Chiral-glass transition and replica symmetry breaking of a three-dimensional Heisenberg spin glass

4 Pages·0.23 MB·English

by  K. Hukushima

Chiral-glass transition and replica symmetry breaking of a three-dimensional
Heisenberg spin glass
K. Hukushima
ISSP, Univ. of Tokyo, Roppongi, Minato-ku, Tokyo 106-8666, Japan
0 H. Kawamura
0 Faculty of Engineering and Design, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan
0 (February 6, 2008)
2
ExtensiveequilibriumMonteCarlosimulationsareperformedforathree-dimensionalHeisenberg
n
spin glass with thenearest-neighbor Gaussian coupling to investigate its spin-glass and chiral-glass
a
orderings. The occurrence of a finite-temperature chiral-glass transition without the conventional
J
spin-glass order is established. Critical exponents characterizing the transition are different from
1
those of the standard Ising spin glass. The calculated overlap distribution suggests theappearance
1
of a peculiar typeof replica-symmetry breaking in thechiral-glass ordered state.

]
n
While experiments have provided convincing evidence the chiral-glass transition as an anomaly in experimen-
n
that spin-glass (SG) magnets exhibit an equilibrium tally accessible quantities.
-
s phase transition at a finite temperature, the true nature Numericalstudiesonthe3dXYspinglasseshavegiven
di ofthe experimentallyobservedSGtransitionandthatof strong support to the occurrence of a finite-temperature
. thelow-temperatureSGphasestillremainopenproblems chiral-glass transition [8–10]. In the Heisenberg case,
t
a [1]. A simple Ising model has widely been used as a “re- while previous numerical works agreed in that the stan-
m alistic”SGmodelinthestudies,e.g.,ofthecriticalprop- dard SG order occurred only at T = 0, the question
- erties ofthe SGtransition,or ofthe issue ofwhether the whether there really occurs a finite-temperature chiral-
d
SGstateexhibitsaspontaneousreplica-symmetrybreak- glassorderhasremainedinconclusive[3,4]. Veryrecently,
n
ing (RSB). One should bear in mind, however, that the an off-equilibrium Monte Carlo simulation by one of the
o
magnetic interactions in many SG materials are nearly authorshas givenevidence for the occurrence of a finite-
c
[ isotropic, being well described by an isotropic Heisen- temperature chiral-glass order in the 3D Heisenberg SG

berg model, in the sense that the magnetic anisotropy is [5]. However,thefullcriticalpropertiesofthechiral-glass
1
considerably weaker than the exchange interaction. In transition as well as the properties of the chiral-glassor-
v
apparentcontrastto experiments, numericalsimulations deredstateitself,particularlythequestionofthepossible
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2 have indicated that the standard spin-glass order occurs RSB,stillremainslargelyunclear. InthisLetter,weper-
1 only at zero temperature in the three-dimensional (3D) form extensive equilibrium Monte Carlo simulations of
1 Heisenberg SG [2–6]. Although the magnetic anisotropy a 3D Heisenberg SG in order to determine the detailed
0 inherenttorealmaterialsis ofteninvokedto explainthis static and dynamic critical properties and to clarify the
0
apparent discrepancy with experiments, it still remains nature of the chiral-glass ordered state.
0
puzzling that no detectable sign of Heisenberg-to-Ising Our model is the classicalHeisenberg model on a sim-
/
t crossoverhasbeenobservedinexperimentswhichisusu- plecubiclatticewithN =L3spinsdefinedbytheHamil-
a
m ally expected to occur if the observedfinite-temperature tonian,
transition is caused by the weak magnetic anisotropy
- H=− J S ·S , (1)
d [1,2]. ij i j
n Inordertosolvethisapparentpuzzle,achiralitymech- Xhiji
o anism of experimentally observed spin-glass transitions where S =(Sx,Sy,Sz) is a three-component unit vec-
c i i i i
: was proposed by one of the authors [3]. This scenario tor, and the sum runs over all nearest-neighbor pairs.
