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arXiv:cond-mat/9907193 · International Journal of Modern Physics A

FUNCTIONAL BOSONIZATION OF NONRELATIVISTIC FERMIONS IN 2+1 DIMENSIONS

Daniel Gustavo Barci, C. A. LINHARES, A. F. DE QUEIROZ, J. F. MEDEIROS NETO

Published Nov 20, 2000• 6 citations
arXivHTMLPDF

Abstract

We analyze the universality of the bosonization rules in nonrelativistic fermionic systems in (2+1)d. We show that, in the case of linear fermionic dispersion relations, a general fermionic theory can be mapped into a gauge theory in such a way that the fermionic density maps into a magnetic flux and the fermionic current maps into a transverse electric field. These are universal rules in the sense that they remain valid whatever the interaction considered. We also show that these rules are universal in the case of nonlinear dispersion relations provided we consider only density–density interactions. We apply the functional bosonization formalism to a nonrelativistic and nonlocal massive Thirring-like model and evaluate the spectrum of collective excitations in several limits. In the large mass limit, we are able to exactly calculate this spectrum for arbitrary density–density and current–current interactions. We also analyze the massless case and show that it has no collective excitations for any density–density potential in the Gaussian approximation. Moreover, the presence of current interactions may induce a gapless mode with a linear dispersion relation.

DOI
10.1142/s0217751x00002032

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We analyze the universality of the bosonization rules in nonrelativistic fermionic systems in (2+1)d. We show that, in the case of linear fermionic dispersion relations, a general fermionic theory can be mapped into a gauge theory in such a way that the fermionic density maps into a magnetic flux and the fermionic current maps into a transverse electric field. These are universal rules in the sense that they remain valid whatever the interaction considered. We also show that these rules are universal in the case of nonlinear dispersion relations provided we consider only density–density interactions. We apply the functional bosonization formalism to a nonrelativistic and nonlocal massive Thirring-like model and evaluate the spectrum of collective excitations in several limits. In the large mass limit, we are able to exactly calculate this spectrum for arbitrary density–density and current–current interactions. We also analyze the massless case and show that it has no collective excitations for any density–density potential in the Gaussian approximation. Moreover, the presence of current interactions may induce a gapless mode with a linear dispersion relation.
We present a collective field formalism for nonrelativistic fermions in one spatial dimension. A bosonization technique is used to convert the quantum mechanical fermionic problem to a bosonic one, which is further described as a second quantized Schrödinger field theory. A formulation in terms of current and density variables gives rise to the collective field representation. Applications of our formalism to the D=1 Hermitian matrix model and the system of one-dimensional fermions in the presence of a weak electromagnetic field are discussed.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
We apply a recently proposed path-integral approach to non-local bosonization to a Thirring-like system modeling non-relativistic massless particles interacting with localized fermionic impurities. We consider forward-scattering processes described by symmetric potentials including interactions between charge, current, spin and spin-current densities. In the general (spin-flipping) problem we obtain an effective action for the collective modes of the model at T = 0, containing WZW-type terms. When spin-flipping processes are disregarded the structure of the action is considerably simplified, allowing us to derive exact expressions for the dispersion relations of collective modes and two-point fermionic correlation functions as functionals of the potentials. Finally, as an example, we compute the momentum distribution for the case in which electrons and impurities are coupled through spin and spin-current densities only. The formulae we get suggest that our formalism could be useful in order to seek for a mechanism able to restore Fermi liquid behavior.
We first present a general method for extracting collective variables out of nonrelativistic fermions by extending the gauge theory of collective coordinates to fermionic systems.We then apply the method to a system of non-interacting flavored fermions confined in a one-dimensional flavor-independent potential.In the limit of a large number of particles we obtain a Lagrangian with the Wess-Zumino-Witten term, which is the wellknown Lagrangian describing the non-Abelian bosonization of chiral fermions on a circle.The result is universal and does not depend on the details of the confining potential.
We develop a general theory of fermion liquids in spatial dimensions greater than 1. The principal method, bosonization, is applied to the cases of short- and long-range longitudinal interactions and to transverse gauge interactions. All the correlation funtions of the system may be obtained with the use of a generating functional. Short-range and Coulomb interactions do not destroy the Landau-Fermi fixed point. Non-Fermi liquid fixed points are found, however, in the cases of a super-long-range longitudinal interaction in two dimensions and transverse gauge interactions in two and three spatial dimensions. We consider in some detail the (2+1)-dimensional problem of a Chern-Simons gauge action combined with a longitudinal two-body interaction V(q)\ensuremath{\propto}\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{\mathit{y}\mathrm{\ensuremath{-}}1}$, which controls the density, and hence gauge, fluctuations. For y0 we find that the gauge interaction is irrelevant and the Landau fixed point is stable, while for y>0 the interaction is relevant and the fixed point cannot be accessed by bosonization. Of special importance is the case y=0 (Coulomb interaction), which describes the Halperin-Lee-Read theory of the half-filled Landau level. We obtain the full quasiparticle propagator, which is of a marginal Fermi-liquid form. Using Ward identities, we show that neither the inclusion of nonlinear terms in the fermion dispersion nor vertex corrections alters our results: the fixed point is accessible by bosonization. As the two-point fermion Green's function is not gauge invariant, we also invetigate the gauge-invariant density response function. Near momentum Q=2kF{\mathit{k}}_{\mathit{F}}kF​, in addition to the Kohn anomaly we find other nonanalytic behavior. In the appendies we present a numerical calculation of the spectral function for a Fermi liquid with Landau parameter f0{\mathit{f}}_{0}f0​\ensuremath{\ne}0. We also show how Kohn's theorem is satisfied within the bosonization framework.
Chapter 31 discussed the generic O(N) non-linear σ-model. We have noticed that the abelian case N = 2 is special because the RG β-function vanishes in two dimensions. The corresponding O(2) invariant spin model is specially interesting: it provides an example of the celebrated Kosterlitz–Thouless phase transition and will be examined in Chapter 33. However, a thorough discussion of this model requires a new technique: we have to establish relations, special to two dimensions, between fermion and boson local field theories, by a method called bosonization. This chapter illustrates the derivation involves several steps with the help of various other two-dimensional models which are physically interesting in their own right. It first studies the free massless boson and fermion fields. It evaluates the determinant of the covariant fermion derivative in the presence of an external gauge field. The result is at the basis of the bosonization technique. The chapter then discusses the sine–Gordon (SG) model. It solves the Schwinger model, QED with massless fermions. Finally, it demonstrates the equivalence between the SG model and two fermion models with current-current interaction: the Thirring model and another model with two species of fermions.