# Observables¶

Operators for commonly used observables.

pytriqs.operators.util.observables.N_op(spin_names, orb_names, off_diag=None, map_operator_structure=None)[source]

Create an operator of the total number of particles.

$\hat N = \sum_{i\sigma} a_{i\sigma}^\dagger a_{i\sigma}.$
Parameters: spin_names : list of strings Names of the spins, e.g. [‘up’,’down’]. orb_names : list of strings or int Names of the orbitals, e.g. [0,1,2] or [‘t2g’,’eg’]. off_diag : boolean Do we have (orbital) off-diagonal elements? If yes, the operators and blocks are denoted by (‘spin’, ‘orbital’), otherwise by (‘spin_orbital’,0). map_operator_structure : dict Mapping of names of GF blocks names from one convention to another, e.g. {(‘up’, 0): (‘up_0’, 0), (‘down’, 0): (‘down_0’,0)}. If provided, the operators and blocks are denoted by the mapping of ('spin', 'orbital'). N : Operator The total number of particles.
pytriqs.operators.util.observables.S_op(component, spin_names, orb_names, off_diag=None, map_operator_structure=None)[source]

Create a component of the spin vector operator.

$\hat S_{x,y,z} = \frac{1}{2}\sum_{i\sigma\sigma'} a^\dagger_{i\sigma} \mathbf{\tau}^{x,y,z}_{\sigma\sigma'} a_{i\sigma'}, \quad\hat S_\pm = \hat S_x \pm i \hat S_y.$
Parameters: component : string Component to be created, one of ‘x’, ‘y’, ‘z’, ‘+’, or ‘-‘. spin_names : list of strings Names of the spins, e.g. [‘up’,’down’]. orb_names : list of strings or int Names of the orbitals, e.g. [0,1,2] or [‘t2g’,’eg’]. off_diag : boolean Do we have (orbital) off-diagonal elements? If yes, the operators and blocks are denoted by (‘spin’, ‘orbital’), otherwise by (‘spin_orbital’,0). map_operator_structure : dict Mapping of names of GF blocks names from one convention to another, e.g. {(‘up’, 0): (‘up_0’, 0), (‘down’, 0): (‘down_0’,0)}. If provided, the operators and blocks are denoted by the mapping of ('spin', 'orbital'). S : Operator The component of the spin vector operator.
pytriqs.operators.util.observables.S2_op(spin_names, orb_names, off_diag=None, map_operator_structure=None)[source]

Create the square of the total spin operator.

$\hat S^2 = \hat S_x^2 + \hat S_y^2 + \hat S_z^2.$
Parameters: spin_names : list of strings Names of the spins, e.g. [‘up’,’down’]. orb_names : list of strings or int Names of the orbitals, e.g. [0,1,2] or [‘t2g’,’eg’]. off_diag : boolean Do we have (orbital) off-diagonal elements? If yes, the operators and blocks are denoted by (‘spin’, ‘orbital’), otherwise by (‘spin_orbital’,0). map_operator_structure : dict Mapping of names of GF blocks names from one convention to another, e.g. {(‘up’, 0): (‘up_0’, 0), (‘down’, 0): (‘down_0’,0)}. If provided, the operators and blocks are denoted by the mapping of ('spin', 'orbital'). S2 : Operator The square of the total spin operator.
pytriqs.operators.util.observables.L_op(component, spin_names, orb_names, off_diag=None, map_operator_structure=None, basis='spherical', T=None)[source]

Create a component of the orbital momentum vector operator.

