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heisenberg_spin.py
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heisenberg_spin.py
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from __future__ import print_function
import numpy as np
from copy import copy
from numpy import sin, cos
import networkx as nx
from pele.potentials import BasePotential
import pele.utils.rotations as rotations
__all__ = ["HeisenbergModel"]
def make3dVector(u):
"""
make a 3d unit vector from (theta, phi)
"""
sinphi = sin(u[1])
vec = np.zeros(3)
vec[0] = sinphi * cos(u[0])
vec[1] = sinphi * sin(u[0])
vec[2] = cos(u[1])
if np.abs(np.linalg.norm(vec) - 1) > 1e-5:
print("make3dVector: vector not normalized", u, vec, np.linalg.norm(vec))
return vec
def make2dVector(u):
"""
make (theta, phi) from a 3d vector
"""
vec = np.zeros(2)
vec[1] = np.arccos(u[2])
vec[0] = np.arctan2(u[1], u[0])
return vec
def coords2ToCoords3(coords2):
if len(np.shape(coords2)) == 1:
nvec = len(coords2) // 2
coords2 = np.reshape(coords2, [nvec, 2])
else:
nvec = len(coords2[:, 0])
coords3 = np.zeros([nvec, 3])
for i in range(nvec):
coords3[i, :] = make3dVector(coords2[i, :])
return coords3
def coords3ToCoords2(coords3):
if len(np.shape(coords3)) == 1:
nvec = len(coords3) / 3
coords3 = np.reshape(coords3, [nvec, 3])
else:
nvec = len(coords3[:, 0])
coords2 = np.zeros([nvec, 2])
for i in range(nvec):
coords2[i, :] = make2dVector(coords3[i, :])
return coords2
def makeGrad2(vec2, grad3):
grad2 = np.zeros(2)
c0 = cos(vec2[0])
c1 = cos(vec2[1])
s0 = sin(vec2[0])
s1 = sin(vec2[1])
grad2[0] = -s0 * grad3[0] + c0 * grad3[1]
grad2[1] = c0 * c1 * grad3[0] + s0 * c1 * grad3[1] - s1 * grad3[2]
grad2[0] *= s1 # I need this to agree with the numerical gradient, but I think it shouldn't be there
return grad2
def grad3ToGrad2(coords2, grad3):
if len(np.shape(grad3)) == 1:
nvec = len(grad3) / 3
grad3 = np.reshape(grad3, [nvec, 3])
else:
nvec = len(grad3[:, 0])
if len(np.shape(coords2)) == 1:
coords2 = np.reshape(coords2, [nvec, 2])
grad2 = np.zeros([nvec, 2])
for i in range(nvec):
grad2[i, :] = makeGrad2(coords2[i, :], grad3[i, :])
return grad2
class HeisenbergModel(BasePotential):
"""
The classical Heisenberg Model of 3d spins on a lattice.
Parameters
----------
dim: list
an array giving the dimensions of the lattice
field_disorder: float
the magnitude of the randomness in the fields
fields: array
use this to explicitly set the values of the field disorder
Notes
-----
The Hamiltonian is::
H = - sum_ij J dot( s_i, s_j )
where s_i are normalized 3d vectors.
This can be generalized for disordered systems to::
H = - sum_ij J_ij dot( s_i, s_j ) - sum_i dot( h_i, s_i )
where h_i are quenched random variables. (h_i is a vector)
"""
def __init__(self, dim=None, field_disorder=1., fields=None):
if dim is None: dim = [4, 4]
self.dim = copy(dim)
self.nspins = np.prod(dim)
self.G = nx.grid_graph(dim, periodic=True)
self.fields = np.zeros([self.nspins, 3])
self.indices = dict()
nodes = sorted(self.G.nodes())
for i, node in enumerate(nodes):
self.indices[node] = i
if fields is None:
self.fields[i, :] = rotations.vec_random() * field_disorder
else:
self.fields[i, :] = fields[i, :]
def getEnergy(self, coords):
"""
coords is a list of (theta, phi) spherical coordinates of the spins
where phi is the azimuthal angle (angle to the z axis)
"""
coords3 = coords2ToCoords3(coords)
E = 0.
for edge in self.G.edges():
u = self.indices[edge[0]]
v = self.indices[edge[1]]
E -= np.dot(coords3[u, :], coords3[v, :])
Efields = -np.sum(self.fields * coords3)
return E + Efields
def getEnergyGradient(self, coords):
"""
coords is a list of (theta, phi) spherical coordinates of the spins
where phi is the azimuthal angle (angle to the z axis)
"""
coords3 = coords2ToCoords3(coords)
coords2 = coords
E = 0.
