117 research outputs found
On the critical exponent of generalized Thue-Morse words
For certain generalized Thue-Morse words t, we compute the "critical
exponent", i.e., the supremum of the set of rational numbers that are exponents
of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer
Science (accepted October 15, 2007
Remarks on the Spectral Properties of Tight Binding and Kronig-Penney Models with Substitution Sequences
We comment on some recent investigations on the electronic properties of
models associated to the Thue-Morse chain and point out that their conclusions
are in contradiction with rigorously proven theorems and indicate some of the
sources of these misinterpretations. We briefly review and explain the current
status of mathematical results in this field and discuss some conjectures and
open problems.Comment: 15,CPT-94/P.3003,tex,
Surface Magnetization of Aperiodic Ising Quantum Chains
We study the surface magnetization of aperiodic Ising quantum chains. Using
fermion techniques, exact results are obtained in the critical region for
quasiperiodic sequences generated through an irrational number as well as for
the automatic binary Thue-Morse sequence and its generalizations modulo p. The
surface magnetization exponent keeps its Ising value, beta_s=1/2, for all the
sequences studied. The critical amplitude of the surface magnetization depends
on the strength of the modulation and also on the starting point of the chain
along the aperiodic sequence.Comment: 11 pages, 6 eps-figures, Plain TeX, eps
Surface Magnetization of Aperiodic Ising Systems: a Comparative Study of the Bond and Site Problems
We investigate the influence of aperiodic perturbations on the critical
behaviour at a second order phase transition. The bond and site problems are
compared for layered systems and aperiodic sequences generated through
substitution. In the bond problem, the interactions between the layers are
distributed according to an aperiodic sequence whereas in the site problem, the
layers themselves follow the sequence. A relevance-irrelevance criterion
introduced by Luck for the bond problem is extended to discuss the site
problem. It involves a wandering exponent for pairs, which can be larger than
the one considered before in the bond problem. The surface magnetization of the
layered two-dimensional Ising model is obtained, in the extreme anisotropic
limit, for the period-doubling and Thue-Morse sequences.Comment: 19 pages, Plain TeX, IOP macros + epsf, 6 postscript figures, minor
correction
Local critical behaviour at aperiodic surface extended perturbation in the Ising quantum chain
The surface critical behaviour of the semi--infinite one--dimensional quantum
Ising model in a transverse field is studied in the presence of an aperiodic
surface extended modulation. The perturbed couplings are distributed according
to a generalized Fredholm sequence, leading to a marginal perturbation and
varying surface exponents. The surface magnetic exponents are calculated
exactly whereas the expression of the surface energy density exponent is
conjectured from a finite--size scaling study. The system displays surface
order at the bulk critical point, above a critical value of the modulation
amplitude. It may be considered as a discrete realization of the Hilhorst--van
Leeuwen model.Comment: 13 pages, TeX file + 6 figures, epsf neede
Subword complexity and power avoidance
We begin a systematic study of the relations between subword complexity of
infinite words and their power avoidance. Among other things, we show that
-- the Thue-Morse word has the minimum possible subword complexity over all
overlap-free binary words and all -power-free binary words, but not
over all -power-free binary words;
-- the twisted Thue-Morse word has the maximum possible subword complexity
over all overlap-free binary words, but no word has the maximum subword
complexity over all -power-free binary words;
-- if some word attains the minimum possible subword complexity over all
square-free ternary words, then one such word is the ternary Thue word;
-- the recently constructed 1-2-bonacci word has the minimum possible subword
complexity over all \textit{symmetric} square-free ternary words.Comment: 29 pages. Submitted to TC
Aperiodic Ising Quantum Chains
Some years ago, Luck proposed a relevance criterion for the effect of
aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In
this article, we show how Luck's criterion can be derived within an exact
renormalisation scheme for Ising quantum chains with coupling constants
modulated according to substitution rules. Luck's conjectures for this case are
confirmed and refined. Among other outcomes, we give an exact formula for the
correlation length critical exponent for arbitrary two-letter substitution
sequences with marginal fluctuations of the coupling constants.Comment: 27 pages, LaTeX, 1 Postscript figure included, using epsf.sty and
amssymb.sty (one error corrected, some minor changes
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