The Bose-Hubbard model with disorder in low-dimensional lattices

In 1958, P.W. Anderson conceived the revolutionary idea that the wave function of a particle in certain random lattices could become localized. [1] Even though the original idea of localization was introduced within the context of spin systems, the subsequent theoretical efforts were mainly concentrated on the localization properties of the electron wave function diffusing into a disordered lattice. In this context, Anderson’s results meant that for sufficiently strong disorder the electron wave function at the chemical potential becomes localized, turning metals into insulators, after him called Anderson insulators. There are by now plenty of experimental evidences of Anderson’s localization phenomenon in amorphous semiconductors, [2] in light-wave experiments, [3, 4] microwaves, [5] sound waves, [6] and electron gases...