Progress of Theoretical Physics Supplement

Non-Monotonic Temperature Dependence of Local Dynamics and Local Energy upon Cooling toward the Ising Spin Glass Transition

Sharon C. Glotzer, Peter H. Poole, Antonio Coniglio A, Naeem Jan

Published May 16, 2013• 0 citations

Abstract

We have performed a detailed analysis of the local dynamics and local energies of the equilibrium, paramagnetic phase of the d = 2 and d = 3 ±J Ising spin glass model. Here we discuss our recently reported observations1) that while the average flip rate and average energy decrease monotonically with decreasing temperature, both the flip rate and energy of an increasing fraction of spins increase as the glass transition is approached on cooling. These findings are consistent with recent experimental results for the frequency-dependent magnetic susceptibility of an insulating spin glass, which showed that the approach to the glass transition could be detected from the high frequency behavior.

DOI: 10.1143/ptp.126.383

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Emergence of Fast Local Dynamics on Cooling toward the Ising Spin Glass Transition

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We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

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Progress of Theoretical Physics Supplement

Non-Monotonic Temperature Dependence of Local Dynamics and Local Energy upon Cooling toward the Ising Spin Glass Transition

Sharon C. Glotzer, Peter H. Poole, Antonio Coniglio A, Naeem Jan

Published May 16, 2013• 0 citations

Publisher

Abstract

DOI: 10.1143/ptp.126.383

Related papers

Emergence of Fast Local Dynamics on Cooling toward the Ising Spin Glass Transition

Peter H. Poole, Sharon C. Glotzer, Antonio Coniglio A +1 more

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Journal of Physics C Solid State PhysicsNov 20, 198460 citations

Phase Transition of Ising Spin Glass Models in Two Dimensions

Michinori Shiomi, Fumitaka Matsubara, Takayuki Shirakura

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

Journal of the Physical Society of JapanSep 1, 200010 citations

We present a detailed Monte Carlo evaluation of the equilibrium distribution of local spin-flip rates and local energies in the paramagnetic phase of the

d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2

and $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3\ifmmode\pm\else\textpm\fi{}J$ Ising spin glass. Both quantities are spatially heterogeneous, and we find that the shapes of the distributions change dramatically with decreasing temperature. In particular as temperature decreases we find that for an increasing fraction of spins the local spin-flip rate and local energy increase as the glass transition is approached.

Slow cooling dynamics of the Ising

p

-spin interaction spin-glass model

We have studied dynamical behaviour of the infinite-range Ising spin glass model with

p

-spin interaction above and below the transition into the non-ergodic phase. The transition is continuous at sufficiently high external magnetic field. The dynamic critical exponent of the power-law decay of the autocorrelation function at the transition point is shown to decrease smoothly to zero as the field approaches the ``tricritical'' point from above; at lower fields the transition is discontinuous. The {\it slow cooling} approach is used to study the nonergodic behavior below the transition at zero external field. It is shown that the anomalous response function

Δ(t,t')

contains

δ

-function as well as regular contributions at {\it any} temperature below the phase transition. No evidence of the second phase transition (known to exist within the static replica solution of the same model) is found. At lower enough temperatures the {\it slow cooling} solution approaches the one known for the standard SK model.

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.

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy

\overline{\Delta F(T)}

and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution,

\overline{\Delta F(T)}

at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution,

\overline{\Delta F(T)}

decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .