Domain-wall free-energy $\delta F$, entropy $\delta S$, and the correlation
function, $C_{\rm temp}$, of $\delta F$ are measured independently in the
four-dimensional $\pm J$ Edwards-Anderson (EA) Ising spin glass. The stiffness
exponent $\theta$, the fractal dimension of domain walls $d_{\rm s}$ and the
chaos exponent $\zeta$ are extracted from the finite-size scaling analysis of
$\delta F$, $\delta S$ and $C_{\rm temp}$ respectively well inside the
spin-glass phase. The three exponents are confirmed to satisfy the scaling
relation $\zeta=d_{\rm s}/2-\theta$ derived by the droplet theory within our
numerical accuracy. We also study bond chaos induced by random variation of
bonds, and find that the bond and temperature perturbations yield the universal
chaos effects described by a common scaling function and the chaos exponent.
These results strongly support the appropriateness of the droplet theory for
the description of chaos effect in the EA Ising spin glasses.Comment: 4 pages, 6 figures; The title, the abstract and the text are changed
  slightl

H. Takayama

H. Yoshino

K. Hukushima

M. Sasaki

P. Nordblad

Physical Review Letters

English

arXiv

4 pages, 6 figures; The title, the abstract and the text are changed slightlyDomain-wall free-energy $\delta F$, entropy $\delta S$, and the correlation function, $C_{\rm temp}$, of $\delta F$ are measured independently in the four-dimensional $\pm J$ Edwards-Anderson (EA) Ising spin glass. The stiffness exponent $\theta$, the fractal dimension of domain walls $d_{\rm s}$ and the chaos exponent $\zeta$ are extracted from the finite-size scaling analysis of $\delta F$, $\delta S$ and $C_{\rm temp}$ respectively well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation $\zeta=d_{\rm s}/2-\theta$ derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses

Sasaki, M.

Hukushima, K.

Yoshino, H.

Takayama, H.

Hal-Diderot

Temperature Chaos and Bond Chaos in the Edwards-Anderson Ising Spin Glass : Domain-Wall Free-Energy Measurements

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Temperature Chaos and Bond Chaos in Edwards-Anderson Ising Spin Glasses: Domain-Wall Free-Energy Measurements

