def update(b, env):
c, t = env
# C insertion
c_tilde = ncon([c, t, t, b, np.conj(b)],
[[1, 2], [2, 3, 4, -4], [-1, 5, 6, 1],
[7, 3, 5, -2, -5], [7, 4, 6, -3, -6]],
[1, 2, 3, 5, 4, 6, 7])
# (D,d,d',R,r,r') -> (D~,R~)
_D, _d, _d_, _R, _r, _r_ = c_tilde.shape
c_tilde = np.reshape(c_tilde, [_D * _d * _d_, _R * _r * _r_])
# T insertion
t_tilde = ncon(
[t, b, np.conj(b)],
[[-1, 1, 2, -6], [3, 1, -2, -4, -7], [3, 2, -3, -5, -8]])
# (L,l,l',d,d',R,r,r') -> (L~,d,d',R~)
_L, _l, _l_, _d, _d_, _R, _r, _r_ = t_tilde.shape
t_tilde = np.reshape(t_tilde, [_L * _l * _l_, _d, _d_, _R * _r * _r_])
# enforce symmetry
c_tilde = _c4v_symmetrise_c(c_tilde)
# find projector
P, _, _ = svd_truncated(c_tilde, chi_max=chi_ctm, cutoff=0.) # (D~,R)
# renormalise
c = np.transpose(P) @ c_tilde @ P
t = ncon([np.conj(P), t_tilde, np.conj(P)],
[[1, -1], [1, -2, -3, 2], [2, -4]])
# enforce symmetry
env = _c4v_symmetrise(c, t)
return env
def _variance(c: np.ndarray, t: np.ndarray) -> float:
"""
Returns the variance |<aa aa>| - <aa>**2 that would be zero for the fixed point of the RG
This version is only checking if c and t are ANY environment, see also `_variance3`
Parameters
----------
c
The corner matrix c(d, r) with `chi`-dimensional legs
t
The half-column / half-row tensor t(l, d, d', r)
Returns
-------
var : float
The variance
"""
# c --- c c --- t --- c c --- t --- t --- c
# lr = | | ltr = | | | lttr = | | | |
# c --- c c --- t --- c c --- t --- t --- c
# if RG fixed point is reached then lttr/lr = <tt> = <t>**2 = (ltr/lr)**2
lr = np.trace(c @ c @ c @ c)
connects = [[1, 6], [6, 7, 8, 5], [5, 4], [4, 3], [2, 1], [2, 7, 8, 3]]
cont_order = [
4,
1,
3,
2,
6,
5,
7,
8,
]
ltr = ncon([c, t, c, c, c, t], connects, cont_order)
connects = [[1, 4], [4, 5, 6, 9], [10, 3], [3, 11], [2, 1], [2, 5, 6, 12],
[9, 7, 8, 10], [12, 7, 8, 11]]
cont_order = [
1,
2,
3,
11,
10,
7,
8,
12,
4,
5,
6,
9,
]
lttr = ncon([c, t, c, c, c, t, t, t], connects, cont_order)
return abs(lttr / lr) - (ltr / lr)**2
def contract_closed_network(tensors,
connects,
contraction_order=None,
check_network=True):
if not all([all([con > 0 for con in tens_cons])
for tens_cons in connects]):
raise ValueError('Not a closed network')
return ncon(tensors, connects, contraction_order, check_network)
def two_layers(bra_tensors, ket_tensors):
lx, ly = len(bra_tensors), len(bra_tensors[0])
assert all(len(bra_tensors[x]) == ly for x in range(lx))
assert len(ket_tensors) == lx
assert all(len(ket_tensors[x]) == ly for x in range(lx))
tensors = [ket_tensors[x][y] for y in range(ly) for x in range(lx)] \
+ [bra_tensors[x][y] for y in range(ly) for x in range(lx)]
connects, order = _get_instructions2(lx, ly)
return ncon(tensors, connects, order)
def _build_evolution_pepo(u_vert,
A,
bc: str,
lx: Optional[int] = None,
ly: Optional[int] = None) -> Operator:
# DOC : lx, ly have no effect for INFINITE
u_bulk = ncon(
[u_vert, A, A, A, A, u_vert],
[[1, -2], [2, 1, -6], [3, 2, -3], [4, 3, -4], [5, 4, -5], [-1, 5]])
if bc == PBC:
if (lx is None) or (ly is None):
raise ValueError
