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 to a complicated renormalized topology for the surviving clusters as soon as $d>1$. In this paper, we propose to include Strong Disorder Renormalization ideas within the more traditional fixed cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent $β$ (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed cell-size procedure to two types of renormalizable fractal lattices (i) the Sierpinski gasket of fractal dimension $D=\ln 3/\ln 2$, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in $d=1$ (ii) a hierarchical diamond lattice of fractal dimension $D=4/3$, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension $d>1$. In both cases, we find that the transition is governed by an Infinite Disorder Fixed Point : besides the measure of the activated exponent $ψ$, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.