Symmetries of a mean-field spin model

TL;DR

This paper classifies the symmetries of a mean-field quantum spin model's equations of motion, revealing how parameters induce invariant subspaces and influence equilibration and equilibrium states.

Contribution

It provides a comprehensive classification of all symmetry-induced invariant subspaces in the thermodynamic limit of a mean-field spin model.

Findings

Parameters induce a structure partitioning Hilbert space into invariant subspaces.

Number of invariant subspaces can be 1, 2, 4, or O(N).

Decoupling of the BBGKY hierarchy affects equilibration and equilibrium.

Abstract

Thermodynamic limit evolution of a closed quantum Heisenberg-type spin model with mean-field interactions is characterized by classifying all the symmetries of the equations of motion. It is shown that parameters of the model induce a structure in the Hilbert space by partitioning it into invariant subspaces, decoupled by the underlying Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. All possible partitions are classified in terms of a $3\times3$ matrix of effective, thermodynamic limit coupling constants. It is found that there are either 1, 2, 4, or O(N) invariant subspaces. The BBGKY hierarchy decouples into the corresponding number of anti-Hermitian operators on each subspace. These findings imply that equilibration and the equilibrium in this model depend on the initial conditions.