Examples of universality in statistical physics of disordered and aperiodic models

Author(s)

Thomás Augusto Santoro Haddad

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2004-07-21

Examining board members

Silvio Roberto de Azevedo Salinas; Jairo Rolim Lopes de Almeida; Joao Carlos Alves Barata; Ronald Dickman; Alessandro Paulo Sérvio de Moura

• Advisor: Silvio Roberto de Azevedo Salinas

Abstract

We present in this work a series of studies on the effects of geometrical perturbations on statistical-physics models with continuous phase transitions. These perturbations are generated by random or aperiodic (deterministic) distributions of fields or microscopic couplings, along the lattices on which the models are defined. In case of non-disordered aperiodic systems, we s: show a wealth of examples of the changes that may be brought about on the critical behavior of Ising, Potts and interacting-polymer models. We employ non-perturbative real-space renormalization group techniques, as well as transfer-matrix methods to study the thermodynamics of such systems in the neighborhood of critical and tricritical points. Our conclusion is that although critical exponents may change appreciably in the presence of aperiodic distributions of couplings, universal classifications are nevertheless still workable. The universality classes associated to different models and the distinct ways of implementing aperiodicity are connected to unexpected attractors in Hamiltonian parameter space, which are thoroughly described. In case of random perturbations that break translational symmetry we argue that some universal classifications should still be possible. First, because these systems are in a sense analogous to the former aperiodic ones, and also because they always seem to be associated with some form of complex dynamics (as the dynamics of vitreous, non-random materials). We make some brief comments on this connection, and present a study of a very simpIe disordered model, whose thermodynamics is completely solvable, and which may hide some signatures of complex dynamics. Finally, we discuss the first steps of the so-called Martin-Siggia-Rose (MSR) method, which may be employed in advanced studies of systems undergoing Langevin-type evolutions, and which was responsible, some decades ago, for a first glimpse into the possible universality of complex dynamical behavior of disordered systems. (AU)