Ground-State and Domain-Wall Energies in the Spin-Glass Region of the 2D $\pm J$ Random-Bond Ising Model

0.3 MB·English

by  Ronald Fisch

Ground-State and Domain-Wall Energies in the Spin-Glass Region
of the 2D J Random-Bond Ising Model
±
Ronald Fisch∗
382 Willowbrook Dr., North Brunswick, NJ 08902
7 Alexander K. Hartmann†
0 Institut fu¨r Theoretische Physik, Universit¨at G¨ottingen,
0 Friedrich-Hund Platz 1, 37077 G¨ottingen, Germany
2 (Dated: February 6, 2008)

n Thestatisticsoftheground-stateanddomain-wallenergiesforthetwo-dimensionalrandom-bond
a Ising model on square lattices with independent, identically distributed bonds of probability p of
J J =−1 and (1−p) of J =+1 are studied. We are able to consider large samples of up to 3202
ij ij
4 spins by using sophisticated matching algorithms. We study L×L systems, but we also consider
2 L×M samples,fordifferentaspectratiosR=L/M. Wefindthatthescalingbehavioroftheground-
state energy and its sample-to-sample fluctuations inside the spin-glass region (pc ≤ p ≤ 1−pc)
] are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusp-like
n
singularity at p . Inside the spin-glass region the average domain-wall energy converges to a finite
n c
nonzero value as the sample size becomes infinite, holding R fixed. Here, large finite-size effects
-
s are visible, which can be explained for all p by a single exponent ω ≈ 2/3, provided higher-order
i
d correctionstoscalingareincluded. Finally,weconfirmthevalidityofaspect-ratioscalingforR→0:
. thedistributionofthedomain-wallenergiesconvergestoaGaussianforR→0,althoughthedomain
t
a walls of neighboring subsystemsof size L×L are not independent.
m
-
d I. INTRODUCTION over nearest neighbors on a simple square lattice of size
n L M. Thestandardmodelforquantizedenergiesisthe
×
o Thespinglass(SG)phase1,2 isnotstableatfinitetem- J model, where we choose each bond Jij to be an in-
c perature in two dimensions (2D). When the energies are ±dependent identically distributed (iid) quenchedrandom
[
not quantized, the behavior at T =0 can be understood variable, with the probability distribution

2 intermsofascalingtheory,3,4 whichisusuallycalledthe
v droplet model. This theory describes the scaling of the P(Jij)=pδ(Jij +1) + (1−p)δ(Jij −1). (3)
9
2 averagedomain-wall(DW) energyEdw withlengthscale Thus we actually set J =1, as usual. The concentration
7 in terms of the stiffness exponent θ, i.e. ofantiferromagneticbondsisp,and(1 p)istheconcen-
−
3 trationof ferromagnetic bonds. With the P(J ) of Eqn.
E Lθ. (1) ij
0 | dw|∼ (3), the EA Hamiltonian is equivalent to the Z2 gauge
6 glass model.12 Wang, Harrington and Preskill9 have ar-
This exponent has a value of about 0.28 for 2D, al-
0
− gued that the anomalous behavior of the J model is
/ most independent of the detailed nature of the bond ±
at distribution5 and describes the scaling of different kinds causedbytopologicallong-rangeorder,asaconsequence
m of excitations like domain walls and droplets, at least of the gauge symmetry.
- if largeenoughsystemsizes are studied.6 The possibility Along the T = 0 axis this model exhibits a phase
d thatquantizationoftheenergiesmightleadtothespecial transition from a ferromagneticly ordered phase13 for
n behaviorθ =0wasfirstpointedoutbyBrayandMoore.7 small concentrations p < pc of the antiferromagnetic
o bonds to a SG critical line at large values p > p
Ithasbecome clearoverthe lastfew yearsthatthereac- c
c
(and p < 1 p due to the bipartite symmetry of the
: tually is a fundamental difference in the behavior of the − c
v squarelattice). Recently, this phase transition was char-
2D Ising spin glass at zero temperature, between those
i acterizedby high-precisionground-statecalculations,9,14
X caseswheretheenergiesarequantizedandthosewhereit
is not.5,8,9 Nevertheless, recent results10,11 indicate that which have in particular yielded pc = 0.103(1). The fer-
r
a in the low-temperature critical scaling regime, the be- romagnetic phase persists at finite temperatures T > 0
for p < p (T), while the spin-glass correlations become
haviorforquantizedandnon-quantizedmodelsmightbe c
long-range only at zero temperature.3,8,15 Interestingly,
very similar.
p (T)>p (0),9,14,16 i.e. theferromagneticphaseisreen-
The Hamiltonian of the Edwards-Anderson model for c c
trant.
Ising spins is
The first hint of the remarkable behavior of the J
±
H = XJijSiSj, (2) model in the SG regime was observed by Wang and
− Swendsen.17 They found that, although for periodic
hiji
boundaryconditionsthereisanenergygapof4J between
whereeachspinS isadynamicalvariablewhichhastwo the ground states (GS) and the first excited states, the
i
allowed states, +1 and 1. The ij indicates a sum specificheatwhenT/J 1appearedtobe proportional
− h i ≪
2
to exp( J/2T).
This−result was questioned by Saul and Kardar.18,19
The behavior of the specific heat was finally demon-
strated in a convincing fashion by Lukic et al.20 Saul
and Kardar also found that the scaling of the DW en-
tropy with size did not appear to agree with the predic-
tion of the droplet model.21 This issue has recently been
clarified,22andtheDWentropyscalinganomalyhasbeen
associated with zero-energy domain walls.
