Magnetic exponents of two-dimensional Ising spin glasses
Magnetic exponents of two-dimensional Ising spin glasses
F. Liers1 and O. C. Martin2
Institut für Informatik, Universität zu Köln, Pohligstraße 1, D-50969 Köln,Germany.
Univ Paris-Sud, UMR8626, LPTMS, Orsay, F-91405; CNRS, Orsay, F-91405, France.
(Dated: October 30, 2018)
The magnetic critical properties of two-dimensional Ising spin glasses are controversial. Using
exact ground state determination, we extract the properties of clusters flipped when increasing con-
tinuously a uniform field. We show that these clusters have many holes but otherwise have statistical
properties similar to those of zero-field droplets. A detailed analysis gives for the magnetization ex-
ponent δ ≈ 1.30 ± 0.02 using lattice sizes up to 80 × 80; this is compatible with the droplet model
prediction δ = 1.282. The reason for previous disagreements stems from the need to analyze both
singular and analytic contributions in the low-field regime.
PACS numbers: 75.10.Nr, 75.40.-s, 75.40.Mg
Spin glasses [1, 2] have been the focus of much interest
because of their many remarkable features: they undergo
a subtle freezing transition as temperature is lowered,
their relaxational dynamics is slow (non-Arrhenius), they
exhibit ageing, memory effects, etc. Although there are
still some heated disputes concerning three-dimensional
spin glasses, the case of two dimensions is relatively
consensual, at least in the absence of a magnetic field.
Indeed, two recent studies [3, 4] found that the ther-
mal properties of two-dimensional Ising spin glasses with
Gaussian couplings agreed very well with the predictions
of the scaling/droplet pictures [5, 6]. Interestingly, the
situation in the presence of a magnetic field remains un-
clear; in particular, some Monte Carlo simulations [7] and
basically all ground state studies [8, 9, 10] seem to go
against the scaling/droplet pictures. Nevertheless, since
spin glasses often have large corrections to scaling, the
apparent disagreement with the droplet picture resulting
from these studies may be misleading and tests in one
dimension give credence to this claim [11].
In this study we use state of the art algorithms for de-
termining exact ground states in the presence of a mag-
netic field and treat significantly larger lattice sizes than
in previous work. By finding the precise points where
the ground states change as a function of the field, we
extract the excitations relevant in the presence of a field
which can then be compared to the zero-field droplets.
Although for small size lattices we agree with previous
studies, at our larger ones a careful analysis, taking into
account both the analytic and the singular terms, gives
excellent agreement with the droplet picture.
The model and its properties — We work on an L×L
square lattice having Ising spins on its sites and couplings
Jij on its bonds. The Hamiltonian is
H({σi}) ≡ −
Jijσiσj −B
σi (1)
The first sum runs over all nearest neighbor sites using
periodic boundary conditions to minimize finite size ef-
fects. The Jij are independent random variables of either
Gaussian or exponential distribution.
It is generally agreed that two-dimensional spin glasses
have a unique critical point at T = B = 0. There, the
free energy is non-analytic and in fact, standard argu-
ments [12] suggest that as T → 0 and B → 0 the free
energy goes as βF (L, β) ∼ βE0+Gs(TLyT , BLyB ) where
E0 is the ground-state energy, β the inverse temperature,
while yT and yB are the thermal and magnetic exponents.
Previous work when B = 0 is compatible with this form
and in fact also agrees with the scaling/droplet picture
of Ising spin glasses in which one has yT = −θ ≈ 0.282.
The stumbling block concerns the behavior when B 6= 0;
there, the droplet prediction in general dimension d is
yB = yT + d/2 (2)
but the numerical evidence for this is muddled at best. It
is thus worth reviewing the hypotheses assumed by the
droplet model 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 charac-
teristic length scale ξB. To see this, consider an infinites-
imal field and zero-field droplets of scale ℓ. These are
expected to be compact. The interfacial energy of such
droplets is O(ℓθ) while their total magnetization goes as
ℓd/2. The magnetic and interfacial energy are then bal-
anced when B reaches a value O(1/ℓd/2−θ): at that value
of the field, some of the droplets will flip and the ground
state will be destabilized. We then see that for each field
strength there is an associated magnetic length scale ξB
ξB ≈ B
−θ (3)
This leads to the identification yB = d/2−θ in agreement
with Eq. 2, giving yb ≈ 1.282 at d = 2.
The droplet model also predicts the scaling of the mag-
netization in the B → 0 limit via the exponent δ:
m(B) ∼ B1/δ (4)
If this form also holds for infinitesimal fields at finite
L, we can consider the field B∗ for which system-size
http://arxiv.org/abs/0704.1626v2
droplets flip; this happens when B = O(1/LyB) and then
the magnetization is O(L−d/2), the droplets having ran-
dom magnetizations. This leads to m(B∗) ∼ L−d/2 and
m(B∗) ∼ [1/LyB ]1/δ so that
dδ = 2yB (5)
Although the droplet model arguments are not proofs,
they seem quite convincing. Nevertheless, the numerical
studies measuring δ do not give good agreement with the
prediction δ = 1.282. For instance, using Monte Carlo at
“low enough” temperatures, Kinzel and Binder [7] find
δ ≈ 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 three independent groups [8, 9, 10] with increasing
power, leading to δ ≈ 1.48, δ ≈ 1.54 and δ ≈ 1.48. Taken
together, these studies show a real discrepancy with the
droplet prediction. To save the droplet model from this
thorny situation, one can appeal to large corrections to
scaling. Such potential effects have been considered [11]
in dimension one where it was shown that ξB was poorly
fitted by a pure power law unless the fields were very
small. Here we revisit the two-dimensional case to reveal
either the size of the corrections to scaling or a cause for
the break down of the droplet reasoning.
Computation of ground states — We determine the
exact ground state of the Hamiltonian (1) by computing
a maximum cut in the graph of interactions [13], a promi-
nent problem in combinatorial optimization. Whereas it
is polynomially solvable on two-dimensional grids with-
out a field and couplings bounded by a polynomial in the
size of the input, it is NP-hard with an external field. In
practice, we rely on a branch-and-cut algorithm [14, 15].
Let the ground state at a field B be denoted as
{σ(G)(B)}. To study the magnetization, we computed
the ground states at increasing values of B, in steps of
size 0.02. When focusing instead on the flipping clusters,
we had to determine the intervals in which the ground
state was constant and in what manner it changed when
going from one interval to the next. In Fig. 1 we show
the associated piecewise constant magnetization curves
for three samples of the disorder variables Jij at L = 10.
To get the sequence of intervals or break points associ-
ated with such a function exactly, we start by computing
the ground state in zero field. By applying postoptimal-
ity analysis from linear programming theory, we deter-
mine [10, 15] a range ∆B such that the ground state at a
field B remains the optimum in the interval [B,B+∆B].
We reoptimize at B+∆B+ ǫ, with ǫ being a sufficiently
small number. By repeatedly applying this procedure,
we get a new ground state configuration, and increase B
until all spins are aligned with the field. This procedure
works for system sizes in which the branch-and-cut pro-
gram can prove optimality without branching, i.e., with-
out dividing the problem into smaller sub-problems. If
the algorithm branches (this occurs only for the largest
0 0.5 1 1.5 2
sample 1
sample 2
sample 3
FIG. 1: Magnetization as a function of B for three typical
L = 10 samples.
system sizes studied here), we apply a divide-and-conquer
strategy for determining {σ(G)(B)} in an interval, say
[B1, B2]. For a fixed configuration the Hamiltonian (1) is
linear in the field, the slope being the system’s magneti-
zation. Let f1, f2 be the two linear functions associated
with {σ(G)(B1)} and {σ(G)(B2)}. If f1 and f2 are equal,
we are done. Otherwise, we determine the field B3 at
which the functions intersect and recursively solve the
problem in the intervals [B1, B3] and [B3, B2].
A typical sample at L = 80 requires about 2 hours of
cpu on a work station for determining the ground states
when B goes through the multiples of 0.02. The more
time consuming computation of the exact break points
takes about 4 hours on typical samples with L = 60,
but less than a minute if L ≤ 30 because the ground-
state determinations are fast and branching almost never
arises. For our work, we considered mainly the case of
Gaussian Jij , analyzing 2500 samples at L = 80, 5000
at L = 70, and from 2000 to 11000 instances for sizes
L = 60, 50, 40, 30, 24, 20, 14. We also analyzed a smaller
number of samples for Jij taken from an exponential
distribution; exponents showed no significant differences
when comparing to the Gaussian case.
The exponent δ — Given the Hamiltonian, it is easy
to see that for each sample the magnetization (density)
mJ(B) =
must be an increasing function of B. (The index J on the
magnetization is to recall that it depends on the disorder
realisation, but in the large L limit mJ is self averag-
ing; also, without loss of generality, we shall work with
B > 0.) At large fields mJ saturates to 1, while at low
fields, its growth law must be above a linear function of
B. Indeed, for continuous Jij , the distribution of local
fields has a finite density at zero and so small clusters
of spins will flip and will lead to a linear contribution to
the magnetization. A more singular behavior is in fact
predicted by the droplet model since δ > 1, indicating
that the system is anomalously sensitive to the magnetic
field perturbation.
If B is not too small, the convergence to the thermo-
dynamic limit (L → ∞) is rapid, and in fact one expects
exponential convergence in L/ξB. We should thus see an
envelope curve m(B) appear as L increases; to make a
power dependence on B manifest, we show in Fig. 2 a
log-log plot of the ratio m(B)/B1/δ where δ is set to its
droplet scaling value of 1.282. For that value of δ there
 0.01  0.1  1
 0.01  0.1  1
 0.01  0.1  1
 0.01  0.1  1
 0.01  0.1  1
 0.01  0.1  1
 0.01  0.1  1
 0.01  0.1  1
 0.01  0.1  1
FIG. 2: Magnetization divided by B1/δ as a function of B; the
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 − χ1B divided
by B1/δ as a function of B. (Same L and symbols as in main
part of the figure.) In both cases, δ is set to its droplet model
value, δ = 1.282.
is not much indication that a flat region is developping
when L increases, while at L = 50 a direct fit to a power
gives δ = 1.45 (cf. line displayed in the figure to guide
the eye), as found in previous work [8, 9, 10]. The prob-
lem with this simple analysis is that m has both analytic
and non-analytic contributions; to lowest order we have
m = χ1B + χSB
1/δ + . . . (7)
Although χ1B is sub-dominant, it is far from negligible in
practice; for instance for it to contribute to less than 10%
of m, one would need B < (0.1χS/χ1)
1/0.282. This could
easily mean B < 10−3 for which there would be huge
finite size effects since L would then be much smaller than
the magnetic length ξB . We thus must take into account
the term χ1B; we have done this, adjusting χ1 so that
(m−χ1B)/B1/δ has an envelope as flat as possible. The
result is displayed in the inset of Fig. 2, showing that the
droplet scaling fits very well the data as long as the χ1B
term is included. In fact, direct fits to the form of Eq. 7
give δ in the range 1.28 to 1.32 depending on the sets of
L’s included in the fits.
The clusters that flip are like zero-field droplets —
The fundamental hypothesis in the droplet argument re-
lating δ or yB to θ is the fact that in an infinitesimal field
one flips droplets defined in zero field, droplets which are
compact and have random (except for the sign) magneti-
zations. We therefore now focus on the properties of the
actual clusters that are flipped at low fields.
At zero field, the droplet of lowest energy almost al-
ways is a single spin (this follows from the large number
of such droplets, in spite of their typically higher energy).
Thus as the field is turned on, the ground state changes
first mainly via single spin flips, and when large clusters
do flip (they finally do so but at larger fields), they nec-
essarily have many “holes” and thus do not correspond
exactly to zero-field droplets. This is not a problem for
the droplet argument as long as these clusters are com-
pact and have random magnetizations. To test this, we
 1.14
 1.16
 1.18
 1.22
 1.24
 10  20  30  40  50  60
FIG. 3: The cluster magnetization divided by the square root
of cluster volume — for the largest cluster flipped in each
sample — is insensitive to L. Inset: The clusters’s mean
surface scales as LdS with dS ≈ 1.32.
consider for each realization of the Jij disorder the largest
cluster that flips during the whole passage from B = 0
to B = ∞. According to the droplet picture, this clus-
ter should contain a number of spins V that scales as L2
(compactness) and have a total magnetization M that
scales as
V (randomness). This is confirmed by our
data where we find M/V ∼ 2/L; in Fig. 3 we plot the
disorder mean ofM/
V for increasing L; manifestly, this
mean is remarkably insensitive to L. Similar conclusions
apply to V/L2. For completeness, we show in the inset of
the figure that the surface of these clusters, defined as the
number of lattice bonds connecting them to their com-
plement, grows as LdS with dS ≈ 1.32; this is to be com-
pared to the value dS = 1.27 for zero-field droplets [16],
in spite of the fact that our clusters have holes. All in
all, we find that the clusters considered have statistical
properties that are completely compatible with those as-
sumed in the droplet scaling argument, thereby directly
validating the associated hypotheses.
The magnetic exponent yB and finite size scaling of the
magnetization — One can also measure the exponent
yB directly via the magnetic length which scales as ξB ∼
B−1/yB . For each sample, define B∗J as that field where
the ground state changes by the largest cluster of spins as
described in the previous paragraph. Since these clusters
involve a number of spins growing as L2, we can identify
J ) with L. Let B
∗ be the disorder average of B∗J ;
then B∗ ∼ L−yB from which we can estimate yB. We
find that a pure power with yB set to its value in the
droplet picture describes the data quite well; in the inset
of Fig. 4 we display the product L1.282B∗ as a function of
 0.01  0.04  0.07
 0.1  1  10
 0.1  1  10
 0.1  1  10
 0.1  1  10
 0.1  1  10
 0.1  1  10
 0.1  1  10
 0.1  1  10
 0.1  1  10
FIG. 4: Inset: Field B∗ times L1/δ as a function of 1/L shows
a limit at large L as expected in the droplet model (δ = 1.282).
Main figure: Data collapse plot exhibiting finite size scaling
of the singular part of the magnetization (L = 10, 14, 20, 24,
30, 40, 50, and 60).
1/L and see that the behavior is compatible with a large
L limit with O(1/L) finite size effects. Direct fits to the
form B∗(L) = uL−yB(1+v/L) give yBs in the range 1.28
to 1.30 depending on the points included in the fit.
Given the magnetic length, one can perform finite size
scaling (FSS) on the magnetization data m(B,L). Since
FSS applies to the singular part of an observable, we
should have a data collapse according to
m(B,L)− χ1B
m(B∗, L)− χ1B∗
= W (B/B∗) (8)
W being a universal function, W (0) = O(1) and W (x) ∼
x1/δ at large x. Using the value of χ1 previously deter-
mined, we display in Fig. 4 the associated data. The
collapse is excellent and we have checked that this also
holds when the Jij are drawn from an exponential distri-
bution. Added to the figure is the function x1/δ to guide
the eye (δ = 1.282 as predicted by the droplet model).
Conclusions — We have investigated the 2d Ising
spin glass with Gaussian and exponential couplings at
zero temperature as a function of the magnetic field.
The magnetization exponent δ can be measured; previ-
ous studies did not find good agreement with the droplet
model prediction δ = 1.282 because the analytic con-
tributions to the magnetization curve were mishandled,
while in this work we found instead 1.28 ≤ δ ≤ 1.32.
We also performed a direct measurement of the magnetic
length, obtaining for the associated exponent 1.28 ≤
yB ≤ 1.30, again in excellent agreement with the droplet
prediction. With this length we showed that finite size
scaling is realized without going to infinitesimal fields or
huge lattices. Finally, we validated the hypotheses un-
derlying the arguments of the droplet model inherent to
the in-field case; we find in particular that in the low-field
limit the spin clusters that are relevant are compact and
have random magnetizations. In summary, by combin-
ing improved computational techniques and greater care
in the analysis, we have lifted the discrepancy on the
magnetic exponents that has existed for over a decade
between numerics and droplet scaling.
We thank T. Jorg for helpful comments. The computa-
tions were performed on the cliot cluster of the Regional
Computing Center and on the scale cluster of E. Speck-
enmeyer’s group, both in Cologne. FL has been sup-
ported by the German Science Foundation in the projects
Ju 204/9 and Li 1675/1 and by the Marie Curie RTN
ADONET 504438 funded by the EU. This work was sup-
ported also by the EEC’s HPP under contract HPRN-
CT-2002-00307 (DYGLAGEMEM).
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