Magnetic exponents of two-dimensional Ising spin glasses

F. Liers1 and O. C. Martin2

1Institut für Informatik, Universität zu Köln, Pohligstraße 1, D-50969 Köln, Germany2Université Paris-Sud, UMR8626, LPTMS, and CNRS, F-91405 Orsay, France

�Received 13 July 2007; published 15 August 2007�

The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using an exactground-state determination, we extract the properties of clusters flipped when a uniform field is increasedcontinuously. We show that these clusters have many holes but otherwise have statistical properties similar tothose of zero-field droplets. A detailed analysis gives for the magnetization exponent ��1.30±0.02 usinglattice sizes up to 80�80; this is compatible with the droplet model prediction, �=1.282. The reason forprevious disagreements stems from the need to analyze both singular and analytic contributions in the low-fieldregime.

DOI: 10.1103/PhysRevB.76.060405 PACS number�s�: 75.10.Nr, 75.40.Mg

Spin glasses1,2 have been the focus of much interest be-cause of their many remarkable features: they undergo asubtle freezing transition as the temperature is lowered, theirrelaxational dynamics is slow �non-Arrhenius�, they exhibitaging, memory effects, etc. Although there are still someheated disputes concerning three-dimensional spin glasses,the case of two dimensions is relatively consensual, at leastin the absence of a magnetic field. Indeed, two recentstudies3,4 found that the thermal properties of two-dimensional Ising spin glasses with Gaussian couplingsagreed very well with the predictions of the scaling anddroplet pictures.5,6 Interestingly, the situation in the presenceof a magnetic field remains unclear; in particular, someMonte Carlo simulations7 and basically all ground-statestudies8–10 seem to go against the scaling and droplet pic-tures. Nevertheless, since spin glasses often have large cor-rections to scaling, the apparent disagreement with the drop-let picture resulting from these studies may be misleadingand tests in one dimension give credence to this claim.11

In this study we use state of the art algorithms for deter-mining exact ground states in the presence of a magneticfield and treat significantly larger lattice sizes than in previ-ous work. By finding the precise points where the groundstates change as a function of the field, we extract the exci-tations relevant in the presence of a field which can then becompared to the zero-field droplets. Although for small sizelattices our results agree with those of previous studies, atour larger ones a careful analysis, taking into account boththe analytic and the singular terms, gives excellent agree-ment with the droplet picture.

The model and its properties. We work on an L�L squarelattice having Ising spins on its sites and couplings Jij on itsbonds. The Hamiltonian is

H���i�� � − ��ij

Jij�i� j − B�i

�i. �1�

The first sum runs over all nearest-neighbor sites using peri-odic boundary conditions to minimize finite-size effects. TheJij are independent random variables of either Gaussian orexponential distribution.

It is generally agreed that two-dimensional spin glasseshave a unique critical point at T=B=0. There, the free

energy is nonanalytic and, in fact, standard arguments12 sug-gest that as T→0 and B→0 the free energy varies as�F�L ,���E0+Gs�TLyT ,BLyB� where E0 is the ground-stateenergy, � is the inverse temperature, and yT and yB are thethermal and magnetic exponents. Previous work when B=0is compatible with this form and in fact also agrees with thescaling and droplet picture of Ising spin glasses, in whichone has yT=−��0.282. The stumbling block concerns thebehavior when B�0; there, the droplet prediction in generaldimension d is

yB = yT + d/2, �2�

but the numerical evidence for this is muddled at best. It isthus worth reviewing the hypotheses assumed by the dropletmodel so that they can be tested directly.

We begin with the fact that, in any dimension d, a mag-netic field destabilizes the ground state beyond a character-istic length scale �B. To see this, consider an infinitesimalfield and zero-field droplets of scale �. These are expected tobe compact. The interfacial energy of such droplets is O����while their total magnetization varies as �d/2. The magneticand interfacial energy are then balanced when B reaches avalue O�1/�d/2−��: at that value of the field, some of thedroplets will flip and the ground state will be destabilized.We then see that for each field strength there is an associatedmagnetic length scale �B,

�B � B−1/�d/2−��. �3�

This leads to the identification yB=d /2−� in agreement withEq. �2�, giving yb�1.282 at d=2.

