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

Abstract

Given a time-independent Hamiltonian $\tilde{H}$, one can construct a time-dependent Hamiltonian $H_t$ by means of the gauge transformation $H_t=U_t\tilde{H}{U}_t^{†}-i{U}_t{\partial }_t{U}_t^{†}$. 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 $\tilde{H}$ is physical (and vice versa). This way we obtain a general method for finding such pairs of physical Hamiltonians $H_t$, $\tilde{H}$ that the driven many-body dynamics governed by $H_t$ can be reduced to the quench dynamics due to the time-independent $\tilde{H}$. 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 $\tilde{H}={\tilde{H}}_t$ is time-dependent but dynamically integrable.

arXiv commentto be published in Proceedings of the Steklov Institute of Mathematics

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Cited by in corpus (1)

• Fermionic transport through a driven quantum point contact: breakdown of Floquet thermalization beyond a critical driving frequency