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

Oleksandr V. Gamayun, Oleg V. Lychkovskiy

Research output: Contribution to journalArticlepeer-review

Abstract

Abstract: Given a time-independent Hamiltonian (Formula presented.), one can construct a time-dependent Hamiltonian H by means of the gauge transformation Ht = UtHU†t− iUt ∂tU†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 (Formula presented.) ensuring that H is physical as long as (Formula presented.) is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians (Formula presented.) and (Formula presented.) such that the driven many-body dynamics governed by H can be reduced to the quench dynamics due to the time-independent (Formula presented.). 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 (Formula presented.) is time-dependent but dynamically integrable.

Original languageEnglish
Pages (from-to)41-51
Number of pages11
JournalProceedings of the Steklov Institute of Mathematics
Volume313
Issue number1
DOIs
Publication statusPublished - Jul 2021

Keywords

  • driven quantum dynamics
  • dynamical integrability
  • gauge transformation

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