Bosonization of the Thirring Model in 2+1 Dimensions

PHYSICAL REVIEW D 101, 076010 (2020)

Bosonization of the Thirring model in 2 + 1 dimensions

† ‡ Rodrigo Corso B. Santos,1,* Pedro R. S. Gomes,1, and Carlos A. Hernaski 1,2, 1Departamento de Física, Universidade Estadual de Londrina, 86057-970 Londrina, PR, Brazil 2Departamento de Física, Universidade Tecnológica Federal do Paraná, 85503-390 Pato Branco, PR, Brazil

(Received 21 October 2019; revised manuscript received 23 February 2020; accepted 30 March 2020; published 10 April 2020)

In this work we provide a bosonized version of the Thirring model in 2 þ 1 dimensions in the case of single fermion species, where we do not have the benefit of large N expansion. In this situation there are very few analytical methods to extract nonperturbative information. Meanwhile, nontrivial behavior is expected to take place precisely in this regime. To establish the bosonization of the Thirring model, we consider a deformation of a basic fermion-boson duality relation in 2 þ 1 dimensions. The bosonized model interpolates between the ultraviolet and infrared regimes, passing several consistency checks and recovering the usual bosonization relation of the web of dualities in the infrared limit. In addition the duality predicts the existence of a nontrivial ultraviolet fixed point in the Thirring model.

DOI: 10.1103/PhysRevD.101.076010

I. INTRODUCTION In addition, this theory is taken at the Wilson-Fisher fixed point [7], so that both theories in (1) are scale invariant. In recent years a confluence of ideas in field theory and The spirit of this work is to take advantage of this condensed matter has led to the discovery of an intricate set relation in order to extend the class of models related by of dualities [1–3] thenceforth called web of dualities. This duality. This is accomplished by considering deformations refers to a set of relations between 2 þ 1-dimensional (3D) of (1) and may be specially useful to explore nonperturba- quantum field theories valid at an infrared stable fixed tive regimes, where usually we do not have much analytical point. In the heart of this web, one finds a relationship of methods at our disposal. A possible deformation of the the bosonization type, namely, duality (1) is to include mass operators in both sides [2,8]. In this case, the duality assumes the form 1 AdA 2 λ 4 ada Ada ψ¯ i=DAψ − ⇔ jDaϕj − jϕj þ þ ; ð1Þ 2 4π 4 4π 2π 1 λ ψ¯ ψ − ψψ¯ − AdA ⇔ j ϕj2 − 2jϕj2 − jϕj4 i=DA M 2 4π Da m 4 from which many other dualities can be derived, including þ ada þ Ada ð Þ the original particle-vortex bosonic duality [4,5] and its 4π 2π : 2 fermionic counterpart proposed in [6]. Da and DA are covariant derivatives acting on charge þ1 fields. The left- hand side corresponds to a free fermion coupled to a Of course the parameters of the two theories must be background field, A, while the right hand side involves a somehow connected. The above relation has been verified complex scalar coupled to a compact dynamical gauge field in the strict limit of infinite masses, reproducing correctly the Hall conductivity (Chern-Simons coefficient) in both with a Chern-SimonsR (CS) term and flux quantization over sides [2,8]. A more precise relation between the masses was the sphere 2 da ¼ 2πZ, which is responsible for imple- S suggested in [8], motivated by an analogy with large N menting the fermion-boson transmutation. The compact studies [9,10]. The idea is to introduce an auxiliary field, σ, field couples to the external field through a BF term Ada. λ 4 which allows us to write the interaction − 4 jϕj as −σjϕj2 þ 1 σ2. The map between operators is then estab- *[email protected] λ † ψψ¯ ∼−σ [email protected] lished in terms of the auxiliary field, according to . ‡ [email protected] Including a mass term for the fermion, −Mψψ¯ , is equivalent to add on the bosonic side the operator Mσ, which Published by the American Physical Society under the terms of in turn implies the identification between the masses in (2) the Creative Commons Attribution 4.0 International license. 2 ∼−λ Further distribution of this work must maintain attribution to as m M. As we shall see, the consistency of this the author(s) and the published article’s title, journal citation, identification emerges as a by-product of the analysis in the and DOI. Funded by SCOAP3. present work.

