def get_aXa(a, op):
# a - on-site tensor
# op - operator
dims_a = a.size()
dims_op = None if op is None else op.size()
if op is None:
# identity
A = einsum('nefgh,nabcd->eafbgchd', a, conj(a))
A = view(contiguous(A),
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 2:
# one-site operator
A = einsum('nefgh,nabcd->eafbgchd', einsum('mefgh,mn->nefgh', a,
op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 3:
# edge operators of some MPO within the transfer matrix
#
# 0 0
# | |
# op--2 ... or ... 2--op
# | |
# 1 1
#
# assume the last index of the op is the MPO dimension.
# It will become the last index of the resulting edge
A = einsum('nefghl,nabcd->eafbgchdl',
einsum('mefgh,mnl->nefghl', a, op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2, -1))
if verbosity > 0: print(f"aXa {A.size()}")
return A
def get_aXa(a, op):
# a - on-site tensor
# op - operator
dims_a = a.size()
dims_op = None if op is None else op.size()
if op is None:
# identity
A = einsum('nefgh,nabcd->eafbgchd', a, conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 2:
# one-site operator
A = einsum('nefgh,nabcd->eafbgchd', einsum('mefgh,mn->nefgh', a,
op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 3:
# edge operators of some MPO
#
# 0 0
# | |
# op--2 ... or ... 2--op
# | |
# 1 1
#
# assume the last index of the op is the MPO dimension.
# It will become the last index of the resulting edge
A = einsum('nefghl,nabcd->eafbgchdl',
einsum('mefgh,mnl->nefghl', a, op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2, -1))
elif len(dims_op) == 4:
# intermediate op of some MPO
#
# 0
# |
# ... 2--op--3 ...
# |
# 1
#
# assume the index 2 is to be contracted with extra (MPO) index
# of edge. The remaining index 3 becomes last index resulting edge
A = einsum('nefghlk,nabcd->eafbgchdlk',
einsum('mefgh,mnlk->nefghlk', a, op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2, dims_op[2], dims_op[3]))
if verbosity > 0: print(f"aXa {A.size()}")
return A
def c2x2_RD_c(*tensors):
C, T1, T2, A = tensors
# 1<-0 0
# 2<-1--T1--2 1--C
C2x2 = contract(C, T1, ([1], [2]))
# 2<-0
# 3<-1--T2
# 2
# 0<-1 0
# 1<-2--T1---C
C2x2 = contract(C2x2, T2, ([0], [2]))
# 2<-0 1<-2
# 3<-1--A--3 3--T2
# 2 |
# 0 |
# 0<-1--T1------C
C2x2 = contract(C2x2, A, ([0, 3], [2, 3]))
# permute 0123->1203
# reshape (12)(03)->01
C2x2 = contiguous(permute(C2x2, (1, 2, 0, 3)))
C2x2 = view(C2x2, (T2.size(0) * A.size(0), T1.size(1) * A.size(1)))
# 0
# |
# 1--C2x2
return C2x2
def c2x2_LU_c(*tensors):
C, T1, T2, A = tensors
# C--10--T1--2
# 0 1
C2x2 = contract(C, T1, ([1], [0]))
# C------T1--2->1
# 0 1->0
# 0
# T2--2->3
# 1->2
C2x2 = contract(C2x2, T2, ([0], [0]))
# C-------T1--1->0
# | 0
# | 0
# T2--3 1 A--3
# 2->1 2
C2x2 = contract(C2x2, A, ([0, 3], [0, 1]))
# permute 0123->1203
# reshape (12)(03)->01
C2x2 = contiguous(permute(C2x2, (1, 2, 0, 3)))
C2x2 = view(C2x2, (T2.size(1) * A.size(2), T1.size(2) * A.size(3)))
# C2x2--1
# |
# 0
return C2x2
def c2x2_RU_c(*tensors):
C, T1, T2, A = tensors
# 0--C
# 1
# 0
# 1--T1
# 2
C2x2 = contract(C, T1, ([1], [0]))
# 2<-0--T2--2 0--C
# 3<-1 |
# 0<-1--T1
# 1<-2
C2x2 = contract(C2x2, T2, ([0], [2]))
# 1<-2--T2------C
# 3 |
# 0 |
# 2<-1--A--3 0--T1
# 3<-2 0<-1
C2x2 = contract(C2x2, A, ([0, 3], [3, 0]))
# permute 0123->1203
# reshape (12)(03)->01
C2x2 = contiguous(permute(C2x2, (1, 2, 0, 3)))
C2x2 = view(C2x2, (T2.size(0) * A.size(1), T1.size(2) * A.size(2)))
# 0--C2x2
# |
# 1
return C2x2
def c2x2_LD_c(*tensors):
C, T1, T2, A = tensors
# 0->1
# T1--2
# 1
# 0
# C--1->0
C2x2 = contract(C, T1, ([0], [1]))
# 1->0
# T1--2->1
# |
# | 0->2
# C--0 1--T1--2->3
C2x2 = contract(C2x2, T2, ([0], [1]))
# 0 0->2
# T1--1 1--A--3
# | 2
# | 2
# C--------T2--3->1
C2x2 = contract(C2x2, A, ([1, 2], [1, 2]))
# permute 0123->0213
# reshape (02)(13)->01
C2x2 = contiguous(permute(C2x2, (0, 2, 1, 3)))
C2x2 = view(C2x2, (T1.size(0) * A.size(0), T2.size(2) * A.size(3)))
# 0
# |
# C2x2--1
return C2x2
def rdm2x1(coord, state, env, sym_pos_def=False, verbosity=0):
r"""
:param coord: vertex (x,y) specifies position of 2x1 subsystem
:param state: underlying wavefunction
:param env: environment corresponding to ``state``
:param verbosity: logging verbosity
:type coord: tuple(int,int)
:type state: IPEPS
:type env: ENV
:type verbosity: int
:return: 2-site reduced density matrix with indices :math:`s_0s_1;s'_0s'_1`
:rtype: torch.tensor
Computes 2-site reduced density matrix :math:`\rho_{2x1}` of a horizontal
2x1 subsystem using following strategy:
1. compute four individual corners
2. construct right and left half of the network
3. contract right and left halt to obtain final reduced density matrix
::
C--T------------T------------------C = C2x2_LU(coord)--C2x2(coord+(1,0))
| | | | | |
T--A^+A(coord)--A^+A(coord+(1,0))--T C2x1_LD(coord)--C2x1(coord+(1,0))
| | | |
C--T------------T------------------C
The physical indices `s` and `s'` of on-sites tensors :math:`A` (and :math:`A^\dagger`)
at vertices ``coord``, ``coord+(1,0)`` are left uncontracted
"""
who="rdm2x1"
#----- building C2x2_LU ----------------------------------------------------
C = env.C[(state.vertexToSite(coord),(-1,-1))]
T1 = env.T[(state.vertexToSite(coord),(0,-1))]
T2 = env.T[(state.vertexToSite(coord),(-1,0))]
dimsA = state.site(coord).size()
a = einsum('mefgh,nabcd->eafbgchdmn',state.site(coord),conj(state.site(coord)))
a = view(contiguous(a),\
(dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# C--10--T1--2
# 0 1
C2x2_LU =contract(C, T1, ([1],[0]))
# C------T1--2->1
# 0 1->0
# 0
# T2--2->3
# 1->2
C2x2_LU =contract(C2x2_LU, T2, ([0],[0]))
# C-------T1--1->0
# | 0
# | 0
# T2--3 1 a--3
# 2->1 2\45
C2x2_LU =contract(C2x2_LU, a, ([0,3],[0,1]))
# permute 012345->120345
# reshape (12)(03)45->0123
# C2x2--1
# |\23
# 0
C2x2_LU= permute(C2x2_LU, (1,2,0,3,4,5))
C2x2_LU= view(contiguous(C2x2_LU), \
(T2.size(1)*a.size(2),T1.size(2)*a.size(3),dimsA[0],dimsA[0]))
if verbosity>0:
print("C2X2 LU "+str(coord)+"->"+str(state.vertexToSite(coord))+" (-1,-1): "+str(C2x2_LU.size()))
#----- building C2x1_LD ----------------------------------------------------
C = env.C[(state.vertexToSite(coord),(-1,1))]
T2 = env.T[(state.vertexToSite(coord),(0,1))]
# 0 0->1
# C--1 1--T2--2
C2x1_LD=contract(C, T2, ([1],[1]))
# reshape (01)2->(0)1
# 0
# |
# C2x1--1
C2x1_LD= view(contiguous(C2x1_LD), (C.size(0)*T2.size(0),T2.size(2)))
if verbosity>0:
print("C2X1 LD "+str(coord)+"->"+str(state.vertexToSite(coord))+" (-1,1): "+str(C2x1_LD.size()))
