Combinatorial ising.py
From Werner KRAUTH
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| - | This page presents the program combinatorial_ising.py, naively implemented for the 2x2 lattice and the 4x4 lattice | + | This page presents the program combinatorial_ising.py, in other words the Kac-Ward matrix for the two-dimensional Ising model, naively implemented for the 2x2 lattice and the 4x4 lattice. |
| __FORCETOC__ | __FORCETOC__ | ||
| =Reference= | =Reference= | ||
| - | [[Krauth_2006|W. Krauth ''Statistical Mechanics: Algorithms and Computations'' Oxford University Press (2006)]] | + | [[SMAC|W. Krauth ''Statistical Mechanics: Algorithms and Computations'' Oxford University Press (2006)]] |
| + | This matrix is discussed in [[ICFP_Stat_Physics_2018#Week_7_.2822_October_2018.29:_Two-dimensional_Ising_model:_From_Kramers_.26_Wannier_to_Kac_.26_Ward_.28Low-_and_high-temperature_expansions.29|Lecture 07]] | ||
| + | of my 2019 ICFP course on Statistical physics. | ||
| =Description= | =Description= | ||
| - | This is a Python implementation of the Kac-Ward matrix that allows to compute the partition function of the two-dimensional Ising model without periodic boundary conditions. | + | This is a Python implementation of the Kac-Ward matrix that computes the partition function of the two-dimensional Ising model without periodic boundary conditions. |
| =Program= | =Program= | ||
| + | import numpy, math | ||
| + | |||
| + | beta = 1.0 | ||
| + | print beta, ' beta' | ||
| + | comp_i = complex(0.0, 1.0) | ||
| + | nu = math.tanh(beta) | ||
| + | alpha = numpy.exp(comp_i * math.pi / 4.0) * math.tanh(beta) | ||
| alphabar = numpy.exp(-comp_i * math.pi / 4.0) * math.tanh(beta) | alphabar = numpy.exp(-comp_i * math.pi / 4.0) * math.tanh(beta) | ||
| - | + | ||
| # 4x4 matrices 'right', 'up', 'left', 'down', 'one', 'zero' | # 4x4 matrices 'right', 'up', 'left', 'down', 'one', 'zero' | ||
| - | + | ||
| r = numpy.mat([ | r = numpy.mat([ | ||
| [nu, alpha, 0.0, alphabar], | [nu, alpha, 0.0, alphabar], | ||
| Line 17: | Line 26: | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| [0.0, 0.0, 0.0, 0.0]], dtype=complex) | [0.0, 0.0, 0.0, 0.0]], dtype=complex) | ||
| - | + | ||
| u = numpy.mat([ | u = numpy.mat([ | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| Line 23: | Line 32: | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| [0.0, 0.0, 0.0, 0.0]], dtype=complex) | [0.0, 0.0, 0.0, 0.0]], dtype=complex) | ||
| - | + | ||
| l = numpy.mat([ | l = numpy.mat([ | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| Line 29: | Line 38: | ||
| [0.0, alphabar, nu, alpha], | [0.0, alphabar, nu, alpha], | ||
| [0.0, 0.0, 0.0, 0.0]], dtype=complex) | [0.0, 0.0, 0.0, 0.0]], dtype=complex) | ||
| - | + | ||
| d = numpy.mat([ | d = numpy.mat([ | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| Line 35: | Line 44: | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| [alpha, 0.0, alphabar, nu]], dtype=complex) | [alpha, 0.0, alphabar, nu]], dtype=complex) | ||
