Sum rules for spin-$1/2$ quantum gases in states with well-defined spins: spin-independent interactions and spin-dependent external fields

TL;DR

This paper derives universal sum rules for matrix elements in spin-$1/2$ quantum gases, enabling simplified analysis of energy spectra under spin-dependent external fields and spin-independent interactions.

Contribution

It provides analytical sum rules for matrix elements over many-body states with fixed total spin, applicable to systems with spin-dependent external fields and spin-independent interactions.

Findings

Sum rules depend on universal factors independent of interaction details.

Analytical expressions are derived for sums over states with given total spin.

Applications include perturbative analysis of energy spectra.

Abstract

Analytical expressions are derived for sums of matrix elements and their squared moduli over 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. For spin-dependent one-body interactions with external fields and spin-independent two-body ones between the particles, the sum dependence on the many-body states is given by universal factors, which are independent of the interaction details and Hamiltonians of non-interacting particles. The sum rules are applied to perturbative analysis of energy spectra.