#----- build left part C2x2_LU--C2x1_LD ------------------------------------
# C2x2_LU--1
# |\23
# 0
# 0
# C2x1_LD--1->0
# TODO is it worthy(performance-wise) to instead overwrite one of C2x2_LU,C2x2_RU ?
left_half= contract(C2x1_LD, C2x2_LU, ([0],[0]))
#----- building C2x2_RU ----------------------------------------------------
vec = (1,0)
shitf_coord = state.vertexToSite((coord[0]+vec[0],coord[1]+vec[1]))
C = env.C[(shitf_coord,(1,-1))]
T1 = env.T[(shitf_coord,(1,0))]
T2 = env.T[(shitf_coord,(0,-1))]
dimsA = state.site(shitf_coord).size()
a= einsum('mefgh,nabcd->eafbgchdmn',state.site(shitf_coord),conj(state.site(shitf_coord)))
a= view(contiguous(a), \
(dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# 0--C
# 1
# 0
# 1--T1
# 2
C2x2_RU =contract(C, T1, ([1],[0]))
# 2<-0--T2--2 0--C
# 3<-1 |
# 0<-1--T1
# 1<-2
C2x2_RU =contract(C2x2_RU, T2, ([0],[2]))
# 1<-2--T2------C
# 3 |
# 45\0 |
# 2<-1--a--3 0--T1
# 3<-2 0<-1
C2x2_RU =contract(C2x2_RU, a, ([0,3],[3,0]))
# permute 012334->120345
# reshape (12)(03)45->0123
# 0--C2x2
# 23/|
# 1
C2x2_RU= permute(C2x2_RU, (1,2,0,3,4,5))
C2x2_RU= view(contiguous(C2x2_RU), \
(T2.size(0)*a.size(1),T1.size(2)*a.size(2), dimsA[0], dimsA[0]))
if verbosity>0:
print("C2X2 RU "+str((coord[0]+vec[0],coord[1]+vec[1]))+"->"+str(shitf_coord)+" (1,-1): "+str(C2x2_RU.size()))
#----- building C2x1_RD ----------------------------------------------------
C = env.C[(shitf_coord,(1,1))]
T1 = env.T[(shitf_coord,(0,1))]
# 1<-0 0
# 2<-1--T1--2 1--C
C2x1_RD =contract(C, T1, ([1],[2]))
# reshape (01)2->(0)1
C2x1_RD = view(contiguous(C2x1_RD), (C.size(0)*T1.size(0),T1.size(1)))
# 0
# |
# 1--C2x1
if verbosity>0:
print("C2X1 RD "+str((coord[0]+vec[0],coord[1]+vec[1]))+"->"+str(shitf_coord)+" (1,1): "+str(C2x1_RD.size()))
#----- build right part C2x2_RU--C2x1_RD -----------------------------------
# 1<-0--C2x2_RU
# |\23
# 1
# 0
# 0<-1--C2x1_RD
right_half =contract(C2x1_RD, C2x2_RU, ([0],[1]))
# construct reduced density matrix by contracting left and right halfs
# C2x2_LU--1 1----C2x2_RU
# |\23->01 |\23
# | |
# C2x1_LD--0 0----C2x1_RD
rdm =contract(left_half,right_half,([0,1],[0,1]))
# permute into order of s0,s1;s0',s1' where primed indices
# represent "ket"
# 0123->0213
# symmetrize and normalize
rdm = contiguous(permute(rdm, (0,2,1,3)))
rdm= _sym_pos_def_rdm(rdm, sym_pos_def=sym_pos_def, verbosity=verbosity, who=who)
return rdm
def apply_TM_2sO_1sChannel(coord,
direction,
state,
env,
edge,
op=None,
verbosity=0):
r"""
:param coord: tuple (x,y) specifying vertex on a square lattice
:param direction: direction in which the transfer operator is applied
:param state: underlying wavefunction
:param env: environment corresponding to ``state``
:param edge: tensor of dimensions :math:`\chi \times D^2 \times \chi`
:param op: two-site operator to be inserted into the two consecutive
transfer matrices
:param verbosity: logging verbosity
:type coord: tuple(int,int)
:type direction: tuple(int,int)
:type state: IPEPS
:type env: ENV
:type edge: torch.tensor
:type op: torch.tensor
:type verbosity: int
:return: ``edge`` with two transfer matrices (and operator ``op``, if any) applied.
The resulting tensor has an identical index structure as the
original ``edge``
:rtype: torch.tensor
Applies two transfer matrices to the ``edge`` tensor, including the two-site operator
``op``. The applied transfer matrices depend on the selected site r=(x,y) and
the ``direction`` of the growth. The following network is contracted::
-----T-------------------T--------------------------
| | |
edge--(a(r)^+ op_l a(r))==(a(r+dir)^+ op_r a(r+dir))--
| | |
-----T-------------------T--------------------------
where the physical indices `s` and `s'` of the on-site tensor :math:`a`
and it's hermitian conjugate :math:`a^\dagger` are contracted with
identity :math:`\delta_{s,s'}` or ``op_l`` and ``op_r`` if ``op`` is supplied.
The ``op_l`` and ``op_r`` are given by the SVD decomposition of the two-site operator
``op``::
0 1 0 1 0 1->0
| | SVD | | | |
| op | = |op_l|--(S--|op^~_r|) = |op_l|--2 2--|op_r|
| | | | | |
2 3 2 3 2->1 3->1
"""
# TODO stronger verification
if op is not None:
assert (len(op.size()) == 4)
# pre-process ``op``
# TODO possibly truncate/compress according to the vanishingly small singular values
dims_op = op.size()
op_mat = view(contiguous(permute(op, (0, 2, 1, 3))),
(dims_op[0]**2, dims_op[0]**2))
op_l, s, op_r = truncated_svd_gesdd(op_mat, op_mat.size(0))
op_l = view(op_l, (dims_op[0], dims_op[0], s.size()[0]))
op_r = conj(transpose(op_r)) * s[:, None]
op_r = contiguous(
permute(view(op_r, (s.size()[0], dims_op[0], dims_op[0])),
(1, 2, 0)))
else:
op_l = None
op_r = None
E = apply_TM_1sO(coord,
direction,
state,
env,
edge,
op=op_l,
verbosity=verbosity)
c1 = (coord[0] + direction[0], coord[1] + direction[1])
E = apply_TM_1sO(c1,
direction,
state,
env,
E,
op=op_r,
verbosity=verbosity)
return E
def apply_TM_2sO_2sChannel(coord,
direction,
state,
env,
edge,
op=None,
verbosity=0):
r"""
:param coord: tuple (x,y) specifying vertex on a square lattice
:param direction: direction in which the transfer operator is applied
:param state: underlying wavefunction
:param env: environment corresponding to ``state``
:param edge: tensor of dimensions :math:`\chi \times (D^2)^2 \times \chi`
:param op: two-site operator to be inserted within the two-site transfer matrix
:param verbosity: logging verbosity
:type coord: tuple(int,int)
:type direction: tuple(int,int)
:type state: IPEPS
:type env: ENV
:type edge: torch.tensor
:type op: torch.tensor
:type verbosity: int
:return: ``edge`` with a single instance of the transfer matrix applied
The resulting tensor has an identical index structure as the
original ``edge``
:rtype: torch.tensor
Applies a single instance of the two-site "transfer matrix" with site r=(x,y) to
the ``edge`` tensor by contracting the following network, or its corresponding
rotation depending on the ``direction``::
direction: right=(1,0) down=(0,1)
-----T-------------------- --------edge-------------
| | | | | |
edge--(a(r)^+ o1 a(r))------ T--(a(r)^+)---(a(r+x)^+)--T
| | \ |\ o1---------o2 |
|----(a(r+y)^+) o2 a(r+y)-- | --a(r)-------a(r+x)-----T
| | | | | |
-----T--------------------
The two-site operator is first decomposed into a simple MPO o1--o2
(TODO case where op comes with extra MPO index)::
s1' s2' s1' s2'
| op | = |o1|-----|o2|
s1 s2 s1 s2
where the physical indices `s` and `s'` of the on-site tensor :math:`a`
and it's hermitian conjugate :math:`a^\dagger` are contracted with
identity :math:`\delta_{s,s'}` or ``o1``, ``o2``. The transfer matrix
is always grown from left to right or from top to bottom.
