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_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_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_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
def test_product_invariant(n_iters, tensorclass, n_qnums, rshape, rqhape,
rdirs, rcharge, rtensor):
"""Generate two invariant tensors, contract them over a random set of legs,
and compare with NumPy.
"""
for iter_num in range(n_iters):
shp1 = rshape(nlow=1) # Shape of the first tensor
# Choose how many indices to contract order, and which indices of
# tensor #1 those should be.
n = np.random.randint(low=1, high=len(shp1) + 1)
if n:
i_list = list(np.random.choice(len(shp1), size=n, replace=False))
else:
i_list = []
# Generate the shape of the second tensor, and which indices it should
# be contracted over.
shp2 = rshape(nlow=n)
if n:
j_list = list(np.random.choice(len(shp2), size=n, replace=False))
else:
j_list = []
# Make sure contracted indices have a dimension of at least 1.
for k in range(n):
dim1 = shp1[i_list[k]]
if np.sum(dim1) < 1:
dim1 = rshape(n=1, chilow=1)[0]
shp1[i_list[k]] = dim1
shp2[j_list[k]] = dim1
# Generate tensor #1.
qhp1 = rqhape(shp1)
qhp2 = rqhape(shp2)
if qhp1 is not None:
for k in range(n):
qhp2[j_list[k]] = qhp1[i_list[k]]
T1 = rtensor(shape=shp1, qhape=qhp1)
T1_orig = T1.copy()
# Generate tensor #2.
if T1.dirs is not None:
dirs2 = rdirs(shape=shp2)
for i, j in zip(i_list, j_list):
dirs2[j] = -T1.dirs[i]
else:
dirs2 = None
T2 = rtensor(shape=shp2, qhape=qhp2, dirs=dirs2)
T2_orig = T2.copy()
# Do the product.
T1_np = T1.to_ndarray()
T2_np = T2.to_ndarray()
T = T1.dot(T2, (i_list, j_list))
assert (T1 == T1_orig).all()
assert (T2 == T2_orig).all()
check_internal_consistency(T)
# Assert that the result has the right shape.
i_list_compl = sorted(set(range(len(shp1))) - set(i_list))
j_list_compl = sorted(set(range(len(shp2))) - set(j_list))
product_shp = [shp1[i] for i in i_list_compl
] + [shp2[j] for j in j_list_compl]
if type(T) == Tensor:
product_shp = Tensor.flatten_shape(product_shp)
assert T.shape == product_shp
# Do the product using NumPy and compare.
T_np = np.tensordot(T1_np, T2_np, (i_list, j_list))
assert np.allclose(T_np, T.to_ndarray())
def truncate_func(
s,
u=None,
v=None,
chis=None,
eps=0,
trunc_err_func=None,
norm_sq=None,
return_error=False,
):
chis = s.matrix_decomp_format_chis(chis, eps)
if hasattr(s, "sects"):
if trunc_err_func is None:
trunc_err_func = fct.partial(type(s).default_trunc_err_func,
norm_sq=norm_sq)
# First, find what chi will be.
s_flat = s.to_ndarray()
s_flat = -np.sort(-np.abs(s_flat))
chi, err = Tensor.find_trunc_dim(s_flat,
chis=chis,
eps=eps,
trunc_err_func=trunc_err_func)
# Find out which values to keep, i.e. how to distribute chi in
# the different blocks.
dim_sum = 0
minusabs_next_els = []
dims = {}
for k, val in s.sects.items():
heapq.heappush(minusabs_next_els, (-np.abs(val[0]), k))
dims[k] = 0
while dim_sum < chi:
try:
minusabs_el_to_add, key = heapq.heappop(minusabs_next_els)
except IndexError:
# All the dimensions are fully included.
break
dims[key] += 1
this_key_els = s[key]
if dims[key] < len(this_key_els):
next_el = this_key_els[dims[key]]
heapq.heappush(minusabs_next_els, (-np.abs(next_el), key))
dim_sum += 1
# Truncate each block and create the dim for the new index.
new_dim = []
todelete = []
for k in s.sects.keys():
d = dims[k]
new_dim.append(d)
if d > 0:
s[k] = s[k][:d]
else:
# Avoiding changing the dictionary during the loop.
todelete.append(k)
for k in todelete:
del s[k]
if u is not None:
todelete = []
for k in u.sects.keys():
klast = k[-1]
d = dims[(klast, )]
if d > 0:
u[k] = u[k][..., :d]
else:
todelete.append(k)
for k in todelete:
del u[k]
if v is not None:
todelete = []
for k in v.sects.keys():
k0 = k[0]
d = dims[(k0, )]
if d > 0:
v[k] = v[k][:d, ...]
else:
todelete.append(k)
for k in todelete:
del v[k]
# Remove zero dimension sectors from qim.
new_qim = s.qhape[0]
new_dim = [d for d in new_dim if d > 0]
for k, d in dims.items():
if d == 0:
new_qim = [q for q in new_qim if q != k[0]]
s.shape = [new_dim]
s.qhape = [new_qim]
if u is not None:
u.shape = u.shape[0:-1] + [new_dim]
u.qhape = u.qhape[0:-1] + [new_qim]
if v is not None:
v.shape = [new_dim] + v.shape[1:]
v.qhape = [new_qim] + v.qhape[1:]
else:
chi, err = type(s).find_trunc_dim(s,
chis=chis,
eps=eps,
trunc_err_func=trunc_err_func)
s = s[:chi]
if u is not None:
u = u[..., :chi]
if v is not None:
v = v[:chi, ...]
retval = (s, )
if u is not None:
retval += (u, )
if v is not None:
retval += (v, )
if return_error:
retval += (err, )
if len(retval) == 1:
retval = retval[0]
return retval