454 research outputs found
Glass models on Bethe lattices
We consider ``lattice glass models'' in which each site can be occupied by at
most one particle, and any particle may have at most l occupied nearest
neighbors. Using the cavity method for locally tree-like lattices, we derive
the phase diagram, with a particular focus on the vitreous phase and the
highest packing limit. We also study the energy landscape via the
configurational entropy, and discuss different equilibrium glassy phases.
Finally, we show that a kinetic freezing, depending on the particular dynamical
rules chosen for the model, can prevent the equilibrium glass transitions.Comment: 24 pages, 11 figures; minor corrections + enlarged introduction and
conclusio
Strengths and Weaknesses of Parallel Tempering
Parallel tempering, also known as replica exchange Monte Carlo, is studied in
the context of two simple free energy landscapes. The first is a double well
potential defined by two macrostates separated by a barrier. The second is a
`golf course' potential defined by microstates having two possible energies
with exponentially more high energy states than low energy states. The
equilibration time for replica exchange is analyzed for both systems. For the
double well system, parallel tempering with a number of replicas that scales as
the square root of the barrier height yields exponential speedup of the
equilibration time. On the other hand, replica exchange yields only marginal
speed-up for the golf course system. For the double well system, the free
energy difference between the two wells has a large effect on the equilibration
time. Nearly degenerate wells equilibrate much more slowly than strongly
asymmetric wells. It is proposed that this difference in equilibration time may
lead to a bias in measuring overlaps in spin glasses. These examples illustrate
the strengths and weaknesses of replica exchange and may serve as a guide for
understanding and improving the method in various applications.Comment: 18 pages, 4 figures. v2: typos fixed and wording changes to improve
clarit
Survey Propagation as local equilibrium equations
It has been shown experimentally that a decimation algorithm based on Survey
Propagation (SP) equations allows to solve efficiently some combinatorial
problems over random graphs. We show that these equations can be derived as
sum-product equations for the computation of marginals in an extended space
where the variables are allowed to take an additional value -- -- when they
are not forced by the combinatorial constraints. An appropriate ``local
equilibrium condition'' cost/energy function is introduced and its entropy is
shown to coincide with the expected logarithm of the number of clusters of
solutions as computed by SP. These results may help to clarify the geometrical
notion of clusters assumed by SP for the random K-SAT or random graph coloring
(where it is conjectured to be exact) and helps to explain which kind of
clustering operation or approximation is enforced in general/small sized models
in which it is known to be inexact.Comment: 13 pages, 3 figure
Matching Kasteleyn Cities for Spin Glass Ground States
As spin glass materials have extremely slow dynamics, devious numerical
methods are needed to study low-temperature states. A simple and fast
optimization version of the classical Kasteleyn treatment of the Ising model is
described and applied to two-dimensional Ising spin glasses. The algorithm
combines the Pfaffian and matching approaches to directly strip droplet
excitations from an excited state. Extended ground states in Ising spin glasses
on a torus, which are optimized over all boundary conditions, are used to
compute precise values for ground state energy densities.Comment: 4 pages, 2 figures; minor clarification
Replica Symmetry Breaking and the Renormalization Group Theory of the Weakly Disordered Ferromagnet
We study the critical properties of the weakly disordered -component
ferromagnet in terms of the renormalization group (RG) theory generalized to
take into account the replica symmetry breaking (RSB) effects coming from the
multiple local minima solutions of the mean-field equations. It is shown that
for the traditional RG flows at dimensions , which are
usually considered as describing the disorder-induced universal critical
behavior, are unstable with respect to the RSB potentials as found in spin
glasses. It is demonstrated that for a general type of the Parisi RSB
structures there exists no stable fixed points, and the RG flows lead to the
{\it strong coupling regime} at the finite scale , where
is the small parameter describing the disorder. The physical concequences
of the obtained RG solutions are discussed. In particular, we argue, that
discovered RSB strong coupling phenomena indicate on the onset of a new spin
glass type critical behaviour in the temperature interval near . Possible relevance of the considered RSB effects for
the Griffith phase is also discussed.Comment: 32 pages, Late
On the structure of correlations in the three dimensional spin glasses
We investigate the low temperature phase of three-dimensional
Edwards-Anderson model with Bernoulli random couplings. We show that at a fixed
value of the overlap the model fulfills the clustering property: the
connected correlation functions between two local overlaps decay as a power
whose exponent is independent of for all . Our findings
are in agreement with the RSB theory and show that the overlap is a good order
parameter.Comment: 5 pages, 5 figure
Phase transition in the Countdown problem
Here we present a combinatorial decision problem, inspired by the celebrated
quiz show called the countdown, that involves the computation of a given target
number T from a set of k randomly chosen integers along with a set of
arithmetic operations. We find that the probability of winning the game
evidences a threshold phenomenon that can be understood in the terms of an
algorithmic phase transition as a function of the set size k. Numerical
simulations show that such probability sharply transitions from zero to one at
some critical value of the control parameter, hence separating the algorithm's
parameter space in different phases. We also find that the system is maximally
efficient close to the critical point. We then derive analytical expressions
that match the numerical results for finite size and permit us to extrapolate
the behavior in the thermodynamic limit.Comment: Submitted for publicatio
Disorder effects in the quantum Heisenberg model: An Extended Dynamical mean-field theory analysis
We investigate a quantum Heisenberg model with both antiferromagnetic and
disordered nearest-neighbor couplings. We use an extended dynamical mean-field
approach, which reduces the lattice problem to a self-consistent local impurity
problem that we solve by using a quantum Monte Carlo algorithm. We consider
both two- and three-dimensional antiferromagnetic spin fluctuations and
systematically analyze the effect of disorder. We find that in three dimensions
for any small amount of disorder a spin-glass phase is realized. In two
dimensions, while clean systems display the properties of a highly correlated
spin-liquid (where the local spin susceptibility has a non-integer power-low
frequency and/or temperature dependence), in the present case this behavior is
more elusive unless disorder is very small. This is because the spin-glass
transition temperature leaves only an intermediate temperature regime where the
system can display the spin-liquid behavior, which turns out to be more
apparent in the static than in the dynamical susceptibility.Comment: 15 pages, 7 figure
New evidence for super-roughening in crystalline surfaces with disordered substrate
We study the behavior of the Binder cumulant related to long distance
correlation functions of the discrete Gaussian model of disordered substrate
crystalline surfaces. We exhibit numerical evidence that the non-Gaussian
behavior in the low- region persists on large length scales, in agreement
with the broken phase being super-rough.Comment: 10 pages and 4 figures, available at
http://chimera.roma1.infn.it/index_papers_complex.html . We have extended the
RG discussion and minor changes in the tex
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