A word on the algorithm
The hybridization expansion approach to solving generalized Anderson impurity models has been combined with a plethora of numerical methods. One of these methods is the bold hybridization expansion [1] in terms of the local atomic propagator \(G(\tau)\), often referred to as the pseudo-particle approach, since it can be derived by introducing a pseudo particle for each many-body state in the impurity local Hilbert space.
In the bold formulation \(G(\tau)\) is self-consistently computed using the Dyson equation
where \(G_0(\tau)\) is the atomic many-body propagator and \(\Sigma(\tau)\) is the pseudo-particle self-energy truncated at a finite expansion order \(n\) in the hybridization function \(\Delta(\tau)\).
Once convergence is reached physical response functions like the single particle Green’s function \(g(\tau) = \langle \mathcal{T} c(\tau) c^\dagger(0) \rangle\) can be evaluted by a separate diagrammatic series (similar to \(\Sigma\)).
The triqs_soehyb solver implements the bold hybridization expansion using the Discrete Lehmann Representation (DLR) [5] [4] for compact representation of propagators in imaginary time \(\tau\) and a separate hybridization function compression approach [3] (based on the famous AAA algorithm) to evaluate the diagram series for \(\Sigma\) with lower computational complexity than standard quadrature integration. [2]