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/0107583 · Physical review. B, Condensed matter

Ferromagnetic–spin-glass transition in a four-dimensional random-bond Ising model

Alexander K. Hartmann

Published Nov 26, 2001• 2 citations
arXivHTMLPDF

Abstract

The four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model is studied using ground-state calculations. System sizes up to ${N=6}^{4}$ spins are considered. Here it is found that the ferromagnetic--spin-glass transition occurs at a critical concentration ${p}_{c}=0.28(1)$ of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent $\ensuremath{\nu}=1.0(1)$ and an order-parameter exponent $\ensuremath{\beta}=0.4(1).$ Thus, this transition is in a different universality class from four-dimensional bond percolation, where $\ensuremath{\nu}=0.678(50)$ and $\ensuremath{\beta}=0.639(20).$

DOI
10.1103/physrevb.64.224430

Related papers

Three-dimensional random-bond Ising model: Phase diagram and critical properties

Joseph D. Reger, Annette Zippelius

A dynamic high-temperature series expansion is used to study the three-dimensional random-bond Ising model as a function of temperature and concentration of ferromagnetic bonds. The phase boundaries for the paramagnetic, the ferromagnetic, and the spin-glass phases are determined. We discuss the static and dynamic critical behavior and verify the dynamic scaling hypothesis at the spin-glass transition.

Physical Review Letters•Dec 22, 1986•24 citations

Domain-wall energies and magnetization of the two-dimensional random-bond Ising model

Carlo Amoruso, Alexander K. Hartmann

We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration $p∊[0,1]$ of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension $L$ up to $700$. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.

arXiv:cond-mat/0401464•Physical Review B•Oct 29, 2004•48 citations

Ferromagnetic-glassy transitions in three-dimensional Ising spin glasses

Giacomo Ceccarelli, Andrea Pelissetto C, Ettore Vicari

We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter $p$, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.

arXiv:1107.3005•Physical Review B•Oct 7, 2011•29 citations

Ground-state behavior of the three-dimensional ±J random-bond Ising model

Alexander K. Hartmann

Large numbers of ground states of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model are calculated for sizes up to ${14}^{3}$ using a combination of a genetic algorithm and cluster-exact approximation. Several quantities are calculated as functions of the concentration p of the antiferromagnetic bonds. The critical concentration where the ferromagnetic order disappears is determined using the Binder cumulant of the magnetization. A value of ${p}_{c}=0.222\ifmmode\pm\else\textpm\fi{}0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\beta}=0.2\ifmmode\pm\else\textpm\fi{}0.1 are calculated. The behavior of the distribution of overlaps $P(q)$ is used to investigate how the spin-glass phase evolves with increasing concentration $p.$ The spin-glass order is characterized by a broad distribution of overlaps which extends down to $q=0.$

arXiv:cond-mat/9808197•Physical review. B, Condensed matter•Feb 1, 1999•53 citations

Numerical studies of Ising spin glasses in two, three, and four dimensions

R. N. Bhatt, A. P. Young

We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at ${T}_{c}$=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero ${T}_{c}$ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

Physical review. B, Condensed matter•Apr 1, 1988•301 citations

Percolation of Fortuin-Kasteleyn clusters for the random-bond Ising model

Hauke Fajen, Alexander K. Hartmann, A. P. Young

We apply generalizations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional ±1 random-bond Ising model. The behavior of the model is determined by the temperature T and the concentration p of negative (antiferromagnetic) bonds. The ground state is ferromagnetic for 0≤p<p_{c}, and a spin glass for p_{c}<p≤0.5 where p_{c}≃0.222. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as a function of temperature for large system sizes up to N=200^{3} spins. Except for p=0 the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for p=1/2 but here we provide evidence for a difference in transition temperatures even for p arbitrarily small. Furthermore, for all values of p>0, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for p=0 where the clusters are strictly tied to Ising correlations so the percolation transition is in the Ising universality class.

arXiv:1905.04220•Physical review. E•Jul 13, 2020•9 citations
The four-dimensional \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ random-bond Ising model is studied using ground-state calculations. System sizes up to N=64{N=6}^{4}N=64 spins are considered. Here it is found that the ferromagnetic--spin-glass transition occurs at a critical concentration pc=0.28(1){p}_{c}=0.28(1)pc​=0.28(1) of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent \ensuremathν=1.0(1)\ensuremath{\nu}=1.0(1)\ensuremathν=1.0(1) and an order-parameter exponent \ensuremathβ=0.4(1).\ensuremath{\beta}=0.4(1).\ensuremathβ=0.4(1). Thus, this transition is in a different universality class from four-dimensional bond percolation, where \ensuremathν=0.678(50)\ensuremath{\nu}=0.678(50)\ensuremathν=0.678(50) and \ensuremathβ=0.639(20).\ensuremath{\beta}=0.639(20).\ensuremathβ=0.639(20).
We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration p∊[0,1]p∊[0,1]p∊[0,1] of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension LLL up to 700700700. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
Large numbers of ground states of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model are calculated for sizes up to 143{14}^{3}143 using a combination of a genetic algorithm and cluster-exact approximation. Several quantities are calculated as functions of the concentration p of the antiferromagnetic bonds. The critical concentration where the ferromagnetic order disappears is determined using the Binder cumulant of the magnetization. A value of ${p}_{c}=0.222\ifmmode\pm\else\textpm\fi{}0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\beta}=0.2\ifmmode\pm\else\textpm\fi{}0.1 are calculated. The behavior of the distribution of overlaps P(q)P(q)P(q) is used to investigate how the spin-glass phase evolves with increasing concentration p.p.p. The spin-glass order is characterized by a broad distribution of overlaps which extends down to q=0.q=0.q=0.
We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at Tc{T}_{c}Tc​=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero Tc{T}_{c}Tc​ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that Tc{T}_{c}Tc​ is nonzero with a long-range-ordered phase below Tc{T}_{c}Tc​. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.
We apply generalizations of the Swendson-Wang and Wolff cluster algorithms, which are based on the construction of Fortuin-Kasteleyn clusters, to the three-dimensional ±1 random-bond Ising model. The behavior of the model is determined by the temperature T and the concentration p of negative (antiferromagnetic) bonds. The ground state is ferromagnetic for 0≤p<p_{c}, and a spin glass for p_{c}<p≤0.5 where p_{c}≃0.222. We investigate the percolation transition of the Fortuin-Kasteleyn clusters as a function of temperature for large system sizes up to N=200^{3} spins. Except for p=0 the Fortuin-Kasteleyn percolation transition occurs at a higher temperature than the magnetic ordering temperature. This was known before for p=1/2 but here we provide evidence for a difference in transition temperatures even for p arbitrarily small. Furthermore, for all values of p>0, our data suggest that the percolation transition is universal, irrespective of whether the ground state exhibits ferromagnetic or spin-glass order, and is in the universality class of standard percolation. This shows that correlations in the bond occupancy of the Fortuin-Kasteleyn clusters are irrelevant, except for p=0 where the clusters are strictly tied to Ising correlations so the percolation transition is in the Ising universality class.