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Journal de Physique I

Simple representation of the eigenstates of the U↦∞ one dimensional Hubbard model

R. G. Dias, J. M. B. Lopes dos Santos

Published Oct 1, 1992• 9 citations
Publisher

Abstract

We have constructed an exact map of the U↦∞ one dimensional Hubbard Hamiltonian onto a system of free fermions related to the charge degrees of freedom (holons). The spin configuration only influences the energy in so far as the fermions move in a loop which is threaded by a fictitious magnetic flux proportional to the spin configuration's total momentum. A rather simple and physically appealing interpretation of some of the features of the Bethe Ansatz solution is possible in this representation. We illustrated its use with a calculation of the t/U corrections to the energy.

DOI
10.1051/jp1:1992252

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We have constructed an exact map of the U↦∞ one dimensional Hubbard Hamiltonian onto a system of free fermions related to the charge degrees of freedom (holons). The spin configuration only influences the energy in so far as the fermions move in a loop which is threaded by a fictitious magnetic flux proportional to the spin configuration's total momentum. A rather simple and physically appealing interpretation of some of the features of the Bethe Ansatz solution is possible in this representation. We illustrated its use with a calculation of the t/U corrections to the energy.
The ground state of the one-dimensional Hubbard Hamiltonian is discussed on the basis of the Bethe ansatz solution of Lieb and Wu. A simple analytical representation of the distribution functions for charge and spin degrees of freedom is derived. The elementary excitations are associated with electron-hole pairs of two pseudo-fermion systems, one for the charge, the other for the magnetic excitations. Spinons and holons correspond to ground state configurations of odd-numbered rings. It is argued that holons will form bound pairs in weakly coupled chains.
We propose a new fermion-spin transformation to implement the charge-spin separation in the large U Hubbard, or the equivalent t-J model. The charge degree of freedom is represented by a spinless fermion while the spin degree of freedom is represented by a hard-core boson. The local constraint for single occupancy is exactly satisfied. Very good agreement with exact solution is obtained for one-dimensional case in the mean field approximation, regarding the total energy, gapless spinon and holon spectra, and the momentum distribution of physical electrons. The same approximation yields good doping dependence of the staggered magnetization in the two-dimensional case.
We propose a novel characterization of the exact solution of the one-dimensional Hubbard model with unparallel boundary magnetic fields. The non-diagonal boundary terms break the U(1) symmetry of the system and the number of electrons with fixed spin is not conserved. Thus it is difficult to study the physical quantities in the thermodynamic limit based on the previously obtained inhomogeneous T−Q relation and Bethe ansatz equations. In order to solve this problem, we propose the t−W scheme. The basic idea is that the eigenvalues of the transfer matrix can be expressed by its zero-roots instead of the usual Bethe roots. This method allows us to study thermodynamic properties of the system such as the ground state, surface energy, charge and spin excitations. The t−W scheme is universal and can be applied to the systems either with or without U(1) symmetry.
We develop a bosonization approach for one-dimensional models based on Bethe ansatz equations. The operator formalism of the exact soluble models in the low energy limit provides a systematic method to calculate the asymptotic correlation functions. As examples with and without internal degrees of freedom, the Calogero-Sutherland (C-S) model and the repulsive Hubbard model are considered respectively. We verify that the low energy behavior of the C-S model is controlled by two classes of c=1 conformal field theories, depending on whether the C-S interactions are among bosons or among fermions. For the Hubbard model, we show the explicit charge-spin separation at low energy for arbitrary U>0. The low energy behavior of the system is described by the (semi-) direct product of two independent Virasoro algebras with c=1.
Abstract We present a simple derivation of the Luttinger liquid relation for the one-dimensional Hubbard model, considering a finite U and in the U = ∞ limit. We describe a simple solution of the Hubbard model in the infinite repulsion limit and use it to calculate the correlators of the model in this limit in a simple and physical way using the bosonization technique. We then calculate the asymptotics of the correlators of the model at arbitrary U through a single parameter, which can be calculated from the Bethe Ansatz solution. Our derivation of the critical exponents is simple and allows one to express different physical operators of the Hubbard model through the charge and spin Bose fields in a direct and physically transparent way.