Strong disorder RG principles within a fixed-cell-size real space renormalization:application to the random transverse field Ising model on various fractal lattices

Abstract

Strong disorder renormalization is an energy-based renormalization that leads toa complicated renormalized topology for the surviving clusters as soon asd > 1. In this paper, we propose to include strong disorder renormalization ideas within the moretraditional fixed-cell-size real space RG framework. We first consider the one-dimensionalchain as a test for this fixed-cell-size procedure: we find that all exactly knowncritical exponents are reproduced correctly, except for the magnetic exponentβ (because it is related to more subtle persistence properties of the full RGflow). We then apply numerically this fixed-cell-size procedure to two types ofrenormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimensionD = ln3/ln2, where there is no underlying classical ferromagnetic transition, sothat the RG flow in the ordered phase is similar to what happens ind = 1; (ii) a hierarchical diamond lattice of fractal dimensionD = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in theordered phase is similar to what happens on hypercubic lattices of dimensiond > 1. In both cases, we find that the transition is governed by an infinitedisorder fixed point: besides the measure of the activated exponentψ, we analyze the RG flow of various observables in the disordered and ordered phases, inorder to extract the ‘typical’ correlation length exponents of these two phases which aredifferent from the finite-size correlation length exponent.