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It is worth emphasizing that the infrared duality that This work is organized as follows. In Sec. II we review follows from (2) can be explicitly derived simply by some useful properties of the Thirring model in 3D. In computing the Hall conductivity in both sides. Both models Sec. III, we discuss how to obtain the bosonized version describe two gapped phases depending on the sign of M of the Thirring model starting from the critical duality. In (m2) on the fermionic (bosonic) side. If the phase transition Sec. IV, some checks of the proposed duality are per- on the bosonic side is of second order, one can conjecture formed. We conclude in Sec. V with a brief summary and that the two models share the same critical point described additional comments. by the duality (1) [2]. By assuming this conjecture we have an exact relation between conformal theories valid, there- II. 3D THIRRING MODEL fore, for arbitrary energies. By adding operators depending on the background field in both sides of the duality and We start by discussing some features of the Thirring 2 þ 1 promoting them to dynamical fields leads to new dualities. model in dimensions given by the action Making this procedure in (2) we get new dualities that must Z g be true in the infrared. However, if this is done in (1) we get S ¼ d3x iψ¯ =∂ψ − Mψψ¯ − ðψγ¯ μψÞ2 : ð3Þ new conjectured dualities valid for arbitrary energy scales. 2 As we shall see, a particularly interesting operator that fits in this perspective is the Thirring interaction, i.e., a four- We are considering the two-component irreducible repre- fermion interaction of the form ðψγ¯ μψÞ2, which is sentation for the Dirac spinors. Most of literature about the extremely important in several contexts. Thirring model consider the reducible four-dimensional Indeed, the 3D Thirring model (more generally, models representation, where generally there is no Chern-Simons with four-fermion interactions) has served over the years as generation in the effective action of fermions coupled to valuable prototype to examine a number of methodological gauge fields. We insist in the case of the irreducible questions, such as large N renormalizability [11–13], representation since this is precisely the case that takes realization of the Weinberg’s asymptotic safety scenario place in the web of dualities, where the Chern-Simons term has a fundamental role. The Thirring model involves [14], dynamical symmetry breaking [15], lattice simula- ½ ¼1 tions [16–20], and the relationship to other models like two dimensionful parameters: the mass, M , and the coupling constant, ½g¼−1. As the coupling constant has QED3 [21,22]. Furthermore, it is relevant for condensed matter systems of great interest, as in the case of high-T negative mass dimension, the Thirring operator is irrel- c N superconductors [23,24] and also in the description of low- evant at weak coupling. On the other hand, large analysis has shown that the model is renormalizable in energy excitations of materials like graphene [25,26]. In this work we investigate the 3D Thirring model with a this framework [12,13] suggesting it has a nonperturba- single fermion field (N ¼ 1), in the light of the web of tive UV fixed point. The essential ingredient in the fermion-boson dualities in dualities. A dual description of the Thirring model in 3D 2 þ 1 dimensions is the particle-vortex nature of the offers in principle the possibility to explore strong coupling mapping between objects of the dual theories. In other regimes, which are difficult to be accessed by analytical 1 words, a local field that creates a particle excitation in one methods like large N expansion. By carrying out simple theory corresponds to a monopole operator in its dual [1,2]. manipulations of the partition functions we are able to A different kind of bosonization duality for the Thirring obtain a bosonized version of the Thirring model from the model has been discovered by Fradkin and Schaposnik in critical duality (1) in the presence of an external field. [30]. In that work the authors show that the strict large mass To support the proposed duality we analyse some limit of the Thirring model is equivalent to the Maxwell- accessible limits of the relation. First, we consider the Chern-Simons (MCS) theory. More precisely, they consider strict limit of infinite mass, where we recover the Fradkin- the energy regime E ∼ 1 ≪ jMj. Since no single fermion Schaposnik map relating the fermionic theory with the g Maxwell-Chern-Simons model [30]. Then we consider state can be excited in the large mass limit, the mapping 2 the limit of large but finite mass, where we perform an actually shows that the bound-state sector of the Thirring expansion in the inverse of mass, matching both sides to the model can be described by a bosonic gauge field governed first leading terms. In particular, this matching confirms the by the MCS dynamics. Therefore the large mass limit of the relation m2 ∼−λM suggested in [8]. 2In the large N expansion, the bound-state condition can be determined exactly by examining the pole structure of the auxiliary vector field Aμ arising when we write the Thirring 1 2 In this context, it is worth to mention an interesting recent μ 2 Aμ μ interaction as −ðg=2Þðψγ¯ ψÞ ¼ − Aμψγ¯ ψ, which exhibits a approach to deal with fermions in 3D in terms of quantum wires 2g 2 4 2 − 4π [27–29], where one spatial dimension is discretized so that the pole satisfying k < M when g> jMj [12]. Thus, in the large 3D problem is transformed into a set of 2D ones, and all the mass limit, for any positive coupling constant g (corresponding to machinery of 2D bosonization can be used. attractive interactions) there is bound-states formation.

