Phase Transitions in Disordered Systems

Introduction

Disordered systems are characterized by two types of variables, configurational degrees of freedom describing the fixed composition of the system under investigation, which usually depends on a preparation process, and dynamical variables proper, by which one denotes those degrees freedom which evolve rapidly on time-scales set by experimental conditions and so attain a thermodynamic equilibrium which depends parametrically on the fixed configuration. Within this project we investigate whether disorder has an influence on the signatures of phase transitions, and if so, to what extent. Currently we are investigating the simplest form of disorder - random spin dilution - and its influence on the transition to ferromagnetic order. One of the questions being investigated is whether disorder will modify the critical exponents of the transition relative to those of the corresponding pure system. A general answer to this question is known for infinitesimally weak disorder. If, on the other hand, disorder is no longer infinitesimally weak, little is known, and the question whether disorder will modify e.g. crititical exponents of the two-dimensional Ising model has remained unsettled for more than 25 years. We are investigating this system using numerical trasfer-matrix and renormalization group methods, in combination with a generalized grand ensemble approach by which disordered systems can be mapped onto ordered ones in a suitably enlarged phase space.

Recent Papers

Large deviations of the finite-time magnetization of the Curie-Weiss random-field Ising model
by Pierre Paga and Reimer Kühn, Phys. Rev. E 96, 022126 (2017) ; also at arXiv:1606.00316
We study the large deviations of the magnetization at some finite time in the Curie-Weiss Random Field Ising Model with parallel updating. While relaxation dynamics in an infinite time horizon gives rise to unique dynamical trajectories (specified by initial conditions and governed by first-order dynamics of the form m(t+1)=f(m(t)), we observe that the introduction of a finite time horizon and the specification of terminal conditions can generate a host of metastable solutions obeying second-order dynamics. We show that these solutions are governed by a Newtonian-like dynamics in discrete time which permits solutions in terms of both the first order relaxation ("forward") dynamics and the backward dynamics m(t+1)= f⁻¹(m(t)). Our approach allows us to classify trajectories for a given final magnetization as stable or metastable according to the value of the rate function associated with them. We find that in analogy to the Freidlin-Wentzell description of the stochastic dynamics of escape from metastable states, the dominant trajectories may switch between the two types (forward and backward) of first-order dynamics.
Gibbs vs. Non-Gibbs in the Equilibrium Ensemble Approach to Disordered Systems

Markov Processes Relat. Fields 10, 523-545 (2004)

We describe the salient ideas of the equilibrium ensemble approach to disordered systems, paying due attention to the appearance of non-Gibbsian measures. A canonical scheme of approximations - constrained annealing - is described and characterised in terms of a Gibbs' variational principle for the free energy functional. It provides a family of increasing exact lower bounds of the quenched free energy of disordered systems, and is shown to avoid the use of non-Gibbsian measures. The connection between the equilibrium ensemble approach and conventional low-concentration expansions or perturbation expansions about ordered reference systems is also explained. Finally applications of the scheme to a number of disordered Ising models and to protein folding are briefly reviewed.

Kühn and Mazzeo Reply
by R. Kühn and Giorgio Mazzeo, Phys. Rev. Lett. 84, 6135 (2000) (pdf-file)

We reply to a comment of Van Enter, Kuelske and Maes Phys. Rev. Lett. 84, 6134 (2000) on our letter "Critical Behavior of the Randomly Spin-Diluted 2-d Ising Model - A Grand Ensemble Approach", Phys. Rev. Lett. 73, 2268-2271 (1994).
Critical Behaviour of the 2-d Spin Diluted Ising Model via the Equilibrium Ensemble Approach
by Giorgio Mazzeo and R. Kühn, Phys. Rev. E 60, 3823-3836 (1999)(pdf-file)
The equilibrium ensemble approach to disordered systems is used to investigate the critical behaviour of the two dimensional Ising model in presence of quenched random site dilution. The numerical transfer matrix technique in semi-infinite strips of finite width, together with phenomenological renormalization and conformal invariance, is particularly suited to put the equilibrium ensemble approach to work. A new method to extract with great precision the critical temperature of the model is proposed and applied. A more systematic finite-size scaling analysis than in previous numerical studies has been performed. A parallel investigation, along the lines of the two main scenarios currently under discussion, namely the logarithmic corrections scenario (with critical exponents fixed in the Ising universality class) versus the weak universality scenario (critical exponents varying with the degree of disorder), is carried out. In interpreting our data, maximum care is constantly taken to be open in both directions. A critical discussion shows that, still, an unambiguous discrimination between the two scenarios is not possible on the basis of the available finite size data.
Equilibrium Ensemble Approach to Disordered Systems I: General Theory, Exact Results
by R. Kühn, Z. Phys. B 100, 231-242 (1996)(pdf)
An outline of Morita's equilibrium ensemble approach to disordered systems is given, and hitherto unnoticed relations to other, more conventional approaches in the theory of disordered systems are pointed out. It is demonstrated to constitute a generalization of the idea of grand ensembles and to be intimately related also to conventional low--concentration expansions as well as to perturbation expansions about ordered reference systems. Moreover, we draw attention to the variational content of the equilibrium ensemble formulation. A number of exact results are presented, among them general solutions for site- and bond- diluted systems in one dimension, both for uncorrelated, and for correlated disorder.
Critical Behaviour of the Randomly Spin-Diluted 2-d Ising Model - A Grand Ensemble Approach
by R. Kühn, Phys. Rev. Lett. 73, 2268-2271 (1994)(pdf-file)
The critical behaviour of the 2--d spin-diluted Ising model is investigated by a new method which combines a grand ensemble approach to disordered systems with phenomenological renormalization. We observe a continuous variation of critical exponents with the density $\rho$ of magnetic impurities, respecting, however, weak universality in the sense that $\eta$ and $\gamma/\nu$ do {\it not\/} depend on $\rho$ while $\gamma$ and $\nu$ separately do. Our results are in complete agreement with a recent Monte--Carlo study.

rk 14.05.2021