Many-body physics with quantum gases in disorder

It is commonly accepted that there are no phase transitions in one-dimensional (1D) systems at a finite temperature, because long-range correlations are destroyed by thermal fluctuations. I will demonstrate that the 1D gas of short-range interacting bosons in the presence of disorder can undergo a finite temperature phase transition between two distinct states: fluid and insulator. None of these states has long-range spatial correlations, but this is a true albeit non-conventional phase transition because transport properties are singular at the transition point. In the fluid phase the mass transport is possible, whereas in the insulator phase it is completely blocked even at finite temperatures. Thus, it is revealed how the interaction between disordered bosons influences their Anderson localization. This key question, first raised for electrons in solids, is now crucial for the studies of atomic bosons where recent experiments have demonstrated Anderson localization. I then consider weakly interacting bosons in a 1D quasiperiodic potential (Aubry-Azbel-Harper model), where all single-particle states are localized if the hopping amplitude in the primary lattice is smaller than half the amplitude of the secondary incommensurate lattice. The interparticle interaction may lead to the many-body localization-delocalization transition, and I will show the finite temperature phase diagram. Counterintuitively, in a wide temperature range an increase in temperature requires a higher interaction strength for delocalization and thus favors the insulator state. In this sense, we have an object that ''gets frozen'' under an increase in temperature.