TY - JOUR

T1 - A Map between Time-Dependent and Time-Independent Quantum Many-Body Hamiltonians

AU - Gamayun, Oleksandr V.

AU - Lychkovskiy, Oleg V.

N1 - Funding Information:
The work of the second author was supported by the Russian Foundation for Basic Research, project no. 18-32-20218.
Publisher Copyright:
© 2021, Pleiades Publishing, Ltd.

PY - 2021/7

Y1 - 2021/7

N2 - 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.

AB - 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.

KW - driven quantum dynamics

KW - dynamical integrability

KW - gauge transformation

UR - http://www.scopus.com/inward/record.url?scp=85110959298&partnerID=8YFLogxK

U2 - 10.1134/S008154382102005X

DO - 10.1134/S008154382102005X

M3 - Article

AN - SCOPUS:85110959298

VL - 313

SP - 41

EP - 51

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

SN - 0081-5438

IS - 1

ER -