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Physical review. B, Condensed matter

sComment on ``Fermions with long-range interaction''

Claudio Castellani, C. Di Castro, A. Maccarone

Published Jan 15, 1997• 9 citations
Publisher

Abstract

In a recent paper by Schmeltzer [Phys. Rev. B 52, 7939 (1995)] it is claimed that a non-Fermi liquid behavior arises in a d-dimensional system of fermions interacting through a long-range potential of the form V(q)=g/|q${\mathrm{|}}^{\mathrm{d}+\mathrm{\ensuremath{\epsilon}}\ensuremath{-}|\mathrm{A}\mathrm{x}\mathrm{\ensuremath{-}}1}$ for 0\ensuremath{\leqslant}1. Schmeltzer arrives at this result using both renormalization group (RG) and bosonization. In this Comment we show that a proper application of the bosonization technique to this problem leads instead to a non-Fermi liquid behavior only for \ensuremath{\epsilon}-hat\ensuremath{\geqslant}d-1, in agreement with several previous analyses. We also give a direct physical interpretation of this result. Finally we comment on the application of RG to this problem.

DOI
10.1103/physrevb.55.2676

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Physical review. B, Condensed matter•Oct 15, 1993•44 citations
In a recent paper by Schmeltzer [Phys. Rev. B 52, 7939 (1995)] it is claimed that a non-Fermi liquid behavior arises in a d-dimensional system of fermions interacting through a long-range potential of the form V(q)=g/|q∣d+\ensuremathϵ\ensuremath−∣Ax\ensuremath−1{\mathrm{|}}^{\mathrm{d}+\mathrm{\ensuremath{\epsilon}}\ensuremath{-}|\mathrm{A}\mathrm{x}\mathrm{\ensuremath{-}}1}∣d+\ensuremathϵ\ensuremath−∣Ax\ensuremath−1 for 0\ensuremath{\leqslant}1. Schmeltzer arrives at this result using both renormalization group (RG) and bosonization. In this Comment we show that a proper application of the bosonization technique to this problem leads instead to a non-Fermi liquid behavior only for \ensuremath{\epsilon}-hat\ensuremath{\geqslant}d-1, in agreement with several previous analyses. We also give a direct physical interpretation of this result. Finally we comment on the application of RG to this problem.
A renormalization-group analysis combined with bosonization is performed for fermions interacting with repulsive long-range interaction of the form V(R)\ensuremath{\simeq}1/${\mathit{R}}^{(1\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\epsilon}}\mathrm{^})}$, 0^\ensuremath{\le}1; we find that in any dimension the system is a non-Fermi liquid with a quasiparticle weight which vanishes like ${\mathit{w}}^{\mathrm{\ensuremath{\epsilon}}\mathrm{^}/4}$ when w\ensuremath{\rightarrow}0. For \ensuremath{\epsilon}^\ensuremath{\rightarrow}0 a marginal Fermi liquid is obtained. A nonperturbative solution is obtained using the bosonization method in d\ensuremath{\ge}1 dimensions. Within the same formalism we investigate also the electron-phonon interaction in dimensions d=3-\ensuremath{\epsilon}. When the coupling constant is imaginary, one obtains a time-dependent potential similar to what is obtained for fermions coupled to transversed U(1) gauge bosons. A non-Fermi-liquid behavior is found for d\ensuremath{\le}3.
We consider systems of nonrelativistic, interacting electrons at finite density and zero temperature in d=2,3,ldots dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi-liquid (LFL) renormalization-group fixed point. We are especially interested in understanding under what circumstances such a system is a marginal Fermi-liquid (MFL) when the dimension of space is d\ensuremath{\geqslant}2. The interacting electron system is analyzed by combining renormalization-group (RG) methods with so called 'Luther-Haldane' bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ , which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant ${\mathrm{g}}_{\ensuremath{\lambda}}$ , which measures the strength of electron interactions at energy scales \ensuremath{\sim}vF{\mathrm{v}}_{\mathrm{F}}vF​ kF{\mathrm{k}}_{\mathrm{F}}kF​ /\ensuremath{\lambda}. For systems with a strictly convex Fermi surface, superconductivity is the only symmetry-breaking instability. Excluding such an instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions, a deviation from LFL behavior is possible: if the exponent \ensuremath{\alpha} parametrizing the singularity of the interaction potential in momentum space by V-hat(|p|)\ensuremath{\sim}1/|p${\mathrm{|}}^{\mathrm{\ensuremath{\alpha}}}$ is greater than or equal to d-1, the results of the bosonization calculation are consistent with a MFL.
We consider systems of nonrelativistic, interacting electrons at finite density and zero temperature in d=2,3,ldots dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi-liquid (LFL) renormalization-group fixed point. We are especially interested in understanding under what circumstances such a system is a marginal Fermi-liquid (MFL) when the dimension of space is d\ensuremath{\geqslant}2. The interacting electron system is analyzed by combining renormalization-group (RG) methods with so called 'Luther-Haldane' bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ , which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant ${\mathrm{g}}_{\ensuremath{\lambda}}$ , which measures the strength of electron interactions at energy scales \ensuremath{\sim}vF{\mathrm{v}}_{\mathrm{F}}vF​ kF{\mathrm{k}}_{\mathrm{F}}kF​ /\ensuremath{\lambda}. For systems with a strictly convex Fermi surface, superconductivity is the only symmetry-breaking instability. Excluding such an instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions, a deviation from LFL behavior is possible: if the exponent \ensuremath{\alpha} parametrizing the singularity of the interaction potential in momentum space by V-hat(|p|)\ensuremath{\sim}1/|p${\mathrm{|}}^{\mathrm{\ensuremath{\alpha}}}$ is greater than or equal to d-1, the results of the bosonization calculation are consistent with a MFL.
We consider systems of non-relativistic, interacting electrons at finite density and zero temperature in d = 2, 3, ... dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi liquid (LFL) renormalization group fixed point. We are especially interested in understanding under what circumstances such a system is a Marginal Fermi liquid (MFL) when the dimension of space is greater or equal than 2 . The interacting electron system is analyzed by combining renormalization group (RG) methods with bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter, λ−1,λ^{-1} ,λ−1, which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant, $ g_λ, $ which measures the strength of electron interactions at corresponding energy scales. Excluding a superconducting instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of $ λ^{-1} $ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions a deviation from LFL behaviour is possible : the results of the bosonization calculation are consistent with a MFL, if the exponent parametrizing the singularity of the interaction potential in momentum space is greater than or equal to d-1.
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 examine the low-energy behavior of two-dimensional electrons with current-current interactions. The interactions are long range, cannot be screened, and result in anomalies in perturbation theory. In the Hartree-Fock theory, we find an instability of the Landau Fermi-liquid state resulting in ``fermion-condensate'' states with a high degeneracy of levels at the Fermi energy. This degeneracy may be removed to give a Fermi-liquid state with vanishing Fermi velocity. As an example, we investigate further the Fermi-liquid state with the quasiparticle spectrum \ensuremath{\epsilon}(p)\ensuremath{\propto}(p-pF{\mathit{p}}_{\mathit{F}}pF​)3/2{)}^{3/2})3/2 proposed by Halperin et al. Application of the recent Haldane scheme of bosonization in higher dimensions to this state leads to coupling between bosons, which has the form required for a breakdown of the Landau Fermi-liquid behavior. A generalized form of the Luttinger liquid is obtained.