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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Hamiltonian approach to kinetic Ising models

V. I. Tokar

Published Feb 1, 1996• 4 citations
Publisher

Abstract

In the framework of the quantum approach to stochastic dynamics, the master equations governing the temporal evolution of the kinetic Ising models (KIM) are cast in the form of imaginary-time Schr\"odinger equations with second-quantized Hermitian Hamiltonians. On the basis of the quantum formalism a classical evolution equation is derived for an effective time-dependent Ising-type Hamiltonian. The grand ensemble corresponding to the latter describes the statistics of the spin configurations evolving under the dynamics of the KIM. The latter approach was used to qualitatively explain the results of recent Monte Carlo simulations of the Kawasaki KIM. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.1411

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In the framework of the quantum approach to stochastic dynamics, the master equations governing the temporal evolution of the kinetic Ising models (KIM) are cast in the form of imaginary-time Schr"odinger equations with second-quantized Hermitian Hamiltonians. On the basis of the quantum formalism a classical evolution equation is derived for an effective time-dependent Ising-type Hamiltonian. The grand ensemble corresponding to the latter describes the statistics of the spin configurations evolving under the dynamics of the KIM. The latter approach was used to qualitatively explain the results of recent Monte Carlo simulations of the Kawasaki KIM. \textcopyright{} 1996 The American Physical Society.
The probability distribution (PD) of spin configurations in kinetic Ising models has been cast in the form of the canonical Boltzmann PD with a time-dependent effective Hamiltonian (EH). It has been argued that in systems with extensive energy EH depends linearly on the number of spins N leading to the exponential dependence of PD on the system size. In macroscopic systems the argument of the exponential function may reach values of the order of the Avogadro number which is impossible to deal with computationally, thus making unusable the linear master equation (ME) governing the PD evolution. To overcome the difficulty, it has been suggested to use instead the nonlinear ME (NLME) for the EH density per spin. It has been shown that in spatially homogeneous systems NLME contains only terms of order unity even in the thermodynamic limit. The approach has been illustrated with the kinetic Husimi-Temperley model (HTM) evolving under the Glauber dynamics. At finite N the known numerical results has been reproduced and extended to broader parameter ranges. In the thermodynamic limit an exact nonlinear partial differential equation of the Hamilton-Jacobi type for EH has been derived. It has been shown that the average magnetization in HTM evolves according to the conventional kinetic mean field equation.
In this paper, we introduce a quantum generalization of classical kinetic Ising models (KIM), described by a certain class of quantum many-body master equations. Similarly to KIMs with detailed balance that are equivalent to certain Hamiltonian systems, our models reduce to a set of Hamiltonian systems determining the dynamics of the elements of the many-body density matrix. The ground states of these Hamiltonians are well described by the matrix product, or pair entangled projected states. We discuss critical properties of such Hamiltonians, as well as entanglement properties of their low-energy states.
The Master equation of the kinetic Ising model with spin-flip and spin-exchange dynamics is written in a second quantized in terms of Pauli operators. The introduction of weighted transition rates allows a general formulation in accordance with the principle of detailed balance. Static and dynamic correlation functions are calculated. The structure factor reveals three different relaxation times already in an extended mean-field approximation. Additionally, we derive a nonlinear evolution equation for the magnetization under the influence of nonthermal external noise. Already in lowest order, the correlation function offers a stretched-exponential behavior with a universal exponent \ensuremath{\sigma}=4(d+2) below the critical dimensionality dc{\mathit{d}}_{\mathit{c}}dc​=2. A one-loop renormalization group approach is used to calculate the critical exponents for the growth of the magnetization. The value of the growth exponent \ensuremath{\beta}=2(2-d)(8+d) is supported by simulations in one dimension. \textcopyright{} 1996 The American Physical Society.
A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.
The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.