Ising dual 4x4.py
From Werner KRAUTH
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| We consider the 4x4 Ising model with periodic boundary conditions. A duality relation was uncovered by Kramers and Wannier, but it does not really apply to finite lattices: | We consider the 4x4 Ising model with periodic boundary conditions. A duality relation was uncovered by Kramers and Wannier, but it does not really apply to finite lattices: | ||
| - | [[Image:Square Lattice duality without pbc.jpg Click here for a High-definition version]]] | + | |
| + | [[Image:Square Lattice duality without pbc.jpg]] | ||
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| [[Image:Ising dual finite difference.jpg Click here for a High-definition version]]] | [[Image:Ising dual finite difference.jpg Click here for a High-definition version]]] | ||
Revision as of 22:14, 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 |
Reference
This program is discussed in Lecture 06 of my ICFP lectures 2016 on Statistical physics. The Lecture is called "Two-dimensional Ising model: From Kramers & Wannier to Kac & Ward".
Description
We consider the 4x4 Ising model with periodic boundary conditions. A duality relation was uncovered by Kramers and Wannier, but it does not really apply to finite lattices:
Image:Ising dual finite difference.jpg Click here for a High-definition version]
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.
Categories: Python | ICFP
