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Physical review. B, Condensed matter

Bosonization with constraints: Application to the one-dimensionalt-Jmodel

D. Schmeltzer, A. R. Bishop

Published Feb 1, 1992• 5 citations
Publisher

Abstract

The bosonization scheme for constrained fermions in one dimension (1D) is constructed. It is shown that the constrained bosonization is similar to the unconstrained one: the only difference is a phase shift simple. This bosonization scheme is applied to the 1D t-J model. For small J/t the model is equivalent to the Luttinger liquid for charge and spin. When J/t\ensuremath{\ge}\ensuremath{\pi} sin(\ensuremath{\pi}\ensuremath{\delta})/cos(2\ensuremath{\pi}\ensuremath{\delta}) (with doping \ensuremath{\delta}) the holon liquid is unstable and one has only a spin liquid. This corresponds to phase separation.

DOI
10.1103/physrevb.45.3168

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The bosonization scheme for constrained fermions in one dimension (1D) is constructed. It is shown that the constrained bosonization is similar to the unconstrained one: the only difference is a phase shift simple. This bosonization scheme is applied to the 1D t-J model. For small J/t the model is equivalent to the Luttinger liquid for charge and spin. When J/t\ensuremath{\ge}\ensuremath{\pi} sin(\ensuremath{\pi}\ensuremath{\delta})/cos(2\ensuremath{\pi}\ensuremath{\delta}) (with doping \ensuremath{\delta}) the holon liquid is unstable and one has only a spin liquid. This corresponds to phase separation.
We develop and describe new approaches to the problem of interacting fermions in spatial dimensions greater than 1. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing fermion liquids. The complicated functional renormalization-group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the bosonization of higher-dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase-space and small-angle scattering processes neglected in our model (but does not, as yet, address questions of stability). The charge sector is equivalent to an exactly solvable Gaussian quantum field theory; the spin sector, however, must be solved semiclassically. Using the Luther-Haldane approach we recover the collective mode equation of Fermi-liquid theory and in three dimensions reproduce the T3{\mathit{T}}^{3}T3ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.
The phase diagram of the one-dimensional t-J model is investigated by analyzing the results of exact diagonalization and the exact solutions at J/t=0 and 2. Phase separation takes place above a critical value of J around Jc{\mathit{J}}_{\mathit{c}}Jc​/t=2.5--3.5 depending on the electron density. In the small-J region, Tomonaga-Luttinger liquid theory holds and its correlation exponents are calculated as a function of J/t and the electron density. Superconducting correlations become dominant in a region between the solvable case (J/t=2) and phase separation. A spin-gap region is also found at low density.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension, and transport in clean quantum wires and in the presence of isolated impurities.
The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d>1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.