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Thirring model is described by a purely bosonic gauge above duality passes some consistency checks and predicts theory an interesting behavior in the UV limit. In the following we shall discuss how to obtain the above relation. g jMj→∞ g bdb To motivate the relation (5), we start with the critical iψ¯ =∂ψ − Mψψ¯ − ðψγ¯ μψÞ2 ⇔ − f2 − ðMÞ ; 2 64π2 b sign 8π 1 ð − Þ2 duality (1) and introduce in both sides the term 2g A B , ð4Þ with B being a new external field. This sets out an energy scale, g−1, in the duality that was previously scale invariant. where the sign of the fermion mass determines the sign of Then, promoting A to a dynamical field and renaming the the CS coefficient in the bosonic description. In [31] a fields as A → eb and B → A, we obtain bosonized version of the Thirring model is obtained that is valid to all orders in an expansion in the inverse of mass. 1 2 2 λ 4 1 ψ¯ ið=∂ − ie=bÞψ þ ðeb − AÞ ⇔ jD ϕj − jϕj þ ada Since no spin and statistical transmutation are involved in 2g a 4 4π the Fradkin-Schaposnik mapping, the compact character of 2 þ e þ e the emergent field, and consequently monopole configu- 2π bda 8π bdb rations, can be ignored. We will show that this mapping can 1 be recovered from the large mass limit of a more complete þ ð − Þ2 ð Þ 2 eb A : 7 bosonized version of the Thirring model for finite mass. It g is important to emphasize that the relation (4) is valid even At this point b is a noncompact field. Notice also that in in the strong coupling limit and thus provides an useful promoting A to a dynamical field we have assigned a consistency check for the duality that we will discuss in the corresponding coupling constant e, which is useful to next sections. properly analyse specific regimes of energy. We see that the left-hand side produces the Thirring interaction after III. THIRRING MODEL FROM integrating out b. Actually, we first make the shift THE CRITICAL DUALITY eb → eb þ A, and then integrate out this redefined field producing the Thirring interaction with a coupling constant In light of the previous discussions, we will see now how g. In order to obtain the proposed duality, the right hand to induce the Thirring operator in the critical duality (1) by side still needs some attention. Let us consider the part of adding operators involving exclusively the external field. In the Lagrangian involving the b field: this way, we propose the following duality relation: 2 1 λ 1 e e 2 ψ¯ ψ − g ðψγ¯ μψÞ2 ⇔ j ϕj2 − jϕj4 þ L½a; b; A ≡ bda þ bdb þ ðeb − AÞ : ð8Þ i=DA 2 Da 4 4π ada 2π 8π 2g 1 g − cdc − f2 In the spirit of the connection between self-dual and MCS 2π 16π2 aþc models [32], we define the interpolating Lagrangian 1 þ ða þ cÞdA; ð5Þ 2π 1 e 1 e L ¼ − cdc þ cdb þ ðeb − AÞ2 þ bda; ð9Þ I 8π 4π 2g 2π where a is an emergent compact gauge field satisfying the Dirac flux quantization condition where c is a new emergent gauge field and similarly to b Z is also noncompact. Integrating over c gives c ¼ eb. da ¼ 2πZ; ð6Þ Plugging this relation back into the Lagrangian (9) gives S2 (8). Therefore, (8) and (9) are equivalent. Alternatively, we can integrate out the field b to get eb ¼ A − g ðdc þ 2daÞ. 2 R2 4π with S being a compactification of the space slices . Plugging this back into the Lagrangian (9) and making the The field c is an independent gauge field with no flux field redefinition c → 2c˜, we obtain quantization condition. Some comments are in order. First, the flux quantization condition ensures that the partition 1 g 1 L ¼ − cd˜ c˜ − f2 þ ða þ c˜ÞdA: ð Þ function for the bosonic model is gauge invariant including 2π 16π2 aþc˜ 2π 10 large gauge transformations of a. In second place, naturally the above relation contains (1). Indeed, by flowing to low Therefore, we can see (9) as an interpolating Lagrangian energies the Thirring operator becomes irrelevant, and we between (8) and (10). Replacing (8) by (10) in (7) and recover the original duality [taking g → 0 we can directly renaming c˜ → c, we obtain the right-hand side of (5). integrate out the gauge field c in the bosonic theory to Now, we analyze the field content of (5) and determine obtain (1)]. In the next section we will see that, in regimes the fermion operator in terms of the bosonic field and the of energy where the Thirring operator is important, the monopole operator (bosonization rule). The charges of the