"""
# TODO stronger verification
op_1, op_2 = None, None
if op is not None:
if len(op.size()) == 4:
# pre-process ``op``
# TODO possibly truncate/compress according to the vanishingly small singular values
dims_op = op.size()
op_mat = view(contiguous(permute(op, (0, 2, 1, 3))),
(dims_op[0]**2, dims_op[0]**2))
op_1, s, op_2 = truncated_svd_gesdd(op_mat, op_mat.size(0))
op_1 = view(op_1, (dims_op[0], dims_op[0], s.size()[0]))
op_2 = conj(transpose(op_2))
op_2 = contiguous(
permute(view(op_2, (s.size()[0], dims_op[0], dims_op[0])),
(1, 2, 0)))
else:
raise ValueError(f"Invalid op: rank {op.size()}")
def shift_coord(c, d=(0, 0)):
c0 = (c[0] + d[0], c[1] + d[1])
return c0, state.vertexToSite(c0)
# Four basic cases of passed op
def get_aXa(a, op):
# a - on-site tensor
# op - operator
dims_a = a.size()
dims_op = None if op is None else op.size()
if op is None:
# identity
A = einsum('nefgh,nabcd->eafbgchd', a, conj(a))
A = view(contiguous(A),
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 2:
# one-site operator
A = einsum('nefgh,nabcd->eafbgchd', einsum('mefgh,mn->nefgh', a,
op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 3:
# edge operators of some MPO within the transfer matrix
#
# 0 0
# | |
# op--2 ... or ... 2--op
# | |
# 1 1
#
# assume the last index of the op is the MPO dimension.
# It will become the last index of the resulting edge
A = einsum('nefghl,nabcd->eafbgchdl',
einsum('mefgh,mnl->nefghl', a, op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2, -1))
if verbosity > 0: print(f"aXa {A.size()}")
return A
dir_r = (1, 0)
dir_b = (0, 1)
c0, c = shift_coord(coord)
if direction == (0, -1): #up
raise ValueError("Direction: " + str(direction) + "not implemented")
elif direction == (-1, 0): #left
raise ValueError("Direction: " + str(direction) + "not implemented")
elif direction == (0, 1): #down
T1 = env.T[(c, (-1, 0))]
# Assume index structure of ``edge`` tensor to be as follows
#
# --edge-- ------
# 0 1->2 2->3 3->4
# 0
# T1--2->1
# 1->0
E = contract(T1, edge, ([0], [0]))
if verbosity > 0: print("E=edgeT " + str(E.size()))
# TODO - more efficent contraction with uncontracted-double-layer on-site tensor
# Possibly reshape indices 1,2 of E, which are to be contracted with
# on-site tensor and contract bra,ket in two steps instead of creating
# double layer tensor
# /
# --A--
# /|s
# X
# s'|/
# --A--
# /
#
# where X is Id or op
a = state.site(c)
A = get_aXa(a, op_1)
# -------edge---- ------
# | 2 3->1 4->2
# | 0
# T1--1 1--A--3->4
# | |\
# 0 3<-2 (4->5)
E = contract(E, A, ([1, 2], [1, 0]))
if verbosity > 0: print("E=edgeTA " + str(E.size()))
c0, c = shift_coord(c0, dir_r)
a = state.site(c)
A = get_aXa(a, op_2)
# ----edge------ ------
# | | 1 2->1
# | | 0
# T1----A--4 1---A--3->4
# | |\ /|
# 0 2<-3 (5)(4) 2->3
E = contract(E,A,([1,4],[0,1])) if op is None else \
contract(E,A,([1,4,5],[0,1,4]))
if verbosity > 0: print("E=edgeTAA " + str(E.size()))
T2 = env.T[(c, (1, 0))]
# ----edge----- -------
# | | | 1
# | | | 0
# T1----A=======A--4 1--T2
# | | | 2->4
# 0 2->1 3->2
E = contract(E, T2, ([1, 4], [0, 1]))
if verbosity > 0: print("E=edgeTAAT " + str(E.size()))
elif direction == (1, 0): #right
T1 = env.T[(c, (0, -1))]
# Assume index structure of ``edge`` tensor to be as follows
#
# -- 0
# edge |-- 1
# |---2
# -- 3
#
# ----0 0--T1--2->1
# | 1->0
# edge--1->2
# |
# ----2->3
# |
# ----3->4
E = contract(T1, edge, ([0], [0]))
if verbosity > 0: print("E=edgeT " + str(E.size()))
# TODO - more efficent contraction with uncontracted-double-layer on-site tensor
# Possibly reshape indices 1,2 of E, which are to be contracted with
# on-site tensor and contract bra,ket in two steps instead of creating
# double layer tensor
# /
# --A--
# /|s
# X
# s'|/
# --A--
# /
#
# where X is Id or op
a = state.site(c)
A = get_aXa(a, op_1)
# ---------T1--1->0
# | 0
# | 0
# edge--2 1--A--3->4
# | 3<-2 \
# ----3->1 (4->5)
# |
# ----4->2
E = contract(E, A, ([0, 2], [0, 1]))
if verbosity > 0: print("E=edgeTA " + str(E.size()))
c0, c = shift_coord(c0, dir_b)
a = state.site(c)
A = get_aXa(a, op_2)
# ---------T1--0
# | |
# edge-------A--4->2
# | | \
# | 3 (5)
# | 0 (4)
# | | /
# ----1 1--A--2->3
# | 3->4
# ----2->1
E = contract(E,A,([1,3],[1,0])) if op is None else \
contract(E,A,([1,3,5],[1,0,4]))
if verbosity > 0: print("E=edgeTAA " + str(E.size()))
T2 = env.T[(c, (0, 1))]
# ---------T1--0
# | |
# edge-------A--2->1
# | |
# ---------A--3->2
# | 3
# | 0
# ----1 1--T2--2->3
E = contract(E, T2, ([1, 3], [1, 0]))
if verbosity > 0: print("E=edgeTAAT " + str(E.size()))
else:
raise ValueError("Invalid direction: " + str(direction))
return E
def absorb_truncate_CTM_MOVE_UP_c(*tensors):
C1, T1, T, T2, C2, A, P2, Pt2, P1, Pt1 = tensors
# 0--C1
# 1
# 0
# 1--T1
# 2
nC1 = contract(C1, T1, ([1], [0]))
# --0 0--C1
# | |
# 0<-2--Pt1 |
# | |
# --1 1--T1
# 2->1
nC1 = contract(Pt1, nC1, ([0, 1], [0, 1]))
# C2--1->0
# 0
# 0
# T2--2
# 1
nC2 = contract(C2, T2, ([0], [0]))
# C2--0 0--
# | |
# | P2--2->1
# | |
# T2--2 1--
# 1->0
nC2 = contract(nC2, P2, ([0, 2], [0, 1]))
# --0 0--T--2->3
# | 1->2
# 1<-2--Pt2
# |
# --1->0
nT = contract(Pt2, T, ([0], [0]))
# -------T--3->1
# | 2
# 0<-1--Pt2 |
# | 0
# --0 1--A--3
# 2
nT = contract(nT, A, ([0, 2], [1, 0]))
# -------T--1 0--
# | | |
# 0--Pt2 | P1--2
# | | |
# -------A--3 1--
# 2->1
nT = contract(nT, P1, ([1, 3], [0, 1]))
nT = contiguous(nT)
# Assign new C,T
#
# C(coord,(-1,-1))-- --T(coord,(0,-1))-- --C(coord,(1,-1))
# | P2-- --Pt2 | P1-- -Pt1 |
# T(coord,(-1,0))--- --A(coord)--------- --T(coord,(1,0))
# | | |
#
# =>
#
# C^new(coord+(0,1),(-1,-1))-- --T^new(coord+(0,1),(0,-1))-- --C^new(coord+(0,1),(1,-1))
# | | |
nC1 = nC1 / nC1.abs().max()
nC2 = nC2 / nC2.abs().max()
nT = nT / nT.abs().max()
return nC1, nC2, nT
def rdm2x2(coord, state, env, sym_pos_def=False, verbosity=0):
