Scaling Study of the Metal-Insulator Transition in one-Dimensional Fermion Systems

TL;DR

This study investigates the scaling behavior of metal-insulator transitions in one-dimensional fermion systems, using numerical solutions of Bethe Ansatz equations to analyze the correlation length and spin stiffness in the XXZ Heisenberg chain and Hubbard model.

Contribution

It introduces a method to determine the correlation length from spin stiffness scaling, applicable to models without exact solutions, enhancing understanding of transport properties in 1D systems.

Findings

Scaling behavior observed in the Hubbard model for certain /N ranges.

Method to extract correlation length from small system size data.

Differences in scaling behavior between the Hubbard model and XXZ chain.

Abstract

We consider the Ising phase of the antiferromagnetic XXZ Heisenberg chain on a finite-size lattice with N sites.We compute the large $N$ behavior of the spin stiffness, obtaining the correlation length \xi. We use our results to discuss the scaling behavior of metal-insulator transitions in 1D systems, taking into account the mapping between the XXZ Heisenberg chain and the spinless fermion model and known results for the Hubbard model. We study the scaling properties of both the Hubbard model and the XXZ Heisenberg chain, by solving numerically the Bethe Ansatz equations. We find that for some range of values of \xi/N the scaling behavior may be observed for the Hubbard model but not for the XXZ Heisenserg chain. We show how \xi can be obtained from the scaling properties of the spin stiffness for small system sizes. This method can be applied to models having not an exact solution, illuminating their transport properties.