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arXiv:cond-mat/0503566 · Journal of Physics A Mathematical and General

Persistence exponents and scaling in two-dimensionalXYmodel and a nematic model

Subhrajit Dutta, Soumen Roy

Published Jun 15, 2005• 4 citations
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Abstract

The persistence exponents associated with the T = 0 quenching dynamics of the two-dimensional XY model and a two-dimensional uniaxial spin nematic model have been evaluated using a numerical simulation. The site persistence or the probability that the sign of a local spin component does not change starting from initial time t = 0 up to a certain time t , is found to decay as L ( t ) −θ ( L ( t ) is the linear domain length scale), with θ = 0.305(±0.020) for the two-dimensional XY model and 0.199(±0.009) for the two-dimensional uniaxial spin nematic model. We have also investigated the scaling (at the late time of phase ordering) associated with the correlated persistent sites in both models. The persistence correlation length was found to grow in the same way as L ( t ).

DOI
10.1088/0305-4470/38/26/002

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The persistence exponents associated with the T = 0 quenching dynamics of the two-dimensional XY model and a two-dimensional uniaxial spin nematic model have been evaluated using a numerical simulation. The site persistence or the probability that the sign of a local spin component does not change starting from initial time t = 0 up to a certain time t , is found to decay as L ( t ) −θ ( L ( t ) is the linear domain length scale), with θ = 0.305(±0.020) for the two-dimensional XY model and 0.199(±0.009) for the two-dimensional uniaxial spin nematic model. We have also investigated the scaling (at the late time of phase ordering) associated with the correlated persistent sites in both models. The persistence correlation length was found to grow in the same way as L ( t ).
Local and global persistence exponents associated with zero-temperature quenched dynamics of two-dimensional XY model and three-dimensional Heisenberg model have been estimated using numerical simulations. The method of block persistence has been used to find the global and local exponents simultaneously (in a single simulation). Temperature universality of both the exponents for three-dimensional Heisenberg model has been confirmed by simulating the stochastic (with noise) version of the equation of motion. The noise amplitudes added were small enough to retain the dynamics below criticality. In the second part of our work we have studied scaling associated with correlated persistence sites in the three-dimensional Heisenberg model in the later stages of the dynamics. The relevant length scale associated with correlated persistent sites was found to behave in a manner similar to the dynamic length scale associated with the phase ordering dynamics.
The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.
The spatial distribution of persistent spins at zero temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, ξ( t )~ t Z , is identified such that for length scales r <<ξ( t ) the persistent spins form a fractal with dimension d f ; for length scales r >>ξ( t ) the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, θ, is found to satisfy the scaling relation θ = Z (2- d f ) with θ = 0.209±0.002 of Jain (Jain S 1999 Phys. Rev. E 59 R2493), Z = 1/2 and d f ~1.58.
We study persistence in one-dimensional ferromagnetic and antiferromagnetic nearest-neighbor Ising models with parallel dynamics. The probability P(t) that a given spin has not flipped up to time t, when the system evolves from an initial random configuration, decays as P(t) approximately 1/t(straight theta(p)) with straight theta(p) approximately 0.75 numerically. A mapping to the dynamics of two decoupled A+A-->0 models yields straight theta(p)=3/4 exactly. A finite size scaling analysis clarifies the nature of dynamical scaling in the distribution of persistent sites obtained under this dynamics.
We use persistent homology and persistence images as an observable of three variants of the two-dimensional XY model to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbor models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behavior and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.
We use persistent homology and persistence images as an observable of three variants of the two-dimensional XY model to identify and study their phase transitions. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a way of computing the persistent homology of lattice spin model configurations and, by considering the fluctuations in the output of logistic regression and k-nearest neighbor models trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behavior and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.