v
is based on the assumption that an isotropic 3D Heisen- The interactionsJ arerandomGaussianvariableswith
ij
i
X bergSGexhibits afinite-temperaturechiral-glass transi- zeromeanandvarianceJ. The localchiralityatthei-th
tion without the conventional spin-glass order, in which site and in the µ-th direction, χ , is defined for three
r iµ
a only spin-reflection symmetry is broken with preserving neighboring spins by the scalar [2,4],
spin-rotation symmetry. Chirality is an Ising-like multi-
χ =S ·(S ×S ), (2)
spinvariablerepresentingthe senseor the handednessof iµ i+eˆµ i i−eˆµ
the noncollinear spin structures induced by spin frustra- where eˆ (µ=x,y,z)denotes a unit lattice vectoralong
µ
tion. In this scenario, all essential features of many of the µ-axis.
real SG transitions and SG ordered states should be de- Monte Carlo simulation is performed based on the ex-
termined by the properties of the chiral-glass transition change MC method, sometimes called “parallel temper-
andofthe chiral-glassstatein the fully isotropic system, ing”, which turns out to be an efficient method for ther-
while the role of the magnetic anisotropy is secondary malizing systems exhibiting slow dynamics [7]. By mak-
which “mixes” the spin and the chirality and “reveals” inguseofthismethod,wehavesucceededinequilibrating
1
the system down to the temperature considerably lower eventocompletelyruleoutthepossibilityofonlyazero-
thanthoseattainedintheprevioussimulations[2,4]. We temperature transition with rapidly growing (e.g., ex-
runtwoindependent sequences ofsystems(replica 1 and ponentially growing) correlation length. Meanwhile, the
2)inparallel,andcomputeanoverlapbetweenthechiral calculated g shows a negative dip whose depth grad-
CG
variables in the two replicas, ually increases with increasing L, whereas its position
gradually shifts toward lower temperature. Note that,
1
q = χ(1)χ(2). (3) if the systems would not exhibit any finite-temperature
χ 3N iµ iµ transition, g (T;L → ∞) should be equal to zero at
iµ CG
X
any T > 0 and to unity at T = 0 (for the nondegen-
In terms ofthis chiraloverlapqχ, the Binder ratio ofthe erate ground state as expected for the present Gaussian
chirality is calculated by coupling). Since g (T;L→∞) is a single-valued func-
CG
tionof T,the existence ofa negative dip growingwith L
1 [hqχ4i] hardlyreconcileswiththeabsenceofafinite-temperature
g (L)= 3− , (4)
CG 2 [hq2i]2 transitionso long as this tendency persists, and suggests
χ !
the chiral-glass transition occurring at T = T > 0
CG
whereh···irepresentsthethermalaverageand[···]repre- at which gCG takes a negative value unlike the standard
sentstheaverageoverbonddisorder. FortheHeisenberg cases of the 3D Edwards-Anderson (EA) Ising model or
spin, one can introduce an appropriate Binder ratio in the infinite-range Sherrington-Kirkpatrick(SK) model.
terms of a tensor overlap q (µ,ν = x,y,z) which has
µν
32 =9 independent components [4],
1.0
L=16
1
q = S(1)S(2) (µ,ν =x,y,z), (5) L=12
µν N iµ iν 0.8 L=10
i L=8
X
L=6
via the relation, 0.6
G
S
1 µ,ν,δ,ρ[hqµ2νqδ2ρi] g0.4
g (L)= 11−9 . (6)
SG 2 ( [hq2 i])2
P µ,ν µν !
0.2 (a) spin
P
The lattice sizes studied are L =6,8,10,12 and 16 with
periodic boundary conditions. Equilibration is checked
0
by monitoringthe stability of the results againstatleast L=16
L=12
three-times longer runs for a subset of samples. Sample 0.8 L=10
average is taken over 1500 (L = 6), 1200 (L = 8), 640 L=8
L=6
(L = 10), 296 (L = 12) and 136 (L = 16) independent 0.6
bond realizations. Note that in the exchange MC simu-
lations the data atdifferent temperatures arecorrelated. CG0.4
g
Errorbarsareestimatedfromstatisticalfluctuationsover
0.2 (b) chirality
the bond realizations.