r"""
:param coord: vertex (x,y) specifies upper left site of 2x2 subsystem
:param state: underlying wavefunction
:param env: environment corresponding to ``state``
:param verbosity: logging verbosity
:type coord: tuple(int,int)
:type state: IPEPS
:type env: ENV
:type verbosity: int
:return: 4-site reduced density matrix with indices :math:`s_0s_1s_2s_3;s'_0s'_1s'_2s'_3`
:rtype: torch.tensor
Computes 4-site reduced density matrix :math:`\rho_{2x2}` of 2x2 subsystem specified
by the vertex ``coord`` of its upper left corner using strategy:
1. compute four individual corners
2. construct upper and lower half of the network
3. contract upper and lower half to obtain final reduced density matrix
::
C--T------------------T------------------C = C2x2_LU(coord)--------C2x2(coord+(1,0))
| | | | | |
T--A^+A(coord)--------A^+A(coord+(1,0))--T C2x2_LD(coord+(0,1))--C2x2(coord+(1,1))
| | | |
T--A^+A(coord+(0,1))--A^+A(coord+(1,1))--T
| | | |
C--T------------------T------------------C
The physical indices `s` and `s'` of on-sites tensors :math:`A` (and :math:`A^\dagger`)
at vertices ``coord``, ``coord+(1,0)``, ``coord+(0,1)``, and ``coord+(1,1)`` are
left uncontracted and given in the same order::
s0 s1
s2 s3
"""
who= "rdm2x2"
#----- building C2x2_LU ----------------------------------------------------
C = env.C[(state.vertexToSite(coord),(-1,-1))]
T1 = env.T[(state.vertexToSite(coord),(0,-1))]
T2 = env.T[(state.vertexToSite(coord),(-1,0))]
dimsA = state.site(coord).size()
a = contiguous(einsum('mefgh,nabcd->eafbgchdmn',state.site(coord),conj(state.site(coord))))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# C--10--T1--2
# 0 1
C2x2_LU = contract(C, T1, ([1],[0]))
# C------T1--2->1
# 0 1->0
# 0
# T2--2->3
# 1->2
C2x2_LU = contract(C2x2_LU, T2, ([0],[0]))
# C-------T1--1->0
# | 0
# | 0
# T2--3 1 a--3
# 2->1 2\45
C2x2_LU = contract(C2x2_LU, a, ([0,3],[0,1]))
# permute 012345->120345
# reshape (12)(03)45->0123
# C2x2--1
# |\23
# 0
C2x2_LU = contiguous(permute(C2x2_LU,(1,2,0,3,4,5)))
C2x2_LU = view(C2x2_LU, (T2.size(1)*a.size(2),T1.size(2)*a.size(3),dimsA[0],dimsA[0]))
if verbosity>0:
print("C2X2 LU "+str(coord)+"->"+str(state.vertexToSite(coord))+" (-1,-1): "+str(C2x2_LU.size()))
#----- building C2x2_RU ----------------------------------------------------
vec = (1,0)
shitf_coord = state.vertexToSite((coord[0]+vec[0],coord[1]+vec[1]))
C = env.C[(shitf_coord,(1,-1))]
T1 = env.T[(shitf_coord,(1,0))]
T2 = env.T[(shitf_coord,(0,-1))]
dimsA = state.site(shitf_coord).size()
a = contiguous(einsum('mefgh,nabcd->eafbgchdmn',state.site(shitf_coord),conj(state.site(shitf_coord))))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# 0--C
# 1
# 0
# 1--T1
# 2
C2x2_RU = contract(C, T1, ([1],[0]))
# 2<-0--T2--2 0--C
# 3<-1 |
# 0<-1--T1
# 1<-2
C2x2_RU = contract(C2x2_RU, T2, ([0],[2]))
# 1<-2--T2------C
# 3 |
# 45\0 |
# 2<-1--a--3 0--T1
# 3<-2 0<-1
C2x2_RU = contract(C2x2_RU, a, ([0,3],[3,0]))
# permute 012334->120345
# reshape (12)(03)45->0123
# 0--C2x2
# 23/|
# 1
C2x2_RU = contiguous(permute(C2x2_RU, (1,2,0,3,4,5)))
C2x2_RU = view(C2x2_RU, (T2.size(0)*a.size(1),T1.size(2)*a.size(2), dimsA[0], dimsA[0]))
if verbosity>0:
print("C2X2 RU "+str((coord[0]+vec[0],coord[1]+vec[1]))+"->"+str(shitf_coord)+" (1,-1): "+str(C2x2_RU.size()))
#----- build upper part C2x2_LU--C2x2_RU -----------------------------------
# C2x2_LU--1 0--C2x2_RU C2x2_LU------C2x2_RU
# |\23->12 |\23->45 & permute |\12->23 |\45
# 0 1->3 0 3->1
# TODO is it worthy(performance-wise) to instead overwrite one of C2x2_LU,C2x2_RU ?
upper_half = contract(C2x2_LU, C2x2_RU, ([1],[0]))
upper_half = permute(upper_half, (0,3,1,2,4,5))
#----- building C2x2_RD ----------------------------------------------------
vec = (1,1)
shitf_coord = state.vertexToSite((coord[0]+vec[0],coord[1]+vec[1]))
C = env.C[(shitf_coord,(1,1))]
T1 = env.T[(shitf_coord,(0,1))]
T2 = env.T[(shitf_coord,(1,0))]
dimsA = state.site(shitf_coord).size()
a = contiguous(einsum('mefgh,nabcd->eafbgchdmn',state.site(shitf_coord),conj(state.site(shitf_coord))))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# 1<-0 0
# 2<-1--T1--2 1--C
C2x2_RD = contract(C, T1, ([1],[2]))
# 2<-0
# 3<-1--T2
# 2
# 0<-1 0
# 1<-2--T1---C
C2x2_RD = contract(C2x2_RD, T2, ([0],[2]))
# 2<-0 1<-2
# 3<-1--a--3 3--T2
# 2\45 |
# 0 |
# 0<-1--T1------C
C2x2_RD = contract(C2x2_RD, a, ([0,3],[2,3]))
# permute 012345->120345
# reshape (12)(03)45->0123
C2x2_RD = contiguous(permute(C2x2_RD, (1,2,0,3,4,5)))
C2x2_RD = view(C2x2_RD, (T2.size(0)*a.size(0),T1.size(1)*a.size(1), dimsA[0], dimsA[0]))
# 0
# |/23
# 1--C2x2
if verbosity>0:
print("C2X2 RD "+str((coord[0]+vec[0],coord[1]+vec[1]))+"->"+str(shitf_coord)+" (1,1): "+str(C2x2_RD.size()))
#----- building C2x2_LD ----------------------------------------------------
vec = (0,1)
shitf_coord = state.vertexToSite((coord[0]+vec[0],coord[1]+vec[1]))
C = env.C[(shitf_coord,(-1,1))]
T1 = env.T[(shitf_coord,(-1,0))]
T2 = env.T[(shitf_coord,(0,1))]
dimsA = state.site(shitf_coord).size()
a = contiguous(einsum('mefgh,nabcd->eafbgchdmn',state.site(shitf_coord),conj(state.site(shitf_coord))))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# 0->1
# T1--2
# 1
# 0
# C--1->0
C2x2_LD = contract(C, T1, ([0],[1]))
# 1->0
# T1--2->1
# |
# | 0->2
# C--0 1--T2--2->3
C2x2_LD = contract(C2x2_LD, T2, ([0],[1]))
# 0 0->2
# T1--1 1--a--3
# | 2\45
# | 2
# C--------T2--3->1
C2x2_LD = contract(C2x2_LD, a, ([1,2],[1,2]))
# permute 012345->021345
# reshape (02)(13)45->0123
# 0
# |/23
# C2x2--1
C2x2_LD = contiguous(permute(C2x2_LD, (0,2,1,3,4,5)))
C2x2_LD = view(C2x2_LD, (T1.size(0)*a.size(0),T2.size(2)*a.size(3), dimsA[0], dimsA[0]))
if verbosity>0:
print("C2X2 LD "+str((coord[0]+vec[0],coord[1]+vec[1]))+"->"+str(shitf_coord)+" (-1,1): "+str(C2x2_LD.size()))
