Effects of chaos in quantum lattice systems

We discuss several quantum lattice systems and show that there may arise chaotic structures in the form of incommensurate or irregular quantum states. As a first example we consider a tight-binding model in which a single electron is strongly coupled with phonons on a one-dimensional (1D) chain of atoms. In the adiabatic approximation the system is described by a discrete nonlinear Schrödinger equation. We have reformulated this equation to the form of a two-dimensional (2D) mapping. By doing this we may investigate a quantum problem in terms usually applied to classical nonlinear dynamic problems. We find three types of solutions: periodic, quasiperiodic and chaotic. The first one are periodic solutions associated with localized deformations of the lattice, like Peierls' charge-density waves or lattice solitons. In the two latter cases the periodicity of solitons breaks down, i.e., the distance between two nearest solitons deviates slightly from the period, that changes randomly, i.e. is unpredictable and therefore chaotic. Thus, we show that the wave function of an electron on a deformable lattice may exhibit incommensurate and irregular structures, analogous to structures arising in classical chaos. We also discuss some other many-body systems including a Kagomé antiferromagnet, where the ground state may have incommensurate structure.