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Physical Review C

Comparison of two methods for calculating finite-temperature ring diagrams

Shen Yang, Meng Jiang, T.T.S. Kuo, P.J. Ellis

Published Aug 1, 1989• 2 citations
Publisher

Abstract

Certain types of ring diagrams of the thermodynamic potential \ensuremath{\Omega} can be summed up to all orders using two apparently very different methods---the eigenvalue method and the transition-amplitude method. A relatively simple derivation of these methods is given, and we show that these two methods are exactly equivalent to each other. To study their numerical applications we have calculated the particle-particle, hole-hole ring diagrams of \ensuremath{\Omega} for nuclear matter with both methods, using a modified Gogny interaction. Our results suggest that the strength-parameter integration involved in the transition-amplitude method can be accurately performed using a small number of mesh points and both methods seem to be suitable for numerical calculations.

DOI
10.1103/physrevc.40.1085

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Certain types of ring diagrams of the thermodynamic potential \ensuremath{\Omega} can be summed up to all orders using two apparently very different methods---the eigenvalue method and the transition-amplitude method. A relatively simple derivation of these methods is given, and we show that these two methods are exactly equivalent to each other. To study their numerical applications we have calculated the particle-particle, hole-hole ring diagrams of \ensuremath{\Omega} for nuclear matter with both methods, using a modified Gogny interaction. Our results suggest that the strength-parameter integration involved in the transition-amplitude method can be accurately performed using a small number of mesh points and both methods seem to be suitable for numerical calculations.
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A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.
We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.
We develop the parquet-diagram summation method for neutron matter interacting via potentials that include spin, tensor, and spin-orbit components. For that purpose, we derive an exact expression for the sum of all ring diagrams in terms of effective local particle-hole interactions involving the above four operators. The parquet equations are closed by deriving the spin-orbit contribution to those particle-hole interactions. We show that many-body correlations screen the bare spin-orbit potential considerably, and the corrections of that screened spin-orbit potential to the other three interaction channels are quite small. We apply our method to the calculation of the response of neutron matter to density and both longitudinal and transverse spin-dependent external fields.
In medium TTT-matrix calculations for symmetric nuclear matter at zero and finite temperatures are presented. The internal energy is calculated from the Galitskii-Koltun's sum rule and from the summation of the diagrams for the interaction energy. The pressure at finite temperature is obtained from the generating functional form of the thermodynamic potential. The entropy at high temperature is estimated and compared to expressions corresponding to a quasiparticle gas.
A method for choosing and deriving a suitable single-particle potential for many-body calculations at finite temperature is suggested. With this potential one can readily determine the self-consistent correlated chemical potential and consequently the resulting equations of state rigorously satisfy a number of basic thermodynamic relations. Application of our method to a Brueckner-Hartree-Fock nuclear matter calculation yields results in remarkably good agreement with the Hugenholtz--Van Hove theorem.