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TABLE I. Charge of the monopole operators and of the scalar For convenience we have made the shift a → a − c, which field. leads to a more direct process of integration over fields. ð1Þ ð1Þ Notice that in this basis, both scalar field and the monopole U a U A operator Ma are charged under c. Nevertheless, the M † a 11combination ϕ M remains uncharged. ϕ a 10Now, we consider the energy regimes where the Thirring operator becomes more important. Actually, the duality can be quantitatively tested in the limit of large but finite mass, monopole operator Ma can be read out from the respective CS and BF coefficients involving the field a. This is i.e., for energies E ≪ jMj, independent of the value of g.In summarized in Table I. The fermion is identified as the this way, the relations that emerge in this limit remain valid operator that is uncharged by the emergent gauge fields a even for the strong-coupling regime. We emphasize that, as and c, being charged only by the electromagnetic field A: the scalar excitations are suppressed in this regime, no spin and statistical transmutation occur and therefore we can † ψ ⇔ ϕ Ma: ð11Þ neglect monopole configurations. On the fermionic side the calculation is straightforward. As the combination of fields a þ c is coupled to the Introducing the mass term in (7) and integrating out the electromagnetic field A, we can read out the corresponding fermion field, we get μ 1 μνρ current ψγ¯ ψ ⇔ 2π ϵ ∂νðaρ þ cρÞ. An immediate consequence of the duality (5) is that it e2sgnðMÞ e2 1 bdb − f2 þ ðeb − AÞ2 þ Oð1=M2Þ: points out to the existence of a nontrivial fixed point in 8π 48πjMj b 2g this the UV regime. Indeed, considering the bosonic dual ð Þ model, a simple dimensional analysis shows that, since λ 14 and g−1 have positive mass dimension, the operators jϕj4, ada, and cdc are negligible at the UV implying that the We consider the alternative Lagrangian bosonic model has a trivial UV fixed point. This, in turn, ð Þ 1 1 must correspond to a strongly coupled fixed point on the sgn M e 2 2 − cdcþ cdb− fc þ ðeb−AÞ : ð15Þ fermionic side, according to the relation 8π 4π 48πjMj 2g

g g ψ¯ i=∂ψ − ðψγ¯ μψÞ2 ⇔ j∂ϕj2 − f2 : ð Þ Integrating c out, we get 2 16π2 aþc 12 e2 μ ¼ ð Þ μ − ϵμνσ b þ ð1 2Þ ð Þ Therefore, the duality predicts an UV nontrivial fixed point c sgn M eb 6j j fνσ O =M ; 16 for the Thirring model. Recent studies employing non- M perturbative methods also provide strong evidence for the existence of nontrivial fixed points in several four-fermion where we have used (16) recursively to write the Maxwell theories in 2 þ 1 dimensions [33–37]. However, a more term for the field c in terms of the field b [the difference ð1 2Þ quantitative comparison with our result is not available being of O =M ]. Plugging this back into (15), we obtain since these previous studies consider the four-dimensional sgnðMÞe2 e2 1 representation for the Dirac spinors, which induces a bdb − f2 þ ðeb − AÞ2 þ Oð1=M2Þ; couple of independent four-fermion interactions, contrarily 8π 48πjMj b 2g to the case of the irreducible representation. ð17Þ

