61 research outputs found
Pure point diffraction and cut and project schemes for measures: The smooth case
We present cut and project formalism based on measures and continuous weight
functions of sufficiently fast decay. The emerging measures are strongly almost
periodic. The corresponding dynamical systems are compact groups and
homomorphic images of the underlying torus. In particular, they are strictly
ergodic with pure point spectrum and continuous eigenfunctions. Their
diffraction can be calculated explicitly. Our results cover and extend
corresponding earlier results on dense Dirac combs and continuous weight
functions with compact support. They also mark a clear difference in terms of
factor maps between the case of continuous and non-continuous weight functions.Comment: 30 page
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
Log-periodic corrections to scaling: exact results for aperiodic Ising quantum chains
Log-periodic amplitudes of the surface magnetization are calculated
analytically for two Ising quantum chains with aperiodic modulations of the
couplings. The oscillating behaviour is linked to the discrete scale invariance
of the perturbations. For the Fredholm sequence, the aperiodic modulation is
marginal and the amplitudes are obtained as functions of the deviation from the
critical point. For the other sequence, the perturbation is relevant and the
critical surface magnetization is studied.Comment: 12 pages, TeX file, epsf, iopppt.tex, xref.tex which are joined. 4
postcript figure
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
Close-packed dimers on the line: diffraction versus dynamical spectrum
The translation action of \RR^{d} on a translation bounded measure
leads to an interesting class of dynamical systems, with a rather rich spectral
theory. In general, the diffraction spectrum of , which is the carrier
of the diffraction measure, live on a subset of the dynamical spectrum. It is
known that, under some mild assumptions, a pure point diffraction spectrum
implies a pure point dynamical spectrum (the opposite implication always being
true). For other systems, the diffraction spectrum can be a proper subset of
the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with
singular continuous diffraction) in \cite{EM}. Here, we construct a random
system of close-packed dimers on the line that have some underlying long-range
periodic order as well, and display the same type of phenomenon for a system
with absolutely continuous spectrum. An interpretation in terms of `atomic'
versus `molecular' spectrum suggests a way to come to a more general
correspondence between these two types of spectra.Comment: 14 pages, with some additions and improvement
Common trends in the critical behavior of the Ising and directed walk models
We consider layered two-dimensional Ising and directed walk models and show
that the two problems are inherently related. The information about the
zero-field thermodynamical properties of the Ising model is contained into the
transfer matrix of the directed walk. For several hierarchical and aperiodic
distributions of the couplings, critical exponents for the two problems are
obtained exactly through renormalization.Comment: 4 pages, RevTeX file + 1 figure, epsf needed. To be published in PR
Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
Delone sets of finite local complexity in Euclidean space are investigated.
We show that such a set has patch counting and topological entropy 0 if it has
uniform cluster frequencies and is pure point diffractive. We also note that
the patch counting entropy is 0 whenever the repetitivity function satisfies a
certain growth restriction.Comment: 16 pages; revised and slightly expanded versio
Interface Fluctuations on a Hierarchical Lattice
We consider interface fluctuations on a two-dimensional layered lattice where
the couplings follow a hierarchical sequence. This problem is equivalent to the
diffusion process of a quantum particle in the presence of a one-dimensional
hierarchical potential. According to a modified Harris criterion this type of
perturbation is relevant and one expects anomalous fluctuating behavior. By
transfer-matrix techniques and by an exact renormalization group transformation
we have obtained analytical results for the interface fluctuation exponents,
which are discontinuous at the homogeneous lattice limit.Comment: 14 pages plain Tex, one Figure upon request, Phys Rev E (in print
Tiling groupoids and Bratteli diagrams
Let T be an aperiodic and repetitive tiling of R^d with finite local
complexity. Let O be its tiling space with canonical transversal X. The tiling
equivalence relation R_X is the set of pairs of tilings in X which are
translates of each others, with a certain (etale) topology. In this paper R_X
is reconstructed as a generalized "tail equivalence" on a Bratteli diagram,
with its standard AF-relation as a subequivalence relation.
Using a generalization of the Anderson-Putnam complex, O is identified with
the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is
built from this sequence, and its set of infinite paths dB is homeomorphic to
X. The diagram B is endowed with a horizontal structure: additional edges that
encode the adjacencies of patches in T. This allows to define an etale
equivalence relation R_B on dB which is homeomorphic to R_X, and contains the
AF-relation of "tail equivalence".Comment: 34 pages, 4 figure
Aperiodicity-Induced Second-Order Phase Transition in the 8-State Potts Model
We investigate the critical behavior of the two-dimensional 8-state Potts
model with an aperiodic distribution of the exchange interactions between
nearest-neighbor rows. The model is studied numerically through intensive Monte
Carlo simulations using the Swendsen-Wang cluster algorithm. The transition
point is located through duality relations, and the critical behavior is
investigated using FSS techniques at criticality. For strong enough
fluctuations of the aperiodic sequence under consideration, a second order
phase transition is found. The exponents and are
obtained at the new fixed point.Comment: LaTeX file with Revtex, 4 pages, 5 eps figures, to appear in Phys.
Rev. Let
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