The droplet model also predicts the scaling of the magne-tization in the B→0 limit via the exponent �:

m�B� B1/�. �4�

If this form also holds for infinitesimal fields at finite L, wecan consider the field B* for which system-size droplets flip;this happens when B=O�1/LyB� and then the magnetizationis O�L−d/2�, the droplets having random magnetizations. Thisleads to m�B*�L−d/2 and m�B*��1/LyB�1/� so that

d� = 2yB. �5�

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Although the droplet model arguments are not proofs,they seem quite convincing. Nevertheless, the numericalstudies measuring � do not give good agreement with theprediction �=1.282. For instance, using Monte Carlo simu-lations at “low enough” temperatures, Kinzel and Binder7

find ��1.39. Since thermalization is difficult at low tem-peratures, it is preferable to work directly with ground states,at least when that is possible. This was done by threeindependent groups8–10 with increasing power, leading to��1.48, ��1.54, and ��1.48. Taken together, these studiesshow a real discrepancy with the droplet prediction. To savethe droplet model from this thorny situation, one can appealto large corrections to scaling. Such potential effects havebeen considered11 in dimension 1, where it was shown that�B was poorly fitted by a pure power law unless the fieldswere very small. Here we reconsider the two-dimensionalcase to reveal either the size of the corrections to scaling or acause for the breakdown of the droplet reasoning.

Computation of ground states. We determine the exactground state of the Hamiltonian �1� by computing a maxi-mum cut in the graph of interactions,13 a prominent problemin combinatorial optimization. Whereas it is polynomiallysolvable on two-dimensional grids without a field and cou-plings bounded by a polynomial in the size of the input, it isNP hard with an external field. �Most computer scientistsbelieve this means that there does not exist a solution algo-rithm running in time that can be bounded polynomially inthe size of the input.� In practice, we rely on a branch-and-cut algorithm.14,15

Let the ground state at a field B be denoted as ���G��B��.To study the magnetization, we computed the ground statesat increasing values of B, in steps of size 0.02. When focus-ing instead on the flipping clusters, we had to determine theintervals in which the ground state was constant and in whatmanner it changed when going from one interval to the next.In Fig. 1 we show the associated piecewise constant magne-tization curves for three samples of the disorder variables Jijat L=10.

To get the sequence of intervals or break points associatedwith such a function exactly, we start by computing theground state in zero field. By applying postoptimality analy-sis from linear programming theory, we determine10,15 a

range �B such that the ground state at a field B remains theoptimum in the interval �B ,B+�B�. We reoptimize atB+�B+�, with � being a sufficiently small number. By re-peatedly applying this procedure, we get a new ground-stateconfiguration, and increase B until all spins are aligned withthe field. This procedure works for system sizes in which thebranch-and-cut program can prove optimality withoutbranching, i.e., without dividing the problem into smallersubproblems. If the algorithm branches �this occurs only forthe largest system sizes studied here�, we apply a divide-and-conquer strategy for determining ���G��B�� in an interval,say, �B1 ,B2�. For a fixed configuration the Hamiltonian �1� islinear in the field, the slope being the system’s magnetiza-tion. Let f1 and f2 be the two linear functions associated with���G��B1�� and ���G��B2��. If f1 and f2 are equal, we are done.Otherwise, we determine the field B3 at which the functionsintersect and recursively solve the problem in the intervals�B1 ,B3� and �B3 ,B2�.

A typical sample at L=80 requires about 2 h of CPU timeon a workstation for determining the ground states when Bgoes through the multiples of 0.02. The more time-consuming computation of the exact break points takes about4 h on typical samples with L=60, but less than a minute ifL30 because the ground-state determinations are fast andbranching almost never arises. For our work, we consideredmainly the case of Gaussian Jij, analyzing 2500 samples atL=80, 5000 at L=70, and from 2000 to 11 000 instances forsizes L=60,50,40,30,24,20,14. We also analyzed a smallernumber of samples for Jij taken from an exponential distri-bution; the exponents showed no significant differenceswhen comparing to the Gaussian case.

The exponent �. Given the Hamiltonian, it is easy to seethat for each sample the magnetization �density�

mJ�B� =

�i

�i�G�

L2 �6�

must be an increasing function of B. �The index J on themagnetization is to recall that it depends on the disorderrealization, but in the large-L limit mJ is self-averaging; also,without loss of generality, we shall work with B0.� Atlarge fields mJ saturates to 1, while at low fields, its growthlaw must be above a linear function of B. Indeed, for con-tinuous Jij, the distribution of local fields has a finite densityat zero, and so small clusters of spins will flip and will leadto a linear contribution to the magnetization. A more singularbehavior is in fact predicted by the droplet model since�1, indicating that the system is anomalously sensitive tothe magnetic field perturbation.

