Quantum Monte Carlo study of the one-dimensional Holstein model of spinless fermions

TL;DR

This study uses Quantum Monte Carlo methods to analyze the ground state and phase transitions of the one-dimensional Holstein model of spinless fermions, revealing a transition from metallic to charge-density-wave insulating phases.

Contribution

First application of Green's function Monte Carlo to the 1D Holstein model, providing detailed insights into phase transitions and Luttinger liquid parameters.

Findings

Identified a metal-insulator transition with charge-density-wave order.

Finite size scaling used to determine Luttinger liquid parameters.

Results agree with known analytics at weak and strong coupling regimes.

Abstract

The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground state energy, first fermionic excited state, density wave correlations, and mean lattice displacement are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of $t=0.1 \omega$, $\omega$, and $10 \omega$ where $\omega$ is the phonon frequency. At a finite fermion-phonon coupling $g$ there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, $u_\rho$ and $K_\rho$, the velocity of charge excitations and the correlation exponent, respectively. For free fermions ($g=0$) and for strong coupling ($g^2 \gg t \omega$) our results agree well with known analytic results. For $t=\omega$ and $t=10\omega$ our results are inconsistent with the metal-insulator transition being a Kosterlitz-Thouless transition.\\