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Physical Review A

Time-dependent behavior of one-dimensional many-fermion models: Comparison with two- and three-dimensional models

M. Howard Lee, J. M. Hong, N. L. Sharma

Published Mar 1, 1984• 20 citations
Publisher

Abstract

Time-dependent behavior of one-dimensional (1D) many-fermion models is obtained by a method of recurrence relations. The Hilbert space of the density-fluctuation operator is two dimensional ($2d$), resulting in a time-independent generalized random force. The relevant Hilbert spaces of 2D and 3D many-fermion models, however, are infinite dimensional and the generalized random forces are consequently time dependent. The structure of these Hilbert spaces provides a picture of time-dependent behavior for 1D models which is fundamentally very different from that for 2D or 3D models.

DOI
10.1103/physreva.29.1561

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Time-dependent behavior of one-dimensional (1D) many-fermion models is obtained by a method of recurrence relations. The Hilbert space of the density-fluctuation operator is two dimensional (2d2d2d), resulting in a time-independent generalized random force. The relevant Hilbert spaces of 2D and 3D many-fermion models, however, are infinite dimensional and the generalized random forces are consequently time dependent. The structure of these Hilbert spaces provides a picture of time-dependent behavior for 1D models which is fundamentally very different from that for 2D or 3D models.
We discuss a new method to perform numerical simulations of one-dimensional systems with fermion and boson degrees of freedom. The method is based on a direct-space, imaginary-time representation of the fermion field. It is fast so that systems having up to 100 sites can easily be simulated. In addition, the method provides an intuitive physical "picture" of the ground state of a one-dimensional many-body system. We discuss in detail how to implement the method and how to compute various physical quantities. In particular, we show how to extend the method to study averages of off-diagonal quantities in an occupation-number representation. To assess the accuracy of our procedure, we apply it to free fermions in one dimension and compare with exact results. We then study a model of spinless interacting fermions and obtain the expected phase structure and behavior of correlation functions. We also consider the extended Hubbard model at various points in its phase diagram and study the behavior of spin-density, charge-density, and pairing correlation functions. We then study the Gross-Neveu model and show how the behavior depends on the number of fermion flavors. Finally, we consider an electron-phonon model and study its behavior both in the one-particle polaron sector and in the half-filled-band case. Along the way we show pictures of the ground-state configurations that give physical insight into the properties of the systems, like charge-density-wave, spin-density-wave, and superconducting states, "fractional charges," and solitons. We conclude by comparing our method with other methods and discuss the possibility of extending it to higher dimensions.
The method of recurrence relations allows one to calculate time dependent correlation functions from first principles. It originates from the Kubo scalar product which realizes abstract Hilbert space of dynamical variables. For this realized space, we show that there exists a unique orthogonalization process by recurrence relation. The method of recurrence relations exploits geometric properties of the structure of the realized Hilbert space such as the dimensionality and shape. These geometric properties depend on models as well as static constraints, e.g., temperature, wave vector, size. This method has been applied to several physical models including the homogeneous dense electron gas, the spin-1/2 XY and transverse Ising models, the spin van der Waals model and others. For these models, we have obtained the relaxation function, response function and memory function.
This document investigates the dynamics of the one-dimensional S=1/2 isotropic XY model and transverse Ising model in the high temperature limit by using the method of recurrence relations. The authors obtain the relaxation functions as well as some Brownian analogs of a generalized Langevin equation for a tagged spin S x over j in these models, namely, the memory functions and the random forces. It is found that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale. Based on the infinite dimensionality of these Hilbert spaces it is also concluded that S x over j is ergodic in both models.
We investigate the dynamics of the one-dimensional S=(1/2) isotropic XY model and transverse Ising model in the high-temperature limit by using the method of recurrence relations. We obtain the relaxation functions as well as some Brownian analogs of the generalized Langevin equation for a tagged spin Sjx{S}_{j}^{x}Sjx​ in these models, namely, the memory functions and the random forces. We find that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale.