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arXiv:hep-lat/9702002

A toy model test of a new algorithm for bozonization of fermion determinants

T. Bakeyev, M. I. Polikarpov, Andrei Alekseevich Slavnov, Alexander I. Veselov

Published Feb 4, 1997• 0 citations
arXivHTMLPDF

Abstract

Different aspects of bozonization algorithm proposed by Slavnov are tested by numerical simulations of a one dimensional toy model.
DOI
10.48550/arxiv.hep-lat/9702002

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We study a system of fermions interacting with a gauge field which can be used to describe either spin liquid or ν=1/2ν=1/2ν=1/2 Quantum Hall state. We propose a generalized model with a dimensionless parameter NNN. We evaluate the properties of the model in both limits N≫1N \gg 1N≫1 and N≪1N \ll 1N≪1 and deduce the properties of the model in the most physically intersting case of N=1,2N=1,2N=1,2. At N≪1N \ll 1N≪1 the motion of the fermions becomes one dimensional. This allows us to obtain the fermion Green function and response functions applying bozonization method in this limit.
It is common in physics to treat 1D system as a toy model in order to test a physical theory because it is easier to work with low dimensional systems and find analytical solutions. One naively will say that one-dimensional models serve as toy models and they do not apply in the real world. But, one-dimensional systems, e.g quantum wires, exists in nature and have quite different physics from higher dimensional systems. It is remarkable that, in 1D, an interacting fermionic problem can be turned to a free bosonic theory. An interesting question to ask is how the system evolves after a change in the systems internal parameters. This is called a quantum quench. In this thesis we investigate quantum quenches in the Luttinger liquid model, which is a 1D model and serves as the analog of the Fermi liquid theory in one dimension. More specifically, we study the kinetic energy, when we periodically drive the internal parameters of the system, and the two point correlation function, after a periodic quench of finite duration.
A comparative study of the efficiency of different algorithms (Metropolis, Langevin, and Hybrid) for the simulation of a one-dimensional spinless fermion model with a half-filled band is presented. The Metropolis updating of the fields is performed with an algorithm introduced by Blankenbecler, Scalapino, and Sugar. The density-density correlation function, the order parameter, and the autocorrelation are calculated. A study of the dependence of the systematic errors on the temperature and the step size for the Langevin and hybrid algorithms is presented. The relation between the accuracy and the speed of the algorithms is studied.