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. :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 eval_corrf_SS(self, state, env_c4v, dist, canonical=False): Sop_zxy = torch.zeros((3, self.phys_dim, self.phys_dim), dtype=self.dtype, device=self.device) Sop_zxy[0, :, :] = self.obs_ops["sz"] Sop_zxy[1, :, :] = 0.5 * (self.obs_ops["sp"] + self.obs_ops["sm"]) Sop_zxy[2, :, :] = -0.5 * (self.obs_ops["sp"] - self.obs_ops["sm"]) # compute vector of spontaneous magnetization if canonical: s_vec_zpm = [] rdm1x1 = rdm_c4v.rdm1x1(state, env_c4v) for label in ["sz", "sp", "sm"]: op = self.obs_ops[label] s_vec_zpm.append(torch.trace(rdm1x1 @ op)) # 0) transform into zxy basis and normalize s_vec_zxy= torch.tensor([s_vec_zpm[0],0.5*(s_vec_zpm[1]+s_vec_zpm[2]),\ 0.5*(s_vec_zpm[1]-s_vec_zpm[2])],dtype=self.dtype,device=self.device) s_vec_zxy = s_vec_zxy / torch.norm(s_vec_zxy) # 1) build rotation matrix R= torch.tensor([[s_vec_zxy[0],-s_vec_zxy[1],0],[s_vec_zxy[1],s_vec_zxy[0],0],[0,0,1]],\ dtype=self.dtype,device=self.device).t() # 2) rotate the vector of operators Sop_zxy = torch.einsum('ab,bij->aij', R, Sop_zxy) # function generating properly rotated operators on every bi-partite site def get_bilat_op(op): rot_op = su2.get_rot_op(self.phys_dim, dtype=self.dtype, device=self.device) op_0 = op op_rot = torch.einsum('ki,kl,lj->ij', rot_op, op_0, rot_op) def _gen_op(r): return op_rot if r % 2 == 0 else op_0 return _gen_op Sz0szR= corrf_c4v.corrf_1sO1sO(state, env_c4v, Sop_zxy[0,:,:], \ get_bilat_op(Sop_zxy[0,:,:]), dist) Sx0sxR = corrf_c4v.corrf_1sO1sO(state, env_c4v, Sop_zxy[1, :, :], get_bilat_op(Sop_zxy[1, :, :]), dist) nSy0SyR = corrf_c4v.corrf_1sO1sO(state, env_c4v, Sop_zxy[2, :, :], get_bilat_op(Sop_zxy[2, :, :]), dist) res = dict({ "ss": Sz0szR + Sx0sxR - nSy0SyR, "szsz": Sz0szR, "sxsx": Sx0sxR, "sysy": -nSy0SyR }) return res
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 eval_obs(self, state, env): r""" :param state: wavefunction :param env_c4v: CTM c4v symmetric environment :type state: IPEPS :type env: ENV_C4V :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 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({"avg_m": 0.}) with torch.no_grad(): rdm1x1 = rdm_c4v.rdm1x1(state, env) rdm1x1 = rdm1x1.view(2, 2, 2, 2, 2, 2, 2, 2) expr_core = 'abc' for r in range(4): expr = expr_core[:r] + 'i' + expr_core[ r:] + expr_core[:r] + 'j' + expr_core[r:] + ',ij' for label, op in self.obs_ops.items(): obs[f"{label}{r}"] = torch.einsum(expr, rdm1x1, op) obs[f"m{r}"] = sqrt( abs(obs[f"sz{r}"]**2 + obs[f"sp{r}"] * obs[f"sm{r}"])) obs["avg_m"] += obs[f"m{r}"] obs["avg_m"] = obs["avg_m"] / 4 # for coord,site in state.sites.items(): # rdm2x1 = rdm_c4v.rdm2x1(coord,state,env) # rdm1x2 = rdm.rdm1x2(coord,state,env) # obs[f"SS2x1{coord}"]= torch.einsum('ijab,ijab',rdm2x1,self.h2) # obs[f"SS1x2{coord}"]= torch.einsum('ijab,ijab',rdm1x2,self.h2) # prepare list with labels and values obs_labels=["avg_m"]+[f"m{r}" for r in range(4)]\ +[f"{lc[1]}{lc[0]}" for lc in list(itertools.product(range(4), 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