Non-existence of quasi-local conserved quantities in higher-dimensional XY/XYZ models

Establish a rigorous non-existence theorem for quasi-local conserved quantities in the S=1/2 XY and XYZ spin models on the d-dimensional hypercubic lattice with d ≥ 2, uniform nearest-neighbor interactions (with X ≠ 0 and Y ≠ 0) and arbitrary uniform magnetic field. Specifically, prove that no nonzero quasi-local operator—defined as a formal sum of Pauli-string operators with suitably decaying coefficients—that commutes with the Hamiltonian exists.

Background

The paper proves that for S=1/2 XY and XYZ models on hypercubic lattices in dimensions d ≥ 2, there are no nontrivial local conserved quantities beyond trivial ones like the Hamiltonian, thereby strongly suggesting non-integrability. To connect this operator-algebraic absence to long-time dynamics, the authors highlight the importance of understanding quasi-local conserved quantities—operators expressed as infinite sums of Pauli strings with decaying coefficients that commute with the Hamiltonian.

In Appendix A, the authors formalize the notion of quasi-local conserved quantities on the infinite lattice and provide a very preliminary no-go result for the XX model that rules out such quantities only under an extremely fast (super-physical) decay assumption. They explicitly indicate the need for fundamentally new ideas to relax this assumption (e.g., to exponential decay) and to obtain a general non-existence theorem analogous to their local case.