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Journal of Physics C Solid State Physics

Theory of the symmetric Wolff model. I. Weak-coupling-long-time-zero-temperature limit

Hans C. Fogedby

Published Aug 14, 1977• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/10/15/021

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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 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 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.
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.
In theoretical studies of two-dimensional (2D) systems, the Mermin-Wagner theorem prevents continuous symmetry breaking at any finite temperature, thus forbidding a Landau phase transition at a critical temperature T_c T c . The difficulty arises when many-body theoretical studies predict a Landau phase transition at finite temperatures, which contradicts the Mermin-Wagner theorem and is termed a pseudo phase transition. To tackle this problem, we systematically develop a symmetrization scheme, defined as averaging physical quantities over all symmetry-breaking states, thus ensuring that it preserves the Mermin-Wagner theorem. We apply the symmetrization scheme to the GW-covariance calculation for the 2D repulsive Hubbard model at half-filling in the intermediate-to-strong coupling regime and at low temperatures, obtaining the one-body Green’s function and spin-spin correlation function, and benchmark them against Determinant Quantum Monte Carlo (DQMC) with good agreement. The spin-spin correlation functions are approached within the covariance theory, a general method for calculating two-body correlation functions from a one-particle starting point, such as the GW formalism used here, which ensures the preservation of the fundamental fluctuation-dissipation relation (FDR) and Ward-Takahashi identities (WTI). With the FDR and WTI satisfied, we conjecture that the \chi χ -sum rule, a fundamental relation from the Pauli exclusion principle, can be used to probe the reliability of many-body methods, and demonstrate this by comparing the GW-covariance and mean-field-covariance approaches. This work provides a novel framework to investigate the strong-coupling and doped regime of the 2D Hubbard model, which is believed to be applicable to real high- T_c T c cuprate superconductors.
The density, spin density and pair correlation functions for the spin-half Tomonaga model are derived analytically within the framework of ordinary many body theory. The author extends and formalises the method of Dzyaloshinskii and Larkin (Sov. Phys. JETP., vol.38, p.202 (1974)). The dynamical constraint imposed by the conservation law is expressed in terms of Wick's theorem and the correlation functions are formulated as functional integrals by means of the Stratonovich-Hubbard transformation (1957-59). The associated one-body problem is explicitly soluble and the correlation functions assume the form of Gaussian integrals which are evaluated analytically.