Critical behavior of spin and polymer models with aperiodic interactions

Abstract

We review and extend some recent investigations of the effects of aperiodic interactions on the critical behavior of ferromagnetic $q$-state Potts models. By considering suitable diamond or necklace hierarchical lattices, and assuming a distribution of interactions according to a class of two-letter substitution rules, the problem can be formulated in terms of recursion relations in parameter space. The analysis of stability of the fixed points leads to an exact criterion to gauge the relevance of geometric fluctuations. For irrelevant fluctuations, the critical behavior remains unchanged with respect to the uniform systems. For relevant fluctuations, there appears a two-cycle of saddle-point character in parameter space. A scaling analysis, supported by direct numerical thermodynamic calculations, shows the existence of novel critical universality classes associated with relevant geometric fluctuations. Also, we show that similar qualitative results are displayed by a simple model of two directed polymers on a diamond hierarchical structure with aperiodic bond interactions.Comment: 3 eps figures (included). Invited talk in STATPHYS21, proceedings to be published in Physica

Efimov effect of triple-stranded DNA: Real-space renormalization group and zeros of the partition function

“…Hierarchical lattices, by virtue of their discrete scaling, allow one to solve many models in statistical mechanics by exact renormalization group transformations [19][20][21][22]. Furthermore, many approximate real-space RGs on real lattices can be viewed as exact real-space RGs on hierarchical lattices.…”

Critical properties of an aperiodic model for interacting polymers

“…where Λ is the leading eigenvalue of the linearized second-iterate of the RG recursion relations about any one of the two points of the attractor. Results of this analysis have already been given elsewhere [1], and will not be repeated here.…”

“…Let us now obtain the structure of the matrices M G for any G. This follows from Eq. (33), from the hierarchical structure of the lattice, and from the detailed discussion of the form of matrices M (i) 1 and M (i) 2 . The first important property related to the structure of the lattice is that, along each branch, there are sites with different connectivities, and they appear according to a well-defined sequence.…”

“…This quantity will be used to characterize the possible universality classes of the model, depending on q, p and the presence of aperiodically distributed interactions. Consider, for example, for p = 2, a layered distribution of interactions [1], ǫ a and ǫ b , chosen according to the two-letter period-doubling sequence,…”

“…In some recent publications [1]- [4], we have used renormalization-group (RG) and transfer-matrix (TM) techniques to investigate the effects of aperiodically distributed (but not disordered) interactions on the critical behavior of ferromagnetic spin models. In a real-space renormalization calculation for simple hierarchical structures, we have written exact recursion relations in order to show that relevant geometric (aperiodic) fluctuations play a very similar role to disorder.…”