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Physical review. B, Condensed matter

Critical-index calculation from Monte Carlo data with use of finite-size scaling and optimization

Y. Gündüç, Okan Tarhan, R. Sever

Published Oct 1, 1993• 0 citations
Publisher

Abstract

A method of critical index calculation is proposed. The method is based on matching the temperature values of two different size lattices by finding the optimized solution of the equation t'=${\mathit{b}}_{\mathit{T}}^{\mathit{Y}}$t, where t and t' are the reduced temperatures of the lattices, b is the ratio of lattice sizes, and ${\mathit{Y}}_{\mathit{T}}$ is the thermal critical index. The values of t and t' are obtained from the fourth cumulant of the order parameter, which satisfies the relation ${\mathit{Q}}_{\mathit{r}}$(t,L)=${\mathit{Q}}_{\mathit{r}}$(t'L') around the critical point. By using this method, the critical indices of the two-dimensional Ising model and the (2+1)-dimensional lattice gauge theory are calculated. The advantage of the method is that one can obtain the critical point and the critical indices with high accuracy by performing a relatively small number of Monte Carlo iterations.

DOI
10.1103/physrevb.48.9851

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A method of critical index calculation is proposed. The method is based on matching the temperature values of two different size lattices by finding the optimized solution of the equation t'=bTY{\mathit{b}}_{\mathit{T}}^{\mathit{Y}}bTY​t, where t and t' are the reduced temperatures of the lattices, b is the ratio of lattice sizes, and YT{\mathit{Y}}_{\mathit{T}}YT​ is the thermal critical index. The values of t and t' are obtained from the fourth cumulant of the order parameter, which satisfies the relation Qr{\mathit{Q}}_{\mathit{r}}Qr​(t,L)=Qr{\mathit{Q}}_{\mathit{r}}Qr​(t'L') around the critical point. By using this method, the critical indices of the two-dimensional Ising model and the (2+1)-dimensional lattice gauge theory are calculated. The advantage of the method is that one can obtain the critical point and the critical indices with high accuracy by performing a relatively small number of Monte Carlo iterations.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.
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Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.
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.
We present a finite size scaling technique to calculate the central charge and some critical exponents of two-dimensional critical models with a Monte Carlo simulation. We use systems with dimensions $L\ifmmode\times\else\texttimes\fi{}M,$ and focus on the scaling behavior in M/L.M/L.M/L. The finite size scaling relation that we use is the universal expression for the stress tensor on the torus. The stress tensor is the operator that governs the anisotropy of the system, and stems for the theory of conformal invariance. We show that a lattice representation of the stress tensor can easily be constructed, such that its expectation value on the torus can be calculated in a Monte Carlo simulation. In doing so, we observe that the stress tensor turns out to be remarkably insensitive to critical slowing down. We show that the typical simulation time scales with the linear system dimension LLL roughly as L4,{L}^{4},L4, and that this scaling holds for the ordinary Metropolis algorithm as well as for more sophisticated cluster algorithms, such that it is fruitless to invoke the latter. We test the method for the Ising model (with central charge c=c=c=12\frac{1}{2}21​), the Ashkin-Teller model (c=1),(c=1),(c=1), and the F model (also c=1c=1c=1).