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Physical review. B, Condensed matter

Bosonization in two dimensions: A space-time tunneling approach

D. Schmeltzer, A. R. Bishop

Published Nov 1, 1994• 7 citations
Publisher

Abstract

Using the space-time curve representation, a bosonization scheme is constructed which includes density excitations around each point on the Fermi surface (FS) and space-time tunneling of charges which belong to different FS points. The space-time tunneling reproduces the two-dimensional Jordan-Wigner transformation. In two dimensions, since the tunneling involves two electrons, the total Jordan-Wigner phase maps boson into boson. Therefore, the tunneling corresponds to instantons in 2e bosonic systems. Using this formalism, a spinless fermion model with short-range repulsive interactions is investigated. We find two components: sound-wave excitations in the normal direction of the FS reproduce the Fermi liquid (FL); the second component corresponds to the cooperon channel, violates charge conservation, and gives rise to two-particle coherent tunneling. As a result, the FL phase is destroyed at temperature T=0, and a tunneling gap for charge 2e is formed with angular momentum l\ensuremath{\ne}0. We apply our formalism to the U(1) gauge field coupled to a spinless fermion and find that at T\ensuremath{\ne}0 the FL is again destroyed. We also make some remarks concerning the t-J model.

DOI
10.1103/physrevb.50.12733

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arXiv:cond-mat/9606100•Jun 14, 1996•0 citations
Using the space-time curve representation, a bosonization scheme is constructed which includes density excitations around each point on the Fermi surface (FS) and space-time tunneling of charges which belong to different FS points. The space-time tunneling reproduces the two-dimensional Jordan-Wigner transformation. In two dimensions, since the tunneling involves two electrons, the total Jordan-Wigner phase maps boson into boson. Therefore, the tunneling corresponds to instantons in 2e bosonic systems. Using this formalism, a spinless fermion model with short-range repulsive interactions is investigated. We find two components: sound-wave excitations in the normal direction of the FS reproduce the Fermi liquid (FL); the second component corresponds to the cooperon channel, violates charge conservation, and gives rise to two-particle coherent tunneling. As a result, the FL phase is destroyed at temperature T=0, and a tunneling gap for charge 2e is formed with angular momentum l\ensuremath{\ne}0. We apply our formalism to the U(1) gauge field coupled to a spinless fermion and find that at T\ensuremath{\ne}0 the FL is again destroyed. We also make some remarks concerning the t-J model.
The t-J model in two dimensions is bosonized using a set of N, coupled two-dimensional Fermi-surface patches. Ignoring tunneling between the patches, the coherent tunneling of holes and the superfluid phase are suppressed. Within this scheme the system remains in the normal phase when temperature T\ensuremath{\rightarrow}0. The main feature of this construction is the absence of screening of the dissipative transversal gauge field generated by the spinons. This dissipative gauge field is responsible for the non-Fermi-liquid behavior, which is manifested in the free energy and single-particle Green function. The deviation from Fermi-liquid behavior is due to the U(1) gauge field, and at long distances a new exponent due to the holes is identified. Experimental consequences are discussed. \textcopyright{} 1996 The American Physical Society.
The Hubbard model in the strong coupling limit is considered in two dimensions. A bosonization method based on the extention of the Jordan-Wigner transformation to particles with spin and no double occupancy is introduced. We find that at T=0 a Fermi liquid exists and superconductivity appears. For a finite temperature and finite doping a normal state populated with vortices exists. As a result, we have a normal state with separated charge- and spin-density-wave excitations. Formally those excitations can be described with the help of the Luttinger-liquid representation. In the second part we parametrize the Fermi surface by a set of N points. Around each point we describe the Fermi surface excitations by holons and spinons and find a set of N-coupled Luttinger channels. At T=0 we have space-time tunneling of electric charges which destroys the Luttinger liquid in favor of a Fermi liquid. At T\ensuremath{\ne}0 or LT\ensuremath{\rightarrow}\ensuremath{\infty} (L is the size of the system) tunneling of electric charges might be prohibited giving rise to gapless excitations. In the last part we consider the case with a commensurate frozen flux in the ground state. As a result we find that the excitations are semions.
Using the Jordan-Wigner transformation and duality relations, a method for bosonization for nonrelativistic spinless fermions is introduced. The same method is used for one- and two-dimensional cases. In one dimension a cubic potential is obtained. After linearization the usual bosonization results for relativistic fermions are reproduced. In two dimensions the nonlocal potential induces vortices which determines the effective potential between the charged particles. We find that the low-energy excitations are acoustic charge-density waves.
The space-time tunneling bosonization has been introduced to study N-coupled Luttinger liquids. The interaction is separated into a forward exchange and a cooperon part. At T=0 the cooperon part is described by tunneling events which form a Coulomb gas in the plasma phase. At T\ensuremath{\ne}0 we ignore the cooperon part and show that the correlation functions show a Luttinger-liquid behavior. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, we extract the two-dimensional behavior and show that for a certain range of parameters the Luttinger liquid survives.
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 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.