Existence of parameter domains with nearly good real-number specifications in general many-body systems with non-commutable Hamiltonians

Determine whether, for general many-body quantum systems whose Hamiltonians include non-commutable terms (such as those arising from odd interspecies spin-coupling channels in multi-species spin-1 boson mixtures), there exist specific domains of the interaction-parameter space in which eigenstates can be specified by a set of nearly good real numbers rather than conserved quantum numbers.

Background

The paper studies a medium-body system of two species of spin-1 bosons where an odd interspecies channel introduces non-commutable terms into the spin-dependent Hamiltonian, causing the combined spins of each species (S_A and S_B) to cease being good quantum numbers. In certain parameter regions, the ground state can instead be described by nearly good real numbers, defined as very narrow intervals for the averages of combined spins whose widths shrink with increasing particle number.

While this behavior is demonstrated numerically for medium-body cases, the authors raise the question of whether analogous domains exist broadly in general many-body systems with non-commutable Hamiltonians, enabling eigenstates to be specified by such nearly good real numbers.