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DAE Symp.Nucl.Phys.

Hugenholtz-Van Hove Theorem for Multi-Component Fermi Systems with Multi-body Forces

Ramesh Chandra Nayak, Subhaswaraj Pattnaik

Published Jan 1, 2011• 2 citations
OpenAlex

Abstract

The second term being the pressure of the system would vanish for a saturating system at ground state i.e. at equilibrium resulting in the Eq. E/A = ǫf . This is a rare theorem in many-body Physics, which has been rigorously shown[1] to be true by its original authors Hugenholtz and Van Hove by taking all orders of perturbation in the frame-work of timeindependent perturbation theory[1–5]. However it should be mentioned here that prior to this rigorous proof Bethe[6] had also visualized the theorem under HF approximation. It is valid for any interacting infinite Fermi system and thereby applicable to liquid He and in particular to nuclear matter. With its

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The second term being the pressure of the system would vanish for a saturating system at ground state i.e. at equilibrium resulting in the Eq. E/A = ǫf . This is a rare theorem in many-body Physics, which has been rigorously shown[1] to be true by its original authors Hugenholtz and Van Hove by taking all orders of perturbation in the frame-work of timeindependent perturbation theory[1–5]. However it should be mentioned here that prior to this rigorous proof Bethe[6] had also visualized the theorem under HF approximation. It is valid for any interacting infinite Fermi system and thereby applicable to liquid He and in particular to nuclear matter. With its
Conserving approximations lead to thermodynamically consistent expressions for Fermi-liquid properties such as compression modulus, symmetry energy and Landau parameters. We will emphasize the need for the precision experiment to determine the density of states at the Fermi surface by discussing nuclear matter approximations in terms of thermodynamic consistency (the Hugenholtz-Van Hove theorem and the virial theorem). The density of states is the primary self-consistent observable to be determined from nuclear many-body approximations. The precise measurement of this observable will establish a rigorous criterion on the validity of nuclear matter approximations.
This paper investigates the criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in many-body systems. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices.Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approximation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem.
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.
We consider many-fermion systems with singular Fermi surfaces, which contain Van Hove points where the gradient of the band function k ↦ e(k) vanishes. In a previous paper, we have treated the case of spatial dimension d ≥ 3. In this paper, we focus on the more singular case d = 2 and establish properties of the fermionic self-energy to all orders in perturbation theory. We show that there is an asymmetry between the spatial and frequency derivatives of the self-energy. The derivative with respect to the Matsubara frequency diverges at the Van Hove points, but, surprisingly, the self-energy is C1 in the spatial momentum to all orders in perturbation theory, provided the Fermi surface is curved away from the Van Hove points. In a prototypical example, the second spatial derivative behaves similarly to the first frequency derivative. We discuss the physical significance of these findings.
New forms of the linked cluster expansion for the ground-state energy of a normal spherical system are given in terms of the exact single-particle propagator and of the associated self-energy operator. These forms are shown to be stationary with respect to variations of the latter quantities. A new concise proof of the Hugenholtz-Van Hove theorem on single-particle energies is then possible. The variational principle also explains the insensitivity of recent binding energy calculations to the choice of single-particle energies. It is shown how all results may be extended to systems with anisotropic features.