions of simple scalar operators in the large-n f limit, at the fixed points with U(N f ) symmetry, are [26]: bqed (tricritical) U(N f )-symmetry bqed + (CP N f 1 model) U(N f )-symmetry [φ φ SU(Nf ) adjoint] = 1 64 [ φ 2 SU(N f ) singlet ] = [φ i φ jφ k φ l traces] = [ φ 4 SU(N f ) singlet ] = [φ φ SU(Nf ) adjoint] = 1 48 [φ i φ jφ k φ l traces] = 2 48 [σ + ] = [ 5 37σ F µν F µν ] = 4 32(4± 37) (2.8) The merging of these two fixed points happens when the φ 4 operator (that at N f = have = 2) in bqed, decreasing N f, become irrelevant, and the σ+ 2 operator (that at N f = has = 4) in bqed + becomes relevant. Actually the operator σ+ 2 mixes strongly with F µν F µν, that also has has = 4 at N f =. The mixing was studied in [49], from which we take the results in

11 4 [F 2 σ 2 +] tricritical bqed [ φ 4 sing] CP N f 1 model /N f (2.9) Imposing that the interactions reach marginality we can estimate N f : [[ φ 4 SU(N f ) singlet] bqed = 3 Nf (2.10) 3π2 [ 0.924σ F µν F µν ] bqed+ = 3 N f 32(4 + 37) 3π (2.11) Another way to estimate the merging point is to impose that the scaling dimension of the singlet bilinear in bqed is equal to the scaling dimension of the Hubbard- Stratonovich field σ + in bqed + : [ φ 2 SU(N f ) singlet] bqed = = [σ + ] bqed+ = N f 9.2 (2.12) Even if these three arguments are not completely independent, it is encouraging to get somewhat consistent results. Let us now move to the fixed points with U(N f /2) 2 symmetry, the scaling dimen- 10

12 sions are [26]: bqed U(N f /2) 2 -symmetry easy plane bqed U(N f /2) 2 -symmetry [φ φ SU(Nf /2) adj, φ φsu(nf /2) adj] = 1 48 [φ i φ j, φ i φ j ] = 1 72 [φ i φ j φ k φl, φ i φ j φ k φ l ] = [ N f /2 i=1 φ i 2 + φ i 2 ] = [( N f /2 i=1 φ i 2 + φ i 2 ) 2 ] = [σ ] = [φ φ SU(Nf /2) adj, φ φsu(nf /2) adj] = 1 32 [φ i φ j, φ i φ j ] = 1 56 [φ i φ j φ k φl, φ i φ j φ k φ l ] = 2 64 [σ ] = [σ + ] = (2.13) Imposing that the singlet bilinear in bqed meets the Hubbard-Stratonovich field σ + in ep-bqed: N f /2 [ φ i 2 + φ i 2 ] bqed = [σ + ] ep bqed Nf 10.3 (2.14) i=1 Unfortunately in this case we do not have scaling dimensions of the pair of operators {σ 2 +, F µν F µν }. From the quartic operator reaching criticality we get N f /2 [( φ i 2 + φ i 2 ) 2 ] bqed = 3 Nf 8.1 (2.15) i=1 2 nd order extrapolation. We are not aware of any exact 2 nd order computation in bosonic QED s. Extrapolating finite-n f numerical simulations, [28] estimated the 2 nd order correction to the adjoint in bqed + to be 8 [φ φ SU(Nf ) adj] = (2). (2.16) Nf 2 Singlet sextic interactions of bosonic tricritical points. At the tricritical fixed point the sextic SU(N f )-singlet operator at infinite N f has = 3. The 1 st order 8 This is taken from η N in the caption of figure 5 of [28], where there seems to by a sign typo. 11

