Segregation in the Falicov--Kimball Model

The Falicov–Kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on-site repulsion; alternatively, it is a model of itinerant electrons and fixed nuclei. It can be seen as a simplification of the Hubbard model; by neglecting the kinetic (hopping) energy of the spin up particles, one gets the Falicov–Kimball model. We show that away from half-filling, i.e. if the sum of the densities of both kinds of particles differs from 1, the particles segregate at zero temperature and for large enough repulsion. In the language of the Hubbard model, this means creating two regions with a positive and a negative magnetization. Our key mathematical results are lower and upper bounds for the sum of the lowest eigenvalues of the discrete Laplace operator in an arbitrary domain, with Dirichlet boundary conditions. The lower bound consists of a bulk term, independent of the shape of the domain, and of a term proportional to the boundary. Therefore, one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary. For the Falicov- Kimball model, this corresponds to having a single “compact” domain that has no heavy particles. Received: 21 June 2001 / Accepted: 4 January 2002     

Enlarged symmetry groups of finite-size clusters with periodic noundary conditions

Any system that approximates an infinite lattice by a family of finite clusters (with periodic boundary conditions) passes through an intermediate region with enlarged (hidden) symmetry as the system size is increased. The hidden symmetry allows for extra degeneracies and level crossings and has application to exact-diagonalization studies, Monte Carlo simulations, lattice gauge theories, and renormalization group calculations.     

Review of the Theoretical Description of Time‐Resolved Angle‐Resolved Photoemission Spectroscopy in Electron‐Phonon Mediated Superconductors

We review recent work on the theory for pump/probe photoemission spectroscopy of electron‐phonon mediated superconductors in both the normal and the superconducting states. We describe the formal developments that allow one to solve the Migdal‐Eliashberg theory in nonequilibrium for an ultrashort laser pumping field, and explore the solutions which illustrate the relaxation as energy is transferred from electrons to phonons. We focus on exact results emanating from sum rules and approximate numerical results which describe rules of thumb for relaxation processes. In addition, in the superconducting state, we describe how Anderson‐Higgs oscillations can be excited due to the nonlinear coupling with the electric field and describe mechanisms where pumping the system enhances superconductivity.