Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach

Abstract

We propose a Lie-algebraic duality approach to analyze non-equilibrium 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 Schrodinger-Bloch equations, which represent a viable alternative to the conventional Schrodinger formulation. These dual Schrodinger-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. 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 non-perturbative dual descriptions of these theories.

25 pages, 1 figure; v2: citations added

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• Barren Plateaus in Variational Quantum Computing
• Driven Electronic States at the Surface of a Topological Insulator
• Integrable Floquet dynamics
• SU(3) semiclassical representation of quantum dynamics of interacting spins
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• Stochastic Approach to Non-Equilibrium Quantum Spin Systems
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• Non-Abelian bosonization in a (3+1)-d Kondo semimetal via quantum anomalies
• Liouville coherent states
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• Time-Evolving Weiss Fields in the Stochastic Approach to Quantum Spins
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• Importance Sampling Scheme for the Stochastic Simulation of Quantum Spin Dynamics
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• Stochastic Representation of the Quantum Quartic Oscillator
• Trajectory-Resolved Weiss Fields for Quantum Spin Dynamics