Mathematical Colloquium: Localization of interacting quantum particles with quasi-random disorder

Vieri Mastropietro
Universita’ di Milano
Monday, May 22, 2017 - 16:00 to 17:00

It is well established at a mathematical level that disorder can produce Anderson localization of the eigenvectors of the single particle Schrödinger equation. Does localization survive in presence of many body interaction? A positive answer to such question would have important physical consequences, related to lack of thermalization in closed quantum systems. Mathematical results on such issue are still rare and a full understanding is a challenging problem. We present an example in which localization can be proved for the ground state of an interacting system of fermionic particles with quasi random Aubry-Andre' potential. The Hamiltonian is given by $N$ coupled almost-Mathieu Schrödinger operators. By assuming Diophantine conditions on the frequency and density, we can establish exponential decay of the ground state correlations. The proof combines methods coming from the direct proof of convergence of KAM Lindstedt series with Renormalization Group methods for many body systems. Small divisors appear in the expansions, whose convergence follows exploiting the Diophantine conditions and fermionic cancellations. The main difficulty comes from the presence of loop graphs, which are the signature of many body interaction and are absent in KAM series. V.Mastropietro. Comm Math Phys 342, 217 (2016); Phys Rev Lett 115, 180401 (2015); Comm. Math. Phys. (2017)

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