def get_initial_tensor_CDL_3d(pars):
delta = np.eye(2, dtype=pars["dtype"])
A = np.einsum(
("ae,fi,jm,nb,cq,rk,lu,vd,gs,to,pw,xh -> abcdefghijklmnopqrstuvwx"),
*((delta,) * 12)
)
return Tensor.from_ndarray(A.reshape((16, 16, 16, 16, 16, 16)))
def get_initial_tensor_CQL_3d(pars):
delta = np.array([[[1, 0], [0, 0]], [[0, 0], [0, 1]]])
A = np.einsum(
("aeu,fiv,gjq,hbr,mcw,nxk,ols,ptd " "-> abcdefghijklmnopqrstuvwx"),
*((delta,) * 8)
)
return Tensor.from_ndarray(A.reshape((16, 16, 16, 16, 16, 16)))
def get_KW_unitary(pars):
""" The unitary that moves the Kramers-Wannier duality defect of the
classical 2D square lattice Ising model.
"""
CZ = Csigma_np("z")
# fmt: off
U = ncon((CZ,
R(np.pi / 4, 'z'), R(np.pi / 4, 'x'),
R(np.pi / 4, 'y')),
([-1, -2, 5, 6],
[-3, 5], [3, 6],
[-4, 3]))
# fmt: on
u = symmetry_bases["ising"]
u_dg = u.T.conjugate()
U = ncon(
(U, u, u_dg, u_dg, u),
([1, 2, 3, 4], [-1, 1], [-2, 2], [3, -3], [4, -4]),
)
U *= -1j
if pars["symmetry_tensors"]:
U = TensorZ2.from_ndarray(
U, shape=[[1, 1]] * 4, qhape=[[0, 1]] * 4, dirs=[1, 1, -1, -1]
)
else:
U = Tensor.from_ndarray(U, dirs=[1, 1, 1, -1, -1, -1])
return U
def get_initial_sixvertex_tensor(pars):
try:
a = pars["sixvertex_a"]
b = pars["sixvertex_b"]
c = pars["sixvertex_c"]
except KeyError:
u = pars["sixvertex_u"]
lmbd = pars["sixvertex_lambda"]
rho = pars["sixvertex_rho"]
a = rho * np.sin(lmbd - u)
b = rho * np.sin(u)
c = rho * np.sin(lmbd)
A_0 = np.zeros((2, 2, 2, 2), dtype=pars["dtype"])
A_0[1, 0, 0, 1] = a
A_0[0, 1, 1, 0] = a
A_0[0, 0, 1, 1] = b
A_0[1, 1, 0, 0] = b
A_0[0, 1, 0, 1] = c
A_0[1, 0, 1, 0] = c
if pars["symmetry_tensors"]:
dim = [1, 1]
qim = [-1, 1]
A_0 = TensorU1.from_ndarray(
A_0, shape=[dim] * 4, qhape=[qim] * 4, dirs=[1, 1, 1, 1]
)
A_0 = A_0.flip_dir(2)
A_0 = A_0.flip_dir(3)
else:
A_0 = Tensor.from_ndarray(A_0)
return A_0
def convertAbe(A):
"""
Convert A to class abeliantensors if it is not
"""
if type(A).__module__.split(".")[0] != 'abeliantensors':
A = Tensor.from_ndarray(A)
return A
def get_initial_impurity(pars, legs=(3,), factor=3, **kwargs):
if kwargs:
pars = pars.copy()
pars.update(kwargs)
A_pure = get_initial_tensor(pars)
model = pars["model"]
impurity = pars["impurity"]
try:
impurity_matrix = impurity_dict[model][impurity](pars)
except KeyError:
msg = "Unknown (model, impurity) combination: ({}, {})".format(
model, impurity
)
raise ValueError(msg)
# TODO The expectation that everything is in the symmetry basis
# clashes with how 2D ising and potts initial tensors are generated.
if not pars["symmetry_tensors"]:
impurity_matrix = Tensor.from_ndarray(impurity_matrix.to_ndarray())
