def GetPairHam_HaldanePhase(J=1, Uzz=0, Bx=0, Bz=0, R=0, dim_spin=3):
'''
:param J: coupling constant.
:param Uzz: spin suppression constant along z direction.
:param Bx: Zeeman field along x.
:param Bz: Zeeman field along z.
:param R: inversion symmetry breaking term constant.
:param dim_spin: Hilbert space dimension of single spin.
:return: single two-site interaction term as a 4-order tensor.
'''
Sz, Sp, Sm, Sx, Id = GenSpinOpr(which_opr='real', dim_spin=dim_spin)
Sz2 = np.dot(Sz, Sz)
R_term = np.kron(np.dot(Sz, Sx), Sx) - np.kron(Sx, np.dot(Sz, Sx))
# H = J * \sum{ S_i * S_j } + Uzz * \sum{ (S^z_i)^2 } + Bx * \sum{ S^x_i }
H_pair = np.kron(Sz, Sz) + 0.5 * J * (np.kron(Sp, Sm) + np.kron(Sm, Sp)) \
+ Uzz * np.kron(Sz2, Id) + Bx * np.kron(Sx, Id) + Bz * np.kron(Sz, Id) + R * R_term
# 0
# / \
# / \
# H H
# \ /
# \ /
# 1
H_pair = np.reshape(H_pair, [dim_spin, dim_spin, dim_spin, dim_spin])
# 0 1
# | |
# H-----H
# | |
# 2 3
return H_pair
def GetSiteHmpo_Heisenberg(J, h, dim_spin=2):
'''
:param J: Heisenberg coupling constant.
:param h: magnetic field strength.
:param dim_spin: Hilbert space dimension of single spin.
:return: single site matrix product operator of Heisenberg model, which is a 4-order tensor. (5,5,2,2).
'''
# generate spin operator
Sp, Sm, Sx, Sy, Sz, Id, S0 = GenSpinOpr(dim_spin=dim_spin)
# H = \sum_<ij> J * kronecker(S_i, S_j) + \sum_i (-h) * S^z_i
mpo_H_single_site_Heisenberg = np.array(
[[Id, S0, S0, S0, S0], [Sp, S0, S0, S0, S0], [Sm, S0, S0, S0, S0],
[Sz, S0, S0, S0, S0], [-h * Sz, J / 2 * Sm, J / 2 * Sp, J * Sz, Id]])
return mpo_H_single_site_Heisenberg
def GetSiteHmpo_TFI(J, h, dim_spin=2):
'''
:param J: Ising coupling constant.
:param h: transverse field strength.
:param dim_spin: Hilbert space dimension of single spin.
:return: single site matrix product operator of the transverse-field Ising (TFI) model.
'''
# generate spin operator
Sp, Sm, Sx, Sy, Sz, Id, S0 = GenSpinOpr(dim_spin=dim_spin)
sigma_z = 2 * Sz
sigma_x = 2 * Sx
# H = (-J) * \sum_<i,i+1>( \sigma^z_i * \sigma^z_j + h * \sigma^x_i)
mpo_H_single_site_TFI = np.array([[Id, S0, S0], [sigma_z, S0, S0],
[-J * h * sigma_x, -J * sigma_z, Id]])
return mpo_H_single_site_TFI
def GetSiteHmpo_HaldanePhase(J, Uzz, Bx=0, dim_spin=3):
'''
:param J: Heisenberg coupling constant.
:params Uzz: spin suppression along z direction.
:param Bx: transverse field strength.
:param dim_spin: Hilbert space dimension of single spin.
:return: single site matrix product operator of generalized spin-1 Heisenberg model, as a 4-order tensor.
'''
# generate spin operator
Sp, Sm, Sx, Sy, Sz, Id, S0 = GenSpinOpr(dim_spin=dim_spin)
# H = J * \sum{ S_i * S_j } + Uzz * \sum{ (S^z_i)^2 } + Bx * \sum{ S^x_i }
Sz2 = np.dot(Sz, Sz)
mpo_H_single_site_HaldanePhase = np.array(
[[Id, S0, S0, S0, S0], [Sp, S0, S0, S0, S0], [Sm, S0, S0, S0, S0],
[Sz, S0, S0, S0, S0],
[Bx * Sx + Uzz * Sz2, J / 2 * Sm, J / 2 * Sp, J * Sz, Id]])
return mpo_H_single_site_HaldanePhase
def GetPairHam_Heisenberg(J, dim_spin):
'''
:param J: coupling constant.
:param dim_spin: Hilbert space dimension of single spin.
:return: single two-site interaction term as a 4-order tensor J * S_i · S_j.
'''
Sz, Sp, Sm, Sx, Id = GenSpinOpr(which_opr='real', dim_spin=dim_spin)
# H = J * \sum{ S_i * S_j }
H_pair = np.kron(Sz, Sz) + 0.5 * J * (np.kron(Sp, Sm) + np.kron(Sm, Sp))
# 0
# / \
# / \
# H H
# \ /
# \ /
# 1
H_pair = np.reshape(H_pair, [dim_spin, dim_spin, dim_spin, dim_spin])
# 0 1
# | |
# H-----H
# | |
# 2 3
return H_pair