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arXiv:0909.0640 · Physical Review E

Exact energy spectrum of a two-temperature kinetic Ising model

I. Mazilu, H. T. Williams

Published Dec 8, 2009• 7 citations
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Abstract

We develop the exact energy spectrum for a two-temperature kinetic Ising spin chain and its dual reaction-diffusion system with spatially alternating pair annihilation and creation rates. We also discuss the symmetries of the system pseudo-Hamiltonian and their role in developing a general solution. The surprisingly simple form for the eigenvalues leads to interesting physical consequences and to a possible numerical analysis of the dynamical properties of the system.

DOI
10.1103/physreve.80.061109

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We develop the exact energy spectrum for a two-temperature kinetic Ising spin chain and its dual reaction-diffusion system with spatially alternating pair annihilation and creation rates. We also discuss the symmetries of the system pseudo-Hamiltonian and their role in developing a general solution. The surprisingly simple form for the eigenvalues leads to interesting physical consequences and to a possible numerical analysis of the dynamical properties of the system.
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We investigate a non-equilibrium reaction - diffusion model and equivalent ferromagnetic spin- spin chain with alternating coupling constant. The exact energy spectrum and the n-point hole correlations are considered with the help of the Jordan - Wigner fermionization and the interparticle distribution function method. Although the Hamiltonian has no explicit translational symmetry, the translational invariance is recovered after a long time due to the diffusion. We see the scaling relations for the concentration and the two-point function in finite-size analysis.
We investigate a non-equilibrium reaction - diffusion model and equivalent ferromagnetic spin- spin chain with alternating coupling constant. The exact energy spectrum and the n-point hole correlations are considered with the help of the Jordan - Wigner fermionization and the interparticle distribution function method. Although the Hamiltonian has no explicit translational symmetry, the translational invariance is recovered after a long time due to the diffusion. We see the scaling relations for the concentration and the two-point function in finite-size analysis.
In this paper the exact solution and correlation functions for a double-chain Ising model with multi-spin interactions and symmetric Hamiltonian density are obtained. The study employs the transfer matrix method to derive fundamental thermodynamic characteristics of the system. The main results include exact expressions for the partition function, free energy, internal energy, specific heat capacity, magnetization, susceptibility, and entropy in a strip of finite length and in the thermodynamic limit. The work provides explicit formulas for the eigenvalues and shows structure of eigenvectors of the transfer matrix. The expression for magnetization in the thermodynamic limit using components of normalized eigenvector corresponding to the maximum eigenvalue is obtained. A detailed analysis is conducted for a special case of interactions involving all kinds of two- and four-spin interactions. This gives the simplified formula for free energy, it is calculated using the root of quadratic equation. Using special relation between parameters the solution for gonihedric model on the plane is obtained. The research reveals properties of the system, including specific features of ground states and phase diagram characteristics. Particular attention is given to the behavior of physical quantities near frustration points and the investigation of spin correlation functions. Plots of physical characteristics, including inverse correlation length, illustrating the obtained results are constructed.
A kinetic one-dimensional Ising model on a ring evolves according to a generalization of Glauber rates, such that spins at even (odd) lattice sites experience a temperature T(e) (T(o)). Detailed balance is violated so that the spin chain settles into a nonequilibrium stationary state, characterized by multiple interactions of increasing range and spin order. We derive the equations of motion for arbitrary correlation functions and solve them to obtain an exact representation of the steady state. Two nontrivial amplitudes reflect the sublattice symmetries; otherwise, correlations decay exponentially, modulo the periodicity of the ring. In the long-chain limit, they factorize into products of two-point functions, in precise analogy to the equilibrium Ising chain. The exact solution confirms the expectation, based on simulations and renormalization group arguments, that the long-time, long-distance behavior of this two-temperature model is Ising-like, in spite of the apparent complexity of the stationary distribution.
The kinetic Glauber–Ising spin chain is one of the very few exactly solvable models of non-equilibrium statistical mechanics. Nevertheless, existing solutions do not yield tractable expressions for two-time correlation and response functions of observables involving products of more than one or two spins. We use a new approach to solve explicitly the full hierarchy of differential equations for the correlation and response functions. From this general solution follow closed expressions for arbitrary multispin two-time correlation and response functions, for the case where the system is quenched from equilibrium at T i > 0 to some arbitrary T ⩾ 0. By way of application, we give the results for two- and four-spin two-time correlation and response functions. From the standard mapping, these also imply new exact results for two-time particle correlation and response functions in one-dimensional diffusion limited annihilation.