Thermodynamic limit evolution of a closed quantum Heisenberg-type spin model with mean-field interactions is characterized by classifying all the symmetries of the equations of motion. It is shown that parameters of the model induce a structure in the Hilbert space by partitioning it into invariant subspaces, decoupled by the underlying Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. All possible partitions are classified in terms of a 3 × 3 matrix of effective, thermodynamic limit coupling constants. It is found that there are either 1, 2, 4, or O ( N ) invariant subspaces. The BBGKY hierarchy decouples into the corresponding number of anti-Hermitian operators on each subspace. These findings imply that equilibration and the equilibrium in this model depend on the initial conditions.
Thermodynamic limit evolution of a closed quantum Heisenberg-type spin model with mean-field interactions is characterized by classifying all the symmetries of the equations of motion. It is shown that parameters of the model induce a structure in the Hilbert space by partitioning it into invariant subspaces, decoupled by the underlying Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. All possible partitions are classified in terms of a 3 × 3 matrix of effective, thermodynamic limit coupling constants. It is found that there are either 1, 2, 4, or O ( N ) invariant subspaces. The BBGKY hierarchy decouples into the corresponding number of anti-Hermitian operators on each subspace. These findings imply that equilibration and the equilibrium in this model depend on the initial conditions.
The time evolution of -spin reduced density operators is studied for a class of Heisenberg-type quantum spin models with long-range interactions. In the framework of the quantum Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy, we introduce an unconventional representation, different from the usual cluster expansion, which casts the hierarchy into the form of a second-order recursion. This structure suggests a scaling of the expansion coefficients and the corresponding time scales in powers of N 1/2 with the system size N , implying a separation of time scales in the large-system limit. For special parameter values and initial conditions, we can show analytically that closing the BBGKY hierarchy by neglecting -spin correlations never leads to equilibration, but gives rise to quasi-periodic time evolution with at most independent frequencies. Moreover, for the same special parameter values and in the large- N limit, we solve the complete recursion relation (the full BBGKY hierarchy), observing a superexponential decay to equilibrium in rescaled time τ = tN − 1/2 .
The statistical thermodynamics of the quantum spin- s XXZ Heisenberg model in the Weiss (mean-field) limit rests on the outer multiplicities of the irreducible representations in the Clebsch–Gordan decomposition of the n -fold spin- s tensor product, which determine the energy-level degeneracies and hence the partition function. Using known closed forms for these multiplicities in terms of generalized binomial coefficients of order 2 s + 1 , we write the exact finite-size partition function of the effective collective Hamiltonian and evaluate its thermodynamic limit by a large-deviation (Laplace) analysis validated against direct summations for systems of up to n = 10 4 spins. The free energy then follows from a two-parameter variational principle in the total spin and the magnetization per site. The ferromagnetic ordering is driven by the rotationally invariant part of the Hamiltonian, so that the Curie temperature T c = 1 3 J s ( s + 1 ) is independent of the anisotropy Δ, and the transition is of second order for every spin s , with classical mean-field exponents. The anisotropy reshapes the phase diagram: for Δ < 1 the field-driven first-order line of the isotropic ( Δ = 1 ) model opens into a dome of easy-plane ferromagnetic order, bounded by a continuous transition that restores the residual U(1) symmetry, and no first-order transitions or metastability occur throughout − 1 ≤ Δ < 1 ; the isotropic Curie-Weiss ferromagnet is recovered as Δ → 1. In two appendices we detail the asymptotics of the outer multiplicities down to the Gaussian-fluctuation prefactor and discuss the q -number and Jacobi-theta structures underlying the model, including the partial-theta-type structure that obstructs a standard theta-function reduction of the residual total-spin sum.
