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Trudy Mat. Inst. Steklova, 2021, Volume 313, Pages 47–58 (Mi tm4159)  

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

Oleksandr V. Gamayuna, Oleg V. Lychkovskiybcd

a Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
b Skolkovo Institute of Science and Technology, Bol'shoi bul. 30, stroenie 1, Moscow, 121205 Russia
c Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
d Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia

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\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.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-32-20218
The work of the second author was supported by the Russian Foundation for Basic Research, project no. 18-32-20218.


DOI: https://doi.org/10.4213/tm4159

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English version:
Proceedings of the Steklov Institute of Mathematics, 2021, 313, 41–51

Bibliographic databases:

UDC: 538.9
Received: July 28, 2020
Revised: October 10, 2020
Accepted: November 18, 2020

Citation: Oleksandr V. Gamayun, Oleg V. Lychkovskiy, “A Map between Time-Dependent and Time-Independent Quantum Many-Body Hamiltonians”, Mathematics of Quantum Technologies, Collected papers, Trudy Mat. Inst. Steklova, 313, Steklov Math. Inst., Moscow, 2021, 47–58; Proc. Steklov Inst. Math., 313 (2021), 41–51

Citation in format AMSBIB
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\by Oleksandr~V.~Gamayun, Oleg~V.~Lychkovskiy
\paper A Map between Time-Dependent and Time-Independent Quantum Many-Body Hamiltonians
\inbook Mathematics of Quantum Technologies
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2021
\vol 313
\pages 47--58
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4159}
\crossref{https://doi.org/10.4213/tm4159}
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\jour Proc. Steklov Inst. Math.
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\vol 313
\pages 41--51
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