13 correction is [26] N f /2 [( (φ i φ i + φ i φ i )) 3 ] = O(1/N 3π 2 f 2 ) (2.17) N f i=1 So the sextic SU(N f ) invariant deformation is irrelevant. Modulo tuning mass and quartic term to zero, tricritical bqed is a stable fixed point. At the merging of the tricritical fixed point with the critical fixed point sextic singlet interactions do not play a role. 3d bosonic gauge theories at the tricritical point (with quartic interactions tuned to zero) were studied in a completely different regime in [68, 69], where they named the model regular boson theory. [68, 69] found that for U(N c ) k Chern-Simons with 1 bosonic flavor, at large N c and large k with N c /k fixed, there is a stable fixed point and possibly (depending on the value of N c /k) an unstable fixed point. Combining these two results, it is natural to suggest that at finite N c, N f, k, bosonic QCD always has a stable tricritical, or regular, fixed point. 2.2 Fermionic QED We consider fermionic QED with N f /2 flavors ψ i plus N f /2 flavors ψ i (each ψ, ψ is a complex two-component 3d fermion). The quartic Gross-Neveu interactions are modeled by Yukawa cubic couplings with two real scalar fields, ρ + and ρ. 9 ρ + and ρ are parity-odd, and all our theories are parity invariant. The Lagrangian reads L = 1 N f /2 4e F µνf µν + ( ψ i /Dψ 2 i + ψi /D ψ i ) + ± i=1 ρ ± N f /2 ( ψ i ψ i ± ψi ψi ) +... (2.18) The... stand for quartic interactions and kinetic terms for the ρ ± fields. The mass terms for ρ ± are relevant at large enough N f but they are tuned to zero. We start discussing the ungauged model, with O(N f ) 2 Z 2 global symmetry, the RG flows between the 4 fixed points are triggered by mass terms for the scalars ρ ±. 9 Much of the existing literature considers QEDs with N four-component Dirac fermions Ψ i, i = 1,..., N, in generic dimension d. In d = 3, the global symmetry can be U(2N) or U(N) 2, depending on the precise form of the Yukawa (or Gross-Neveu-Yukawa) couplings. In terms of two-component 3d fermions Ψ i = (ψ i, ψ i ) and Ψ i = ( ψ i, ψi ). So N Ψ i=1 i Ψ i = N i=1 ( ψ i ψ i ψi ψi ) is a U(N)-singlet in d 3, but it is part of the SU(2N)-adjoint in d = 3. On the other hand N Ψ i=1 i Γ 5 Ψ i = N i=1 ( ψ i ψ i + ψi ψi ) is a SU(2N)-singlet in d = 3. Often, what is called QED-Gross-Neveu has L int = σ N Ψ i=1 i Ψ i, with U(N) 2 global symmetry in d = 3. We instead named this model QED-GN. On the other hand [5] calls QED-Gross-Neveu the model that we named QED-GN +, with d = 3 global symmetry U(2N). i=1 12

14 There are 4 fixed points, similar to the bosonic case: a free theory, a decoupled fixed point with both ρ + and ρ (renaming ρ ± = ρ ± ρ, it splits into two decoupled O(N f )-invariant Gross-Neveu models), a Gross-Neveu fixed point with only ρ and O(N f ) 2 Z e 2-symmetry, and a Gross-Neveu fixed point with only ρ + and O(2N f )- symmetry. Gross-Neveu ρ 2 2N f Free Majorana fermions ρ 2 + O(N f ) 2 Z 2 ρ 2 + ρ 2 O(2N f )-Gross-Neveu Two decoupled O(N f )-Gross-Neveu s (2.19) We now gauge the U(1) symmetry. The global symmetry becomes (SU(N f /2) SU(N f /2) U(1) b U(1) top ) Z e 2 Z Nf Z C 2, (2.20) where Z Nf is a gauge transformation acting as {ψ i, ψ i, M bare } {e 2πi/N f ψ i, e 2πi/N f ψi, M bare } (2.21) The quotient acts also on U(1) top because bare monopoles M bare operators are not gauge invariant. The gauge invariant monopole operators with minimal topological charge, M ±1, are dressed with N f /2 fermionic zero-modes and are invariant under 2.21 [70]. U(1) b acts as {ψ i, ψ i } {e iα ψ i, e iα ψi }. Z e 2 acts as ψ i ψ i. Z C 2 is the charge-conjugation symmetry. There is also a time-reversal (or parity) symmetry that satisfies a non-trivial algebra, see [70] for pure QED, adding Yukawa interactions does not change their results. As in the bosonic case, if N f is large enough, gauging the U(1) symmetry triggers 13

15 an RG flow from 2.19 to the following 4 interacting fermionic fixed points: QED-GN U(N f /2) 2 ρ 2 + fqed (standard QED) U(N f ) ρ 2 ρ 2 + ρ 2 QED-GN + U(N f ) QED-NJL U(N f /2) 2 (2.22) The global symmetry at the fixed points fqed and QED-GN + is enhanced to SU(N f ) U(1) top Z Nf Z C 2. (2.23) The scaling dimensions of simple scalar operators to leading order in the large-n f limit are [26, 51, 53 55]: fqed U(N f ) QED-GN U(N f /2) 2 [ ψψ SU(Nf ) adj] = (28 3π2 ) 9π 4 N 2 f [ ψψ SU(Nf ) singlet] = [ ψ 4 [0,1,0...,0,1] ] = [ ψ 4 [2,0,...,0,2] ] = [{( ψ 2 singlet )2, F µν F µν }] = (2± 7) [ ψψ SU(Nf /2) adj] = (100 9π2 ) 9π 4 N 2 f [ ψ i ψj, ψi ψ j ] = 2 72 [ N f /2 i=1 ( ψ i ψ i + ψi ψi )] = [ρ ] = ( π2 ) 9π 4 N 2 f [ρ 2 ] = N f 0.47 N 2 f N f N 2 f N f N 2 f (2.24) The quartic fermionic operators in fqed were computed in [53, 54], we indicated the Dinkyn labels of the SU(N f ) representation under which they transform. The mixing between the quartic singlet and F µν F µν is strong also here, and was solved in [54], the lowest eigenvalue of the singlets does not seem to run fast enough to hit = 3. 14

