The thermodynamic limit of quantum Coulomb systems Part II. Applications

In a previous paper [C. Hainzl, M. Lewin, J.P. Solovej, The thermodynamic limit of quantum Coulomb systems. Part I. General theory, Adv. Math. (2009), doi:10.1016/j.aim.2008.12.010 (this issue)], we have developed a general theory of thermodynamic limits. We apply it here to three different Coulomb quantum systems, for which we prove the convergence of the free energy per unit volume.The first system is the crystal for which the nuclei are classical particles arranged periodically in space and only the electrons are quantum particles. We recover and generalize a previous result of Fefferman. In the second example, both the nuclei and the electrons are quantum particles, submitted to a periodic magnetic field. We thereby extend a seminal result of Lieb and Lebowitz. Finally, in our last example we take again classical nuclei but optimize their position. To our knowledge such a system was never treated before.The verification of the assumptions introduced in [C. Hainzl, M. Lewin, J.P. Solovej, The thermodynamic limit of quantum Coulomb systems. Part I. General theory, Adv. Math. (2009), doi:10.1016/j.aim.2008.12.010 (this issue)] uses several tools which have been introduced before in the study of large quantum systems. In particular, an electrostatic inequality of Graf and Schenker is one main ingredient of our new approach.