Nature of the phase transition of the three-dimensional isotropic Heisenberg spin glass

a r X i v :c o n d -m a t /0504016v 2 [c o n d -m a t .d i s -n n ] 11 A p r 2005

Nature of the phase transition of the three-dimensional isotropic Heisenberg spin glass

Koji Hukushima 1,?and Hikaru Kawamura 2,?

1

Department of Basic Science,University of Tokyo,3-8-1Komaba,Meguro-ku,Tokyo 153-8902,Japan

2

Department of Earth and Space Science,Faculty of Science,Osaka University,Toyonaka,Osaka 560-0043,Japan

(Dated:February 2,2008)

Equilibrium properties of the three-dimensional isotropic Heisenberg spin glass are studied by extensive Monte Carlo simulations,with particular attention to the nature of its phase transition.A ?nite-size-scaling analysis is performed both for the spin-glass (SG)and the chiral-glass (CG)orders.Our results suggest that the model exhibits the CG long-range order at ?nite temperatures without accompanying the conventional SG long-range order,in contrast to some of the recent works claiming the simultaneous SG and CG transition.Typical length and time scales which represent a crossover from the spin-chirality coupling regime at short scales to the spin-chirality decoupling regime at long scales are introduced and examined in order to observe the true asymptotic transition behavior.On the basis of these crossover scales,discussion is given concerning the cause of the discrepancy between our present result and those of other recent numerical works.

PACS numbers:75.10.Nr,05.10.Ln,05.70.Fh,64.60.Fr

I.INTRODUCTION

Spin glasses (SGs)have attracted the attention of re-searchers both in experiments and theory as a prototype of complex systems with quenched randomness.1SGs are random magnets in which magnetic ions interact with each other either ferromagnetically or antiferromagneti-cally,depending on their positions.Most of theoretical works have been so far devoted to the minimal SG model,i.e.,the Ising Edwards and Anderson (EA)model.After a discussion early in the 1980s,it is now widely believed that a three-dimensional (3D)Ising SG model exhibits a SG phase transition at a ?nite https://www.sodocs.net/doc/297143073.html,rge-scale Monte Carlo (MC)simulations presented an evidence for the ?nite-temperature SG ordering.2,3Subsequently,the critical exponents evaluated by MC simulations were con-sistently compared with those evaluated experimentally for the Ising-like SG compound FeMnTiO 3.

Compared to the Ising case,the nature of the phase transition of continuous spin systems such as XY and Heisenberg SGs are still poorly understood.Since many SG magnets including canonical SG possess only weak magnetic anisotropy,an isotropic Heisenberg SG model,rather than the strongly anisotropic Ising model,is ex-pected to be a realistic model of SG magnets.Exper-imentally,an equilibrium SG phase transition has been established in real SG materials via measurements of the divergent non-linear susceptibility,etc .In sharp con-trast to experiments,earlier theoretical studies on the Heisenberg SG model indicated that the standard SG long-range order occurred only at zero temperature in three dimensions.4,5,6,7,8

In order to solve this apparent puzzle,a chirality mech-anism of experimentally observed SG transitions was pro-posed by Kawamura 9,10.This scenario is based on the assumption that an isotropic 3D Heisenberg SG exhibits a ?nite-temperature chiral-glass (CG)transition without

the conventional SG order.In terms of symmetry,among the global symmetries of the isotropic Hamiltonian,only the Z 2spin-re?ection (or spin-inversion)symmetry as-sociated with the chirality is spontaneously broken with keeping the SO (3)spin-rotation symmetry.Indeed,some numerical studies 11,12claimed that the standard SG or-der associated with the freezing of the Heisenberg spin occurred at a temperature lower than the CG transition temperature,i.e.,T SG

In this chirality scenario of experimental SG transi-tions,essential features of many of the real SG tran-sition and of the SG ordered state are determined by the properties of the CG transition and of the CG state of the fully isotropic system .The role of the magnetic anisotropy is secondary which re-couples the spin to the chirality and reveals the CG transition in the chiral sector as an anomaly in experimentally accessible spin-related quantities.The scenario successfully explained the phase diagram under magnetic ?elds observed by the recent nu-merical simulation 13,14and experiments.15

More recently,however,some researchers argued a pos-sibility that in the 3D Heisenberg SG model the spin ordered at a ?nite temperature simultaneously with the chirality,i.e.,T SG =T CG >0.16,17,18,19,20Thus,the na-ture of the ordering of the 3D Heisenberg SG,as well as the validity of the chirality scenario,is now under de-bate.Under such circumstances,it is highly interesting to perform further extensive numerical studies of the 3D Heisenberg SG in order to clarify the true nature of its or-dering.In the present study,we investigate both the SG and CG orderings of the model by means of a large-scale equilibrium MC simulation.

Interestingly,recent experiments reported on a quali-

2 tative di?erence in aging phenomena between a canonical

Heisenberg-like SG and an Ising-like SG.21We also ex-

pect that the full understanding of the equilibrium prop-

erties of the3D Heisenberg SG will also give a valuable

insight into these o?-equilibrium properties of SGs.

The article is organized as follows.In Sec.II,we give a

background of the present numerical study.In Sec.II A,

we explain?rst the basics of the chirality mechanism.

In Sec.II B,we introduce the crossover length and time

scales beyond which the spin and the chirality are decou-

pled with each other.These length and time scales are

crucially important in the chirality mechanism,and are

also essential in properly interpreting the numerical data

of MC simulations.In Sec.III,we explain the model

and the MC method employed.In Sec.IV,we introduce

various physical quantities measured in our MC simula-

tions,while the results of our simulations are presented

in Sec.V.In view of our MC results,we examine and

discuss in Sec.VI the recent numerical results on the3D

Heisenberg SG by other authors.Finally,we present a

brief summary of the results in section VII.

II.BACKGROUND

In this section,we wish to give a background of the

present numerical study of the3D isotropic Heisenberg

SG.First,we explain the basics of the chirality mecha-

nism of experimental SG transition as proposed in Refs.9

and10.Then,we explain the notion of the spin-chirality

decoupling,together with the crossover length and time

scales which play a crucially important role in the chi-

rality mechanism and are also essential in properly inter-

preting the numerical data.

A.Chirality mechanism

Chirality is an Ising-like multi-spin variable represent-

ing the sense or the handedness of noncollinear spin struc-

tures induced by spin frustration.In frustrated magnets

with continuous spins,the chirality often plays an essen-

tial role in their magnetic ordering.The local chirality

χiμat the i th site in theμ-direction may be de?ned by

χiμ= S i+?e

μ·( S i× S i??e

μ

),(1)

?eμ(μ=x,y,z)being a unit lattice vector along theμaxis. This quantity is often called a scalar chirality:It takes a non-zero value only when the three neighboring spins take the non-coplanar con?guration in spin space,while it vanishes for the collinear or the coplanar spin con?g-uration.The chirality de?ned above is a pseudoscalar variable since it is invariant under the global SO(3)spin rotations but changes its sign under the global Z2spin re?ections or inversions.

The chirality mechanism of Refs.9and10takes the fol-

lowing two-step strategy in explaining the real SG transi-tion:The?rst step concerns with the property of the fully isotropic Heisenberg SG,an idealization of experimental SG materials.The chirality scenario claims that the fully isotropic Heisenberg SG exhibits a?nite-temperature CG transition without the conventional SG long-range or-der.The CG transition breaks only the Z2spin-re?ection symmetry with keeping the SO(3)spin-rotational sym-metry.The occurrence of the CG transition necessarily entails the spin-chirality decoupling.

Obviously,such a scenario does not apply to the in?-nite dimensional limit,i.e.,to the mean-?eld Heisenberg Sherrington-Kirkpatrick(SK)model,in which the spin itself,not the chirality,behaves as an order parameter of the transition.Due to the noncoplanar nature of the spin con?guration in the SG state,the SG long-range order trivially accompanies the CG long-rage order,whereas the opposite is not necessarily true.One should note that,in the conventional case where the spin variable is a proper order parameter of the transition as in the case of the SK model,the chirality,which is given by the multiple of the spin,exhibits a less singular behavior than the spin at the SG transition.In fact,the chirality shows only moderate behavior at the SG transition of the mean-?eld Heisenberg SK model in which the spin,not the chirality,is the order parameter of the transition.22 In contrast to the mean-?eld model or the high-dimensional Heisenberg SG models,the problem could be very non-trivial in lower dimensions where the order-parameter?uctuation might change the nature of order-ing dramatically.At present,there seems to be no con-sensus about the lower critical dimension d LCD

SG

of the SG order,while the corresponding upper critical dimension is expected to be six.The CG order,if exists,may emerge

slightly above,at,or below d LCD

SG

.

It has been proved that the SG long-range order does not exist at any?nite temperature in the two di-mensional Heisenberg SG.23The numerical domain-wall renormalization-group calculation as well as the MC sim-ulation suggested that both the spin and the chirality or-dered only at zero temperature9.Interestingly,however, the estimated SG and CG correlation-length exponents at this T=0transition di?er signi?cantly from each other,i.e.,νCG>νSG.24This implies that in2D the spin and the chirality are decoupled at long length scale, the chirality dominating the long-length behavior.

