Chapter 7: Summary of Results and
Comparison with Published Results

Simulations were performed (as described in Section 2.2) using the pure Ising square model and Glauber dynamics to compare simulation results for internal energy per spin (U/N) and specific heat per spin (C/N) for temperatures from 1.0 to 5.0.

The results for the two sources for U/N are:

Temperature       3.0      3.5      4.0      4.5      5.0
Simulation     -0.817   -0.660   -0.557   -0.484   -0.428
Series exp.    -0.815   -0.660   -0.557   -0.484   -0.428

The results for the two sources for C/N are:

Temperature      3.0       3.5      4.0      4.5      5.0
Simulation     0.405     0.245    0.171    0.125    0.097
Series exp.    0.392     0.247    0.171    0.126    0.098

These results show that the simulation program gives results which are in accord with those from high temperature series expansion.

Section 3.3 describes the determination of the critical temperature of the pure 2d Ising model on three different lattice geometries: square, triangular and honeycomb. The simulation results (see Figures 3.3.1, 3.3.2 and 3.3.3) are as follows:

Lattice       Simulation    Expected    Source
geometry      result        result

Honeycomb     1.52(1)       1.519       High temperature series expansion
                                        (Fisher, 1967, p.671)
Square        2.27(1)       2.269       Onsager's analytical solution
                                        2/ln(1+√2) = 2.269185
Triangular    3.64(2)       3.641       High temperature series expansion
                                        (Fisher, 1967, p.671)

The critical temperature is roughly proportional to the coordination number n of the lattice, since the expression Tc = 0.7n - 0.57 gives 1.53, 2.23 and 3.63 for coordination numbers 3, 4 and 6 respectively, but the linear relation is not exact.

Section 3.4 describes the determination of the critical temperature of the pure 3d Ising model on the cubic lattice and the diamond lattice. The simulation results (see Figures 3.4.5 and 3.4.6) are as follows:

Lattice       Simulation    Expected    Source
geometry      result        result

Diamond       2.700(25)     2.7040      High temperature series expansion
                                        (Fisher, 1967, p.671)
Cubic         4.515(25)     4.5115(1)   Monte Carlo simulation (Heuer, 1993)
                                        High temperature series expansion
                                        Series expansion (see Section 3.4(a))

The diamond lattice has the same coordination number as the square lattice (4), but the critical temperatures are different (2.269 and 2.704). Similarly the cubic lattice has the same coordination number as the triangular lattice (6) but Tc is different (3.641 and 4.512).

Section 3.5 describes the determination of the critical temperature of the pure 4d Ising model on the hypercubic lattice. The simulation result (see Figure 3.5.1) is:

Lattice         Simulation    Expected    Source
geometry        result        result

4d hypercubic   6.68(3)       6.682(2)    High temperature series expansion
                                          (Gaunt, Sykes and McKenzie, 1979)

Section 4.2 describes the determination of the critical temperature of the dilute (p=0.8) cubic Ising model. The simulation result (see Figure 4.2.1) is as follows:

Lattice    Simulation    Expected    Source
geometry   result        result

Cubic      3.50(1)       3.4992(5)   Monte Carlo simulation (Heuer, 1993)

Sections 4.3 and 4.4 present details of the dependence of the critical temperature of a dilute square lattice on the degree of dilution (for both site-diluted and bond-diluted lattices). Published data with which to compare the numerical results obtained by the simulation program was not found. The results do suggest, however, that the Binder cumulant method for determining critical temperatures breaks down as the degree of dilution approaches the site or bond percolation threshold.

Sections 5.2 through 5.4 describe the extraction of β, the critical exponent of the magnetization, for the pure Ising 2d model on various lattice geometries: square, triangular and honeycomb. In each case the exact value for 1/β (obtained by Yang 1952), is 8. The simulation results are:

                                 Square        8.03(21)
                                 Triangular    7.96(21)
                                 Honeycomb     8.31(09)

As was pointed out in Section 5.1, the method used to obtain these results for β is based on an approximation, which may account for the divergence of the result for the honeycomb lattice from the expected value of 8. Nevertheless these results confirm that the 2d Ising models on the square, triangular and honeycomb lattices fall into the same universality class1, and support the hypothesis that the value of β is the same for all 2d planar lattices regardless of the lattice geometry.

Section 5.5 describes the extraction of β for the pure 3-state Potts model on the square lattice. The exact value for 1/β is believed to be 9 (Wu, 1982). The simulation result is 9.32 ± 1.24, confirming that the 3-state Potts model is not in the same universality class as the Ising model.

Sections 6.1 and 6.2 describe a method for estimating θ, the critical exponent characteristic of the "initial critical slip" observed by Janssen, Schaub and Schmittmann (1989). Okano et al. (1997) obtained estimates of θ for the Ising model and for the 3-state Potts model on a square lattice.

The following tables compare results obtained by Okano et al. for simulations on the Ising and 3-state Potts models on the square lattice using Glauber dynamics (see Sections 6.2 and 6.3) with those obtained using the simulation program (numbers of samples are given in square brackets).

For the Ising model the slope of the log-log plot of the magnetization against time was measured using a square lattice of size 128 and initial magnetization of 0.02 at the critical temperature, and the slopes of the log-log plots of the autocorrelation and the 2nd moment of the magnetization were obtained using a square lattice of size 256 and zero initial magnetization at the critical temperature.

Slope of log-log plot     Okano et al.            This study           Figure
for quantity

Magnetization             0.187(1) [300,000]      0.189(1) [15,000]    6.2.1
Autocorrelation          -0.737(1)  [35,000]     -0.739(2)  [2,800]    6.2.2
2nd moment of mag'n       0.817(7)  [35,000]      0.823(3)  [2,800]    6.2.3

For the 3-state Potts model the slope of the log-log plot of the magnetization against time was measured using a square lattice of size 72 and initial magnetization of 0.02 at the critical temperature, and the slopes of the log-log plots of the autocorrelation and the 2nd moment of the magnetization were obtained using a square lattice of size 144 and zero initial magnetization at the critical temperature.

Slope of log-log plot     Okano et al.            This study            Figure
for quantity

Magnetization             0.084(3) [600,000]2      0.074(3) [50,000]     6.3.1
Autocorrelaton           -0.839(1) [-]3           -0.842(2)  [6,000]     6.3.3
2nd moment of mag'n       0.789(2)                 0.789(2)  [6,000]     6.3.4

Section 6.4 applies the method of Section 6.3 to the 4-state Potts model. The simulation data implies θ = -0.046(5). For the dynamic critical exponent a value of z = 2.24(7) was obtained, which is significantly smaller than some older published results.

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