3 research outputs found
Symmetry breaking in the periodic Thomas--Fermi--Dirac--von Weizs{\"a}cker model
We consider the Thomas--Fermi--Dirac--von~Weizs{\"a}cker model for a system
composed of infinitely many nuclei placed on a periodic lattice and electrons
with a periodic density. We prove that if the Dirac constant is small enough,
the electrons have the same periodicity as the nuclei. On the other hand if the
Dirac constant is large enough, the 2-periodic electronic minimizer is not
1-periodic, hence symmetry breaking occurs. We analyze in detail the behavior
of the electrons when the Dirac constant tends to infinity and show that the
electrons all concentrate around exactly one of the 8 nuclei of the unit cell
of size 2, which is the explanation of the breaking of symmetry. Zooming at
this point, the electronic density solves an effective nonlinear Schr\"odinger
equation in the whole space with nonlinearity . Our results
rely on the analysis of this nonlinear equation, in particular on the
uniqueness and non-degeneracy of positive solutions