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Physical Review A

Vertex Renormalization for a System of Interacting Bosons at Nonzero Temperatures

K. K. Singh

Published Nov 1, 1970• 1 citation
Publisher

Abstract

The problem of vertex renormalization at nonzero temperatures for a system of bosons interacting via a strong short-ranged repulsive potential and a weak long-ranged attractive potential is examined. It is shown that with the assumption of low temperatures and densities, one obtains expressions for the thermodynamic functions which corroborate the results of an earlier paper.

DOI
10.1103/physreva.2.2159

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The problem of vertex renormalization at nonzero temperatures for a system of bosons interacting via a strong short-ranged repulsive potential and a weak long-ranged attractive potential is examined. It is shown that with the assumption of low temperatures and densities, one obtains expressions for the thermodynamic functions which corroborate the results of an earlier paper.
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The problem of a system of bosons at zero temperature, interacting via a strong short-ranged repulsive interaction and a weak long-ranged attraction, is treated by the methods of quantum-field theory. It is shown that if the parameters of the attractive part of the interaction are suitably restricted, an approximation to the self-energy parts of the Green's functions, similar to that used by Beliaev for the case of purely repulsive interactions, leads to an excitation spectrum which in the limit of small momenta represents phonons. The approximation also yields an NNN-body bound state corresponding to zero pressure. Apart from minor differences, these results are in agreement with an earlier calculation of Huang.
The thermodynamics of a homogeneous dilute Bose gas with an arbitrarily strong repulsion between particles is investigated on the basis of the exact relation connecting the pair correlation function with the in-medium pair wave functions and occupation numbers. It is shown that the effective-interaction scheme, which is reduced to the Bogoliubov model with the effective pairwise potential, is not acceptable for investigating the short-range particle correlations in a dilute strongly interacting Bose gas. In contrast to this scheme, our model is thermodynamically consistent and free of the ultraviolet divergences due to accurate treatment of the short-range boson correlations. An equation for the in-medium scattering amplitude is derived that makes it possible to find the in-medium renormalization for the pair wave functions at short boson separations. Low-density expansions for the main thermodynamic quantities are reinvestigated on the basis of this equation. In addition, the expansions are found for the interaction and kinetic energies per particle. It is demonstrated that for a many-boson system of hard spheres the interaction energy is equal to zero for any boson density. The exact relationship between the chemical potential and in-medium pair wave functions is also established.
Abstract: The one-loop effective potential for non-relativistic bosons with a delta function repulsive potential is calculated for a given chemical potential using functional methods. After renormalization and at zero temperature it reproduces the standard ground state energy and pressure as function of the particle density. At finite temperatures it is found necessary to include ring corrections to the one-loop result in order to satisfy the Goldstone theorem. It is natural to introduce an effective chemical potential directly related to the order parameter and which uniformly decreases with increasing temperatures. This is in contrast to the the ordinary chemical potential which peaks at the critical temperature. The resulting thermodynamics in the condensed phase at very low temperatures is found to be the same as in the Bogoliubov approximation where the degrees of freedom are given by the Goldstone bosons. At higher temperatures the ring corrections dominate and result in a critical temperature unaffected by the interaction. 1
We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential V(r)=gr2V(r)=\frac{g}{{r}^{2}}V(r)=r2g​ with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant ggg, as a determinant of order $N\ifmmode\times\else\texttimes\fi{}N$. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.
The well{known results concerning a dilute Bose gas with the short{range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method as basic to much of the present understanding of this subject and nonrelevant behaviour of the pair distribution function at small boson separations.The aim of our paper is to propose a new way of treating the dilute Bose gas with an arbitrary strong interaction.Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials like the Lennard{Jones one.All the consideration concerns the zero temperature.