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arXiv:0911.0624 · New Journal of Physics

Quantum kinetic Ising models

Remigiusz Augusiak, Fernando Cucchietti, Fritz Haake, Maciej Lewenstein

Published Feb 26, 2010• 13 citations
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Abstract

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.

DOI
10.1088/1367-2630/12/2/025021

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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.
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.
Editor's note Kinetic Ising models in 1D provide a gallery of exactly solvable systems with nontrivial dynamics. The emphasis has traditionally been on their exact solvability, although much attention has also been devoted to models with conservation laws that have to be treated by numerical and approximation methods. Chapter 4 reviews these models with emphasis on steady states and the approach to steady-state behavior. Chapter 5 puts the simplest 1D kinetic Ising models into a wider framework of the evaluation of dynamical critical behavior, analytically, in 1D, and numerically, for general dimension. Finally, Ch. 6 describes low-temperature nonequilibrium properties such as domain growth and freezing. For a general description of dynamical critical behavior, not limited to 1D, as well as an excellent review and classification of various types of dynamics, the reader is directed to the classical work [1]. Certain probabilistic cellular automata are equivalent to kinetic Ising models.
We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of d-level systems. The ground state energy eigenvalue and its derivatives, whose nonanalyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signaled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.
We demonstrate an efficient method that allows for simultaneous determination of the ground state, low energy excitation properties and excitation gap in quantum many body systems. To this aim we first use the \textit{ab-initio} optimization principle of tensor networks (TN), to show that the infinite density matrix renormalization group (iDMRG) in the real space is associated in a natural manner to the infinite time-evolving block decimation (iTEBD) implemented on a continuous matrix product state (MPS), and defined in imaginary time. We illustrate this association showing that the (imaginary) time matrix product state (MPS) in iTEBD reproduces accurately the properties of the two-dimensional (2D) classical Ising model, verifying in this way that the time MPS corresponds to a well-defined physical state. We apply then our scheme to the one-dimensional (1D) quantum Ising chain, where the time MPS is defined in continuous imaginary time. It is found that the time MPS at or close to the critical point is always maximally entangled, but gives essentially different correlations than those obtained from iDMRG; we argue that these correlations describe the dynamic correlations for the ground state.
We introduce quantum versions of the Kac Master Equation and the Kac Boltzmann Equation. We study the steady states of each of these equations, and prove a propagation of chaos theorem that relates them. The Quantum Kac Master Equation (QKME) describes a quantum Markov semigroup, while the Kac Boltzmann Equation describes a non-linear evolution of density matrices on the single particle state space. All of the steady states of the NNN particle quantum system described by the QKME are separable, and thus the evolution described by the QKME is entanglement breaking. The results set the stage for a quantitative study of approach to equilibrium in quantum kinetic theory, and a quantitative study the rate of destruction of entanglement in a class of quantum Markov semigroups describing binary interactions.