#----- build lower part C2x2_LD--C2x2_RD -----------------------------------
# 0 0->3 0 3->1
# |/23->12 |/23->45 & permute |/12->23 |/45
# C2x2_LD--1 1--C2x2_RD C2x2_LD------C2x2_RD
# TODO is it worthy(performance-wise) to instead overwrite one of C2x2_LD,C2x2_RD ?
lower_half = contract(C2x2_LD, C2x2_RD, ([1],[1]))
lower_half = permute(lower_half, (0,3,1,2,4,5))
# construct reduced density matrix by contracting lower and upper halfs
# C2x2_LU------C2x2_RU
# |\23->01 |\45->23
# 0 1
# 0 1
# |/23->45 |/45->67
# C2x2_LD------C2x2_RD
rdm = contract(upper_half,lower_half,([0,1],[0,1]))
# permute into order of s0,s1,s2,s3;s0',s1',s2',s3' where primed indices
# represent "ket"
# 01234567->02461357
# symmetrize and normalize
rdm= contiguous(permute(rdm, (0,2,4,6,1,3,5,7)))
rdm= _sym_pos_def_rdm(rdm, sym_pos_def=sym_pos_def, verbosity=verbosity, who=who)
return rdm
def absorb_truncate_CTM_MOVE_RIGHT_c(*tensors):
C1, T1, T, T2, C2, A, P2, Pt2, P1, Pt1 = tensors
# 0->1 0
# 2<-1--T1--2 1--C1
nC1 = contract(C1, T1, ([1], [2]))
# 2->0
# __Pt1_
# 1 0
# 1 0
# 1<-2--T1----C1
nC1 = contract(Pt1, nC1, ([0, 1], [0, 1]))
# 1<-0--T2--2 0--C2
# 2<-1 0<-1
nC2 = contract(C2, T2, ([0], [2]))
# 0<-1--T2----C2
# 2 0
# 1 0
# |__P2_|
# 2->1
nC2 = contract(nC2, P2, ([0, 2], [0, 1]))
# 1<-2
# ___Pt2__
# 0<-1 0
# 0
# 2<-1--T
# 3<-2
nT = contract(Pt2, T, ([0], [0]))
# 0<-1
# ___Pt2__
# 0 |
# 0 |
# 2<-1--A--3 2--T
# 3<-2 1<-3
nT = contract(nT, A, ([0, 2], [0, 3]))
# 0
# ___Pt2__
# | |
# | |
# 1<-2--A-------T
# 3 1
# 1 0
# |___P1__|
# 2
nT = contract(nT, P1, ([1, 3], [0, 1]))
nT = contiguous(nT)
# Assign new C,T
#
# --T(coord,(0,-1))--C(coord,(1,-1)) =>--C^new(coord+(-1,0),(1,-1))
# |______ ________| |
# P2
# |
#
# |
# ______Pt2
# | | |
# --A(coord)--T(coord,(1,0)) --T^new(coord+(-1,0),(1,0))
# |______ _| |
# P1
# |
#
# |
# ______Pt1______
# | | |
# --T(coord,(0,1))--C(coord,(1,1)) --C^new(coord+(-1,0),(1,1))
nC1 = nC1 / nC1.abs().max()
nC2 = nC2 / nC2.abs().max()
nT = nT / nT.abs().max()
return nC1, nC2, nT
def absorb_truncate_CTM_MOVE_DOWN_c(*tensors):
C1, T1, T, T2, C2, A, P2, Pt2, P1, Pt1 = tensors
# 0->1
# T1--2->2
# 1
# 0
# C1--1->0
nC1 = contract(C1, T1, ([0], [1]))
# 1->0
# T1--2 1--
# | |
# | Pt1--2->1
# | |
# C1--0 0--
nC1 = contract(nC1, Pt1, ([0, 2], [0, 1]))
# 1<-0
# 2<-1--T2
# 2
# 0
# 0<-1--C2
nC2 = contract(C2, T2, ([0], [2]))
# 0<-1
# --1 2--T2
# | |
# 1<-2--P2 |
# | |
# --0 0--C2
nC2 = contract(nC2, P2, ([0, 2], [0, 1]))
# --1->0
# |
# 1<-2--P1
# | 0->2
# --0 1--T--2->3
nT = contract(P1, T, ([0], [1]))
# 0->2
# --0 1--A--3
# | 2
# 0<-1--P1 |
# | 2
# -------T--3->1
nT = contract(nT, A, ([0, 2], [1, 2]))
# 2->1
# -------A--3 1--
# | | |
# 0--P1 | Pt2--2
# | | |
# -------T--1 0--
nT = contract(nT, Pt2, ([1, 3], [0, 1]))
nT = contiguous(permute(nT, (1, 0, 2)))
# Assign new C,T
#
# | | |
# T(coord,(-1,0))-- --A(coord)-------- --T(coord,(1,0))
# | Pt1-- --P1 | Pt2-- --P2 |
# C(coord,(-1,1))-- --T(coord,(0,1))-- --C(coord,(1,1))
#
# =>
#
# | | |
# C^new(coord+(0,-1),(-1,1))-- --T^new(coord+(0,-1),(0,1))-- --C^new(coord+(0,-1),(1,1))
nC1 = nC1 / nC1.abs().max()
nC2 = nC2 / nC2.abs().max()
nT = nT / nT.abs().max()
return nC1, nC2, nT
def absorb_truncate_CTM_MOVE_LEFT_c(*tensors):
C1, T1, T, T2, C2, A, P2, Pt2, P1, Pt1 = tensors
# C1--1 0--T1--2
# | |
# 0 1
nC1 = contract(C1, T1, ([1], [0]))
# C1--1 0--T1--2->1
# | |
# 0 1
# 0 1
# |___Pt1__|
# 2->0
nC1 = contract(Pt1, nC1, ([0, 1], [0, 1]))
# 0 0->1
# C2--1 1--T2--2
nC2 = contract(C2, T2, ([1], [1]))
# 2->0
# ___P2___
# 0 1
# 0 1
# C2-----T2--2->1
nC2 = contract(P2, nC2, ([0, 1], [0, 1]))
# 2->1
# ___P1__
# 0 1->0
# 0
# T--2->3
# 1->2
nT = contract(P1, T, ([0], [0]))
# 1->0
# ___P1____
# | 0
# | 0
# T--3 1--A--3
# 2->1 2
nT = contract(nT, A, ([0, 3], [0, 1]))
# 0
# ___P1___
# | |
# | |
# T-------A--3->1
# 1 2
# 0 1
# |___Pt2_|
# 2
nT = contract(nT, Pt2, ([1, 2], [0, 1]))
nT = contiguous(permute(nT, (0, 2, 1)))
# Assign new C,T
#
# C(coord,(-1,-1))--T(coord,(0,-1))-- => C^new(coord+(1,0),(-1,-1))--
# |________ ______| |
# Pt1
# |
#
# |
# _________P1______
# | | |
# T(coord,(-1,0))--A(coord)-- T^new(coord+(1,0),(-1,0))--
# |________ _____| |
# Pt2
# |
#
# |
# ________P2_______
# | | |
# C(coord,(-1,1))--T(coord,(0,1))-- C^new(coord+(1,0),(-1,1))
nC1 = nC1 / nC1.abs().max()
nC2 = nC2 / nC2.abs().max()
nT = nT / nT.abs().max()
return nC1, nC2, nT
def rdm1x2(coord, state, env, sym_pos_def=False, verbosity=0):
r"""
:param coord: vertex (x,y) specifies position of 1x2 subsystem
:param state: underlying wavefunction
:param env: environment corresponding to ``state``
:param verbosity: logging verbosity
:type coord: tuple(int,int)
:type state: IPEPS
:type env: ENV
:type verbosity: int
:return: 2-site reduced density matrix with indices :math:`s_0s_1;s'_0s'_1`
:rtype: torch.tensor
Computes 2-site reduced density matrix :math:`\rho_{1x2}` of a vertical
1x2 subsystem using following strategy:
1. compute four individual corners
2. construct upper and lower half of the network
3. contract upper and lower halt to obtain final reduced density matrix
::
C--T------------------C = C2x2_LU(coord)--------C1x2(coord)
| | | | |
T--A^+A(coord)--------T C2x2_LD(coord+(0,1))--C1x2(coord+0,1))
| | |
T--A^+A(coord+(0,1))--T
| | |
C--T------------------C
The physical indices `s` and `s'` of on-sites tensors :math:`A` (and :math:`A^\dagger`)
at vertices ``coord``, ``coord+(0,1)`` are left uncontracted
"""
who="rdm1x2"
#----- building C2x2_LU ----------------------------------------------------
C = env.C[(state.vertexToSite(coord),(-1,-1))]
T1 = env.T[(state.vertexToSite(coord),(0,-1))]
T2 = env.T[(state.vertexToSite(coord),(-1,0))]
dimsA = state.site(coord).size()
a= einsum('mefgh,nabcd->eafbgchdmn',state.site(coord), conj(state.site(coord)))
a= view(contiguous(a), \
(dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# C--10--T1--2
# 0 1
C2x2_LU =contract(C, T1, ([1],[0]))
# C------T1--2->1
# 0 1->0
# 0
# T2--2->3
# 1->2
C2x2_LU =contract(C2x2_LU, T2, ([0],[0]))
# C-------T1--1->0
# | 0
# | 0
# T2--3 1 a--3
# 2->1 2\45
C2x2_LU =contract(C2x2_LU, a, ([0,3],[0,1]))
# permute 012345->120345
# reshape (12)(03)45->0123
# C2x2--1
# |\23
# 0
C2x2_LU= permute(C2x2_LU, (1,2,0,3,4,5))
C2x2_LU= view(contiguous(C2x2_LU), \
(T2.size(1)*a.size(2),T1.size(2)*a.size(3),dimsA[0],dimsA[0]))
if verbosity>0:
print("C2X2 LU "+str(coord)+"->"+str(state.vertexToSite(coord))+" (-1,-1): "+str(C2x2_LU.size()))
#----- building C1x2_RU ----------------------------------------------------
C = env.C[(state.vertexToSite(coord),(1,-1))]
T1 = env.T[(state.vertexToSite(coord),(1,0))]
# 0--C
# 1
# 0
# 1--T1
# 2
C1x2_RU =contract(C, T1, ([1],[0]))
# reshape (01)2->(0)1
# 0--C1x2
# 23/|
# 1
C1x2_RU= view(contiguous(C1x2_RU), (C.size(0)*T1.size(1),T1.size(2)))
if verbosity>0:
print("C1X2 RU "+str(coord)+"->"+str(state.vertexToSite(coord))+" (1,-1): "+str(C1x2_RU.size()))