16 We also included the order 1/Nf 2 contributions, when known [51, 55].10 The 2 nd order corrections to the adjoint are quite small, while ρ receives a big contribution at 2 nd order, from which [71] estimated chiral symmetry breaking below Nf 5.7 in fqed. The conjectural merging of the two fixed points fqed and QED-GN happens when, decreasing N f, the lowest quartic fermion ψ 4 operator becomes relevant and the mass term of the Hubbard-Stratonovich field ρ + becomes relevant: [ ψ 4 [0,1,0...,0,1]] fqed = 3 Nf π2 (2.25) [ρ 2 ] QED GN = 3 Nf π2 (2.26) Another estimate of the point of merging comes equating the adjoint in fqed with ρ in QED-GN, using the 2 nd order anomalous dimensions we get [ ψψ SU(Nf ) adj] fqed = [ρ ] QED GN N f 5.7 (2.27) If we use the 1 st order anomalous dimensions, we would get Nf 3.9, so the second order corrections in fermionic QEDs seem to increase the value of the merging point. We now move to the last fixed points. The scaling dimensions for QED-GN + and QED-NJL are [26]: QED-GN + U(N f ) QED-NJL U(N f /2) 2 [ ψψ SU(Nf ) adj] = 2 48 [ρ + ] = [ρ 2 +] = [ ψψ SU(Nf /2) adj] = 2 32 [ ψ i ψj, ψi ψ j ] = 2 56 [ρ ] = [ρ + ] = [ρ + ρ ] = 2 32 [ρ (4 17)ρ 2 ] = 2 16(3 17±5) (2.28) 10 [51] studies pure QED and eq. 27 gives the scaling dimension of N Ψ i=1 i Ψ i = N i=1 ( ψ i ψ i ψi ψi ) which is part of the SU(N f =2N)-adjoint in d = 3. See footnote 9. The QED-GN results are given eqs. 4.4 and 4.6 of [55], which studies a model (referred to as QED- Gross-Neveu in [55]) with L int = σ N Ψ i=1 i Ψ i. When d = 3 this model is what we call QED-GN, with U(N) 2 3d global symmetry. So the results of [55] are valid for our QED-GN with N f =2N flavors. Moreover, [55] reports the dimension of N Ψ i=1 i Ψ i, an operator which vanishes on-shell because of the equation of motion of σ. We report the scaling dimension of σ, using the relation [σ] = 3 [ N Ψ i=1 i Ψ i ]. 15

17 We can estimate N f in two ways. First, imposing that the adjoint in QED-GN + meets the singlet in QED-NJL: [ ψψ SU(Nf ) adj] = 2 48 = [ρ ] = N f 2.7 (2.29) It is conceivable that, as above, including 2 nd order anomalous dimensions moves this estimate up significantly. Second, looking at when ρ 2 (after having solved the mixing with ρ 2 +) hits criticality: [ρ ρ 2 +] QED NJL = (3 17 5) = 3 N 3π 2 f 16(3 17 5) 4 N f 3π 2 (2.30) 3 Boson fermion dualities for QED 3 with 2 flavors [4, 5] discovered that 3d QED s with two fermionic or two bosonic flavors satisfy two different boson fermion dualities. We start from the duality between N f =2 QED- GN + and bosonic QED + (CP 1 model) and from this duality we obtain the other three dualities. Even if N f = 2 is very small, we will try to compare large N f scaling dimensions for mesonic and monopole operators with numerical results and expectations from dualities. It turns out the the monopole scaling dimensions agree surprisingly well with expectations from duality and numerical results. As for the mesons, the operators transforming in the adjoint seem to behave relatively well, consistently with the fact that the adjoint receives small corrections from the N f = value. On the other hand, operators which are singlets of the global symmetry group do not agree with expectations from duality and numerical simulations. This is related to the fact that both 1 st and 2 nd order corrections in 1/N f seems to be large for singlets. The large N f monopole scaling dimensions are available only for bqed + [58, 72] and fqed [57, 73], it would be very interesting to study monopoles also in the other six models considered in this paper, they might provide many new constraints on the values of Nf. Let us mention that if the CFT s are complex, the anomalous dimensions will have an imaginary part. Such an imaginary part cannot be seen in a numerical computation in an obvious way, since they probe the real RG flow passing between the complex CFT s. Also the perturbative large N f anomalous dimensions are real. 16

