Renormalization of Sparse Disorder in the Ising Model

Abstract

We consider the renormalization of quenched bond disorder in the Ising model in the limit that it is sparse -- highly localized and vanishing in the thermodynamic limit. We begin in 1D with arbitrary disorder assigned to a finite number of bonds and study how the system renormalizes, finding non-trivial fixed point structure for any given bond with a separatrix at zero bond strength, equivalent to inserting a break in the chain. Either side of this critical line, renormalization group (RG) trajectories flow towards one of two attractors on which the disordered bonds settle onto ferromagnetic or anti-ferromagnetic (AF) couplings of equal and opposite magnitude. Bonds that settle on an AF attractor are equivalent to inserting a twist in the chain at the location of the bond, implying a multi-kink ground state solution. Qualitatively different behavior emerges at the RG step when bonds start to coalesce, with the chain `untwisting' whenever two AF bonds coalesce. Our findings generalize to higher dimensions for codimension one defects that are sparse from the perspective of the orthogonal complement lattice. In 2D, the $\mathbb Z_2$ symmetry of the model has an IR manifestation where one can construct field strengths and Wilson loops (which characterize frustration) from fundamental plaquette variables. In the non-sparse limit where the disorder parameter is drawn from an arbitrary but homogeneously assigned probability distribution function, we recover previously found fixed distributions as special cases, but find only the trivial paramagnetic distribution to be an attractor. For non-homogeneously assigned disorder that is sufficiently dilute however, we find the Edwards-Anderson model with equal probability $\pm J$ bonds to also be an IR attractor.