def energy_1x1(self, state, env):
r"""
:param state: wavefunction
:param env_c4v: CTM c4v symmetric environment
:type state: IPEPS
:type env: ENV_C4V
:return: energy per site
:rtype: float
For 1-site invariant c4v iPEPS it's enough to construct a 1-site reduced
density matrix :py:func:`ctm.one_site_c4v.rdm_c4v.rdm1x1`, effectively
representing a 2x2 plaquette, and 2-site reduced
density matrix :py:func:`ctm.one_site_c4v.rdm_c4v.rdm2x1` which represents
interaction between two plaquettes of the underlying physical system:
.. math::
e = \langle h1 \rangle_{\rho_{1x1}} + \langle h2 \rangle_{\rho_{2x1}}
"""
rdm1x1 = rdm_c4v.rdm1x1(state, env)
rdm2x1 = rdm_c4v.rdm2x1(state, env)
e1s = torch.einsum('ij,ij', rdm1x1, self.h1)
e2s = torch.einsum('ijab,ijab', rdm2x1, self.h2)
energy_per_site = (e1s + e2s) / 4
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. magnetization
2. :math:`\langle S^z \rangle,\ \langle S^+ \rangle,\ \langle S^- \rangle`
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*}
"""
# TODO optimize/unify ?
# expect "list" of (observable label, value) pairs ?
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[f"m"] = sqrt(abs(obs[f"sz"]**2 + obs[f"sp"] * obs[f"sm"]))
rdm2x1 = rdm_c4v.rdm2x1(state, env_c4v)
obs[f"SS2x1"] = torch.einsum('ijab,ijab', rdm2x1, self.h2_rot)
# prepare list with labels and values
obs_labels = [f"m"] + [f"{lc}"
for lc in self.obs_ops.keys()] + [f"SS2x1"]
obs_values = [obs[label] for label in obs_labels]
return obs_values, obs_labels
def energy_1x1(self,state,env_c4v):
rdm2x1 = rdm_c4v.rdm2x1(state, env_c4v)
energy = torch.einsum('ijab,ijab',rdm2x1,self.h2_rot)
return energy