Physical Review Letters

Nature of Ordering in Spin-Glasses

Jayanth R. Banavar, Marek Cieplak

Published Mar 22, 1982• 155 citations

Abstract

A new equilibrium characterization of the spin-glass-paramagnet transition is presented. It is based on the size dependence of the sensitivity of the free energy to the boundary conditions. Results of numerical studies of a three-dimensional Heisenberg spin-glass at $T=0$ are consistent with an algebraic decay of this sensitivity and suggest a zero-temperature phase transition.

DOI: 10.1103/physrevlett.48.832

Related papers

Lower critical dimensionality of Heisenberg spin-glasses

Marek Cieplak, Jayanth R. Banavar

Physical review. B, Condensed matterJan 1, 198429 citations

Sensitivity to boundary conditions of Ising spin-glasses

Marek Cieplak, Jayanth R. Banavar

Physical review. B, Condensed matterJan 1, 198319 citations

Domain-wall scaling near the Ising spin-glass transition

Andrew T. Ogielski

Physical review. B, Condensed matterNov 1, 19864 citations

Dynamical critical phenomena in three-dimensional Heisenberg spin glasses

Mitsuhiro Matsumoto, Koji Hukushima, Hajime Takayama

arXiv:cond-mat/0204225Physical review. B, Condensed matterSep 4, 200222 citations

Dynamics of three-dimensional Ising spin glasses in thermal equilibrium

Andrew T. Ogielski

Physical review. B, Condensed matterDec 1, 1985773 citations

Search for a transition in the three-dimensional J Ising spin-glass

R. N. Bhatt, A. P. Young

Physical Review LettersMar 4, 1985392 citations

Physical Review Letters

Nature of Ordering in Spin-Glasses

Jayanth R. Banavar, Marek Cieplak

Published Mar 22, 1982• 155 citations

Publisher

Abstract

T=0

are consistent with an algebraic decay of this sensitivity and suggest a zero-temperature phase transition.

DOI: 10.1103/physrevlett.48.832

Related papers

Lower critical dimensionality of Heisenberg spin-glasses

Marek Cieplak, Jayanth R. Banavar

Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions for two-dimensional and four-dimensional Heisenberg spin-glasses at $T=0$ are presented. The results are consistent with an algebraic size dependence of this sensitivity. They suggest that the two-dimensional system has a zero-temperature phase transition and that the four-dimensional system behaves as if it were at its lower critical dimensionality.Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions for two-dimensional and four-dimensional Heisenberg spin-glasses at $T=0$ are presented. The results are consistent with an algebraic size dependence of this sensitivity. They suggest that the two-dimensional system has a zero-temperature phase transition and that the four-dimensional system behaves as if it were at its lower critical dimensionality.

Physical review. B, Condensed matterJan 1, 198429 citations

Sensitivity to boundary conditions of Ising spin-glasses

Marek Cieplak, Jayanth R. Banavar

Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions in two random two-dimensional Ising systems at $T=0$ are presented. The spin-glass with a Gaussian distribution of exchange couplings is characterized by an algebraic-size dependence of this sensitivity. The results may be interpreted to yield a lower critical dimensionality of 4 for this system. For the system with a $\ifmmode\pm\else\textpm\fi{}J$ probability distribution for the exchange couplings we expect an exponential-size dependence of the sensitivity but our data are inconclusive.Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions in two random two-dimensional Ising systems at $T=0$ are presented. The spin-glass with a Gaussian distribution of exchange couplings is characterized by an algebraic-size dependence of this sensitivity. The results may be interpreted to yield a lower critical dimensionality of 4 for this system. For the system with a $\ifmmode\pm\else\textpm\fi{}J$ probability distribution for the exchange couplings we expect an exponential-size dependence of the sensitivity but our data are inconclusive.

Physical review. B, Condensed matterJan 1, 198319 citations

Domain-wall scaling near the Ising spin-glass transition

Andrew T. Ogielski

The distribution of energy and free energy of domain walls near the spin-glass transition is studied numerically in the three-dimensional Ising spin-glass model with discrete $\ifmmode\pm\else\textpm\fi{}J$ interactions. Temperature dependence of the width of the free-energy distribution clearly shows a phase transition to a low-temperature phase with long-range spin correlations at $\frac{k{T}_{g}}{J}\ensuremath{\simeq}1.2$. Simple scaling analysis is consistent with previous studies of this system.The distribution of energy and free energy of domain walls near the spin-glass transition is studied numerically in the three-dimensional Ising spin-glass model with discrete $\ifmmode\pm\else\textpm\fi{}J$ interactions. Temperature dependence of the width of the free-energy distribution clearly shows a phase transition to a low-temperature phase with long-range spin correlations at $\frac{k{T}_{g}}{J}\ensuremath{\simeq}1.2$. Simple scaling analysis is consistent with previous studies of this system.

