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Journal of Physics A Mathematical and General

Comment on `Large-Ntheory of strongly commensurate dirty bosons: absence of a transition in two dimensions'

M. B. Hastings

Published Mar 4, 2002• 0 citations
Publisher

Abstract

We show that the strongly commensurate dirty boson system must have a phase transition in two dimensions, and argue that the low-lying states have a diverging localization length.
DOI
10.1088/0305-4470/35/10/401

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Trivedi and Makivic Reply

Nandini Trivedi, M. S. Makivić

In response to a recent comment [Phys. Rev. Lett. 74 1038 (1995)], the authors further their Monte Carlo study of the two[minus]dimensional dirty boson problem. (AIP)

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The spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).
The large-NNN $\ensuremath{\sigma}$ model, in D<4D<4D<4 space-time dimensions, with disorder a function of d space dimensions, is analyzed via a renormalization-group treatment. Critical exponents for average quantities are calculated, first to lowest order and then to all orders, in $\ensuremath{\epsilon}=D\ensuremath{-}2\ensuremath{-}d/2.$ In particular, it is found that $\ensuremath{\nu}d=2.$ When D=d+1,D=d+1,D=d+1, this model is equivalent to a large-NNN limit of the strongly commensurate dirty boson problem.
The large-NNN commensurate dirty boson model, in both the weakly and strongly commensurate cases, is considered via a perturbative renormalization-group treatment. In the weakly commensurate case, there exists a fixed line under renormalization-group flow, with varying amounts of disorder along the line. Including 1/N1/N1/N corrections causes the system to flow to strong disorder, indicating that the model does not have a phase transition perturbatively connected to the Mott insulator-superfluid (MI-SF) transition. I discuss the qualitative effects of instantons on the low-energy density of excitations. In the strongly commensurate case, a fixed point found previously is considered and results are obtained for higher moments of the correlation functions. To lowest order, correlation functions have a log-normal distribution. Finally, I prove two interesting theorems for large-NNN vector models with disorder, relevant to the problem of replica symmetry breaking and frustration in such systems.