def energy_2x2_1site_BP(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 1x1 iPEPS which tiles the lattice with a bipartite pattern composed
of two tensors A, and B=RA, where R rotates approriately the physical Hilbert space
of tensor A on every "odd" site::
1x1 C4v => rotation P => BIPARTITE
A A A A A B A B
A A A A B A B A
A A A A A B A B
A A A A B A B A
A single reduced density matrix :py:func:`ctm.rdm.rdm2x2` of a 2x2 plaquette
is used to evaluate the energy.
"""
if not (hasattr(self, 'h2x2_nn_rot') or hasattr(self, 'h2x2_nn_nrot')):
s2 = su2.SU2(self.phys_dim, dtype=self.dtype, device=self.device)
rot_op= s2.BP_rot()
self.h2x2_nn_rot= torch.einsum('irtlaxyd,jr,kt,xb,yc->ijklabcd',\
self.h2x2_nn,rot_op,rot_op,rot_op,rot_op)
self.h2x2_nnn_rot= torch.einsum('irtlaxyd,jr,kt,xb,yc->ijklabcd',\
self.h2x2_nnn,rot_op,rot_op,rot_op,rot_op)
tmp_rdm= rdm.rdm2x2((0,0),state,env)
energy_nn= torch.einsum('ijklabcd,ijklabcd',tmp_rdm,self.h2x2_nn_rot)
energy_nnn= torch.einsum('ijklabcd,ijklabcd',tmp_rdm,self.h2x2_nnn_rot)
energy_per_site = 2.0*(self.j1*energy_nn/4.0 + self.j2*energy_nnn/2.0)
return energy_per_site
def energy_2x2_2site(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 2x1 unit cell containing two tensors A, B. We can
tile the square lattice in two ways::
BIPARTITE STRIPE
A B A B A B A B
B A B A A B A B
A B A B A B A B
B A B A A B A B
Taking reduced density matrix :math:`\rho_{2x2}` of 2x2 cluster with indexing
of sites as follows :math:`\rho_{2x2}(s_0,s_1,s_2,s_3;s'_0,s'_1,s'_2,s'_3)`::
s0--s1
| |
s2--s3
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
1--A--3
2
Ex.1 unit cell A B, with BIPARTITE tiling
A3--1B, B3--1A, A, B, A3 , B3 , 1A, 1B
2 0 \ \ / /
0 2 \ \ / /
B A 1A 1B A3 B3
Ex.2 unit cell A B, with STRIPE tiling
A3--1A, B3--1B, A, B, A3 , B3 , 1A, 1B
2 0 \ \ / /
0 2 \ \ / /
A B 1B 1A B3 A3
"""
# A3--1B B3 1A
# 2 \/ 2 2 \/ 2
# 0 /\ 0 0 /\ 0
# B3--1A & A3 1B
# A3--1B B3--1A
# 2 \/ 2 2 \/ 2
# 0 /\ 0 0 /\ 0
# A3--1B & B3--1A
energy_nn = 0
energy_nnn = 0
for coord in state.sites.keys():
tmp_rdm = rdm.rdm2x2(coord, state, env)
energy_nn += torch.einsum('ijklabcd,ijklabcd', tmp_rdm,
self.h2x2_nn)
energy_nnn += torch.einsum('ijklabcd,ijklabcd', tmp_rdm,
self.h2x2_nnn)
energy_per_site = 2.0 * (self.j1 * energy_nn / 8.0 +
self.j2 * energy_nnn / 4.0)
energy_per_site = _cast_to_real(energy_per_site)
return energy_per_site
def energy_2x2_1site(self, state, env):
rdm2x2 = rdm.rdm2x2((0, 0), state, env)
energy_per_site = torch.einsum("ijklabcd,ijklabcd", rdm2x2, self.Ham)
return energy_per_site