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We consider the Thomas--Fermi--Dirac--von~Weizs{\"a}cker model for a systemcomposed of infinitely many nuclei placed on a periodic lattice and electronswith 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 theDirac constant is large enough, the 2-periodic electronic minimizer is not1-periodic, hence symmetry breaking occurs. We analyze in detail the behaviorof the electrons when the Dirac constant tends to infinity and show that theelectrons all concentrate around exactly one of the 8 nuclei of the unit cellof size 2, which is the explanation of the breaking of symmetry. Zooming atthis point, the electronic density solves an effective nonlinear Schr\"odingerequation in the whole space with nonlinearity $u^{7/3}-u^{4/3}$. Our resultsrely on the analysis of this nonlinear equation, in particular on theuniqueness and non-degeneracy of positive solutions.

Symmetry breaking in the periodic Thomas--Fermi--Dirac--von Weizs{\"a}cker model