| - | + | ||
| o = numpy.mat([ | o = numpy.mat([ | ||
| [1.0, 0.0, 0.0, 0.0], | [1.0, 0.0, 0.0, 0.0], | ||
| Line 41: | Line 50: | ||
| [0.0, 0.0, 1.0, 0.0], | [0.0, 0.0, 1.0, 0.0], | ||
| [0.0, 0.0, 0.0, 1.0]], dtype=complex) | [0.0, 0.0, 0.0, 1.0]], dtype=complex) | ||
| - | + | ||
| z = numpy.mat([ | z = numpy.mat([ | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| Line 47: | Line 56: | ||
| [0.0, 0.0, 0.0, 0.0], | [0.0, 0.0, 0.0, 0.0], | ||
| [0.0, 0.0, 0.0, 0.0]], dtype=complex) | [0.0, 0.0, 0.0, 0.0]], dtype=complex) | ||
| - | + | ||
| U2x2 = numpy.bmat([ | U2x2 = numpy.bmat([ | ||
| [o, r, u, z], | [o, r, u, z], | ||
| Line 53: | Line 62: | ||
| [d, z, o, r], | [d, z, o, r], | ||
| [z, d, l, o]]) | [z, d, l, o]]) | ||
| - | + | ||
| L = 2 | L = 2 | ||
| n_edge = 2 * L * (L - 1) | n_edge = 2 * L * (L - 1) | ||
| N = L ** 2 | N = L ** 2 | ||
| - | + | ||
| print 2 ** N * math.cosh(beta) ** n_edge * \ | print 2 ** N * math.cosh(beta) ** n_edge * \ | ||
| numpy.sqrt(numpy.real(numpy.linalg.det(U2x2))), ' Z_2x2 from Kac-Ward' | numpy.sqrt(numpy.real(numpy.linalg.det(U2x2))), ' Z_2x2 from Kac-Ward' | ||
| print 2 ** 4 * math.cosh(beta) ** 4 * (1.0 + math.tanh(beta) ** 4), \ | print 2 ** 4 * math.cosh(beta) ** 4 * (1.0 + math.tanh(beta) ** 4), \ | ||
| ' Z_2x2 from SMAC eq. (5.11)' | ' Z_2x2 from SMAC eq. (5.11)' | ||
| - | + | ||
| U4x4 = numpy.bmat([ | U4x4 = numpy.bmat([ | ||
| #0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | #0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ||
| Line 81: | Line 90: | ||
| [z, z, z, z, z, z, z, z, z, z, d, z, z, l, o, r], # line 14 | [z, z, z, z, z, z, z, z, z, z, d, z, z, l, o, r], # line 14 | ||
| [z, z, z, z, z, z, z, z, z, z, z, d, z, z, l, o]]) # line 15 | [z, z, z, z, z, z, z, z, z, z, z, d, z, z, l, o]]) # line 15 | ||
| - | + | ||
| L = 4 | L = 4 | ||
| n_edge = 2 * L * (L - 1) | n_edge = 2 * L * (L - 1) | ||
| Line 100: | Line 109: | ||
| 1 * math.tanh(beta) ** 20), ' Z_4x4 from SMAC eq. (5.10)' | 1 * math.tanh(beta) ** 20), ' Z_4x4 from SMAC eq. (5.10)' | ||
| - | |||
| =Version= | =Version= | ||
| See history for version information. | See history for version information. | ||
| - | [[Category:Python]] [[Category:Honnef_2015]] | + | [[Category:Python]] [[Category:ICFP]] |
Current revision
This page presents the program combinatorial_ising.py, in other words the Kac-Ward matrix for the two-dimensional Ising model, naively implemented for the 2x2 lattice and the 4x4 lattice.
Contents |
[edit]
Reference
W. Krauth Statistical Mechanics: Algorithms and Computations Oxford University Press (2006) This matrix is discussed in Lecture 07 of my 2019 ICFP course on Statistical physics.
[edit]
Description
This is a Python implementation of the Kac-Ward matrix that computes the partition function of the two-dimensional Ising model without periodic boundary conditions.
