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arXiv:cond-mat/0509296

Spin and Charge Dynamics in a Renormalised Perturbation Theory

Alex C. Hewson

Published Sep 12, 2005• 0 citations
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Abstract

We calculate the spin and charge dynamical susceptibilities of a strongly correlated impurity model in a renormalised perturbation theory. The irreducible for vertices for the quasiparticle scattering are deduced from the renormalised parameters, which have been calculated by fitting of the low-lying levels of a numerical renormalisation group (NRG) calculation to those of an effective Anderson model. The susceptibilities are asymptotically exact in the low frequency limit and satisfy the Korringa-Shiba relation. By comparing the results with those calculated from a direct NRG calculation, we show that the renormalised perturbation theory (RPT) description gives a very good description of spin dynamics for all values of the local interaction U, not only at low frequencies but over the whole relevant frequency range. In the presence of a magnetic field the approach can be generalised using field dependent renormalised parameters to calculate the transverse and parallel spin dynamic susceptibilities. The RPT results give accurate results for the spin dynamics over the whole frequency range and for all values of the magnetic field.

DOI
10.48550/arxiv.cond-mat/0509296

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We calculate the spin and charge dynamical susceptibilities of a strongly correlated impurity model in a renormalised perturbation theory. The irreducible for vertices for the quasiparticle scattering are deduced from the renormalised parameters, which have been calculated by fitting of the low-lying levels of a numerical renormalisation group (NRG) calculation to those of an effective Anderson model. The susceptibilities are asymptotically exact in the low frequency limit and satisfy the Korringa-Shiba relation. By comparing the results with those calculated from a direct NRG calculation, we show that the renormalised perturbation theory (RPT) description gives a very good description of spin dynamics for all values of the local interaction U, not only at low frequencies but over the whole relevant frequency range. In the presence of a magnetic field the approach can be generalised using field dependent renormalised parameters to calculate the transverse and parallel spin dynamic susceptibilities. The RPT results give accurate results for the spin dynamics over the whole frequency range and for all values of the magnetic field.
We calculate the renormalized parameters for the quasiparticles and their interactions for the Hubbard model in the paramagnetic phase as deduced from the low-energy Fermi-liquid fixed point using the results of a numerical renormalization-group calculation (NRG) and dynamical mean-field theory (DMFT). Even in the low-density limit there is significant renormalization of the local quasiparticle interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$, in agreement with estimates based on the two-particle scattering theory of J. Kanamori [Prog. Theor. Phys. 30, 275 (1963)]. On the approach to the Mott transition we find a finite ratio for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$, where $2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$ is the renormalized bandwidth, which is independent of whether the transition is approached by increasing the on-site interaction UUU or on increasing the density to half filling. The leading ${\ensuremath{\omega}}^{2}$ term in the self-energy and the local dynamical spin and charge susceptibilities are calculated within the renormalized perturbation theory (RPT) and compared with the results calculated directly from the NRG-DMFT. We also suggest, more generally from the DMFT, how an approximate expression for the $\mathbf{q},\ensuremath{\omega}$ spin susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ can be derived from repeated quasiparticle scattering with a local renormalized scattering vertex.
We extend the renormalized quasiparticle description of the symmetric Anderson model in a magnetic field HHH, developed in earlier work, to the nonsymmetric model. The renormalized parameters are deduced from the low energy numerical renormalization group (NRG) fixed point for arbitrary field values. We find quasiparticle resonance widths, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}}_{\ensuremath{\sigma}}(H)$, which depend in general on the spin $\ensuremath{\sigma}$ as well as HHH. The low temperature static properties can be expressed completely in terms of these parameters, which can also be used as inputs for a renormalized perturbation theory. We show that taking into account repeated quasiparticle scattering gives results for the longitudinal and transverse dynamic spin susceptibilities which are in very good agreement with those obtained from direct NRG calculations.
We present dynamical mean-field theory (DMFT) results for the local spectral densities of the one- and two-particle response functions for the infinite-dimensional Hubbard model in a magnetic field. We look at the different regimes corresponding to half-filling, near half-filling, and well away from half-filling, for intermediate and strong values of the local interaction UUU. The low energy results are analyzed in terms of quasiparticles with field-dependent parameters. These renormalized parameters are determined by two different methods, both based on numerical renormalization group (NRG) calculations, and we find good agreement. Away from half-filling, the quasiparticle weights, ${z}_{\ensuremath{\sigma}}(H)$, differ according to the spin type $\ensuremath{\sigma}=\ensuremath{\uparrow}$ or $\ensuremath{\sigma}=\ensuremath{\downarrow}$. Using the renormalized parameters, we show that DMFT-NRG results for the local longitudinal and transverse dynamic spin susceptibilities in an arbitrary field can be understood in terms of repeated scattering of these quasiparticles. We also check Luttinger's theorem for the Hubbard model and find it to be satisfied in all parameter regimes and for all values of the magnetic field.
In recent years the numerical renormalization group (NRG) method has been extended to the calculation of dynamic response functions and transport properties of magnetic impurity models. The approach can now be applied more widely to lattice models of strongly correlated electron systems by the use of dynamical mean field theory (DMFT), in which the lattice problem is transformed into one for an effective impurity with an additional self-consistency constraint. We review these developments and assess the potential for further applications of this approach. We also discuss an alternative approach to renormalization, renormalized perturbation theory, in which the leading asymptotically exact results for the low temperature regime for a number of magnetic impurity models can be obtained within finite order perturbation theory.
We show that the functional renormalization group is a numerically cheap method to obtain the low-energy behavior of the Anderson impurity model describing a localized impurity level coupled to a bath of conduction electrons. Our approach uses an external magnetic field as the flow parameter, partial bosonization of the transverse spin fluctuations, and frequency-independent interaction vertices determined by Ward identities. The magnetic field serves also as a regulator for the bosonized spin fluctuations, which are suppressed at large field. We calculate the quasiparticle residue and spin susceptibility in the particle-hole symmetric case and obtain excellent agreement with the Bethe ansatz results for arbitrary coupling.
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 (ω/ω K ) and field ( H /ω 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.