Low-energy tail of the spectral density for a particle interacting with a quantum phonon bath

We describe two approximation methods designed to capture the leading behavior of the low-energy tail of the momentum-dependent spectral density A(k, E) and the tunneling density of states D(E) for an injected particle, such as an electron or an exciton, interacting with a bath of phonons at a nonzero initial temperature T, including quantum corrections due to the nonzero frequencies of the relevant phonons. In our imaginary-time-dependent Hartree (ITDH) approximation, we consider a situation in which the particle is injected into a specified coherent state of the phonon system, and we show how one can use the ITDH approximation to obtain the correlation function C(v) for that initial state. The thermal average C(v) is obtained, in principle, by integrating the result over all possible initial phonon coherent states, weighted by a thermal distribution. However, in the low-energy tail, one can obtain a good first approximation by considering only initial states near the one that maximizes the integrand. Our second approximation, the fixed-wave-function (FWF) approximation, assumes that the wave function of the injected particle evolves instantaneously to a wave function which then is independent of time, while the phonon system continues to evolve due to interaction with the particle. We discuss how to invert the Laplace transform and how to obtain A(k, E) as well as D(E) from the imaginary-time analysis. The FWF approximation is used to calculate D(E) for a one-dimensional continuum model of a particle interacting with acoustic phonons, and effects due to the quantum motion of phonons are observed. In the classical phonon limit, where the nuclear mass is taken to infinity while the elastic constants and other parameters are held fixed, the dominant behaviors of both the ITDH and FWF approximations in the low-energy tail reduce to that found in the past for a particle in a random potential with a Gaussian statistical distribution.

Automated summary

We present two approximation methods for describing the leading behavior of the low-energy tail of the momentum-dependent spectral density A(k, E) and the tunneling density of states D(E) in a system of injected particles interacting with a phonon bath. The imaginary-time-dependent Hartree (ITDH) approximation considers the injection of particles into coherent states of the phonon system and calculates the correlation function C(v) for the initial state. The fixed-wave-function (FWF) approximation assumes that the wave function of the injected particle becomes time-independent while the phonon system continues to evolve. These approximations are used to study the effects of quantum motion of phonons on the calculated D(E) in a one-dimensional continuum model.