42 research outputs found
Broken symmetry and the variation of critical properties in the phase behaviour of supramolecular rhombus tilings
The degree of randomness, or partial order, present in two-dimensional
supramolecular arrays of isophthalate tetracarboxylic acids is shown to vary
due to subtle chemical changes such as the choice of solvent or small
differences in molecular dimensions. This variation may be quantified using an
order parameter and reveals a novel phase behaviour including random tiling
with varying critical properties as well as ordered phases dominated by either
parallel or non-parallel alignment of neighbouring molecules, consistent with
long-standing theoretical studies. The balance between order and randomness is
driven by small differences in the intermolecular interaction energies, which
we show, using numerical simulations, can be related to the measured order
parameter. Significant variations occur even when the energy difference is much
less than the thermal energy highlighting the delicate balance between entropic
and energetic effects in complex self-assembly processes
Topology by Design in Magnetic nano-Materials: Artificial Spin Ice
Artificial Spin Ices are two dimensional arrays of magnetic, interacting
nano-structures whose geometry can be chosen at will, and whose elementary
degrees of freedom can be characterized directly. They were introduced at first
to study frustration in a controllable setting, to mimic the behavior of spin
ice rare earth pyrochlores, but at more useful temperature and field ranges and
with direct characterization, and to provide practical implementation to
celebrated, exactly solvable models of statistical mechanics previously devised
to gain an understanding of degenerate ensembles with residual entropy. With
the evolution of nano--fabrication and of experimental protocols it is now
possible to characterize the material in real-time, real-space, and to realize
virtually any geometry, for direct control over the collective dynamics. This
has recently opened a path toward the deliberate design of novel, exotic
states, not found in natural materials, and often characterized by topological
properties. Without any pretense of exhaustiveness, we will provide an
introduction to the material, the early works, and then, by reporting on more
recent results, we will proceed to describe the new direction, which includes
the design of desired topological states and their implications to kinetics.Comment: 29 pages, 13 figures, 116 references, Book Chapte
Conjectures on exact solution of three - dimensional (3D) simple orthorhombic Ising lattices
We report the conjectures on the three-dimensional (3D) Ising model on simple
orthorhombic lattices, together with the details of calculations for a putative
exact solution. Two conjectures, an additional rotation in the fourth curled-up
dimension and the weight factors on the eigenvectors, are proposed to serve as
a boundary condition to deal with the topologic problem of the 3D Ising model.
The partition function of the 3D simple orthorhombic Ising model is evaluated
by spinor analysis, by employing these conjectures. Based on the validity of
the conjectures, the critical temperature of the simple orthorhombic Ising
lattices could be determined by the relation of KK* = KK' + KK'' + K'K'' or
sinh 2K sinh 2(K' + K'' + K'K''/K) = 1. For a simple cubic Ising lattice, the
critical point is putatively determined to locate exactly at the golden ratio
xc = exp(-2Kc) = (sq(5) - 1)/2, as derived from K* = 3K or sinh 2K sinh 6K = 1.
If the conjectures would be true, the specific heat of the simple orthorhombic
Ising system would show a logarithmic singularity at the critical point of the
phase transition. The spontaneous magnetization and the spin correlation
functions of the simple orthorhombic Ising ferromagnet are derived explicitly.
The putative critical exponents derived explicitly for the simple orthorhombic
Ising lattices are alpha = 0, beta = 3/8, gamma = 5/4, delta = 13/3, eta = 1/8
and nu = 2/3, showing the universality behavior and satisfying the scaling
laws. The cooperative phenomena near the critical point are studied and the
results obtained based on the conjectures are compared with those of the
approximation methods and the experimental findings. The 3D to 2D crossover
phenomenon differs with the 2D to 1D crossover phenomenon and there is a
gradual crossover of the exponents from the 3D values to the 2D ones.Comment: 176 pages, 4 figure
Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman
In the introduction to this volume, we discuss some of the highlights of the
research career of Chuck Newman. This introduction is divided into two main
sections, the first covering Chuck's work in statistical mechanics and the
second his work in percolation theory, continuum scaling limits, and related
topics.Comment: 38 pages (including many references), introduction to Festschrift in
honor of C.M. Newma
On the Permanent of Certain Circulant Matrices
In this paper we first review the basic computational properties of permanents, and then address some problems concerning permanents of (0; 1) circulant matrices. In particular we analyze their role at the boundary between computational tractability and intractability, showing that (i) a generic circulant matrix contains large arbitrary submatrices, a fact which casts some doubt on the tractability of its permanent; (ii) a very sparse circulant has special properties which can be exploited to efficiently compute its permanent