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 present a structured Pad\'e approximation scheme in the momentum space to determine
g(r). The calculated
g(r) for distinguishable particles interacting with Lennard-Jones and hard-sphere potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path-integral Monte Carlo simulation. 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.
Abstract The Wigner formulation of quantum mechanics is used to derive a new path integral representation of quantum density of states (DOSs). A path integral Monte Carlo approach is developed for the numerical investigation of DOSs, internal energy and spin-resolved radial distribution functions for a 2D system of strongly correlated soft-sphere fermions. The peculiarities of the DOSs and internal energy distributions depending on the hardness of the soft-sphere potential and particle density are investigated and explained. In particular, at high enough densities the DOSs rapidly tends to a constant value, as for an ideal system of 2D fermions.
The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.
A trial solution constructed from two-particle functions is applied to the problem of the
N-particle system with strong interactions. In a variational treatment based on this wave function the expectation values, which cannot be factored into single-particle integrals, are evaluated by a cluster development in powers of the particle density. The procedure is illustrated by a calculation of the pair distribution function and zero-point energy of the hard sphere gas, for Bose and Fermi statistics.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of
(Nε−1)O(logΔ), where
N is the system size,
ε the desired precision, and
Δ the interaction-graph degree. This becomes particularly severe in long-range systems, where
Δ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime
O~(N4ε−3). For quantum partition functions at temperatures
β≤βsampling≡O(1), this improves to
O~(N3+1/Dε−3), where
D is the lattice dimension. The framework directly applies to long-range quantum systems with interactions
∝1/rα, including
2-local models with
α>2D.
The formalism based on correlated basis functions and the cluster-expansion technique has been recently employed to derive an effective interaction from a realistic nuclear Hamiltonian. To gauge the reliability of this scheme, we perform a systematic comparison between the results of its application to the Fermi hard-sphere system and the predictions obtained from low-density expansions, as well as from other many-body techniques. The analysis of a variety of properties, including the ground-state energy, the effective mass, and the momentum distribution, shows that the effective-interaction approach is quite accurate, thus suggesting that it may be employed to achieve a consistent description of the structure and dynamics of nuclear matter in the density region relevant to astrophysical applications.