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arXiv:2307.07479 · Physical review. B./Physical review. B

Rigorous Wilsonian renormalization group for impurity models with a spectral gap

Peter Zalom

Published Nov 13, 2023• 10 citations
arXivHTMLPDF

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 $\mathrm{\ensuremath{\Delta}}$ 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 developing a numerical RG procedure that employs a discretization of the gapped tunneling densities of states into patches that accumulate at the gap edges. This reveals an unusual double scaling, which is a shared behavior between 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 the log-gap numerical RG, we also rigorously confirm the phenomenon of in-gap states escaping into the continuum. The gapped AIM is then tackled in an exact numerical RG approach and confirms quantitatively the assertions based on models with auxiliary metallic leads, but reveals that the other common approach used, for example, in the work of Moca and Roman [Phys. Rev. B 81, 235106 (2010)] is of strictly approximate nature.

DOI
10.1103/physrevb.108.195123

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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 \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ 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 developing a numerical RG procedure that employs a discretization of the gapped tunneling densities of states into patches that accumulate at the gap edges. This reveals an unusual double scaling, which is a shared behavior between 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 the log-gap numerical RG, we also rigorously confirm the phenomenon of in-gap states escaping into the continuum. The gapped AIM is then tackled in an exact numerical RG approach and confirms quantitatively the assertions based on models with auxiliary metallic leads, but reveals that the other common approach used, for example, in the work of Moca and Roman [Phys. Rev. B 81, 235106 (2010)] is of strictly approximate nature.
We use the numerical renormalization group method to investigate the spectral properties of a single-impurity Anderson model with a gap $\ensuremath{\delta}$ across the Fermi level in the conduction-electron spectrum. For any finite $\ensuremath{\delta}>0$, at half filling the ground state of the system is always a doublet. Away from half filling a quantum phase transition (QPT) occurs as function of the gap value $\ensuremath{\delta}$, and the system evolves from the strong-coupling (SC) Kondo-type state, corresponding to $\ensuremath{\delta}<{\ensuremath{\delta}}_{C}$ toward a localized moment (LM) regime for $\ensuremath{\delta}>{\ensuremath{\delta}}_{C}$. The opening of the gap leads to the formation of one (two) bound states when the system is in the SC (LM) regime. The evolution across the QPT of their positions and the corresponding weights together with the dynamic properties of the model are investigated.
We use the numerical renormalization group method (NRG) to investigate a single-impurity Anderson model with a coupling of the impurity to a superconducting host. Analysis of the energy flow shows that, contrary to previous belief, NRG iterations can be performed up to a large number of sites, corresponding to energy differences far below the superconducting gap Δ. This allows us to calculate the impurity spectral function A(ω) very accurately for frequencies |ω|∼Δ, and to resolve, in a certain parameter regime, sharp peaks in A(ω) close to the gap edge.
A simple impurity solver is shown to capture impurity-induced superconducting subgap states in quantitative agreement with the numerical renormalization group and quantum Monte Carlo simulations. The solver is based on the exact diagonalization of a single-impurity Anderson model with discretized superconducting reservoirs including only a small number of effective levels. Their energies and couplings to the impurity ddd level are chosen so as to best reproduce the Matsubara frequency dependence of the hybridization function. We provide a number of critical benchmarks and demonstrate the solver's efficiency in combination with the reduced basis method [V. V. Baran and D. R. Nichita, Phys. Rev. B 107, 144503 (2023)] by calculating the phase diagram for an interacting three-terminal junction.
Quantum impurity problems can be solved using the numerical renormalization group (NRG), which involves discretizing the free conduction electron system and mapping to a ``Wilson chain.'' It was shown recently that Wilson chains for different electronic species can be interleaved by use of a modified discretization, dramatically increasing the numerical efficiency of the RG scheme [Phys. Rev. B 89, 121105(R) (2014)]. Here we systematically examine the accuracy and efficiency of the ``interleaved'' NRG (iNRG) method in the context of the single impurity Anderson model, the two-channel Kondo model, and a three-channel Anderson-Hund model. The performance of iNRG is explicitly compared with ``standard'' NRG (sNRG): when the average number of states kept per iteration is the same in both calculations, the accuracy of iNRG is equivalent to that of sNRG but the computational costs are significantly lower in iNRG when the same symmetries are exploited. Although iNRG weakly breaks SU(NNN) channel symmetry (if present), both accuracy and numerical cost are entirely competitive with sNRG exploiting full symmetries. iNRG is therefore shown to be a viable and technically simple alternative to sNRG for high-symmetry models. Moreover, iNRG can be used to solve a range of lower-symmetry multiband problems that are inaccessible to sNRG.
We investigate the real-space spectral properties of strongly correlated multi-impurity arrays in the Kondo insulator regime. Employing a recently developed mapping onto an effective correlated cluster problem makes the problem accessible to the numerical renormalization group. The evolution of the spectrum as a function of cluster size and cluster site is studied. We applied the extended Lieb-Mattis theorem to predict whether the spectral function must vanish at the Fermi energy developing a true pseudogap or whether the spectral function remains finite at $\ensuremath{\omega}=0$. Our numerical renormalization group spectra confirm the predictions of the theorem and shows a metallic behavior at the surface of a cluster prevailing in arbitrary spatial dimensions. We present a conventional minimal extension of a particle-hole symmetric Anderson lattice model at U=0U=0U=0 that leads to a gapped bulk band but a surface band with mainly fff-orbital character for weak and moderate hybridization strength. The change in the site-dependent spectra upon introducing a Kondo hole in the center of the cluster is presented as a function of the hole-orbital energy. In particular, the spectral signatures across the Kosterlitz-Thouless-type quantum phase transition from a singlet to a local moment fixed point are discussed.
We propose a versatile strategy for numerical renormalization group (NRG) solution of general channel-mixing Kondo and Anderson impurity models beyond previous reach. We illustrate the strategy by investigating the quantum phase transitions in models of Anderson impurities coupled to sss- and ddd-wave superconducting baths. We discuss the effects of nontrivial channel-mixing in such models. Our strategy opens the door toward broad applications of NRG as impurity solver in cluster dynamical mean field theory for strongly correlated electron systems.