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 energy_2x1_1x2(self, state, env):
r"""
:param state: wavefunction
:param env: CTM environment
:type state: IPEPS
:type env: ENV
:return: energy per site
:rtype: float
We assume iPEPS with 2x2 unit cell containing four tensors A, B, C, and D with
simple PBC tiling::
A B A B
C D C D
A B A B
C D C D
Taking the reduced density matrix :math:`\rho_{2x1}` (:math:`\rho_{1x2}`)
of 2x1 (1x2) cluster given by :py:func:`rdm.rdm2x1` (:py:func:`rdm.rdm1x2`)
with indexing of sites as follows :math:`s_0,s_1;s'_0,s'_1` for both types
of density matrices::
rdm2x1 rdm1x2
s0--s1 s0
|
s1
and without assuming any symmetry on the indices of individual tensors a following
set of terms has to be evaluated in order to compute energy-per-site::
0 0 0
1--A--3 1--B--3 1--A--3
2 2 2
0 0 0
1--C--3 1--D--3 1--C--3
2 2 2 A--3 1--B, A B C D
0 0 B--3 1--A, 2 2 2 2
1--A--3 1--B--3 C--3 1--D, 0 0 0 0
2 2 , terms D--3 1--C, and C, D, A, B
"""
energy = 0.
for coord, site in state.sites.items():
rdm2x1 = rdm.rdm2x1(coord, state, env)
rdm1x2 = rdm.rdm1x2(coord, state, env)
ss = einsum('ijab,ijab', rdm2x1, self.h2)
energy += ss
if coord[1] % 2 == 0:
ss = einsum('ijab,ijab', rdm1x2, self.h2)
else:
ss = einsum('ijab,ijab', rdm1x2, self.alpha * self.h2)
energy += ss
# return energy-per-site
energy_per_site = energy / len(state.sites.items())
energy_per_site = _cast_to_real(energy_per_site)
return energy_per_site
def SS(self, xyz=(1., 1., 1.)):
r"""
:param xyz: coefficients of anisotropy of spin-spin interaction
xyz[0]*(S^z S^z) + xyz[1]*(S^x S^x) + xyz[2]*(S^y S^y)
:type xyz: tuple(float)
:return: spin-spin interaction as rank-4 for tensor
:rtype: torch.tensor
"""
pd = self.J
expr_kron = 'ij,ab->iajb'
# spin-spin interaction \vec{S}_1.\vec{S}_2 between spins on sites 1 and 2
# First as rank-4 tensor
SS = xyz[0]*einsum(expr_kron,self.SZ(),self.SZ()) \
+ 0.5*(einsum(expr_kron,self.SP(),self.SM()) \
+ einsum(expr_kron,self.SM(),self.SP()))
return SS
def conjugate_op(op):
#rot_op= su2.get_rot_op(self.phys_dim, dtype=self.dtype, device=self.device)
rot_op = torch.eye(self.phys_dim,
dtype=self.dtype,
device=self.device)
op_0 = op
op_rot = einsum('ki,kj->ij', rot_op, mm(op_0, rot_op))
def _gen_op(r):
#return op_rot if r%2==0 else op_0
return op_0
return _gen_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
def get_h(self):
s2 = su2.SU2(self.phys_dim, dtype=self.dtype, device=self.device)
expr_kron = 'ij,ab->iajb'
SS = einsum(expr_kron,s2.SZ(),s2.SZ()) + 0.5*(einsum(expr_kron,s2.SP(),s2.SM()) \
+ einsum(expr_kron,s2.SM(),s2.SP()))
return SS
def eval_obs(self, state, env):
r"""
:param state: wavefunction
:param env: CTM environment
:type state: IPEPS
:type env: ENV
:return: expectation values of observables, labels of observables
:rtype: list[float], list[str]
Computes the following observables in order
1. average magnetization over the unit cell,
2. magnetization for each site in the unit cell
3. :math:`\langle S^z \rangle,\ \langle S^+ \rangle,\ \langle S^- \rangle`
for each site in the unit cell
4. :math:`\mathbf{S}_i.\mathbf{S}_j` for all non-equivalent nearest neighbour
bonds
where the on-site magnetization is defined as
.. math::
\begin{align*}
m &= \sqrt{ \langle S^z \rangle^2+\langle S^x \rangle^2+\langle S^y \rangle^2 }
=\sqrt{\langle S^z \rangle^2+1/4(\langle S^+ \rangle+\langle S^-
\rangle)^2 -1/4(\langle S^+\rangle-\langle S^-\rangle)^2} \\
&=\sqrt{\langle S^z \rangle^2 + 1/2\langle S^+ \rangle \langle S^- \rangle)}
\end{align*}
Usual spin components can be obtained through the following relations
.. math::
\begin{align*}
S^+ &=S^x+iS^y & S^x &= 1/2(S^+ + S^-)\\
S^- &=S^x-iS^y\ \Rightarrow\ & S^y &=-i/2(S^+ - S^-)
\end{align*}
"""
obs = dict({"avg_m": 0.})
with torch.no_grad():
for coord, site in state.sites.items():
rdm1x1 = rdm.rdm1x1(coord, state, env)
for label, op in self.obs_ops.items():
obs[f"{label}{coord}"] = einsum('ij,ji', rdm1x1, op)
obs[f"m{coord}"] = sqrt(
abs(obs[f"sz{coord}"]**2 +
obs[f"sp{coord}"] * obs[f"sm{coord}"]))
obs["avg_m"] += obs[f"m{coord}"]
obs["avg_m"] = obs["avg_m"] / len(state.sites.keys())
for coord, site in state.sites.items():
rdm2x1 = rdm.rdm2x1(coord, state, env)
rdm1x2 = rdm.rdm1x2(coord, state, env)
SS2x1 = einsum('ijab,ijab', rdm2x1, self.h2)
SS1x2 = einsum('ijab,ijab', rdm1x2, self.h2)
obs[f"SS2x1{coord}"] = SS2x1.real if SS2x1.is_complex(
) else SS2x1
obs[f"SS1x2{coord}"] = SS1x2.real if SS1x2.is_complex(
) else SS1x2
# prepare list with labels and values
obs_labels=["avg_m"]+[f"m{coord}" for coord in state.sites.keys()]\
+[f"{lc[1]}{lc[0]}" for lc in list(itertools.product(state.sites.keys(), self.obs_ops.keys()))]
obs_labels += [f"SS2x1{coord}" for coord in state.sites.keys()]
obs_labels += [f"SS1x2{coord}" for coord in state.sites.keys()]
obs_values = [obs[label] for label in obs_labels]
return obs_values, obs_labels
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 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 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 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 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 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
