Effective single particle picture for anharmonic lattice dynamics: a Rosetta stone for electronic and ionic response

Abstract

We establish a theoretical framework for the dynamics of a lattice of ions in a mean-field approach, where anharmonicity is included via self-consistency. In this picture, the many-body dynamics of a system of $N$ atoms in three dimensions is mapped onto two kinds of $6N$-dimensional vectors: the phonon condensate, describing the evolution of the average atomic positions, and the phonon spinors, describing the evolution of the atomic elastic constants. The phonon spinors are classified by a quantum number that behaves as a spin: the phonon pseudospin. The many-body Liouville equation is replaced by two wave equations equivalent to the time-dependent Schrödinger equation for the electronic wave function in density functional theory. Exploiting this parallelism, we formulate the response of the anharmonic lattice in one-to-one correspondence with time-dependent density functional theory for electrons. In complete analogy with the electronic case, we express the ionic response in terms of matrix elements of operators representing external fields and forces. We show how anharmonicity screens external perturbations through a phonon analogue of the Hartree-exchange-correlation kernel. We provide expressions for the lattice optical and thermal conductivity, showing how thermal conductivity depends on the phonon pseudospin. By approximating the density matrix as a Gaussian, we recover the equations of the time-dependent self-consistent harmonic approximation. In this case, the linear-response equations are formulated in terms of an anharmonic kernel including three- and four-phonon scattering. By translating anharmonic lattice dynamics into the language of density functional theory, this work shows how theoretical and computational advances in modeling the dynamical response of interacting electrons can be directly applied to interacting ions.