In view of these transition behaviors of the Heisenberg SGs,it appears likely that the principal player in long-scale phenomena changes from the spin to the chirality as the spacial dimensionality is decreased.Thus,the be-havior in dimension three is the current issue,which is the subject of the present work.

The second step of the chirality mechanism concerns with the e?ect of the random magnetic anisotropy which inevitably exists in real SG magnets.The random anisotropy energetically breaks the SO(3)spin-rotation symmetry in the Hamiltonian,retaining the Z2inversion

3

c o r r e l a t i o n l e n g t h /t i m e

Temperature

L × or t ×

CG SG ×Chiral Glass Spin Glass

FIG.1:A schematic ?gure of the crossover between the spin-glass and chiral-glass correlation lengths (correlation times)expected in the chirality mechanism.According to the chiral-ity mechanism,the CG correlation length (correlation time)diverges toward the CG transition temperature T =T CG ,while the SG one diverges at a lower temperature T =T SG

symmetry only.When the anisotropic system exhibits the CG long-range order with spontaneously breaking the Z 2inversion symmetry,there no longer remains any global symmetry degree of freedom to leave the system in the paramagnetic phase.Hence,once the CG order oc-curs in the presence of the random anisotropy,the spin degree of freedom also behaves like the chirality.This is the spin-chirality recoupling due to the random magnetic anisotropy.

Such an anomaly revealed in the spin sector via the random magnetic anisotropy can be detected experimen-tally by standard magnetic measurements e.g.,as a di-vergence of the nonlinear susceptibility etc ,whereas the CG long-range order is di?cult to observe experimen-tally.

We note that in this mechanism the anisotropy plays only a secondary role:The anisotropy certainly reduces the symmetry of the Hamiltonian relative to the fully isotropic system,but does not change the broken sym-metry of the transition .The critical properties of the CG transition and of the low-temperature CG phase are expected to be not a?ected by the magnetic anisotropy,which,however,are now directly observable via the stan-dard spin-related quantities.This chirality scenario pre-dicts that experimentally observed SG transitions belong to the same universality class as that of the the CG tran-sition of the fully isotropic model.It is thus highly inter-esting to clarify the nature of the phase transition of the ideal isotropic Heisenberg SG.

B.Spin-chirality decoupling/coupling scenario

As mentioned above,in lower dimensions,a relevant degree of freedom which dominates the long-scale phe-nomena might well change from the spin to the chirality.The chirality scenario expects that in 3D there exists a crossover temperature T ×which separates the two tem-perature regimes,as illustrated in Fig.1.In the higher temperature regime,the SG correlation length is longer than the CG correlation length,dominating the long-scale phenomena.This is simply due to the fact that the sensible de?nition of the local chirality requires the development of the spin short-range order of at least a few lattice spacings.As the temperature is decreased,both the SG and CG correlation lengths grow,but at di?erent rates,so that the CG correlation length eventu-ally outgrows the SG correlation length at the crossover temperature T ×.An example of such a crossover behav-ior between the spin and the chiral correlations can be seen explicitly in a certain toy model:See Fig.10of Ref.25.Then,the relevant degree of freedom for the long-scale behavior changes at T ×from the spin to the chirality.Below T ×,the long-scale phenomena are gov-erned by the CG correlation,not by the SG one.This is the spin-chirality decoupling expected to occur in the fully isotropic model.

Let us discuss in some detail the ?nite-size e?ect in-herent to the simulation data in the critical region.The situation here is not simple because the system has two length scales,each associated with the spin and with the chirality.Suppose that the CG transition occurs at T =T CG without the conventional SG long-range or-der.Then,the crossover temperature T ×at which the CG correlation length outgrows the SG correlation length should be located somewhat above T CG :See Fig.1.A necessary condition for detecting the spin-chirality de-coupling is that the measurement temperature lies be-low T ×.It is,however,not enough.At a temperature below T ×,one needs to probe the system beyond the crossover length above which the spin-chirality decou-pling becomes eminent.Thus,a large-size simulation exceeding the crossover length is required in order to detect the spin-chirality decoupling.Unfortunately,the crossover length scale is unknown a priori ,and is to be in-vestigated by numerical simulations.A natural criterion might be that it is given by the SG correlation length at the crossover temperature T ×as shown in Fig.1.Even in the CG ordered phase,the spin-chirality decoupling might hardly be observable at the length scale below the crossover length.Rather,it is natural to expect that the trivial spin-chirality coupling is observed below the crossover length scale because the chirality is a composite operator of the spin on the short scale of lattice spacing not independent of the spin,roughly being χ≈S 3.

In the CG critical region,the chirality-related quanti-ties should exhibit the true asymptotic critical behavior,e.g.,a power-law singularity characterized by the asso-ciated CG exponents.At short length scales below the

4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1a u t o c o r r e l a t i o n f u n c t i o n

time

Spin Glass

Chiral Glass

FIG.2:A schematic ?gure of the time evolution of the spin-glass and chiral-glass two-time autocorrelation functions.

crossover length scale,due to the trivial coupling between

the spin and the chirality,even the spin-related quanti-ties are expected to exhibit the similar critical behavior to the chirality-related https://www.sodocs.net/doc/297143073.html,ly,up to the crossover length scale,it seems as if the SG order devel-oped as a long-range order.It is intrinsically di?cult at shorter length scales to distinguish such a pseudo-critical behavior induced by the CG long-range order from the true SG long-range order.Hence,it is crucially impor-tant to estimate the crossover length scale and to study the long-scale behavior of the system beyond this length scale.

Essentially the same argument also applies to the case of the temporal scale.As an example,we discuss here the behavior of the autocorrelation functions based on the notion of the crossover time scale.Figure 2shows the schematic representation of the behavior of the SG and CG autocorrelation functions below T CG expected from the spin-chirality coupling/decoupling picture.In the long-time limit,the chirality autocorrelation func-tion is expected to saturate to a certain ?nite value after an initial fast decay,while the spin autocorrelation func-tion is expected to decay toward zero asymptotically.A comment is in order concerning the transient behavior of the spin autocorrelation function at time scales shorter than the crossover time scale:At shorter time scales,the spin autocorrelation function might exhibit the pseudo-ordering feature dictated by the CG one through the trivial spin-chirality coupling.This might well lead to a hump-like pseudo-ordering structure in the time depen-dence of the spin autocorrelation function as shown in Fig.2,which,however,does not persist in the long-time limit beyond the crossover time scale.This means that,in order to properly discuss the true asymptotic behav-ior of the dynamics of the model,a particular attention should be paid to the crossover time scale.

III.

THE MODEL AND THE MONTE CARLO

METHOD

We study a classical Heisenberg model de?ned by the Hamiltonian,

H ( S

)=?

ij

J ij S

i · S j ,(2)

where S i =(S x i ,S y i

,S z i )is a three-component unit vector,and the summation runs over all nearest-neighbor pairs.The lattice is a simple-cubic lattice with the total number of N =L 3sites.The nearest-neighbor couplings J ij take the values ±J randomly with equal probability.Periodic boundary conditions are imposed for all the directions.The lattice sizes studied are L =8,12,16and 20,where the sample average is taken over 976(L =8),964(L =12),280(L =16),and 32(L =20)independent bond realizations.

We perform an equilibrium MC simulation of the model.In our simulation,we make use of the exchange MC method,26which is also called parallel tempering.27In the exchange MC method,one MC step consists of two elementary updates,a standard single-spin heat-bath ?ip 6and an exchange trial of spin con?gurations at neigh-boring temperatures.The latter reduces the slow re-laxation at low temperatures with the help of the high-temperature fast dynamics.The method has turned out to be quit e?cient in thermalizing a wide class of hardly-relaxing systems such as SG systems and proteins.We ensure equilibration by checking that various observables attain stable values,no longer changing with the amount of MC steps:See Refs.22and 28for further details of the equilibration procedure.Our MC simulations have been performed up to the size L =20and up to the tem-perature T/J =0.15.This could be achieved only by using the exchange MC method.The numbers of tem-perature points used in our exchange MC method are 32for L =8,12,16,and 48for L =20.

Error bars are estimated via sample-to-sample ?uctu-ations for the linear quantities such as the order parame-ters,and by the jackknife method for the non-linear quan-tities such as the Binder parameter and the correlation length mentioned below.

IV.PHYSICAL OBSER V ABLES

In the present section,we introduce various physical quantities observed in our simulations,and discuss some of their basic properties.

In glassy systems,it is often convenient to de?ne as an order parameter an overlap variable between two in-dependent systems with the same Hamiltonian.For the Heisenberg spin,the overlap may be de?ned as a tensor variable between the μand νcomponents (μ,ν=x,y,z )

5 of the Heisenberg spin by

qμν=

1

3N iμχ(1)iμχ(2)iμ.(4)

The squared SG order parameter is then given by

q(2)SG= μνq2μν ,(5)

where ··· denotes a thermal average and[···]denotes

an average over the bond disorder.The corresponding

squared CG order parameter is de?ned by

q(2)

CG

= q2χ χ4,(6)

which is normalized by the mean-square amplitude of the

local chirality,

3N

N

i μ[ χ2iμ ].(7)

The local chirality amplitude remains non-zero only when the spins have a non-coplanar structure locally.This quantity weakly depends on the temperature,in contrast to the Heisenberg spin variable whose amplitude is?xed to be unity by de?nition.In the high-temperature sym-metric phase,these SG and CG order parameters are essentially equivalent to the associated SG and CG sus-

ceptibilities de?ned byχSG=Nq(2)

SG andχCG=3Nq(2)

CG

,

respectively.

