A Map between Time-Dependent and Time-Independent Quantum Many-Body Hamiltonians

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\kern 1pt \widetilde H \kern 1pt U^\dagger _t-i\kern 1pt U_t\kern 1pt\partial _t U_t^\dagger $. Here $U_t$ is the unitary transformation that relates the solutions of the corresponding Schrödinger 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 pairs of physical Hamiltonians $H_t$ and $\widetilde H$ such 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 a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (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.