WFU Physics Colloquium

TITLE: Constrained Variational Calculation of Second-Order Reduced Density Matrices for Hubbard Rings

SPEAKER: William B. Hodge,

TIME: Monday July 21, 2008 at 1:00 PM

PLACE: Room 101 in Olin Physical Laboratory

The quantum-mechanical wave function of a system contains more information than is necessary to calculate many of its properties. For a system with only one- and two-particle interactions almost all properties can be calculated from just the reduced density matrices for one and two particles. This fact presents the question "Can we just calculate the two-particle density matrix in place of the full N-particle wave function?" (The one-particle matrix is obtained by a partial trace on the two-particle matrix.) An exact calculation requires solving the N-particle Schrödinger equation, which is exactly what we don't want to do. Instead we employ a variational method to obtain approximate results. A naive search through the space of two-particle matrices encounters a difficult problem called the "N-representability problem", which we will explain. This problem is overcome by imposing constraints on the search, based on known properties of the reduced density matrix; for example, it must be positive- semidefinite. The more constraints that can be imposed, the better the result. The limitation then becomes the computing power available, and we overcome that as best we can by use of an efficient semidefinite programming algorithm. We compare our variational results for the energy and other properties of the ground state with our results obtained by exact diagonalization of the Hubbard Hamiltonian. This method results in a reasonable lower bound for the energy of the ground state.