Renormalized phonons in nonlinear lattices: A variational approach

TL;DR

This paper introduces a variational method to analyze renormalized phonons in nonlinear lattices, accounting for pressure and potential asymmetry, providing accurate predictions of phonon properties and sound velocity.

Contribution

It develops a new variational approach extending the Gibbs-Bogoliubov inequality to study phonons in nonlinear lattices with asymmetric potentials and pressure effects.

Findings

Accurately predicts phonon properties in various nonlinear lattice models.

Recovers known results for symmetric potentials at zero pressure.

Extends analysis to asymmetric potentials and pressure conditions.

Abstract

We propose a variational approach to study renormalized phonons in momentum conserving nonlinear lattices with either symmetric or asymmetric potentials. To investigate the influence of pressure to phonon properties, we derive an inequality which provides both the lower and upper bound of the Gibbs free energy as the associated variational principle. This inequality is a direct extension to the Gibbs-Bogoliubov inequality. Taking the symmetry effect into account, the reference system for the variational approach is chosen to be harmonic with an asymmetric quadratic potential which contains variational parameters. We demonstrate the power of this approach by applying it to one dimensional nonlinear lattices with a symmetric or asymmetric Fermi-Pasta- Ulam type potential. For a system with a symmetric potential and zero pressure, we recover existing results. For other systems which beyond the scope of existing theories, including those having the symmetric potential and pressure, and those having the asymmetric potential with or without pressure, we also obtain accurate sound velocity.