Real-space renormalization group calculations for spin glasses and dilute magnets

Cluster percolation in the two-dimensional Ising spin glass

Suitable cluster definitions have allowed researchers to describe many ordering transitions in spin systems as geometric phenomena related to percolation. For spin glasses and some other systems with quenched disorder, however, such a connection has not been fully established, and the numerical evidence remains incomplete. Here we use Monte Carlo simulations to study the percolation properties of several classes of clusters occurring in the Edwards-Anderson Ising spin-glass model in two dimensions. The Fortuin-Kasteleyn-Coniglio-Klein clusters originally defined for the ferromagnetic problem do percolate at a temperature that remains nonzero in the thermodynamic limit. On the Nishimori line, this location is accurately predicted by an argument due to Yamaguchi. More relevant for the spin-glass transition are clusters defined on the basis of the overlap of several replicas. We show that various such cluster types have percolation thresholds that shift to lower temperatures by increasing the system size, in agreement with the zero-temperature spin-glass transition in two dimensions. The overlap is linked to the difference in density of the two largest clusters, thus supporting a picture where the spin-glass transition corresponds to an emergent density difference of the two largest clusters inside the percolating phase.

Ordering behavior of the two-dimensional Ising spin glass with long-range correlated disorder

The standard short-range two-dimensional Ising spin glass is numerically well accessible, in particular, because there are polynomial-time ground-state algorithms. On the other hand, in contrast to higher dimensional spin glasses, it does not exhibit a rich behavior, i.e., no ordered phase at finite temperature. Here, we investigate whether long-range correlated bonds change this behavior. This would still keep the model numerically well accessible while exhibiting a more interesting behavior. The bonds are drawn from a Gaussian distribution with a two-point correlation for bonds at distance r that decays as (1+r^{2})^{-a/2}, a≥0. We study numerically with exact algorithms the ground-state and domain-wall excitations. Our results indicate that the inclusion of bond correlations still does not lead to a spin-glass order at any finite temperature. A further analysis reveals that bond correlations have a strong effect at local length scales, inducing ferro- and antiferromagnetic domains into the system. The length scale of ferro- and antiferromagnetic order diverges exponentially as the correlation exponent approaches a critical value, a→a_{crit}=0. Thus, our results suggest that the system becomes a ferro- or antiferromagnet only in the limit a→0.

Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices

We study the wetting transition and the directed polymer delocalization transition on diamond hierarchical lattices. These two phase transitions with frozen disorder correspond to the critical points of quadratic renormalizations of the partition function. (These exact renormalizations on diamond lattices can also be considered as approximate Migdal-Kadanoff renormalizations for hypercubic lattices.) In terms of the rescaled partition function z=Z/Z(typ) , we find that the critical point corresponds to a fixed point distribution with a power-law tail P(c)(z) ~ Phi(ln z)/z(1+mu) as z-->+infinity [up to some subleading logarithmic correction Phi(ln z)], so that all moments z(n) with n>mu diverge. For the wetting transition, the first moment diverges z=+infinity (case 0<mu<1 ), and the critical temperature is strictly below the annealed temperature T(c)<T(ann). For the directed polymer case, the second moment diverges z(2)=+infinity (case 1<mu<2 ), and the critical temperature is strictly below the exactly known transition temperature T2 of the second moment. We then consider the correlation length exponent nu : the linearized renormalization around the fixed point distribution coincides with the transfer matrix describing a directed polymer on the Cayley tree, but the random weights determined by the fixed point distribution P(c)(z) are broadly distributed. This induces some changes in the traveling wave solutions with respect to the usual case of more narrow distributions.

Fractals, phase transitions and criticality

Analogies between behaviour in fractal media and at phase transitions suggest deeper connections. These arise from the scale invariance of configurations at continuous phase transitions, making fractal view-points useful there and making scaling techniques necessary in both areas. These concepts, inter-relations, and techniques are illustrated with the percolation problem and the Ising spin system, which provide simple examples of geometrical and thermal transitions respectively. Diluted magnets involve both geometrical and thermal effects and, when prepared at concentrations near the percolation threshold, exhibit the influence of geometrical self-similarity on such varied processes as cooperative thermal behaviour and dynamics. Phase transitions in these systems are briefly discussed. The anomalous dynamical behaviour of self-similar systems is introduced, and illustrated by the critical dynamics of Heisenberg and Ising magnets diluted to the percolation threshold. These involve linear (spin wave), and nonlinear (activated) dynamical processes on the underlying fractal percolation structure, leading in the linear case to the magnetic analogue of ‘fracton’ dynamics and in the case of activated dynamics to a highly singular critical behaviour.