ABSTRACT

In this chapter, we will discuss the conditions under which ground states of quantum systems fail to develop the classical-field component and enter an insulating state. There are several reasons for this to happen. If interparticle interactions are strong and particles are heavy, bosons localize into the crystalline structure. One may argue that a solid with small zero-point fluctuations is essentially a classical state. However, in this case, we employ the classical-particle picture, which, somewhat ironically, cannot be connected to the classical-field picture without invoking quantum mechanics in a non-perturbative way. Bosons may be localized into an insulator also by a deep external lattice potential if their number is commensurate with the lattice period (in a commensurate state, the number of particles per unit lattice cell-the so-called filling factor ν-is integer). For integer ν, an insulating state that does not break any of the lattice symmetries is called a Mott insulator (MI); otherwise, the most common insulating structure is a lattice solid. Finally, an insulating state may result from placing bosons in a strong disordered potential. The generic name for such a state is the Bose glass (BG). The crucial difference between the BG and MI is that the former is compressible, while the latter has a gap for creating particle or hole excitations. Under special circumstances (e.g., in the limit of extremely large [but integer] site occupation numbers and disorder in the interaction potential or hopping amplitudes [the so-called off-diagonal disorder]), the system Hamiltonian becomes particle-hole symmetric. This symmetry leads to a new disordered insulating state, the Mott glass (MG), which has no gap for exciting particles and holes and yet has zero compressibility. Finally, superfluidity and crystalline order may coexist in one and the same material, leading to the supersolid state.