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arXiv:cond-mat/0310139

On the applicability of an equation of motion method at low-temperatures: comments on cond-mat/0309458 and cond-mat/0308413

Akira Oguri

Published Oct 7, 2003• 0 citations
arXivHTMLPDF

Abstract

The equation of motion method (EOM) is one of the approximations to calculate transport coefficients of interacting electron systems. The method is known to be useful to examine high-temperature properties. However, sometimes a naive application of the EOM fails to capture an important physics at low-energy scale, and it happens in recent preprints cond-mat/0309458 and cond-mat/0308413 which study a series of quantum dots. These preprints concluded that a unitarity-limit transport due to the Kondo resonance, which has been deduced from a Fermi-liquid behavior of the self-energy at T=0, $ω=0$ [A.O., PRB {\bf 63}, 115305 (2001)], does not occur. We show that the EOM self-energy obtained with a finite cluster has accidentally a singular $1/ω$ dependence around the Fermi energy, and it misleads one to the result incompatible with a Fermi-liquid ground state.

DOI
10.48550/arxiv.cond-mat/0310139

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The equation of motion method (EOM) is one of the approximations to calculate transport coefficients of interacting electron systems. The method is known to be useful to examine high-temperature properties. However, sometimes a naive application of the EOM fails to capture an important physics at low-energy scale, and it happens in recent preprints cond-mat/0309458 and cond-mat/0308413 which study a series of quantum dots. These preprints concluded that a unitarity-limit transport due to the Kondo resonance, which has been deduced from a Fermi-liquid behavior of the self-energy at T=0, ω=0ω=0ω=0 [A.O., PRB {\bf 63}, 115305 (2001)], does not occur. We show that the EOM self-energy obtained with a finite cluster has accidentally a singular 1/ω1/ω1/ω dependence around the Fermi energy, and it misleads one to the result incompatible with a Fermi-liquid ground state.
The equations-of-motion (EOM) hierarchy satisfied by the Green functions of a quantum dot embedded in an external mesoscopic network is considered within a high-order decoupling approximation scheme. Exact analytic solutions of the resulting coupled integral equations are presented in several limits. In particular, it is found that at the particle-hole symmetric point the EOM Green function is temperature independent due to a discontinuous change in the imaginary part of the interacting self-energy. However, this imaginary part obeys the Fermi liquid unitarity requirement away from this special point, at zero temperature. Results for the occupation numbers, the density of states, and the local spin susceptibility are compared with exact Fermi liquid relations and the Bethe ansatz solution. The approximation is found to be very accurate far from the Kondo regime. In contrast, the description of the Kondo effect is valid on a qualitative level only. In particular, we find that the Friedel sum-rule is considerably violated, up to 30%, and the spin susceptibility is underestimated. We show that the widely used simplified version of the EOM method, which does not account fully for the correlations on the network, fails to produce the Kondo correlations even qualitatively.
In a recent preprint (cond-mat/9905415), Fujimoto has used the Bethe ansatz to compute the finite temperature, zero frequency Drude weight of spin transport in the quantum O(3) non-linear sigma model in a magnetic field H≠0H \neq 0H=0. We show here that, contrary to his claims, the results are in accord with earlier semiclassical results (Sachdev and Damle, cond-mat/9610115). We also comment on his 1/N expansion, and show that it does not properly describe the long-time correlations.
We study the nonequilibrium transport properties of fully (exactly) screened Kondo quantum dots subject to a finite bias voltage or a finite temperature. First, we calculate the Fermi-liquid coefficients of the conductance for models with arbitrary spin, i.e., its leading behavior for small bias voltages or temperatures. Second, we determine the low-temperature behavior of the static susceptibility from the exactly known Bethe ansatz results for the magnetization. Third, we study the crossover from strong to weak coupling in the spin-1/2 and the spin-1 models coupled to one or two screening channels, respectively. Using a real-time renormalization group method we calculate the static and dynamical spin-spin correlation functions for the spin-1/2 model as well as the linear and differential conductance and the static susceptibility for the spin-1 model. We define various Kondo scales and discuss their relations. We assess the validity of the renormalization-group treatment by comparing with known results for the temperature dependence of the linear conductance and static susceptibility as well as the Fermi-liquid behavior at low energies.
We study the single impurity Anderson model (SIAM) using the equations of motion method (EOM), the non-crossing approximation (NCA), the one-crossing approximation (OCA), and Wilson's numerical renormalization group (NRG). We calculate the density of states and the linear conductance focusing on their dependence on the chemical potential and on the temperature paying special attention to the Kondo and Coulomb blockade regimes for a large range of model parameters. We report that some standard approximations based on the EOM technique display a rather unexpected poor behavior in the Coulomb blockade regime even at high temperatures. Our study offers a critical comparison between the different methods as well as a detailed compilation of the shortcomings and limitations due the approximations involved in each technique, thus allowing for a cost-benefit analysis of the different solvers that considers both numerical precision and computational performance.
In a series of papers we combined the Bethe ansatz with a kinetic (Boltzmann) equation in order to compute exactly transport properties in a Luttinger liquid with an impurity. In a recent comment [cond-mat/9708163], Skorik claimed that while our results were correct in linear response, there was a ``serious flaw'' in our calculation at finite voltage. We explain here why Skorik's arguments are inappropriate, and that there is no flaw in our work [cond-mat/9408068; cond-mat/9505031; cond-mat/9503172].
For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. Recent progress in understanding 1D quantum fluids beyond the low-energy limit is reviewed, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi-edge singularity and the Fermi-liquid theory, perturbation theory in the interaction strength, and new ways of treating finite-size properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. Existing results for various dynamic correlation functions are reviewed, in particular, the dynamic structure factor and the spectral function. Moreover, it is shown how a dispersion nonlinearity leads to finite particle lifetimes and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and the relation between the two is explained.