In this work we explore the behavior of the J model
± FIG. 1: (color online) 2D Ising spin glass with all spins up
inside the full SG phase p p 0.5 (the behavior
c ≤ ≤ (left, up spins not shown). Straight lines are ferromagnetic,
0.5 p 1 p is equivalent due to the symmetry of
≤ ≤ − c jagged lines are anti-ferromagnetic bonds. The dashed lines
the model). In particular we study the finite-size scaling
connect frustrated plaquettes (crosses). The bonds crossed
behavior of the GS energy, of the fluctuations of the GS
by the dashed lines are unsatisfied. In the right part the
energy, and of the domain-wall energy. We also employ GS with three spins pointing down (all others up) is shown,
aspect-ratio (AR) scaling,23,24 i.e. we study rectangular corresponding to theminimum numberof unsatisfied bonds.
lattices of width L and height M, the aspect ratio being
R=L/M. Carter,BrayandMoore25 haveextendedAR
scaling to the Ising SG, and demonstrated that study- the figure: all dashed lines start or end at frustrated
ing the scaling as a function of R is an effective method plaquettes and each frustrated plaquette is connected to
forcalculatingtheexponentθ. Here,welookatthelimit exactly one other frustrated plaquette by a dashed line.
R 0andshowthatARindeedworksforthe J model Each pair of plaquettes is then said to be matched. In
as→well, in contrast to previous attempts.26 T±he entire general, closed loops of broken bonds unrelated to frus-
probability distribution P(E ) has a simple Gaussian trated plaquettes can also appear, but this is possible
dw
formforsmallR,although,aswewillshow,theunderly- only for excited states. Now, one can consider the frus-
ing assumption of independent contributions to the DW trated plaquettes as the vertices and all possible pairs of
energy is not strictly valid. connections as the edges of a (dual) graph. The dashed
lines are selected from the edges connecting the vertices
and called a perfect matching, since all plaquettes are
II. METHODS matched. One can assign weights to the edges in the
dualgraph,theweightsareequaltothesumoftheabso-
We define domain walls for the SG as it was done in lutevaluesofthebondscrossedbythedashedlines. The
the seminal work of McMillan.15 We look at differences weight Λ of the matching is defined as the sum of the
in the GS energy between two samples with the same weights of the edges contained in the matching. As we
set of bonds, and the same boundary conditions in one haveseen, Λ counts the brokenbonds, hence, the energy
direction, but different boundary conditions in the other of the configuration is given by
direction. In particular we use periodic (p) or antiperi-
odic (ap) boundary conditions along the x-axis and free E = X Jij +2Λ, (5)
− | |
boundaryconditionsalongthey-axis. Foranexampleof hi,ji
a DW created in such a way, cf. Fig. 11. For each set of
with J = 1 from Eqn. (3). Note that this holds for
bonds, the DW energy Edw is then defined to be the dif- any co|nifij|gurationofthe spins, if one alsoincludes closed
ference in the GS energies of the two different boundary
loopsinΛ,sinceacorrespondingmatchingalwaysexists.
conditions:
Obtaining a GS means minimizing the total weight
of the broken bonds (see right panel of Fig. 1). This
E =E E . (4)
dw p ap
− automatically forbids closed loops of broken bonds, so
We useaso-calledmatching algorithm tocalculatethe one is looking for a minimum-weight perfect matching.
GS.27,28 Let us now sketch just the basic ideas of the This problem is solvable in polynomial time. The al-
matching algorithm. For the details, see Refs. 27,29, gorithms for minimum-weight perfect matchings32,33 are
30,31. The algorithm allows us to find ground states among the most complicated algorithms for polynomial
for lattices which are planar graphs. This is the reason problems. Fortunately the LEDA library offers a very
why we apply (anti-)periodic boundary conditions only efficient implementation,34 which we have applied here.
in one direction, x, while the other direction, y, has free Withfreeboundaryconditionsinatleastonedirection,
boundary conditions. In the left part of Fig. 1 a small E is a multiple of 2 (since J = 1). When periodic or
dw
2Dsystemwith(forsimplicity)freeboundaryconditions antiperiodicboundaryconditionsareappliedinboththe
in both directions is shown. All spins are assumed to x and y directions, however, E becomes a multiple
dw
be “up”, hence all antiferromagnetic bonds are not sat- of 4 if the number of 1 bonds is even. If the number
−
isfied. If one draws a dashed line perpendicular to each of 1 bonds is odd, E takes on values (4n+2) with
dw
−
broken bond, one ends up with the situation shown in these boundary conditions. When M is odd, changing
3
the boundaryconditionsinthe xdirectionfromperiodic samples used varies with size, ranging between typically
to antiperiodic changes the number of 1 bonds from 100000 (L=4) to typically 10000 (L=320).
−
odd to even, or vice versa.
The behavior of the DW energy for the J model -1
±
with R > 1 was discussed previously.26 For the bound-
ary conditions we are using, the average E goes to
dw
h| |i -1.1
zero exponentially as R becomes much greaterthan one. L 0.6
Here ... denotes an average over the ensemble of ran- e)0
dom bhonid distributions for an L M lattice. We can L)-
understand this result by thinking×of the system as con- -1.2 (e(00.5
sisting of blocks of size M M pasted together along )
× L
the x direction. The probability of having a zero-energy e(0-1.3 0.4
domain wall in each block is almost independent of the 0 0.05 0.1 0.15
1/L
otherblocks. ThesamethinghappenswhenM isevenif
the boundary conditions in the y direction are periodic -1.4 p=0.5
or antiperiodic. However, if M is odd and the boundary p=0.25
p=0.2
conditionsintheydirectionareperiodicorantiperiodic,
p=0.17
Edw goesto2atlargeR,becausethenEdw =0isnot -1.5 p=0.15
ah|llowe|di. Sinceacriticalexponentshouldbe independent p=0.05
of boundary conditions, this is a(nother) demonstration 1 10 100 1000
that θ =0 for the J model in 2D. L
±
As pointed out earlier26, these results for large R do
notagreewiththe scalinglaw predictionofCarter,Bray FIG. 2: (color online) The symbols show the average ground
and Moore,25 due to the special role of zero-energy do- stateenergye0(L)forthesquaresystems(R=1)asfunction
ofthesystemsizeLforselectedvaluesofp. Thelinesgivethe
main walls. On the other hand, in the limit R 0 with
→ results to the fits (L≥16) (see text). The data for p=0.05
our boundary conditions, the prediction of Carter, Bray
look similar, but is outside the frame (e0 = −1.8025). The
and Moore is
inset shows the same data, rescaled as (e0(L)−e0)×L so
that theexpected behavioryieldsstraight lineswhen plotted
Edw Lθ(M/L)(d−1)/2, M L. (6) as function of 1/L. The data for p = 0.05 have been shifted
h| |i∼ ≫
downwards by 0.2 for bettervisibility.
Thislimithasnotbeenstudiedbeforeforthe J model.
When θ = 0, h|Edw|i should scale as R−1/2 i±n 2D. For In Fig. 2, the GS energy per spin e0(L) is shown as a
smallRwemaythinkofthelatticeasconsistingofL L- function of the system size for selected values of p. This
×
sized subsystems stacked in the y direction, with Edw finite-size scaling behavior yields particular insight into
being the sum of the DW energies of the subsystems. the physics and seems to be related35,36 to the stiffness
Therefore, by the central limit theorem, we anticipate exponent θ. It has been argued35 that for the case of
thattheprobabilitydistributionofEdw shouldapproach open boundary conditions in exactly one direction and
a Gaussian distribution in the limit of small R. In the periodic boundary conditions in the other direction the
spin-glassregionof the phase diagram,the center of this GS energy per spin follows to lowest orders the form
limiting Gaussian will approach zero as L increases. We
willshowthatournumericalresultsforthesmallRlimit b c d
e (L)=e + + + . (7)
are indeed in good agreement with these expectations, 0 0 L L2 L2−θ
even though the subsystems are not fully independent.
Note that the arguments used in Ref. 35 should apply
for the ferromagnetic phase p 0.103 as well. Since we
≤
expect θ = 0 everywhere in the SG phase (see below)
III. NUMERICAL RESULTS
and θ = 1 for the ferromagnetic phase, we can restrict
ourselves to the contributions e , b/L and c/L2. When
0
A. Ground-state energy fitting39 the data to this e (L), fits with very high qual-
0
ity result, as shown by lines in Fig. 2. This is confirmed
We begin the study of the T = 0 behavior of when plotting the data and the fitting functions in the
the J random bond model by studying the GS form(e (L) e )Lasafunctionof1/L. Thisshouldgive,
0 0
± −
energy for different concentrations p of the antiferro- accordingto Eqn.(7), a linearbehaviorwith slopec and
magnetic bonds [p = 0.05 (ferromagnetic phase), p = ordinate intersection b. We see that indeed the data fol-
0.11,0.12,0.13,0.14,0.15,0.17,0.2,0.25,0.3,0.35,0.4,0.45 lowsthefunctionalformwell,althoughevenhigherorder
and p = 0.5] and different system sizes L = 4,...,320 corrections seem to be present. They become significant
for square samples N = L L, i.e. aspect ratio R = 1. at very small sizes, but we cannot quantify them within
×
All results are averages over many different realizations thestatisticalaccuracyofthedata. Weonlyobservethat
of the disorder. The minimum number of independent this correction term seems to have an opposite sign for
4
0.4
0.4
0.38
L)| p=0.15
(
L)/|e00.36 p=0.17 c(p)σ
(
σ
L
p=0.20
0.34
p=0.25
p=0.05
p=0.50
0.32
1 10 100 1000 0.3
0.1 0.2 0.3 0.4 0.5
L
p
FIG. 3: (color online) The symbols show the width σ of the
FIG. 4: The symbols show the limiting value c (p) =
distributions of theGS energy e0 for square systems (R=1) σ
asfunctionofsystemsizeLforselectedvaluesofp≥p . The limL→∞Lσ/e0(L) for the width σ of the distributions of the
c GS energy (as shown in Fig. 3) as a function of p. The solid
data is rescaled with the average GS energy e0(L) and with
line represents a fit according toEqn. (8), see text.
theleadingbehaviorL. Additionally,thedashedlinedisplays
the result for p = 0.05. The full lines show fits to constants
at larger system sizes.
transition. Notethatthevalueofthe scalingexponentis
very close to the exponent ω = 2/3, which describes the
small (p p 0.25) compared to larger values of p. finite-size scaling of the DW energy E ; see below.
c dw
Furthermo≤re,th≤e behaviorofthec/L2 termis quite sim- Nevertheless,thebasicobservationofthe(almost)uni-
ilar inside the SG phase for different values of p, while versalbehaviorinsidetheSGregion,whichwefindwhen
the behavior in the ferromagnetic phase (p = 0.05) is looking at the GS energy, becomes even more apparent,
very different. The value of the ordinate intercept b is whenstudying the behaviorofthe DW energy,which we
monotonic in p, as expected.35 do in the next section.
Next, we consider the width σ of the distribution of
GSenergiesperspin. Forshort-ranged-dimensionalspin
glasses, it has been proven37 that σ L−d/2, i.e. σ
B. Domain-wall energy
L−1 in this case. To remove finite-siz∼e effects caused b∼y
our special choice of boundary conditions, we normalize
For all samples considered in the previous section,
σ by the finite-size GS energy. Thus, we plot Lσ/e (L)
| 0 | we have also calculated the DW energy as defined in
as a function of system size and expect a horizontal line
Eqn. (4). In Fig. 5 the average value E of the DW
at cσ(p) = limL→∞Lσ/e0(L). As seen in Fig. 3, this is energy is shown as a function of systemh sdizweifor selected
indeed the case. Only corrections for very small system-
values ofp. Closeto the phase transitionp =0.103,the
sizes become visible. We have also included the data c
systems exhibit a high degree of ferromagnetic order for
for p = 0.05, which behave in the same way. Hence,
small sizes, leading to relatively large values of the av-
here the ferromagnetic phase and the SG phase cannot
erage DW energy. This explains why very large system
be distinguished just from the asymptotic behavior, in
sizes were needed in Ref. 14 to determined the location
contrast to the behavior of the mean alone.
