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model.py
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model.py
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""" Construction of cMPO for several models.
T: cMPO. The structure of cMPO:
-- --
| I + dtau Q -- sqrt(dtau) R -- |
| |
| | |
| sqrt(dtau) L P |
| | |
-- --
W: the unitary gate U that connects the cMPO to its transpose is given by
-- --
| 1 0 .. 0|
| 0 |
U = | : W |
| 0 |
-- --
this can save some time since <l| is related to |r> by U.
set W = None if U doesn't exist or you don't want to find it
ph_leg: the physical dimension of the cMPO
d: the virtual dimension of the cMPO minus 1
"""
import torch
from cmpo import cmpo
import numpy as np
torch.manual_seed(42)
class spin_half(object):
""" The Pauli matrices and S+, S-
"""
def __init__(self, dtype, device):
# Pauli matrices
self.Id = torch.eye(2, dtype=dtype, device=device)
self.X = torch.tensor([[0.0, 1.0], [1.0, 0.0]], dtype=dtype, device=device)
self.iY = torch.tensor([[0.0, 1.0], [-1.0, 0.0]], dtype=dtype, device=device)
self.Z = torch.tensor([[1.0, 0.0], [0.0, -1.0]], dtype=dtype, device=device)
self.Sp = torch.tensor([[0.0, 1.0], [0.0, 0.0]], dtype=dtype, device=device)
self.Sm = torch.tensor([[0.0, 0.0], [1.0, 0.0]], dtype=dtype, device=device)
class ising(object):
""" transverse field Ising model
H = -J \sum_{<i,j>} Z_i Z_j - Gamma \sum_i X_i
"""
def __init__(self, Gamma, J, dtype, device):
s = spin_half(dtype, device)
Q = Gamma * s.X
L = np.sqrt(J) * s.Z.view(1,2,2)
R = np.sqrt(J) * s.Z.view(1,2,2)
P = torch.zeros(1,1,2,2, dtype=dtype, device=device)
self.T = cmpo(Q, L, R, P)
self.W = torch.diag(torch.tensor([1], dtype=dtype, device=device))
self.ph_leg = 2
self.d = 1
class xxz_spm(object):
""" XXZ model
H = \sum_{<i,j>} (-Jxy Sx_i Sx_j - Jxy Sy_i Sy_j + Jz Sz_i Sz_j)
The cMPO is implemented by S+, S-, and Sz
"""
def __init__(self, Jz, Jxy, dtype, device):
s = spin_half(dtype, device)
Jxy_sign = np.sign(Jxy)
Jxy_abs = np.abs(Jxy)
Jz_sign = np.sign(Jz)
Jz_abs = np.abs(Jz)
Q = torch.zeros(2,2, dtype=dtype, device=device)
L = torch.cat((
s.Sp.view(1,2,2)*np.sqrt(Jxy_abs/2),
s.Sm.view(1,2,2)*np.sqrt(Jxy_abs/2),
np.sqrt(Jz_abs)*s.Z.view(1,2,2)/2
), dim=0)
R = torch.cat((
s.Sm.view(1,2,2)*np.sqrt(Jxy_abs/2)*Jxy_sign,
s.Sp.view(1,2,2)*np.sqrt(Jxy_abs/2)*Jxy_sign,
np.sqrt(Jz_abs)*s.Z.view(1,2,2)/2 * (-Jz_sign)
), dim=0)
P = torch.zeros(3,3,2,2, dtype=dtype, device=device)
self.T = cmpo(Q, L, R, P)
self.W = torch.tensor([[0, Jxy_sign, 0], [Jxy_sign, 0, 0], [0, 0, -Jz_sign]], dtype=dtype, device=device)
self.ph_leg = 2
self.d = 3
class xxz(object):
""" XXZ model
H = \sum_{<i,j>} (-Jxy Sx_i Sx_j - Jxy Sy_i Sy_j + Jz Sz_i Sz_j)
Alternative construction of the cMPO with Sx, iSy, Sz
"""
def __init__(self, Jz, Jxy, dtype, device):
s = spin_half(dtype, device)
Jxy_sign = np.sign(Jxy)
Jxy_abs = np.abs(Jxy)
Jz_sign = np.sign(Jz)
Jz_abs = np.abs(Jz)
Q = torch.zeros(2,2, dtype=dtype, device=device)
L = torch.cat((
np.sqrt(Jxy_abs)/2 * s.X.view(1,2,2),
np.sqrt(Jxy_abs)/2 * s.iY.view(1,2,2),
np.sqrt(Jz_abs)/2 * s.Z.view(1,2,2)
), dim=0 )
R = torch.cat((
np.sqrt(Jxy_abs)/2 * s.X.view(1,2,2)*Jxy_sign,
-np.sqrt(Jxy_abs)/2 * s.iY.view(1,2,2)*Jxy_sign,
-Jz_sign * np.sqrt(Jz_abs)/2 * s.Z.view(1,2,2)
), dim=0 )
P = torch.zeros(3,3,2,2, dtype=dtype, device=device)
self.T = cmpo(Q, L, R, P)
self.W = torch.diag(torch.tensor([Jxy_sign,-Jxy_sign,-Jz_sign], dtype=dtype, device=device))
self.ph_leg = 2
self.d = 3
class ising_NNN(object):
""" TFIM with next-nearest-neighboring interaction
