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

• URL: http://arxiv.org/abs/2009.13873v4
• Date: Thu, 26 Nov 2020 08:01:15 GMT
• Title: A map between time-dependent and time-independent quantum many-body Hamiltonians
• Authors: Oleksandr Gamayun and Oleg Lychkovskiy
• Abstract summary: Given a time-independent Hamiltonian $widetilde H$, one can construct a time-dependent Hamiltonian $H_t$ by means of the gauge transformation $H_t=U_t widetilde H, Udagger_t-i, U_t, partial_t U_tdagger$. Here $U_t$ is the unitary transformation that relates the solutions of the corresponding Schrodinger equations.
• Score: 23.87373187143897
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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