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Physical review. B, Condensed matter

Path-integral approach to the one-dimensional large-UHubbard model

Zheng-Yu Weng, D. N. Sheng, C. S. Ting, Z. B. Su

Published Apr 1, 1992• 36 citations
Publisher

Abstract

A path-integral formalism for the one-dimensional Hubbard model in the strong-coupling regime, which is equivalent to the t-J model in t/U expansion but without any explicit constraint, is developed. Based on this formalism, the zero-temperature properties of the Hubbard chain are systematically studied. In the infinite-U limit, the charge and spin degrees of freedom are shown to be completely separated. Such a separation at U=\ensuremath{\infty} provides a controllable perturbative scheme to study the strong-coupling case. In the large-U regime, the spin degree of freedom is represented by a ``squeezed'' Heisenberg chain. We solve the (squeezed) Heisenberg chain by introducing a soliton representation. Both the charge and spin excitations are found to agree well with the exact solution. The bare electron (hole) is shown to be a composite particle of these basic excitations, i.e., holon and spinon, together with a nonlocal string field. This string field makes the electron behave like a ``semion'' and plays an important role in determining various correlation functions. We analytically derive the asymptotic forms of the spin-spin and density-density correlation functions as well as the single-electron and the pair-electron Green's functions. The implications of the present work to the two-dimensional model are discussed.

DOI
10.1103/physrevb.45.7850

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A path-integral formalism for the one-dimensional Hubbard model in the strong-coupling regime, which is equivalent to the t-J model in t/U expansion but without any explicit constraint, is developed. Based on this formalism, the zero-temperature properties of the Hubbard chain are systematically studied. In the infinite-U limit, the charge and spin degrees of freedom are shown to be completely separated. Such a separation at U=\ensuremath{\infty} provides a controllable perturbative scheme to study the strong-coupling case. In the large-U regime, the spin degree of freedom is represented by a squeezed'' Heisenberg chain. We solve the (squeezed) Heisenberg chain by introducing a soliton representation. Both the charge and spin excitations are found to agree well with the exact solution. The bare electron (hole) is shown to be a composite particle of these basic excitations, i.e., holon and spinon, together with a nonlocal string field. This string field makes the electron behave like a semion'' and plays an important role in determining various correlation functions. We analytically derive the asymptotic forms of the spin-spin and density-density correlation functions as well as the single-electron and the pair-electron Green's functions. The implications of the present work to the two-dimensional model are discussed.
The momentum distribution function n(k) and spin-correlation function S(k) are determined for the one-dimensional large-U Hubbard model with various electron densities. The present work is featured with an application of the Bethe-ansatz wave function, which has a simple form in the large-U limit for any electron density. Namely, its charge degrees of freedom are expressed as a Slater determinant of spinless fermions, while its spin degrees of freedom are equivalent to the one-dimensional S=(1/2 Heisenberg model. The singularity of n(k) at k=kF{k}_{F}kF​ is analyzed from the system size dependence. In addition to the kF{k}_{F}kF​ singularity, n(k) has a weak singularity at k=3kF{k}_{F}kF​; however, no detectable singularity is present at 2kF{k}_{F}kF​, which one might expect from the spinless fermion wave function. The singularity of S(k) at 2kF{k}_{F}kF​ is also examined in detail. It is concluded from the size dependence that, when the system is away from half-filling, the nature of the singularity at 2kF{k}_{F}kF​ is different from that in the Heisenberg model. The results are compared with the behavior in the weak-correlation regime examined with the perturbation calculation, the prediction of g-ology, and recent Monte Carlo calculations.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.
The one-dimensional Hubbard model is studied for general electron densities and large U. On the basis of the coupled integral equations of Lieb and Wu exact asymptotic expansions are derived to fifth order in ${U}^{\mathrm{\ensuremath{-}}1}$ for the ground-state energy, the kinetic energy, and the number of doubly occupied sites. A closed-form expression for the distribution functions of charge and spin degrees of freedom is proposed which yields the same large-U expansion as that derived directly from the Lieb-Wu equations. Our analytical results are in excellent agreement with previous numerical studies.
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.