A standard?nite-size scaling of the second-order tran-sition for the equilibrium SG and CG order parameters takes the form

q(2)～L?(1+η)f |T?T c|L1/ν ,(8) whereνis the exponent of the correlation length,andηis the exponent describing the decay of the correlation function at the critical point T=T c.At T c,the order parameter decays as a power law with the size L,

q(2)∝L?(1+η).(9) One often uses the Binder parameter to estimate the critical temperature.In the Heisenberg SG,the Binder parameter for the SG order is de?ned by

g SG=1

[ q2 ]2 ,q2= μ,νq2μν,(10)

while that for the CG oder is de?ned by

g CG=

1

[ q2χ ]2 .(11)

In the thermodynamic limit,these Binder parameters

are normalized to unity in the non-degenerate ordered

state,and to zero in the high-temperature disordered

state.Since the Binder parameter is a dimensionless

quantity,and the dimensionless quantity should be size-

independent at the critical temperature T c,the Binder

parameters of di?erent system sizes plotted as a function

of temperature should yield a crossing or merging point

at T c.

In terms of the k-dependent overlap variable,one can

de?ne the Fourier-transformed two-point CG and SG cor-

relation functions.For the CG,the k-dependent chiral-

overlap is de?ned by

qχ( k)=

1

N

N

i=1S(1)iμS(2)iνexp(i k· r i),(14)

whereas the Fourier-transformed SG correlation function

is de?ned by

q(2)

SG

( k)= μν|qμν( k)|2 .(15)

Via these CG and SG Fourier-transformed correlation

functions,the associated CG and SG?nite-system corre-

lation lengths are de?ned by

ξ=

1q(2)( 0)

6 the overlap.The chiral-overlap distribution is de?ned by

P(q′χ)= δ(q′χ?qχ) .(17)

The squared CG order parameter q(2)CG de?ned above

is the second moment of the chiral-overlap distribution

function.

The spin-overlap distribution is de?ned originally in

the tensor space with3×3=9components.To make

the quantity more easily tractable,one may de?ne the

diagonal spin overlap which is the trace of the original

tensor overlap,and introduce the associated diagonal-

spin-overlap distribution by

P(q diag)= δ q diag? μ=x,y,z qμμ .(18)

This distribution function is symmetric with respect to

q diag=0,and is expected to be a Gaussian distribu-

tion around q diag=0in the high-temperature disor-

dered phase.Re?ecting the fact that the diagonal-spin-

overlap is not invariant under the global O(3)spin rota-

tion,P(q diag)in the possible SG ordered phase develops a

nontrivial shape,not just consisting of the delta-function

peaks related to q EA,even when the ordered state is

a trivial one simply described by a self-overlap q EA.22

If the possible SG ordered state accompanies a replica-

symmetry breaking(RSB),further nontrivial structures

would be added to P(q diag).Meanwhile,it is recently

shown in Ref.22that,in the possible SG ordered state,

the diverging peaks corresponding to the self-overlap nec-

essarily appears in P(q diag)at q diag=±1

[ q2χ ]2

,(19)

while for the SG order,

A SG(T)≡

[ q2 2]?[ q2 ]2

[ q4χ ]?[ q2χ ]2,(21)

while for the SG order,

G SG(T)≡

[ q2 2]?[ q2 ]2

9

(1?g SG)G SG.(24)

These relations indicate that,so long as the Binder pa-

rameter g takes any value di?erent from unity in the or-

dered phase,a non-zero A necessarily means a non-zero

G.By contrast,if the Binder parameter g takes a value

unity in the ordered phase,a non-zero A may or may not

mean a non-zero G.

Information about the equilibrium dynamics can be

obtained from the spin and chiral two-time autocorrela-

tion functions de?ned by

C s(t)=

1

3N iμ[ χiμ(t0)χiμ(t+t0) ],(26)

where the time evolution in our MC simulation is made

according to the standard heat-bath updating not accom-

panying the exchange process.Initial spin con?gurations

at t=t0are taken from equilibrium spin con?gurations

generated in our exchange MC runs.Below the transition

temperature T c(if any),these autocorrelation functions

converge in the long time limit to the Edwards-Anderson

SG and CG order parameters,whereas above T c these au-

tocorrelation functions decay exponentially toward zero

with a characteristic correlation time,which diverges as

the temperature T approaches T c.Just at T c,the auto-

correlation functions exhibit a power-law decay,

C(t)～t?β/zν,(27)

where z is the dynamical critical exponent.These fea-

tures are described by the standard bulk dynamical scal-

ing form,

C(t)～|T?T c|βf(t|T?T c|zν),(28)

where f(x)is a scaling function whose asymptotic forms

for x?1and x?1are x?β/zνand exp(?x),respec-

tively.

V.THE NUMERICAL RESULTS

In the present section,we show our MC results of the

three-dimensional isotropic Heisenberg SG model.

7

-0.4-0.2 0 0.2 0.4 0.6

0.8 1 0.1 0.15

0.2

0.25

0.3

0.35

0.4 0.45

0.5

g C G

(a) Chirality

L=8L=12L=16L=20

0 0.2 0.4 0.6 0.8 1

0.1

0.15 0.2 0.25

0.3 0.35 0.4 0.45

g S G

T/J

(b) Spin

L=8L=12L=16

L=20

FIG.3:The temperature and size dependence of the chiral-glass Binder parameter;upper ?gure (a),and of the spin-glass Binder parameter;lower ?gure (b),of the 3D ±J Heisenberg SG.

A.Binder parameter

In Fig.3,we show the temperature and size depen-dence of the Binder parameters both for the chirality;upper ?gure (a),and for the spin;lower ?gure (b).As can be seen from Fig.3(a),a crossing of the CG Binder parameter g CG of di?erent L is observed on the negative side of g CG ,not on the positive side as in the standard cases.With increasing L ,the crossing temperature grad-ually shifts toward lower temperatures.A behavior sim-ilar to this has also been observed in the Binder param-eter of other models,including the Heisenberg SG 12,22and the mean-?eld SG 33,34.In particular,in a class of mean-?eld SG models exhibiting a one-step RSB,the Binder parameter at the transition point T c takes a neg-ative value,sometimes even negatively divergent.33,34It implies that the temperature at which the Binder pa-rameter for ?nite L takes a minimum,a dip tempera-ture T dip (L ),approaches the critical temperature,i.e.,T dip (L )→T c as L →∞.Recently,this method of estimating the bulk transition temperature was success-fully applied to the Heisenberg SG.13,22In Fig.4,we plot

00.050.10.150.20.25

0.30.350.4T d i p

1/L

FIG.4:The dip temperature of the chiral-glass Binder pa-rameter g CG is plotted against 1/L .The solid line represents a linear ?t of the data.Its extrapolation to the L →∞limit gives an estimate of the bulk chiral-glass transition tempera-ture,T CG /J =0.194(5).

T dip (L )against 1/L .An extrapolation to the thermody-namic limit 1/L →0gives us an estimate of the bulk CG transition temperature,T CG /J =0.194(5).

By contrast,as shown in Fig.3(b),the SG Binder pa-rameter g SG monotonically decreases toward zero with increasing L at all temperatures studied.There is no signature of the transition in the investigated tempera-ture range,no negative dip nor the crossing,in contrast to the CG Binder parameter.This suggests that the SG transition temperature,if any,is located at a temperature lower than the temperature range studied here.Fig.3(b)reveals,however,that an anomalous bend appears in g SG for larger sizes L ≥16at around T/J ?0.22,close to the CG transition temperature,although g SG never becomes size-invariant at any temperature,as it should have been in a second-order transition.The reason why g SG ex-hibits such an anomalous bend around T CG might be understood as follows:At the CG transition,a re?ection symmetry is spontaneously broken and the entire phase space is divided into ergodic components,in each of which a proper-rotational symmetry is still preserved.As a re-sult,the ordering behavior of the Heisenberg spin would change at T CG ,though the spin itself does not order even below T CG .We note that a similar bend in g SG has also been observed in the two-dimensional Heisenberg SG 24where the absence of a ?nite-temperature SG transition has been well established.23

It is sometimes argued in the literature that the Binder-parameter analysis might not work in the SG problem.Such a suspicion might partly be based on the observation that only weak merging behavior was observed at the SG transition temperature of the three-dimensional EA Ising model which is believed to exhibit a ?nite-temperature SG transition.2,3,35As long as the SG long-rage order really sets in at ?nite temperatures,however,it is hardly conceivable that the Binder pa-

8

rameter for asymptotically large lattices exhibits a non-singular behavior only.In particular,the Binder param-eter should become scale-invariant at the SG transition point,so long as the transition is continuous.Indeed,in a recent MC simulation of the mean-?eld Heisenberg SG,22which is known to exhibit a non-zero SG long-range order below T SG ,a clear crossing of the SG Binder parameter g SG has been

observed at T SG ,in sharp contrast to our present data of Fig.3(b).

