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arXiv:0803.1251 · Journal of Physics Condensed Matter

Numerical renormalization group calculation of near-gap peaks in spectral functions of the Anderson model with superconducting leads

T. Hecht, Andreas Weichselbaum, Jan von Delft, R. Bulla

Published Jun 4, 2008• 57 citations
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Abstract

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.

DOI
10.1088/0953-8984/20/27/275213

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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.
We introduce an alternative approach for obtaining smooth finite-temperature spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique. It is based on calculating first the Green's function on the imaginary-frequency axis, followed by an analytic continuation to the real-frequency axis using Pad\'e approximants. The arbitrariness in choosing a suitable kernel in the conventional broadening approach is thereby removed and, furthermore, we find that the Pad\'e method is able to resolve fine details in spectral functions with less artifacts on the scale of $\ensuremath{\omega}\ensuremath{\sim}T$. We discuss the convergence properties with respect to the NRG calculation parameters (discretization $\ensuremath{\Lambda}$, zzz-averaging, truncation cutoff) and the number of Matsubara points taken into account in the analytic continuation. We test the technique on the single-impurity Anderson model and the Hubbard model (within the dynamical mean-field theory). For the Anderson impurity model, we discuss the shape of the Kondo resonance and its temperature dependence. For the Hubbard model, we discuss the inner structure of the Hubbard bands in metallic and insulating solutions at half-filling, as well as in the doped Mott insulator. Based on these test cases, we conclude that the Pad\'e approximant approach provides improved results for spectral functions at low-frequency scales of $\ensuremath{\omega}\ensuremath{\lesssim}T$ and that it is capable of resolving sharp spectral features also at high frequencies.
Using the functional renormalization group (FRG) and the numerical renormalization group (NRG), we calculate the spectral function of the Anderson impurity model at zero and finite temperatures. In our FRG scheme spin fluctuations are treated nonperturbatively via a suitable Hubbard-Stratonovich field, but vertex corrections are neglected. A comparison with our highly accurate NRG results shows that this FRG scheme provides a good description of the spectral line shape at zero and finite temperatures both in the weak- and strong-coupling regimes, although at zero temperature the FRG is not able to reproduce the known exponential narrowing of the Kondo resonance at strong coupling.
Using a recently developed perturbative renormalization group (RG) scheme, we calculate analytically the spectral function of a Kondo impurity for either large frequencies $\ensuremath{\omega}$ or large magnetic field B and arbitrary frequencies. For large $\ensuremath{\omega}\ensuremath{\gg}\mathrm{max}[{B,T}_{K}]$ the spectral function decays as $1/{\mathrm{ln}}^{2}[\ensuremath{\omega}{/T}_{K}]$ with prefactors which depend on the magnetization. The spin-resolved spectral function displays a pronounced peak at $\ensuremath{\omega}\ensuremath{\sim}B$ with a characteristic asymmetry. In a detailed comparison with results from numerical renormalization group (NRG) and bare perturbation theory in next-to-leading logarithmic order, we show that our perturbative RG scheme is controlled by the small parameter $1/\mathrm{ln}[\mathrm{max}(\ensuremath{\omega}{,B)/T}_{K}].$ Furthermore, we assess the ability of the NRG to resolve structures at finite frequencies.
We study the symmetric Anderson-Holstein (AH) model at zero temperature with Wilson's numerical renormalization-group (NRG) technique to investigate the interplay between the electron-electron and electron-phonon interactions. An improved method for calculating the phonon propagator using the NRG technique is presented. It is more accurate and reliable than the previous methods in that it is written in the form of the Dyson equation which calculates the phonon renormalization explicitly, and satisfies the boson sum rule better. The method is applied to calculate the renormalized phonon propagators along with the electron propagators as the on-site Coulomb repulsion U and electron-phonon coupling constant g are varied. As g is increased from 0, the phonon mode gets softened (hardened) for $U\ensuremath{\gtrsim}(\ensuremath{\lesssim}{)U}_{s}(\ensuremath{\Delta})$ depending on the parameters of the AH model, U, and the hybridization $\ensuremath{\Delta}.$ And, for $g\ensuremath{\gtrsim}{g}_{\mathrm{co}},$ one crosses over to the regime where the phonon mode splits into two components, one of which developing into a soft mode and the other approaching back to the bare frequency as g is further increased. Correlated with the emergence of the soft mode, the central peak of the electron spectral function is severely suppressed. These NRG calculations are compared with the standard Green's-function results for the weak-coupling regime to understand the phonon renormalization and soft mode.
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.
The symmetric Anderson impurity model with a hybridization vanishing at the Fermi level, Δ I ∝|ω| r , is studied via the numerical renormalization group (NRG) at T = 0; and detailed comparison made with predictions arising from the local moment approach (LMA), a recently developed many-body theory which is found to provide a remarkably successful description of the problem. Results for the `normal' ( r = 0) impurity model are obtained as a specific case, and likewise compared. Particular emphasis is given both to single-particle excitation dynamics, and to the transition between the strong-coupling (SC) and local moment (LM) phases of the model. Scaling characteristics and asymptotic behaviour of the SC/LM phase boundaries are considered. Single-particle spectra D (ω) are investigated in some detail, for the SC phase in particular. Here, in accordance with a recently established result, the modified spectral functions ℱ(ω)∝|ω| r D (ω) are found to contain a generalized Kondo resonance that is ubiquitously pinned at the Fermi level; and which exhibits a characteristic low-energy Kondo scale, ω K ( r ), that narrows progressively upon approach to the SC→LM transition, where it vanishes. Universal scaling of the spectra as the transition is approached thus results. The scaling spectrum characteristic of the normal Anderson model is recovered as a particular case, that exemplifies behaviour characteristic of the SC phase generally, and which is captured quantitatively by the LMA. In all cases the r -dependent scaling spectra are found to possess characteristic low-energy asymptotics, but to be dominated by generalized Doniach-Sunjic tails, in agreement with LMA predictions.