1,051 research outputs found
Exact Solutions for Loewner Evolutions
In this note, we solve the Loewner equation in the upper half-plane with
forcing function xi(t), for the cases in which xi(t) has a power-law dependence
on time with powers 0, 1/2 and 1. In the first case the trace of singularities
is a line perpendicular to the real axis. In the second case the trace of
singularities can do three things. If xi(t)=2*(kappa*t)^1/2, the trace is a
straight line set at an angle to the real axis. If xi(t)=2*(kappa*(1-t))^1/2,
the behavior of the trace as t approaches 1 depends on the coefficient kappa.
Our calculations give an explicit solution in which for kappa<4 the trace
spirals into a point in the upper half-plane, while for kappa>4 it intersects
the real axis. We also show that for kappa=9/2 the trace becomes a half-circle.
The third case with forcing xi(t)=t gives a trace that moves outward to
infinity, but stays within fixed distance from the real axis. We also solve
explicitly a more general version of the evolution equation, in which xi(t) is
a superposition of the values +1 and -1.Comment: 20 pages, 7 figures, LaTeX, one minor correction, and improved
hyperref
Conducting-angle-based percolation in the XY model
We define a percolation problem on the basis of spin configurations of the
two dimensional XY model. Neighboring spins belong to the same percolation
cluster if their orientations differ less than a certain threshold called the
conducting angle. The percolation properties of this model are studied by means
of Monte Carlo simulations and a finite-size scaling analysis. Our simulations
show the existence of percolation transitions when the conducting angle is
varied, and we determine the transition point for several values of the XY
coupling. It appears that the critical behavior of this percolation model can
be well described by the standard percolation theory. The critical exponents of
the percolation transitions, as determined by finite-size scaling, agree with
the universality class of the two-dimensional percolation model on a uniform
substrate. This holds over the whole temperature range, even in the
low-temperature phase where the XY substrate is critical in the sense that it
displays algebraic decay of correlations.Comment: 16 pages, 14 figure
Exact valence bond entanglement entropy and probability distribution in the XXX spin chain and the Potts model
By relating the ground state of Temperley-Lieb hamiltonians to partition
functions of 2D statistical mechanics systems on a half plane, and using a
boundary Coulomb gas formalism, we obtain in closed form the valence bond
entanglement entropy as well as the valence bond probability distribution in
these ground states. We find in particular that for the XXX spin chain, the
number N_c of valence bonds connecting a subsystem of size L to the outside
goes, in the thermodynamic limit, as = (4/pi^2) ln L, disproving a recent
conjecture that this should be related with the von Neumann entropy, and thus
equal to 1/(3 ln 2) ln L. Our results generalize to the Q-state Potts model.Comment: 4 pages, 2 figure
Shear banding of colloidal glasses - a dynamic first order transition?
We demonstrate that application of an increasing shear field on a glass leads
to an intriguing dynamic first order transition in analogy to equilibrium
transitions. By following the particle dynamics as a function of the driving
field in a colloidal glass, we identify a critical shear rate upon which the
diffusion time scale of the glass exhibits a sudden discontinuity. Using a new
dynamic order parameter, we show that this discontinuity is analogous to a
first order transition, in which the applied stress acts as the conjugate field
on the system's dynamic evolution. These results offer new perspectives to
comprehend the generic shear banding instability of a wide range of amorphous
materials.Comment: 4 pages, 4 figure
Angular spectrum of quantized light beams
We introduce a generalized angular spectrum representation for quantized
light beams. By using our formalism, we are able to derive simple expressions
for the electromagnetic vector potential operator in the case of: {a)}
time-independent paraxial fields, {b)} time-dependent paraxial fields, and {c)}
non-paraxial fields. For the first case, the well known paraxial results are
fully recovered.Comment: 3 pages, no figure
End to end distance on contour loops of random gaussian surfaces
A self consistent field theory that describes a part of a contour loop of a
random Gaussian surface as a trajectory interacting with itself is constructed.
The exponent \nu characterizing the end to end distance is obtained by a Flory
argument. The result is compared with different previuos derivations and is
found to agree with that of Kondev and Henley over most of the range of the
roughening exponent of the random surface.Comment: 7 page
Criticality in Dynamic Arrest: Correspondence between Glasses and Traffic
Dynamic arrest is a general phenomenon across a wide range of dynamic
systems, but the universality of dynamic arrest phenomena remains unclear. We
relate the emergence of traffic jams in a simple traffic flow model to the
dynamic slow down in kinetically constrained models for glasses. In kinetically
constrained models, the formation of glass becomes a true (singular) phase
transition in the limit . Similarly, using the Nagel-Schreckenberg
model to simulate traffic flow, we show that the emergence of jammed traffic
acquires the signature of a sharp transition in the deterministic limit \pp\to
1, corresponding to overcautious driving. We identify a true dynamical
critical point marking the onset of coexistence between free flowing and jammed
traffic, and demonstrate its analogy to the kinetically constrained glass
models. We find diverging correlations analogous to those at a critical point
of thermodynamic phase transitions.Comment: 4 pages, 4 figure
Intersecting Loop Models on Z^D: Rigorous Results
We consider a general class of (intersecting) loop models in D dimensions,
including those related to high-temperature expansions of well-known spin
models. We find that the loop models exhibit some interesting features - often
in the ``unphysical'' region of parameter space where all connection with the
original spin Hamiltonian is apparently lost. For a particular n=2, D=2 model,
we establish the existence of a phase transition, possibly associated with
divergent loops. However, for n >> 1 and arbitrary D there is no phase
transition marked by the appearance of large loops. Furthermore, at least for
D=2 (and n large) we find a phase transition characterised by broken
translational symmetry.Comment: LaTeX+elsart.cls; 30 p., 6 figs; submitted to Nucl. Phys. B; a few
minor typos correcte
A constrained Potts antiferromagnet model with an interface representation
We define a four-state Potts model ensemble on the square lattice, with the
constraints that neighboring spins must have different values, and that no
plaquette may contain all four states. The spin configurations may be mapped
into those of a 2-dimensional interface in a 2+5 dimensional space. If this
interface is in a Gaussian rough phase (as is the case for most other models
with such a mapping), then the spin correlations are critical and their
exponents can be related to the stiffness governing the interface fluctuations.
Results of our Monte Carlo simulations show height fluctuations with an
anomalous dependence on wavevector, intermediate between the behaviors expected
in a rough phase and in a smooth phase; we argue that the smooth phase (which
would imply long-range spin order) is the best interpretation.Comment: 61 pages, LaTeX. Submitted to J. Phys.
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