Rigorous Wilsonian Renormalization Group for impurity models with a spectral gap

Abstract

The Anderson impurity model (AIM) has long served as a cornerstone in the study of correlated electron systems. While numerical renormalization group (RG) offers great flexibility for metallic reservoirs, it becomes impossible in an unbiased way when a spectral gap $\Delta$ opens up in the tunneling density of states. The only known exception is provided by the superconducting bath. In this paper, we lift these limitations by a novel numerical RG procedure that employs a discretization of the gapped tunneling densities of states into patches which accumulate at the gap edges. This reveals an unusual double scaling which is a shared behavior by the superconducting and the scalar gapped AIMs. Moreover, it requires a special iterative diagonalization procedure with an alternating scheme for discarding states only every second iteration. The discretization and the diagonalization scheme form together, what we refer to as, the log-gap numerical RG. It is successfully applied to the superconducting and to the scalar gapped AIM. Consequently, it reveals that both models belong to the same RG equivalence class which manifests physically in common singlet-doublet quantum phase transitions accompanied by in-gap bound states of given parities. While superconducting AIM is mainly used for benchmarking of the log-gap numerical RG, we also rigorously confirm the phenomenon of in-gap states escaping into the continuum, which was recently indirectly considered in Ref. [1]. The gapped AIM is then tackled in a first ever exact numerical RG approach and confirms quantitatively assertions based on models with auxiliary metallic leads [2-5]. Moreover, it reveals that calculations performed in Refs. [6-8] are of strictly approximate nature.

16 pages, 10 figures

References in corpus (9)

• Continuous-time Monte Carlo methods for quantum impurity models
• The numerical renormalization group method for quantum impurity systems
• Kondo effect in quantum dots
• Density matrix numerical renormalization group for non-Abelian symmetries
• Adaptive logarithmic discretization for numerical renormalization group methods
• Analytic impurity solver with the Kondo strong-coupling asymptotics
• Anderson impurity in a semiconductor
• Quantum phase transition in a gapped Anderson model: A numerical renormalization group study
• Tunable reentrant Kondo effect in quantum dots coupled to metal-superconducting hybrid reservoirs

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• Renormalized and iterative formalism of the Andreev levels within large multi-parametric space
• Quantum phase transition in a double quantum dot Josephson junction driven by electron-electron interactions
• Magnetic flux controlled current phase relationship in double Quantum Dot Josephson junction
• Effects of electron-electron interaction and spin-orbit coupling on Andreev pair qubits in quantum dot Josephson junctions