tensors = [[u_bulk for _ in range(ly)] for _ in range(lx)]
return Pepo(tensors, bc=PBC, hermitian=False)
elif bc == OBC:
if (lx is None) or (ly is None):
raise ValueError
u_edge = ncon([u_vert, A, A, A, u_vert],
[[1, -2], [2, 1, -5], [3, 2, -3], [4, 3, -4], [-1, 4]])
u_corner = ncon([u_vert, A, A, u_vert],
[[1, -2], [2, 1, -3], [3, 2, -4], [-1, 3]])
tensors = [[u_corner] + [u_edge for _ in range(ly - 2)] + [u_corner]] \
+ [[u_edge] + [u_bulk for _ in range(ly - 2)] + [u_edge] for _ in range(lx - 2)] \
+ [[u_corner] + [u_edge for _ in range(ly - 2)] + [u_corner]]
return Pepo(tensors, bc=OBC, hermitian=False)
elif bc == INFINITE:
return Ipepo([u_bulk],
unit_cell=np.array([[0]]),
symmetry='c4v',
hermitian=False)
raise ValueError(f'Unknown boundary conditions: {bc}')
def _get_env(tensors, connects, contraction_order, check_network, n):
assert 0 <= n < len(tensors)
# remove tensors[n]
tensors = tensors[:n] + tensors[n + 1:]
# adjust connects. deep copy to avoid list-mutation problems
connects = [sublist[:] for sublist in connects]
# replace labels of legs that would be contracted with tensors[n] by open labels
cons_to_be_removed = connects.pop(n)
for n, con in enumerate(cons_to_be_removed):
connects = [[-(n + 1) if c == con else c for c in sublist]
for sublist in connects]
# remove labels from order
if contraction_order:
contraction_order = [
i for i in contraction_order if (i not in cons_to_be_removed)
]
env = ncon(tensors, connects, contraction_order, check_network)
return env
def expval(env_tensors, ipeps_tensors, operator, operator_geometry):
ipeps_tensor, = ipeps_tensors
c, t = env_tensors
chi_ctm = c.shape[0]
chi_peps = ipeps_tensor.shape[1]
d = operator.shape[0]
n_sites = len(operator.shape) // 2
assert c.shape == (chi_ctm, chi_ctm)
assert t.shape == (chi_ctm, chi_peps, chi_peps, chi_ctm)
assert ipeps_tensor.shape == (d, ) + 4 * (chi_peps, )
assert operator.shape == (d, ) * (2 * n_sites)
assert np.all(operator_geometry < n_sites)
if operator_geometry.shape == (1, 1): # single-site operator
# /
# c - t - c - a -
# | | | | / |
# t - B - t - B - = OP
# | | | | | /
# c - t - c - a -
# /
tensors = [
c, c, c, c, t, t, t, t, ipeps_tensor,
np.conj(ipeps_tensor), operator
]
connects = [[6, 1], [7, 2], [3, 8], [4, 5], [1, 13, 18, 7],
[2, 12, 17, 3], [8, 14, 15, 4], [5, 11, 16, 6],
[9, 13, 11, 14, 12], [10, 18, 16, 15, 17], [10, 9]]
cont_order = [
4,
10,
6,
8,
5,
16,
15,
7,
2,
1,
3,
18,
17,
11,
14,
9,
13,
12,
]
val = ncon(tensors, connects,
cont_order) # tensortrace: expval_1_1.ttp network 1
tensors = [c, c, c, c, t, t, t, t, ipeps_tensor, np.conj(ipeps_tensor)]
connects = [[6, 1], [7, 2], [3, 8], [4, 5], [1, 10, 11, 7],
[2, 16, 15, 3], [8, 13, 14, 4], [5, 9, 12, 6],
[17, 10, 9, 13, 16], [17, 11, 12, 14, 15]]
cont_order = [
5,
7,
8,
4,
14,
12,
1,
2,
6,
11,
15,
3,
10,
16,
13,
9,
17,
]
normalisation = ncon(
tensors, connects,
cont_order) # tensortrace: expval_1_1.ttp network 2
return val / normalisation
elif operator_geometry.shape in [(2, 1),
(1, 2)]: # nearest-neighbour operator
