Ferromagnetic Ordering of Energy Levels for $U_q(\mathfrak{sl}_2)$ Symmetric Spin Chains

TL;DR

This paper establishes conditions under which quantum spin chains with $U_q(rak{sl}_2)$ symmetry exhibit ferromagnetic ordering of energy levels, linking positivity of interactions to spectral properties.

Contribution

It introduces a new basis of $U_q(rak{sl}_2)$ intertwiners and characterizes the cone of positive interactions ensuring ferromagnetic ordering.

Findings

Positivity of interactions in the dual canonical basis implies ferromagnetic ordering.

The cone of positive interactions is a simplicial cone generated by cascade operators.

Applications to interacting particle processes are discussed.

Abstract

We consider the class of quantum spin chains with arbitrary $U_q(\mathfrak{sl}_2)$-invariant nearest neighbor interactions, sometimes called $\textrm{SU}_q(2)$ for the quantum deformation of $\textrm{SU}(2)$, for $q>0$. We derive sufficient conditions for the Hamiltonian to satisfy the property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the property that the ground state energy restricted to a fixed total spin subspace is a decreasing function of the total spin. Using the Perron-Frobenius theorem, we show sufficient conditions are positivity of all interactions in the dual canonical basis of Lusztig. We characterize the cone of positive interactions, showing that it is a simplicial cone consisting of all non-positive linear combinations of "cascade operators," a special new basis of $U_q(\mathfrak{sl}_2)$ intertwiners we define. We also state applications to interacting particle processes.