Communications in Mathematical Physics

, Volume 227, Issue 2, pp 243–279 | Cite as

Segregation in the Falicov--Kimball Model

  • James K. Freericks
  • Elliott H. Lieb
  • Daniel Ueltschi


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.


Dirichlet Boundary Lattice Model Laplace Operator Dirichlet Boundary Condition Hubbard Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • James K. Freericks
    • 1
  • Elliott H. Lieb
    • 2
  • Daniel Ueltschi
    • 3
  1. 1.Department of Physics, Georgetown University, Washington, DC 20057, USA.¶E-mail: freericks@physics.georgetown.eduUS
  2. 2.Departments of Mathematics and Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA.¶E-mail: lieb@math.princeton.eduUS
  3. 3.Department of Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA.¶E-mail: ueltschi@math.ucdavis.eduUS

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