18 3.1 QED-GN + bosonic QED (CP 1 model) The duality we start from was discussed in detail in [5]. On the fermionic side the model is called QED-Gross-Neveu. On the bosonic side the model is also called noncompact CP 1 model. U(1) + 2 ψ s V = ρ + ( ψ 1 ψ 1 + ψ 2 ψ 2 ) U(1) + 2 φ s V = σ + ( φ φ 2 2 ) (3.1) We denote the fermionic flavors ψ 1, ψ 2 and the bosonic flavors φ 1, φ of the simplest operators, which is soon going to be very useful for us, is { } { } ρ + φ 1 2 φ 2 2 ψ 1 ψ 1 ψ 2 ψ 2 σ + The mapping (3.2) This map [5] follows from the structure of massive deformations of the fixed point, or deriving the duality from the basic bosonization duality. The global symmetries SU(2) U(1) top Z 2 Z C 2 and ( SU(2) Z 2 U(1) top ) Z C 2 enhance at the fixed point to SO(5) [5]. It is possible to argue for the enhanced SO(5) symmetry using a self-duality of the CP 1 model, self-duality that follows from the old particle vortex duality [42]. The 4 monopoles of QED-GN + combine with ρ + to form a 5 of SO(5) and map to the monopoles plus the 3 adjoint mesons in bqed + : 12 {M +1 ψ 1, M +1 ψ 2, M 1 ψ 1, M 1 ψ 2, ρ + } {M +1, M 1, φ 1φ 2, φ 2φ 1, φ 1 2 φ 2 2 } (3.3) M q denotes the monopole of topological charge q. The monopole scaling dimension in large-n f bosonic QED + [58, 72] (the 2 nd order in 1/N f [58] result reproduces well the lattice results of [27, 28] at small N f ) give [M ±1 ] = The large-n f results for the mesonic operators instead give [ρ + ] (2.28), 3π 2 2 and [φ φ spin 1 ] (2.16). It is encouraging that adding the 3π nd -order correction 2.16 provides a seemingly correct value for the scaling dimension of φ φ spin 1. Higher degree operators organize into the symmetric traceless, the 14, of SO(5): { M ±2 ψψ, M±1 ψ ρ +, ψψ } { } spin 1, ρ 2 + M ±2, M ± φ φ spin 1, (φ φ) 2 spin 2, σ + (3.4) 11 Integrating out the Hubbard-Stratonovich real scalar σ + we would get V = ( φ φ 2 2 ) 2, and the second mapping in 3.2 would be ψ 1 ψ 1 ψ 2 ψ 2 φ φ To be more precise, M is a complex operator, with M = M 1, and the operators appearing in the 5, which is a real irrep of SO(5), are Re[M] and Im[M]. Same comment for M ψ, φ 1φ 2, φ 2φ 1,

19 The large-n f anomalous dimensions of spin-0 operators ρ +, ρ 2 +, σ +, e.g. [ρ 2 +] π 2 2 (2.28), [σ +] π 2 2 (2.8), are unphysical when N f = 2. On the other hand, both in the 5 and in the 14, the monopoles [58], the spin-1 and spin-2 operators 2.28, 2.8 agree pretty well with the expectations of the duality: [M ±1 ] 0.63 [φ φ spin 1 ] 0.64 (3.5) [M ±2 ] [ ψψ spin 1 ] [(φ φ) 2 spin 2] π (3.6) 3.2 fermionic QED easy plane CP 1 model We now obtain other dualities starting from the above one. The strategy is to introduce scalar fields, couple these scalar fields to the simple operators 13 appearing in 3.9 and flow to new theories on both sides of the duality. Let us introduce a scalar field σ and couple it to the first line in 3.2: δv = ρ + σ δv = σ ( φ 1 2 φ 2 2 ) (3.7) On the left hand side both ρ + and σ become massive and can be integrated out. We get a new duality: U(1) + 2 ψ s U(1) + 2 φ s V = 0 V = ± σ ±( φ 1 2 ± φ 2 2 ) { ψ1 ψ 1 + ψ } { } 2 ψ 2 σ ψ 1 ψ 1 ψ 2 ψ 2 σ + (3.8) (3.9) The duality relates pure N f =2 fermionic QED with N f =2 bosonic QED with an easy plane potential. This process involves only one field in the 5 of SO(5), so it breaks the global SO(5) symmetry of the fixed point to O(4). The charge-±1 monopoles of fqed transform in the vector of O(4) and map to monopoles and bifundamental mesons in ep-bqed: {M +1 ψ 1, M +1 ψ 2, M 1 ψ 1, M 1 ψ 2, } {M +1, M 1, φ 1φ 2, φ 2φ 1 } (3.10) The fqed monopoles M +1 ψ α, M 1 ψ α that, on the fqed side, the four monopoles M ±1 ψ are dressed by a charged fermion. Let us emphasize are degenerate even if chiral symmetry 13 This procedure is known under the name of flipping in the supersymmetric literature. It was recently applied to minimally supersymmetric 3d theories in [22, 44, 45]. 18