An accurate description of the nonequilibrium dynamics of systems with coupled spin and bosonic degrees of freedom remains theoretically challenging, especially for large system sizes and in higher than one dimension. Phase-space methods such as the truncated Wigner approximation (TWA) have the advantage of being easily scalable and applicable to arbitrary dimensions. In this work we adapt the TWA to generic spin-boson models by making use of recently developed algorithms for discrete phase spaces [J. Schachenmayer, A. Pikovski, and A. M. Rey, Phys. Rev. X 5, 011022 (2015)]. Furthermore we go beyond the standard TWA approximation by applying a scheme based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations to our coupled spin-boson model. This allows us, in principle, to study how systematically adding higher-order corrections improves the convergence of the method. To test various levels of approximation we study an exactly solvable spin-boson model, which is particularly relevant for trapped-ion arrays. Using TWA and its BBGKY extension we accurately reproduce the time evolution of a number of one- and two-point correlation functions in several dimensions and for an arbitrary number of bosonic modes.
Abstract We investigate the classical-quantum correspondence of non-Hermitian Spin-orbit (SO)-coupled bosonic junctions, where an effective decay term is introduced in one of the two wells. Starting from the normalized two-point functions, we analytically demonstrate that the mean-field system has a classical Hamiltonian structure, and we successfully derive a non-Hermitian discrete nonlinear Schrödinger (Gross–Pitaevskii) equation. We discover that near the symmetry-breaking phase transition point, the correspondence between classical (mean-field) and quantum dynamics is more likely to break down. Our numerical simulations of the full quantum model, limited to a finite number of particles due to computational constraints, reveal that when the effective spin–orbit coupling (SOC) strength takes half-integer values, atomic self-trapping in a non-lossy well consistently occurs regardless of the system parameters, and the corresponding quantum dynamics exhibit remarkable insensitivity to the number of particles in this scenario. Additionally, we reveal that in both the mean-field and many-particle models, the SOC effects can greatly promote the synchronous periodic oscillations between the spin-up and spin-down components, and this synchronous dynamic is closely related to symmetry.
A system of N spin-1/2 particles interacting with a thermal reservoir is used as a pedagogical example for advanced undergraduate and graduate students. We introduce and illustrate some methods, approximations, and phenomena related to dissipation and nonlinearity in many-particle physics. We start our analysis from the dynamical Sudarshan-Lindblad quantum master equation for the density operator of a system S interacting with a thermal reservoir R. We derive the quantum version of the so-called Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations such that the master equation can be decomposed in a hierarchical set of N - 1 equations (N > 1). The hierarchy is broken by introducing the mean-field approximation and reducing the problem to a nonlinear single particle system. In this scenario, the Hamiltonian is nonlinear (i.e., it depends on the state of <S), although the superoperator responsible for the dissipation and decoherence of S remains unaffected. To provide a useful tool to students: (1) we discuss the physical approximations involved, (2) we derive the analytical solution to the mean values equations of motion resulting from the Hamiltonian, (3) we solve analytically the master equation in the stationary regime, (4) we obtain and discuss the solution of the nonlinear master equation, numerically, and finally, (5) we discuss the master equation beyond the mean-field approximation and show how to introduce higher order quantum correlations that have been previously neglected.
The general structure and existence of equilibrium and ground states of infinite spin systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)] is investigated. The form and the role of the ‘‘Bogoliubov–Haag’’ Hamiltonians is cleared. The analysis is based on a recent mathematically correct definition of the corresponding time automorphism group τQ (specified by a classical Hamiltonian Q) of an appropriately defined ‘‘physical’’ C*-algebra of observables 𝒞 containing an abelian subalgebra 𝒩 of classical macroscopic quantities as well as the conventional ‘‘bare’’ quasilocal (sub-) algebra 𝒜, 𝒞=𝒜⊗𝒩 [P. Bóna, J. Math. Phys. 29, 2223 (1988)]. A Lie group G of ‘‘hidden symmetries’’ acting nontrivially (also) on 𝒩 plays an essential role in the presented theory. The applications to the quasispin strong-coupling version of the BCS model of superconductivity, and to the corresponding model of the Josephson junction illustrate possibilities of the developed formalism as well as an easy manipulation with it. A simple mathematical description of ‘‘symmetry breaking’’ is sketched for the considered class of mathematical models of large quantal systems.