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arXiv:1304.5210 · Physical Review Letters

Yucesoy, Katzgraber, and Machta Reply:

Burcu Yücesoy, Helmut G. Katzgraber, Jonathan L. Machta

Published May 24, 2013• 12 citations
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Abstract

In Phys. Rev. Lett. 110, 219701 (2013) [arXiv:1211.0843] Billoire et al. criticize the conclusions of our Letter [Phys. Rev. Lett. 109, 177204 (2012), arxiv:1206.0783]. They argue that considering the Edwards-Anderson and Sherrington-Kirkpatrick models at the same temperature is inappropriate and propose an interpretation based on the replica symmetry breaking theory. Here we show that the theory presented in the Comment does not explain our data on the Edwards-Anderson spin glass and we stand by our assertion that the low-temperature behavior of the Edwards-Anderson spin glass model does not appear to be mean-field like.

DOI
10.1103/physrevlett.110.219702

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In Phys. Rev. Lett. 110, 219701 (2013) [arXiv:1211.0843] Billoire et al. criticize the conclusions of our Letter [Phys. Rev. Lett. 109, 177204 (2012), arxiv:1206.0783]. They argue that considering the Edwards-Anderson and Sherrington-Kirkpatrick models at the same temperature is inappropriate and propose an interpretation based on the replica symmetry breaking theory. Here we show that the theory presented in the Comment does not explain our data on the Edwards-Anderson spin glass and we stand by our assertion that the low-temperature behavior of the Edwards-Anderson spin glass model does not appear to be mean-field like.
A recent interesting paper [Yucesoy et al. Phys. Rev. Lett. 109, 177204 (2012), arXiv:1206:0783] compares the low-temperature phase of the 3D Edwards-Anderson (EA) model to its mean-field counterpart, the Sherrington-Kirkpatrick (SK) model. The authors study the overlap distributions P_J(q) and conclude that the two models behave differently. Here we notice that a similar analysis using state-of-the-art, larger data sets for the EA model (generated with the Janus computer) leads to a very clear interpretation of the results of Yucesoy et al., showing that the EA model behaves as predicted by the replica symmetry breaking (RSB) theory.
The three-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied via large-scale Monte Carlo simulations at low temperatures, deep within the spin-glass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states.
We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of L3{L}^{3}L3 spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions p(q)p(q)p(q) of the overlap parameter qqq at very low-temperature TTT. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.
The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.
A mean-field spherical model with random couplings between pairs, quartets, and possibly higher multiplets of spins is considered. It has the same critical behavior as the Sherrington-Kirkpatrick model. It thus exhibits replica symmetry breaking. The order parameter function is solved exactly in the whole low temperature phase. The zero-field-cooled susceptibility remains finite at low TTT. Next a quantum version of the system is considered. Whereas the magnetic properties are not altered qualitatively, the thermodynamics is now regular at small temperatures.