Exchange symmetry in a system of nonrelativistic spin-1/2 fermions in the Feynman quantum statistics representation

Abstract

A compact representation is obtained for the quantum statistical sum of indistinguishable nonrelativistic spin-1/2 fermions in the form of Feynman path integrals which can be used as the basis to develop a fundamentally exact method of computer modeling for systems of strongly interacting electrons at nonzero temperature. A basis of symmetrized wave functions is constructed using Young symmetry operators. An exact permutation symmetrization procedure leads to an avalanche-like multiplication in the number of diagrams of linked Feynman integrals of the order of N!. The partition function can be simplified without introducing any approximations and this is performed numerically by computer by direct sorting of diagrams. The control tables obtained, containing combinatorial weights of diagrams, direct the Markov random walk process in virtual trajectory space which is achieved numerically by computer. The equilibrium characteristics of the quantum system are calculated by averaging. This approach is an expansion of the Monte Carlo-Metropolis method to systems of quantum indistinguishable particles with spin. Demonstration numerical calculations using this method were made for the simplest exchange systems, for a hydrogen molecule, a Be+ ion, and a Li atom. The ground state of the hydrogen molecule is reproduced with a statistical error of 0.2%. Exchange-correlation effects lead to nontrivial structural changes in the thermally excited electron shells of ions in a state of strong plasma compression.