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Physical Review A

Theory of Single-Particle Time-Correlation Functions

Shoon K. Kim, Robert S. Wilson

Published Apr 1, 1973• 17 citations
Publisher

Abstract

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 $a$ and creation ${a}^{\ifmmode\dagger\else\textdagger\fi{}}(t)$ at time $t$. 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 $T$. As a possible application of the results the electrical conductivity due to a localized mode is discussed.

DOI
10.1103/physreva.7.1396

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Physical review. B, Condensed matter•Jan 1, 1984•65 citations
We present a general theory for the calculation of the single-particle time-correlation function \ensuremathα(t)=〈[a,a\ifmmode†\else\textdagger\fi(t)]〉\ensuremath{\alpha}(t)=〈[a,{a}^{\ifmmode\dagger\else\textdagger\fi{}}(t)]〉\ensuremathα(t)=〈[a,a\ifmmode†\else\textdagger\fi(t)]〉 which is the canonical average of the commutator between the particle annihilation aaa and creation a\ifmmode†\else\textdagger\fi(t){a}^{\ifmmode\dagger\else\textdagger\fi{}}(t)a\ifmmode†\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 \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ and the width function \stackrel{^}{\ensuremath{\gamma}}(\ensuremath{\omega}). From the analytical property of \stackrel{^}{\ensuremath{\alpha}}(\ensuremath{\omega}) in the complex \ensuremathω\ensuremath{\omega}\ensuremathω plane for the weak-coupling limit, the long-time behavior of the correlation function \ensuremathα(t)\ensuremath{\alpha}(t)\ensuremathα(t) and the relaxation functions is obtained. For a harmonic oscillator immersed in a heat bath, the perturbation calculations for \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ 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 \ensuremathω\ensuremath{\omega}\ensuremathω and the temperature TTT. As a possible application of the results the electrical conductivity due to a localized mode is discussed.
A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where H\mathcal{H}H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order mmm, and also to satisfy the appropriate boson commutators and fermion anticommutators to order mmm, then the problem is solved to order mmm. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of H\mathcal{H}H and are orthonormal, all to order mmm. A procedure is otained for finding the (m+1m+1m+1)-order contributions to the particle creation energies, in terms of the mmm-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.
A fermion system with a simple attractive interaction is discussed with the aid of time-dependent correlation functions. Although perturbation theory is inapplicable, a sequence of correlation approximations described in the first paper of this series can be employed. The lowest approximation in the sequence expresses the two-particle correlation function in terms of single-particle functions and leads to the Hartree approximation; the second expresses three-particle correlation function in terms of one- and two-particle correlation functions and leads to the time-dependent correlation functions that characterize the super-conducting model of Bardeen, Cooper, and Schrieffer. In the second section of this paper these correlation functions are determined and the thermodynamic properties of the superconductor are calculated from them.In the third section of the paper, the electromagnetic effects of the superconductor predicted by the Bardeen-Cooper-Schrieffer time-dependent correlation functions are considered. Their unsatisfactory description of current conservation is indicated and overcome in the fourth section by a more accurate solution valid at nonvanishing temperature. This solution predicts different diffusive properties but the same Meissner effect and superconductive behavior, since the longitudinal current correlation function is modified while the transverse current correlation functions is not.The fifth section of the paper is devoted to the properties of a pure superconductor which depend on the lifetimes of the single-particle excitations. The effect of these lifetimes on the static electrical conductivity is determined, and it is shown that they do not destroy supercurrents although they eliminate a gap in the single-particle excitation spectrum. Their effect on the thermal conductivity is also calculated using heat current correlation functions. It is shown that a model which treats the lifetime of the single-particle excitation due to lattice interactions as constant yields results in agreement with observed thermal conductivities.
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.
We explore transport properties in a disordered nonlinear chain of classical harmonic oscillators, and thereby identify a regime exhibiting behavior analogous to that seen in quantum many-body-localized systems. Through extensive numerical simulations of this system connected at its ends to heat baths at different temperatures, we computed the heat current and the temperature profile in the nonequilibrium steady state as a function of system size N, disorder strength Δ, and temperature T. The conductivity κ_{N}, obtained for finite length (N), saturates to a value κ_{∞}>0 in the large N limit, for all values of disorder strength Δ and temperature T>0. We show evidence that for any Δ>0 the conductivity goes to zero faster than any power of T in the (T/Δ)→0 limit, and find that the form κ_{∞}∼e^{-B|ln(CΔ/T)|^{3}} fits our data. This form has earlier been suggested by a theory based on the dynamics of multioscillator chaotic islands. The finite-size effect can be κ_{N}<κ_{∞} due to boundary resistance when the bulk conductivity is high (the weak disorder case), or κ_{N}>κ_{∞} due to direct bath-to-bath coupling through bulk localized modes when the bulk is weakly conducting (the strong disorder case). We also present results on equilibrium dynamical correlation functions and on the role of chaos on transport properties. Finally, we explore the differences in the growth and propagation of chaos in the weak and strong chaos regimes by studying the classical version of the out-of-time-ordered commutator.
We present new results on the time-dependent correlation functions ${\ensuremath{\Xi}}_{n}(t)=4〈{S}_{0}^{\ensuremath{\xi}}(t){S}_{n}^{\ensuremath{\xi}}〉$, $\ensuremath{\xi}=x,y$, at zero temperature of the one-dimensional S=12S=\frac{1}{2}S=21​ isotropic XY\mathrm{XY}XY model ($h=\ensuremath{\gamma}=0$) and of the transverse Ising (TI) model at the critical magnetic field ($h=\ensuremath{\gamma}=1$). Both models are characterized by special cases of the Hamiltonian $H=\ensuremath{-}J\ensuremath{\Sigma}{l}^{}[(1+\ensuremath{\gamma}){S}_{l}^{x}{S}_{l+1}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{l}^{y}{S}_{l+1}^{y}+h{S}_{1}^{z}]$. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions ${\ensuremath{\Xi}}_{0}(0)$ and on the singularities of their frequency-dependent Fourier transforms ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$. We have also determined the latter functions by high-precision numerical calculations. The functions ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$, $\ensuremath{\xi}=x,y$, have singularities at the infinite sequence of frequencies $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$, m=0,1,2,3,…m=0,1,2,3,\dots{}m=0,1,2,3,…, where ${\ensuremath{\omega}}_{0}=J$ for the XY\mathrm{XY}XY model and ${\ensuremath{\omega}}_{0}=2J$ for the TI model. In the TI case, the leading singularities in ${\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})$ are alternately one-sided and two-sided power-law singularities, the first two of which (at $\ensuremath{\omega}=0,2J$) are divergent. The dominant singularities in the XY\mathrm{XY}XY case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at $\ensuremath{\omega}=0,J$) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at $\ensuremath{\omega}\ensuremath{\ne}0$ are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the XY\mathrm{XY}XY-like substances Cs2{\mathrm{Cs}}_{2}Cs2​CoCl4{\mathrm{Cl}}_{4}Cl4​ and PrCl3{\mathrm{Cl}}_{3}Cl3​ and the S=12S=\frac{1}{2}S=21​ Ising-like substance CsCoCl3{\mathrm{Cl}}_{3}Cl3​${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.