Bootstrapping Bloch bands

Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.

One-particle spectral densities and phase diagrams of one-dimensional proton conductors

The equilibrium states of one-dimensional proton conductors in the systems with hydrogen bonds are investigated. Our extended hard-core boson lattice model includes short-range interactions between hydrogen ions, their transfer along the hydrogen bonds with two-minima local anharmonic potential, as well as their inter-bond hopping, and the modulating field is taken into account. The exact diagonalization method for finite one-dimensional system with periodic boundary conditions is used. The existence of various phases of the system at T = 0, depending on the values of short-range interactions between particles and the modulating field strength, is established by analyzing the character of the obtained frequency dependence of one-particle spectral density; the phase diagrams are built.