# TODO This was commented in before 2019-09-06. Why? It's clearly wrong
# for 3D Ising U-impurity and id-impurity.
# impurity_matrix *= -1
A_impure = 0
if 0 in legs:
A_impure += ncon(
(A_pure, impurity_matrix), ([1, -2, -3, -4, -5, -6], [1, -1])
)
if 1 in legs:
A_impure += ncon(
(A_pure, impurity_matrix), ([-1, 2, -3, -4, -5, -6], [2, -2])
)
if 2 in legs:
A_impure += ncon(
(A_pure, impurity_matrix.transpose()),
([-1, -2, 3, -4, -5, -6], [3, -3]),
)
if 3 in legs:
A_impure += ncon(
(A_pure, impurity_matrix.transpose()),
([-1, -2, -3, 4, -5, -6], [4, -4]),
)
if 4 in legs:
A_impure += ncon(
(A_pure, impurity_matrix), ([-1, -2, -3, -4, 5, -6], [5, -5])
)
if 5 in legs:
A_impure += ncon(
(A_pure, impurity_matrix.transpose()),
([-1, -2, -3, -4, -5, 6], [6, -6]),
)
# TODO This was commented in before 2019-09-06. Why? It's clearly wrong
# for 3D Ising U-impurity and id-impurity.
# A_impure *= factor
return A_impure
def get_initial_tensor_CDL_3d_v2(pars):
delta = np.eye(2, dtype=pars["dtype"])
A = ncon(
(delta,) * 12,
# fmt: off
(
[-11, -21], [-12, -41], [-13, -51], [-14, -61],
[-31, -22], [-32, -42], [-33, -52], [-34, -62],
[-23, -63], [-64, -43], [-44, -53], [-54, -24],
),
# fmt: on
)
return Tensor.from_ndarray(A.reshape((16, 16, 16, 16, 16, 16)))
def get_initial_tensor_ising_3d(pars):
beta = pars["beta"]
ham = ising3d_ham(beta)
A_0 = np.einsum(
"ai,aj,ak,al,am,an -> ijklmn", ham, ham, ham, ham, ham, ham
)
if pars["symmetry_tensors"]:
cls, dim, qim = TensorZ2, [1, 1], [0, 1]
A_0 = cls.from_ndarray(
A_0, shape=[dim] * 6, qhape=[qim] * 6, dirs=[1, 1, -1, -1, 1, -1]
)
else:
A_0 = Tensor.from_ndarray(A_0)
return A_0
def get_initial_dilute_sixvertex_tensor(pars):
# Copied and adapted from Roman
# 1
# |
# 4 -- A -- 2
# |
# 3
# "dilute six-vertex model",
# states : 0:empty, 1:top arrow, 1:bot arrow. Time flows toward N-E
w = np.exp(1j * np.pi / 8.0)
z = 0.57 # Close to critical
# z = 1.0 # XXZ universality class
A = np.zeros((3, 3, 3, 3), dtype=np.complex)
A[1, 1, 1, 1] = 1.0
A[2, 1, 1, 2] = z * w
A[1, 2, 2, 1] = z / w
A[0, 1, 1, 0] = z / w
A[1, 0, 0, 1] = z * w
A[2, 0, 1, 1] = z / w
A[0, 2, 1, 1] = z * w
A[1, 1, 2, 0] = z * w
A[1, 1, 0, 2] = z / w
A[2, 2, 2, 2] = z ** 2
A[0, 0, 0, 0] = z ** 2
A[2, 0, 2, 0] = z ** 2
A[0, 2, 0, 2] = z ** 2
A[2, 0, 0, 2] = (z ** 2) * (w ** 2 + 1.0 / w ** 2)
A[0, 2, 2, 0] = (z ** 2) * (w ** 2 + 1.0 / w ** 2)
# U(1) charge : Q=(-1,0,+1) Q1+Q2=Q3+Q4
if pars["symmetry_tensors"]:
dim = [1, 1, 1]
qim = [-1, 0, 1]
A = TensorU1.from_ndarray(
A, shape=[dim] * 4, qhape=[qim] * 4, dirs=[1, 1, -1, -1]
)
else:
A = Tensor.from_ndarray(A)
