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arXiv:cond-mat/9805385 · Journal of Physics A Mathematical and General

Slow coarsening in an Ising chain with competing interactions

S. Redner, P. L. Krapivsky

Published Nov 20, 1998• 20 citations
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Abstract

We investigate the zero-temperature coarsening dynamics of a chain of Ising spins with a nearest-neighbour ferromagnetic and an n th-neighbour antiferromagnetic interactions. For sufficiently large antiferromagnetic interaction, the ground state consists of n consecutive up spins followed by n down spins, etc. We show that the asymptotic coarsening into this ground state can be described in terms of a multispecies reactive gas of elementary excitations. The basic elementary excitations are identified and each decays at a different power-law rate in time. The dominant excitations are domains of n + 1 spins which diffuse freely and disappear through processes which are effectively governed by ( n + 1)-particle annihilation. This leads to a slow temporal approach to the ground state.

DOI
10.1088/0305-4470/31/46/013

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Journal of the Physical Society of Japan•Dec 15, 1989•16 citations
We investigate the zero-temperature coarsening dynamics of a chain of Ising spins with a nearest-neighbour ferromagnetic and an n th-neighbour antiferromagnetic interactions. For sufficiently large antiferromagnetic interaction, the ground state consists of n consecutive up spins followed by n down spins, etc. We show that the asymptotic coarsening into this ground state can be described in terms of a multispecies reactive gas of elementary excitations. The basic elementary excitations are identified and each decays at a different power-law rate in time. The dominant excitations are domains of n + 1 spins which diffuse freely and disappear through processes which are effectively governed by ( n + 1)-particle annihilation. This leads to a slow temporal approach to the ground state.
We investigate the zero-temperature coarsening dynamics of a chain of Ising spins with a nearest-neighbour ferromagnetic and an nth-neighbour antiferromagnetic interactions. For sufficiently large antiferromagnetic interaction, the ground state consists of n consecutive up spins followed by n down spins, etc. We show that the asymptotic coarsening into this ground state can be described in terms of a multispecies reactive gas of elementary excitations. The basic elementary excitations are identified and each decays at a different power-law rate in time. The dominant excitations are domains of n + 1 spins which diffuse freely and disappear through processes which are effectively governed by (n + 1)-particle annihilation. This leads to a slow temporal approach to the ground state.
If quenched to zero temperature, the one-dimensional Ising spin chain undergoes coarsening, whereby the density of domain walls decays algebraically in time. We show that this coarsening process can be interrupted by exerting a rapidly oscillating periodic field with enough strength to compete with the spin-spin interaction. By analyzing correlation functions and the distribution of domain lengths both analytically and numerically, we observe nontrivial correlation with more than one length scale at the threshold field strength.
We present a lattice spin model that mimics a system of interacting particles through a short-range repulsive potential and a long-range attractive power-law decaying potential. We perform a detailed analysis of the general equilibrium phase diagram of the model at finite temperature, showing that the only possible equilibrium phases are the ferromagnetic and the antiferromagnetic ones. We then study the nonequilibrium behavior of the model after a quench to subcritical temperatures, in the antiferromagnetic region of the phase diagram region, where the pair interaction potential behaves in the same qualitative way as in a Lennard-Jones gas. We find that even in the absence of quenched disorder or geometric frustration, the competition between interactions gives rise to nonequilibrium disordered structures at low enough temperatures that strongly slow down the relaxation of the system. This nonequilibrium state presents several features characteristic of glassy systems such as subaging, nontrivial fuctuation dissipation relations, and possible logarithmic growth of free-energy barriers to coarsening.
Abstract While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.
We consider a linear quench from the paramagnetic to ferromagnetic phase in the quantum Ising chain interacting with a static spin environment. Both decoherence from the environment and nonadiabaticity of the evolution near a critical point excite the system from the final ferromagnetic ground state. For weak decoherence and relatively fast quenches the excitation energy, proportional to the number of kinks in the final state, decays like an inverse square root of a quench time, but slow transitions or strong decoherence makes it decay in a much slower logarithmic way. We also find that fidelity between the final ferromagnetic ground state and a final state after a quench decays exponentially with a size of a chain, with a decay rate proportional to average density of excited kinks and a proportionality factor evolving from 1.3 for weak decoherence and fast quenches to approximately 1 for slow transitions or strong decoherence. Simultaneously, correlations between kinks randomly distributed along the chain evolve from a near-crystalline antibunching to a Poissonian distribution of kinks in a number of isolated Anderson localization centers randomly scattered along the chain.
Excitation spectrum as well as ground states is studied in detail in the region near the Ising limit with small XY-component in a Heisenberg ferromagnetic chain with competing antiferromagnetic second-neighbor interaction. A spin-fluid-like phase with gapless excitation spectrum, which exhibits ferromagnetic long-range order, is derived in the intermediate region between the saturated ferromagnetic and (2, 2) antiphase states, by mapping onto a system of Bose-like particles with hard core of finite radius. Expressions for the magnetization, the susceptibility and the dispersion relalion of the excitation spectrum are obtained in the intermediate phase. This mapping is expected to be exact in the region near the degenerate point.