SASAKI, Munetaka

Hukushima, Koji

Yoshino, Hajime

Takayama, Hajime

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Temperature Chaos and Bond Chaos inEdwards-Anderson Ising Spin Glasses:Domain-Wall Free-Energy Measurements著者 SASAKI  Munetaka, Hukushima  Koji, Yoshino Hajime, Takayama  Hajimejournal orpublication titlePhysical review lettersvolume 95number 26page range 267203-1-267203-4year 2005URL http://hdl.handle.net/10097/35058doi: 10.1103/PhysRevLett.95.267203PRL 95, 267203 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2005Temperature Chaos and Bond Chaos in Edwards-Anderson Ising Spin Glasses:Domain-Wall Free-Energy MeasurementsM. Sasaki,1 K. Hukushima,2 H. Yoshino,3,4 and H. Takayama11Institute for Solid State Physics, University of Tokyo, Kashiwa-no-ha 5-1-5, Kashiwa, 277-8581, Japan2Department of Basic Science, University of Tokyo, Tokyo 153-8902, Japan3Department of Earth and Space, Osaka University, Toyonaka 560-0043, Japan4Laboratoire de Physique The´orique et Hautes Energies, Jussieu, 75252 Paris Cedex 05, France(Received 5 November 2004; published 20 December 2005)0031-9007=Domain-wall free energy F, entropy S, and the correlation function Ctemp of F are measuredindependently in the four-dimensional J Edwards-Anderson (EA) Ising spin glass. The stiffnessexponent , the fractal dimension of domain walls ds, and the chaos exponent  are extracted from thefinite-size scaling analysis of F, S, and Ctemp, respectively, well inside the spin-glass phase. The threeexponents are confirmed to satisfy the scaling relation   ds=2  derived by the droplet theory withinour numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that thebond and temperature perturbations yield the universal chaos effects described by a common scalingfunction and the chaos exponent. These results strongly support the appropriateness of the droplet theoryfor the description of chaos effect in the EA Ising spin glasses.DOI: 10.1103/PhysRevLett.95.267203 PACS numbers: 75.10.Nr, 05.10.Ln, 75.40.MgS SL RFIG. 1. Model for the boundary flip MC.In randomly frustrated systems such as spin glasses,directed polymer in random media (DPRM), and vortexglasses, the equilibrium ordered state could be completelyreorganized by an infinitesimally small change in environ-ment [1–7]. This curious property called the chaos effecthas attracted much attention since it was found in the1980s. Especially chaos induced by temperature variation(temperature chaos) is now of great interest because of itspotential relevance for rejuvenation caused by temperaturevariation [8]. However, the issue of temperature chaos stillremains far from being resolved. In particular, concerninglow-dimensional Edwards-Anderson (EA) Ising spin-glassmodels, the situation is very controversial because numeri-cal studies done so far provide the evidence both for andagainst temperature chaos [5,7,9–11].In the present work, we examine temperature chaos bynumerical measurements of the domain-wall free energyF, the difference in the free energy between the systemwith the periodic boundary condition (BC) and that withthe antiperiodic BC. This F relates to the effective cou-pling Jeff between the two boundary spins SL and SR (seeFig. 1) as Jeff  F=2 [12,13]. We find F of eachsample exhibits oscillations along the temperature axisproviding direct evidence of the temperature chaos.Furthermore, we find from simultaneous observations ofthe domain-wall energy E and so entropy S that E andTS are large but they cancel with each other in the leadingorder to yield significantly small F  E TS. Suchintriguing behavior is indeed predicted by the droplettheory [14]. For a quantitative check of the droplet theorywe focus on the anticipated scaling relation  ds=2  (1)derived from it, where the stiffness exponent  is extracted05=95(26)=267203(4)$23.00 26720from F  L, the fractal dimension of domain walls froms  Lds , and the so-called chaos exponent  fromCtempL; T; T    fLT1= 	. Here F and S arethe standard deviations of F and S, respectively, Ctemp isthe correlation function of F’s defined by Eq. (4) below,and fx is a certain scaling function. We find the threefundamental exponents thus extracted indeed satisfyEq. (1) well inside the spin-glass phase whose thermody-namic properties are dominantly governed by the T  0fixed point.We also study bond chaos by measuring how F varieswith changes in couplings. The result evidently shows theexistence of bond chaos. Moreover, the scaling analysis oftwo correlation functions associated with temperature andbond perturbations reveals quantitatively that not only thechaos exponent but also the scaling function are commonto both the perturbations. This universal aspect of chaoseffect anticipated from the droplet theory [14] is alsoobserved in the Migdal-Kadanoff spin glasses [15,16]and the DPRM [6]. All of our numerical results, particu-larly the quantitative check of Eq. (1), are strong evidencenot only for the existence of chaos in the EA Ising spinglasses but also for the appropriateness of the droplettheory for its description.The boundary flip MC method.