20 breaking U(2) U(1) 2 occurs, because the discrete symmetries Z e 2 (ψ 1 ψ 2 ) and Z C 2 (ψ α ψ α ) are unbroken. Large-N f monopole scaling dimension in fqed [57, 73] give [M ±1 ψ ] = gives [φ 1φ 2, φ 1 φ 2] π Higher degree operators are organized in the symmetric-traceless of O(4): 14 {M ±2 ψψ, ψψ spin 1 } {M ±2, M ±1 φ 1 φ 2, M ±1 φ 2 φ 1, (φ 2 φ 1) 2, (φ 1 φ 2) 2, σ + } (3.11) Also in this case we get qualitative agreement in the non-singlet sector: large-n f monopoles give [57] [M ±2 ψψ ] = Large-N f mesons give [ ψψ spin 1 ] (2.24) and [(φ φ 1) 2, (φ 1 φ 2) 2 ] Lattice computations 3π 2 2 in fqed [31] give [ ψψ spin 1 ] = 1 ± 0.2. The singlet operator [σ + ] π 2 2 (2.13) is instead unphysical. 3.3 QED-GN bosonic QED Applying the same logic, we can propose two new dualities. 3.9: getting From duality 3.8 we introduce a scalar field ρ and couple it to the second line in δv = ρ ( ψ 1 ψ 1 ψ 2 ψ 2 ) δv = ρ σ + (3.12) U(1) + 2 ψ s V = ρ ( ψ 1 ψ 1 ψ 2 ψ 2 ) ρ U(1) + 2 φ s V = σ ( φ 1 2 φ 2 2 ) with mapping { ψ1 ψ 1 + ψ } { } 2 ψ 2 σ φ φ 2 2 (3.13) (3.14) In this case both UV global symmetries are only U(1) 2, but the four monopoles in QED-GN must still be degenerate, because of the discrete symmetries Z e 2 Z C 2. {M +1 ψ 1, M +1 ψ 2, M 1 ψ 1, M 1 ψ 2 } {M +1, M 1, φ 1φ 2, φ 2φ 1 } (3.15) Duality implies symmetry enhancement to O(4). Higher degree operators are organized in the symmetric-traceless of O(4): {M ±2 ψψ, ψ 1 ψ 2, ψ 2 ψ 1, ρ } {M ±2, M ±1 φ 1 φ 2, M ±1 φ 2 φ 1, (φ 2 φ 1) 2, (φ 1 φ 2) 2, φ φ 2 2 } (3.16) The large-n f scaling dimensions are [ ψ 1 ψ 2, ψ 2 ψ 1 ] (2.24), [(φ 3π 2 2 2φ 1) 2, (φ 1 φ 2) 2 ] (2.13) π From the symmetrize product of the 4 we need to remove the trace, which is M 2 + φ 1 2 φ

21 3.4 QED-NJL tricritical bosonic QED The fourth and last duality relates what bqed with QED-Nambu-Jona-Lasinio. We start from 3.1 and introduce ρ and couple it to the first line in 3.2: δv = ρ ( ψ 1 ψ 1 ψ 2 ψ 2 ) δv = ρ σ +. (3.17) Notice that in this way we are not breaking the UV SU(2) symmetry on the bosonic side. On the r.h.s. both ρ and σ + becomes massive, and we flow to a new duality: U(1) + 2 ψ s V = ± ρ ±( ψ 1 ψ 1 ± ψ 2 ψ 2 ) U(1) + 2 φ s V = 0 (3.18) The mapping of the mass operators is { } { } ρ + φ 1 2 φ 2 2 φ φ 2 2 ρ (3.19) On the l.h.s., because of the Z 2 symmetries, the 4 monopole operators are still degenerate, and they map as in duality 3.1: {M +1 ψ 1, M +1 ψ 2, M 1 ψ 1, M 1 ψ 2, ρ + } {M +1, M 1, φ 1φ 2, φ 2φ 1, φ 1 2 φ 2 2 } (3.20) We conclude that the CFT enjoies SO(5) global symmetry, the above operators forming the 5-dimensional representation of SO(5). Higher order operators organize into the symmetric traceless, the 14, of SO(5): { M ±2 ψψ, M±1 ψ ρ +, ψ 1 ψ 2, ψ } { 2 ψ 1, ρ, ρ 2 + M ±2, M ± φ φ spin 1, (φ φ) 2 spin 2, φ φ 2 2} (3.21) The large-n f scaling dimensions are [ ψ 1 ψ 2, ψ 2 ψ 1 ] , (2.24), 3π 2 2 [(φ φ) 2 spin 2] (2.8). 3π 2 2 Acknowledgments We are grateful to Francesco Benini, Pasquale Calabrese and Silviu Pufu for useful discussions, and to Diego Rodriguez-Gomez for comments on the draft. S. B. is indebted with Andrea Guerrieri for an old collaboration on related topics. This work is supported in part by the MIUR-SIR grant RBSI1471GJ Quantum Field Theories at Strong Coupling: Exact Computations and Applications. S.B. is partly supported by the INFN Research Projects GAST and ST&FI. 20

