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arXiv:cond-mat/9810145 · Journal of Physics Condensed Matter

Semiclassical description of Heisenberg models via spin-coherent states

John Schliemann, Franz G Mertens

Published Feb 9, 1998• 17 citations
arXivHTMLPDF

Abstract

We use spin-coherent states as the basis of a time-dependent variational ansatz for a semiclassical description of a large family of Heisenberg models. In addition to following common approaches we also evaluate the square variance of the Hamiltonian in terms of coherent states. This quantity turns out to have a natural interpretation with respect to time-dependent solutions of the equations of motion and allows an estimate of quantum fluctuations in a semiclassical regime to be made. The general results are applied to solitons, instantons and vortices in several one- and two-dimensional models.

DOI
10.1088/0953-8984/10/5/016

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We use spin-coherent states as the basis of a time-dependent variational ansatz for a semiclassical description of a large family of Heisenberg models. In addition to following common approaches we also evaluate the square variance of the Hamiltonian in terms of coherent states. This quantity turns out to have a natural interpretation with respect to time-dependent solutions of the equations of motion and allows an estimate of quantum fluctuations in a semiclassical regime to be made. The general results are applied to solitons, instantons and vortices in several one- and two-dimensional models.
We use coherent states as a time-dependent variational ansatz for a semiclassical treatment of the dynamics of anharmonic quantum oscillators. In this approach the square variance of the Hamiltonian within coherent states is of particular interest. This quantity turns out to have a natural interpretation with respect to time-dependent solutions of the semiclassical equations of motion. Moreover, our approach allows for an estimate of the decoherence time of a classical object due to quantum fluctuations. We illustrate our findings with the example of the Toda chain.
We introduce a variational method for simulating the dynamics of interacting open quantum spin systems. The method is based on the spin phase-space representation and variationally targets the Husimi-QQQ function with an ansatz based on a multi-dimensional mixture of spin-coherent states. Crucially, the mixture coefficients are allowed to take negative values, enabling the faithful capture of quantum correlations beyond semiclassical descriptions. The resulting equations of motion are derived from the Dirac-Frenkel variational principle and can be evaluated efficiently without resorting to Monte Carlo sampling by exploiting the analytical structure of the ansatz. As a first application, we demonstrate that this approach accurately captures both the full quantum dynamics and the non-equilibrium steady states of the transverse-field quantum Ising model, in excellent agreement with exact diagonalization. Furthermore, we show that the method scales efficiently to large two-dimensional lattices, a regime that remains challenging for other techniques.
Generalized Coherent States (GCS) are constructed (and discussed) in order to study quasiclassical behaviour of quantum spin models of the Heisenberg type. Several such models are taken to their semiclassical limits, whose form depends on the spin value as well as the Hamiltonian symmetry. In the continuum approximation, SU(2)/U(1) GCS when applied give rise to the well known Landau - Lifshitz classical phenomenology. For arbitrary spin values one obtains a lattice of coupled nonlinear oscillators. Corresponding classical continuum models are described as well.
Variational methods are of fundamental importance and widely used in theoretical physics, especially for strongly interacting systems. In this work, we present a set of variational equations of state (VES) for pure states of an interacting quantum Hamiltonian. The VES can be expressed in terms of the variation of the density operators or static correlation functions. We derive the algebraic relationship between a known pure state density matrix and its variation, and obtain the VES by applying this relation to the averaged Heisenberg-equations-of-motion for the exact density matrix. Additionally, we provide a direct expression of the VES in terms of correlation functions to make it computable. We present three nontrivial applications of the VES: a perturbation calculation of correlation functions of the transverse field Ising model in arbitrary spatial dimensions, a study of a longitudinal field perturbation to the one-dimensional transverse field Ising model at the critical point and variational calculation of magnetization and ground state energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice. For the second one, our results not only recover the scaling limit, but also indicate the possibility of continuous tuning of the critical exponents by adjusting the longitudinal fields differently from the scaling limit. For the Heisenberg model, we obtained results numerically comparable to established results with simple calculations. The VES approach provides a powerful and versatile tool for studying interacting quantum systems.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.