The size and temperature dependence of the Binder
0
ratios of the spin and of the chirality, g and g , are
SG CG
shown in Fig. 1(a) and (b), respectively. As can be seen
-0.2
from Fig. 1(a), gSG constantly decreases with increasing 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
T/J
L at all temperatures studied, suggesting that the con-
ventional spin-glass order occurs only at zero tempera- FIG.1. Temperature and size dependence of the Binder
ture, consistentwith the previousresults [2–6]. Fig.1(a) ratios of thespin (a) and of thechirality (b).
reveals that g for larger lattices (L = 10,12,16) ex-
SG
hibits an anomalous bending around T/J ≃ 0.15, sug-
More unambiguous estimate of T can be obtained
CG
gestingachangeintheorderingbehaviorinthistemper-
from the equilibrium dynamics of the model. Thus,
aturerange. Ascanbe seenfromFig.1(b),thecurvesof
we calculate both the spin and chirality autocorrelation
g fordifferentLtendtomergeforlargerLinthetem-
CG functions defined by
peraturerangewherethecurvesofg exhibitananoma-
SG
lousbending. Furthermore,onincreasingL,themerging
points gradually move toward higher temperatures, sug- 1
C (t)= [hS (t )·S (t+t )i], (7)
gestingthatthechiral-glasstransitionindeedoccursata s N i 0 i 0
i
finitetemperature. Sinceg fordifferentLdonotcross X
CG 1
hereatgCG >0,however,itisnotnecessarilyeasytoun- Cχ(t)= [hχiµ(t0)χiµ(t+t0)i], (8)
3N
ambiguously locate the chiral-glass transition point, or
i,µ
X
2
whereMCsimulationisperformedaccordingtothestan- The behavior of the chiral-glass order parameter, or
dard heat-bath updating here. The starting spin con- theassociatedchiral-glasssusceptibilityχ =3N[hq2i],
CG χ
figuration at t = t is taken from the equilibrium spin turns out to be consistent with this. In the inset of
0
configuration generated in our exchange MC runs. Fig. 3, we show the reduced chiral-glass susceptibility
χ˜ ≡ χ /χ¯4, normalized by the amplitude of the lo-
Monte Carlo time dependence of the calculated C (t) CG CG
s
calchiralityχ¯2 ≡(1/3N) [hχ2 i],versusthereduced
and Cχ(t) are shown in Fig. 2 on log-log plots at several i,µ iµ
temperatures for L=16. We found no significant differ- temperaturet≡|(T−TCG)/TCG |onalog-logplot. From
P
enceinthedataofL=12and16,andthefinite-sizeeffect the asymptotic slope of the data, the susceptibility ex-
is negligible in our time window. As can be seen from ponent is estimated to be γCG = 1.5±00..13. Note that the
Fig. 2(a), C (t) shows a downward curvature at all tem- estimated susceptibility exponent is significantly smaller
s
perature studied, suggesting an exponential-like decay thanthatofthestandard3DIsingEAmodel,γ ≃4[1d].
