Condensed Matter Seminar

Large Hermitian eigenvalue problems appear frequently in electronic structure theory. The underlying quantum many-body problem is often referred to as "incomputable" since it involves Hilbert spaces whose dimensionality grows exponentially with the number of particles. This complexity is, in fact, key to the idea of quantum computer, since a moderately large quantum many-body system possesses an amount of information greater than any conventional computer can handle. Yet, exact diagonalization of finite clusters using conventional computers is a valuable tool that helps us understand many physical phenomena.

In this presentation I will review the Lanczos recursion and related Krylov subspace methods that allow us to solve ultra-large Hermitian eigenvalue problems. I will also discuss a complementary classical problem that leads to Hamiltonian (non-Hermitian) eigenvalue equation, and an intricate relation that exists between its eigenvalues and the eigenvalue differences, or excitation energies, of the quantum system.

I will present a generalization of Rayleigh-Ritz minimum principle and of Lanczos recursion to this class of problems and discuss implications to time-dependent (TD) quantum ansatz methods, such as TD density functional theory (TD-DFT), TD Hartree Fock (TDHF), etc.