Sum rules for spin-$1/2$ quantum gases in states with well-defined spins: II. Spin-dependent two-body interactions

TL;DR

This paper derives analytical sum rules for matrix elements of spin-dependent two-body interactions in many-body spin-$1/2$ quantum gases, facilitating spectral analysis and correlation calculations.

Contribution

It extends previous work by providing explicit sum rules for non-maximal total spin states, aiding perturbative and correlation studies in spinor gases.

Findings

Sum rules are expressed in universal factors independent of non-interacting Hamiltonians.

Analytical formulas are derived for sums of matrix elements and their products.

Applications include perturbative energy spectrum analysis and local correlation calculations.

Abstract

Sums of matrix elements of spin-dependent two-body momentum-independent interactions and sums of their products are calculated analytically in the basis of many-body states with given total spin --- the states built from spin and spatial wavefunctions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. As in the first part of the series [V. A. Yurovsky, Phys. Rev. A 91, 053601 (2015)], the sum dependence on the many-body states is given by universal factors, which are independent of the Hamiltonians of non-interacting particles. The sum rules are applied to perturbative analysis of energy spectra and to calculation of two-body spin-dependent local correlations.