IV. TESTING THE DUALITY which is equivalent to (14) up to Oð1=M2Þ terms. In order to make some quantitative tests of the duality, it Now, integrating out the b field in (15), we have ¼ − g is convenient to extend it to the massive case. With this eb A 4π dc. Plugging this back into (15), we finally get purpose, we assume that the massive form of relation (2) remains valid in the present case, so that we consider the sgnðMÞ 1 g 1 − cdc þ cdA − þ f2: ð18Þ following relation 8π 4π 64π2 48πjMj c g ψ¯ ψ − ψψ¯ − ðψγ¯ μψÞ2 On the bosonic side things are slightly more subtle. The i=DA M 2 strategy is similar to the fermionic case, in the sense that we λ 1 2 2 2 4 will integrate out the scalar field in the large mass limit. ⇔ jDa−cϕj − m jϕj − jϕj þ ada 4 4π The resulting effective action is then organized in a loop 1 1 1 g 2 expansion (with internal lines involving only the scalar − cdc − adc − f þ adA: ð Þ 4π 2π 16π2 a 2π 13 1 λ field) with parameter κ ≡ 3 j j. The difference, however, ð4πÞ2 m

076010-4 BOSONIZATION OF THE THIRRING MODEL IN 2 þ 1 … PHYS. REV. D 101, 076010 (2020) is that we have to consider the two phases of the model 3 m2 ¼ − Mλ; ð24Þ separately. 4π 2 0 Let us examine firstly the Higgs phase, where m < ,so 2 that the combination (a − c) acquires a mass. Expanding which is valid in the two phases, i.e., M>0 ⇒ m < 0 qffiffiffiffiffiffiffiffiffiffi 0 ⇒ 2 0 2m2 (Higgs phase) and M< m > (symmetric phase). around the new vacuum v ¼hϕi¼ − λ on the right- λ Comparison of the Maxwell term fixes the value of jmj, hand side (RHS) of (7) and integrating out the massive scalar mode, we obtain λ 2π ¼ : ð25Þ jmj 3 2m2 1 1 − ð1 þ OðκÞÞða − cÞ2 − cdc þ ada λ 4π 4π This is the specific point at which the duality is valid in the 1 g 1 regime of energy E ≪ jmj. The fact that the coupling − adc − f2 þ Ada: ð Þ 2π ð4πÞ2 a 2π 19 constant λ is not a free parameter is a consistency require- ment, since the fermionic side does not contain any λ μ μ λ μνσ λ2 dependence on . This also happens in the original duality, Integrating c out, we have c ¼ a − 2 ϵ ∂νaσ þOð 4Þ, 4πm m which holds at the Wilson-Fisher fixed point. Furthermore, where we have used the equation of motion for c recur- we see that the value (25) leads to a properly perturbative sively to replace dc by da. Using this relation, we finally parameter κ, get 1 λ 1 κ ≡ 3 ¼ 1 ∼ 0.047; ð26Þ 1 λ g 1 ð4πÞ2 jmj 12π2 − ada − − þ f2 þ Ada: ð20Þ 2π 16π2m2 16π2 a 2π consistent with the initial assumption that κ is small. Comparing the Chern-Simons coefficient of (20) with (18) It is interesting to notice that the matching of the (after rescaling a → a=2), we see that this phase corre- Maxwell terms is automatic for the Thirring coupling g. sponds to the case with M>0 on the fermionic side. This indicates that our calculation also gives a matching for 1 Furthermore, matching the Maxwell term gives the relation the original duality (2) to the order of =M if we turn off g. On the other hand, in the strict limit jMj → ∞, it gives the 3 mapping m2 ¼ − jMjλ: ð21Þ 4π j j→∞ ð Þ ψ¯ ψ − ψψ¯ − g ðψγ¯ μψÞ2 M⇔ − sgn M i =DA M 2 2π ada This result is consistent with the mechanism proposed in 1 g [8] to deform the critical duality by the addition of mass þ − 2 2π Ada 2 fa; terms in both sides, which leads to a relation of the 16π type m2 ∼−Mλ. ð27Þ In the symmetric phase, by integrating out the scalar field gives which, after the rescaling a → a=2, is precisely the Fradkin-Schaposnik mapping [30] shown in (4). There is a massive excitation appearing on the right-hand side with 1 2 1 1 − ð1 þ OðκÞÞf − − cdc þ ada ∼1 2 96πjmj a c 4π 4π mass =g . This corresponds to a fermion-antifermion 1 1 bound-state on the left hand side. In the UV limit, where the − − g 2 þ ð Þ g → ∞ adc 2 fa Ada: 22 Thirring model is strongly coupled, , the bound state 2π ð4πÞ 2π becomes massless.