[edit]
Program
import numpy, math beta = 1.0 print beta, ' beta' comp_i = complex(0.0, 1.0) nu = math.tanh(beta) alpha = numpy.exp(comp_i * math.pi / 4.0) * math.tanh(beta) alphabar = numpy.exp(-comp_i * math.pi / 4.0) * math.tanh(beta) # 4x4 matrices 'right', 'up', 'left', 'down', 'one', 'zero' r = numpy.mat([ [nu, alpha, 0.0, alphabar], [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]], dtype=complex) u = numpy.mat([ [0.0, 0.0, 0.0, 0.0], [alphabar, nu, alpha, 0.0], [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]], dtype=complex) l = numpy.mat([ [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0], [0.0, alphabar, nu, alpha], [0.0, 0.0, 0.0, 0.0]], dtype=complex) d = numpy.mat([ [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0], [alpha, 0.0, alphabar, nu]], dtype=complex) o = numpy.mat([ [1.0, 0.0, 0.0, 0.0], [0.0, 1.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0], [0.0, 0.0, 0.0, 1.0]], dtype=complex) z = numpy.mat([ [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]], dtype=complex) U2x2 = numpy.bmat([ [o, r, u, z], [l, o, z, u], [d, z, o, r], [z, d, l, o]]) L = 2 n_edge = 2 * L * (L - 1) N = L ** 2 print 2 ** N * math.cosh(beta) ** n_edge * \ numpy.sqrt(numpy.real(numpy.linalg.det(U2x2))), ' Z_2x2 from Kac-Ward' print 2 ** 4 * math.cosh(beta) ** 4 * (1.0 + math.tanh(beta) ** 4), \ ' Z_2x2 from SMAC eq. (5.11)' U4x4 = numpy.bmat([ #0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [o, r, z, z, u, z, z, z, z, z, z, z, z, z, z, z], # line 0 [l, o, r, z, z, u, z, z, z, z, z, z, z, z, z, z], # line 1 [z, l, o, r, z, z, u, z, z, z, z, z, z, z, z, z], # line 2 [z, z, l, o, z, z, z, u, z, z, z, z, z, z, z, z], # line 3 [d, z, z, z, o, r, z, z, u, z, z, z, z, z, z, z], # line 4 [z, d, z, z, l, o, r, z, z, u, z, z, z, z, z, z], # line 5 [z, z, d, z, z, l, o, r, z, z, u, z, z, z, z, z], # line 6 [z, z, z, d, z, z, l, o, z, z, z, u, z, z, z, z], # line 7 [z, z, z, z, d, z, z, z, o, r, z, z, u, z, z, z], # line 8 [z, z, z, z, z, d, z, z, l, o, r, z, z, u, z, z], # line 9 [z, z, z, z, z, z, d, z, z, l, o, r, z, z, u, z], # line 10 [z, z, z, z, z, z, z, d, z, z, l, o, z, z, z, u], # line 11 [z, z, z, z, z, z, z, z, d, z, z, z, o, r, z, z], # line 12 [z, z, z, z, z, z, z, z, z, d, z, z, l, o, r, z], # line 13 [z, z, z, z, z, z, z, z, z, z, d, z, z, l, o, r], # line 14 [z, z, z, z, z, z, z, z, z, z, z, d, z, z, l, o]]) # line 15 L = 4 n_edge = 2 * L * (L - 1) N = L ** 2 print 2 ** N * math.cosh(beta) ** n_edge * \ math.sqrt(numpy.real(numpy.linalg.det(U4x4))), \ ' Z_4x4 from Kac-Ward' print 2 ** N * math.cosh(beta) ** n_edge * ( 1 + 9 * math.tanh(beta) ** 4 + 12 * math.tanh(beta) ** 6 + 50 * math.tanh(beta) ** 8 + 92 * math.tanh(beta) ** 10 + 158 * math.tanh(beta) ** 12 + 116 * math.tanh(beta) ** 14 + 69 * math.tanh(beta) ** 16 + 4 * math.tanh(beta) ** 18 + 1 * math.tanh(beta) ** 20), ' Z_4x4 from SMAC eq. (5.10)'
[edit]
Version
See history for version information.
Categories: Python | ICFP
