250,986 research outputs found
Criticality and quenched disorder: rare regions vs. Harris criterion
We employ scaling arguments and optimal fluctuation theory to establish a
general relation between quantum Griffiths singularities and the Harris
criterion for quantum phase transitions in disordered systems. If a clean
critical point violates the Harris criterion, it is destabilized by weak
disorder. At the same time, the Griffiths dynamical exponent diverges upon
approaching the transition, suggesting unconventional critical behavior. In
contrast, if the Harris criterion is fulfilled, power-law Griffiths
singularities can coexist with clean critical behavior but saturates at a
finite value. We present applications of our theory to a variety of systems
including quantum spin chains, classical reaction-diffusion systems and
metallic magnets; and we discuss modifications for transitions above the upper
critical dimension. Based on these results we propose a unified classification
of phase transitions in disordered systems.Comment: 4.5 pages, 1 eps figure, final version as publishe
The Harris-Luck criterion for random lattices
The Harris-Luck criterion judges the relevance of (potentially) spatially
correlated, quenched disorder induced by, e.g., random bonds, randomly diluted
sites or a quasi-periodicity of the lattice, for altering the critical behavior
of a coupled matter system. We investigate the applicability of this type of
criterion to the case of spin variables coupled to random lattices. Their
aptitude to alter critical behavior depends on the degree of spatial
correlations present, which is quantified by a wandering exponent. We consider
the cases of Poissonian random graphs resulting from the Voronoi-Delaunay
construction and of planar, ``fat'' Feynman diagrams and precisely
determine their wandering exponents. The resulting predictions are compared to
various exact and numerical results for the Potts model coupled to these
quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one
figure added for clarification, minor re-wordings and typo cleanu
Two-dimensional Site-Bond Percolation as an Example of Self-Averaging System
The Harris-Aharony criterion for a statistical model predicts, that if a
specific heat exponent , then this model does not exhibit
self-averaging. In two-dimensional percolation model the index .
It means that, in accordance with the Harris-Aharony criterion, the model can
exhibit self-averaging properties. We study numerically the relative variances
and for the probability of a site belongin to the
"infinite" (maximum) cluster and the mean finite cluster size . It was
shown, that two-dimensional site-bound percolation on the square lattice, where
the bonds play the role of impurity and the sites play the role of the
statistical ensemble, over which the averaging is performed, exhibits
self-averaging properties.Comment: 15 pages, 5 figure
Self-Averaging in the Three Dimensional Site Diluted Heisenberg Model at the critical point
We study the self-averaging properties of the three dimensional site diluted
Heisenberg model. The Harris criterion \cite{critharris} states that disorder
is irrelevant since the specific heat critical exponent of the pure model is
negative. According with some analytical approaches \cite{harris}, this implies
that the susceptibility should be self-averaging at the critical temperature
(). We have checked this theoretical prediction for a large range of
dilution (including strong dilution) at critically and we have found that the
introduction of scaling corrections is crucial in order to obtain
self-averageness in this model. Finally we have computed critical exponents and
cumulants which compare very well with those of the pure model supporting the
Universality predicted by the Harris criterion.Comment: 11 pages, 11 figures, 14 tables. New analysis (scaling corrections in
the g2=0 scenario) and new numerical simulations. Title and conclusions
chang
Harris Corners in the Real World: A Principled Selection Criterion for Interest Points Based on Ecological Statistics
In this report, we consider whether statistical regularities in natural images might be exploited to provide an improved selection criterion for interest points. One approach that has been particularly influential in this domain, is the Harris corner detector. The impetus for the selection criterion for Harris corners, proposed in early work and which remains in use to this day, is based on an intuitive mathematical definition constrained by the need for computational parsimony. In this report, we revisit this selection criterion free of the computational constraints that existed 20 years ago, and also importantly, taking advantage of the regularities observed in natural image statistics. Based on the motivating factors of stability and richness of structure, a selection threshold