A map between time-dependent and time-independent quantum many-body Hamiltonians

TL;DR

This paper develops a method to relate time-dependent and time-independent quantum many-body Hamiltonians through gauge transformations, enabling simplification of driven dynamics to static quench dynamics in various systems.

Contribution

It formulates conditions under which gauge transformations map physical Hamiltonians to each other, simplifying the analysis of driven many-body quantum systems.

Findings

Mapped spin systems with magnetic fields to field-free systems

Eliminated time-dependent magnetic fields in fermionic systems

Applied the method to quantum Ising and spin-boson models

Abstract

Given a time-independent Hamiltonian $\widetilde H$, one can construct a time-dependent Hamiltonian $H_t$ by means of the gauge transformation $H_t=U_t \widetilde H \, U^\dagger_t-i\, U_t\, \partial_t U_t^\dagger$. Here $U_t$ is the unitary transformation that relates the solutions of the corresponding Schrodinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as "physical". We formulate sufficient conditions on $U_t$ ensuring that $H_t$ is physical as long as $\widetilde H$ is physical (and vice versa). This way we obtain a general method for finding such pairs of physical Hamiltonians $H_t$, $\widetilde H$ that the driven many-body dynamics governed by $H_t$ can be reduced to the quench dynamics due to the time-independent $\widetilde H$. We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to the time-independent system without a magnetic field [F. Yan, L. Yang, B. Li, Phys. Lett. A 251, 289 (1999); Phys. Lett. A 259, 207 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where $\widetilde H = \widetilde H_t$ is time-dependent but dynamically integrable.