# / /
# c - t - t - c - a - a -
# | | | | | | / | / |
# t - **B** - t - **B** - = **OP*
# | | | | | | | / | /
# c - t - t - c - a - a -
# / /
# computed as horizontal, but due to rotational symmetry, vertical is the same
tensors = [
c, c, c, c, t, t, t, t, t, t, ipeps_tensor, ipeps_tensor,
np.conj(ipeps_tensor),
np.conj(ipeps_tensor), operator
]
connects = [[10, 1], [3, 4], [5, 6], [8, 9], [1, 22, 23, 2],
[2, 26, 27, 3], [4, 17, 18, 5], [6, 25, 28, 7],
[7, 21, 24, 8], [9, 15, 20, 10], [11, 22, 15, 21, 16],
[13, 26, 16, 25, 17], [12, 23, 20, 24, 19],
[14, 27, 19, 28, 18], [12, 14, 11, 13]]
cont_order = [
5,
8,
3,
6,
17,
25,
1,
10,
18,
28,
4,
26,
27,
13,
14,
23,
20,
7,
2,
9,
12,
24,
19,
22,
15,
16,
11,
21,
]
val = ncon(tensors, connects,
cont_order) # tensortrace: expval_2_1.ttp
tensors = [
c, c, c, c, t, t, t, t, t, t, ipeps_tensor, ipeps_tensor,
np.conj(ipeps_tensor),
np.conj(ipeps_tensor)
]
connects = [[10, 1], [3, 4], [5, 6], [8, 9], [1, 18, 19, 2],
[2, 22, 23, 3], [4, 13, 14, 5], [6, 21, 24, 7],
[7, 17, 20, 8], [9, 11, 16, 10], [26, 18, 11, 17, 12],
[25, 22, 12, 21, 13], [26, 19, 16, 20, 15],
[25, 23, 15, 24, 14]]
cont_order = [
8,
5,
6,
14,
24,
10,
3,
13,
21,
25,
4,
22,
23,
7,
1,
15,
20,
9,
16,
19,
2,
11,
18,
12,
17,
26,
]
normalisation = ncon(
tensors, connects,
cont_order) # tensortrace: expval_2_1_normalisation.ttp
return val / normalisation
else:
raise NotImplementedError
def _variance3(c: np.ndarray, t: np.ndarray, a: np.ndarray) -> float:
"""
Returns the variance |<aa aa>| - <aa>**2 that would be zero for the fixed point of the RG
This version also checks if `c` and `t` are an infinite environment of `aa`
Parameters
----------
c
The corner matrix c(d, r) with `chi`-dimensional legs
t
The half-column / half-row tensor t(l, d, d', r)
a
A iPEPS tensor a(p, u, l, d, r)
Returns
-------
var : float
The variance
"""
# c --- c c --- t --- c c --- t --- t --- c
# | | | | | | | | |
# lr = t --- t ltr = t --- aa--- t lttr = t --- aa--- aa--- t
# | | | | | | | | |
# c --- c c --- t --- c c --- t --- t --- c
# if RG fixed point is reached then lttr/lr = <tt> = <t>**2 = (ltr/lr)**2
connects = [[1, 5], [2, 7, 8, 1], [5, 4], [3, 6], [6, 2], [3, 7, 8, 4]]
cont_order = [
6,
5,
3,
4,
1,
2,
7,
8,
]
lr = ncon([c, t, c, c, c, t], connects, cont_order)
connects = [[1, 10], [2, 14, 17, 1], [11, 4], [3, 13], [12, 2],
[3, 15, 16, 4], [13, 8, 9, 12], [10, 6, 7, 11],
[5, 6, 14, 8, 15], [5, 7, 17, 9, 16]]
cont_order = [
4,
3,
11,
1,
2,
12,
16,
7,
10,
17,
13,
9,
14,
8,
15,
6,
5,
]
ltr = ncon([c, t, c, c, c, t, t, t, a, np.conj(a)], connects, cont_order)
connects = [[1, 15], [2, 21, 26, 1], [17, 4], [3, 20], [18, 2],
[3, 23, 24, 4], [19, 8, 9, 18], [15, 6, 7, 16],
[5, 6, 21, 8, 22], [5, 7, 26, 9, 25], [20, 13, 14, 19],
[16, 11, 12, 17], [10, 11, 22, 13, 23], [10, 12, 25, 14, 24]]
cont_order = [
18,
15,
20,
1,
3,
7,
26,
17,
14,
24,
13,
23,
10,
4,
11,
12,
19,
16,
2,
9,
25,
6,
21,
5,
22,
8,
]
lttr = ncon([c, t, c, c, c, t, t, t, a,
np.conj(a), t, t, a,
np.conj(a)], connects, cont_order)
return abs(lttr / lr) - (ltr / lr)**2