Diagrammatic description of a system coupled strongly to a bosonic bath

TL;DR

This paper develops a formalism for describing strongly coupled system-bath interactions, especially with low-frequency spectral weight, using a unitary transformation and an expansion on the Keldysh contour, extending beyond traditional approximations.

Contribution

It introduces a method to derive a formally exact master equation expansion for strongly coupled systems with bosonic baths, including higher-order corrections beyond NIBA.

Findings

Higher-order diagrams reveal two classes of contributions.

Convergence depends on the spectral function's low-frequency behavior.

Lowest-order matches the $P(E)$-theory/NIBA approximation.

Abstract

We study a system-bath description in the strong coupling regime where it is not possible to derive a master equation for the reduced density matrix by a direct expansion in the system-bath coupling. A particular example is a bath with significant spectral weight at low frequencies. Through a unitary transformation it can be possible to find a more suitable small expansion parameter. Within such approach we construct a formally exact expansion of the master equation on the Keldysh contour. We consider a system diagonally coupled to a bosonic bath and expansion in terms of a non-diagonal hopping term. The lowest-order expansion is equivalent to the so-called $P(E)$-theory or non-interacting blip approximation (NIBA). The analysis of the higher-order contributions shows that there are two different classes of higher-order diagrams. We study how the convergence of this expansion depends on the form of the spectral function with significant weight at zero frequency.