Toward a theory of quantum integrability in finite size systems

Abstract

OF THE DISSERTATION Toward a theory of quantum integrability in finite size systems by Haile Owusu Dissertation Director: Prof. Emil Yuzbashyan Classical integrability is known to be well defined, but there remains no comparable quantum analogue. Nevertheless, condensed matter models – BCS, Hubbard, Gaudin, XXZ models, etc. – exhibiting unusual properties like the existence of non-trivial conservation laws, exact solvability, and violation of the Wigner-von Neumann noncrossing rule are widely considered to be quantum integrable. In this thesis we propose a well-defined simple notion of quantum integrability in finite size systems based on the parameter dependence of a Hamiltonian and its integrals of motion. We require that integrable Hamiltonians be linear in some system coupling, e.g. u, and that their integrals of motion be similarly linear in u. With this definition of integrability alone we are able to classify all integrable systems according to their number of conservation laws and, for a large subset, parameterize and exactly solve them. In this thesis we classify families of N x N commuting Hamiltonians according to their Type, whereby a Type M family has N −M + 1 members. For Type 1 matrices, having a maximum N such matrices, we derive a parameterization and exact solution of these maximal operators. We further show that these maximal operators are equivalent to a sector of the Gaudin magnets and their spectra necessarily violate the Wigner-von