Linked Cluster Expansion of the Many-Body Path-Integral

In this thesis we have combined the idea of cluster expansion and path integration to develop the quantum version of cluster and virial expansion. We derive a diagrammatic series expansion for different thermodynamic quantities like free energy, chemical potential and pair distribution function and show that the diagrammatic expansion is linked. This expansion in n-body clusters can also be thought of as a power series expansion in the particle density. We first present the results for the pair distribution function by evaluating it up to a first few orders in particle density and show that there is systematic order-by-order improvement. Second, we use a Pade` resummation scheme in momentum space to extrapolate to infinite order. This scheme is constructed in such a way so that it yields the calculated order by order expansion terms and the classical limit correctly. We have also used our proposed quantum version of Hypernetted-chain (HNC) equations to calculate the observables by solving self-consistently a set of integral equations which sum a certain class of contributing diagrams which resemble a “hypernetted” network. We have tested our summation schemes on a Lennard-Jones and a hard-sphere system containing distinguishable particles and our results agree very well with those obtained from the path-integral Monte Carlo simulation. We have also demonstrated the applicability to a system of identical particles by applying it to the bosonic system of 4He particles. Our method is easily applicable to the case of a short-range singular potential where the established analytical and semi-analytical tools of many-body perturbation theory and quantum statistical mechanics cannot be applied in a straightforward manner.