# To translate between Roman's convention and mine.
A = A.transpose((3, 0, 1, 2))
return A
def get_initial_tensor(pars, **kwargs):
if kwargs:
pars = pars.copy()
pars.update(kwargs)
model_name = pars["model"].strip().lower()
if model_name == "dilute_sixvertex":
return get_initial_dilute_sixvertex_tensor(pars)
elif model_name == "sixvertex":
return get_initial_sixvertex_tensor(pars)
elif model_name == "ising3d":
return get_initial_tensor_ising_3d(pars)
elif model_name == "potts33d":
return get_initial_tensor_potts33d(pars)
elif model_name == "complexion_qising":
ham = get_ham(pars, model="qising")
complexion = build_complexion(ham, pars)
return complexion
elif model_name == "complexion_qising_tricrit":
ham = get_ham(pars, model="qising_tricrit")
complexion = build_complexion(ham, pars)
return complexion
elif model_name == "complexion_sq_qising":
ham = get_ham(pars, model="qising")
complexion = build_complexion(ham, pars, square_hamiltonian=True)
return complexion
else:
ham = hamiltonians[model_name](pars)
boltz = np.exp(-pars["beta"] * ham)
A_0 = np.einsum("ab,bc,cd,da->abcd", boltz, boltz, boltz, boltz)
u = symmetry_bases[model_name]
u_dg = u.T.conjugate()
A_0 = ncon(
(A_0, u, u, u_dg, u_dg),
([1, 2, 3, 4], [-1, 1], [-2, 2], [3, -3], [4, -4]),
)
if pars["symmetry_tensors"]:
cls, dim, qim = symmetry_classes_dims_qims[model_name]
A_0 = cls.from_ndarray(
A_0, shape=[dim] * 4, qhape=[qim] * 4, dirs=[1, 1, -1, -1]
)
else:
A_0 = Tensor.from_ndarray(A_0)
return A_0
def get_initial_tensor_potts33d(pars):
beta = pars["beta"]
Q = potts_Q(beta, 3)
A = np.einsum(
"ai,aj,ak,al,am,an -> ijklmn",
Q,
Q,
Q.conjugate(),
Q.conjugate(),
Q,
Q.conjugate(),
)
if np.linalg.norm(np.imag(A)) < 1e-12:
A = np.real(A)
if pars["symmetry_tensors"]:
cls, dim, qim = symmetry_classes_dims_qims["potts3"]
A = cls.from_ndarray(
A, shape=[dim] * 6, qhape=[qim] * 6, dirs=[1, 1, -1, -1, 1, -1]
)
else:
A = Tensor.from_ndarray(A)
return A
def get_KW_tensor(pars):
""" The Kramers-Wannier duality defect of the classical 2D
square lattice Ising model.
"""
eye = np.eye(2, dtype=np.complex_)
ham = hamiltonians["ising"](pars)
B = np.exp(-pars["beta"] * ham)
H = np.array([[1, 1], [1, -1]], dtype=np.complex_) / np.sqrt(2)
y_trigged = np.ndarray((2, 2, 2), dtype=np.complex_)
y_trigged[:, :, 0] = eye
y_trigged[:, :, 1] = sigma("y")
D_sigma = np.sqrt(2) * np.einsum(
"ab,abi,ic,ad,adk,kc->abcd",
B,
y_trigged,
H,
B,
y_trigged.conjugate(),
H,
)
u = symmetry_bases["ising"]
u_dg = u.T.conjugate()
D_sigma = ncon(
(D_sigma, u, u, u_dg, u_dg),
([1, 2, 3, 4], [-1, 1], [-2, 2], [3, -3], [4, -4]),
)
if pars["symmetry_tensors"]:
D_sigma = TensorZ2.from_ndarray(
D_sigma,
shape=[[1, 1]] * 4,
qhape=[[0, 1]] * 4,
dirs=[1, 1, -1, -1],
)
else:
D_sigma = Tensor.from_ndarray(D_sigma, dirs=[1, 1, -1, -1])
return D_sigma