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

Physical Review Letters

Nature of the Griffiths phase

Alan J. Bray

Published Aug 3, 1987• 377 citations
Publisher

Abstract

Arguments are given that, for random spin systems, the density of states \ensuremath{\rho}(\ensuremath{\mu}) of the inverse of the susceptibility matrix vanishes as \ensuremath{\rho}(\ensuremath{\mu})\ensuremath{\sim}exp(-A/\ensuremath{\mu}), for \ensuremath{\mu}\ensuremath{\rightarrow}0, throughout the ``Griffiths phase.'' The amplitude A vanishes at the onset of magnetic long-range order, and diverges at the transition between ``Griffiths'' and ``paramagnetic'' phases. For an O(m) spin system, with m\ensuremath{\rightarrow}\ensuremath{\infty}, the spin autocorrelation function C(t) is found to have the ``stretched-exponential'' form, lnC(t)\ensuremath{\sim}-(At${)}^{1/2}$, in the Griffiths phase.

DOI
10.1103/physrevlett.59.586

Related papers

Absence of phase transitions in one-dimensional long-range spin systems with random Hamiltonian

Konstantin Mikhailovich Khanin

Theoretical and Mathematical Physics•May 1, 1980•22 citations

Dynamics of dilute magnets above Tc

Alan J. Bray

Nonperturbative contributions are shown to produce anomalously slow relaxation in dilute ferromagnets (or antiferromagnets), both in the paramagnetic and ``Griffiths'' phases. In the paramagnetic phase, the pseudoscaling form C(t)\ensuremath{\sim}exp{-${\mathit{t}}^{\mathit{d}/(\mathit{d}+\mathit{z})}$f(t/${\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}$)} is predicted for the spin autocorrelation function, with ${\ensuremath{\xi}}_{\mathit{p}}$ the correlation length of the nondilute system. In the Griffiths phase, C(t)\ensuremath{\sim}exp{-A1nt${)}^{\mathit{d}/(\mathit{d}\mathrm{\ensuremath{-}}1)}$} and exp{-(Bt${)}^{1/2}$} are predicted for Ising Heisenberg systems, respectively, for t\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}$.

Physical Review Letters•Feb 22, 1988•101 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

Revealing quantum Griffiths singularities inside the ferromagnetic phase

Adane Gebretsadik, Ruizhe Wang, Sara Ubaid-Kassis +8 more

Bulletin of the American Physical Society•Mar 13, 2017•0 citations

Dynamics at a smeared phase transition

Bernard J. Fendler, Rastko Sknepnek, Thomas Vojta

We investigate the effects of rare regions on the dynamics of Ising magnets with planar defects, i.e., disorder perfectly correlated in two dimensions. In these systems, the magnetic phase transition is smeared because static long-range order can develop on isolated rare regions. We first study an infinite-range model by numerically solving local dynamic mean-field equations. Then we use extremal statistics and scaling arguments to discuss the dynamics beyond mean-field theory. In the tail region of the smeared transition the dynamics is even slower than in a conventional Griffiths phase: the spin autocorrelation function decays like a stretched exponential at intermediate times before approaching the exponentially small equilibrium value following a power law at late times.

arXiv:cond-mat/0409688•Journal of Physics A Mathematical and General•Mar 3, 2005•3 citations

The Phase Transition of the Spin‐1/2 Ising Model with a Random Field

A. Bobák, J. Karaba, Ľ. Tóth

physica status solidi (b)•Oct 1, 1989•20 citations
Arguments are given that, for random spin systems, the density of states \ensuremath{\rho}(\ensuremath{\mu}) of the inverse of the susceptibility matrix vanishes as \ensuremath{\rho}(\ensuremath{\mu})\ensuremath{\sim}exp(-A/\ensuremath{\mu}), for \ensuremath{\mu}\ensuremath{\rightarrow}0, throughout the Griffiths phase.'' The amplitude A vanishes at the onset of magnetic long-range order, and diverges at the transition between Griffiths'' and paramagnetic'' phases. For an O(m) spin system, with m\ensuremath{\rightarrow}\ensuremath{\infty}, the spin autocorrelation function C(t) is found to have the stretched-exponential'' form, lnC(t)\ensuremath{\sim}-(At)1/2{)}^{1/2})1/2, in the Griffiths phase.
Nonperturbative contributions are shown to produce anomalously slow relaxation in dilute ferromagnets (or antiferromagnets), both in the paramagnetic and ``Griffiths'' phases. In the paramagnetic phase, the pseudoscaling form C(t)\ensuremath{\sim}exp{-td/(d+z){\mathit{t}}^{\mathit{d}/(\mathit{d}+\mathit{z})}td/(d+z)f(t/${\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}$)} is predicted for the spin autocorrelation function, with ${\ensuremath{\xi}}_{\mathit{p}}$ the correlation length of the nondilute system. In the Griffiths phase, C(t)\ensuremath{\sim}exp{-A1nt${)}^{\mathit{d}/(\mathit{d}\mathrm{\ensuremath{-}}1)}$} and exp{-(Bt)1/2{)}^{1/2})1/2} are predicted for Ising Heisenberg systems, respectively, for t\ensuremath{\gg}${\ensuremath{\xi}}_{\mathit{p}}^{\mathit{d}+\mathit{z}}$.
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.
We investigate the effects of rare regions on the dynamics of Ising magnets with planar defects, i.e., disorder perfectly correlated in two dimensions. In these systems, the magnetic phase transition is smeared because static long-range order can develop on isolated rare regions. We first study an infinite-range model by numerically solving local dynamic mean-field equations. Then we use extremal statistics and scaling arguments to discuss the dynamics beyond mean-field theory. In the tail region of the smeared transition the dynamics is even slower than in a conventional Griffiths phase: the spin autocorrelation function decays like a stretched exponential at intermediate times before approaching the exponentially small equilibrium value following a power law at late times.