$\begin{split}\hat L_{z,+,-} &= \sum_{ii'\sigma} a^\dagger_{i\sigma} L^{z,+,-}_{ii'} a_{i'\sigma},\\ \hat L_x &= \frac{1}{2}(\hat L_+ + \hat L_-),\ \hat L_y = \frac{1}{2i}(\hat L_+ - \hat L_-),\\ L^z_{ii'} &= i\delta_{i,i'}, \ L^+_{ii'} = \delta_{i,i'+1}\sqrt{l(l+1)-i'(i'+1)}, \ L^+_{ii'} = \delta_{i,i'-1}\sqrt{l(l+1)-i'(i'-1)}.\end{split}$
Parameters: component : string Component to be created, one of ‘x’, ‘y’, ‘z’, ‘+’, or ‘-‘. spin_names : list of strings Names of the spins, e.g. [‘up’,’down’]. orb_names : list of strings or int Names of the orbitals, e.g. [0,1,2] or [‘t2g’,’eg’]. off_diag : boolean Do we have (orbital) off-diagonal elements? If yes, the operators and blocks are denoted by (‘spin’, ‘orbital’), otherwise by (‘spin_orbital’,0). map_operator_structure : dict Mapping of names of GF blocks names from one convention to another, e.g. {(‘up’, 0): (‘up_0’, 0), (‘down’, 0): (‘down_0’,0)}. If provided, the operators and blocks are denoted by the mapping of ('spin', 'orbital'). basis : string, optional The basis in which the interaction matrix should be computed. Takes the values ‘spherical’: spherical harmonics, ‘cubic’: cubic harmonics (valid only for the integer orbital momenta, i.e. for odd sizes of orb_names), ‘other’: other basis type as given by the transformation matrix T. T : real/complex numpy array, optional Transformation matrix for basis change. Must be provided if basis=’other’. L : Operator The component of the orbital momentum vector operator.
pytriqs.operators.util.observables.L2_op(spin_names, orb_names, off_diag=None, map_operator_structure=None, basis='spherical', T=None)[source]

Create the square of the orbital momentum operator.

$\hat L^2 = \hat L_x^2 + \hat L_y^2 + \hat L_z^2.$
Parameters: spin_names : list of strings Names of the spins, e.g. [‘up’,’down’]. orb_names : list of strings or int Names of the orbitals, e.g. [0,1,2] or [‘t2g’,’eg’]. off_diag : boolean Do we have (orbital) off-diagonal elements? If yes, the operators and blocks are denoted by (‘spin’, ‘orbital’), otherwise by (‘spin_orbital’,0). map_operator_structure : dict Mapping of names of GF blocks names from one convention to another, e.g. {(‘up’, 0): (‘up_0’, 0), (‘down’, 0): (‘down_0’,0)}. If provided, the operators and blocks are denoted by the mapping of ('spin', 'orbital'). basis : string, optional The basis in which the interaction matrix should be computed. Takes the values ‘spherical’: spherical harmonics, ‘cubic’: cubic harmonics (valid only for the integer orbital momenta, i.e. for odd sizes of orb_names), ‘other’: other basis type as given by the transformation matrix T. T : real/complex numpy array, optional Transformation matrix for basis change. Must be provided if basis=’other’. L2 : Operator The square of the orbital momentum operator.
pytriqs.operators.util.observables.LS_op(spin_names, orb_names, off_diag=None, map_operator_structure=None, basis='spherical', T=None)[source]

Create a spin-orbital coupling operator.

$\mathbf{\hat L\cdot\hat S} = \hat L_x \hat S_x + \hat L_y \hat S_y + \hat L_z \hat S_z.$
Parameters: spin_names : list of strings Names of the spins, e.g. [‘up’,’down’]. orb_names : list of strings or int Names of the orbitals, e.g. [0,1,2] or [‘t2g’,’eg’]. off_diag : boolean Do we have (orbital) off-diagonal elements? If yes, the operators and blocks are denoted by (‘spin’, ‘orbital’), otherwise by (‘spin_orbital’,0). map_operator_structure : dict Mapping of names of GF blocks names from one convention to another, e.g. {(‘up’, 0): (‘up_0’, 0), (‘down’, 0): (‘down_0’,0)}. If provided, the operators and blocks are denoted by the mapping of ('spin', 'orbital'). basis : string, optional The basis in which the interaction matrix should be computed. Takes the values ‘spherical’: spherical harmonics, ‘cubic’: cubic harmonics (valid only for the integer orbital momenta, i.e. for odd sizes of orb_names), ‘other’: other basis type as given by the transformation matrix T. T : real/complex numpy array, optional Transformation matrix for basis change. Must be provided if basis=’other’. LS : Operator The spin-orbital coupling operator.