Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2008-07-02
2009 J. Phys. A: Math. Theor. 42 035206
Physics
Condensed Matter
Statistical Mechanics
33 pages, 10 figures, minor typos corrected, reference added, model generalized beyond real symmetric to Hermitian operators
Scientific paper
10.1088/1751-8113/42/3/035206
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.
Owusu Haile K.
Wagh Kshitij
Yuzbashyan Emil A.
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