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arXiv:1003.3188 · Physical Review B

Spectral function of the Anderson impurity model at finite temperatures

Aldo Isidori, David Roosen, Lorenz Bartosch, Walter Hofstetter, Peter Kopietz

Published Jun 17, 2010• 20 citations
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Abstract

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.

DOI
10.1103/physrevb.81.235120

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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.
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.
We investigate nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening $\ensuremath{\Gamma}/2$ can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions $U\ensuremath{\lesssim}3\ensuremath{\Gamma}$. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.
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.
The Anderson Impurity Model is a generic model to describe systems in which one or more energy levels are coupled to an electronic host. On the energy levels the electrons are subject to a Coulomb interaction. Possible applications of this model are magnetic atoms in a metal or quantum dot systems. Already for one impurity it displays very interesting physics including the well known Kondo effect. For two and more impurities the properties of the system in the local moment regime are determined by an interplay of the Kondo effect and the coupling between the impurity spins. Due to the interplay of multiple energy scales, this model poses a serious task for any computational tool. For a single impurity the model can be very efficiently treated using Wilson"s numerical renormalization group (NRG). Since the numerical effort of NRG calculations rises exponentially with the number of impurities, generically it cannot be applied to systems with more than two impurities. The functional renormalization group (fRG) has been very successfully applied to quantum impurity problems in the past. In this work it will be used to calculate the single-particle dynamics for Anderson-like quantum impurity systems in equilibrium. A comparison to NRG in the case of a single impurity will serve as a benchmark before systems with two impurities and quantum dot systems with two and three dots will be studied. The interplay between the Kondo effect and magnetic and orbital ordering will be investigated in these systems. Although the truncation in the presented fRG scheme is motivated by second order perturbation theory, it will be shown that fRG works also in those cases where perturbation theory fails. Generally it will turn out that fRG is capable of treating a broad variety of Anderson impurity systems within the same framework, including asymmetric impurity levels and interaction between multiple impurities.
Numerical renormalization group (NRG) is formulated for nonequilibrium steady-state by converting finite-lattice many-body eigenstates into scattering states. Extension of the full-density-matrix NRG for a biased Anderson impurity model, simplified by formulating with the original orbital basis as the Hamiltonian, enables detailed studies of the sub-Kondo spectral evolution in the zero-temperature limit, confirming the double-resonance structure at bias of the Kondo energy scale TKT_KTK​. The distribution shows distinct multi-scale spectral features at energy ωωω below the Kondo scale (ω≲TKω\lesssim T_Kω≲TK​) and near the bias (ω≳Vω\gtrsim Vω≳V), leading to the nonequilibrium temperature TlocT_{\rm loc}Tloc​ local to the Kondo dot scaling as kBTloc≈Vk_BT_{\rm loc}\approx VkB​Tloc​≈V for V≫TKV\gg T_KV≫TK​. The current-voltage relation in the low-temperature limit (T≪TKT\ll T_KT≪TK​) deviates from the unitary limit as the bias exceeds the Kondo scale (V/2≳TKV/2\gtrsim T_KV/2≳TK​) and reaches the current saturation regime.
We introduce a method to obtain the specific heat of quantum impurity models via a direct calculation of the impurity internal energy requiring only the evaluation of local quantities within a single numerical renormalization group (NRG) calculation for the total system. For the Anderson impurity model we show that the impurity internal energy can be expressed as a sum of purely local static correlation functions and a term that involves also the impurity Green function. The temperature dependence of the latter can be neglected in many cases, thereby allowing the impurity specific heat Cimp{C}_{\mathrm{imp}}Cimp​ to be calculated accurately from local static correlation functions; specifically via ${C}_{\mathrm{imp}}=\frac{\ensuremath{\partial}{E}_{\mathrm{ionic}}}{\ensuremath{\partial}T}+\frac{1}{2}\frac{\ensuremath{\partial}{E}_{\mathrm{hyb}}}{\ensuremath{\partial}T}$, where Eionic{E}_{\mathrm{ionic}}Eionic​ and Ehyb{E}_{\mathrm{hyb}}Ehyb​ are the energies of the (embedded) impurity and the hybridization energy, respectively. The term involving the Green function can also be evaluated in cases where its temperature dependence is non-negligible, adding an extra term to Cimp{C}_{\mathrm{imp}}Cimp​. For the nondegenerate Anderson impurity model, we show by comparison with exact Bethe ansatz calculations that the results recover accurately both the Kondo induced peak in the specific heat at low temperatures as well as the high-temperature peak due to the resonant level. The approach applies to multiorbital and multichannel Anderson impurity models with arbitrary local Coulomb interactions. An application to the Ohmic two-state system and the anisotropic Kondo model is also given, with comparisons to Bethe ansatz calculations. The approach could also be of interest within other impurity solvers, for example, within quantum Monte Carlo techniques.