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

arXiv:cond-mat/9607091 · Journal of Physics A Mathematical and General

Griffiths singularities in the two-dimensional diluted Ising model

Juan Jesus Ruiz-Lorenzo

Published Jan 21, 1997• 17 citations
arXivHTMLPDF

Abstract

We study numerically the probability distribution of the Yang - Lee zeroes inside the Griffiths phase for the two-dimensional site diluted Ising model and we check that the shape of this distribution is that predicted in previous analytical works. By studying the finite-size scaling of the averaged smallest zero at the phase transition we extract, for two values of the dilution, the anomalous dimension, , which agrees very well with the previous estimated values.

DOI
10.1088/0305-4470/30/2/014

Related papers

In and beyond the Griffiths phase: A large-deviation study of the magnetic susceptibility of the two-dimensional bond-diluted Ising model

Lambert Münster, Alexander K. Hartmann, Martin Weigel

The Griffiths phase in systems with quenched disorder occurs below the ordering transition of the pure system down to the ordering transition of the actual disordered system. While it does not exhibit long-range order, large fluctuations in the disorder degrees of freedom result in exponentially rare, long-range ordered states and hence the occurrence of broad distributions in response functions. Inside the Griffiths phase of the two-dimensional bond-diluted Ising model the distribution of the magnetic susceptibility is expected to have such a broad, exponential tail. A large-deviation Monte Carlo algorithm is used to sample this distribution and the exponential tail is extracted over a wide range of the support down to very small probabilities of the order of 10^{-300}. We study the behavior of the susceptibility distribution across the full phase diagram, from the paramagnetic state through the Griffiths phase to the ferromagnetically ordered system and down to the zero-temperature point. We extract the rate function of large-deviation theory as well as its finite-size scaling behavior and we reveal interesting differences and similarities between the cases. A connection between the fraction of ferromagnetic bonds in a given disorder sample and the size of the magnetic susceptibility is demonstrated numerically.

arXiv:2405.02889•Physical review. E•Nov 13, 2024•5 citations

Revisiting the Lee-Yang singularities in the four-dimensional Ising model: a tribute to the memory of Ralph Kenna

Juan Jesus Ruiz-Lorenzo

We have studied numerically the Lee-Yang singularities of the four dimensional Ising model at criticality, which is believed to be in the same universality class as the φ44 scalar field theory. We have focused in the numerical characterization of the logarithmic corrections to the scaling of the zeros of the partition function and its cumulative probability distribution, finding a very good agreement with the predictions of the renormalization group computation on the φ44 scalar field theory. To obtain these results, we have extended a previous study [R. Kenna, C. B. Lang, Nucl. Phys., 1993, B393, 461] in which there were computed numerically the first two zeros for L≤24 lattices, to the computation of the first four zeros for L≤64 lattices.

arXiv:2402.03932•Condensed Matter Physics•Sep 24, 2024•3 citations

Griffiths-McCoy singularities, Lee-Yang zeros, and the cavity method in a solvable diluted ferromagnet

Chris R. Laumann, Antonello Scardicchio, Shivaji L. Sondhi

We study the diluted Ising ferromagnet on the Bethe lattice as a case study for the application of the cavity method to problems with Griffiths-McCoy singularities. Specifically, we are able to make much progress at infinite coupling where we compute, from the cavity method, the density of Lee-Yang zeros in the paramagnetic Griffiths region as well as the properties of the phase transition to the ferromagnet. This phase transition is itself of a Griffiths-McCoy character albeit with a power law distribution of cluster sizes.

arXiv:0712.3540•Physical Review E•Jun 27, 2008•6 citations

Griffiths-McCoy Singularities in the Transverse Field Ising Model on the Randomly Diluted Square Lattice

Tohru Ikegami, Seiji Miyashita, Heiko Rieger

The site-diluted transverse field Ising model in two dimensions is studied with Quantum-Monte-Carlo simulations. Its phase diagram is determined in the transverse field (Gamma) and temperature (T) plane for various (fixed) concentrations (p). The nature of the quantum Griffiths phase at zero temperature is investigated by calculating the distribution of the local zero-frequency susceptibility. It is pointed out that the nature of the Griffiths phase is different for small and large Gamma.

arXiv:cond-mat/9803270•Journal of the Physical Society of Japan•Aug 15, 1998•22 citations

Griffiths singularities in random magnets: Results for a soluble model

Alan J. Bray, Deng Huifang

A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling ${K}_{C}$(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.

