Phase transition and phase diagram at a general filling in the spinless one-dimensional Holstein Model

TL;DR

This paper derives an exact second-order transition condition for the spinless 1D Holstein model, revealing that charge-density-wave order occurs in a more restricted parameter space away from half-filling, correcting mean-field predictions.

Contribution

It provides a novel, exact second-order transition condition for the Holstein model using a blocked perturbative approach, refining understanding of CDW formation at general fillings.

Findings

CDW occurs in a more restricted parameter space away from half-filling.

The transition condition is derived exactly to second order in a new perturbative method.

Mean-field predictions are corrected by considering dynamic susceptibility instead of static.

Abstract

Among the mechanisms for lattice structural deformation, the electron-phonon interaction mediated Peierls charge-density-wave (CDW) instability in single band low-dimensional systems is perhaps the most ubiquitous. The standard mean-field picture predicts that the CDW transition occurs at all fillings and all values of the electron-phonon coupling $g$ and the adiabaticity parameter $t/\omega_0$. Here, we correct the mean-field expression for the Peierls instability condition by showing that the non-interacting static susceptibility, at twice the Fermi momentum, should be replaced by the dynamic one. We derive the Luttinger liquid (LL) to CDW transition condition, {\it exact to second order in a novel blocked perturbative approach}, for the spinless one-dimensional Holstein model in the adiabatic regime. The small parameter is the ratio $g \omega_0/t$. We present the phase diagram at non-half-filling by obtaining the surprising result that the CDW occurs in a more restrictive region of a two parameter ($g^2 \omega_0/t$ and $t/\omega_0$) space than at half-filling.