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A map between time-dependent and time-independent quantum many-body Hamiltonians | | Oleksandr Gamayun
; Oleg Lychkovskiy
; | | Date: |
29 Sep 2020 | | Abstract: | Given a time-independent Hamiltonian $widetilde H$, one can construct a
time-dependent Hamiltonian $H_t$ by means of the gauge transformation $H_t=U_t
widetilde H_t , U^dagger_t-i, U_t, partial_t U_t^dagger$. 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. | | Source: | arXiv, 2009.13873 | | Services: | Forum | Review | PDF | Favorites | |
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