Ising dual 4x4.py
From Werner KRAUTH
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| - | This page presents the program Ising_dual_4x4.py, that checks the Kramers-Wannier duality for a finite 4x4 Ising model with periodic boundary conditions. | + | This page presents the program Ising_dual_4x4.py, that checks Kramers-Wannier duality for a finite 4x4 Ising model with periodic boundary conditions. |
| __FORCETOC__ | __FORCETOC__ | ||
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| This program is discussed in [[ICFP_Stat_Physics_2015|Lecture 4]] of my ICFP course 2015 on Statistical physics. The Lectures is called "Exact computations in the two-dimensional Ising model (Kac-Ward)". | This program is discussed in [[ICFP_Stat_Physics_2015|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= | =Description= | ||
| - | The program inputs the density of states of the 4x4 lattice with periodic boundary conditions (that can be computed by [[enumerate_ising_dos.py|the enumerate-Ising program]]. For any beta it computes a beta_tilde, and then compares Z and Z_tilde. At low temperature, where only short loops come up, everything is OK, but at higher temperature, the winding paths create a problem. This is discussed in the lecture notes. | + | 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= | =Program= | ||
| - | import math | ||
| - | |||
| 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 + 0.1 * i for i in range(80)] | + | list_T = [0.1 + 0.2 * i for i in range(20)] |
| + | print 'T, Z factor / Z_tilde' | ||
| for T in list_T: | for T in list_T: | ||
| beta = 1.0 / T | beta = 1.0 / T | ||
| - | beta_tilde = math.atanh(math.exp(-2 * beta)) | + | beta_tilde = math.atanh(math.exp(-2.0 * beta)) |
| - | Z = 0.0 | + | T_tilde = 1.0 / beta_tilde |
| - | Z_tilde = 0.0 | + | |
| factor = 2 ** (N - 1) * (math.cosh(beta_tilde)) ** (2 * N) * \ | factor = 2 ** (N - 1) * (math.cosh(beta_tilde)) ** (2 * N) * \ | ||
| (math.tanh(beta_tilde)) ** N | (math.tanh(beta_tilde)) ** N | ||
| - | for E in dos.keys(): | + | Z = sum(math.exp(- beta * E) * dos[E] for E in dos.keys()) |
| - | weight = math.exp(- beta * E) * dos[E] | + | Z_tilde = sum(math.exp(- beta_tilde * E) * dos[E] for E in dos.keys()) |
| - | weight_tilde = math.exp(- beta_tilde * E) * dos[E] | + | print T, (Z * factor) / Z_tilde |
| - | Z += weight | + | |
| - | Z_tilde += weight_tilde | + | |
| - | print T, (Z * factor) / Z_tilde, ' Temperature, Z * factor / Z_tilde' | + | |
| + | |||
| =Version= | =Version= | ||
Revision as of 21:59, 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 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.
Categories: Python | ICFP
