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arXiv:1012.2873 · Physical Review A

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

Victor M. Galitski

Published Jul 26, 2011• 40 citations
arXivHTMLPDF

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 Schr\"odinger-Bloch equations, which represent a viable alternative to the conventional Schr\"odinger formulation. These dual Schr\"odinger-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

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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 Schr"odinger-Bloch equations, which represent a viable alternative to the conventional Schr"odinger formulation. These dual Schr"odinger-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.
The manifold of one-particle operators has a rich group structure. It is the Lie algebra of the complex Lie group GL(2s,), (2s = dimension of one-particle spin orbital basis) realized as second-quantized operators acting in fermion fock space. This representation is reduced by the subspaces referring to N particles (N = 0, 1, 2,…). One can associate families of coherent states within each of these subspaces with real forms, Gr, of GL(2s,). These families have isotropy groups, Hr, which are associated with subgroups of these real forms (isotropy groups are the groups of transformations that leave a state invariant). The cosets GL(2s,)/Hr can be given an exponential parameterization that labels the coherent states. This label space is a complex vector space which is in correspondence with the nonlinear manifold of coherent states in the hilbert space ℋ︁N. These complex variables can be made time dependent, and by utilizing the time-dependent variation principle one can obtain a nonlinear approximation to the time-dependent Schrödinger equation which has the form of classical equations of motion in a generalized phase space 𝒫 = Grc/Hrc. This classical dynamics can be given a Hilbert space description, where the Hilbert space is a subspace of square integrable holomorphic functions f: ↦ 𝒫 → and is isomorphic with ℋ︁N. Different pairs {Gr,Hr} are associated with different families of coherent states. Some examples of these constructions are given, and the structure of the approximate time evolution is discussed and related to time-dependent approximation schemes and generalized random phase approximations (RPA).
We develop a dynamical symmetry approach to path integrals for general interacting quantum spin systems. The time-ordered exponential obtained after the Hubbard-Stratonovich transformation can be disentangled into the product of a finite number of the usual exponentials. This procedure leads to a set of stochastic differential equations on the group manifold, which can be further formulated in terms of the supersymmetric effective action. This action has the form of the Witten topological field theory in the continuum limit. As a consequence, we show how it can be used to obtain the exact results for a specific quantum many-body system which can be otherwise solved only by the Bethe ansatz. This represents an example of a many-body system treated exactly using the path-integral formulation. Moreover, our method can deal with time-dependent parameters, which we demonstrate explicitly.
An infinite quantum system with correctly defined dynamics τQ as an automorphism group of a C*-algebra 𝒞 of observables is determined by any continuous unitary representation U(G) of a connected Lie group G, as well as by an arbitrary differentiable real function Q on the dual space 𝔤* to the Lie algebra 𝔤 of G with the canonically defined Poisson flow φQ on 𝔤*. For specific choices of Q and G, the system can be obtained as the thermodynamic limit of a net of finite lattice systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)]. A simple nontrivial model of this type is the quasispin BCS model of superconductivity in the strong coupling limit, or a corresponding model of the Josephson junction. A peculiar feature of the considered models is τQ noninvariance of the usually considered C*-algebra 𝒜 of quasilocal observables, as well as an important role of classical dynamics of a set of macroscopic (intensive) observables Qφ in the description of τQ. The work is restricted here to norm-continuous representations U(G), in which case 𝒞 is isomorphic to the tensor product 𝒜⊗𝒩, where 𝒩 is the commutative algebra of classical (intensive) observables of the considered infinite quantum system.
Abstract The microscopic dynamics for a class of long range interacting multi-lattice quantum systems is constructed in the thermodynamical limit by means of operator algebraic concepts. By direct estimations the existence of the limiting Schrödinger dynamics is proven for a set of states, which comprises also globally non-equilibrium situations. The expectation values of the classical observables in the pure phase states are shown to satisfy a set of coupled non-linear differential equations. The limiting Heisenberg dynamics is derived as a W*-automorphism group in the partially universal von Neumann algebra which corresponds to the selected set of states; it is in general, however, not σ-weakly continuous in the time parameter.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place. We establish a direct link between dynamics in the time-dependent Krylov subspace and the underlying Lie-algebraic structure of the Hamiltonian. We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra. In particular, we identify the minimal conditions under which the exact time-dependent Krylov dynamics is naturally determined by the interaction-picture Hamiltonian and governed by an embedded sl(2,C)\mathfrak{sl}(2,\mathbb{C})sl(2,C) subalgebra. We further show that an exact single-exponential representation of the time-evolution operator gives rise to a distinct time-independent Krylov dynamics in a unitarily related basis, from which the exact time-dependent Krylov dynamics can nevertheless be recovered. We also extend the framework to the oscillator algebra as the simplest extension of the nilpotent Heisenberg--Weyl algebra, and provide further examples, including the translated and dilated harmonic oscillator, systems governed by closed Virasoro subalgebras, a spin in a rotating magnetic field, and higher-dimensional generalizations for multi-level systems. In addition, we introduce a new quantum speed limit to the complexity growth rate generated by a time-dependent generator and show that, for evolutions governed by a Lie algebra, it retains the same functional form as in the time-independent case. Remarkably, saturation of this bound is strongly affected by temporal driving and persists only when the Hamiltonian commutes with itself at different times. These results establish a unified framework for characterizing operator growth and Krylov complexity in time-dependent quantum systems with underlying Lie-algebraic structures.