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Journal of Applied Physics

Application of the spreading of damage technique to the S=1/2 Ising thin film

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

Published May 15, 1994• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.355576

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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 numerically investigate the critical behavior of layered S=1/2 Ising ferromagnetic thin films with an enhanced surface coupling constant by using the spreading of damage technique through the Metropolis dynamics. We obtain that near the bulk critical spreading temperature, the Hamming distance profile changes its shape from a positive curvature profile for Js/J<1.4 to a negative curvature profile for Js/J>1.4, which is a signature of the reminiscent special surface-bulk transition which occurs in the semi-infinite system. Furthermore, we show that the temporal evolution of the damage front exhibits a typical exponential growth above critically while in the critical region the behavior is diffusive.
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 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.
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.
We investigate the damage or Hamming distance between two configurations of Ising spins. We find an exact relation between the difference of the two possible types of damage and the spin-spin correlation function, which is generally valid. For the specific case of ferromagnetic interactions, heat bath dynamics and same sequence of random numbers, this relation involves only one type of damage. The susceptibility and the magnetization can also be expressed in terms of the damage. Numerical determination of the damage for the 2 d Ising model is not only an efficient way to calculate correlation functions but also gives access to spin fluctuations visualized as clusters of damaged sites which have a fractal dimension d −β/ν at T c and whose size distribution is also related to static exponents.
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.