#----- build upper part C2x2_LU--C1x2_RU -----------------------------------
# C2x2_LU--1 0--C1x2_RU
# |\23 |
# 0->1 1->0
upper_half =contract(C1x2_RU, C2x2_LU, ([0],[1]))
#----- building C2x2_LD ----------------------------------------------------
vec = (0,1)
shitf_coord = state.vertexToSite((coord[0]+vec[0],coord[1]+vec[1]))
C = env.C[(shitf_coord,(-1,1))]
T1 = env.T[(shitf_coord,(-1,0))]
T2 = env.T[(shitf_coord,(0,1))]
dimsA = state.site(shitf_coord).size()
a= einsum('mefgh,nabcd->eafbgchdmn',state.site(shitf_coord),conj(state.site(shitf_coord)))
a= view(contiguous(a), \
(dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# 0->1
# T1--2
# 1
# 0
# C--1->0
C2x2_LD =contract(C, T1, ([0],[1]))
# 1->0
# T1--2->1
# |
# | 0->2
# C--0 1--T2--2->3
C2x2_LD =contract(C2x2_LD, T2, ([0],[1]))
# 0 0->2
# T1--1 1--a--3
# | 2\45
# | 2
# C--------T2--3->1
C2x2_LD =contract(C2x2_LD, a, ([1,2],[1,2]))
# permute 012345->021345
# reshape (02)(13)45->0123
# 0
# |/23
# C2x2--1
C2x2_LD= permute(C2x2_LD, (0,2,1,3,4,5))
C2x2_LD= view(contiguous(C2x2_LD), \
(T1.size(0)*a.size(0),T2.size(2)*a.size(3), dimsA[0], dimsA[0]))
if verbosity>0:
print("C2X2 LD "+str((coord[0]+vec[0],coord[1]+vec[1]))+"->"+str(shitf_coord)+" (-1,1): "+str(C2x2_LD.size()))
#----- building C2x2_RD ----------------------------------------------------
C = env.C[(shitf_coord,(1,1))]
T2 = env.T[(shitf_coord,(1,0))]
# 0
# 1--T2
# 2
# 0
# 2<-1--C
C1x2_RD =contract(T2, C, ([2],[0]))
# permute 012->021
# reshape 0(12)->0(1)
C1x2_RD = view(contiguous(permute(C1x2_RD,(0,2,1))), \
(T2.size()[0],C.size()[1]*T2.size()[1]))
# 0
# |
# 1--C1x2
if verbosity>0:
print("C1X2 RD "+str((coord[0]+vec[0],coord[1]+vec[1]))+"->"+str(shitf_coord)+" (1,1): "+str(C1x2_RD.size()))
#----- build lower part C2x2_LD--C1x2_RD -----------------------------------
# 0->1 0
# |/23 |
# C2x2_LD--1 1--C1x2_RD
lower_half =contract(C1x2_RD, C2x2_LD, ([1],[1]))
# construct reduced density matrix by contracting lower and upper halfs
# C2x2_LU------C1x2_RU
# |\23->01 |
# 1 0
# 1 0
# |/23 |
# C2x2_LD------C1x2_RD
rdm =contract(upper_half,lower_half,([0,1],[0,1]))
# permute into order of s0,s1;s0',s1' where primed indices
# represent "ket"
# 0123->0213
# symmetrize and normalize
rdm = contiguous(permute(rdm, (0,2,1,3)))
rdm= _sym_pos_def_rdm(rdm, sym_pos_def=sym_pos_def, verbosity=verbosity, who=who)
return rdm
def rdm1x1(coord, state, env, sym_pos_def=False, verbosity=0):
r"""
:param coord: vertex (x,y) for which reduced density matrix is constructed
:param state: underlying wavefunction
:param env: environment corresponding to ``state``
:param verbosity: logging verbosity
:type coord: tuple(int,int)
:type state: IPEPS
:type env: ENV
:type verbosity: int
:return: 1-site reduced density matrix with indices :math:`s;s'`
:rtype: torch.tensor
Computes 1-site reduced density matrix :math:`\rho_{1x1}` centered on vertex ``coord`` by
contracting the following tensor network::
C--T-----C
| | |
T--A^+A--T
| | |
C--T-----C
where the physical indices `s` and `s'` of on-site tensor :math:`A` at vertex ``coord``
and it's hermitian conjugate :math:`A^\dagger` are left uncontracted
"""
who= "rdm1x1"
# C(-1,-1)--1->0
# 0
# 0
# T(-1,0)--2
# 1
rdm = contract(env.C[(coord,(-1,-1))],env.T[(coord,(-1,0))],([0],[0]))
if verbosity>0:
print("rdm=CT "+str(rdm.size()))
# C(-1,-1)--0
# |
# T(-1,0)--2->1
# 1
# 0
# C(-1,1)--1->2
rdm = contract(rdm,env.C[(coord,(-1,1))],([1],[0]))
if verbosity>0:
print("rdm=CTC "+str(rdm.size()))
# C(-1,-1)--0
# |
# T(-1,0)--1
# | 0->2
# C(-1,1)--2 1--T(0,1)--2->3
rdm = contract(rdm,env.T[(coord,(0,1))],([2],[1]))
if verbosity>0:
print("rdm=CTCT "+str(rdm.size()))
# TODO - more efficent contraction with uncontracted-double-layer on-site tensor
# Possibly reshape indices 1,2 of rdm, which are to be contracted with
# on-site tensor and contract bra,ket in two steps instead of creating
# double layer tensor
# /
# --A--
# /|s
#
# s'|/
# --A--
# /
#
dimsA = state.site(coord).size()
a = contiguous(einsum('mefgh,nabcd->eafbgchdmn',state.site(coord),conj(state.site(coord))))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2, dimsA[0], dimsA[0]))
# C(-1,-1)--0
# |
# | 0->2
# T(-1,0)--1 1--a--3
# | 2\45(s,s')
# | 2
# C(-1,1)-------T(0,1)--3->1
rdm = contract(rdm,a,([1,2],[1,2]))
if verbosity>0:
print("rdm=CTCTa "+str(rdm.size()))
# C(-1,-1)--0 0--T(0,-1)--2->0
# | 1
# | 2
# T(-1,0)--------a--3->2
# | |\45->34(s,s')
# | |
# C(-1,1)--------T(0,1)--1
rdm = contract(env.T[(coord,(0,-1))],rdm,([0,1],[0,2]))
if verbosity>0:
print("rdm=CTCTaT "+str(rdm.size()))
# C(-1,-1)--T(0,-1)--0 0--C(1,-1)
# | | 1->0
# | |
# T(-1,0)---a--2
# | |\34(s,s')
# | |
# C(-1,1)---T(0,1)--0->1
rdm = contract(env.C[(coord,(1,-1))],rdm,([0],[0]))
if verbosity>0:
print("rdm=CTCTaTC "+str(rdm.size()))
# C(-1,-1)--T(0,-1)-----C(1,-1)
# | | 0
# | | 0
# T(-1,0)---a--2 1------T(1,0)
# | |\34->23(s,s') 2->0
# | |
# C(-1,1)---T(0,1)--1
rdm = contract(env.T[(coord,(1,0))],rdm,([0,1],[0,2]))
if verbosity>0:
print("rdm=CTCTaTCT "+str(rdm.size()))
# C(-1,-1)--T(0,-1)--------C(1,-1)
# | | |
# | | |
# T(-1,0)---a--------------T(1,0)
# | |\23->12(s,s') 0
# | | 0
# C(-1,1)---T(0,1)--1 1----C(1,1)
rdm = contract(rdm,env.C[(coord,(1,1))],([0,1],[0,1]))
if verbosity>0:
print("rdm=CTCTaTCTC "+str(rdm.size()))
# symmetrize and normalize
rdm= _sym_pos_def_rdm(rdm, sym_pos_def=sym_pos_def, verbosity=verbosity, who=who)
return rdm
def apply_TM_1sO(coord, direction, state, env, edge, op=None, verbosity=0):
r"""
:param coord: tuple (x,y) specifying vertex on a square lattice
:param direction: direction in which the transfer operator is applied
:param state: underlying wavefunction
:param env: environment corresponding to ``state``
:param edge: tensor of dimensions :math:`\chi \times D^2 \times \chi (\times d_{MPO})`,
potentially with 4th index beloning to some MPO
:param op: operator to be inserted into transfer matrix
:param verbosity: logging verbosity
:type coord: tuple(int,int)
:type direction: tuple(int,int)
:type state: IPEPS
:type env: ENV
:type edge: torch.tensor
:type op: torch.tensor
:type verbosity: int
:return: ``edge`` with a single instance of the transfer matrix applied
The resulting tensor has an identical index structure as the
original ``edge``
:rtype: torch.tensor
Applies a single instance of the "transfer matrix" of site r=(x,y) to
the ``edge`` tensor by contracting the following network, or its corresponding
rotation depending on the ``direction``::
direction: right=(1,0) down=(0,1)
-----T---------------- --------edge--------
| | | | |\
edge--(a(r)^+ op a(r))-- T--(a(r)^+ op a(r))--T \
| | | | | (MPO)
-----T----------------
(\-----------------MPO)
where the physical indices `s` and `s'` of the on-site tensor :math:`a`
and it's hermitian conjugate :math:`a^\dagger` are contracted with
identity :math:`\delta_{s,s'}` or ``op`` (if supplied). Potentially,
the ``edge`` can carry an extra MPO index.
