Linked cluster expansion of the many-body path integral

TL;DR

The paper introduces a linked cluster expansion approach for the many-body path integral, deriving diagrammatic rules and generalizing the hypernetted-chain equation to quantum systems, with results matching Monte Carlo simulations.

Contribution

It develops a systematic linked cluster expansion for quantum many-body path integrals and extends classical HNC equations to quantum cases, enabling accurate calculations.

Findings

g(r) matches Monte Carlo results at high densities

Method applicable to various quantum many-body problems

Extension to fermions is feasible

Abstract

We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free-energy and the pair distribution function $g(r)$ as a systematic power series expansion in the particle density. We also generalize the hypernetted-chain (HNC) equation for $g(r)$, known from its application to classical statistical mechanics, to a set of quantum HNC equations (QHNC) for the quantum case. The calculated $g(r)$ for distinguishable particles interacting with a Lennard-Jones potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path integral Monte Carlo simulation even for densities as high as the equilibrium density of the strongly correlated liquid $^4$He. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.