p = 0.103 of the phase transition precisely. For larger
On the other hand, as visible in Fig. 3, the value of c
values p 0.2, the ferromagnetic correlation length is
cσ(p) increases when approaching the phase transition small. ≥
p p . For a more detailed analysis, we have plotted
c
→ InFig.6theaverageabsolutevalue E oftheDW
cσ(p) as function of p inside the SG phase in Fig. 4, also h| dw|i
energy is shown as a function of system size for selected
forvaluesofpclosertop thanthoseshowninFig.3. One
c
valuesofp. Forverysmallvalues ofp, closetothe phase
can see a strong increase when approaching the critical
transition,thisDWenergyincreaseswithsystemsizefor
concentration. Fitting a power law (with p =0.103)
c
small system sizes. Hence, if only slow algorithm were
c (p)=C +K(p p )κ (8) available, the system would look like exhibiting an or-
σ c c
− dered phase (i.e. θ > 0). When going to larger system
inthe rangep [0.1...0.17]yieldsvaluesC =0.422(3), sizes, it becomes apparent that this is only a finite-size
c
∈
K = 0.38(4) and κ = 0.64(5), i.e. a cusp at the phase effect. For intermediate values of p, the DW energy de-
−
5
3.375
p=0.12 p=0.12
0 p=0.13 p=0.13
10
p=0.15 p=0.15
p=0.17
p=0.2
p=0.2
p=0.25
10-1 2.25 p=0.25
p=0.5
p=0.5
> >
<Edw10-2 <|E|dw
1.5
-3
10
-4
10
1
10 100 1000
10 100
L
L
FIG. 5: The average value of the DW energy as function of
FIG. 6: The average absolute value of the DW energy as
systemsizeforselectedvaluesofp. Errorbarsareverysmall
a function of system size for selected values of p. Note the
for values hE i > 0.1 but relatively large where the DW
dw double-logarithmic axes. Lines are guides to theeye.
energy is small. Thus, they are omitted for readability. Note
thedouble-logarithmic axes. Lines are guides to the eye.
0.6
creases with growing system size, but no saturation of
E is visible on the accessible length scales, hence
dw 0.5
ht|heda|tialooksimilartotheresultsforGaussiansystems8
with θ < 0. When looking at the results for p > 0.25,
vitalbueec.omThesisaipspdaureenttotthhaetqhu|Eandtwi|ziedconnavteurrgee5sotfothaefipnoitse- =0)w0.4
sible values for Edw. Therefore, it is clear that also for Ed
(
intermediate values of p, the DW energy converges to P p=0.5
a finite plateau-value as well, but for much larger sys- p=0.25
tem sizes. Hence, the model provides a striking example 0.3 p=0.2
that finite-size corrections can persist up to huge length p=0.17
scales. Note that the convergence to a non-zero plateau p=0.15
value is compatible with the results from the previous
section, where we have found that inside the SG phase 0.2
the scaling function Eqn. (7) with θ = 0 describes the 1 10 100 1000
L
data well everywhere.
Due to the quantization of the possible DW energies,
FIG.7: TheprobabilityP(E =0)forazero-energydomain
a value E close to zero means that many system dw
h| dw|i wall as function of system size for selected values of p. All
exhibit actual zero DW energy E = 0. We have
dw errorbarsareat most ofthesymbolsize. Linesareguidesto
also studied the finite-size dependence of the probabil- theeye.
ity P(E = 0) for a zero-energy DW as a function of
dw
the system size, as shown in Fig. 7. One observes that
zero-energydomain walls are very common. Their prob- The exponent ω⋆ describes the rate of convergence. It
ability of occurrence increases with the concentration p dependsonthe fractionpoftheantiferromagneticbonds
of the antiferromagnetic bonds (until p=0.5 due to the and on the aspect ratio R, as well as the constants A⋆
symmetry of the model in p,(1 p)) and with the sys- andB⋆. We haveincludedthe dependence onR because
−
tem size leading to a convergence to a limiting value for belowwealsostudy valuesofR<1,butforthemoment
L . we stick to R=1.
→∞
Thesimplestassumptionfortheconvergenceof Edw When fitting Eqn. (9) to the DW energy data for
h| |i
and P(Edw = 0), at given values of p, to their limiting p 0.15, i.e. far enough away from pc, and for sizes
≥
values for L is a power law L 10, we obtain results as shown in Fig. 8. For small
→∞ con≥centrations p, the exponent ω⋆ is less negative than
F⋆(R,p,L)=A⋆(p,R)+B⋆(p,R)L−ω⋆(p,R). (9) for p close to 0.5, corresponding to needing larger sizes
6
0 explainedalso. Wehavechosenthevalueω =2/3,which
1 is similar to the values used previously in other work
-0.1 on this model,8,9 empirically. For a critical point with
0.9 T >0,the generaltheoryoffinite-sizescaling38 requires
c
-0.2 A* thatω =1/ν,whereν isthecorrelationlengthexponent.
0.8 That result does not apply here, however, since T =0.
c
-0.3
0.7
) 0.1 0.2 0.3 0.4 0.5
p
∗ω(-0.4 1.30(cid:13) (a)(cid:13) 2DISG free perp BC(cid:13)
p=0.50(cid:13)
-0.5 1.25(cid:13)
R=1(cid:13)
-0.6 1.20(cid:13)
1.15(cid:13)
-0.7
>(cid:13)
-0.8 |<|E(cid:13)dw(cid:13)1.10(cid:13)
0.1 0.2 0.3 0.4 0.5
p 1.05(cid:13)
FIG. 8: The results of fitting Eqn. (9) to the data (R = 1): 1.00(cid:13)
exponent ω⋆ (main plot) and limiting value A⋆ (inset).
