Single-particle dynamics of the Anderson model: a local moment approach

A non-perturbative local moment approach to single-particle dynamics of the general asymmetric Anderson impurity model is developed. The approach encompasses all energy scales and interaction strengths. It captures thereby strong coupling Kondo behaviour, including the resultant universal scaling behaviour of the single-particle spectrum; as well as the mixed valent and essentially perturbative empty orbital regimes. The underlying approach is physically transparent and innately simple, and as such is capable of practical extension to lattice-based models within the framework of dynamical mean-field theory.Comment: 26 pages, 9 figure

Dynamics and transport properties of heavy fermions: theory

The paramagnetic phase of heavy fermion systems is investigated, using a non-perturbative local moment approach to the asymmetric periodic Anderson model within the framework of dynamical mean field theory. The natural focus is on the strong coupling Kondo-lattice regime wherein single-particle spectra, scattering rates, dc transport and optics are found to exhibit w/w_L,T/w_L scaling in terms of a single underlying low-energy coherence scale w_L. Dynamics/transport on all relevant (w,T)-scales are encompassed, from the low-energy behaviour characteristic of the lattice coherent Fermi liquid, through incoherent effective single-impurity physics likewise found to arise in the universal scaling regime, to non-universal high-energy scales; and which description in turn enables viable quantitative comparison to experiment.Comment: 27 pages, 12 figure

Field-dependent dynamics of the Anderson impurity model

Single-particle dynamics of the Anderson impurity model in the presence of a magnetic field $H$ are considered, using a recently developed local moment approach that encompasses all energy scales, field and interaction strengths. For strong coupling in particular, the Kondo scaling regime is recovered. Here the frequency ($\omega/\omega_{\rm K}$) and field ($H/\omega_{\rm K}$) dependence of the resultant universal scaling spectrum is obtained in large part analytically, and the field-induced destruction of the Kondo resonance investigated. The scaling spectrum is found to exhibit the slow logarithmic tails recently shown to dominate the zero-field scaling spectrum. At the opposite extreme of the Fermi level, it gives asymptotically exact agreement with results for statics known from the Bethe ansatz. Good agreement is also found with the frequency and field-dependence of recent numerical renormalization group calculations. Differential conductance experiments on quantum dots in the presence of a magnetic field are likewise considered; and appear to be well accounted for by the theory. Some new exact results for the problem are also established

Anderson impurities in gapless hosts: comparison of renormalization group and local moment approaches

The symmetric Anderson impurity model, with a soft-gap hybridization vanishing at the Fermi level with power law r > 0, is studied via the numerical renormalization group (NRG). Detailed comparison is 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. 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 are investigated in some detail, for the SC phase in particular. Here, the modified spectral functions 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 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, and 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.Comment: 26 pages, 14 figures, submitted for publicatio

A Spectral Method for Elliptic Equations: The Neumann Problem

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation $-\Delta u+\gamma u=f$ over $\Omega$ with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball $B$, and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $u_{n}$ of degree $\leq n$ that is convergent to $u$. The transformation from $\Omega$ to $B$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}(\overline{\Omega})$ and assuming $\partial\Omega$ is a $C^{\infty}$ boundary, the convergence of $\Vert u-u_{n}\Vert_{H^{1}}$ to zero is faster than any power of $1/n$. Numerical examples in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ show experimentally an exponential rate of convergence.Comment: 23 pages, 11 figure

Local-Ansatz Approach with Momentum Dependent Variational Parameters to Correlated Electron Systems

A new wavefunction which improves the Gutzwiller-type local ansatz method has been proposed to describe the correlated electron system. The ground-state energy, double occupation number, momentum distribution function, and quasiparticle weight have been calculated for the half-filled band Hubbard model in infinite dimensions. It is shown that the new wavefunction improves the local-ansatz approach (LA) proposed by Stollhoff and Fulde. Especially, calculated momentum distribution functions show a reasonable momentum dependence. The result qualitatively differs from those obtained by the LA and the Gutzwiller wavefunction. Furthermore, the present approach combined with the projection operator method CPA is shown to describe quantitatively the excitation spectra in the insulator regime as well as the critical Coulomb interactions for a gap formation in infinite dimensions.Comment: To be published in Phys. Soc. Jpn. 77 No.11 (2008