Physical review. B, Condensed matter•Oct 1, 1989•53 citations

Scaling analysis of the site-diluted Ising model in two dimensions

Ralph Kenna B, Juan Jesus Ruiz-Lorenzo

A combination of recent numerical and theoretical advances are applied to analyze the scaling behavior of the site-diluted Ising model in two dimensions, paying special attention to the implications for multiplicative logarithmic corrections. The analysis focuses primarily on the odd sector of the model (i.e., that associated with magnetic exponents), and in particular on its Lee-Yang zeros, which are determined to high accuracy. Scaling relations are used to connect to the even (thermal) sector, and a first analysis of the density of zeros yields information on the specific heat and its corrections. The analysis is fully supportive of the strong scaling hypothesis and of the scaling relations for logarithmic corrections.

arXiv:0807.4628•Physical Review E•Sep 29, 2008•35 citations
We study numerically the probability distribution of the Yang - Lee zeroes inside the Griffiths phase for the two-dimensional site diluted Ising model and we check that the shape of this distribution is that predicted in previous analytical works. By studying the finite-size scaling of the averaged smallest zero at the phase transition we extract, for two values of the dilution, the anomalous dimension, , which agrees very well with the previous estimated values.
The Griffiths phase in systems with quenched disorder occurs below the ordering transition of the pure system down to the ordering transition of the actual disordered system. While it does not exhibit long-range order, large fluctuations in the disorder degrees of freedom result in exponentially rare, long-range ordered states and hence the occurrence of broad distributions in response functions. Inside the Griffiths phase of the two-dimensional bond-diluted Ising model the distribution of the magnetic susceptibility is expected to have such a broad, exponential tail. A large-deviation Monte Carlo algorithm is used to sample this distribution and the exponential tail is extracted over a wide range of the support down to very small probabilities of the order of 10^{-300}. We study the behavior of the susceptibility distribution across the full phase diagram, from the paramagnetic state through the Griffiths phase to the ferromagnetically ordered system and down to the zero-temperature point. We extract the rate function of large-deviation theory as well as its finite-size scaling behavior and we reveal interesting differences and similarities between the cases. A connection between the fraction of ferromagnetic bonds in a given disorder sample and the size of the magnetic susceptibility is demonstrated numerically.
We have studied numerically the Lee-Yang singularities of the four dimensional Ising model at criticality, which is believed to be in the same universality class as the φ44 scalar field theory. We have focused in the numerical characterization of the logarithmic corrections to the scaling of the zeros of the partition function and its cumulative probability distribution, finding a very good agreement with the predictions of the renormalization group computation on the φ44 scalar field theory. To obtain these results, we have extended a previous study [R. Kenna, C. B. Lang, Nucl. Phys., 1993, B393, 461] in which there were computed numerically the first two zeros for L≤24 lattices, to the computation of the first four zeros for L≤64 lattices.
We study the diluted Ising ferromagnet on the Bethe lattice as a case study for the application of the cavity method to problems with Griffiths-McCoy singularities. Specifically, we are able to make much progress at infinite coupling where we compute, from the cavity method, the density of Lee-Yang zeros in the paramagnetic Griffiths region as well as the properties of the phase transition to the ferromagnet. This phase transition is itself of a Griffiths-McCoy character albeit with a power law distribution of cluster sizes.
A soluble, but nontrivial, model of a dilute Ising ferromagnet is studied, with infinite-range interactions but finite average connectivity c. The density of (Yang-Lee) zeros of the partition function in the complex z=exp(-2H) plane (where H is the external magnetic field in units of the temperature) is calculated explicitly in the high-temperature phase for large but finite c and small \ensuremath{\Vert}H\ensuremath{\Vert}. The density of zeros on the unit circle H=i\ensuremath{\theta} has the form \ensuremath{\rho}(\ensuremath{\theta})\ensuremath{\sim}exp{-[cf(K)/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}]ln/B (1/\ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert})} for \ensuremath{\Vert}\ensuremath{\theta}\ensuremath{\Vert}\ensuremath{\rightarrow}0. The function f(K) (K=J/T) vanishes at the critical coupling KC{K}_{C}KC​(c). Heuristic arguments are given for the form of \ensuremath{\rho}(\ensuremath{\theta}) expected for systems with short-range interactions.
A combination of recent numerical and theoretical advances are applied to analyze the scaling behavior of the site-diluted Ising model in two dimensions, paying special attention to the implications for multiplicative logarithmic corrections. The analysis focuses primarily on the odd sector of the model (i.e., that associated with magnetic exponents), and in particular on its Lee-Yang zeros, which are determined to high accuracy. Scaling relations are used to connect to the even (thermal) sector, and a first analysis of the density of zeros yields information on the specific heat and its corrections. The analysis is fully supportive of the strong scaling hypothesis and of the scaling relations for logarithmic corrections.