Princeton University

Title: Extracting Hidden Symmetry from the Energy Spectrum

Abstract:

I will revisit the problem of hidden symmetry and discuss the connection between integrability and degeneracy in quantum systems. I will consider two contemporary examples of hidden symmetry. The first example comes from recent experiments on spin relaxation in alkali-metal vapors. The model Hamiltonian has a nontrivial degeneracy that shows up as resonances in the spin relaxation rate. I extract the symmetry responsible for this degeneracy and use this example to outline a general approach for determining hidden symmetry from the energy spectrum. As a second example I will discuss 1d Hubbard chains. These systems display many degeneracies of levels of the same coupling independent symmetry. I will show that this apparent violation of the Wigner-von Neumann noncrossing rule follows directly from the existence of conservation laws specific to 1d and is a characteristic signature of quantum integrability. I will also discuss multiply degenerate eigenstates of the Hubbard Hamiltonian. The wave functions of many of these states do not depend on the coupling constant, which suggests the existence of an additional coupling independent symmetry of the Hubbard model.