0.95(cid:13)
0.00(cid:13) 0.05(cid:13) 0.10(cid:13) 0.15(cid:13) 0.20(cid:13) 0.25(cid:13) 0.30(cid:13) 0.35(cid:13) 0.40(cid:13) 0.45(cid:13)
toreachthe asymptoticvalueA⋆. Onthe otherhandA⋆ L(cid:13)-2/3(cid:13)
itself seems to depend only weakly on p, confirming the
notionthateverywhereinside the SG regionthe DW en-
ergy reaches a non-zero plateau value. The dependence (b)(cid:13)
2DISG free perp BC(cid:13)
ofω⋆ appearslikeastrongviolationofuniversalityinside 0.56(cid:13) p=0.50(cid:13)
the SG region. To investigate this assumption, whether R=1(cid:13)
it is a true non-universality or just a finite-size scaling 0.54(cid:13)
effect, we have performed fits which also include correc-
tions to scaling. We do this by fitting the data to a form 0.52(cid:13)
>(cid:13)
0)
=
F(R,p,L)=A(R,p)+ Xm∗ Bm(R,p)L−mω. (10) <P(E(cid:13)dw(cid:13)0.50(cid:13)
m=1
0.48(cid:13)
F(R,p,L) represents any function of P(E ). Two ex-
dw
amplesthatwechosetostudywere E andtheprob- 0.46(cid:13)
dw
h| |i
ability P(E =0).
dw
The number of correction-to-scaling terms included, 0.00(cid:13) 0.05(cid:13) 0.10(cid:13) 0.15(cid:13) 0.20(cid:13) 0.25(cid:13) 0.30(cid:13) 0.35(cid:13) 0.40(cid:13) 0.45(cid:13)
m∗,waschosenaccordingtotheamountofdatatobefit. L(cid:13)-2/3(cid:13)
Here we have tried 2 and 3. Of course, the values of the
coefficientsAandB whicharefoundbythefitsdepend
m
somewhatonthe choiceofm∗. The computedstatistical FIG. 9: (color online) Finite-size scaling fits for p = 0.5 and
errorsdo notinclude any allowancefor systematic errors R=1: (a) h|Edw|i vs. L−2/3; (b) probability of Edw =0 vs.
duetothechoiceoftheformofthescalingfunction. For- L−2/3. The error bars show one standard deviation.
tunately, it turned out that the value of A is relatively
insensitive to the choice of m∗. As an example, our finite-size scaling fits (using m∗ =
In general, whether we chose m∗ to be 2 or 3, the 3) to the data using Eqn. (10) for the case p = 0.5 and
computedB coefficientsdidnotallhavethe samesign. R = 1 are shown in Fig. 9, the data for all system sizes
m
Thiseffectiscausedbythe behavioratsmallL. Forthis was included in the fits. In the figures, we show F(R,L)
reason, there is no well defined prescription for deciding as a function of L−2/3. One observes a straight line for
what the best choice of the exponent ω is. We have not L−2/3 0, showing that the choice of the exponent is
→
included ω in the set of free fitting parameters. Since it compatible with the data. Note that the same R = 1
is the purpose of this section to show that the behavior data was originally8 fit (only for the case p = 0.5) by
inside the full SG region is universal, we have selected a making different assumptions which are not consistent
value for ω, such that the data for all values of p can be with Eqn. (10).
fit. Thus the strong finite-size effects close to p can be Fig. 10(a) shows the fit coefficients for E , as de-
c dw
h| |i
7
2D ISG <|E(cid:13) |>(cid:13)
15.0(cid:13) (a)(cid:13) Finite-size scadlwin(cid:13) g(cid:13) E d w =2 E d w =0
in powers of L(cid:13)-2/3(cid:13)
10.0(cid:13)
5.0(cid:13)
s(cid:13)
nt
e
ci 0.0(cid:13)
effi
o
c
Fit -5.0(cid:13)
A(cid:13)
-10.0(cid:13) B1(cid:13)
B2(cid:13)
B3(cid:13)
-15.0(cid:13)
0.10(cid:13) 0.15(cid:13) 0.20(cid:13) 0.25(cid:13) 0.30(cid:13) 0.35(cid:13) 0.40(cid:13) 0.45(cid:13) 0.50(cid:13) 0.55(cid:13)
p(cid:13)
FIG. 11: Domain walls of subsystems are not independent:
Comparison of a minimum-energy DW for a system of size
6.0(cid:13)
5.0(cid:13) (b)(cid:13) F2DinIiSteG-s i z Pe( sEc(cid:13)dawl(cid:13)=in0g)(cid:13)(cid:13) 100×200 (left) having Edw =2 and of the same system cut
into two systems of size 100×100 (right), having E = 0.
4.0(cid:13) in powers of L(cid:13)-2/3(cid:13) Notethatintheleftfigure,theDWwrapsaroundthedswystem
3.0(cid:13) in thex-direction. Notealso that theDWsare highlydegen-
2.0(cid:13) erate and the algorithm does not find DWs in a controlled
ents(cid:13) 1.0(cid:13) willauys.trAat“esjuwmhpa”taintythpeicDalWDWatltohoeksrilgikhet.isTvhiesiibmlep.orTthanistfipgouinret
ci 0.0(cid:13)
oeffi -1.0(cid:13) hereis thedifference in theDW energies.