Physical review. B, Condensed matterNov 1, 19864 citations

Dynamical critical phenomena in three-dimensional Heisenberg spin glasses

Mitsuhiro Matsumoto, Koji Hukushima, Hajime Takayama

Spin-glass (SG) and chiral-glass (CG) orderings in three-dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size-scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a lower temperature, which is compatible with zero. Our results of the anisotropic case support the chirality scenario for the phase transitions in the 3D Heisenberg spin glasses.Spin-glass (SG) and chiral-glass (CG) orderings in three-dimensional (3D) Heisenberg spin glass with and without magnetic anisotropy are studied by using large-scale off-equilibrium Monte Carlo simulations. A characteristic time of relaxation, which diverges at a transition temperature in the thermodynamic limit, is obtained as a function of the temperature and the system size. Based on the finite-size-scaling analysis for the relaxation time, it is found that in the isotropic Heisenberg spin glass, the CG phase transition occurs at a finite temperature, while the SG transition occurs at a lower temperature, which is compatible with zero. Our results of the anisotropic case support the chirality scenario for the phase transitions in the 3D Heisenberg spin glasses.

arXiv:cond-mat/0204225Physical review. B, Condensed matterSep 4, 200222 citations

Dynamics of three-dimensional Ising spin glasses in thermal equilibrium

Andrew T. Ogielski

I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈 ${S}_{x}$ (0) ${S}_{x}$ (t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes ( ${8}^{3}$ , ${16}^{3}$ , ${32}^{3}$ , and ${64}^{3}$ ). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈${S}_{x}$(0)${S}_{x}$(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (${8}^{3}$, ${16}^{3}$, ${32}^{3}$, and ${64}^{3}$). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.

Physical review. B, Condensed matterDec 1, 1985773 citations

Search for a transition in the three-dimensional J Ising spin-glass

R. N. Bhatt, A. P. Young

The three-dimensional Ising spin-glass in zero field with nearest-neighbor interactions having \ifmmode\pm\else\textpm\fi{}J distribution is studied by Monte Carlo simulations for samples of linear dimension L with 3\ensuremath{\le}L\ensuremath{\le}20. Results for the probability distribution, ${\mathrm{P}}_{\mathrm{L}}$ (q), of the spin-glass order parameter are analyzed by finite-size scaling. Data for T\ensuremath{\ge}1.2 are consistent with a conventional phase transition at ${\mathrm{T}}_{\mathrm{c}}$ ${=1.2}_{\mathrm{\ensuremath{-}}0.2}^{+0.1}$, with exponents \ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\eta}=-0.3\ifmmode\pm\else\textpm\fi{}0.2. However, results at lower temperature indicate marginal behavior and suggest that the lower critical dimensional is close to three.The three-dimensional Ising spin-glass in zero field with nearest-neighbor interactions having \ifmmode\pm\else\textpm\fi{}J distribution is studied by Monte Carlo simulations for samples of linear dimension L with 3\ensuremath{\le}L\ensuremath{\le}20. Results for the probability distribution, ${\mathrm{P}}_{\mathrm{L}}$(q), of the spin-glass order parameter are analyzed by finite-size scaling. Data for T\ensuremath{\ge}1.2 are consistent with a conventional phase transition at ${\mathrm{T}}_{\mathrm{c}}$${=1.2}_{\mathrm{\ensuremath{-}}0.2}^{+0.1}$, with exponents \ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\eta}=-0.3\ifmmode\pm\else\textpm\fi{}0.2. However, results at lower temperature indicate marginal behavior and suggest that the lower critical dimensional is close to three.

Physical Review LettersMar 4, 1985392 citations

T=0

are consistent with an algebraic decay of this sensitivity and suggest a zero-temperature phase transition.

Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions for two-dimensional and four-dimensional Heisenberg spin-glasses at

T=0

are presented. The results are consistent with an algebraic size dependence of this sensitivity. They suggest that the two-dimensional system has a zero-temperature phase transition and that the four-dimensional system behaves as if it were at its lower critical dimensionality.

Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions in two random two-dimensional Ising systems at

T=0

are presented. The spin-glass with a Gaussian distribution of exchange couplings is characterized by an algebraic-size dependence of this sensitivity. The results may be interpreted to yield a lower critical dimensionality of 4 for this system. For the system with a $\ifmmode\pm\else\textpm\fi{}J$ probability distribution for the exchange couplings we expect an exponential-size dependence of the sensitivity but our data are inconclusive.

I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈

{S}_{x}

(0)

{S}_{x}

(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (

{8}^{3}

{16}^{3}

{32}^{3}

, and

{64}^{3}

). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.

{\mathrm{P}}_{\mathrm{L}}

(q), of the spin-glass order parameter are analyzed by finite-size scaling. Data for T\ensuremath{\ge}1.2 are consistent with a conventional phase transition at

{\mathrm{T}}_{\mathrm{c}}

${=1.2}_{\mathrm{\ensuremath{-}}0.2}^{+0.1}$, with exponents \ensuremath{\nu}=1.3\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\eta}=-0.3\ifmmode\pm\else\textpm\fi{}0.2. However, results at lower temperature indicate marginal behavior and suggest that the lower critical dimensional is close to three.