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Physical review. B, Condensed matter

Spreading of damage in the quantum S= 12 Heisenberg ferromagnet

I. V. Rojdestvenski, M. L. Lyra, U.M.S. Costa

Published Aug 1, 1997• 6 citations
Publisher

Abstract

We suggest a tool to investigate the processes of the ferromagnetic ordering in the quantum Heisenberg model. We introduce the spreading of a damage technique based on the Handscomb Monte Carlo method and apply it to the three-dimensional simple cubic Heisenberg ferromagnet. Particularly, we study the thermal dependence of the graph Hamming distance and employ a finite-size scaling analysis of its size dependence. It is obtained that a properly defined Hamming distance displays a maximum at a characteristic temperature ${T}_{s},$ below which it becomes roughly size independent. We discuss the properties of these dependences in relevance to other thermodynamic functions of the model.

DOI
10.1103/physrevb.56.2698

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Demage Spreading in the Ising Model with a special Metropolis Dynamics Approach

LIUCe-Jun, HUJia-Zhen

The time evolution of the Hamming distance (damage spreading)for the S=1/2 and S=1 Ising models on the square lattice is performed with a special metropolis dynamics algorithm.Two distinct regimes are observed according to the temperature range for both models:a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size;a high- temperature one where the distance vanishes in the long-time limit.Using the finite size scaling method,the dynamical phase transition(damage spreading transition) temperature is obtained as Tc ≈1.675±0.025 for the S=1 lsing model.

理论物理通讯：英文版•Jan 1, 2001•0 citations
We suggest a tool to investigate the processes of the ferromagnetic ordering in the quantum Heisenberg model. We introduce the spreading of a damage technique based on the Handscomb Monte Carlo method and apply it to the three-dimensional simple cubic Heisenberg ferromagnet. Particularly, we study the thermal dependence of the graph Hamming distance and employ a finite-size scaling analysis of its size dependence. It is obtained that a properly defined Hamming distance displays a maximum at a characteristic temperature Ts,{T}_{s},Ts​, below which it becomes roughly size independent. We discuss the properties of these dependences in relevance to other thermodynamic functions of the model.
In this work we extend a recently proposed prescription for studying the dynamical properties of quantum Heisenberg ferromagnets which is based on the spreading of the damage technique. Within the Handscomb Monte Carlo method, we introduce a Hamming distance defined as a difference in the cyclic structure of Mayers diagrams in such a way that the quantum nature of the Hamiltonian is accounted for. We show that the cyclic Hamming distance is closely related to the static spin susceptibility and illustrate our results by comparing them with previous two-loop renormalization group calculations of the susceptibility low-temperature behavior in two-dimensional quantum Heisenberg models.
We apply the spreading of the damage technique to a S=1/2 Ising ferromagnetic thin film. We use Metropolis dynamics to calculate the temperature dependence of the Hamming distance as well as the Hamming distance distributions along the Markov chain. We suggest a modification of the algorithm for calculating the Hamming distance in the case of Metropolis dynamics and we discuss the results for the two-dimensional model obtained by the modified approach. For a film with three layers we present the thermal dependences of the above quantities. We also suggest a new way to introduce damage to the system as the difference in the boundary conditions. We discuss the results of this new approach for the two-dimensional model.
We investigate the critical relaxational dynamics of the S=1/2S=1/2S=1/2 Heisenberg ferromagnet on a simple cubic lattice within the Handscomb prescription, on which a diagrammatic series expansion of the partition function is computed by means of a Monte Carlo procedure. Using a phenomenological renormalization-group analysis of graph quantities related to the spin susceptibility and order parameter, we obtain precise estimates of the critical exponents relations $\ensuremath{\gamma}/\ensuremath{\nu}=1.98\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.512\ifmmode\pm\else\textpm\fi{}0.002$ and of the Curie temperature ${k}_{B}{T}_{c}/J=1.6778\ifmmode\pm\else\textpm\fi{}0.0002.$ The critical correlation time ${\ensuremath{\tau}}_{\mathrm{int}}$ of both energy and susceptibility is also computed. We found the number of Monte Carlo steps needed to generate uncorrelated diagram configurations scales with the system's volume. We estimate the efficiency of the Handscomb method, comparing its ability in dealing with the critical slowing down to that of other quantum and classical Monte Carlo prescriptions.
The time evolution of the Hamming distance (damage spreading) for the and Ising models on the square lattice is performed with a special metropolis dynamics algorithm. Two distinct regimes are observed according to the temperature range for both models: a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size; a high-temperature one where the distance vanishes in the long-time limit. Using the finite size scaling method, the dynamical phase transition (damage spreading transition) temperature is obtained as for the Ising model.
The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.
The time evolution of the Hamming distance (damage spreading)for the S=1/2 and S=1 Ising models on the square lattice is performed with a special metropolis dynamics algorithm.Two distinct regimes are observed according to the temperature range for both models:a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size;a high- temperature one where the distance vanishes in the long-time limit.Using the finite size scaling method,the dynamical phase transition(damage spreading transition) temperature is obtained as Tc ≈1.675±0.025 for the S=1 lsing model.