—Let us first describe theboundary flip Monte Carlo (MC) method [17,18] whichenables us to measure the domain-wall free energy. We3-1 © 2005 The American Physical Society-20246-50510-10-50510-6-3036-404812-10-5051015-20-1001020-40-20020401.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0T T TδF δE TδSL=4L=6L=8L=10L=4L=6L=8L=10L=4L=6L=8L=10FIG. 2 (color online). F (left), E (middle), and TS (right)vs temperature for 5 samples. L  4, 6, 8, and 10 from top tobottom.010014 6 8 01σF, σS, σE, σSbondLσFσSσEσSdnobx08.1 86.0x192.0 07.1x67.4 07.18.02.16.18.0 2.1 6.1θ(eff) , d s(eff) /2Tθ )ffe(ds)ffe( 2/FIG. 3 (color online). Size dependences of F, E, S, andbondS at T  0:6J. See text for their definitions. The two straightlines for F and S are obtained by linear least-square fits oflnF= lnS against lnL. The line for bondS has the sameslope as that for S. The inset shows the data for eff anddeffs =2.PRL 95, 267203 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2005consider a model which consists of Ising spins on ad-dimensional hyper-cubic lattice of Ld and two boundaryIsing spins SL and SR (see Fig. 1). The usual periodic BC isapplied for the directions along which the two boundaryspins do not lie. The Hamiltonian isH  PhijiJijSiSj,where the sum is over all the nearest neighbor pairs in-cluding those consisting of one of the two boundary spinsand a spin on the surfaces of the lattice. In our boundaryflip MC simulation, the two boundary spins are also up-dated according to a standard MC procedure. For each spinconfiguration simulated, we regard the BC as periodic(antiperiodic) when SL and SR are in parallel (antiparallel).Since the probability PPAPT for finding the periodic(antiperiodic) BC is proportional to expFPAPT=T	,where FPAPT is the free energy with the periodic (anti-periodic) BC, we obtain, with PAP  1 PP,FT  FPT  FAPT kBTflogPPT	  log1 PPT	g: (2)We also measure the thermally averaged energy EPAPTwhen the two boundary spins are in parallel (antiparallel).It enables us to estimate the domain-wall energy ET EPT  EAPT. Then, the domain-wall entropy S isevaluated either from S  E F=T or S  @F@T . We have checked that both the estimations yieldidentical results within our numerical accuracy.We study the four-dimensional J Ising spin glasses inthe present work. In four dimensions the value of the stiff-ness exponent  is significantly large [18,19], which en-ables us to make scaling analyses rather easily as comparedin three dimensions. The values of fJijg are taken from abimodal distribution with equal weights at Jij  J. Weuse the exchange MC method [20] to accelerate the equili-bration. The temperature range we investigate is between0:6J and 4:5J, whereas the critical temperature of themodel is around 2:0J [21]. The sizes we study are L  4,5, 6, 7, 8, and 10. The number of samples is 824 for L  10and 1500 for the others. The period for thermalization andthat for measurement are set sufficiently (at least 5 times)larger than the ergodic time, which is defined by theaverage MC step for a specific replica to move from thelowest to the highest temperature and return to the lowestone.Temperature chaos.—In Fig. 2, we show temperaturedependence of F, E, and TS for 5 samples. Oscilla-tions of the three observables become stronger with in-creasing L. We, in fact, see that F of some sampleschanges its sign, meaning that the favorable BC with thelower free energy changes with temperature. We also seethat, as predicted by the droplet theory [14], ET andTST exhibit very similar temperature dependence andcancel each other in the leading order to yield relativelysmall F.26720In Fig. 3, the standard deviations, F, E, and S, atT  0:6J are plotted as a function of L. Interestingly, S,which gives the amplitude of j @F@T j, increases more rap-idly than F, i.e., the amplitude of F. See Ref. [11] for asimilar observation in the three-dimensional EA model. Asargued by Banavar and Bray [15], this result naturally leads3-2PRL 95, 267203 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2005us to the conclusion that F in the limit L ! 