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Journal of Physics C Solid State Physics

Time-dependent phase shift analysis of the symmetric Wolff model

Hans C. Fogedby

Published Jun 28, 1976• 3 citations
Publisher

Abstract

The local single-particle correlation function for the symmetric Wolff model is evaluated at zero temperature in the weak-coupling-long-time limit by a stationary-phase argument. The leading behaviour is given by an asymptotic expansion in the coupling strength. The correlation function furthermore has a weak essential singularity at vanishing coupling strength, corresponding to the exponential decay of transient Fermi surface effects, and is scale invariant under a simultaneous change of time scale and coupling-strength scale. In the intermediate and strong coupling regime the calculation breaks down due to many-body effects. Comments are made on some frequently used operator representations and some work by Mattis.

DOI
10.1088/0022-3719/9/12/003

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Physical Review A•Apr 1, 1973•17 citations
The local single-particle correlation function for the symmetric Wolff model is evaluated at zero temperature in the weak-coupling-long-time limit by a stationary-phase argument. The leading behaviour is given by an asymptotic expansion in the coupling strength. The correlation function furthermore has a weak essential singularity at vanishing coupling strength, corresponding to the exponential decay of transient Fermi surface effects, and is scale invariant under a simultaneous change of time scale and coupling-strength scale. In the intermediate and strong coupling regime the calculation breaks down due to many-body effects. Comments are made on some frequently used operator representations and some work by Mattis.
The local single-particle and density correlation functions for the symmetric Wolff model are derived analytically. The scattering channels are separated by means of the Stratonovich-Hubbard transformation and the associated one-body problem is solved explicitly in the long-time limit. A stationary phase argument yields expressions for the correlation functions in the weak-coupling limit. The single-particle correlation function acquires a time-dependent phase shift due to the readjustment of the spin-up and spin-down Fermi systems. The spin-up and spin-down Fermi systems relax on a timescale determined by the inverse bandwidth. In the intermediate- and strong-coupling regimes many-body effects intervene and the calculation breaks down.
For pt.I see ibid., vol.10, p.2869 (1977). The local single-particle and density correlation functions for the symmetric Wolff model (1961) are studied in the asymptotic weak-coupling-long-time-low-temperature limit. The scattering channels are separated by means of the Stratonovich-Hubbard transformation and the associated one-body problem is solved explicitly in the low-temperature limit. A steepest-descent argument yields expressions for the correlation functions in the weak-coupling limit. The correlation functions are characterised by a time dependent phase shift which decays on a time scale of the order of the bandwidth. In the intermediate and strong coupling regime many-body effects intervene and the calculation breaks down; in that context comments are made on some recent work by Mattis (1976).
The longitudinal and transverse space and time dependent spin correlation functions for the one-dimensional Heisenberg-Ising model are derived asymptotically in the weak-coupling-low-frequency-long-wavelength limit within the framework of ordinary many-body theory (field theory). By means of the Jordan-Wigner transformation and using methods developed previously for the Tomonaga model (1950), the zero-temperature correlation functions is expressed as functional integrals. The associated one-body problem is explicitly soluble in the asymptotic regions and the correlation functions assume the form of Gaussian integrals which are evaluated analytically in the weak-coupling-low-frequency-long-wavelength limit. Apart from a linear correction to the transverse exchange constant agreement with Luther and Peschel (1975) is obtained.
A local moment approach is developed for single-particle excitations of a symmetric Anderson impurity model (AIM) with a soft-gap hybridization vanishing at the Fermi level: I | | r , with r >0. Local moments are introduced explicitly from the outset, and a two-self-energy description is employed in which 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 where it is perturbatively exact for those r -domains in which perturbation theory in U is non-singular. While the primary emphasis is on single-particle dynamics, the quantum phase transition between strong-coupling (SC) and local moment (LM) phases can also be addressed directly; for the spin-fluctuation regime in particular a number of asymptotically exact results are thereby obtained, notably for the behaviour of the critical U c ( r ) separating SC/LM states and the Kondo scale m ( r ) characteristic of the SC phase. Results for both single-particle spectra and SC/LM phase boundaries are found to agree well with recent numerical renormalization group (NRG) studies; and a number of further testable predictions are made. Single-particle spectra are examined systematically for both SC and LM states; in particular, for all 0 r <½, 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 wings of the Kondo resonance, as well as characteristic low-energy Fermi liquid behaviour and the exponential diminution with U of the Kondo scale itself. The normal AIM is found to represent a particular case of more generic behaviour characteristic of the r >0 SC phase which, in agreement with conclusions drawn from recent NRG work, may be viewed as a non-trivial but natural generalization of Fermi liquid physics.
We present a general theory for the calculation of the single-particle time-correlation function $\ensuremath{\alpha}(t)=〈[a,{a}^{\ifmmode\dagger\else\textdagger\fi{}}(t)]〉$ which is the canonical average of the commutator between the particle annihilation aaa and creation ${a}^{\ifmmode\dagger\else\textdagger\fi{}}(t)$ at time ttt. The theory is based on the projection-operator method. The complex spectral function $\stackrel{^}{\ensuremath{\alpha}}(\ensuremath{\omega})$ is expressed in terms of the natural frequency of oscillation $\ensuremath{\Omega}$ and the width function $\stackrel{^}{\ensuremath{\gamma}}(\ensuremath{\omega})$. From the analytical property of $\stackrel{^}{\ensuremath{\alpha}}(\ensuremath{\omega})$ in the complex $\ensuremath{\omega}$ plane for the weak-coupling limit, the long-time behavior of the correlation function $\ensuremath{\alpha}(t)$ and the relaxation functions is obtained. For a harmonic oscillator immersed in a heat bath, the perturbation calculations for $\ensuremath{\Omega}$ and $\stackrel{^}{\ensuremath{\gamma}}(\ensuremath{\omega})$ are given in the power of the coupling constant. By means of this series, the spectral function $\stackrel{^}{\ensuremath{\alpha}}(\ensuremath{\omega})$ for a single normal mode of an anharmonic system is explicitly calculated as a function of the frequency $\ensuremath{\omega}$ and the temperature TTT. As a possible application of the results the electrical conductivity due to a localized mode is discussed.