c
Fit -2.0(cid:13) A(cid:13)
-3.0(cid:13) B1(cid:13)
C. Aspect-ratio scaling
B2(cid:13)
-4.0(cid:13)
B3(cid:13)
-5.0(cid:13)
The mainassumptionwhenusing AR scalingEqn.(6)
-6.0(cid:13)
0.10(cid:13) 0.15(cid:13) 0.20(cid:13) 0.25(cid:13) 0.30(cid:13) 0.35(cid:13) 0.40(cid:13) 0.45(cid:13) 0.50(cid:13) 0.55(cid:13) in the limit R 0 is that different subsystems of size
→
p(cid:13) M M are independent of each other, which leads to
×
a Gaussian distribution of the DW energies E in the
dw
limit R 0. Nevertheless, one can easily imagine that
→
FIG. 10: (color online) Fit coefficients (defined by Eqn. 10) the DWs in different parts of the system are not truly
for R=1: (a) h|Edw|i vs. p;(b) P(Edw =0) vs. p. independent of each other. This is demonstrated for a
sample system in Fig. 11, where the DW energy of each
subsystem (1)/(2) of size 100 100 is E(1)/(2) = 0, but
× dw
the DW energy of the full system is E =2. Also, in a
dw
previousstudy26, the validity ofARscalingcouldnotbe
established for the J model. Here, we will show that
±
AR scaling indeed works also for this case, i.e. the dis-
fined by Eqn. (10), as a function of p for R = 1. The tribution of the DW energies becomes indeed Gaussian,
quality of fit was good everywhere, proving that the ap- despite the non-independencies of the domain walls in-
parent non-universality visible in the dependence of the side the different subsystems. Because the shape of the
effectiveexponentω⋆ isduetothepresenceofcorrections P(Edw) distribution is changing with R, we expect that
toscaling. WeagainseethatthevalueofA,theestimate the AR scaling will not be perfect in the range of our
for L , changes very little over this range of p; the data. As we shall see, however, the deviations from the
values→are∞for p 0.3 very similar to the fit according to Gaussian fit become smaller than our statistical errors
Eqn. (9), but th≥ey are larger in the present fit for small for R 1/4. Thus we are able to verify that we are
≤
concentrations,p<0.3. Wealsonotethattherearesub- approaching the predicted scaling limit.
stantial variations in the B coefficients. In particular, First,weconcentrateonp=1/2wherethedistribution
m
B alternates in sign for the smaller values of p. In Fig. P(J ) of Eqn. (3) is symmetric about zero. Thus in
3 ij
10(b) the results of fitting P(E =0), which shows the this case the distribution P(E ) is (ignoring statistical
dw dw
same type of behavior, are given. In this case the signs fluctuations) also symmetric around zero, for any values
of B and B alternate. of L and M. Therefore, we used p= 1/2 to collect data
2 3
8
for sets of random lattices with sequences of L and M 1000
having the aspect ratios R = 1, 1/2, 1/4, 1/8, 1/16 and R=1
p=0.5
1/32. ForeachvalueofR,wethenusedfinite-sizescaling R=1/32
to extrapolate to the large lattice limit.
100
10-1 ndf
0.25 2χ/
10-2
10
0.2 10-3
E)dw0.15 10-4
P( 10-5 1
1 10 100 1000
0.1 L
10-6
-8 -6 -4 -2 0 2 4 6 8
FIG. 13: Quality of the Gaussian fit for the DW energy dis-
0.05 p=0.5, L=40, R=1 tribution measured by χ2/ndf as a function of the block size
L for p=0.5 (circles: R=1, squares: R=1/32).
0
-5 0 5 10 15 20
E 0.04
dw
FIG.12: ThesymbolsdisplaythedistributionofDWenergies
for the case L=M =40 (i.e. R=1) and p=0.5, the lines 10-2
0.03
showstheresultofafittoaGaussian. Strongdeviationsfrom
aGaussiandistributionarevisibleatP(E =0)(mainplot)
dw
and in the“tail” of thedistribution (inset).
)w 10-3
Ed0.02
We start by studying L L samples (R = 1), to es- P(
×
tablish that in this case the distribution of the DW en-
ergies is indeed not Gaussian. (Otherwise, it would not 10-4
0.01 -20 0 20
be surprising if it is also Gaussian for R 1.) The
distribution for L = 40 is shown in Fig. 12≪. This dis- p=0.5, L=40, R=1/32
tribution is obtained from an average over 39000 real-
izations. We have fit39 a Gaussian to the data. (Note 0
-40 -20 0 20 40 60 80
that the fitting procedure ignores the fact that the en-
E
dw
ergies are quantized.) Strong deviations from Gaussian
behavior are visible at P(E = 0) (main plot) and in
dw FIG. 14: The symbols show the distribution of DW energies
the “tail” of the distribution (inset). Correspondingly
for the case L= 40,M =1280 (i.e. R= 1/32) and p = 0.5,
the quality of the fit, i.e. χ2 per degree of freedom, thelinesshowstheresultoffittingtoaGaussian. Main plot:
as given by our fitting program39 in this case, is very linear ordinate, inset: logarithmic ordinate scale. The data
high: χ2/ndf= 78. Even larger deviations are expected match the Gaussian very well.
in the tail. To observe these, more sophisticated tech-
niques would be needed40,41, which is beyond the scope
ofthiswork. ThestrongdeviationfromGaussianbehav- InFig.14weshowtheprobabilitydistributionP(E )
dw
iorisnotafinite-sizeeffect,asmaybe seenfromFig.13, for a set of lattices with p=1/2, R=1/32 and L=40.
where the circles display χ2/ndf as a function of L for Inthiscase,thereare90,000randomsamplesinthedata
the R=1 case. set. Again,thedistributionwasfittoaGaussian39. This
The reader should also note that we find no evidence fit is quite good, as demonstrated by the fact that the
for any ”sawtooth” structure in the data. This is in value of χ2/ndf areis close to one. Note that the precise
marked contrast to the case of periodic or antiperiodic value of χ2/ndf depends on the number of points in the
boundary conditions in the y direction, which causes al- tailofthe distributionthat areincludedin the fit, which
ternating values of E to have zero probability. There- is somewhat arbitrary. The Gaussian behavior here is
dw
foreP(E )does notbecome independentofthe bound- not a finite-size effect either. This can be seen from Fig.
dw
ary conditions even in the limit of large lattices. This 13, where the square symbols display χ2/ndf as function
is an aspect of the topological long-range order9 which of L for the R=1/32 case.
exists in this model when the boundary conditions are Whenp=0.5theconfigurationaveragesofalltheodd
periodic along both x and y. moments of P(E ) must vanish. Then the dominant
dw
9
TABLE II: Probability of E = 0 for p= 0.5, extrapolated
0.6(cid:13) dw
2D ISG(cid:13) to L=∞, as a function of the aspect ratio, R. The m∗ and
p=0.5(cid:13) A are defined by Eqn. (10), and the eA are statistical error
0.5(cid:13)
estimates.
0.4(cid:13) R m∗ A eA A(R)/A(2R)
1.0 3 0.55921 0.00294
3(cid:13) 0.3(cid:13) 0.5 3 0.33215 0.00261 0.594
2(cid:13) )(cid:13)- m(cid:13)2(cid:13) 0.2(cid:13) R=1(cid:13) 00..21525 22 00..2114671396 00..0000113359 00..665812
/(m(cid:13)4(cid:13) R=1/2(cid:13) 0.0625 2 0.10264 0.00144 0.697
R=1/4(cid:13)
0.1(cid:13) R=1/8(cid:13) 0.03125 1 0.07053 0.00069 0.687
R=1/16(cid:13)
0.0(cid:13) R=1/32(cid:13)
Again, for R < 1/4 the agreement is about as good as
-0.1(cid:13)
4(cid:13) 6(cid:13) 8(cid:13)10(cid:13) 20(cid:13) 40(cid:13) 60(cid:13)801(cid:13)00(cid:13) 200(cid:13) 400(cid:13) could be expected.
L(cid:13) We have also studied the AR approach for p = 0.15
and found similar results (not shown). For aspect ratios
R < 1/4 the assumptions of a Gaussian distribution of
FIG. 15: (color online) Kurtosis vs. L (p=0.5.) The x axis the DW energies is again valid. Hence, we believe that
is scaled logarithmically. insidetheentireSGregionthebehaviorisconsistentwith
the assumptions of the AR approach.
TABLE I: h|E |i for p = 0.5, extrapolated to L = ∞, as a
dw
functionoftheaspectratio,R. Them∗ andAaredefinedby
D. Discussion of AR scaling
Eqn.(10), and theeA are statistical error estimates.
R m∗ A eA A(R)/A(2R) AttheendofSectionIIwearguedthatforsmallRthe
1.0 3 0.96025 0.00792 system may be thought of as a set of L L subsystems
0.5 2 1.81967 0.00471 1.895 ×
stacked in the y direction. This implies that the behav-
0.25 2 2.89085 0.00671 1.589
ior of E in this limit should obey Eqn. (6) with θ =0.
0.125 2 4.28859 0.01013 1.484 dw
As we have seen, our numerical results are indeed con-
0.0625 2 6.21247 0.02387 1.449
sistent with this argument. However, the hand-waving
0.03125 1 8.98954 0.03402 1.447
argument does not specify how the subsystems are to
be connected to each other. Before actually seeing the
numerical results, we were far from confident that this
contribution to non-Gaussian behavior will be the kur- argument would correctly predict what was found.
tosis, which simplifies to m4/(m2)2 3, where mi is the There is another apparently reasonable argument,
−
ith moment of P(Edw), under these conditions. Fig. 15 which leads one to expect that the results for small R
shows the behavior of the kurtosis of the P(Edw) distri- shouldnotgivethisanswer. AsR 0,weknowthatthe
butions at p = 0.5 as a function of L and R. It shows width of the E distribution wil→l diverge. This means
dw
that for R 1/4 the kurtosis is negligible, except when that for most of our sample lattices E 1 in this
L < 8, qua≤ntifying the convergence towards a Gaussian limit. One might have thought, there|fordew,|t≫hat the fact
distribution for small R. thatE is quantizedwouldno longermatter. Thus one
dw
Thisconvergencecanbealsoseenwhenfitting Edw mighthaveguessedthatthesystemwouldbehaveinthis
h| |i
to Eqn. (10), now for aspect ratios R < 1. In Table I limit as if θ 0.28, the value for an unquantized dis-
≈ −
we show the results of the finite-size scaling fits at p = tribution.
0.5. According to Eqn. (6), the prediction of AR scaling Therefore, our results indicate that a Bohr correspon-
theory is that the ratio A(R)/A(2R) should approach dence principle, that in the limit of large quantum num-
√2 1.414forsmallR. Consideringthestatisticalerrors berstheresultsshouldbeindistinguishablefromthoseof
≈
and the corrections to scaling, the agreement is quite anunquantized system, does not apply. How can we un-
satisfactory. HereweobservethatforR<1/4thescaling derstand this? In our opinion, what we learn from this
assumption is reasonably accurate. is that θ = 0 is directly connected to the existence of
Table II shows the results of the finite-size scaling fits topological long-range order9 at T =0.
