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Physical Review

Correlation Functions in Ring - Diagram Approximation

A. Isihara

Published Aug 5, 1968• 11 citations
Publisher

Abstract

The singlet and pair correlation functions of particles in quantum-mechanical many-body systems are evaluated in ring diagram approximation; thereby consideration is also given to chain diagrams. The results, expressed in terms of the eigenvalues characteristic of the topologically different diagrams which enter the theory, agree with what we expect from Montroll and Ward's results on the grand partition function.

DOI
10.1103/physrev.172.166

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The singlet and pair correlation functions of particles in quantum-mechanical many-body systems are evaluated in ring diagram approximation; thereby consideration is also given to chain diagrams. The results, expressed in terms of the eigenvalues characteristic of the topologically different diagrams which enter the theory, agree with what we expect from Montroll and Ward's results on the grand partition function.
Quantum field theoretical methods are discussed to evaluate the spatial distribution in many-body systems in equilibrium. The pair and triplet distribution functions are explicitly evaluated for noninteracting cases. A general method to evaluate the pair distribution function in the chain and ring diagram approximations for interacting systems is presented in detail. The result is expressed in terms of the eigenvalues of the propagators characteristic of the diagrams. This result generalized the Gell-Mann and Brueckner formula for the equation of state, the derivation of which is also given. The eigenvalues are evaluated for Boltzmann, Bose and Fermi statistics. Application of the result to a hard-sphere Bose gas is made, and the effect of condensation is studied. It is found that the phonon velocity depends on the number of particles in the lowest state. The distribution functions of a degenerate Fermi system are also calculated.
A systematic generalization of the Mayer cluster integral theory has been developed to deal with the quantum statistics of interacting particles. The grand partition function appears in a natural way and the cluster integrals are integrals over propagators which are derived from the Green's function solution of the Bloch equation (which follows from the Schroedinger equation by replacing it/ℏ by β = 1/kT). Every cluster integral can be represented by a hybrid of a Mayer graph and a Feynman diagram in (β, r) space. The generalization of classical ring cluster integrals has been analyzed. It is shown that in the case of the electron gas the classical limit of the contribution of these integrals to the grand partition function yields the Debye-Huckel theory while the low temperature limit leads to the Gell-Mann—Brueckner equation for the correlation energy of the ground state. A prescription is given for the construction of the cluster integral associated with any given diagram.
A general method of calculation is described for quantum statistical mechanics. It is based on a simplification of the Laplace transform of the density matrix which follows from a theorem due to Hugenholtz. The basic result is that an element of the density matrix can be written as a sum over graphs, with the contribution of each graph factored into contributions from connected or linked graphs. Applied to the grand partition function, the exponential formula of Bloch and DeDominicis is obtained in a simple way. A similar formula is then derived for the canonical ensemble for the case of a nondegenerate gas. In this way the familiar result of Uhlenbeck and Beth is obtained for the second virial coefficient. Techniques are also introduced for evaluating ensemble averages of operators. In this connection, some care must be exercised in the case of diagonal operators. Finally, these methods are used to calculate the pair-correlation function for a system of fermions interacting through short-range forces.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.
We study the connections between the chain summations of variational theories of Fermi fluids and those generated by ring-diagram summations in the correlated-basis-functions (CBF) perturbation theory. The effective interactions of CBF theory and the random-phase-approximation (RPA) equations in the correlated basis are rewritten in an irreducible form. We discuss local approximations to the irreducible vertex which are sympathetic with the use of variational wave functions. They allow contact to be made with conventional formulations of the RPA, to weak-coupling, pseudopotential, or "local-screening" theories. Our studies lead also to consistency requirements between variational calculations of ground-state properties and the effective interactions used in CBF theory.