If B is not too small, the convergence to the thermody-namic limit �L→�� is rapid, and in fact one expects expo-nential convergence in L /�B. We should thus see an envelopecurve m�B� appear as L increases; to make a power depen-dence on B manifest, we show in Fig. 2 a log-log plot of theratio m�B� /B1/� where � is set to its droplet scaling value of1.282.

For that value of � there is not much indication that a flatregion is developing when L increases, while at L=50 a di-

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0 0.5 1 1.5 2

m

B

sample 1sample 2sample 3

FIG. 1. Magnetization as a function of B for three typicalL=10 samples.

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rect fit to a power law gives �=1.45 �see the line displayed inthe figure to guide the eye�, as found in previous work.8–10

The problem with this simple analysis is that m has bothanalytic and nonanalytic contributions; to lowest order wehave

m = �1B + �SB1/� + ¯ . �7�

Although �1B is subdominant, it is far from negligible inpractice; for instance for it to contribute to less than 10% ofm, one would need B �0.1�S /�1�1/0.282. This could easilymean B 10−3, for which there would be huge finite-sizeeffects, since L would then be much smaller than the mag-netic length �B. We thus must take into account the term �1B;we have done this, adjusting �1 so that �m−�1B� /B1/� has anenvelope as flat as possible. The result is displayed in theinset of Fig. 2, showing that the droplet scaling fits the datavery well as long as the �1B term is included. In fact, directfits to the form of Eq. �7� give � in the range 1.28 to 1.32depending on the sets of L’s included in the fits.

The clusters that flip are like zero-field droplets. The fun-damental hypothesis in the droplet argument relating � or yBto � is the fact that in an infinitesimal field one flips dropletsdefined in zero field, droplets which are compact and haverandom �except for the sign� magnetizations. We thereforenow focus on the properties of the actual clusters that areflipped at low fields.

At zero field, the droplet of lowest energy almost alwaysis a single spin �this follows from the large number of suchdroplets, in spite of their typically higher energy�. Thus asthe field is turned on, the ground state changes first mainlyvia single spin flips, and when large clusters do flip �theyfinally do so but at larger fields�, they necessarily have many“holes” and thus do not correspond exactly to zero-fielddroplets. This is not a problem for the droplet argument aslong as these clusters are compact and have random magne-tizations.

To test this, we consider for each realization of the Jijdisorder the largest cluster that flips during the whole pas-sage from B=0 to �. According to the droplet picture, this

cluster should contain a number of spins V that scales as L2

�compactness� and have a total magnetization M that scalesas V �randomness�. This is confirmed by our data where wefind M /V2/L; in Fig. 3 we plot the disorder mean ofM / V for increasing L; manifestly, this mean is remarkablyinsensitive to L. Similar conclusions apply to V /L2. For com-pleteness, we show in the inset of the figure that the surfaceof these clusters, defined as the number of lattice bonds con-necting them to their complement, grows as LdS with dS�1.32; this is to be compared to the value dS=1.27 for zero-field droplets,16 in spite of the fact that our clusters haveholes. All in all, we find that the clusters considered havestatistical properties that are completely compatible withthose assumed in the droplet scaling argument, thereby di-rectly validating the associated hypotheses.

The magnetic exponent yB and finite-size scaling of themagnetization. One can also measure the exponent yB di-rectly via the magnetic length which scales as �BB−1/yB.For each sample, define BJ

* as that field where the groundstate changes by the largest cluster of spins as described inthe previous paragraph. Since these clusters involve a num-ber of spins growing as L2, we can identify �B�BJ

*� with L.Let B* be the disorder average of BJ

*; then B*L−yB fromwhich we can estimate yB. We find that a pure power lawwith yB set to its value in the droplet picture describes thedata quite well; in the inset of Fig. 4 we display the productL1.282B* as a function of 1/L and see that the behavior iscompatible with a large-L limit with O�1/L� finite-size ef-fects. Direct fits to the form B*�L�=uL−yB�1+v /L� give yB’sin the range 1.28 to 1.30 depending on the points included inthe fit.

Given the magnetic length, one can perform finite-sizescaling �FSS� on the magnetization data m�B ,L�. Since FSSapplies to the singular part of an observable, we should havea data collapse according to

m�B,L� − �1B

m�B*,L� − �1B* = W�B/B*� , �8�

W being a universal function, W�0�=O�1� and W�x�x1/� atlarge x. Using the value of �1 previously determined, we

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FIG. 2. Magnetization divided by B1/� as a function of B; them=B1/1.45 line is to guide the eye. From top to bottom, L=14, 20,24, 30, 40, 50, 60, 70, and 80. Inset: m−�1B divided by B1/� as afunction of B. �Same L and symbols as in main part of the figure.�In both cases, � is set to its droplet model value, �=1.282.