¼ − þ 1 þ ð1 2Þ Integrating c out, we have c a 12jmj da O =m . V. CONCLUSIONS Plugging this back into (22), we obtain We have provided a bosonization of the Thirring model in 2 þ 1 dimensions for the case of a single fermion specie 1 1 1 − g þ 2 þ þ ð Þ with arbitrary mass. Therefore it extends previous works on 2 fa ada Ada; 23 16π 24πjmj 2π 2π the Thirring model that relied mostly on large N techniques or large mass limit. This was obtained from the bosoniza- which can be compared to (18) after rescaling a → a=2. tion duality of a Dirac fermion coupled to a background We see that this corresponds to the fermionic phase with gauge field, which lies in the heart of the web of dualities. M<0. Therefore we can write an identification between By carrying out manipulations of the corresponding par- masses in an unified way, tition functions we were able to produce the Thirring model

076010-5 SANTOS, GOMES, and HERNASKI PHYS. REV. D 101, 076010 (2020) on one side and the corresponding bosonic theory on the In the above context, it is interesting to make further other. A number of consistency checks were performed, contact with the work [8], which suggests that the UV fixed including the infinite mass limit where we recover the point of the Gross-Neveu model is described in terms of a known Fradkin-Schaposnik map, placing it in the context free scalar theory. Up to a decoupled Maxwell term, this is of the web of dualities. We have also explicitly shown the the type of theory that emerges in the UV limit of our validity of the duality for large but finite mass, which duality for the Thirring model. We believe that this is a encompasses automatically the original duality. As a by- reflection of the fact that the Thirring interaction in 3D with product of our analysis, we confirmed the relation implied a single spinor in the irreducible representation is equiv- in [8] connecting the mass parameters of the models and the alent to the Gross-Neveu interaction. In this sense, the coupling constant of the quartic bosonic interaction. UV limit of our duality is in compliance with that one The duality we have proposed for the Thirring model proposed in [8]. involves on the bosonic side a combination of gauge fields, We conclude by saying that further checks of the duality where the four-fermion coupling constant is directly tied to proposed in this work are currently under investigation and the Maxwell term. Thus the highly nontrivial task of shall be reported elsewhere. Nevertheless, we expect that analyzing the strong coupling properties of the Thirring discussions in this work could be helpful in the under- model can become manageable using its dual version since standing of strongly interacting four-fermion theories in it corresponds to a quadratic term. Indeed, a simple 2 þ 1 dimensions. dimensional analysis shows that the bosonic model has a trivial fixed point in the UV. The duality relation then ACKNOWLEDGMENTS implies that this should correspond to a nontrivial UV fixed point in the Thirring model. This is an important We are grateful to Holger Gies for the useful correspon- physical information that is extremely hard to be extracted dence concerning the existence of the UV nontrivial fixed when we do not have benefits of the large N or large mass point in the Thirring model. We acknowledge the financial expansions. support of Brazilian agencies CAPES and CNPq.

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