1 is totallytemperature chaotic.The inset of Fig. 3 shows T and dsT=2 estimated bylinear least-square fits of lnF and lnS against lnL ateach temperature. As expected from the droplet theorywhich is constructed around the T  0 fixed point, thetwo exponents converge to a certain value at low tempera-tures. By averaging over the lowest five temperatures, weobtain  0:69 0:03; ds  3:42 0:06: (3)Our  is compatible with other estimations [18,19], whileour ds is somewhat smaller than other ones [22]. Theapparent temperature dependence of T and dsT athigher temperatures is considered to be due to the criticalfluctuation associated with the unstable fixed point at Tc,combined with the finite-size effect. Its detailed quantita-tive analysis is, however, beyond the scope of the presentwork.We next examine the correlation function defined byCtempL; T; T T  FL; TFL; T  TFL; TFL; T  T ; (4)where    is the sample average. A similar correlationfunction was first introduced by Bray and Moore to studybond chaos [4]. The inset of Fig. 4 shows the raw data of1 Ctemp at T  0:6J. Ctemp approaches zero rapidly withincreasing L. From the prediction of the overlap length bythe droplet theory, over which the configurations at the twotemperatures are unrelated, we expect one parameter scal-ing of Ctemp  fLT1=  whose test is shown in the mainframe of Fig. 4. We see that the scaling works nicely. Thevalue of  is evaluated to be 1:12 0:05 by the fitting.Quite interestingly, this value is consistent with the value101 1-01 2-01 3-01 4-1 011-C temp(L,T=0.6,T+∆T)(L ∆ )T /1 ζζ 21.1=01=L8=L7=L6=L5=L4=Lx2ζ101 1-01 2-01 3-1.0 1∆TFIG. 4 (color online). A scaling plot of 1 CtempL; T; T T at T  0:6J against LT1= with   1:12. The line isproportional to x2 . In the inset, 1 Ctemp for L  4, 5, 6, 7, 8,and 10 (from bottom to top) are plotted as a function of T.26720  1:02 0:06 obtained by substituting Eq. (3) intoEq. (1) predicted by the droplet theory. This is one of themain results of the present work. We also see that the dataare consistent with the expected asymptotic behavior in thelimit LT ! 0, 1 Ctemp / LT2 [4], as depicted bythe line.Bond chaos and universality.—We also study bondchaos by comparing two systems with correlated couplingsets. The perturbed couplings fJ0ijg are obtained from theunperturbed ones fJijg by changing the sign of Jij withprobability p. Since simulation for bond chaos costs muchmore time than that for temperature chaos, we only exam-ined L  4; 6; 8 for bond chaos.Now let us consider an observable Sbond   F0FJ ,where J  pp and F (F0) is the domain-wall freeenergy of the unperturbed (perturbed) system. Sbondhere and S discussed above are similar in a sense thatthe both are the increment ratios of F against the pertur-bations. The ratio against J, not p itself, is considered tocompare temperature perturbation and bond perturbationproperly [5]. In Fig. 3, the standard deviation of Sbond,denoted as bondS , is also shown. Sbond is estimated withJ  0:03, which corresponds to a small value of p 0:0009. The line for bondS and that for S have the sameslope, which suggests that temperature and bond perturba-tions belong to the same universality class. The coefficientof bondS is, however, about 16.4 times as large as that of S.In the inset of Fig. 5 we show the raw data of the correla-tion function for bond perturbation defined byCbondL; T; p  FL; TF0L; TFL; T0FL; T: (5)Again, the correlation decays faster with increasing L. Inthe main frame of Fig. 5, we test a similar scaling to that in1.012.0 11-C bond(L,T=0.6,p)(L ∆ )J /1 ζζ 01.1=8=L6=L4=Lx2ζ1.0101 3- 01 2-pFIG. 5 (color online). A scaling plot of 1 CbondL; T; p atT  0:6J against LJ1= with   1:10, where J  pp . Theline is proportional to x2 . In the inset, the raw data for L  4, 6,and 8 (from bottom to top) are shown as a function of p.3-3101 1-01 2-01 3-01 4-1 011-C temp, 1-C bond(L ∆ )T /1 ζ 5.71 , × (L ∆ )J /1 ζζ 21.1=pmet 01=Lpmet 8=Lpmet 7=Lpmet 6=Lpmet 5=Lpmet 4=Ldnob 8=Ldnob 6=Ldnob 4=Lx2ζ02.04.06.08.010 5 01 51 02 52 03C temp, C bondFIG. 