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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 λà 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 equiv- alent 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. 2470-0010=2020=101(7)=076010(7) 076010-1 Published by the American Physical Society SANTOS, GOMES, and HERNASKI PHYS. REV. D 101, 076010 (2020) 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.
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