Many-Polaron Effects in the Holstein Model

TL;DR

This paper derives an exact second-order effective polaronic Hamiltonian for the one-dimensional Holstein model, maps it to a spin model, and analyzes phase transitions and correlations, extending the approach to higher dimensions.

Contribution

It provides a novel derivation of a polaronic Hamiltonian exact to second order and maps it onto a spin model, offering new insights into phase transitions in the Holstein model.

Findings

Identified the conditions for the validity of the perturbation expansion.

Mapped the polaronic Hamiltonian to an anisotropic Heisenberg spin model.

Analyzed the Luttinger liquid to CDW transition and correlation lengths.

Abstract

We derive an effective polaronic interaction Hamiltonian, {\it exact to second order in perturbation}, for the spinless one-dimensional Holstein model. The small parameter is given by the ratio of the hopping term ($t$) to the polaronic energy ($g^2 \omega_0$) in all the region of validity for our perturbation; however, the exception being the regime of extreme anti-adiabaticity ($t/\omega_0 \le 0.1$) and small electron-phonon coupling ($g < 1$) where the small parameter is $t/\omega_0$. We map our polaronic Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg spin model. By studying the mass gap and the power-law exponent of the spin-spin correlation function for our Heisenberg spin model, we analyze the Luttinger liquid to charge-density-wave transition at half-filling in the effective polaronic Hamiltonian. We calculate the structure factor at all fillings and find that the spin-spin correlation length decreases as one deviates from half-filling. We also extend our derivation of polaronic Hamiltonian to $d$-dimensions.