Dual theory of the superfluid-Bose-glass transition in the disordered Bose-Hubbard model in one and two dimensions

Abstract

I study the zero temperature phase transition between superfluid and insulating ground states of the Bose-Hubbard model in a random chemical potential and at large integer average number of particles per site. Duality transformation maps the pure Bose-Hubbard model onto the sine-Gordon theory in one dimension (1D), and onto the three dimensional Higgs electrodynamics in two dimensions (2D). In 1D the random chemical potential in dual theory couples to the space derivative of the dual field, and appears as a random magnetic field along the imaginary time direction in 2D. I show that the transition from the superfluid state in both 1D and 2D is always controlled by the random critical point. This arises due to a coupling constant in the dual theory with replicas which becomes generated at large distances by the random chemical potential, and represents a relevant perturbation at the pure superfluid-Mott insulator fixed point. At large distances the dual theory in 1D becomes equivalent to the Haldane's macroscopic representation of disordered quantum fluid, where the generated term is identified with random backscattering. In 2D the generated coupling corresponds to the random mass of the complex field which represents vortex loops. I calculate the critical exponents at the superfluid-Bose glass fixed point in 2D to be \nu=1.38 and z=1.93, and the universal conductivity at the transition \sigma_c = 0.26 e_{*}^2 /h, using the one-loop field-theoretic renormalization group in fixed dimension.