From Werner KRAUTH

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Line 16:		Line 16:
	N = 16		N = 16
-	dos = { -32: 2, -24: 32, -20: 64, -16: 424, -12: 1728, -8: 6688,	+	dos = { -32: 2, -24: 32, -20: 64, -16: 424, -12: 1728, -8: 6688, -4: 13568, 0: 20524, 4: 13568, 8: 6688, 12: 1728, 16: 424, 20: 64, 24: 32, 32: 2}
-	-4: 13568, 0: 20524, 4: 13568, 8: 6688, 12: 1728, 16: 424, 20: 64, 24: 32, 32: 2}	+
	list_T = [0.1 + 0.2 * i for i in range(20)]		list_T = [0.1 + 0.2 * i for i in range(20)]
	print 'T, Z factor / Z_tilde'		print 'T, Z factor / Z_tilde'
Line 29:		Line 28:
	Z_tilde = sum(math.exp(- beta_tilde * E) * dos[E] for E in dos.keys())		Z_tilde = sum(math.exp(- beta_tilde * E) * dos[E] for E in dos.keys())
	print T, (Z * factor) / Z_tilde		print T, (Z * factor) / Z_tilde
-
-
	=Version=		=Version=

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Revision as of 22:00, 14 October 2016

This page presents the program Ising_dual_4x4.py, that checks Kramers-Wannier duality for a finite 4x4 Ising model with periodic boundary conditions.

Contents 1 Reference 2 Description 3 Program 4 Version

Reference

This program is discussed in Lecture 4 of my ICFP course 2015 on Statistical physics. The Lectures is called "Exact computations in the two-dimensional Ising model (Kac-Ward)".

Description

We consider the 4x4 Ising model with periodic boundary conditions. Using the density of states (the dictionary "dos" is from SMAC Table 5.2), we can compute the partition functions Z and Z_tilde at inverse temperatures beta and beta_tilde that are related by the Kramers-Wannier duality relation.

At low temperatures, the corresponding partition functions are related to each other by the factor that stems from duality.

At higher temperatures, this is no longer true as there are terms in the high-temperature expansion that do not appear in the low-temperature expansion. Note that taking the transformation from beta -> beta_tilde a second time brings us back to beta in other words: beta_tilde_tilde = beta

Program

N = 16
dos = { -32: 2, -24: 32, -20: 64, -16: 424, -12: 1728, -8: 6688, -4: 13568, 0: 20524, 4: 13568, 8: 6688, 12: 1728, 16: 424, 20: 64, 24: 32, 32: 2}
list_T = [0.1 + 0.2 * i for i in range(20)]
print 'T, Z factor / Z_tilde'
for T in list_T:
    beta = 1.0 / T
    beta_tilde = math.atanh(math.exp(-2.0 * beta))
    T_tilde = 1.0 / beta_tilde
    factor = 2 ** (N - 1) * (math.cosh(beta_tilde)) ** (2 * N) * \
            (math.tanh(beta_tilde)) ** N
    Z = sum(math.exp(- beta * E) * dos[E] for E in dos.keys())
    Z_tilde = sum(math.exp(- beta_tilde * E) * dos[E] for E in dos.keys())
    print T, (Z * factor) / Z_tilde

Version

See history for version information.