Then,one might argue that the ?nite-size e?ect would be signi?cant here in g SG and the large-L asymptote might still be far away.One sees from Fig.3(a),however,that the CG Binder parameter g CG for our two largest sizes L =16and 20exhibits an almost size-invariant behavior at and below T CG .If the Heisenberg spin or-ders simultaneously with the chirality,and if the spin is the order parameter of the transition and the chirality is only composite (secondary),it seems a bit hard to under-stand why the chirality exhibits an almost scale-invariant near-critical behavior for L ≥16,while the Heisenberg spin still exhibits an o?-critical scale-dependent behavior.Hence,the behavior of g SG observed in Fig.3(b)remains to be resolved if the occurrence of a ?nite-temperature SG transition is to be accepted in the investigated tem-perature range.

B.Order parameter

In Fig.5,we show the size dependence of the squared

CG and SG order parameters,q (2)

CG and q (2)SG ,for vari-ous temperatures.In the upper ?gure (a),a double-logarithmic plot of the CG order parameter q (2)

CG is shown against the system size L .One generally expects that at

T c the data of q (2)

CG should lie on a straight line.In fact,

the data of q (2)

CG show a clear straight-line behavior around T/J =0.19,which is close to the CG transition temper-ature obtained by our analysis of g CG .The critical-point decay exponent ηCG can be estimated from the slope of this straight line,yielding 1+ηCG ～1.8.At higher tem-peratures,a deviation from the straight line,a downward trend,is observed indicative of the disordered phase.At lower temperatures,particularly at our lowest tempera-ture simulated T/J =0.15,the data of q (2)

CG show a clear upward trend.This suggests that this temperature is in-deed below T CG ,and that the low-temperature phase is

a rigid one characterized by a non-zero q (2)

CG ,not likely to be a critical phase like the Kosterlitz-Thouless (KT)phase.

In Fig.5(b),a double-logarithmic plot of the corre-sponding SG order parameter q (2)

SG is shown against the system size L .Again,the data are expected to lie on a straight line at the critical SG transition temperature,if any.Such a straight-line behavior,however,is found only at our lowest temperature studied T/J =0.15,whereas q (2)

SG never exhibits an upward trend characteristic of the

10?3

10?2

10?1

q (2)C G

L

10?1

20

16

12

8

6

q (2)S G

L

FIG.5:Double-logarithmic plot of the squared chiral-glass

order parameter q (2)

CG ;upper ?gure (a),and of the squared

spin-glass order parameter q (2)

SG ;lower ?gure (b),as a function of the system size L for several temperatures around the ex-pected chiral-glass transition temperature.Straight lines are drawn by connecting the two data points of L =8and 12at each temperature.

long-range ordered phase at any temperature studied ,in

sharp contrast to the behavior of q (2)

CG .At temperatures higher than T/J =0.15,including the one at the CG

transition temperature T/J ?0.19,the data of q (2)

SG show a linear behavior for smaller sizes,which gradu-ally changes into a downward trend for larger sizes.This can simply be interpreted as a size-crossover which oc-curs around the length scale of the SG correlation length at each temperature.We note that such a size-crossover is clearly discernible even at a temperature T/J =0.17which is below T CG .The length scale of the crossover,comparable to the spin correlation length,grows as T decreases,and it is considered to exceed our largest size L =20at around T/J =0.15.

This observation strongly suggests that the standard SG transition temperature of the model is lower than T/J =0.15,and that,at least within the temperature

range 0.15<～T <～0.19,solely the CG long-range order

9

-0.5-0.4-0.3-0.2-0.100.10.2

0.30.40.5

T/J

FIG.6:The curvatures of the L -dependence of the squared

spin-glass order parameter q (2)

SG (open symbols),and of the

squared chiral-glass order parameter q (2)

CG (?lled symbols),are plotted versus the temperature.The curvature is expected to be zero at the respective transition temperature.

exists without the standard SG long-range order,i.e.,one has T CG >T SG .

To make the situation more pronounced,we esti-mate following Refs.36and 37the curvatures of the L -dependence of the two order parameters,q (2)SG and q (2)

CG ,via second-order polynomial ?ts to the data of Figs.5.The curvature is expected to be zero at the respective transition temperature.As shown in Fig.6,the curva-ture for the CG crosses the zero-axis around T/J ?0.19,while that for the SG does not cross the zero-axis there,but marginally touches on it at a lower temperature,T/J ?0.15.The result indicates that the two transi-tion temperatures,T SG and T CG ,are well separated.

C.

Finite-size scaling of the order parameter

In order to estimate the correlation-length exponent νassociated with the CG transition,we apply the stan-dard ?nite-size scaling analysis to the squared CG or-der parameter q (2)

CG based on Eq.(8).By taking |T ?T CG |/T CG L 1/νas the scaling variable,the best data col-lapse is obtained with T CG /J =0.19,νCG =1.2and ηCG =0.8.As shown in Fig.7,the data both below and above T CG scale fairly well.If |1/T ?1/T CG |T CG L 1/νis taken as the scaling variable,on the other hand,a slightly larger value of ν,i.e.,νCG =1.4and ηCG =0.7,is preferred.The observed di?erence in the best values of the exponents might be due to the correction to scaling.Thus,we ?nally quote T CG /J =0.19(1),νCG =1.3(2)and ηCG =0.8(2).The error bar is estimated by ex-amining the quality of the ?ts with varying the scal-ing parameters.The estimated values of critical expo-nents are compatible with the previous values obtained for the Heisenberg SG model but with the Gaussian bond

0.1

1

10

-0.8-0.6-0.4-0.2

0.2

0.4 0.6 0.8 1

q (2)C G

L 1+η(T ?T CG )L

1/ν

T CG /J=0.19ν=1.2η=0.8

Chirality

L=8L=12L=16L=20

FIG.7:Finite-size-scaling plot of the squared chiral-glass order parameter.The best scaling is obtained with T CG /J =0.19,ν=1.2and 1+η=1.8.

distribution,12and also with those for the ±J Heisenberg SG under external ?elds 13,38.

After establishing the occurrence of a ?nite-temperature CG transition,we next wish to re-examine via the ?nite-size scaling analysis the issue whether the standard SG order occurs at the same temperature with the CG order or not.In Fig.8,we show a ?nite-size

scaling plot of the SG order parameter q (2)

SG ,assuming a simultaneous CG and SG transition with a common correlation length exponent,i.e.,we set T SG /J =0.19and νSG =1.2.Although the data turn out to scale well for smaller sizes,a signi?cant deviation from the scaling is seen for larger sizes and at lower temperatures.The quality of the scaling is not improved if one tries to ad-just νto somewhat larger values.A similar poor scaling behavior is also observed even when one instead chooses |1/T ?1/T CG |T CG L 1/νas the scaling variable,and tries to adjust the scaling parameters around νSG =1.4.The data for smaller sizes turn out to scale best with choos-ing ηSG =?0.1.These parameter values νSG =1.2and ηSG =?0.1are close to the values reported by Matsub-ara et al in Ref.39.Hence,for the SG order parameter,we have observed a pseudo-critical behavior for smaller sizes,as well as a systematic deviation from the scaling for larger sizes.If the observed deviations were due to the correction-to-scaling,the scaling should be better for larger sizes,which is opposite to our present observa-tion.Therefore,we do not consider the apparent scaling obtained for smaller sizes with T SG /J =0.19to be ac-ceptable as a true asymptotic scaling.

In Fig.9,we show a ?nite-size-scaling plot of the SG

order parameter q (2)

SG using the same data as in Fig.8,but now assuming a zero-temperature SG transition,i.e.,T SG =0and ηSG =?1.The value η=?1is generically expected for a zero-temperature transition with the non-degenerate ground state.As shown in Fig.9,the best data collapse is obtained by choosing νSG =2.2.If one

10

0.2 0.4 0.6 0.8 1 1.2

1.4 1.6-0.6

-0.4

-0.2

0.2 0.4 0.6

0.8

1

q (2)S G L

1+η(T ?T SG )L 1/ν

Spin

L=8

L=12L=16L=20

FIG.8:Finite-size-scaling plot of the squared spin-glass or-der parameter,assuming a simultaneous chiral-glass and spin-glass transition with a common correlation length exponent,i.e.,T SG /J =0.19and νSG =1.2.The best scaling is ob-tained with ηSG =?0.1.

0.04

0.06 0.08 0.1

0.12 0.14

0.6

0.8

1

q (2)S G

L 1/ν

T/J

T SG =0νSG =2.2ηSG =?1

Spin

L=12L=16L=20

0.02

0.06

0.1 0.14

0.5

0.7

0.9

1.1

L=8

L=12L=16L=20

FIG.9:Finite-size-scaling plot of the squared spin-glass order parameter with assuming a zero-temperature spin-glass tran-sition T SG =0.In the main panel,only the data for larger sizes and at lower temperatures,i.e.,those at T /J ≤0.19and with L ≥12,are plotted.In the inset,the same scaling plot using all the data is shown.

uses in the scaling plot the data at low temperatures,lower than the CG transition temperature T/J ≤0.19,and the data for larger sizes L ≥12,the scaling turns out to work well:See the main panel.By contrast,If one includes in the scaling plot the data for the small-est size L =8and at high temperatures T/J ≥0.19,a signi?cant deviation from the scaling is observed for these data:See the inset.In sharp contrast to the scal-ing plot of Fig.8with T SG =T CG ,we have observed here a better scaling for larger sizes,and a systematic deviation from the scaling for smaller lattices.In that sense,the present ?nite-size scaling analysis is fully con-sistent with the occurrence of a T =0SG transition,

as has long been believed in the community.4,5,6,7,8,9,12,40Furthermore,the exponent associated with the possible T =0SG transition happens to be rather close to the previous estimates based on the numerical domain-wall method.4,5,9

Of course,as discussed above,the CG transition occur-ring at T/J ?0.19would necessarily a?ect the nature of the SG ordering,even if the Heisenberg spin itself does not order at T =T CG .Thus,even if the SG transition oc-curs only at T =0,an intrinsic SG critical phenomenon associated with this T =0transition should set in at low temperatures below T CG ,whereas the data at and above T CG would be “contaminated”by the CG transi-tion which might well change the ordering behavior of the Heisenberg spin via the associated phase-space nar-rowing.

