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Monte Carlo study of the phase boundary of the Ising antiferromagnet
Volume 65A, number 2
PHYSICS LETTERS
20 February 1978
MONTE CARLO STUDY OF THE PHASE BOUNDARY OF THE ISING ANTIFERROMAGNET D.C. RAPAPORT Physics Department, Bar-ilan University, Ramat-Gan, Israel Received 7 November 1977
A conjectured expression for the field dependence of the square Ising antiferromagnet critical temperature is shown to agree with the results of Monte Carlo analysis.
Recently, a technique for computing the interface free energy of the square lattice Ising antiferromagnet was described [1]. The method involved the study of only a restricted class of interface and, on the assumption that the vanishing of this interface free energy corresponds to the critical point Tc(If), the following expression for the boundary separating antiferromagnetic and paramagnetic phases was derived (1) cosh[H/T~(H)] = sinh[2Ji/Tc(J-f)] siIth[2J2ITc(I1)1. Here His the applied field, and J1,2 the anisotropic nearest neighbor interactions. Eq. (1) reduces to known results in the H 0 and Tc(R) 0 limits, and for H small and J1 = J2 it is consistent with the predictions of high temperature series analysis [2]. In this letter we present the results of a test of the isotropic form of eq. (1), namely 2 [2JITc(II)1 (2) C05h[HITc(H)1 = sinh using the Monte Carlo (MC) method. Since MC techniques only involve systems of finite size, the critical point divergences of the specific heat C and staggered susceptibility x~’ which occur at Tc(R) in the infinite system, are replaced by finite height peaks whose maxima are shifted away from Tc(If). At H = 0 it is known [3] that the location Tmax of the specific heat —*
-~
,
maximum of an N X N Ising system is given by Tmax = Tc(O) + 0.82/N + O(N2),
sumably holds also for H 0. We have applied the MC technique [4] to a lattice of size 60 X 61 sites with toroidal boundary conditions. Runs of 2400 MC steps per spin (the first 800 steps were not used in the analysis in order to allow convergence) were performed for various values ofH and T, and a search made for those values of Tat which the maxima of C and x~occurred. Once the approximate location of a maximum was found, further runs were made to determine the shape of the peak and thereby provide an estimate of the uncertainty in the value of Tmax. Each data point used in the final estimation of Tmax represents the average of as many
H/J 43
-
2-
I
I
(3)
thus the MC study of a lattice of moderate size (e.g. N = 60) should be capable of yielding Tc estimates accurate to within 1%. A result similar to eq. (3) pre-
~
T~(H ) /J Fig. 1. The full curve represents the solution of eq. (2). The horizontal bars indicate the location of Tmax; for each H the bar on the left corresponds to C, the bar on the right to x
5• 147
Volume 65A, number 2
PHYSICS LETTERS
20 February 1978
as eight separate MC runs using different starting configurations and random number sequences. The results are summarized in fig. 1. The full curve represents the solution of eq. (2), and the horizontal line segments are the estimates of T~(JI)based on the locations of the maxima of C and x5~In the case of
To summarize, on the basis of the MC analysis it would appear that the conjectured form of the phase boundary, eq. (2), is correct. The general anisotropic result in eq. (1) is probably also correct, although it has not as yet been tested.
the specific heat it is evident that eq. (2) fits the data remarkably well. The x5 maxima are seen to occur at consistently higher temperatures than the C maxima; a reasonable explanation is that the finiteness of the lattice affects x more strongly than C, and as N ~ Tmax(Xs) converges to the limiting value T~(J])at a slower rate than Tmax(C). A result analogous to eq. (3) does not exist for x5 however.
References
148
~ij E. MOfler-Hartman and J. Zittartz, Z. Phys. B27 (1977) 261. [2] D.C. Rapaport Domb, J. Phys. (1971) 131 A.E. Ferdinandand andC.M.E. Fisher, Phys.C4 Rev. 185 2684. (1969) 832. [4] C-P. Yang, Proc. Symp. Applied mathematics 15 (1963) 351.
