4,618 research outputs found
Statistical mechanics of glass transition in lattice molecule models
Lattice molecule models are proposed in order to study statistical mechanics
of glass transition in finite dimensions. Molecules in the models are
represented by hard Wang tiles and their density is controlled by a chemical
potential. An infinite series of irregular ground states are constructed
theoretically. By defining a glass order parameter as a collection of the
overlap with each ground state, a thermodynamic transition to a glass phase is
found in a stratified Wang tiles model on a cubic lattice.Comment: 18 pages, 8 figure
Quasiperiodic Heisenberg antiferromagnets in two dimensions
This is a review of the properties of 2d quantum quasiperiodic
antiferromagnets as reported in studies that have been carried out in the last
decade. Many results have been obtained for perfectly ordered as well as for
disordered two dimensional bipartite quasiperiodic tilings. The theoretical
methods used include spin wave theory, and renormalization group along with
Quantum Monte Carlo simulations. These methods all show that the ground state
of these unfrustrated antiferromagnets have N\'eel type order but with a highly
complex spatial distribution of local staggered magnetization. The ground state
properties, excitation energies and spatial dependence, structure factor, and
local susceptibilities are presented. The effects of introducing geometrical
disorder on the magnetic properties are discussed.Comment: 21 pages, 29 figure
Modelling quasicrystals at positive temperature
We consider a two-dimensional lattice model of equilibrium statistical
mechanics, using nearest neighbor interactions based on the matching conditions
for an aperiodic set of 16 Wang tiles. This model has uncountably many ground
state configurations, all of which are nonperiodic. The question addressed in
this paper is whether nonperiodicity persists at low but positive temperature.
We present arguments, mostly numerical, that this is indeed the case. In
particular, we define an appropriate order parameter, prove that it is
identically zero at high temperatures, and show by Monte Carlo simulation that
it is nonzero at low temperatures
Repetitive Delone Sets and Quasicrystals
This paper considers the problem of characterizing the simplest discrete
point sets that are aperiodic, using invariants based on topological dynamics.
A Delone set whose patch-counting function N(T), for radius T, is finite for
all T is called repetitive if there is a function M(T) such that every ball of
radius M(T)+T contains a copy of each kind of patch of radius T that occurs in
the set. This is equivalent to the minimality of an associated topological
dynamical system with R^n-action. There is a lower bound for M(T) in terms of
N(T), namely N(T) = O(M(T)^n), but no general upper bound.
The complexity of a repetitive Delone set can be measured by the growth rate
of its repetitivity function M(T). For example, M(T) is bounded if and only if
the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and
densely repetitive if M(T) = O(N(T))^{1/n}). We show that linearly repetitive
sets and densely repetitive sets have strict uniform patch frequencies, i.e.
the associated topological dynamical system is strictly ergodic. It follows
that such sets are diffractive. In the reverse direction, we construct a
repetitive Delone set in R^n which has
M(T) = O(T(log T)^{2/n}(log log log T)^{4/n}), but does not have uniform
patch frequencies. Aperiodic linearly repetitive sets have many claims to be
the simplest class of aperiodic sets, and we propose considering them as a
notion of "perfectly ordered quasicrystal".Comment: To appear in "Ergodic Theory and Dynamical Systems" vol.23 (2003). 37
pages. Uses packages latexsym, ifthen, cite and files amssym.def, amssym.te
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
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