Variational cluster approach to the single-impurity Anderson model

Abstract

We study the single impurity Anderson model by means of cluster perturbation theory and the variational cluster approach (VCA). An expression for the VCA grand potential for a system in a non interacting bath is presented. Results for the single-particle dynamics in different parameter regimes are shown to be in good agreement with established renormalization group results. We aim at a broad and comprehensive overview of the capabilities and shortcomings of the methods. We address the question to what extent the elusive low energy properties of the model are reproducible within the framework of VCA. These are furthermore benchmarked against continuous time quantum Monte Carlo calculations. We also discuss results obtained by an alternative, i.e. self consistent formulation of VCA, which was introduced recently in the context of non equilibrium systems.

Steady-state spectra, current, and stability diagram of a quantum dot: A nonequilibrium variational cluster approach

“…This has been already observed in Ref. 77. However, there this result was deduced from the convergence of results with increasing cluster size and not with a benchmark comparison with alternative numerical methods.…”

“…Reasonable results for the SIAM in equilibrium have been obtained previously by CPT as well as VCA. 77 Both cluster methods are approximate but they yield fairly reliable results and are therefore interesting due to their great flexibility and versatility. They are computationally not very demanding and allow in principle treating a wide range of fermionic/bosonic lattice Hamiltonians out of equilibrium.…”

“…Although the length of Kondo correlations scales exponentially in interaction strength, it has been shown in Ref. 77 that the VCA approximation, similarly to other approximate methods such as FRG or Gutzwiller wave functions, is capable of retaining qualitatively most features of the ongoing Kondo physics, although, possibly, with renormalized scales. The steady-state current can be evaluated on any link either within the central region or between the central and the neighboring regions, since we find that the continuity equation is fulfilled within at least 10 −6 relative to the steady-state current amplitude.…”

“…This is because these methods fulfill the Friedel sum rule in equilibrium. 77 Actually, nCPT tends to stay in the linear response regime longer than nVCA and therefore overestimates the current for low (but not very low) bias voltages. We model the leads by one-dimensional tight-binding chains, yielding a semicircular density of states [Eq.…”

“…32 and compared to the traditional VCA criterion in Ref. 77. It is based on the idea of starting from a system which is as similar as possible to the total, original system in terms of physically observable quantities.…”