def energy_1x1(self, state, env_c4v):
r"""
:param state: wavefunction
:param env_c4v: CTM c4v symmetric environment
:type state: IPEPS
:type env_c4v: ENV_C4V
:return: energy per site
:rtype: float
We assume 1x1 C4v 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
Due to C4v symmetry it is enough to construct a single reduced density matrix
:py:func:`ctm.one_site_c4v.rdm_c4v.rdm2x2` of a 2x2 plaquette. Afterwards,
the energy per site `e` is computed by evaluating a single plaquette term :math:`h_p`
containing two nearest-neighbour terms :math:`\bf{S}.\bf{S}` and `h4_p` as:
.. math::
e = \langle \mathcal{h_p} \rangle = Tr(\rho_{2x2} \mathcal{h_p})
"""
rdm2x2 = rdm_c4v.rdm2x2(state, env_c4v)
energy_per_site = torch.einsum('ijklabcd,ijklabcd', rdm2x2,
self.hp_rot)
return energy_per_site
def energy_1x1_plaqette(self, state, env_c4v):
r"""
:param state: wavefunction
:param env_c4v: CTM c4v symmetric environment
:type state: IPEPS
:type env_c4v: ENV_C4V
:return: energy per site
:rtype: float
For 1-site invariant c4v iPEPS it's enough to construct a single reduced
density matrix of a 2x2 plaquette. Afterwards, the energy per site `e` is
computed by evaluating individual terms in the Hamiltonian through
:math:`\langle \mathcal{O} \rangle = Tr(\rho_{2x2} \mathcal{O})`
.. math::
e = -(\langle h2_{<\bf{0},\bf{x}>} \rangle + \langle h2_{<\bf{0},\bf{y}>} \rangle)
+ q\langle h4_{\bf{0}} \rangle - h_x \langle h4_{\bf{0}} \rangle
"""
rdm2x2 = rdm_c4v.rdm2x2(state, env_c4v)
energy_per_site = torch.einsum('ijklabcd,ijklabcd', rdm2x2, self.hp)
# From individual contributions
# eSx= torch.einsum('ijklajkl,ia',rdm2x2,self.sx)
# eSzSz= torch.einsum('ijklabkl,ijab',rdm2x1,self.szsz)
# eSzSzSzSz= torch.einsum('ijklabcd,ijklabcd',rdm2x2,self.szszszsz)
# energy_per_site = -2*eSzSz - self.hx*eSx + self.q*eSzSzSzSz
return energy_per_site
def eval_obs(self, state, env_c4v):
r"""
:param state: wavefunction
:param env_c4v: CTM c4v symmetric environment
:type state: IPEPS
:type env_c4v: ENV_C4V
:return: expectation values of observables, labels of observables
:rtype: list[float], list[str]
Computes the following observables in order
1. :math:`\langle 2S^z \rangle,\ \langle 2S^x \rangle` for each site in the unit cell
TODO 2site observable SzSz
"""
obs = dict()
with torch.no_grad():
rdm1x1 = rdm_c4v.rdm1x1(state, env_c4v)
for label, op in self.obs_ops.items():
obs[f"{label}"] = torch.trace(rdm1x1 @ op)
obs["sx"] = 0.5 * (obs["sp"] + obs["sm"])
rdm2x2 = rdm_c4v.rdm2x2(state, env_c4v)
obs["SzSzSzSz"] = torch.einsum('ijklabcd,ijklabcd', rdm2x2,
self.szszszsz)
# prepare list with labels and values
obs_labels = [lc for lc in ["sz", "sx"]]
obs_labels += ["SzSzSzSz"]
obs_values = [obs[label] for label in obs_labels]
return obs_values, obs_labels
def energy_1x1(self, state, env_c4v, **kwargs):
r"""
:param state: wavefunction
:param env_c4v: CTM c4v symmetric environment
:type state: IPEPS
:type env_c4v: ENV_C4V
:return: energy per site
:rtype: float
We assume 1x1 C4v 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
Due to C4v symmetry it is enough to construct a single reduced density matrix
:py:func:`ctm.one_site_c4v.rdm_c4v.rdm2x2` of a 2x2 plaquette. Afterwards,
the energy per site `e` is computed by evaluating a single plaquette term :math:`h_p`
containing two nearest-nighbour terms :math:`\bf{S}.\bf{S}` and two next-nearest
neighbour :math:`\bf{S}.\bf{S}`, as:
.. math::
e = \langle \mathcal{h_p} \rangle = Tr(\rho_{2x2} \mathcal{h_p})
"""
rdm2x2= rdm_c4v.rdm2x2(state,env_c4v,sym_pos_def=True,\
verbosity=cfg.ctm_args.verbosity_rdm)
energy_per_site = torch.einsum('ijklabcd,ijklabcd', rdm2x2, self.hp)
if abs(self.j3) > 0:
rdm3x1= rdm_c4v.rdm3x1(state,env_c4v,sym_pos_def=True,\
force_cpu=False,verbosity=cfg.ctm_args.verbosity_rdm)
ss_3x1 = torch.einsum('ijab,ijab', rdm3x1, self.SS)
energy_per_site = energy_per_site + 2 * self.j3 * ss_3x1
energy_per_site = _cast_to_real(energy_per_site)
return energy_per_site
def energy_1x1(self, state, env_c4v, **kwargs):
r"""
:param state: wavefunction
:param env_c4v: CTM c4v symmetric environment
:type state: IPEPS
:type env_c4v: ENV_C4V
:return: energy per site
:rtype: float
We assume 1x1 C4v 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
Due to C4v symmetry it is enough to construct a single reduced density matrix
:py:func:`ctm.one_site_c4v.rdm_c4v.rdm2x2` of a 2x2 plaquette. Afterwards,
the energy per site `e` is computed by evaluating a single plaquette term :math:`h_p`
containing two nearest-nighbour terms :math:`\bf{S}.\bf{S}` and two next-nearest
neighbour :math:`\bf{S}.\bf{S}`, as:
.. math::
e = \langle \mathcal{h_p} \rangle = Tr(\rho_{2x2} \mathcal{h_p})
"""
rdm2x2 = rdm_c4v.rdm2x2(state, env_c4v, cfg.ctm_args.verbosity_rdm)
energy_per_site = torch.einsum('ijklabcd,ijklabcd', rdm2x2, self.hp)
# id2= torch.eye(4,dtype=self.dtype,device=self.device)
# id2= id2.view(2,2,2,2).contiguous()
# print(f"rdm2x1 {torch.einsum('ijklabcd,ijab,klcd',rdm2x2,self.h2_rot,id2)}"\
# + f" rdm1x2 {torch.einsum('ijklabcd,ikac,jlbd',rdm2x2,self.h2_rot,id2)}"\
# + f" rdm_0cc3_diag {torch.einsum('ijklabcd,ilad,jkbc',rdm2x2,self.h2,id2)}"\
# + f" rdm_c12c_diag {torch.einsum('ijklabcd,jkbc,ilad',rdm2x2,self.h2,id2)}")
return energy_per_site