characteristic of the disordered phase, consistently with If we combine the present estimate of γCG with the
the absence of the standard spin-glass order. In sharp estimate of βCG from the off-equilibrium simulation of
contrast to this, C (t) shows either a downward curva- Ref. [5] and use the scaling relations,various chiral-glass
χ
ture characteristicofthe disorderedphase,oranupward exponents can be estimated to be α ≃ −1.7, βCG ≃ 1.1,
curvature characteristicof the long-rangeorderedphase, γCG ≃ 1.5, νCG ≃ 1.2 and ηCG ≃ 0.8. The dy-
dependingonwhetherthetemperatureishigherorlower namical exponent is estimated to be zCG ≃ 4.7 by us-
thanT/J ≃0.16,while justatT/J ≃0.16the linearbe- ing the estimated value of λ and the scaling relation
haviorcorrespondingtothepower-lawdecayisobserved: λ = βCG/zCGνCG. While the dynamical exponent zCG
See Fig. 2(b). Hence, our dynamical data indicates that comes rather close to the z of the 3D EA model, the
the chiral-glassorderwithoutthe standardspin-glassor- obtained static exponents differ significantly from those
der takes place at T /J = 0.160±0.005, below which of the 3D Ising EA model β ≃ 0.55, γ ≃ 4.0, ν ≃ 1.7
CG
a finite chiral EA order parameter qEA > 0 develops. and η ≃ −0.35 [1d], suggesting that the universality
CG
From the slope of the data at T = T , the exponent class of the chiral-glass transition of the 3D Heisenberg
CG
λ characterizing the power-law decay of C (t) ≈ t−λ is spinglassdiffers fromthatof the standard3DIsing spin
χ
estimatedto beλ=0.193±0.005. TheestimatedT is glass. Possible long-range and/or many-body nature of
CG
ingoodagreementwiththe previousestimateofRef.[5], the chirality-chirality interaction might be the cause of
T /J =0.157±0.01. this deviation.
CG
0.35
0.3 0.3
10
T/J=
0.100 0.25 G
) 0.2 0.109 ~χC
(Cts 00..112311 0.2 slope ~ 1.50
0.139 G
0.151 0.15
0.1 0.160 L=16 1
0.173 L=12 0.1 t 1
0.187 (a) spin 0.1 L=10
0.202 L=8
0.1 0.05
0
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
T/J
0.100 FIG.3. TemperatureandsizedependenceoftheGparam-
()Ctχ 0.01 00..110291 eter of the chirality defined by Eq. (9). The inset represents
0.131 a log-log plot of the reduced chiral-glass susceptibility versus
0.139 thereduced temperature t≡|(T −TCG)/TCG |.
0.151
0.160
0.173 (b) chirality Furtherevidenceofaphasetransitionisobtainedfrom
0.187
the behavior of the G parameter of the chirality defined
0.202
0.001 by
1 10 100 1000
t [MCS]
[hq2i2]−[hq2i]2
FIG.2. Log-log plots of thetime dependenceof theequi- G(L)= χ χ . (9)
librium spin (a) and chirality (b) autocorrelation functions [hqχ4i]−[hqχ2i]2
at several temperatures. The lattice size is L = 16 averaged
over 64 samples. In (b), the best straight-line fit is obtained While this quantity was originally introduced to repre-
at T/J =0.16, represented by thebroken line. sent the non-self-averaging character of the system [11],
3
Bokil et al argued that it was not necessarily so [12]. We note that, if P(q ) has a nontrivial structure as sug-
χ
Still, a crossing of G(L) for different L, if it occurs, can gested from Fig. 4, the denominator of Eq. (9) should
beusedtoidentifythetransition[12]. AsshowninFig.3, remainnonzeroat0<T <T . Then,ourdataofG(L)
CG
for T > T G(L) decreases with increasing L tending in Fig. 3 indicates that the chiral-glass state is non-self-
CG
to zero, while for T < T it tends to increase with L, averaging.
CG
thus lending further support to the existence of a phase In summary, spin-glass and chiral-glass orderings of
transition at T =TCG. the 3D Heisenberg SG are studied by Monte Carlo sim-
ulations. Our observation both on statics and dynamics
strongly suggests the existence of a stable chiral-glass
0.045
L=16 phase at finite temperatures without the conventional
0.04 L=12
L=10 spin-glass order. This fact strengthens the plausibility
0.035 T/J=0.10 L=8 of the chirality mechanism for experimentally observed
L=6
0.03 spin-glass transitions. The nature of the chiral-glass or-
dered state as well as of the critical phenomena are dif-
)qχ 0.025 ferentfromthoseofthe3DIsingSG,withstrongsimilar-
(P 0.02 ities to the system showing the one-step-like RSB, while
0.015 its exact nature and physical originhave remained to be
understood.