Hence,although our present data are fully consistent with the occurrence of the T =0SG transition,in order to see such a behavior clearly,one has to choose the scal-ing region carefully.Inclusion of the data of smaller sizes and at higher temperatures in the analysis would easily deteriorate the quality of the scaling plot,leading to the opposite conclusion.We believe that this is indeed the situation of the recent study of Ref.39,in which a simul-taneous spin and chiral transition at a ?nite temperature T SG =T CG was concluded.

D.Correlation length

The temperature dependence of the normalized SG and CG correlation lengths,ξSG /L and ξCG /L ,for various sizes are shown in Fig.10.In contrast to the Binder parameter shown in Fig.3,the normalized correlation lengths of L =8and 12shown in in Fig.10(a)exhibit a clear crossing at a temperature around T/J ?0.2for both cases of the SG and the CG.The observed behavior is consistent with the behavior recently reported by Lee and Young 20for the 3D Heisenberg SG model with the Gaussian coupling for the sizes up to L =12.

We now extend the system size up to L =20,and the result is presented in Fig.10(b).While the crossing temperature for larger sizes L =16and 20shifts to-ward lower temperature for both cases of the SG and the CG,the CG correlation length still has a clear crossing around T/J ?0.19,very close to the estimate of T CG in the previous subsections,with a ?nite crossing angle .On the other hand,for the SG correlation length,the cross-ing becomes weaker and almost fades https://www.sodocs.net/doc/297143073.html,ly,the curves of L =16and 20merge nearly tangentially with a vanishing crossing angle .The L =16and 20curves of ξSG /L stay on top of each other in an entire tem-perature region studied below T/J ?0.19,as if they were in the critical KT-like phase.Hence,for the SG,with increasing L ,not simply the crossing temperature shifts toward lower temperature,but the crossing-angle becomes smaller and almost vanishes.This is in contrast to the behavior of the CG correlation length where the

11

0.1 0.2 0.3 0.4 0.5

0.6 0.7 0.16

0.18

0.2

0.22

0.24

ξC G /L , ξS G /L

T/J

0.1 0.2 0.3 0.4 0.5

0.6 0.7 0.16

0.18

0.2

0.22

0.24

ξC G /L , ξS G /L

T/J

FIG.10:The correlation length divided by the linear size L plotted against the temperature for the chiral-glass ξCG /L (?lled symbols)and for the spin-glass ξSG /L (open symbols).The data for smaller sizes (L =8and 12)are shown in the upper ?gure,and those for larger sizes (L =16and 20)are shown in the lower ?gure.

crossing-angle remains ?nite with increasing L .It thus seems possible that the further increase of L eventually leads to the disappearance of the crossing for ξSG /L ,at least in the temperature range studied here.

This would be consistent with the size-crossover ex-pected from the spin-chirality coupling/decoupling pic-ture,and with our observation in Sec.V B that the de-coupling length scale is about L =20.Unfortunately,at present,we cannot go to lattices larger than L =20due to the limitation of our computation capability.We certainly expect,however,that the crossing of ξSG /L eventually disappears,or at least shifts to a temperature considerably lower than the CG transition temperature T CG /J ?0.19,if we could study lattices considerably larger than L =20.For now,we only mention that,although the recent data of the normalized correlation length for smaller lattices of L ≤12might look rather conclusive at ?rst sight,20in view of our present data for larger sizes presented in Fig.10(b),it is still di?cult to draw a de?nite conclusion about the ordering nature of

the model only through the correlation-length measure-ments.

E.Overlap distribution

In Fig.11,we show the chiral-overlap distribution func-tion;upper ?gure (a),and the diagonal-spin-overlap dis-tribution function;lower ?gure (b),at a temperature T/J =0.15below the CG transition temperature.One sees from Fig.11(a)of the chiral-overlap distribution function P (q χ)that,with increasing L ,the side peaks

corresponding to the CG EA order parameter q EA

CG grow and sharpen,which indicates the occurrence of the CG long-range order.In addition,a central peak at q χ=0shows up for L ≥12,which also grows and sharpens with increasing L .The existence of this central peak co-existing with the side peaks suggests the occurrence of a one-step-like RSB in the CG ordered state.This feature is also consistent with the existence of a negative dip in the CG Binder parameter g CG and with the crossing of g CG occurring on the negative side,as was discussed in Sec.V A.The behavior of P (q χ)observed here is similar to the previous reports for the 3D Heisenberg SG with the Gaussian coupling 12and the related Heisenberg SG models.13,22,38By contrast,such a one-step-like feature of the overlap distribution has never seen in the Ising SGs both with the short-range 41and in?nite-range 42interac-tions,nor in the Heisenberg SG with the in?nite-range interaction 22.

Fig.11(b)represents the size dependence of the diagonal-spin-overlap distribution function P (q diag )de-?ned by Eq.(18).For larger L ≥16,the distribution function P (q diag )has only a single peak at q diag =0,which grows with increasing L ,without any other diver-gent peak.This is in sharp contrast to the triple-peak structure observed in the chiral-overlap distribution func-tion P (q χ)of Fig.11(a),peaked at q χ=0and ±q EA

CG .It is also in contrast to the double-peak structure observed in P (q diag )of the mean-?eld Heisenberg SK model,peaked at q =±1

3q EA should arise in P (q diag )

in the possible SG ordered state with a non-zero EA SG order parameter,22the absence of any divergent peak at non-zero q diag for larger L strongly suggests that the sys-tem is in the SG disordered state even at this low tem-perature T/J =0.15.Interestingly,a closer inspection of Fig.11(b)reveals that a weak double-peak structure can be seen for smaller sizes corresponding to the spin-chirality coupling regime,L =8and 12.However,such a double-peak structure in P (q diag )tends to disappear for larger sizes corresponding to the spin-chirality decou-pling regime,L ≥16.Again,this could be interpreted as the size-crossover from the small-size pseudo SG order to the large-size SG disorder,as is naturally expected from the spin-chirality coupling/decoupling picture.

12

0.005

0.01

0.015

0.02 0.025 0.03 0.035-0.06

-0.04

-0.02

0.02

0.04

0.06

P (q χ)

q χ

(a) Chirality

8 12 16 20

0.005

0.01

0.015

0.02

-1-0.5 0 0.5 1

P (q d i a g )

q diag

(b) Spin

8 12 16 20

FIG.11:The size dependence of the chiral-overlap distri-bution function;upper ?gure (a),and of the diagonal-spin-overlap distribution function;lower ?gure (b),at the lowest temperature of the present simulation,T /J =0.15.

F.Equilibrium auto-correlation functions

Next,we discuss the ordering behavior of the model by studying its equilibrium dynamics.In Fig.12,we show the MC time dependence of the chiral and spin autocorrelation functions for our two largest sizes L =16and 20.Here,the time is measured in units of the standard heat-bath MC steps without the temperature-exchange procedure.

In the chiral autocorrelation function C χ(t )shown in Fig.12(a),no appreciable di?erence is observed between the data of L =16and 20in the time window of t ≤

10

?2

10?1

100

101

102

103

104

C s (t )

t [MCS]

(b) Spin

0.4

0.6

0.8

10

100

1000

10?3

10?2

10?1

101

102

103

104

C χ(t )

t [MCS]

(a) Chirality

T/J=0.24 0.21 0.19 0.17 0.15

FIG.12:The Monte Carlo time dependence of the chiral autocorrelation function C χ(t );upper ?gure (a),and of the spin autocorrelation function C s (t );lower ?gure (b),at vari-ous temperatures both below and above T CG /J ?0.19.The system size is L =16(given by symbols)and L =20(given by thin lines).The inset is an enlarged view of C s (t )in the short-time region where the ?nite-size e?ect is negligible.

104,beyond which a weak size e?ect is appreciable.The spin autocorrelation function C s (t ),by contrast,is more susceptible to the ?nite-size e?ect,as can be seen from Fig.12(b).Even in this case,however,the data in the time window t ≤103shows a negligible size e?ect as shown in the inset.

As can be seen from Fig.12(a),C χ(t )shows a down-ward trend above T/J =0.19,an upward trend below T/J =0.19,and a near linear behavior at T/J =0.19.In order to quantify this,we ?t the data at each tem-perature to the form (27)and plot the χ2-deviation of the ?t in Fig.13as a function of the temperature around the expected CG transition temperature.The plot has a minimum around T/J =0.19(1),at which the data are optimally ?tted to a power law.This estimate of T CG based on the chiral autocorrelation function agrees well with those obtained from the CG Binder parameter and the CG order parameter.

We also test a dynamical scaling analysis of the chi-

13

-0.35-0.3

-0.25-0.2-0.15

-0.1

-0.05 0

0.2

0.4 0.6 0.8 1

1.2 1.4 1.6 1.8 2β/z ν

χ2

T/J

FIG.13:Residuals per degrees of freedom associated with the χ2-?tting of the chiral autocorrelation function (marked by ?lled circle),and an estimated e?ective exponent β/zν(marked by open circle)plotted against the temperature.

ral

autocorrelation function C χ(t ).As can be seen from Fig.14,the dynamical scaling works well both above and below T CG ,with the scaling parameters T CG /J =0.195,βCG =0.8and z CG νCG =5.0.The present estimate of βCG is slightly smaller than,but is not far from the pre-vious estimate of Ref.40for the 3D Heisenberg SG with the Gaussian coupling βCG ?1.1.

As can be seen from Fig.12(b),by contrast,the spin autocorrelation C s (t )shows a downward trend at longer times at any temperature studied,suggesting an exponential-like decay characteristic of the disordered phase.A closer inspection of the data of C s (t )reveals that the data below T CG exhibit a weak hump-like struc-ture at short times t ?102,though this hump eventually gives way to the down-bending trend characteristic of the

disordered phase at longer times t >～103

:See the inset of Fig.12(b).This hump-like structure observed in C s (t )at short times might be a manifestation of the trivial spin-chirality coupling expected at short time scales,and is not likely to be an indication of the SG long-range or-der ,since the downward trend is recovered at longer time scales:See Sec.VI for further details.Hence,from our dynamical data,we conclude again that the CG transi-tion occurs at T CG /J =0.19(1),without accompanying the simultaneous SG order.

G.

Equilibrium correlation time

Generally,the temporal decay of the autocorrelation is characterized by the temperature-dependent characteris-tic time scale,the correlation time τ(T ),which represents a dynamical crossover from the short-time critical behav-ior to the long-time relaxation.The correlation time τ(T )diverges as τ(T )?|T ?T c |?zνwhen the temperature approaches T c from above.One promising method of es-timating τfrom the autocorrelation function has been

10?310?2

10?1

100

101

10?1010?910?810?710?610?510?410?3

C χ(t )|T ?T C G |?βC G

t|T ?T CG |z CG νCG

T/J=0.25

0.220.210.200.180.170.15

FIG.14:Dynamical scaling plot of the chiral autocorrela-tion function.The best scaling is obtained with choosing T CG =0.19,βCG =0.8and z CG νCG =5.4.The upper branch represents the scaling of the data below T CG ,while the lower branch represents that above T CG .

0.2 0.4 0.6

0.8 1

0.01 0.1 1 10

R χ(t )

t/τCG

0.2 0.4 0.6

0.8 1

0.01

0.1

1

10

R S (t )

t/τSG

FIG.15:Scaling plot of the chiral ratio function;upper ?gure (a),and of the spin ratio function;lower ?gure (b),at various temperatures for the sizes L =16and 20.

14

proposed by Bhatt and Young43,who employed a scaling analysis of the dynamical ratio function.For the spin autocorrelation,this reads as

R s(t)=

C s(t)

1

3N iμχiμ(t0)χiμ(t+t0) 2 .(30) Because the

ratio function is dimensionless,the prefactor t?β/zνin Eq.(28)is canceled out.The dynamical scaling form of R(t)is then given as a single-variable function of t/τ,

R(t)=

R is a scaling function.If one appropriately chooses the scaling parameterτwhich depends on the temperature and the system size,the ratio functions should be scaled on to a single https://www.sodocs.net/doc/297143073.html,ing this method, Bhatt and Young43successfully estimated the correlation time of the short-range Ising EA model and the mean-?eld Ising SK model.Subsequently,this method has been extended to non-equilibrium relaxation,where the ratio function depends not only on the measurement time t but also on the waiting time t w.The o?-equilibrium method was applied recently by Matsumoto,Hukushima and Takayama to the3D±J Heisenberg SG.44

Here we use this method to estimate the correlation times both for the spin and for the chirality.In compari-son with the previous o?-equilibrium study,44the present equilibrium study has an advantage that one needs not extrapolate to an equilibrium limit,i.e.,needs not take the t w→∞limit.In Fig.15,we show the scaling plot of the chiral and spin ratio functions.We note that both the spin and chiral scaling functions are described roughly by an exponential form.

In order to compare the spin and chiral correlation times,denoted byτSG andτCG,respectively,we plot them in Fig.16as a function of the temperature.In the?gure,we combine the data with those obtained in a wider temperature range by o?-equilibrium simu-lation of Ref.44.As can be seen from Fig.16,the chiral correlation time is shorter than the spin correla-tion time at higher temperatures,similarly to the be-havior of the correlation length discussed in Sec.V D. As the temperature is decreased,the chiral correlation timeτCG grows faster than the spin correlation timeτSG, and eventually exceedsτSG at a certain characteristic temperature T=T×(L).The size dependence of this crossover temperature T=T×(L)is apparently weak: We get T=T×?0.24both for L=16and20.It strongly suggests that,even in the thermodynamic limit,

102

103

104

105

106

107

c

o

r

r

e

l

a

t

i

o

n

t

i

m

e

τ

T/J

FIG.16:The temperature dependence of the the chiral and spin correlation times for the sizes L=16and20.The cor-responding data obtained from the o?-equilibrium simulation of Ref.44are also included.

the chiral correlation time exceeds the spin correlation time at a crossover temperature T×(∞),which is lo-cated somewhat above the CG transition temperature T CG/J?0.19.It means that,with decreasing the tem-perature,the relevant degree of freedom dominating the long-time ordering behavior changes from the spin to the chirality at T=T×.In order to further illustrate this changeover,we show in Fig.17the time-dependence of the spin and chiral ratio functions at two representative temperatures T/J=0.25and0.20,each above and below T×.As can be clearly seen from the?gure,with decreas-ing the temperature across T×,the temporal decay of the chiral ratio function becomes much slower than that of the spin ratio function:Compare the two arrows in the ?gure.

The time scale associated with such a crossover,t×,is roughly estimated to be105～106MCS.For more precise estimate of t×,more quantitative analysis of the size de-pendence of the crossover time scale would be necessary. Naturally,this crossover time t×gives a measure of the time scale above which the spin-chirality decoupling can be observed in dynamics.Thus,the spin-chirality decou-pling in the dynamics would be eminent only at temper-atures lower than T×/J?0.24and at times longer than t×?105?106MCS.This crossover time scale is rather long,yet,is?nite.It is important to realize that,in order to resolve the controversy concerning the presence or the absence of the spin-chirality decoupling in the Heisenberg SG,one has to probe the equilibrium dynamics beyond this crossover time scale105?106MCS at temperatures lower than T×/J?0.24,about some20～30%above T CG.45

As argued in Sec.II,from the spin-chirality cou-pling/decoupling picture,a similar phenomenon is ex-pected also in the spatial correlation of the model in terms of the length https://www.sodocs.net/doc/297143073.html,ly,one expects that at a certain crossover temperature T′×,which is prob-

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.8 0.9 1

R χ(t ), R S (t )

t [MCS]

FIG.17:Temporal decay of the chiral and spin ratio func-tions at temperatures T /J =0.25and 0.20.The system size is L =16.With decreasing the temperature from T /J =0.25to 0.20,the chiral relaxation slows down much more slowly than the spin relaxation,as illustrated by the arrows.

ably close to the dynamical crossover temperature T ×discussed above,the CG correlation length ξCG exceeds the SG correlation length ξSG .This changeover of the dominant length scale gives a crossover length scale L ×above which the spin-chirality decoupling is eminent in spatial correlations.Unfortunately,unlike the case of the correlation time,the limitation of the available sys-tem size prevents us from directly estimating L ×.In Fig.18,we plot the temperature dependence of the CG and SG correlation lengths for the sizes L =16and 20.For these sizes,the crossing of ξSG and ξCG occurs at a temperature lower than the CG transition temperature,in contrast to the case of the correlation time.Neverthe-less,the crossover temperature at which ξSG and ξCG of ?nite L cross,tends to increase with increasing L .If we roughly estimate the crossover length scale of ?nite sys-tems by extrapolating the data of Fig.18,we tentatively get L ×?11(L =16)and L ×?14(L =20).These results are certainly not inconsistent with our estimate of L ×?20based on the behaviors of the SG order param-eter,the dimensionless correlation length ξ/L and other quantities.

H.

The A and G parameters

We have also calculated the A and G parameters de-?ned in Sec.IV both for the CG and SG orders.In Fig.19,the temperature and size dependence of the A and G parameters for th CG order,A CG and G CG ,is shown.Although the data are rather noisy due to the large sample-to-sample ?uctuations,the A parameter of di?erent L show a crossing and a peak around the ex-pected CG transition point T/J ?0.19,as can be seen from Fig.19(a).In particular,with increasing L ,A CG stays non-zero below T CG ,indicating that the CG or-

2

4 6 8 10 12

14 16 18ξC G , ξS G

T/J

FIG.18:The temperature dependence of the chiral and spin correlation lengths for ?nite systems.The system size is L =16and 20.The data are the same as those shown in Fig.10,but not divided by L here.The curves are polynomial ?ts of the data which are extrapolated to lower temperatures to deduce the crossing temperature given in the text.