for Harris corners is proposed that is optimal with respect to the structure observed in natural images. Following the protocol proposed by Mikolajczyk et al. \cite{miko2005} we demonstrate that the proposed approach produces interest points that are more stable across various image deformations and are more distinctive resulting in improved matching scores. Finally, the proposal may be shown to generalize to provide an improved selection criterion for other types of interest points. As a whole, the report affords an improved selection criterion for Harris corners which might foreseeably benefit any system that employs Harris corners as a constituent component, and additionally presents a general strategy for the selection of interest points based on any measure of local image structure
Rare regions and Griffiths singularities at a clean critical point: The five-dimensional disordered contact process
We investigate the nonequilibrium phase transition of the disordered contact
process in five space dimensions by means of optimal fluctuation theory and
Monte Carlo simulations. We find that the critical behavior is of mean-field
type, i.e., identical to that of the clean five-dimensional contact process. It
is accompanied by off-critical power-law Griffiths singularities whose
dynamical exponent saturates at a finite value as the transition is
approached. These findings resolve the apparent contradiction between the
Harris criterion which implies that weak disorder is renormalization-group
irrelevant and the rare-region classification which predicts unconventional
behavior. We confirm and illustrate our theory by large-scale Monte-Carlo
simulations of systems with up to sites. We also relate our results to a
recently established general relation between the Harris criterion and
Griffiths singularities [Phys. Rev. Lett. {\bf 112}, 075702 (2014)], and we
discuss implications for other phase transitions.Comment: 10 pages, 5 eps figures included, applies the optimal fluctuation
theory of arXiv:1309.0753 to the contact proces
Effective critical behaviour of diluted Heisenberg-like magnets
In agreement with the Harris criterion, asymptotic critical exponents of
three-dimensional (3d) Heisenberg-like magnets are not influenced by weak
quenched dilution of non-magnetic component. However, often in the experimental
studies of corresponding systems concentration- and temperature-dependent
exponents are found with values differing from those of the 3d Heisenberg
model.
In our study, we use the field--theoretical renormalization group approach to
explain this observation and to calculate the effective critical exponents of
weakly diluted quenched Heisenberg-like magnet. Being non-universal, these
exponents change with distance to the critical point as observed
experimentally. In the asymptotic limit (at ) they equal to the critical
exponents of the pure 3d Heisenberg magnet as predicted by the Harris
criterion.Comment: 15 pages, 4 figure
Criticality and Quenched Disorder: Harris Criterion Versus Rare Regions
We employ scaling arguments and optimal fluctuation theory to establish a general relation between quantum Griffiths singularities and the Harris criterion for quantum phase transitions in disordered systems. If a clean critical point violates the Harris criterion, it is destabilized by weak disorder. At the same time, the Griffiths dynamical exponent z\u27 diverges upon approaching the transition, suggesting unconventional critical behavior. In contrast, if the Harris criterion is fulfilled, power-law Griffiths singularities can coexist with clean critical behavior, but z\u27 saturates at a finite value. We present applications of our theory to a variety of systems including quantum spin chains, classical reaction-diffusion systems and metallic magnets, and we discuss modifications for transitions above the upper critical dimension. Based on these results we propose a unified classification of phase transitions in disordered systems
Quantum phase transitions in disordered dimerized quantum spin models and the Harris criterion
We use quantum Monte Carlo simulations to study effects of disorder on the
quantum phase transition occurring versus the ratio g=J/J' in square-lattice
dimerized S=1/2 Heisenberg antiferromagnets with intra- and inter-dimer
couplings J and J'. The dimers are either randomly distributed (as in the
classical dimer model), or come in parallel pairs with horizontal or vertical
orientation. In both cases the transition violates the Harris criterion,
according to which the correlation-length exponent should satisfy nu >= 1. We
do not detect any deviations from the three-dimensional O(3) universality class
obtaining in the absence of disorder (where nu = 0.71). We discuss special
circumstances which allow nu<1 for the type of disorder considered here.Comment: 4+ pages, 3 figure
- …