0.01
The numerical calculation was performed on the Fu-
0.005
jitsuVPP500atthesupercomputercenter,ISSP,Univer-
0 sityofTokyo,ontheHITACHISR-2201atthesupercom-
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 puter center, University of Tokyo, and on the CP-PACS
q
χ
computeratthe CenterforComputationalPhysics,Uni-
FIG. 4. Chiral-overlap distribution function below TCG.
versity of Tsukuba.
The temperature is T/J =0.1.
In Fig. 4, we display the distribution function of the
chiral-overlap defined by P(q′) = [hδ(q −q′)i] calcu-
χ χ χ
lated at a temperature T/J =0.1, well below the chiral-
glasstransitiontemperature. Theshapeofthecalculated
P(qχ)issomewhatdifferentfromtheoneobservedinthe [1] Forreviews on spin glasses, see e. g., (a) K.Binderand
standard Ising-like models such as the 3D EA model or A. P. Young, Rev. Mod. Phys. 58 (1986) 801; (b) K. H.
the mean-field SK model. As usual, P(qχ) has standard FischerandJ.A.Hertz,SpinGlassesCambridgeUniver-
“side-peaks”correspondingtotheEdwards-Andersonor- sity Press (1991); (c) J.A.Mydosh,Spin Glasses Taylor
der parameter ±qEA, which grow and sharpen with in- &Francis(1993);(d)A.P.Young(ed.),Spinglasses and
CG
creasing L. The extracted value of ±qEA coincides with random fields World Scientific,Singapore (1997).
CG
that evaluated from the relaxation of C (t). In addi- [2] J.A.Olive,A.P.YoungandD.Sherrington,Phys.Rev.
χ
tion to the side peaks, an unexpected “central peak” B34, 6341 (1986).
at q = 0 shows up for larger L, which also grows and [3] H. Kawamura, Phys. Rev. Lett. 68 (1992) 3785.; Int.
χ
sharpens with increasing L. This latter aspect, i.e., the Jour. Mod. Phys. 7, 345 (1996).
existence of a central peak, is a peculiar feature of the [4] H. Kawamura, J. Phys.Soc. Jpn. 64, 26 (1995).
chiral-glass ordered state never observed in the EA or [5] H. Kawamura, Phys.Rev.Lett. 80, 5421 (1998).
SK models. Since wedo notfind anysignofafirst-order [6] F. Matsubara, T. Iyota, and S. Inawashiro, Phys. Rev.
Lett. 67, 1458 (1991).
transition such as a discontinuity in the energy, the spe-
cific heat nor the order parameter qEA, this feature is [7] K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65,
CG 1604 (1996).
likely to be relatedto a nontrivialstructure in the phase
[8] H. Kawamura and M. Tanemura, J. Phys. Soc. Jpn. 60
space associated with the chirality. We note that this
(1991) 608.
peculiar feature is reminiscent of the behavior charac-
[9] H. Kawamura, Phys.Rev. B 51 (1995)12398.
teristic of some mean-field models showing the so-called
[10] J. Maucourt and D. R. Grempel, Phys. Rev. Lett. 80
one-step RSB[1b]. Indeed, the existence of a negative
(1998) 770.
dip in the Binder ratio g and the absence of a cross-
CG [11] E. Marinari, C. Naitza, G. Parisi, M. Picco, F. Ritort
ing of gCG at gCG > 0 are consistent with the occur- and F. Zuliani, Phys. Rev.Lett. 81, 1698 (1998).
rence of such one-step-like RSB [13]. Our data of P(q )
χ [12] H. Bokil, A.J. Bray, B. Drossel and M.A. Moore, cond-
are also not incompatible with the existence of a con- mat/9811080.
tinuous plateau between [−qCEGA,qCEGA] in addition to the [13] Wehavecheckedbynumericalsimulationsthatthesefea-
delta-function peaks. According to the chirality mech- tures are indeed seen in the three-state mean-field Potts
anism, such novel one-step-like RSB should be realized glass, amodelexactlyknowntoexhibitaone-stepRSB;
in the spin ordering of real Heisenberg-like spin glasses. K. Hukushimaand H. Kawamura, unpublished.
4