100

10

S(L

)

L

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S(L

)

L

1.14

1.16

1.18

1.2

1.22

1.24

10 20 30 40 50 60

M/V

1/2

L

FIG. 3. The cluster magnetization divided by the square root ofcluster volume—for the largest cluster flipped in each sample—isinsensitive to L. Inset: The clusters’ mean surface scales as LdS withdS�1.32.

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display in Fig. 4 the associated data. The collapse is excellentand we have checked that this also holds when the Jij aredrawn from an exponential distribution. Added to the figureis the function x1/� to guide the eye ��=1.282 as predicted bythe droplet model�.

Conclusions. We have investigated the two-dimensionalIsing spin glass with Gaussian and exponential couplings atzero temperature as a function of the magnetic field. Themagnetization exponent � can be measured; previous studies

did not find good agreement with the droplet model predic-tion �=1.282 because the analytic contributions to the mag-netization curve were mishandled, while in this work wefound instead 1.28�1.32. We also performed a directmeasurement of the magnetic length, obtaining for the asso-ciated exponent 1.28yB1.30, again in excellent agree-ment with the droplet prediction. With this length we showedthat finite-size scaling is realized without going to infinitesi-mal fields or huge lattices. Finally, we validated the hypoth-eses underlying the arguments of the droplet model inherentin the in-field case; we find in particular that in the low-fieldlimit the spin clusters that are relevant are compact and haverandom magnetizations. In summary, by combining im-proved computational techniques and greater care in theanalysis, we have lifted the discrepancy in the magnetic ex-ponents that has existed for over a decade between numericsand droplet scaling.

We thank T. Jorg for helpful comments. The computationswere performed on the cliot cluster of the Regional Comput-ing Center and on the scale cluster of E. Speckenmeyer’sgroup, both in Cologne. F.L. has been supported by the Ger-man Science Foundation in the projects Ju 204/9 and Li1675/1 and by the Marie Curie RTN ADONET 504438funded by the EU. This work was supported also by theEEC’s HPP under contract HPRN-CT-2002-00307�DYGLAGEMEM�.

1 M. Mézard, G. Parisi, and M. A. Virasoro, Spin-Glass Theory andBeyond, Lecture Notes in Physics Vol. 9 �World Scientific, Sin-gapore, 1987�.

2 Spin Glasses and Random Fields, edited by A. Young �WorldScientific, Singapore, 1998�.

3 H. G. Katzgraber, L. W. Lee, and A. P. Young, Phys. Rev. B 70,014417 �2004�.

4 J. Houdayer and A. K. Hartmann, Phys. Rev. B 70, 014418�2004�.

5 A. J. Bray and M. A. Moore, in Heidelberg Colloquium on GlassyDynamics, edited by J. L. van Hemmen and I. Morgenstern,Lecture Notes in Physics Vol. 275 �Springer, Berlin, 1986�, pp.121–153.

6 D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 �1986�.7 W. Kinzel and K. Binder, Phys. Rev. Lett. 50, 1509 �1983�.

8 N. Kawashima and M. Suzuki, J. Phys. A 25, 1055 �1992�.9 F. Barahona, Phys. Rev. B 49, 12864 �1994�.

10 H. Rieger, L. Santen, U. Blasum, M. Diehl, M. Jünger, and G.Rinaldi, J. Phys. A 29, 3939 �1996�.

11 A. Carter, A. Bray, and M. Moore, J. Phys. A 36, 5699 �2003�.12 J. Cardy, Scaling and Renormalization in Statistical Physics

�Cambridge University Press, Cambridge, U.K., 1996�.13 F. Barahona, J. Phys. A 15, 3241 �1982�.14 F. Barahona, M. Grötschel, M. Jünger, and G. Reinelt, Oper. Res.

36, 493 �1988�.15 F. Liers, M. Jünger, G. Reinelt, and G. Rinaldi, in New Optimi-

zation Algorithms in Physics, edited by A. Hartmann and H.Rieger �Wiley-VCH, Berlin, 2004�.

16 A. K. Hartmann and A. P. Young, Phys. Rev. B 66, 094419�2002�.

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FIG. 4. Inset: Field B* times L1/� as a function of 1/L shows alimit at large L as expected in the droplet model ��=1.282�. Mainfigure: Data collapse plot exhibiting finite-size scaling of the singu-lar part of the magnetization �L=10, 14, 20, 24, 30, 40, 50, and 60�.

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