Proof of absence of local conserved quantities in two- and higher-dimensional quantum Ising models

TL;DR

This paper proves that higher-dimensional quantum Ising models with transverse and longitudinal fields lack local conserved quantities other than the Hamiltonian, extending previous one-dimensional results to more complex lattice structures.

Contribution

It introduces a method to demonstrate the absence of local conserved quantities in higher-dimensional and ladder quantum Ising models, generalizing prior one-dimensional findings.

Findings

No local conserved quantities other than the Hamiltonian in higher dimensions

Results apply to hypercubic lattices and ladder systems

Technique extends to systems with nonzero transverse and longitudinal fields

Abstract

We prove that the Ising models with transverse and longitudinal fields on the hypercubic lattices with dimensions higher than one have no local conserved quantities other than the Hamiltonian. This holds for any value of the longitudinal field, including zero, as far as the transverse field and the Ising interactions are nonzero. The conserved quantity considered here is ``local'' in a very weak sense: it can be written as a linear combination of operators whose side lengths of the supports in one direction do not exceed half the system size, while the side lengths in the other directions are arbitrary. We also prove that the above result holds even in the ladder system. Our results extend the recently developed technique of the proof of absence of local conserved quantities in one-dimensional systems to higher dimensions and to the ladder.