H = -J \sum_{<i,j>} Z_i Z_j -J2 \sum_{<<i,j>>} Z_i Z_j - Gamma \sum_i X_i
"""
def __init__(self, Gamma, J, J2, dtype, device):
s = spin_half(dtype, device)
Q = Gamma * s.X
L = torch.cat((
np.sqrt(J/2) * s.Z.view(1,2,2),
np.sqrt(J/2) * s.Z.view(1,2,2),
), dim=0 )
R = torch.cat((
np.sqrt(J/2) * s.Z.view(1,2,2),
np.sqrt(J/2) * s.Z.view(1,2,2),
), dim=0 )
P0 = torch.tensor([[0, 2*J2/J], [0, 0]], dtype=dtype, device=device)
P = torch.einsum('mn,ab->mnab', P0, s.Id)
self.T = cmpo(Q, L, R, P)
self.W = torch.tensor([[0, 1], [1, 0]], dtype=dtype, device=device)
self.ph_leg =2
self.d = 2
class ising_expLR(object):
""" TFIM with exponentially decaying long-range interaction
H = -J \sum_{i,j} \exp(-alpha * |i-j|) Z_i Z_j - Gamma \sum_i X_i
"""
def __init__(self, Gamma, J, alpha, dtype, device):
s = spin_half(dtype, device)
Q = Gamma * s.X
L = np.sqrt(J) * np.exp(-alpha/2) * s.Z.view(1,2,2)
R = np.sqrt(J) * np.exp(-alpha/2) * s.Z.view(1,2,2)
P = np.exp(-alpha) * s.Id.view(1,1,2,2)
self.T = cmpo(Q, L, R, P)
self.W = torch.diag(torch.tensor([1], dtype=dtype, device=device))
self.ph_leg =2
self.d = 1
class ising_powLR(object):
""" TFIM with power-law decaying long-range interaction
H = -J \sum_{i,j} |j-i|^{-alpha} Z_i Z_j - Gamma \sum_i X_i
the power-law decaying interaction is approximated with a sum of exponentials
|j-i|^{-alpha} = \sum_k^K mu_k exp(-l_k |j-i|)
initial guess for mu, l , and choice of K is suggested by arXiv:physics/0605149
"""
def __init__(self, Gamma, J, alpha, dtype, device):
K = int(np.log(500)/np.log(3.87))
eta = np.float_power(500, 1/K)
mu0 = 1/np.sum(
[np.float_power(eta, -ix*alpha) * np.exp(-alpha/eta**ix) for ix in range(K)]
)
mu_vec0 = np.float_power(eta, -np.arange(K) * alpha) * mu0
l_vec0 = alpha / np.float_power(eta, np.arange(K))
mu_vec = torch.nn.Parameter(torch.tensor(mu_vec0, dtype=dtype, device=device))
l_vec = torch.nn.Parameter(torch.tensor(l_vec0, dtype=dtype, device=device))
# save parameters (borrow the class from cmps)
from cmpo import data_cmps, datasave, dataload
para_data = data_cmps(mu_vec, l_vec)
path='power_fit_parameters_alpha{:.2f}.pt'.format(alpha)
# optimization obtained over range of 200 sites
fa = lambda x, alpha: 1/np.float_power(x, alpha)
fb = lambda x, mu_vec, l_vec: mu_vec @ torch.exp(-l_vec * x)
def func(mu_vec, l_vec):
y = 0
for j in np.arange(200)+1:
y += (fa(j, alpha) - fb(j, mu_vec, l_vec))**2
return y
try:
dataload(para_data, path)
print('parameters loaded successfully')
except:
print('parameters failed to load, fitting')
optimizer0 = torch.optim.LBFGS([mu_vec, l_vec], max_iter = 20, tolerance_grad=0, tolerance_change=0, line_search_fn="strong_wolfe")
def closure0():
optimizer0.zero_grad()
loss = func(mu_vec, l_vec)
loss.backward()
return loss
counter = 0
loss0 = 9.99e99
while counter < 2:
loss = optimizer0.step(closure0)
print('--> ' + '{:.12f}'.format(loss.item()), end=' \r')
if np.isclose(loss.item(), loss0, rtol=1e-9, atol=1e-9):
counter += 1
loss0 = loss.item()
datasave(para_data, path)
mu_vec, l_vec = mu_vec.detach(), l_vec.detach()
self.mu_vec = mu_vec
self.l_vec = l_vec
print('final loss', func(mu_vec, l_vec).item())
s = spin_half(dtype, device)
Q = Gamma * s.X
mu_abs_vec = torch.abs(mu_vec)
mu_sgn_vec = torch.sign(mu_vec)
L = torch.einsum('m,ab->mab', torch.exp(-l_vec/2)*torch.sqrt(J*mu_abs_vec), s.Z)
R = torch.einsum('m,ab->mab', torch.exp(-l_vec/2)*torch.sqrt(J*mu_abs_vec)*mu_sgn_vec, s.Z)
P = torch.einsum('m,mn,ab->mnab',torch.exp(-l_vec), torch.eye(K, dtype=dtype, device=device), s.Id)
self.T = cmpo(Q, L, R, P)
self.W = torch.diag(mu_sgn_vec)
self.ph_leg = 2
self.d = K