22 References [1] V. Gorbenko, S. Rychkov and B. Zan, Walking, Weak first-order transitions, and Complex CFTs, JHEP 1810, 108 (2018) doi: /jhep10(2018)108 [arxiv: [hep-th]]. [2] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, M.P.A. Fisher, Deconfined quantum critical points, Science 303, 1490 (2004) doi: /science [arxiv:cond-mat/ [cond-mat.str-el]]. [3] T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M.P. A. Fisher, Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm, Phys. Rev. B 70, doi: /physrevb [arxiv:cond-mat/ [cond-mat.str-el]]. [4] A. Karch and D. Tong, Particle-Vortex Duality from 3d Bosonization, Phys. Rev. X 6, no. 3, (2016) doi: /physrevx [arxiv: [hep-th]]. [5] C. Wang, A. Nahum, M. A. Metlitski, C. Xu and T. Senthil, Deconfined quantum critical points: symmetries and dualities, Phys. Rev. X 7, no. 3, (2017) doi: /physrevx [arxiv: [cond-mat.str-el]]. [6] O. Aharony, G. Gur-Ari and R. Yacoby, d=3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories, JHEP 1203, 037 (2012) doi: /jhep03(2012)037 [arxiv: [hep-th]]. [7] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia and X. Yin, Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J. C 72, 2112 (2012) doi: /epjc/s [arxiv: [hep-th]]. [8] O. Aharony, G. Gur-Ari and R. Yacoby, Correlation Functions of Large N Chern-Simons-Matter Theories and Bosonization in Three Dimensions, JHEP 1212, 028 (2012) doi: /jhep12(2012)028 [arxiv: [hep-th]]. [9] D. T. Son, Is the Composite Fermion a Dirac Particle?, Phys. Rev. X 5, no. 3, (2015) doi: /physrevx [arxiv: [cond-mat.mes-hall]]. [10] O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, JHEP 1602, 093 (2016) doi: /jhep02(2016)093 [arxiv: [hep-th]]. [11] N. Seiberg, T. Senthil, C. Wang and E. Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics, Annals Phys. 374, 395 (2016) doi: /j.aop [arxiv: [hep-th]]. 21

23 [12] A. Karch, B. Robinson and D. Tong, More Abelian Dualities in 2+1 Dimensions, JHEP 1701, 017 (2017) doi: /jhep01(2017)017 [arxiv: [hep-th]]. [13] M. A. Metlitski, A. Vishwanath and C. Xu, Duality and bosonization of (2+1) -dimensional Majorana fermions, Phys. Rev. B 95, no. 20, (2017) doi: /physrevb [arxiv: [cond-mat.str-el]]. [14] P. S. Hsin and N. Seiberg, Level/rank Duality and Chern-Simons-Matter Theories, JHEP 1609, 095 (2016) doi: /jhep09(2016)095 [arxiv: [hep-th]]. [15] O. Aharony, F. Benini, P. S. Hsin and N. Seiberg, Chern-Simons-matter dualities with SO and USp gauge groups, JHEP 1702, 072 (2017) doi: /jhep02(2017)072 [arxiv: [cond-mat.str-el]]. [16] F. Benini, P. S. Hsin and N. Seiberg, Comments on global symmetries, anomalies, and duality in (2 + 1)d, JHEP 1704, 135 (2017) doi: /jhep04(2017)135 [arxiv: [cond-mat.str-el]]. [17] F. Benini, Three-dimensional dualities with bosons and fermions, JHEP 1802, 068 (2018) doi: /jhep02(2018)068 [arxiv: [hep-th]]. [18] K. Jensen, A master bosonization duality, JHEP 1801, 031 (2018) doi: /jhep01(2018)031 [arxiv: [hep-th]]. [19] Z. Komargodski and N. Seiberg, A symmetry breaking scenario for QCD 3, JHEP 1801, 109 (2018) doi: /jhep01(2018)109 [arxiv: [hep-th]]. [20] J. Gomis, Z. Komargodski and N. Seiberg, Phases Of Adjoint QCD 3 And Dualities, SciPost Phys. 5, 007 (2018) doi: /scipostphys [arxiv: [hep-th]]. [21] V. Bashmakov, J. Gomis, Z. Komargodski and A. Sharon, Phases of N = 1 theories in dimensions, JHEP 1807, 123 (2018) doi: /jhep07(2018)123 [arxiv: [hep-th]]. [22] F. Benini and S. Benvenuti, N =1 dualities in 2+1 dimensions, arxiv: [hep-th]. [23] C. Choi, M. Rocek and A. Sharon, Dualities and Phases of 3DN = 1 SQCD, JHEP 1810, 105 (2018) doi: /jhep10(2018)105 [arxiv: [hep-th]]. [24] C. Choi, D. Delmastro, J. Gomis and Z. Komargodski, Dynamics of QCD 3 with Rank-Two Quarks And Duality, arxiv: [hep-th]. [25] T. Senthil, D. T. Son, C. Wang and C. Xu, Duality between (2 + 1)d Quantum Critical Points, arxiv: [cond-mat.str-el]. 22