6 (color online). A scaling plot of 1 C by using all thedata in Figs. 4 and 5. The scaling variable for the data oftemperature perturbation is LT1= , while that for the dataof bond perturbation is 17:5 LJ1= , where   1:12. Theline is proportional to x2 . The inset shows the same plot for C.PRL 95, 267203 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending31 DECEMBER 2005Fig. 4 by assuming that the overlap length of the bondperturbation scales as J1= . All the data again collapseinto a single curve. The chaos exponent  is evaluated to be1:10 0:10 by the fitting.To compare the two scaling functions for temperatureand bond perturbations, we plot in Fig. 6 all the data ofboth Ctemp and Cbond by using the same chaos exponent.Here we use the value   1:12 in Fig. 4, and multiply thescaling variable LJ1= by a factor of 17.5. All the dataroughly merge into a single curve, indicating that the chaosexponent and the scaling function for temperature chaosare the same as those for bond chaos. Lastly, by estimatingthe overlap length ‘ as the value of L for which C  0:5,we obtain‘temp  11:5T1= ; ‘bond  0:657J1= ; (6)where   1:12. We see that the overlap length of the bondperturbation is much shorter than that of the temperatureone. This result, as well as the factor of 16.4 in the bondSscaling mentioned above, is the quantitative description ofthe well-known fact that temperature chaos is much moredifficult to observe than bond chaos [5,11].Conclusion.—In the present work, we have studied thefour-dimensional EA Ising spin glass with a focus on thechaos effect. As a consequence, many nontrivial predic-tions of the droplet theory, such as the F oscillation alongthe temperature axis and the cancellation of E and TS,are found in this model. Most importantly, the scalingrelation of Eq. (1) and the universal aspect of temperatureand bond chaos effects are quantitatively confirmed wellinside its spin-glass phase whose thermodynamic proper-ties are dominantly governed by the T  0 fixed point.26720These results are certainly strong evidences for the appro-priateness of the droplet theory for the description of chaoseffect in the EA Ising spin glasses. On the other hand,recent work by Rizzo and Crisanti [23] indicates the ex-istence of similar chaos effects in the Sherrington-Kirkpatrick model. Whether our results are consistentwith the mean field viewpoint or not is an interestingopen problem.We would like to thank Dr. Katzgraber for fruitful dis-cussion and useful suggestions. M. S. acknowledges finan-cial support from the Japan Society for the Promotion ofScience. The present work is supported by Grant-in-Aidfor Scientific Research Program (No. 14540351,No. 14084204, No. 14740233) and NAREGI Nano-science Project from the MEXT. The present simulationshave been performed on SGI 2800/384 at the Super-computer Center, Institute for Solid State Physics, Univer-sity of Tokyo.3-4[1] S. R. McKay, A. N. Berker, and S. Kirkpatrick, Phys. Rev.Lett. 48, 767 (1982).[2] H. Kitatani, S. Miyashita, and M. Suzuki, J. Phys. Soc.Jpn. 55, 865 (1986).[3] F. Ritort, Phys. Rev. B 50, 6844 (1994).[4] A. J. Bray and M. A. Moore, Phys. Rev. Lett. 58, 57(1987).[5] M. Ney-Nifle, Phys. Rev. B 57, 492 (1998).[6] M. Sales and H. Yoshino, Phys. Rev. E 65, 066131 (2002).[7] D. A. Huse and L.-F. Ko, Phys. Rev. B 56, 14 597 (1997).[8] P. Nordblad and P. Svedlindh, in Spin Glasses and RandomFields, edited by A. P. Young (World Scientific, Singapore,1998); V. Dupuis et al., Phys. Rev. B 64, 174204 (2001).[9] A. Billoire and E. Marinari, J. Phys. A 33, L265 (2000).[10] K. Hukushima and Y. Iba, cond-mat/0207123.[11] T. Aspelmeier, A. J. Bray, and M. A. Moore, Phys. Rev.Lett. 89, 197202 (2002).[12] A. J. Bray and M. A. Moore, J. Phys. C 17, L463 (1984).[13] W. L. McMillan, Phys. Rev. B 31, 340 (1985).[14] D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601(1986); Phys. Rev. B 38, 386 (1988).[15] J. R. Banavar and A. J. Bray, Phys. Rev. B 35, 8888 (1987).[16] M. Nifle and H. J. Hilhorst, Phys. Rev. Lett. 68, 2992(1992).[17] M. Hasenbusch, J. Phys. I (France) 3, 753 (1993).[18] K. Hukushima, Phys. Rev. E 60, 3606 (1999).[19] A. K. Hartmann, Phys. Rev. E 60, 5135 (1999).[20] K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65, 1604(1996).[21] E. Marinari and F. Zuliani, J. Phys. A 32, 7447 (1999).[22] M. Palassini and A. P. Young, Phys. Rev. Lett. 85, 3017(2000); H. G. Katzgraber, M. Palassini, and A. P. Young,Phys. Rev. B 63, 184422 (2001).[23] T. Rizzo and A. Crisanti, Phys. Rev. Lett. 90, 137201(2003).

Temperature Chaos and Bond Chaos in the Edwards-Anderson Ising Spin
  Glass : Domain-Wall Free-Energy Measurements

Temperature Chaos and Bond Chaos in the Edwards-Anderson Ising Spin Glass : Domain-Wall Free-Energy Measurements

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