"""
# Check if edge has MPO index
contains_mpo = len(edge.size()) == 4
# Four basic cases of passed op
def get_aXa(a, op):
# a - on-site tensor
# op - operator
dims_a = a.size()
dims_op = None if op is None else op.size()
if op is None:
# identity
A = einsum('nefgh,nabcd->eafbgchd', a, conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 2:
# one-site operator
A = einsum('nefgh,nabcd->eafbgchd', einsum('mefgh,mn->nefgh', a,
op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2))
elif len(dims_op) == 3:
# edge operators of some MPO
#
# 0 0
# | |
# op--2 ... or ... 2--op
# | |
# 1 1
#
# assume the last index of the op is the MPO dimension.
# It will become the last index of the resulting edge
A = einsum('nefghl,nabcd->eafbgchdl',
einsum('mefgh,mnl->nefghl', a, op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2, -1))
elif len(dims_op) == 4:
# intermediate op of some MPO
#
# 0
# |
# ... 2--op--3 ...
# |
# 1
#
# assume the index 2 is to be contracted with extra (MPO) index
# of edge. The remaining index 3 becomes last index resulting edge
A = einsum('nefghlk,nabcd->eafbgchdlk',
einsum('mefgh,mnlk->nefghlk', a, op), conj(a))
A= view(contiguous(A), \
(dims_a[1]**2, dims_a[2]**2, dims_a[3]**2, dims_a[4]**2, dims_op[2], dims_op[3]))
if verbosity > 0: print(f"aXa {A.size()}")
return A
c = state.vertexToSite(coord)
if direction == (0, -1): #up
T1 = env.T[(c, (-1, 0))]
# Assume index structure of ``edge`` tensor to be as follows
#
# 0
# T1--2->1
# 1
# 0 1->2 2->3 (3->4)
# --edge-- -----
E = contract(T1, edge, ([1], [0]))
if verbosity > 0: print("E=edgeT " + str(E.size()))
# TODO - more efficent contraction with uncontracted-double-layer on-site tensor
# Possibly reshape indices 1,2 of E, which are to be contracted with
# on-site tensor and contract bra,ket in two steps instead of creating
# double layer tensor
# /
# --A--
# /|s
# X
# s'|/
# --A--
# /
#
# where X is Id or op
a = state.site(c)
A = get_aXa(a, op)
# edge with no MPO edge with MPO index
#
# 0 2<-0 (4) 0 2<-0 ---------(4)(5->4)
# | |/ | |/ |
# T1--1 1--A--3 T1--1 1--A--3 |
# | 2 | 2 |
# | 2 3->1 | 2 3->1 (4)
# --------edge-- --------edge-- ------
E = contract(E,A,([1,2,4],[1,2,4])) if contains_mpo else \
contract(E,A,([1,2],[1,2]))
if verbosity > 0: print("E=EA " + str(E.size()))
# 0 1<-2 (4->2) 0->2 (0->3)
# | |/ |
# T1----A--3 1------T2
# | | 2
# | | 1
# -----edge---------
T2 = env.T[(c, (1, 0))]
E = contract(E, T2, ([1, 3], [2, 1]))
# if we have extra MPO dimension, permute it to the last index
if len(E.size()) == 4: E = contiguous(permute(E, (0, 1, 3, 2)))
if verbosity > 0: print("E=ET " + str(E.size()))
elif direction == (-1, 0): #left
T1 = env.T[(c, (0, -1))]
# Assume index structure of ``edge`` tensor to be as follows
#
# 0 --
# 1 --| edge
# 2 --
#
# 0--T1--2 0----
# 1 |
# 2<-1--edge
# |
# 3<-2----|
# (4<-3)---
E = contract(T1, edge, ([2], [0]))
if verbosity > 0: print("E=edgeT " + str(E.size()))
# TODO - more efficent contraction with uncontracted-double-layer on-site tensor
# Possibly reshape indices 1,2 of E, which are to be contracted with
# on-site tensor and contract bra,ket in two steps instead of creating
# double layer tensor
# /
# --A--
# /|s
# X
# s'|/
# --A--
# /
#
# where X is Id or op
a = state.site(c)
A = get_aXa(a, op)
# 0--T1--------- 0--T1---------
# 1 | 1 |
# 0 | 0 |
# 2<-1--A----3 2--edge 2<-1--A----3 2--edge
# / 2->3 | / 2->3 |
# | 1<-3--| (4) 1<-3--
# (4<-5)(4)-------(4)--
E = contract(E,A,([1,2,4],[0,3,4])) if contains_mpo else \
contract(E,A,([1,2],[0,3]))
if verbosity > 0: print("E=EA " + str(E.size()))
# 0--T1-------
# | |
# | |
# 1<-2--A-------edge
# / 3 |
# (2<-4) 0 |
# (3<-1) 2<-1--T2--2 1--
T2 = env.T[(c, (0, 1))]
E = contract(E, T2, ([1, 3], [2, 0]))
if len(E.size()) == 4: E = contiguous(permute(E, (0, 1, 3, 2)))
if verbosity > 0: print("E=ET " + str(E.size()))
elif direction == (0, 1): #down
T1 = env.T[(c, (-1, 0))]
# Assume index structure of ``edge`` tensor to be as follows
#
# --edge-- ------
# 0 1->2 2->3 (3->4)
# 0
# T1--2->1
# 1->0
E = contract(T1, edge, ([0], [0]))
if verbosity > 0: print("E=edgeT " + str(E.size()))
# TODO - more efficent contraction with uncontracted-double-layer on-site tensor
# Possibly reshape indices 1,2 of E, which are to be contracted with
# on-site tensor and contract bra,ket in two steps instead of creating
# double layer tensor
# /
# --A--
# /|s
# X
# s'|/
# --A--
# /
#
# where X is Id or op
a = state.site(c)
A = get_aXa(a, op)
# -------edge---- -------edge---- ------
# | 2 3->1 | 2 3->1 (4)
# | 0 | 0 |
# T1--1 1--A--3 T1--1 1--A--3 |
# | |\ | |\ |
# 0 2 (4) 0 2 -----------(4)(5->4)
E = contract(E,A,([1,2,4],[1,0,4])) if contains_mpo else \
contract(E,A,([1,2],[1,0]))
if verbosity > 0: print("E=EA " + str(E.size()))
# ----edge-------
# | | 1
# | | 0
# T1----A--3 1----T2
# | |\ |
# 0 1<-2 (4->2) 2 (2->3)
T2 = env.T[(c, (1, 0))]
E = contract(E, T2, ([1, 3], [0, 1]))
if len(E.size()) == 4: E = contiguous(permute(E, (0, 1, 3, 2)))
if verbosity > 0: print("E=ET " + str(E.size()))
elif direction == (1, 0): #right
T1 = env.T[(c, (0, -1))]
# Assume index structure of ``edge`` tensor to be as follows
#
# -- 0
# edge |-- 1
# -- 2
#
# ----0 0--T1--2->1
# | 1->0
# edge--1->2
# |
# ----2->3
# |
# ----(3->4)
E = contract(T1, edge, ([0], [0]))
if verbosity > 0: print("E=edgeT " + str(E.size()))
# TODO - more efficent contraction with uncontracted-double-layer on-site tensor
# Possibly reshape indices 1,2 of E, which are to be contracted with
# on-site tensor and contract bra,ket in two steps instead of creating
# double layer tensor
# /
# --A--
# /|s
# X
# s'|/
# --A--
# /
#
# where X is Id or op
a = state.site(c)
A = get_aXa(a, op)
# ---------T1--1->0 ---------T1--1->0
# | 0 | 0
# | 0 | 0
# edge--2 1--A--3 edge--2 1--A--3
# | 2 \ | 2
# ----3->1 | ----3->1
# | |
# ----4 ---- (4)(5->4)
E = contract(E,A,([0,2,4],[0,1,4])) if contains_mpo else \
contract(E,A,([0,2],[0,1]))
if verbosity > 0: print("E=EA " + str(E.size()))