24 [26] S. Benvenuti and H. Khachatryan, The large N f limit of easy-plane QED 3 s, to appear. [27] J. Lou, A.W. Sandvik and N. Kawashima, Antiferromagnetic to valence-bond-soild transitions in two-dimensional SU(N) Heisenberg models with multi-spin interactions, Phys. Rev. B 80, (R) doi: /physrevb [arxiv: [cond-mat.str-el]]. [28] R. K. Kaul and A. W. Sandvik, Lattice Model for the SU(N) Nèel to Valence-Bond Solid Quantum Phase Transition at Large N, Phys. Rev. Lett. 108, no. 13, (2012) doi: /physrevlett [arxiv: [cond-mat.str-el]]. [29] J. D Emidio and R. K. Kaul, New easy-plane CP N 1 fixed points, Phys. Rev. Lett. 118, no. 18, (2017) doi: /physrevlett [arxiv: [cond-mat.str-el]]. [30] X. F. Zhang, Y. C. He, S. Eggert, R. Moessner and F. Pollmann, Continuous Easy-Plane Deconfined Phase Transition on the Kagome Lattice, Phys. Rev. Lett. 120, no. 11, (2018) doi: /physrevlett [arxiv: [cond-mat.str-el]]. [31] N. Karthik and R. Narayanan, Scale-invariance of parity-invariant three-dimensional QED, Phys. Rev. D 94, no. 6, (2016) doi: /physrevd [arxiv: [hep-th]]. [32] Y. Nakayama and T. Ohtsuki, Conformal Bootstrap Dashing Hopes of Emergent Symmetry, Phys. Rev. Lett. 117, no. 13, (2016) doi: /physrevlett [arxiv: [cond-mat.str-el]]. [33] D. Simmons-Duffin, unpublished. [34] L. Iliesiu, talk at Simons Center for Geometry and Physics: The Nèel-VBA quantum phase transition and the conformal bootstrap, [35] D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, arxiv: [hep-th]. [36] R. D. Pisarski, Chiral Symmetry Breaking in Three-Dimensional Electrodynamics, Phys. Rev. D 29, 2423 (1984). doi: /physrevd [37] K. Kaveh and I. F. Herbut, Chiral symmetry breaking in QED(3) in presence of irrelevant interactions: A Renormalization group study, Phys. Rev. B 71, (2005) doi: /physrevb [cond-mat/ ]. [38] L. Di Pietro, Z. Komargodski, I. Shamir and E. Stamou, Quantum Electrodynamics 23

25 in d=3 from the ɛ-expansion, Phys. Rev. Lett. 116, no. 13, (2016) doi: /physrevlett [arxiv: [hep-th]]. [39] L. Di Pietro and E. Stamou, Scaling dimensions in QED 3 from the ɛ-expansion, JHEP 1712, 054 (2017) doi: /jhep12(2017)054 [arxiv: [hep-th]]. [40] S. Giombi, I. R. Klebanov and G. Tarnopolsky, Conformal QED d, F -Theorem and the ɛ Expansion, J. Phys. A 49, no. 13, (2016) doi: / /49/13/ [arxiv: [hep-th]]. [41] A. Nahum, J. T. Chalker, P. Serna, M. Ortuno and A. M. Somoza, Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models, Phys. Rev. X 5, no. 4, (2015) doi: /physrevx [arxiv: [cond-mat.str-el]]. [42] A. Nahum, P. Serna, J. T. Chalker, M. Ortuno and A. M. Somoza, Emergent SO(5) Symmetry at the Nèel to Valence-Bond-Solid Transition, Phys. Rev. Lett. 115, no. 26, (2015) doi: /physrevlett [arxiv: [cond-mat.str-el]]. [43] S. Benvenuti and H. Khachatryan, Supersymmetric QED 3 s: large N f limits and dualities, to appear. [44] D. Gaiotto, Z. Komargodski and J. Wu, Curious Aspects of Three-Dimensional N = 1 SCFTs, JHEP 1808, 004 (2018) doi: /jhep08(2018)004 [arxiv: [hep-th]]. [45] F. Benini and S. Benvenuti, N = 1 QED in 2+1 dimensions: Dualities and enhanced symmetries, arxiv: [hep-th]. [46] J. Y. Lee, C. Wang, M. P. Zaletel, A. Vishwanath and Y. C. He, Emergent Multi-flavor QED3 at the Plateau Transition between Fractional Chern Insulators: Applications to graphene heterostructures, Phys. Rev. X 8, no. 3, (2018) doi: /physrevx [arxiv: [cond-mat.str-el]]. [47] X. Y. Song, C. Wang, A. Vishwanath and Y. C. He, Unifying Description of Competing Orders in Two Dimensional Quantum Magnets, arxiv: [cond-mat.str-el]. [48] X. Y. Song, Y. C. He, A. Vishwanath and C. Wang, From spinon band topology to the symmetry quantum numbers of monopoles in Dirac spin liquids, arxiv: [cond-mat.str-el]. [49] A.N. Vasil ev, M.Yu. Nalimov, The CP N 1 model: Calculation of anomalous 24

