The Link between Integrability, Level Crossings, and Exact Solution in Quantum Models

TL;DR

This paper explores how integrability in quantum models guarantees energy level crossings and exact solutions, linking the existence of conserved quantities to spectral properties and explicitly constructing such Hamiltonians.

Contribution

It demonstrates that the commutation of Hamiltonians linear in coupling leads to exact solutions and level crossings, connecting integrability to Gaudin magnets.

Findings

Energy level crossings are guaranteed in maximally integrable Hamiltonians.

Explicit construction of integrable Hamiltonians equivalent to Gaudin magnets.

Fewer conservation laws do not ensure level crossings.

Abstract

We investigate the connection between energy level crossings in integrable systems and their integrability, i.e. the existence of a set of non-trivial integrals of motion. In particular, we consider a general quantum Hamiltonian linear in the coupling u, H(u) = T + uV, and require that it has the maximum possible number of nontrivial commuting partners also linear in u. We demonstrate how this commutation requirement alone leads to: (1) an exact solution for the energy spectrum and (2) level crossings, which are always present in these Hamiltonians in violation of the Wigner-von Neumann non-crossing rule. Moreover, we construct these Hamiltonians explicitly by resolving the above commutation requirement and show their equivalence to a sector of Gaudin magnets (central spin Hamiltonians). In contrast, fewer than the maximum number of conservation laws does not guarantee level crossings.