Quantum mechanics with a time-dependent random unitary Hamiltonian: A perturbative study of the nonlinear Keldysh sigma-model

TL;DR

This paper studies a perturbative approach to quantum mechanics with time-dependent random Hamiltonians, simplifying calculations by canceling higher-order vertices and verifying the cancellation of four-loop corrections.

Contribution

It introduces a reduction of the sigma-model to a matrix  model and confirms the cancellation of certain high-order corrections, supporting theoretical conjectures.

Findings

Vertices of order higher than four cancel in the perturbative series.

The energy-diffusion constant calculation reduces to a matrix  model.

Four-loop corrections to the diffusion constant cancel in the high-velocity limit.

Abstract

We analyze the perturbative series of the Keldysh-type sigma-model proposed recently for describing the quantum mechanics with time-dependent Hamiltonians from the unitary Wigner-Dyson random-matrix ensemble. We observe that vertices of orders higher than four cancel, which allows us to reduce the calculation of the energy-diffusion constant to that in a special kind of the matrix \phi^4 model. We further verify that the perturbative four-loop correction to the energy-diffusion constant in the high-velocity limit cancels, in agreement with the conjecture of one of the authors.