Quantumtoclassical correspondence and HubbardStratonovich dynamical systems: A Liealgebraic approach
Abstract
We propose a Liealgebraic duality approach to analyze nonequilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of HubbardStratonovich dynamical systems). The first part of the paper utilizes a geometric Hilbertspaceinvariant formulation of unitary time evolution, where a quantum Hamiltonian is viewed as a trajectory in an abstract Lie algebra, while the soughtafter evolution operator is a trajectory in a dynamic group, generated by the algebra via exponentiation. The evolution operator is uniquely determined by the timedependent dual generators that satisfy a system of differential equations, dubbed here dual SchroedingerBloch equations, which represent a viable alternative to the conventional Schroedinger formulation. These dual SchroedingerBloch 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 quantumtoclassical correspondence. In the second part of the paper, we further extend the Liealgebraic approach to a wide class of interacting manyparticle lattice models. A generalized HubbardStratonovich transform is proposed and it is used to show that the thermodynamicmore »
 Authors:

 Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 207424111 (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 10502947
 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; MANYBODY PROBLEM; PARTITION FUNCTIONS
Citation Formats
Galitski, Victor. Quantumtoclassical correspondence and HubbardStratonovich dynamical systems: A Liealgebraic approach. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.012118.
Galitski, Victor. Quantumtoclassical correspondence and HubbardStratonovich dynamical systems: A Liealgebraic approach. United States. doi:10.1103/PHYSREVA.84.012118.
Galitski, Victor. Fri .
"Quantumtoclassical correspondence and HubbardStratonovich dynamical systems: A Liealgebraic approach". United States. doi:10.1103/PHYSREVA.84.012118.
@article{osti_22038601,
title = {Quantumtoclassical correspondence and HubbardStratonovich dynamical systems: A Liealgebraic approach},
author = {Galitski, Victor},
abstractNote = {We propose a Liealgebraic duality approach to analyze nonequilibrium evolution of closed dynamical systems and thermodynamics of interacting quantum lattice models (formulated in terms of HubbardStratonovich dynamical systems). The first part of the paper utilizes a geometric Hilbertspaceinvariant formulation of unitary time evolution, where a quantum Hamiltonian is viewed as a trajectory in an abstract Lie algebra, while the soughtafter evolution operator is a trajectory in a dynamic group, generated by the algebra via exponentiation. The evolution operator is uniquely determined by the timedependent dual generators that satisfy a system of differential equations, dubbed here dual SchroedingerBloch equations, which represent a viable alternative to the conventional Schroedinger formulation. These dual SchroedingerBloch 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 quantumtoclassical correspondence. In the second part of the paper, we further extend the Liealgebraic approach to a wide class of interacting manyparticle lattice models. A generalized HubbardStratonovich transform is proposed and it is used to show that the thermodynamic partition function of a generic manybody quantum lattice model can be expressed in terms of traces of singleparticle evolution operators governed by the dynamic HubbardStratonovich fields. The corresponding HubbardStratonovich dynamical systems are generally nonunitary, which yields a number of notable complications, including breakdown of the global exponential representation. Finally, we derive HubbardStratonovich dynamical systems for the BoseHubbard model and a quantum spin model and use the Liealgebraic approach to obtain new nonperturbative dual descriptions of these theories.},
doi = {10.1103/PHYSREVA.84.012118},
journal = {Physical Review. A},
issn = {10502947},
number = 1,
volume = 84,
place = {United States},
year = {2011},
month = {7}
}