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physica status solidi (b)

Non‐interacting Symmetric Single‐Impurity Anderson Model on a Lattice at Finite Temperatures

Zakaria M. M. Mahmoud, Florian Gebhard

Published Feb 7, 2019• 1 citation
Publisher

Abstract

We study the non‐interacting single‐impurity Anderson model (resonant level model) on a lattice at finite temperature as an illustrative example for an exactly solvable quantum‐mechanical problem, and derive the free energy, various thermodynamic potentials (internal energy, entropy, magnetization), and response functions (specific heat, zero‐field magnetic susceptibility). We calculate the magnetic screening cloud, and derive the corresponding correlation length in one dimension beyond which the correlations decay exponentially. The present results remain qualitatively applicable for the interacting single‐impurity Anderson model when the energy scale Г of the resonant‐level model is replaced by the Kondo scale TK.

DOI
10.1002/pssb.201800670

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We study the non‐interacting single‐impurity Anderson model (resonant level model) on a lattice at finite temperature as an illustrative example for an exactly solvable quantum‐mechanical problem, and derive the free energy, various thermodynamic potentials (internal energy, entropy, magnetization), and response functions (specific heat, zero‐field magnetic susceptibility). We calculate the magnetic screening cloud, and derive the corresponding correlation length in one dimension beyond which the correlations decay exponentially. The present results remain qualitatively applicable for the interacting single‐impurity Anderson model when the energy scale Г of the resonant‐level model is replaced by the Kondo scale TK.
The Anderson single impurity model, including its Kondo limit, can be seen as the hydrogen atom for correlated systems. Its most striking feature is the universal scaling of its low-energy properties governed by the appearance of a single emergent scale, the Kondo temperature TK{T}_{K}TK​. In this work, we demonstrate the emergence of a second independent energy scale in a quantum impurity model, the interacting resonant level model, which is equivalent to the anisotropic Kondo model. We study the local density of states at the impurity site in the ground state (zero temperature limit) of the model. We see collapse of data onto universal curves at low energies, however this is only achieved by defining a second low-energy scale with a different power law dependence on model parameters than the Kondo temperature. We provide an exact expression for this second critical exponent. We also report on a splitting of the central resonance as the interaction strength is increased in the absence of any external magnetic field in the model.
We present exact numerical results for the thermodyamic properties of a 4-level Anderson model. Ground state and excited states energies, average occupation numbers, susceptibility, specific heat, spin and charge fluctuations, spin-spin correlation functions are derived. The results are presented as a function of the temperature for several values of the Coulomb repulsion integral. Drastic changes of the calculated quantities appear in correspondence of the excited state energies for some critical values of the parameters. The results are in qualitative agreement with some other Anderson model calculations.
In this work we consider several exactly solvable models of magnetic impurities in critical quantum antiferromagnetic spin chains and multichannel Kondo impurities. Their ground-state properties are studied, and the finite set of nonlinear integral equations which exactly describe the thermodynamics of the models is constructed. We obtain several analytical low-energy expressions for the temperature, magnetic field, and frequency dependences of important characteristics of exactly solvable disordered quantum spin models and disordered multichannel Kondo impurities with essential many-body interactions. We show that the only low-energy parameter that gets renormalized is the velocity of the low-lying excitations (or the effective crossover scale connected with each impurity); the others appear to be universal. In our study several kinds of strong disorder important for experiments were used. Some of them produce low divergences in certain characteristics of our strongly disordered critical systems (compared with finite values for the homogeneous case or a single impurity). For weak disorder, or for narrow distributions of the local Kondo temperatures, our exact results reveal the presence of Kondo screening of disordered ensembles of magnetic impurities by low-lying excitations of the host. We point out that our results qualitatively coincide with the data of experiments on real disordered quasi-one-dimensional antiferromagnetic systems and with the similar behavior of some heavy metallic alloys.
We analyze the ground-state energy, magnetization, magnetic susceptibility, and Kondo screening cloud of the symmetric single-impurity Anderson model (SIAM) that is characterized by the bandwidth WWW, the impurity interaction strength UUU, and the local hybridization VVV. We compare Gutzwiller variational and magnetic Hartree-Fock results in the thermodynamic limit with numerically exact data from the density-matrix renormalization group (DMRG) method on large rings. To improve the DMRG performance, we use a canonical transformation to map the SIAM onto a chain with half the system size and open boundary conditions. We compare to Bethe ansatz results for the ground-state energy, magnetization, and spin susceptibility that become exact in the wide-band limit. Our detailed comparison shows that the field-theoretical description is applicable to the SIAM on a ring for a broad parameter range. Hartree-Fock theory gives an excellent ground-state energy and local moment for intermediate and strong interactions. However, it lacks spin fluctuations and thus cannot screen the impurity spin. The Gutzwiller variational energy bound becomes very poor for large interactions because it does not describe properly the charge fluctuations. Nevertheless, the Gutzwiller approach provides a qualitatively correct description of the zero-field susceptibility and the Kondo screening cloud. The DMRG provides excellent data for the ground-state energy and the magnetization for finite external fields. At strong interactions, finite-size effects make it extremely difficult to recover the exponentially large zero-field susceptibility and the mesoscopically large Kondo screening cloud.
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.
A magnetic moment in a metal or in a quantum dot is, at low temperatures, screened by the conduction electrons through the mechanism of the Kondo effect. This gives rise to spin-spin correlations between the magnetic moment and the conduction electrons, which can have a substantial spatial extension. We study this phenomenon, the so-called Kondo cloud, by means of the density matrix renormalization group method for the case of the single-impurity Anderson model. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. For several mechanisms---the gate potential and a magnetic field---which destroy the Kondo effect, we investigate the behavior of the screening cloud induced by these perturbations.