Spectral properties of a narrow-band Anderson model

We consider single-particle spectra of a symmetric narrow-band Anderson impurity model, where the host bandwidth $D$ is small compared to the hybridization strength $\Delta_{0}$. Simple 2nd order perturbation theory (2PT) in $U$ is found to produce a rich spectral structure, that leads to rather good agreement with extant Lanczos results and offers a transparent picture of the underlying physics. It also leads naturally to two distinct regimes of spectral behaviour, $\Delta_{0}Z/D\gg 1$ and $\ll 1$ (with $Z$ the quasi-particle weight), whose existence and essential characteristics are discussed and shown to be independent of 2PT itself. The self-energy $\Sigma_{i\omega}$ is also examined beyond the confines of PT. It is argued that on frequency scales of order $\omega\sim\sqrt{Delta_{0}D}$, the self-energy in {\em strong} coupling is given precisely by the 2PT result, and we point out that the resultant poles in $\Sigma_{i\omega}$ connect continuously to that characteristic of the atomic limit. This in turn offers a natural rationale for the known inability of the skeleton expansion to capture such behaviour, and points to the intrinsic dangers of partial infinite-order summations that are based on PT in $U$.Comment: 10 pages, 2 Postscript figures, uses RevTex 3.1; accepted for publication in Phys. Rev. B1

A Local Moment Approach to magnetic impurities in gapless Fermi systems

A local moment approach is developed for the single-particle excitations of a symmetric Anderson impurity model (AIM), with a soft-gap hybridization vanishing at the Fermi level with a power law r > 0. Local moments are introduced explicitly from the outset, and a two-self-energy description is employed in which the single-particle excitations are coupled dynamically to low-energy transverse spin fluctuations. The resultant theory is applicable on all energy scales, and captures both the spin-fluctuation regime of strong coupling (large-U), as well as the weak coupling regime. While the primary emphasis is on single particle dynamics, the quantum phase transition between strong coupling (SC) and (LM) phases can also be addressed directly; for the spin-fluctuation regime in particular a number of asymptotically exact results are thereby obtained. Results for both single-particle spectra and SC/LM phase boundaries are found to agree well with recent numerical renormalization group (NRG) studies. A number of further testable predictions are made; in particular, for r < 1/2, spectra characteristic of the SC state are predicted to exhibit an r-dependent universal scaling form as the SC/LM phase boundary is approached and the Kondo scale vanishes. Results for the `normal' r = 0 AIM are moreover recovered smoothly from the limit r -> 0, where the resultant description of single-particle dynamics includes recovery of Doniach-Sunjic tails in the Kondo resonance, as well as characteristic low-energy Fermi liquid behaviour.Comment: 52 pages, 19 figures, submitted to Journal of Physics: Condensed Matte

A spectral method for elliptic equations: the Dirichlet problem

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.Comment: This is latex with the standard article style, produced using Scientific Workplace in a portable format. The paper is 22 pages in length with 8 figure

For Whom, and for What, is Experience Sampling More Accurate Than Retrospective Report?

The experience sampling method (ESM) is often used in research, and promoted for clinical use, with the rationale that it avoids problematic inaccuracies and biases that attend retrospective measures of mental phenomena. Research suggests that averaged scores from ESM data are more accurate than retrospective ratings. However, it is not known how well individuals can remember information about momentary (rather than averaged) mental states, nor how accurately they estimate the dynamic covariation of these states. Individual differences in retrospective accuracy are also poorly understood. In two pre-registered studies, we examined differences between retrospective memory for stress and self-esteem and data gathered via experience sampling and examined whether alexithymia predicted accuracy. Results of both studies revealed substantial discrepancies between retrospective ratings and ESM ratings, especially for momentary states and their covariation. Alexithymia was positively related to recognition of stress means and variability but unrelated to recall of either stress or self-esteem, their variability, or their covariation. These findings suggest that experience sampling may be more useful than self-report when precise information is needed about the timing of mental states and dynamics among them