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Physical review. B, Condensed matter

Monte Carlo study of Ising spin dynamics on a fractal lattice

Erling Pytte

Published Aug 1, 1986• 13 citations
Publisher

Abstract

The dynamics of Ising spins on a two-dimensional fractal lattice has been studied by Monte Carlo simulations. Both the limits $L>>\ensuremath{\xi}$ and $L<<\ensuremath{\xi}$ were studied where $L$ is the size of the sample and $\ensuremath{\xi}$ the correlation length. Exact transfer matrix calculations were also carried out to check that the correct equilibrium values were reached. In both limits the predicted breakdown of conventional dynamic scaling was obtained. Using finite-size scaling $\ensuremath{\tau}\ensuremath{\sim}{L}^{z}$ for $L<<\ensuremath{\xi}$, an effective temperature-dependent critical exponent $z$ was obtained consistent with the recent prediction $z\ensuremath{\propto}\frac{1}{T}$ for small $T$.

DOI
10.1103/physrevb.34.2060

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The dynamics of Ising spins on a two-dimensional fractal lattice has been studied by Monte Carlo simulations. Both the limits L>>\ensuremathξL>>\ensuremath{\xi}L>>\ensuremathξ and L<<\ensuremathξL<<\ensuremath{\xi}L<<\ensuremathξ were studied where LLL is the size of the sample and \ensuremathξ\ensuremath{\xi}\ensuremathξ the correlation length. Exact transfer matrix calculations were also carried out to check that the correct equilibrium values were reached. In both limits the predicted breakdown of conventional dynamic scaling was obtained. Using finite-size scaling \ensuremathτ\ensuremath∼Lz\ensuremath{\tau}\ensuremath{\sim}{L}^{z}\ensuremathτ\ensuremath∼Lz for L<<\ensuremathξL<<\ensuremath{\xi}L<<\ensuremathξ, an effective temperature-dependent critical exponent zzz was obtained consistent with the recent prediction z\ensuremath∝1Tz\ensuremath{\propto}\frac{1}{T}z\ensuremath∝T1​ for small TTT.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
An extensive Monte Carlo study of the two-dimensional Ising model is made to investigate the statistical behavior of spin clusters and interfaces as a function of temperature, T . We use a tie-breaking rule to define interfaces of spin clusters on a square lattice with strip geometry and show that such definitions are consistent with conformally invariant properties of interfaces at the critical temperature, T c . The effective fractal dimensions of spin clusters and interfaces ( d c and d I , respectively) are obtained as a function of temperature. We find that the effective fractal dimension of the spin clusters varies almost linearly with temperature in three different regimes. It is also found that the effective fractal dimension of the interfaces undergoes a sharp crossover around T c , between the values 1 and 1.75 at low and high temperatures, respectively. We also check the finite-size scaling hypothesis for the percolation probability, and the average mass of the largest spin cluster is in a good agreement with the theoretical predictions.
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.
The short-time critical relaxation of an Ising model on a Sierpinski carpet is investigated using Monte Carlo simulation. We find that when the system is quenched from high temperature to the critical temperature, the evolution of the order parameter and its persistence probability, the susceptibility, and the autocorrelation function all show power-law scaling behavior at the short-time regime. The results suggest that the spatial heterogeneity and the fractal nature of the underlying structure do not influence the scaling behavior of the short-time critical dynamics. The critical temperature, dynamic exponent z, and other equilibrium critical exponents beta and nu of the fractal spin system are determined accurately using conventional Monte Carlo simulation algorithms. The mechanism for short-time dynamic scaling is discussed.
Finite-size scaling calculations are performed for the critical dynamics of the ferromagnetic Ising model on fractal lattices. We confirm the predictions of Henley's heuristic theory of singular dynamic scaling and compare our results with those recently obtained by Kutasov et al. using Monte Carlo methods. It is found that fractal lattices which have odd coordination have an exponentially large number of metastable states but that these states do not necessarily play a significant role in the critical dynamics.