26 dimensions and the mixing matrices in the order 1/N, Theor. Math. Phys. (1983) 56: [50] J. A. Gracey, Gauged Nambu-Jona-Lasinio model at O(1/N) with and without a Chern-Simons term, Mod. Phys. Lett. A 8, 2205 (1993) doi: /s [hep-th/ ]. [51] J. A. Gracey, Electron mass anomalous dimension at O(1/(Nf(2)) in quantum electrodynamics, Phys. Lett. B 317, 415 (1993) doi: / (93)91017-h [hep-th/ ]. [52] R. K. Kaul and S. Sachdev, Quantum criticality of U(1) gauge theories with fermionic and bosonic matter in two spatial dimensions, Phys. Rev. B 77, (2008) doi: /physrevb [arxiv: [cond-mat.str-el]]. [53] C. Xu, Renormalization group studies on four-fermion interaction instabilities on algebraic spin liquids, Phys. Rev. B 78, doi: /physrevb [arxiv: [hep-th]]. [54] S. M. Chester and S. S. Pufu, Anomalous dimensions of scalar operators in QED 3, JHEP 1608, 069 (2016) doi: /jhep08(2016)069 [arxiv: [hep-th]]. [55] J. A. Gracey, Fermion bilinear operator critical exponents at O(1/N 2 ) in the QED-Gross-Neveu universality class, Phys. Rev. D 98, no. 8, (2018) doi: /physrevd [arxiv: [hep-th]]. [56] I. R. Klebanov, S. S. Pufu, S. Sachdev and B. R. Safdi, Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors, JHEP 1205, 036 (2012) doi: /jhep05(2012)036 [arxiv: [hep-th]]. [57] S. S. Pufu, Anomalous dimensions of monopole operators in three-dimensional quantum electrodynamics, Phys. Rev. D 89, no. 6, (2014) doi: /physrevd [arxiv: [hep-th]]. [58] E. Dyer, M. Mezei, S. S. Pufu and S. Sachdev, Scaling dimensions of monopole operators in the CP Nb 1 theory in dimensions, JHEP 1506, 037 (2015) Erratum: [JHEP 1603, 111 (2016)] doi: /jhep03(2016)111, /JHEP06(2015)037 [arxiv: [hep-th]]. [59] K. Diab, L. Fei, S. Giombi, I. R. Klebanov and G. Tarnopolsky, On C J and C T in the Gross-Neveu and O(N) models, J. Phys. A 49, no. 40, (2016) doi: / /49/40/ [arxiv: [hep-th]]. 25

27 [60] S. Giombi, G. Tarnopolsky and I. R. Klebanov, On C J and C T in Conformal QED, JHEP 1608, 156 (2016) doi: /jhep08(2016)156 [arxiv: [hep-th]]. [61] S. M. Chester, L. V. Iliesiu, M. Mezei and S. S. Pufu, Monopole Operators in U(1) Chern-Simons-Matter Theories, JHEP 1805, 157 (2018) doi: /jhep05(2018)157 [arxiv: [hep-th]]. [62] J. Braun, H. Gies, L. Janssen and D. Roscher, Phase structure of many-flavor QED 3, Phys. Rev. D 90, no. 3, (2014) doi: /physrevd [arxiv: [hep-ph]]. [63] L. Janssen and Y. C. He, Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality, Phys. Rev. B 96, no. 20, (2017) doi: /physrevb [arxiv: [cond-mat.str-el]]. [64] B. Ihrig, L. Janssen, L. N. Mihaila and M. M. Scherer, Deconfined criticality from the QED 3 -Gross-Neveu model at three loops, Phys. Rev. B 98, no. 11, (2018) doi: /physrevb [arxiv: [cond-mat.str-el]]. [65] N. Zerf, P. Marquard, R. Boyack and J. Maciejko, Critical behavior of the QED 3 -Gross-Neveu-Yukawa model at four loops, Phys. Rev. B 98, no. 16, (2018) doi: /physrevb [arxiv: [cond-mat.str-el]]. [66] F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision Islands in the Ising and O(N) Models, JHEP 1608, 036 (2016) doi: /jhep08(2016)036 [arxiv: [hep-th]]. [67] P. Calabrese, A. Pelissetto and E. Vicari, Multicritical phenomena in O(n(1)) + O(n(2)) symmetric theories, Phys. Rev. B 67, (2003) doi: /physrevb [cond-mat/ ]. [68] O. Aharony, S. Jain and S. Minwalla, Flows, Fixed Points and Duality in Chern-Simons-matter theories, arxiv: [hep-th]. [69] A. Dey, I. Halder, S. Jain, L. Janagal, S. Minwalla and N. Prabhakar, Duality and an exact Landau-Ginzburg potential for quasi-bosonic Chern-Simons-Matter theories, JHEP 1811, 020 (2018) doi: /jhep11(2018)020 [arxiv: [hep-th]]. [70] C. Cordova, P. S. Hsin and N. Seiberg, Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)d, SciPost Phys. 5, 006 (2018) doi: /scipostphys [arxiv: [cond-mat.str-el]]. [71] V. P. Gusynin and P. K. Pyatkovskiy, Critical number of fermions in 26