Python API Reference

This section contains the complete Python API reference for the triqs_xca package, including all modules, classes, and compiled extensions.

Main Module

The main triqs_xca module provides access to all submodules and the high-level Solver interface.

High-Level Solver Interface

class triqs_xca.block_sparse_solver.BlockSparseSolver(H_loc, beta, w_max, eps, gf_struct, conserved_operators='automatic', dlr_symmetrize=False, timer=None, atom_diag=None, verbose=True)[source]

Solver class for triqs_xca using the block sparse algorithm.

Parameters:

H_loctriqs.operators.Operator: Local Hamiltonian, including both quadratic and quartic terms (\(H_{\textrm{loc}} = \sum c^\dagger c + \sum c^\dagger c^\dagger c c\)).
betafloat: Inverse temperature \(\beta\).
w_maxfloat: Imaginary time DLR discretization frequency cutoff \(\omega_{\text{max}}\).
epsfloat: Imaginary time DLR accuracy tolerance \(\epsilon\).
gf_structdict: Triqs style Green’s function structure, e.g. [('up', 1), ('down', 1)], matching the operator content of H_loc.
conserved_operatorslist, optional: List of conserved operators (triqs.operators.Operator instances). Default is 'automatic' using the autopartition algorithm of triqs.atom_diag. To disable symmetries and use the dense diagram evaluator (DenseDiagramEvaluator), pass an empty list [].
timerTimer, optional: Timer for performance measurements. Default: None (will be setup automatically).
atom_diagAtomDiag, optional: Triqs AtomDiag object triqs.atom_diag.AtomDiag. Default: None. If not provided, it will be initialized automatically using the provided local Hamiltonian and conserved operators.
verbosebool/int, optional: Verbosity flag controlling level of printouts. Default: True.

Methods

`expectation_value`(operator)	Expectation value \(\langle \hat{O} \rangle\) of an operator \(\hat{O}\), computed from the pseudo particle Green's function as
`partition_function`()	Partition function \(Z\) of the impurity model, computed from the pseudo particle Green's function as
`pseudo_particle_chemical_potential`()	Pseudo particle chemical potential \(\eta\).
`set_dynamic_interactions`(dynint_ops, dynint_tau)	Set retarded (dynamical) interactions of the impurity problem.
`solve`(max_order[, tol, maxiter, mix, ...])	Solve the impurity problem using pseudo particle self-consistent perturbation theory.

Notes

The hybridization function is available as the Delta_tau attribute, which is a block Green’s function in imaginary time. It has to be set by the user before calling solve(). If not set by the user, it will be initialized to zero.

The main method of the class is solve(), which runs the pseudo particle self-consistent perturbation theory until convergence.

The single particle Green’s function is available as the G_tau attribute after calling solve(). It is a block Green’s function in imaginary time.

expectation_value(operator)[source]

Expectation value \(\langle \hat{O} \rangle\) of an operator \(\hat{O}\), computed from the pseudo particle Green’s function as

\[\langle \hat{O} \rangle = -\mathrm{Tr}[ \hat{O} G(\beta) ]\]

Parameters:

operatortriqs.operators.Operator: Operator \(\hat{O}\) for which the expectation value is computed. It has to be compatible with the operator content of the local Hamiltonian and the Green’s function structure.

Returns:

float: Expectation value \(\langle \hat{O} \rangle\) of the operator \(\hat{O}\).

partition_function()[source]

Partition function \(Z\) of the impurity model, computed from the pseudo particle Green’s function as

\[Z = -\mathrm{Tr}[ G(\beta) ] \cdot e^{-\beta \eta}\]

Returns:

float: Partition function \(Z\) of the impurity model.

pseudo_particle_chemical_potential()[source]

Pseudo particle chemical potential \(\eta\).

The pseudo particle chemical potential \(\eta\) is a shift of the pseudo particle energies used to enforce the normalization condition \(\textrm{Tr}[G(\beta)] = -1\) on the pseudo particle Green’s function \(G\).

Returns:

float: Pseudo particle chemical potential \(\eta\).

set_dynamic_interactions(dynint_ops, dynint_tau)[source]

Set retarded (dynamical) interactions of the impurity problem.

The interactions enter the impurity action as the term

\[\iint_0^\beta d\tau d\tau' \sum_{ij} O_i^\dagger(\tau) \, D_{ij}(\tau - \tau') \, O_j(\tau')\]

where \(O_i\) are the operators in dynint_ops and \(D_{ij}(\tau)\) is the kernel dynint_tau. The retarded interaction lines are expanded on the same footing as the hybridization lines, i.e. the expansion order max_order of solve() counts both kinds of lines.

Parameters:

dynint_opslist or tuple of triqs.operators.Operator: Local many-body operators \(O_i\) coupled by the retarded interaction, e.g. [n('up', 0)] for a density-density interaction. Each operator has to map every atom_diag subspace to at most one subspace.
dynint_tautriqs.gfs.Gf: Retarded interaction kernel \(D_{ij}(\tau)\) with target shape [len(dynint_ops)]*2, on a DLR imaginary time mesh (MeshDLRImTime) of either statistic, with the same beta as the solver, w_max smaller than or equal to, and eps larger than or equal to those of the solver. If the mesh differs from the solver mesh mesh_tau, the kernel is interpolated onto mesh_tau.

Returns:

None

Notes

The action term has no prefactor \(\frac{1}{2}\). A density-density interaction written as \(\frac{1}{2} \iint d\tau d\tau' \, n(\tau) \mathcal{U}(\tau - \tau') n(\tau')\) is therefore set with dynint_ops=[n] and dynint_tau equal to \(\frac{1}{2} \mathcal{U}(\tau)\).

The interaction kernel is fitted together with the hybridization function when calling solve(), so this method has to be called before solve(). Calling it again replaces the previously set interactions.

See the tutorial on the one-boson retarded interaction for a complete example.

solve(max_order, tol=0.0001, maxiter=10, mix=1.0, hyb_tol=None, hyb_comp=True, normalization='classic', spgf_max_order=None, verbose=True)[source]

Solve the impurity problem using pseudo particle self-consistent perturbation theory.

Parameters:

max_orderint: Maximum order of the perturbation theory.
tolfloat, optional: Tolerance for the convergence criterion.
maxiterint, optional: Maximum number of iterations.
mixfloat, optional: Mixing parameter for the self-consistent iteration.
hyb_tolfloat, optional: Tolerance for the hybridization function compression. Defaults to 0.1 * tol if not provided.
hyb_compbool, optional: Whether to compress the hybridization function. Default: True. When set to False, the hybridization function is represented using the full DLR basis.
spgf_max_orderint, optional: Maximum order of the single particle Green’s function diagram expansion. Default: max_order, is the Luttinger-Ward-consistent choice. However, using spgf_max_order = max_order + 1 improves convergence order by one, while giving up the conserving property of Luttinger-Ward.
normalizationstr, optional: Normalization method for the pseudo particle Green’s function. Default: 'classic'.

Returns:

None

Notes

Available normalization methods:

'classic': Classical normalization (Default)
'root': Root normalization
'ode+classic': ODE with classic normalization
'ode+root': ODE with root normalization
'odeG+classic': ODE on G with classic normalization
'odeG+root': ODE on G with root normalization