PHYSICS LETTERS
20 February 1978
MONTE CARLO STUDY OF THE PHASE BOUNDARY OF THE ISING ANTIFERROMAGNET D.C. RAPAPORT Physics Department, Bar-ilan University, Ramat-Gan, Israel Received 7 November 1977
A conjectured expression for the field dependence of the square Ising antiferromagnet critical temperature is shown to agree with the results of Monte Carlo analysis.
Recently, a technique for computing the interface free energy of the square lattice Ising antiferromagnet was described [1]. The method involved the study of only a restricted class of interface and, on the assumption that the vanishing of this interface free energy corresponds to the critical point Tc(If), the following expression for the boundary separating antiferromagnetic and paramagnetic phases was derived (1) cosh[H/T~(H)] = sinh[2Ji/Tc(J-f)] siIth[2J2ITc(I1)1. Here His the applied field, and J1,2 the anisotropic nearest neighbor interactions. Eq. (1) reduces to known results in the H 0 and Tc(R) 0 limits, and for H small and J1 = J2 it is consistent with the predictions of high temperature series analysis [2]. In this letter we present the results of a test of the isotropic form of eq. (1), namely 2 [2JITc(II)1 (2) C05h[HITc(H)1 = sinh using the Monte Carlo (MC) method. Since MC techniques only involve systems of finite size, the critical point divergences of the specific heat C and staggered susceptibility x~’ which occur at Tc(R) in the infinite system, are replaced by finite height peaks whose maxima are shifted away from Tc(If). At H = 0 it is known [3] that the location Tmax of the specific heat —*
-~
,
maximum of an N X N Ising system is given by Tmax = Tc(O) + 0.82/N + O(N2),
sumably holds also for H 0. We have applied the MC technique [4] to a lattice of size 60 X 61 sites with toroidal boundary conditions. Runs of 2400 MC steps per spin (the first 800 steps were not used in the analysis in order to allow convergence) were performed for various values ofH and T, and a search made for those values of Tat which the maxima of C and x~occurred. Once the approximate location of a maximum was found, further runs were made to determine the shape of the peak and thereby provide an estimate of the uncertainty in the value of Tmax. Each data point used in the final estimation of Tmax represents the average of as many
H/J 43
-
2-
I
I
(3)
thus the MC study of a lattice of moderate size (e.g. N = 60) should be capable of yielding Tc estimates accurate to within 1%. A result similar to eq. (3) pre-
~
T~(H ) /J Fig. 1. The full curve represents the solution of eq. (2). The horizontal bars indicate the location of Tmax; for each H the bar on the left corresponds to C, the bar on the right to x
5• 147
Volume 65A, number 2
PHYSICS LETTERS
20 February 1978
as eight separate MC runs using different starting configurations and random number sequences. The results are summarized in fig. 1. The full curve represents the solution of eq. (2), and the horizontal line segments are the estimates of T~(JI)based on the locations of the maxima of C and x5~In the case of
To summarize, on the basis of the MC analysis it would appear that the conjectured form of the phase boundary, eq. (2), is correct. The general anisotropic result in eq. (1) is probably also correct, although it has not as yet been tested.
the specific heat it is evident that eq. (2) fits the data remarkably well. The x5 maxima are seen to occur at consistently higher temperatures than the C maxima; a reasonable explanation is that the finiteness of the lattice affects x more strongly than C, and as N ~ Tmax(Xs) converges to the limiting value T~(J])at a slower rate than Tmax(C). A result analogous to eq. (3) does not exist for x5 however.
References
148
~ij E. MOfler-Hartman and J. Zittartz, Z. Phys. B27 (1977) 261. [2] D.C. Rapaport Domb, J. Phys. (1971) 131 A.E. Ferdinandand andC.M.E. Fisher, Phys.C4 Rev. 185 2684. (1969) 832. [4] C-P. Yang, Proc. Symp. Applied mathematics 15 (1963) 351.