0 0.05 0.1 0.15 0.2 0.25 0.3

0.35 0.4 0.45 0.5A C G

T/J

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4G C G

T/J

FIG.19:The temperature and size dependence of the A parameter of the chirality;upper ?gure (a),and of the G parameter of the chirality;lower ?gure (b).

16

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

A S G

T/J

0 0.05 0.1 0.15 0.2 0.25 0.3

0.35 0.4 0.45

G S G

T/J

FIG.20:The temperature and size dependence of the A pa-rameter of the spin;upper ?gure (a),and of the G parameter of the spin;lower ?gure (b).

dered state is non-self-averaging.These ?ndings com-bined with the peculiar shape of P (q χ)shown in Sec.V E suggest that the CG ordered phase accompanies an RSB with a non-self-averageness.For the corresponding G parameter,the crossing is not so clear,as is shown in Fig.19(b).

The temperature and size dependence of the A and G parameters for th SG order,A SG and G SG ,is shown in Fig.20.As shown in Fig.20(a),the A parameter ex-hibits a crossing,although the crossing temperature is located considerably above T c for this range of L .One might be tempted to interpret such a crossing of A SG as an unambiguous evidence of the occurrence of the stan-dard SG transition.However,one has to be careful here:Although the crossing of A SG is certainly a signature of some sort of phase transition occurring there,it does not necessarily mean the occurrence of the standard SG tran-sition characterized by a non-zero SG order parameter.A non-zero A SG persisting in the L →∞limit simply

means that the SG order parameter q (2)

SG ,or the SG sus-ceptibility χSG =Nq (2)

SG ,is non-self-averaging.Below the CG transition temperature,one expects that the SG

order parameter is still Gaussian-distributed around zero with a width corresponding to the ?nite SG susceptibility χSG ,while the width exhibits sample-to-sample ?uctua-tions leading to the non-self-averaging χSG .The latter is a natural consequence of the phase-space narrowing which should inevitably accompany the CG transition with the one-step-like RSB.Hence,the crossing of A SG ,and a ?nite A SG remaining in the L →∞limit below T CG ,are compatible with the absence of the standard SG long-range order,and is entirely consistent with the the CG transition not accompanying the standard SG long-range order.

As shown in Fig.20(b),the G parameter of the spin exhibits a crossing around T CG .The relation Eq.(24),combined with our observation in Fig.3(b),indicates that G SG also takes a non-zero value below T CG .Thus,the observed crossing of G SG is just as one expects for the CG transition.In other words,one cannot interpret the crossing of G SG as an indicator of the onset of the standard SG long-range order.

VI.

DISCUSSIONS

In this section,in view of our MC results presented in the previous section,we wish to examine and discuss the recent numerical studies on the 3D Heisenberg SG.Many of these studies suggested,contrary to our present study,that the spin and the chirality ordered simultaneously at a ?nite temperature with a common correlation length exponent νSG =νCG ,i.e.,no spin-chirality decoupling in the 3D Heisenberg SG.16,18,19,20,39,46Below,we wish to make some comments on these numerical works from the standpoint of the spin-chirality coupling/decoupling picture.

A.

Sti?ness method

First,we wish to discuss the analyses based on the sti?ness method.16,17,47In this method,one computes by some numerical means the change of the ground-state energy of ?nite systems of size L under the appropriate change of boundary conditions imposed on the system.This energy is called a sti?ness energy (or a domain-wall energy),?E L ,which gives a measure of an energy scale of low-energy excitations of size L .For large L ,?E L is expected to behave as a power-law,?E L ≈L θ,θbeing a sti?ness exponent.If θ<0,the system remains in the disordered state at any nonzero temperature,whereas if θ>0the system possesses a ?nite long-range order at low enough temperatures with T c >0.Here,we discuss this method ?rst in conjunction with the detection of the standard SG order,leaving the detection of the CG order later.

The nontrivial part of this sti?ness method concerns with the choice of the boundary conditions employed in computing the sti?ness energy.There could be various

17

choices,and the behavior of?E L might in principle de-

pend on these choices particularly for small sizes accessi-ble in numerical simulations.The most standard choice

is the combination of the periodic and the antiperiodic boundary conditions(P/AP).In the case of the Heisen-

berg SG,the P/AP combination necessarily accompanies a?ipping of the chirality(remember that the chirality

of the Heisenberg spin is odd under the spin inversion S→? S),so that the P/AP combination should detect the chiral order for large enough L.9In order to detect

the standard SG order independently of the CG order

by this sti?ness method,Ref.9introduced the“rota-tion”boundary conditions(ROT),which imposed aπrotation on the boundary spins without?ipping the chi-rality,which was combined with the standard P bound-ary conditions in calculating the sti?ness energy.9Such a P/ROT combination applied to the3D Heisenberg SG yielded a negativeθ,i.e.,θ～?0.51for the Gaussian coupling,andθ～?0.49for the±J coupling,which im-plied the absence of the standard SG order at nonzero temperature.9

By contrast,Matsubara,Endoh and Shirakura pro-

posed a di?erent choice of boundary conditions in com-puting?E L,i.e.,to use the free(open)boundary con-ditions as a reference and impose the rotational-twist to such“optimized”spin con?gurations obtained under the free boundary conditions in which the stress at the boundary is released.16,17These authors observed that the sti?ness exponent evaluated in this way was largely positive,close to the spin-wave exponentθ=1,and argued that the3D Heisenberg SG exhibited a?nite-temperature SG transition.The method similar in spirit to the one used in Refs.16and17was also applied to the XY SG by Kosterlitz and Akino,47leading to the similar conclusion.Thus,the result obtained by applying the free/twisted-free(F/TF)boundary conditions,θ>0im-plying T SG>0,is in sharp contrast to the result obtained by applying the P/ROT boundary conditions,θ<0im-plying T SG=0.Discrepancy between the sti?ness ex-ponents evaluated by the di?erent choices of boundary conditions was also observed in other SG models,e.g.,in the2D Ising SG.48

Although the authors of Refs.16and17argued that

their“optimized”boundary conditions were superior to the other choices,a theoretical basis of such a claim seems not so obvious.For example,one might make the follow-ing counter-argument that one should not optimize the boundary conditions in calculating?E L:In the spirit of the domain-wall renormalization-group idea by Bray and Moore49,?E(L)represents an energy scale associated with the interaction between the coarse-grained blocks of size L in an in?nite SG sample.Since these blocks are necessarily subject to the strong frustration e?ect caused by the interaction with the neighboring blocks surround-ing them,an optimization of their energy,independently at each block ignoring the inevitable frustration e?ect due to the interaction with the neighboring blocks,is hardly compatible with the original RG idea.One may thus argue that,in calculating?E L in SGs,one should not make the optimization of boundary conditions referring to the particular bond realization of each sample. Concerning the apparent di?erence of the sti?ness ex-ponents arising from the di?erent choices of boundary conditions,there generally exist two possibilities:Either, (i)the observed di?erence is a?nite-size e?ect where there is a single sti?ness exponent for large enough lat-tices independent of applied boundary conditions,or,(ii) the observed di?erence is a bulk e?ect which persists even in large enough lattices.In the case of the2D Ising SG, Carter,Bray and Moore numerically observed that,al-though both the P/AP and F/TF boundary conditions yielded the same sti?ness exponent asymptotically for large L,i.e.,the possibility(i)above,the?nite-size ef-fect was much reduced in the P/AP than in the F/TF.48 For vector SG,there so far exists no convincing evidence which of the above(i)and(ii)is really the case.In any case,a practical question we face with is which set of boundary conditions gives a true asymptotic answer from smaller sizes accessible in simulations.

One plausible criterion might be that,among all pos-sible excitations in the system,the one giving the low-est excitation energy?E L,or equivalently,the one giv-ing the smallest sti?ness exponentθ,should be chosen. The reason is simply because among all possible excita-tions the one with the lowest excitation energy should be most e?cient in destroying the order so long as it has non-negligible weight in the thermodynamics,and would dominate the low-energy dynamics of the model. Under this criterion,when the di?erent choices of bound-ary conditions yield di?erentθvalues,the one giving the smallestθ,or the most negativeθ,should be chosen.In particular,when one set of boundary conditions yields a positiveθwhile the other yields a negativeθ,the one giving a negativeθshould be chosen.If so,in the case of the3D Heisenberg SG of our interest,the P/ROT com-bination without any optimization procedure should be chosen since it gives the lowestθ(negativeθ)reported so far.9This suggests that the standard SG order in the 3D Heisenberg SG occurs only at T=0.Of course,it is still possible that some other type of boundary condi-tions might yield a still smallerθ,but it does not change the conclusion that the SG order occurs only at T=0. It should also be remembered that the types of low-energy excitations generated via a particular choice of boundary conditions is only a subset of all possible exci-tations in the system:They are basically wall-like exci-tations,not including more complex excitations like,say, a“vortex”excitation which is possible in the Heisenberg SG re?ecting the SO(3)nature of its order parameter space,50or a”sponge”excitation which is closely related to the RSB structure of the ordered state.51,52Unfortu-nately,we have little knowledge concerning what is the most relevant low-energy excitation governing the order-ing of the system,and hence,have no well-based criterion to choose one set of boundary conditions from the others as superior.Although we feel that our argument above

18

speaks for a zero-temperature SG transition in the3D Heisenberg SG,it would be fair to say at present that no de?nitive conclusion can be drawn solely based on this sti?ness method.

