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Progress of Theoretical Physics

The Asymptotic Behavior of Perturbation Expansions in Many-Body Problem and Superconducting State

Kenji Fukushima, Nobuyuki Fukuda

Published Nov 1, 1962• 4 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.28.809

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Physical review. B, Condensed matter•May 1, 1991•12 citations
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.
An N-body system with interparticle forces that are attractive at short range collapses in the limit N→∞, namely in this limit the ground-state energy per particle diverges to negative infinity. If instead the forces are sufficiently repulsive at short range, in the limit N→∞ saturation occurs, leading to a finite ground-state energy per particle ε. This quantity depends, among other things, on the ’’coupling constant’’ g (entering as a factor that multiplies the interaction), and the above remark clearly implies that it is defined in the N→∞ limit only for positive values of g (although, for finite N, it is defined both for positive and negative g). This fact is generally taken to imply that the function ε (g) is nonanalytic in g at g=0 and, therefore, that the perturbative expansion of ε (g), being a power expansion in g, is necessarily nonconvergent (although it might be asymptotic). The purpose of this paper is to demonstrate the lack of cogency of this argument. It is therefore concluded that nonconvergence of the perturbative expansion for ε (g) is thus far an unproven hypothesis. The lack of cogency of analogous current arguments concerning the equilibrium density of many-body systems is also pointed out.
A strongly coupled system---the limiting case of a highly degenerate many-fermion system for which the variation of the kinetic energy is neglected, and the interaction restricted to a region of momentum space neighboring the Fermi surface---has been analyzed in a manner not dependent upon assumptions about the convergence of power series expansions or on partial summations of infinite series. The vacuum expectation value of the resolvent operator, ${〈\frac{1}{(H\ensuremath{-}z)}〉}_{0}$, is expressed as the Laplace transform of the exponential of a function linearly dependent on the volume of the system. It is shown that the linked-cluster expansion of the vacuum expectation value of the resolvent operator has a zero radius of convergence as a power series in the coupling constant. The most serious physical consequence of this is that a nontrivial interaction never results in a "normal" system.
In the conventional perturbation treatment of the many-body problem of interacting particles, the zero-order Hamiltonian corresponds to independent particles moving in a static over-all potential. A discussion of the effects of particle interactions or `correlations' shows that if one starts instead with a Hamiltonian representing noninteracting particles in a velocity-dependent over-all potential, deeper for slow, and shallower for fast particles, then part of the correlation effect is included already in zero order. In addition, a velocity-dependent over-all potential may be called for by velocity dependent interparticle forces or by exchange forces. The degree of the improvement in the convergence of a perturbation expansion based on a Hamiltonian with a velocity-dependent over-all potential is discussed and illustrated by a simple example in which the velocity dependence of the potential gives rise to a reduced "effective mass" of the particles.The many-body problem of a large, uniform system of interacting particles (e.g., the case of a heavy nucleus without surface effects) is formulated in detail in perturbation theory, starting with a velocity-dependent potential constant in space. The work involved in such a calculation turns out to be essentially the same as with a velocity-independent potential, the effect of the velocity dependence being to reduce each term in the perturbation expansion by a constant factor raised to a power equal to the number of "energy denominators" ${E}_{m}\ensuremath{-}{E}_{n}$ in the term in question. A simple equation is deduced for the optimum degree of velocity dependence of the over-all potential which ensures the most rapid convergence.
Convergence of the Schwinger --- DeWitt expansion for the evolution operator kernel for special class of potentials is studied. It is shown, that this expansion, which is in general case asymptotic, converges for the potentials considered (widely used, in particular, in one-dimensional many-body problems), and besides, convergence takes place only for definite discrete values of the coupling constant. For other values of the charge divergent expansion determines the kernels having essential singularity at origin (beyond usual δδδ-function). If one consider only this class of potentials then one can avoid many problems, connected with asymptotic expansions, and one get the theory with discrete values of the coupling constant that is in correspondence with discreteness of the charge in the nature. This approach can be transmitted into the quantum field theory.
An approximation method developed previously to deal with many particles in strong interaction is examined in further detail. It is shown that the series giving the interaction energy is a development in a sequence of linked or irreducible cluster terms each of which gives a contribution to the energy proportional to the total number of particles. Consequently the convergence of the expansion is independent of the total number of particles. The origin of this simple feature is illustrated by showing that a similar situation exists in the expansion of standard perturbation theory. The numerical convergence of the expansion is quantitatively discussed for the nuclear problem where it is shown that the correction arising from the first cluster term involving three particles is less than the leading term by a factor of about ${10}^{\ensuremath{-}4}$. The smallness of the correction is largely a result of the action of the exclusion principle.
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.