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Physical review. B, Condensed matter

Bosonization theory of fermions interacting via a Coulomb potential in the limit e2→∞

Tai Kai Ng

Published Sep 15, 2000• 7 citations
Publisher

Abstract

We study in this paper the properties of a gas of fermions interacting via a scalar potential $v(q)=4\ensuremath{\pi}{e}^{2}{/q}^{2}$ for $q<\ensuremath{\Lambda}\ensuremath{\ll}{k}_{F}$ at dimensions larger than 1, where $\ensuremath{\Lambda}\ensuremath{\ll}{k}_{F}$ is a high-momentum cutoff and ${k}_{F}$ is the Fermi wave vector. In particular, we shall consider the ${e}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, where charge fluctuations at wave vectors $q<\ensuremath{\Lambda}$ are forbidden. Within a bosonization approximation, effective actions describing the low-energy physics of the system are constructed, where we show that the system can be described as a Fermi liquid formed by chargeless quasiparticles, which has vanishing wave function overlap with the bare fermions in the system.

DOI
10.1103/physrevb.62.7019

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We study in this paper the properties of a gas of fermions interacting via a scalar potential v(q)=4\ensuremathπe2/q2v(q)=4\ensuremath{\pi}{e}^{2}{/q}^{2}v(q)=4\ensuremathπe2/q2 for q<\ensuremathΛ\ensuremath≪kFq<\ensuremath{\Lambda}\ensuremath{\ll}{k}_{F}q<\ensuremathΛ\ensuremath≪kF​ at dimensions larger than 1, where \ensuremathΛ\ensuremath≪kF\ensuremath{\Lambda}\ensuremath{\ll}{k}_{F}\ensuremathΛ\ensuremath≪kF​ is a high-momentum cutoff and kF{k}_{F}kF​ is the Fermi wave vector. In particular, we shall consider the e2\ensuremath→\ensuremath∞{e}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}e2\ensuremath→\ensuremath∞ limit, where charge fluctuations at wave vectors q<\ensuremathΛq<\ensuremath{\Lambda}q<\ensuremathΛ are forbidden. Within a bosonization approximation, effective actions describing the low-energy physics of the system are constructed, where we show that the system can be described as a Fermi liquid formed by chargeless quasiparticles, which has vanishing wave function overlap with the bare fermions in the system.
We discuss in this paper the low energy properties of a liquid of fermions coupling to a U(1) gauge field at wavevectors q<Λ≪k F at dimensions larger than one, where Λ≪k F is a high momentum cutoff and k F is the Fermi wave vector. In particular, we shall consider the e2→∞ limit where charge and current fluctuations at wave vectors q<Λ are forbidden, and the problem reduces to the problem of imposing constraint that no charge and current fluctuations are allowed in the liquid of fermions. Within a bosonization approximation, we show that the low energy properties of the system can be described as a Fermi liquid of chargeless quasiparticles which has vanishing wavefunction overlap with the bare fermion's in the system. The case of a two component system (t–J model) will also be discussed.
In this paper we study the properties of a gas of fermions coupling to a U(1) gauge field at wave vectors $q<\ensuremath{\Lambda}\ensuremath{\ll}{k}_{F}$ at dimensions larger than 1, where $\ensuremath{\Lambda}\ensuremath{\ll}{k}_{F}$ is a high-momentum cutoff and kF{k}_{F}kF​ is the Fermi wave vector. In particular, we shall consider the ${e}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, where charge and current fluctuations at wave vectors $q<\ensuremath{\Lambda}$ are forbidden. By generalizing a method reported in a previous paper [Phys. Rev. B 62, 7019 (2000)], we show that the system can be described as a ``marginal'' Fermi liquid of spin 1/2;1/2;1/2; chargeless fermions with vanishing wave functions overlap with the bare fermions in the system.
We investigate the effects of the Coulomb two-body interaction on Fermi liquids via bosonization in two and three spatial dimensions. The Coulomb interaction is singular in the limit of low momentum transfer, and recent interest in the possibility that some singular interactions might destroy the Fermi liquid state motivate us to reexamine it. We calculate the exact boson correlation function to show that the Fermi liquid state is retained in the case of Coulomb interactions in two and three dimensions. Spin and charge degrees of freedom propagate together at the same velocity and collective charge excitations (plasmons) exhibit the expected energy gap in three dimensions. Non-Fermi liquid behaviour occurs in two dimensions, however, for a super-long-range interaction studied recently by Bares and Wen.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (qc{\mathit{q}}_{\mathit{c}}qc​/kF{\mathit{k}}_{\mathit{F}}kF​)d{)}^{\mathit{d}})dF0{\mathit{F}}_{0}F0​ [1+F0{\mathit{F}}_{0}F0​${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here qc{\mathit{q}}_{\mathit{c}}qc​ is the cutoff of the interaction in wave-vector space, kF{\mathit{k}}_{\mathit{F}}kF​ is the Fermi wave vector, \ensuremath{\mu} is the chemical potential, F0{\mathit{F}}_{0}F0​ is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff qc{\mathit{q}}_{\mathit{c}}qc​ is small compared with kF{\mathit{k}}_{\mathit{F}}kF​, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.
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.
We present a consistent bosonization treatment for interacting fermion systems in dimensions higher than one. In our analysis, all physically relevant excitations corresponding to forward scattering are included and physical results do not depend on the cutoff parameter introduced in the bosonization process. We calculate the fermion momentum distribution function, from which a condition for the appearance of a Luttinger-liquid state is derived. We find that, for nonsingular interactions, the zero-frequency particle-hole pair excitations directly destroy the orthogonality catastrophe so that there is no Luttinger-liquid fixed point. We further consider the unscreened Coulomb interaction and find that the system remains in a Fermi-liquid state. We observe that the class of singular interactions which are expected to generate a Luttinger-liquid state is very tightly constrained by these zero-frequency excitations.