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Title: Quantum-to-classical correspondence and Hubbard-Stratonovich dynamical systems: A Lie-algebraic approach

Abstract

We propose a Lie-algebraic duality approach to analyze nonequilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems). The first part of the paper utilizes a geometric Hilbert-space-invariant formulation of unitary time evolution, where a quantum Hamiltonian is viewed as a trajectory in an abstract Lie algebra, while the sought-after evolution operator is a trajectory in a dynamic group, generated by the algebra via exponentiation. The evolution operator is uniquely determined by the time-dependent dual generators that satisfy a system of differential equations, dubbed here dual Schroedinger-Bloch equations, which represent a viable alternative to the conventional Schroedinger formulation. These dual Schroedinger-Bloch equations are derived and analyzed on a number of specific examples. It is shown that deterministic dynamics of a closed classical dynamical system occurs as action of a symmetry group on a classical manifold and is driven by the same dual generators as in the corresponding quantum problem. This represents quantum-to-classical correspondence. In the second part of the paper, we further extend the Lie-algebraic approach to a wide class of interacting many-particle lattice models. A generalized Hubbard-Stratonovich transform is proposed and it is used to show that the thermodynamicmore » partition function of a generic many-body quantum lattice model can be expressed in terms of traces of single-particle evolution operators governed by the dynamic Hubbard-Stratonovich fields. The corresponding Hubbard-Stratonovich dynamical systems are generally nonunitary, which yields a number of notable complications, including breakdown of the global exponential representation. Finally, we derive Hubbard-Stratonovich dynamical systems for the Bose-Hubbard model and a quantum spin model and use the Lie-algebraic approach to obtain new nonperturbative dual descriptions of these theories.« less

Authors:
 [1]
  1. Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742-4111 (United States)
Publication Date:
OSTI Identifier:
22038601
Resource Type:
Journal Article
Journal Name:
Physical Review. A
Additional Journal Information:
Journal Volume: 84; Journal Issue: 1; Other Information: (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; ALGEBRA; BLOCH EQUATIONS; DIFFERENTIAL EQUATIONS; HAMILTONIANS; HILBERT SPACE; HUBBARD MODEL; LIE GROUPS; MANY-BODY PROBLEM; PARTITION FUNCTIONS

Citation Formats

Galitski, Victor. Quantum-to-classical correspondence and Hubbard-Stratonovich dynamical systems: A Lie-algebraic approach. United States: N. p., 2011. Web. doi:10.1103/PHYSREVA.84.012118.
Galitski, Victor. Quantum-to-classical correspondence and Hubbard-Stratonovich dynamical systems: A Lie-algebraic approach. United States. doi:10.1103/PHYSREVA.84.012118.
Galitski, Victor. Fri . "Quantum-to-classical correspondence and Hubbard-Stratonovich dynamical systems: A Lie-algebraic approach". United States. doi:10.1103/PHYSREVA.84.012118.
@article{osti_22038601,
title = {Quantum-to-classical correspondence and Hubbard-Stratonovich dynamical systems: A Lie-algebraic approach},
author = {Galitski, Victor},
abstractNote = {We propose a Lie-algebraic duality approach to analyze nonequilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems). The first part of the paper utilizes a geometric Hilbert-space-invariant formulation of unitary time evolution, where a quantum Hamiltonian is viewed as a trajectory in an abstract Lie algebra, while the sought-after evolution operator is a trajectory in a dynamic group, generated by the algebra via exponentiation. The evolution operator is uniquely determined by the time-dependent dual generators that satisfy a system of differential equations, dubbed here dual Schroedinger-Bloch equations, which represent a viable alternative to the conventional Schroedinger formulation. These dual Schroedinger-Bloch equations are derived and analyzed on a number of specific examples. It is shown that deterministic dynamics of a closed classical dynamical system occurs as action of a symmetry group on a classical manifold and is driven by the same dual generators as in the corresponding quantum problem. This represents quantum-to-classical correspondence. In the second part of the paper, we further extend the Lie-algebraic approach to a wide class of interacting many-particle lattice models. A generalized Hubbard-Stratonovich transform is proposed and it is used to show that the thermodynamic partition function of a generic many-body quantum lattice model can be expressed in terms of traces of single-particle evolution operators governed by the dynamic Hubbard-Stratonovich fields. The corresponding Hubbard-Stratonovich dynamical systems are generally nonunitary, which yields a number of notable complications, including breakdown of the global exponential representation. Finally, we derive Hubbard-Stratonovich dynamical systems for the Bose-Hubbard model and a quantum spin model and use the Lie-algebraic approach to obtain new nonperturbative dual descriptions of these theories.},
doi = {10.1103/PHYSREVA.84.012118},
journal = {Physical Review. A},
issn = {1050-2947},
number = 1,
volume = 84,
place = {United States},
year = {2011},
month = {7}
}