Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

On the violation of the Harris criterion by RSRG methods

Heber R. da Cruz, Robin B. Stinchcombe

Published Jul 10, 1986• 9 citations
Publisher

Abstract

The authors elucidate the violation of the Harris criterion that often occurs in real-space renormalization group (RSRG) treatments of quenched random systems. The system studied is the bond-diluted q-state Potts model on the diamond Berker lattice, since it allows either sign for the pure specific heat exponent alpha which is calculated exactly and satisfies a generalised hyperscaling relation involving the fractal dimension. It is argued that it is necessary to allow the bond probability distribution to evolve, under length scaling, from its initial binary form. The width of the distribution (excluding the weight of the missing bonds) is found to play a crucial role and is used as an additional parameter. These conclusions are confirmed by actual calculation for this system of the RG flow in a generalised parameter space, in which the random fixed point is located. The associated fixed-point distributions are discussed, and it is demonstrated that the generalised view allows the Harris criterion to be satisfied.

DOI
10.1088/0022-3719/19/19/012

Related papers

Fisher renormalisation and the application of RSRG methods to annealed dilute systems

Heber R. da Cruz, Robin B. Stinchcombe

The application of RSRG methods to annealed random systems is studied with special reference to the violation of Fisher renormalisation by approximate techniques. An RSRG cluster approximation is first applied to the annealed dilute Ising model to obtain flowlines and fixed points. Other possible forms of scaling are considered, and how they may be tested against the scaling of equivalent pure models. It is argued that to satisfy Fisher renormalisation it is necessary to allow the probability distribution of the annealed system to evolve, under length scaling, from its initial simple form. The evolution is investigated for the annealed dilute q-state Potts model on the diamond Berker lattice (which can have either sign for the pure specific heat exponent alpha ). By using the width of the distribution as a crucial additional scaling parameter, it is shown how the flow in the increased parameter space can satisfy the Fisher renormalisation criterion and allow the proper appearance of the random fixed point. The treatment is less complete than that for the quenched case because of the need to approximate a relation between indicator functions and the probability distribution.

Journal of Physics A Mathematical and General•Aug 21, 1986•2 citations

Scale-invariant quenched disorder and its stability criterion at random critical points

David Andelman, A. Nihat Berker

The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for $n$-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.

Physical review. B, Condensed matter•Mar 1, 1984•104 citations

Influence of quenched disorder on absorbing phase transitions in the conserved lattice gas model

Sang Bub Lee

Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.

Physical Review E•Oct 18, 2011•8 citations

Critical phenomena in systems with long-range-correlated quenched disorder

Abel Weinrib, B. I. Halperin

As a model for a phase transition in an inhomogeneous system, we consider a system where the local transition temperature varies in space, with a correlation function obeying a power law $\ensuremath{\sim}{x}^{\ensuremath{-}a}$ for large separations $x$. We extend the Harris criterion for this case, finding that for $a<d$ (where $d$ is the spatial dimension) the disorder is irrelevant if $a\ensuremath{\nu}\ensuremath{-}2>0$, while if $a>d$ we recover the usual Harris criterion: The disorder is irrelevant if $d\ensuremath{\nu}\ensuremath{-}2=\ensuremath{-}\ensuremath{\alpha}>0$. An $m$-vector system of this type is studied with the use of a renormalization-group expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and $\ensuremath{\delta}=4\ensuremath{-}a$. We find a new long-range-disorder fixed point in addition to the short-range-disorder and pure fixed points found previously. The crossover between fixed points is found to follow the extended Harris criterion. The new fixed point has complex eigenvalues, leading to oscillating corrections to scaling, and has a correlation-length exponent $\ensuremath{\nu}=\frac{2}{a}$. We argue that this new scaling relation is exact and applies more generally than just to the specific model. We show that the extended Harris criterion also applies to percolation with long-range-correlated site or bond-occupation probabilities, so that the scaling law should be obeyed by such systems. Results for the percolation properties of the triangular Ising model are in agreement with these predictions.