grad3 = np.zeros([self.nspins, 3])
for edge in self.G.edges():
u = self.indices[edge[0]]
v = self.indices[edge[1]]
E -= np.dot(coords3[u, :], coords3[v, :])
grad3[u, :] -= coords3[v, :]
grad3[v, :] -= coords3[u, :]
Efields = -np.sum(self.fields * coords3)
grad3 -= self.fields
grad2 = grad3ToGrad2(coords2, grad3)
grad2 = np.reshape(grad2, self.nspins * 2)
return E + Efields, grad2
def normalize_spins(v3):
v = v3.reshape([-1, 3])
norms = np.sqrt((v * v).sum(1))
v = v / norms[:, np.newaxis]
v = v.reshape(-1)
v3[:] = v[:]
def test_basin_hopping(pot, angles): # pragma: no cover
from pele.basinhopping import BasinHopping
from pele.takestep.displace import RandomDisplacement
from pele.takestep.adaptive import AdaptiveStepsize
takestep = RandomDisplacement(stepsize=np.pi / 4)
takestepa = AdaptiveStepsize(takestep, frequency=20)
bh = BasinHopping(angles, pot, takestepa, temperature=1.01)
bh.run(20)
# def test():
# pi = np.pi
# L = 8
# nspins = L**2
#
# #phases = np.zeros(nspins)
# pot = HeisenbergModel( dim = [L,L], field_disorder = 1.) #, phases=phases)
#
# coords = np.zeros([nspins, 2])
# for i in range(nspins):
# vec = rotations.vec_random()
# coords[i,:] = make2dVector(vec)
# coords = np.reshape(coords, [nspins*2])
# if False:
# normfields = np.copy(pot.fields)
# for i in range(nspins): normfields[i,:] /= np.linalg.norm(normfields[i,:])
# coords = coords3ToCoords2( np.reshape(normfields, [nspins*3] ) )
# coords = np.reshape(coords, nspins*2)
# #print np.shape(coords)
# coordsinit = np.copy(coords)
#
# #print "fields", pot.fields
# print coords
#
# if False:
# coords3 = coords2ToCoords3(coords)
# coords2 = coords3ToCoords2(coords3)
# print np.reshape(coords, [nspins,2])
# print coords2
# coords3new = coords2ToCoords3(coords2)
# print coords3
# print coords3new
#
# e = pot.getEnergy(coords)
# print "energy ", e
# if False:
# print "numerical gradient"
# ret = pot.getEnergyGradientNumerical(coords)
# print ret[1]
# if True:
# print "analytical gradient"
# ret2 = pot.getEnergyGradient(coords)
# print ret2[1]
# print ret[0]
# print ret2[0]
# print "ratio"
# print ret2[1] / ret[1]
# print "inverse sin"
# print 1./sin(coords)
# print cos(coords)
#
#
# print "try a quench"
# from pele.optimize import mylbfgs
# ret = mylbfgs(coords, pot, iprint=1)
#
# print "quenched e = ", ret.energy, "funcalls", ret.nfev
# print ret.coords
# with open("out.spins", "w") as fout:
# s = coords2ToCoords3( ret.coords )
# h = pot.fields
# c = coords2ToCoords3( coordsinit )
# for node in pot.G.nodes():
# i = pot.indices[node]
# fout.write( "%g %g %g %g %g %g %g %g %g %g %g\n" % (node[0], node[1], \
# s[i,0], s[i,1], s[i,2], h[i,0], h[i,1], h[i,2], c[i,0], c[i,1], c[i,2] ) )
#
# coords3 = coords2ToCoords3( ret.coords )
# m = np.linalg.norm( coords3.sum(0) ) / nspins
# print "magnetization after quench", m
#
# test_basin_hopping(pot, coords)
#
#def test_potential():
# L=4
# nspins=L*L
# pot = HeisenbergModel( dim = [L,L], field_disorder = 1.) #, phases=phases)
#
# coords = np.zeros([nspins, 2])
# for i in range(nspins):
# vec = rotations.vec_random()
# coords[i,:] = make2dVector(vec)
# coords = np.reshape(coords, [nspins*2])
#
# coords[1] = 2.*np.pi - coords[1]
#
# pot.test_potential(coords)
#
#def test_potential_constrained():
# L=4
# nspins=L*L
# pot = HeisenbergModelConstraint( dim = [L,L], field_disorder = 1.) #, phases=phases)
#
# coords = np.zeros([nspins, 3])
# for i in range(nspins):
# coords[i,:] = rotations.vec_random()
# coords = coords.reshape(-1)
#
# pot.test_potential(coords)
#
#
#if __name__ == "__main__":
# test_potential_constrained()
## test()