def GenerateHamiltonian( generation, lattice = 'cactus', periodic = True ):
"""
Generate the linearized hamiltonian from the adjacency list for the Husimi
cactus or triangular lattice
"""
if lattice == 'cactus':
N = HusimiCactus.numNodes( generation )
adjacencyList = HusimiCactus.AdjacencyList( generation, periodic )
elif lattice == 'triangle':
N = TriangleLattice.numNodes( generation )
adjacencyList = TriangleLattice.AdjacencyList( generation )
# Trim the adjacency list of redundant pairs
i = 0
while (i < len(adjacencyList)):
x,y = adjacencyList[i]
if x <= y:
i += 1
else:
tossVal = adjacencyList.pop(i)
H = scipy.zeros( (2*N, 2*N) )
h = scipy.zeros( N )
J_ij = -1.0
cosTheta_ij = -.5
for pair in adjacencyList:
i,j = pair
# theta_i = h_i * ( - y_i + J_ij * y_j )
H[ i , j + N ] += J_ij
H[ j , i + N ] += J_ij
# y_i = h_i * ( theta_i - J_ij * cos(theta_ij) )
H[ i + N, j ] += -1.0 * J_ij * cosTheta_ij
H[ j + N, i ] += -1.0 * J_ij * cosTheta_ij
# h_i = Z_i / 2
h[i] += .5
h[j] += .5
# Distributing the h_i terms
for i in range( int(N) ):
H[ i , i + N ] += -1.0
H[ i + N, i ] += 1.0
h_i = h[i]
H[ i , : ] *= h_i
H[ i + N, : ] *= h_i
return H
def HusimiHamiltonian( generation, periodic = True ):
"""
Construct second-order (in Holstein-Primakoff bosons) contribution to the
hamiltonian for the heisenberg interaction on the Husimi cactus of
generation 'generation'
Some conventions for indexes in H:
a_i^d * b_j^d --> [ i , j + N ]
a_i^d * b_j --> [ i , j ]
a_i * b_j^d --> [ i + N, j + N ]
a_i * b_j --> [ i + N, j ]
"""
N = HusimiCactus.numNodes( generation )
adjacencyList = HusimiCactus.AdjacencyList( generation, periodic )
# Trim the adjacency list of redundant pairs
i = 0
while (i < len(adjacencyList)):
x,y = adjacencyList[i]
if x <= y:
i += 1
else:
tossVal = adjacencyList.pop(i)
H = scipy.zeros( (2*N, 2*N) )
for pair in adjacencyList:
i,j = pair
H[ i , j ] += 1.0 # s+ t
H[ j , i ] += 1.0 # t+ s
H[ i + N, j + N ] += 1.0 # s t+
H[ j + N, i + N ] += 1.0 # t s+
H[ i , i ] += 2.0 # s+ s
H[ j , j ] += 2.0 # t+ t
H[ i + N, i + N ] += 2.0 # s s+
H[ j + N, j + N ] += 2.0 # t t+
H[ i , j + N ] += -3.0 # s+ t+
H[ j , i + N ] += -3.0 # t+ s+
H[ i + N, j ] += -3.0 # s t
H[ j + N, i ] += -3.0 # s t
#H *= S / 8.
H *= 1 / 8.
return H