# -------T1--0
# | |
# | |
# edge-----A--3->1
# | 2 \(4->2)
# | 0
# --1 1--T2--2 (2->3)
T2 = env.T[(c, (0, 1))]
E = contract(E, T2, ([1, 2], [1, 0]))
if len(E.size()) == 4: E = contiguous(permute(E, (0, 1, 3, 2)))
if verbosity > 0: print("E=ET " + str(E.size()))
else:
raise ValueError("Invalid direction: " + str(direction))
return E
def init_from_ipeps_pbc(state, env, verbosity=0):
if verbosity > 0:
print("ENV: init_from_ipeps")
for coord, site in state.sites.items():
for rel_vec in [(-1, -1), (1, -1), (1, 1), (-1, 1)]:
env.C[(coord, rel_vec)] = torch.zeros(env.chi,
env.chi,
dtype=env.dtype,
device=env.device)
# Left-upper corner
#
# i = C--1
# j--A--3 0
# /\
# 2 m
# \ i
# j--A--3
# /
# 2
vec = (-1, -1)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('mijef,mijab->eafb', A, conj(A)))
a = view(a, (dimsA[3]**2, dimsA[4]**2))
a = a / a.abs().max()
env.C[(coord,vec)][:min(env.chi,dimsA[3]**2),:min(env.chi,dimsA[4]**2)]=\
a[:min(env.chi,dimsA[3]**2),:min(env.chi,dimsA[4]**2)]
# right-upper corner
#
# i = 0--C
# 1--A--j 1
# /\
# 2 m
# \ i
# 1--A--j
# /
# 2
vec = (1, -1)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('miefj,miabj->eafb', A, conj(A)))
a = view(a, (dimsA[2]**2, dimsA[3]**2))
a = a / a.abs().max()
env.C[(coord,vec)][:min(env.chi,dimsA[2]**2),:min(env.chi,dimsA[3]**2)]=\
a[:min(env.chi,dimsA[2]**2),:min(env.chi,dimsA[3]**2)]
# right-lower corner
#
# 0 = 0
# 1--A--j 1--C
# /\
# i m
# \ 0
# 1--A--j
# /
# i
vec = (1, 1)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('mefij,mabij->eafb', A, conj(A)))
a = view(a, (dimsA[1]**2, dimsA[2]**2))
a = a / a.abs().max()
env.C[(coord,vec)][:min(env.chi,dimsA[1]**2),:min(env.chi,dimsA[2]**2)]=\
a[:min(env.chi,dimsA[1]**2),:min(env.chi,dimsA[2]**2)]
# left-lower corner
#
# 0 = 0
# i--A--3 C--1
# /\
# j m
# \ 0
# i--A--3
# /
# j
vec = (-1, 1)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('meijf,maijb->eafb', A, conj(A)))
a = view(a, (dimsA[1]**2, dimsA[4]**2))
a = a / a.abs().max()
env.C[(coord,vec)][:min(env.chi,dimsA[1]**2),:min(env.chi,dimsA[4]**2)]=\
a[:min(env.chi,dimsA[1]**2),:min(env.chi,dimsA[4]**2)]
# upper transfer matrix
#
# i = 0--T--2
# 1--A--3 1
# /\
# 2 m
# \ i
# 1--A--3
# /
# 2
vec = (0, -1)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('miefg,miabc->eafbgc', A, conj(A)))
a = view(a, (dimsA[2]**2, dimsA[3]**2, dimsA[4]**2))
a = a / a.abs().max()
env.T[(coord, vec)] = torch.zeros((env.chi, dimsA[3]**2, env.chi),
dtype=env.dtype,
device=env.device)
env.T[(coord,vec)][:min(env.chi,dimsA[2]**2),:,:min(env.chi,dimsA[4]**2)]=\
a[:min(env.chi,dimsA[2]**2),:,:min(env.chi,dimsA[4]**2)]
# left transfer matrix
#
# 0 = 0
# i--A--3 T--2
# /\ 1
# 2 m
# \ 0
# i--A--3
# /
# 2
vec = (-1, 0)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('meifg,maibc->eafbgc', A, conj(A)))
a = view(a, (dimsA[1]**2, dimsA[3]**2, dimsA[4]**2))
a = a / a.abs().max()
env.T[(coord, vec)] = torch.zeros((env.chi, env.chi, dimsA[4]**2),
dtype=env.dtype,
device=env.device)
env.T[(coord,vec)][:min(env.chi,dimsA[1]**2),:min(env.chi,dimsA[3]**2),:]=\
a[:min(env.chi,dimsA[1]**2),:min(env.chi,dimsA[3]**2),:]
# lower transfer matrix
#
# 0 = 0
# 1--A--3 1--T--2
# /\
# i m
# \ 0
# 1--A--3
# /
# i
vec = (0, 1)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('mefig,mabic->eafbgc', A, conj(A)))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[4]**2))
a = a / a.abs().max()
env.T[(coord, vec)] = torch.zeros((dimsA[1]**2, env.chi, env.chi),
dtype=env.dtype,
device=env.device)
env.T[(coord,vec)][:,:min(env.chi,dimsA[2]**2),:min(env.chi,dimsA[4]**2)]=\
a[:,:min(env.chi,dimsA[2]**2),:min(env.chi,dimsA[4]**2)]
# right transfer matrix
#
# 0 = 0
# 1--A--i 1--T
# /\ 2
# 2 m
# \ 0
# 1--A--i
# /
# 2
vec = (1, 0)
A = state.site((coord[0] + vec[0], coord[1] + vec[1]))
dimsA = A.size()
a = contiguous(einsum('mefgi,mabci->eafbgc', A, conj(A)))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[3]**2))
a = a / a.abs().max()
env.T[(coord, vec)] = torch.zeros((env.chi, dimsA[2]**2, env.chi),
dtype=env.dtype,
device=env.device)
env.T[(coord,vec)][:min(env.chi,dimsA[1]**2),:,:min(env.chi,dimsA[3]**2)]=\
a[:min(env.chi,dimsA[1]**2),:,:min(env.chi,dimsA[3]**2)]
def run(state,
env,
conv_check=None,
ctm_args=cfg.ctm_args,
global_args=cfg.global_args):
r"""
:param state: wavefunction
:param env: environment
:param conv_check: function which determines the convergence of CTM algorithm. If ``None``,
the algorithm performs ``ctm_args.ctm_max_iter`` iterations.
:param ctm_args: CTM algorithm configuration
:param global_args: global configuration
:type state: IPEPS
:type env: ENV
:type conv_check: function(IPEPS,ENV,list[float],CTMARGS)->bool
:type ctm_args: CTMARGS
:type global_args: GLOBALARGS
Executes directional CTM algorithm for generic iPEPS starting from the intial environment ``env``.
TODO add reference
"""
# 0) Create double-layer (DL) tensors, preserving the same convenction
# for order of indices
#
# / /
# --A^dag-- = --a--
# /| /
# |/
# --A--
# /
#
sitesDL = dict()
for coord, A in state.sites.items():
dimsA = A.size()
a = contiguous(einsum('mefgh,mabcd->eafbgchd', A, conj(A)))
a = view(a, (dimsA[1]**2, dimsA[2]**2, dimsA[3]**2, dimsA[4]**2))
sitesDL[coord] = a
stateDL = IPEPS(sitesDL, state.vertexToSite)
# 1) perform CTMRG
t_obs = t_ctm = 0.
history = None
for i in range(ctm_args.ctm_max_iter):
t0_ctm = time.perf_counter()
for direction in ctm_args.ctm_move_sequence:
ctm_MOVE(direction, stateDL, env, ctm_args=ctm_args, global_args=global_args, \
verbosity=ctm_args.verbosity_ctm_move)
t1_ctm = time.perf_counter()
t0_obs = time.perf_counter()
if conv_check is not None:
# evaluate convergence of the CTMRG procedure
converged, history = conv_check(state,
env,
history,
ctm_args=ctm_args)
if ctm_args.verbosity_ctm_convergence > 1: print(history[-1])
if converged:
if ctm_args.verbosity_ctm_convergence > 0:
print(
f"CTMRG converged at iter= {i}, history= {history[-1]}"
)
break
t1_obs = time.perf_counter()
t_ctm += t1_ctm - t0_ctm
t_obs += t1_obs - t0_obs
return env, history, t_ctm, t_obs