Proof of the absence of local conserved quantities in the XYZ chain with a magnetic field

Abstract

We rigorously prove that the spin-1/2 XYZ chain with a magnetic field has no local conserved quantity. Any nontrivial conserved quantity of this model is shown to be a sum of operators supported by contiguous sites with at least half of the entire system. We establish that the absence of local conserved quantity in concrete models is provable in a rigorous form.

Figures (1)

• Figure 1: We employ a working definitions of quantum integrability and non-integrability: the presence and absence of local conserved quantities. Integrable systems have sufficiently many local conserved quantities, and their eigenenergies and eigenstates are exactly solvable. Non-integrable systems have no local conserved quantities. Systems with local conserved quantities do not thermalize, and systems with no local conserved quantity are considered to thermalize. Two schematics draw the role of local conserved quantities in the time-evolution of states. If a system has local conserved quantities, the system can evolve only in subspace with fixed local conserved quantities (the tubes in the left figure). Although almost all physical systems in nature are considered to be non-integrable, maybe surprisingly no concrete model has been proven to be non-integrable.