for the probability that E =0 at p=0.5. Since, from One might try to argue that R = 1/32 is not small
dw
thecentrallimittheorem,forthesumΣofK suitablyiid enough, and that for even smaller R one would indeed
integers,P(Σ=0) K−0.5,viaK R−1 theprediction find a crossover to θ 0.28. Such a viewpoint is sug-
of AR scaling theo∼ry for this quan∼tity is that the ratio gested by the recent ≈wo−rk of J¨org et al.10 On the other
A(R)/A(2R)shouldapproach1/√2 0.707forsmallR. hand a more detailed analysis11 of the low temperature
≈
10
behavior of the specific heat of the J model does not plateauvalue everywhereinside the SGphase. This con-
±
support this interpretation. vergence can be described, again universally inside the
SG phase, by a single exponent ω 2/3, just by taking
≈
into account higher-order corrections to scaling. Finally,
IV. SUMMARY we have studied L M rectangular lattices with aspect
×
ratio R between 1/32 and 1. We have demonstrated by
Wehavestudiedthestatisticsandfinite-sizescalingbe- an example that one assumption underlying AR scaling,
havior of GS and DW energies for the 2D Ising random- i.e.the assumedindependence ofthe DWenergiesofdif-
bond spin glass with an mixture of +1 and 1 bonds, ferent blocks, is not strictly valid. Nevertheless, we find
where p is the concentrationof antiferromagne−ticbonds. thatfor largelattices the probabilitydistributionof Edw
By using sophisticated matching algorithms, we can cal- in the SG region of the phase diagram approaches for
culate the GS energy and DW energies for quite large R 1/4 a Gaussian centered at Edw =0. Hence, in the
≤
systems exactly, whichallowsus to study the scalingbe- small R limit the behavior obeys the AR scaling predic-
havior very precisely. We find that the scaling behavior tions of Carter, Bray and Moore.25
of the GS energycan be described everywhereinside the
SG region by the same simple scaling function Eqn. (7),
as predicted by Ref. 35. Furthermore, when looking at
Acknowledgments
the fluctuations of the GS energy, we find, after care-
fullytakingintoaccountfinite-sizecorrections,thatthey
follow the predicted simple L−1-scaling with an ampli- RF is grateful to S. L. Sondhi, F. D. M. Haldane
tude which has a cusp singularity at the ferromagnetic- and D. A. Huse for helpful discussions, and to Prince-
SG transition p . The singularity is described by an ex- ton University for providing use of facilities. AKH
c
ponent which is close to 2/3. acknowledges financial support from the VolkswagenS-
The behavior of the DW energy is characterized close tiftung(Germany)withinthe program“Nachwuchsgrup-
top ,byhugefinite-sizeeffects. Nevertheless,theresults pen an Universit¨aten” and from the DYGLAGEMEN
c
are compatible with a convergence of E to a finite programfunded by the EU.
dw
h| |i
∗ [email protected] (2004).
† [email protected] 15 W. L. McMillan, Phys.Rev.B 29, 4026 (1984).
1 S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 16 F.MerzandJ.T.Chalker,Phys.Rev.B65,054425(2002).
(1975). 17 J.-S. Wang and R. H. Swendsen, Phys. Rev. B 38, 4840
2 Reviews on spin glasses can be found in: K. Binder and (1988).
A.P.Young,Rev.Mod.Phys.58,801(1986);M.Mezard, 18 L. Saul and M. Kardar, Phys. Rev.E 48, R3221 (1993).
G.Parisi,M.A.Virasoro,SpinGlass TheoryandBeyond, 19 L. Saul and M. Kardar, Nucl. Phys. B 432, 641 (1994).
(WorldScientific,Singapore1987);K.H.FischerandJ.A. 20 J. Lukic, A. Galluccio, E. Marinari, O. C. Martin and G.
Hertz, Spin Glasses, (Cambridge University Press, Cam- Rinaldi, Phys. Rev.Lett. 92, 117202 (2004).
bridge1991);A.P.Young(ed.),SpinGlassesandRandom 21 D.S.FisherandD.A.Huse,Phys.Rev.B38,386(1988).
Fields, (World Scientific,Singapore 1998). 22 R.Fisch, J. Stat. Phys.125, 793 (2006).
3 A.J.BrayandM.A.Moore,Phys.Rev.B31,631(1985). 23 A. E. Ferdinand and M. E. Fisher, Phys. Rev. 185, 832
4 D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 (1969).
(1986). 24 H. Park and M. den Nijs, Phys.Rev.B 38, 565 (1988).
5 C. Amoruso, E. Marinari, O. C. Martin and A. Pagnani, 25 A. C. Carter, A. J. Bray and M. A. Moore, Phys. Rev.
Phys. Rev.Lett. 91, 087201 (2003). Lett. 88, 077201 (2002).
6 A. K. Hartmann and M. A. Moore, Phys. Rev. Lett. 90, 26 A. K. Hartmann, A. J. Bray, A. C. Carter, M. A. Moore
127201 (2003) and A. P. Young,Phys.Rev. B 66, 224401 (2002).
7 A. J. Bray and M. A. Moore, Heidelberg Colloquium on 27 A. K. Hartmann and H. Rieger, Optimization Algorithms
Glassy Dynamics, J. L. van Hemmen and I. Morgenstern, in Physics, (Wiley-VCH,Berlin, 2001).
(ed.), (Springer, Berlin, 1986), pp.121-153. 28 A. K. Hartmann and H. Rieger, New Optimization Algo-
8 A. K. Hartmann and A. P. Young, Phys. Rev. B 64, rithms in Physics, (Wiley-VCH,Berlin, 2004).
180404(R) (2001). 29 I. Bieche, R. Maynard, R. Rammal, and J. P. Uhry, J.
9 C.Wang,J.HarringtonandJ.Preskill,Ann.Phys.(N.Y.) Phys. A 13, 2553 (1980).
303, 31 (2003). 30 F.Barahona, R.Maynard,R.Rammal,andJ.P.Uhry,J.
10 T. J¨org, J. Lukic, E. Marinari and O. C. Martin, Phys. Phys. A15, 673 (1982).
Rev.Lett. 96, 237205 (2006) 31 U. Derigs and A. Metz, Math. Prog. 50, 113 (1991).
11 R. Fisch, cond-mat/0607622. 32 W. J. Cook, W. H. Cunningham, W. R. Pulleyblank, and
12 Y. Ozekiand H.Nishimori, J. Phys. A 26, 3399 (1993). A. Schrijver, Combinatorial Optimization, (John Wiley &
13 S. Kirkpatrick,Phys. Rev.B 16, 4630 (1977) Sons, New York 1998).
14 C.AmorusoandA.K.Hartmann,Phys.Rev.B70,134425 33 B.KorteandJ.Vygen,CombinatorialOptimization-The-