95 research outputs found
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
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
Multidimensional Gaussian sums arising from distribution of Birkhoff sums in zero entropy dynamical systems
A duality formula, of the Hardy and Littlewood type for multidimensional
Gaussian sums, is proved in order to estimate the asymptotic long time behavior
of distribution of Birkhoff sums of a sequence generated by a skew
product dynamical system on the torus, with zero Lyapounov
exponents. The sequence, taking the values , is pairwise independent
(but not independent) ergodic sequence with infinite range dependence. The
model corresponds to the motion of a particle on an infinite cylinder, hopping
backward and forward along its axis, with a transversal acceleration parameter
. We show that when the parameter is rational then all
the moments of the normalized sums , but the second, are
unbounded with respect to n, while for irrational , with bounded
continuous fraction representation, all these moments are finite and bounded
with respect to n.Comment: To be published in J. Phys.
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
Anomalous Diffusion in Aperiodic Environments
We study the Brownian motion of a classical particle in one-dimensional
inhomogeneous environments where the transition probabilities follow
quasiperiodic or aperiodic distributions. Exploiting an exact correspondence
with the transverse-field Ising model with inhomogeneous couplings we obtain
many new analytical results for the random walk problem. In the absence of
global bias the qualitative behavior of the diffusive motion of the particle
and the corresponding persistence probability strongly depend on the
fluctuation properties of the environment. In environments with bounded
fluctuations the particle shows normal diffusive motion and the diffusion
constant is simply related to the persistence probability. On the other hand in
a medium with unbounded fluctuations the diffusion is ultra-slow, the
displacement of the particle grows on logarithmic time scales. For the
borderline situation with marginal fluctuations both the diffusion exponent and
the persistence exponent are continuously varying functions of the
aperiodicity. Extensions of the results to disordered media and to higher
dimensions are also discussed.Comment: 11 pages, RevTe
Surface Magnetization and Critical Behavior of Aperiodic Ising Quantum Chains
We consider semi-infinite two-dimensional layered Ising models in the extreme
anisotropic limit with an aperiodic modulation of the couplings. Using
substitution rules to generate the aperiodic sequences, we derive functional
equations for the surface magnetization. These equations are solved by
iteration and the surface magnetic exponent can be determined exactly. The
method is applied to three specific aperiodic sequences, which represent
different types of perturbation, according to a relevance-irrelevance
criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant
perturbation, the surface magnetization vanishes with a square root
singularity, like in the homogeneous lattice. For the period-doubling sequence,
the perturbation is marginal and the surface magnetic exponent varies
continuously with the modulation amplitude. Finally, the Rudin-Shapiro
sequence, which corresponds to the relevant case, displays an anomalous surface
critical behavior which is analyzed via scaling considerations: Depending on
the value of the modulation, the surface magnetization either vanishes with an
essential singularity or remains finite at the bulk critical point, i.e., the
surface phase transition is of first order.Comment: 8 pages, 7 eps-figures, uses RevTex and epsf, minor correction
Crossover between aperiodic and homogeneous semi-infinite critical behaviors in multilayered two-dimensional Ising models
We investigate the surface critical behavior of two-dimensional multilayered
aperiodic Ising models in the extreme anisotropic limit. The system under
consideration is obtained by piling up two types of layers with respectively
and spin rows coupled via nearest neighbor interactions and
, where the succession of layers follows an aperiodic sequence. Far
away from the critical regime, the correlation length is smaller
than the first layer width and the system exhibits the usual behavior of an
ordinary surface transition. In the other limit, in the neighborhood of the
critical point, diverges and the fluctuations are sensitive to the
non-periodic structure of the system so that the critical behavior is governed
by a new fixed point. We determine the critical exponent associated to the
surface magnetization at the aperiodic critical point and show that the
expected crossover between the two regimes is well described by a scaling
function. From numerical calculations, the parallel correlation length
is then found to behave with an anisotropy exponent which
depends on the aperiodic modulation and the layer widths.Comment: LaTeX file, 9 pages, 8 eps figures, to appear in Phys. Rev.
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,
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