Physical review. B, Condensed matter•Jan 1, 1983•437 citations

Critical properties of random Potts models

Wolfgang Kinzel, Eytan Domany

The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.

Physical review. B, Condensed matter•Apr 1, 1981•131 citations

Critical behavior with axially correlated random bonds

David Andelman, Amnon Aharony A

Critical properties are studied in systems with quenched bond disorder that is correlated along ${d}_{1}$ of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+${d}_{1}$\ensuremath{\nu} is used. For ${d}_{1}$, we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For ${d}_{1}$=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for ${d}_{1}$=1 are relevant to random quantum systems.

Physical review. B, Condensed matter•Apr 1, 1985•31 citations
The authors elucidate the violation of the Harris criterion that often occurs in real-space renormalization group (RSRG) treatments of quenched random systems. The system studied is the bond-diluted q-state Potts model on the diamond Berker lattice, since it allows either sign for the pure specific heat exponent alpha which is calculated exactly and satisfies a generalised hyperscaling relation involving the fractal dimension. It is argued that it is necessary to allow the bond probability distribution to evolve, under length scaling, from its initial binary form. The width of the distribution (excluding the weight of the missing bonds) is found to play a crucial role and is used as an additional parameter. These conclusions are confirmed by actual calculation for this system of the RG flow in a generalised parameter space, in which the random fixed point is located. The associated fixed-point distributions are discussed, and it is demonstrated that the generalised view allows the Harris criterion to be satisfied.
The application of RSRG methods to annealed random systems is studied with special reference to the violation of Fisher renormalisation by approximate techniques. An RSRG cluster approximation is first applied to the annealed dilute Ising model to obtain flowlines and fixed points. Other possible forms of scaling are considered, and how they may be tested against the scaling of equivalent pure models. It is argued that to satisfy Fisher renormalisation it is necessary to allow the probability distribution of the annealed system to evolve, under length scaling, from its initial simple form. The evolution is investigated for the annealed dilute q-state Potts model on the diamond Berker lattice (which can have either sign for the pure specific heat exponent alpha ). By using the width of the distribution as a crucial additional scaling parameter, it is shown how the flow in the increased parameter space can satisfy the Fisher renormalisation criterion and allow the proper appearance of the random fixed point. The treatment is less complete than that for the quenched case because of the need to approximate a relation between indicator functions and the probability distribution.
The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for nnn-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.
Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.
As a model for a phase transition in an inhomogeneous system, we consider a system where the local transition temperature varies in space, with a correlation function obeying a power law $\ensuremath{\sim}{x}^{\ensuremath{-}a}$ for large separations xxx. We extend the Harris criterion for this case, finding that for a<da<da<d (where ddd is the spatial dimension) the disorder is irrelevant if $a\ensuremath{\nu}\ensuremath{-}2>0$, while if a>da>da>d we recover the usual Harris criterion: The disorder is irrelevant if $d\ensuremath{\nu}\ensuremath{-}2=\ensuremath{-}\ensuremath{\alpha}>0$. An mmm-vector system of this type is studied with the use of a renormalization-group expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and $\ensuremath{\delta}=4\ensuremath{-}a$. We find a new long-range-disorder fixed point in addition to the short-range-disorder and pure fixed points found previously. The crossover between fixed points is found to follow the extended Harris criterion. The new fixed point has complex eigenvalues, leading to oscillating corrections to scaling, and has a correlation-length exponent $\ensuremath{\nu}=\frac{2}{a}$. We argue that this new scaling relation is exact and applies more generally than just to the specific model. We show that the extended Harris criterion also applies to percolation with long-range-correlated site or bond-occupation probabilities, so that the scaling law should be obeyed by such systems. Results for the percolation properties of the triangular Ising model are in agreement with these predictions.
The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.
Critical properties are studied in systems with quenched bond disorder that is correlated along d1{d}_{1}d1​ of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+d1{d}_{1}d1​\ensuremath{\nu} is used. For d1{d}_{1}d1​, we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For d1{d}_{1}d1​=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for d1{d}_{1}d1​=1 are relevant to random quantum systems.