Proof of absence of local conserved quantities in the mixed-field Ising chain

TL;DR

This paper rigorously proves that the mixed-field Ising chain with all nonzero couplings lacks nontrivial local conserved quantities, confirming its nonintegrability and implications for thermalization.

Contribution

It provides the first rigorous proof that the mixed-field Ising chain has no local conserved quantities besides trivial ones, extending understanding of nonintegrability in spin models.

Findings

No nontrivial local conserved quantities for nonzero couplings.

Proof applies to both periodic and open boundary conditions.

Supports the model's nonintegrability and thermalization properties.

Abstract

Absence of local conserved quantities is often required, such as for thermalization or for the validity of response theory. Although many studies have discussed whether thermalization occurs in the Ising chain with longitudinal and transverse fields, rigorous results on local conserved quantities of this model have still been lacking. Here, we rigorously prove that, if all coupling constants are nonzero, this model has no conserved quantity spanned by local operators with support size up to half of the system size other than a trivial one, i.e., a linear combination of the Hamiltonian and the identity. The proof is given not only for the periodic boundary condition but also for the open boundary condition. We also discuss relation to the integrability of the model where the longitudinal field is set to zero. Our results provide the second example of spin models whose nonintegrability is rigorously proved.