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Physical review. B, Condensed matter

Perturbation expansions of the response function and the asymptotic regime

J. Besprosvany

Published May 1, 1991• 12 citations
Publisher

Abstract

Perturbation expansions of the response function S(q,\ensuremath{\omega}) of a many-body system in orders of any chosen component of the Hamiltonian are obtained. The response S(q,\ensuremath{\omega}) describes the inclusive scattering of a projectile interacting weakly with a target, where q and \ensuremath{\omega} are the momentum and energy transfer. Each term of a given order is written as an expectation value of products of paired Green's functions representing each initial- and final-state effects. For several choices of the perturbing operator, close expressions are obtained for the first orders of these expansions. The resulting terms represent partial summations over the Gersch series, whose relation to Green's-function expansions is established. Also, they are finite for singular potentials and are useful in the asymptotic regime of large q. In this regime they can give the dominant contribution to the response and initial- and final-state interactions of the hit particle and other particles in the system may be fully accounted for. Contributions of the n-particle density matrix can be separated for some expansions. In particular, an expression for the complete two-body contribution involving the hit particle is given.

DOI
10.1103/physrevb.43.10070

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Perturbation expansions of the response function S(q,\ensuremath{\omega}) of a many-body system in orders of any chosen component of the Hamiltonian are obtained. The response S(q,\ensuremath{\omega}) describes the inclusive scattering of a projectile interacting weakly with a target, where q and \ensuremath{\omega} are the momentum and energy transfer. Each term of a given order is written as an expectation value of products of paired Green's functions representing each initial- and final-state effects. For several choices of the perturbing operator, close expressions are obtained for the first orders of these expansions. The resulting terms represent partial summations over the Gersch series, whose relation to Green's-function expansions is established. Also, they are finite for singular potentials and are useful in the asymptotic regime of large q. In this regime they can give the dominant contribution to the response and initial- and final-state interactions of the hit particle and other particles in the system may be fully accounted for. Contributions of the n-particle density matrix can be separated for some expansions. In particular, an expression for the complete two-body contribution involving the hit particle is given.
We discuss a basis set developed to calculate perturbation coefficients in an expansion of the general N -body problem. This basis has two advantages. First, the basis is complete order-by-order for the perturbation series. Second, the number of independent basis tensors spanning the space for a given order does not scale with N , the number of particles, despite the generality of the problem. At first order, the number of basis tensors is 25 for all N , i.e. the problem scales as N 0 , although one would initially expect an N 6 scaling at first order. The perturbation series is expanded in inverse powers of the spatial dimension. This results in a maximally symmetric configuration at lowest order which has a point group isomorphic with the symmetric group, S N . The resulting perturbation series is order-by-order invariant under the N ! operations of the S N point group which is responsible for the slower than exponential growth of the basis. In this paper, we demonstrate the completeness of the basis and perform the first test of this formalism through first order by comparing to an exactly solvable fully interacting problem of N particles with a two-body harmonic interaction potential.
The problem of nonlinear response is considered by employing a general time-evolution equation, and a Green's function which is the transition or conditional probability density for an unperturbed system. Expansion of the Green's function in terms of orthonormal functions enables us to express the distribution function describing the nonlinear behavior by means of matrix products whose elements are composed of correlation functions in the absence of the perturbation. In other words, it is shown how the distribution function induced by a strong perturbation may be calculated by knowing the Green's function without the perturbation. As the special case of the linear response, we have obtained Kubo's relation. The Laplace-transform technique with respect to time is found quite useful in developing the present theory in which the transient effect is also taken into account. As an application of the theory, a new relation valid in the region of the second-order perturbation connecting the transient rise and decay with the stationary alternating perturbations has been obtained.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.
We especially need imagination in science. aria Mitchell Summary The many-body problem consists of two parts: the first is the non-interacting system in a materials-specific external potential; the second is the Coulomb interaction that makes the problem so hard to solve. The most straightforward idea is to use perturbation theory, with the Coulomb interaction as perturbation. This is conceptually simple, but it turns out to be difficult in practice, since the Coulomb interaction is often not small compared with typical energy differences, it is long-ranged and in the thermodynamic limit there is an infinite number of particles, contributing with an infinite number of mutual interaction processes. The present chapter outlines how one can deal with this problem. It contains an overview of facts that one can also find in many standard textbooks on the many-body problem, but that are useful to keep in mind in order to look at later chapters from a sound and well-established perspective. The many-body problem is a tough one, and it has many facets. Sorting it out is like putting together a huge puzzle. The eight introductory chapters of this book provide pieces of the puzzle, and ideas on what one might do about it. In the present chapter we choose to go in one of the possible directions, in order to arrive at something tangible. The chapter gives the general framework and the main ideas; specific approximations are the topic of Chs. 10–15. The idea is to start from an independent-particle problem and add the Coulomb interaction as a perturbation. This is not easy: first, the interaction is responsible for a rich variety of phenomena that are completely absent otherwise, such as the finite lifetime of quasiparticles, or additional structures in spectra due to the fact that a quasi-particle excitation may transfer its energy to other elementary excitations, for example plasmons. Second, because of the two-body Coulomb interaction, the problem scales badly with the number of electrons, and straightforward perturbation theory for the many-body hamiltonian with the Coulomb interaction as perturbation rapidly becomes intractable or even useless, especially in large systems. To get started, Sec. 9.1 recalls why things are not so easy. The following sections try to solve one problem after the other, starting from Sec. 9.2 where the Green's function is reformulated in a way that is appropriate for a perturbation expansion.
We extend the self-consistent Green's functions formalism to take into account three-body interactions. We analyze the perturbative expansion in terms of Feynman diagrams and define effective one- and two-body interactions, which allows for a substantial reduction of the number of diagrams. The procedure can be taken as a generalization of the normal ordering of the Hamiltonian to fully correlated density matrices. We give examples up to third order in perturbation theory. To define nonperturbative approximations, we extend the equation-of-motion method in the presence of three-body interactions. We propose schemes that can provide nonperturbative resummation of three-body interactions. We also discuss two different extensions of the Koltun sum rule to compute the ground state of a many-body system.
The convergence problem of perturbation expansions in the many-body problem is discussed by employing some simple models. The expectation value of rhe resolvent operator is analytic around the origin of the coupling constant so long as the system is enclosed in a finite volume, but as the volume tends to infinity with the density of particles kept constant, the question arises whether this limit is allowed to be taken prior to the sum of the perturbation series. Even if this is the case, the resulting series turns out to be usually divergent, but in some models we consider we can find a function whose asymptotic expansion coincides with this series. It is shown that for the repulsive interactions the function thus obtained gives a correct answer, while for the attractive interactions which lead to a superconducting state it is far from the true one. The asymptotic behavior of the perturbation series in the latter case is examined in some detail.