We?nally wish to refer to the sti?ness method in de-tecting the chiral order.As mentioned,since the sign of the chirality is?ipped by changing the boundary condi-tions from P to AP,the most standard P/AP combina-tion could be used in detecting the chiral order,at least for large enough L.In practice,however,the applica-tion of the re?ecting(R)boundary is more e?cient in detecting the chiral order,as shown in Ref.9.The chiral sti?ness exponent of the3D Heisenberg SG determined in this way turned out to be positive implying a CG transi-tion occurring at a nonzero temperature.9Other authors also reported a positive value for the chiral sti?ness ex-ponent both for the3D XY SG47and the3D Heisenberg SG17.

B.Equilibrium dynamics Matsubara,Shirakura and Endoh reported a further evidence of the simultaneous spin and chiral transition in the3D Heisenberg SG by investigating the equilibrium spin dynamics18.In order to eliminate the e?ect of global spin rotations inherent to?nite systems,Matsubara et al introduced an arti?cial global-rotation correction in the spin dynamics of the model.They observed that the modi?ed spin autocorrelations adjusted by the global-rotation correction exhibited at lower temperatures a ten-dency to approach a nonzero value at longer times,which was interpreted as an evidence of a?nite SG long-range order.

It should be noticed here that,when one looks at a quantity which is even under the symmetry transforma-tion of the Hamiltonian like the modi?ed spin autocorre-lation function of Ref.18,one needs to examine its size dependence carefully.As is well-known,an even quan-tity in?nite systems always takes a nonzero value even above T c due to the?nite-size e?ect,where this nonzero value decreases with the size L,eventually vanishing as L→∞above T c.(Indeed,in an extreme occasion of a single spin,the modi?ed spin autocorrelation function as computed by Matsubara et al does not decay at all even at an in?nite temperature!)The ordering behavior of the modi?ed spin autocorrelation as observed by Mat-subara et al might possibly be caused by the?nite-size e?ect.In order to refute such suspicion,one needs to study its size dependence carefully,whereas the analysis of Ref.18was limited to a?xed size L=16.

It should be stressed that,even within the spin-chirality coupling/decoupling scheme,it is still possible that the spin autocorrelation function C s(t)of an in?nite system exhibits below T CG a hump-like weak structure at short times as illustrated in Fig.2,which is an echo of the plateau-like structure of the chiral autocorrelation function.In the temperature range T SG>t×Indeed,as was shown in the inset of Fig.12,such a hump-like weak structure of the spin autocorrelation was discernible in our present data of C s(t)at short times t?102,which,however, eventually decayed toward zero at longer times. Berthier and Young also observed in their recent o?-equilibrium simulation of the3D Heisenberg SG a weak hump-like structure in the spin autocorrelation in the time range t<～104,which corresponded to the quasi-equilibrium regime46.These authors interpreted the ob-served hump as an evidence of a nonzero SG long-range order at that temperature.As noted above,however, such a hump is also consistent with the the spin-chirality coupling/decoupling picture as long as the hump is ob-served only at shorter times t

C.Nonequilibrium dynamics Nakamura and Endoh applied a non-equilibrium method to study the SG and the CG orderings of the3D ±J Heisenberg SG.19Analyzing the time dependence of the initial growth of the SG and the CG susceptibilities with use of a dynamical scaling,these authors concluded that the spin and the chirality ordered simultaneously at a?nite temperature T/J=0.21～0.22.While the lattice size studied L≤59was rather large,the crucial question to be addressed is whether the long-time limit t→∞was safely taken justifying the use of the dynami-cal scaling.In other words,although the nominal lattice size studied was large,the equilibrated length scale ac-tually probed in these o?-equilibrium simulations might be rather short.In fact,their non-equilibrium method is uncontrolled with respect to the time scale toward equilibrium.Since the equilibration time could easily become a huge number in SG,care has to be taken as regards the equilibrated length scale actually probed by this type of non-equilibrium simulation.As one judges from the maximum values of the SG and the CG sus-ceptibilities reached by the o?-equilibrium simulation of Ref.19,the“dynamical correlation length”still remained rather short:Namely,even around the transition tem-perature T CG/J?0.2,it reached around10lattice spac-ings for the spin,and only1or2lattice spacings for the chirality.The dynamical chiral correlation length stayed particularly short.This is consistent with a re-cent o?-equilibrium simulation by Berthier and Young in which the dynamical chiral correlation length stayed much shorter than the dynamical spin correlation length in the investigated time range46.Here note that,irre-spective of the question of whether the CG transition accompanies the simultaneous SG transition or not,the chiral correlation length in equilibrium should diverge at and below the CG transition temperature T CG/J?0.2. Hence,the observation above simply tells that,even at

19

the maximum simulation time of Refs.19and46,the sys-tem still stayed in an extreme initial time regime.In or-der to deduce the equilibrium ordering properties from these o?-equilibrium data,one is forced to extrapolate the behavior aroundξCG～1toξCG=∞,which could be dangerous in the present model since the model might possess the characteristic crossover length scale at around 20.

One may feel that the dynamical spin correlation length reached in the o?-equilibrium simulation of Ref.19,ξ≈10,might be reasonably large for deduc-ing the ordering properties of the spin.However,we feel it is not enough.This length scale of10is still not large enough compared with the crossover length scale esti-mated in the present work L×≈20.Remember that the spin-chirality decoupling,if any,is a long scale phenom-ena observable at length scale longer than L×.Second, in the o?-equilibrium simulations of Refs.19and46,even when the dynamical SG correlation length grows around 10lattice spacings,the Z2chiral degree of freedom was not equilibrated at all at this length scale,in sharp con-trast to the fully equilibrated simulation as was done in the present paper.In other words,at the length scale of 10lattice spacings,the chiralities are little thermalized and are virtually frozen in the non-equilibrium pattern, while only the SG correlation grows modestly in such a nonequilibrium chiral environment.After all,however, we have to understand the spin dynamics at long enough length scales at which the Z2chiral degree of freedom is also fully thermalized.Thus,the spin dynamics as ob-served in the o?-equilibrium situations of Refs.19and 46may not faithfully represent the close-to-equilibrium critical dynamics of the original model.

A similar dynamical simulation on the3D±J Heisen-berg SG was performed by Matsumoto,Hukushima and Takayama.44They also made a dynamical scaling analy-sis taking the e?ect of global spin rotations into account. In this study,the time scale toward equilibrium was con-trolled via the analysis of the waiting-time dependence of the results.In contrast to Refs.19and46,Matsumoto et al suggested that their data were consistent with the separate spin and chiral transition,i.e.,T SG

In Ref.46,the aging phenomena were persistently ob-served at lower temperatures,not only for the chirality but also for the spin,which was interpreted as an evi-dence of the occurrence of simultaneous spin and chiral transition.46Again,this cannot be taken as an unam-biguous indicator of a?nite-temperature SG transition, since the aging phenomena could arise simply when the time scale of measurements becomes comparable to the longest relaxation time in the system which could be ex-tremely long in SG even in the paramagnetic phase.For example,in the2D Ising SG which is known to exhibit no ?nite-temperature SG transition,clear aging phenomena have been observed both in numerical simulations53,54 and in experiments55,56.

D.Correlation length

In Ref.20,Lee and Young calculated by means of equi-librium simulations both the SG and the CG correlation lengths of the3D Heisenberg SG with the Gaussian cou-pling in the range of sizes4≤L≤12.Lee and Young ob-served a crossing of the dimensionless correlation lengths ξ/L for di?erent L for both cases of the spin and the chi-rality,and concluded that the spin and the chirality or-dered simultaneously at a?nite temperature T/J?0.15. The behavior ofξ/L observed in Ref.20turned out to be quite di?erent from that of some other dimension-less quantities,e.g.,the Binder ratio,whereas Lee and Young argued that the correlation length might be the most trustable quantity to look at.Generally speaking, however,ξ/L is also subject to signi?cant?nite-size ef-fects,sometimes no better than other quantities.37

We note that the numerical data of Ref.20are basi-cally consistent with our present data for smaller sizes L≤12:See Fig.10(a).As emphasized in Sec.V D of our present paper,however,the crossing behavior of the dimensionless SG correlation lengthξSG/L tends to change for larger lattices L>12:The crossing becomes weaker and weaker,andξSG/L of L=16and that of L=20do not quite cross with a?nite crossing angle as occurs for smaller lattices L≤12,but instead,merge al-most tangentially and stay on top of each other at lower temperatures:See Fig.10(b).In contrast,the dimen-sionless CG correlation lengthξCG/L of L=16and that of L=20persistently exhibits a clear crossing.If such a tendency continues for larger lattices,the crossing of ξSG/L might no longer occur for large enough lattices, at least at the crossing temperature ofξCG/L.We thus suspect that the crossing behavior ofξSG/L as reported in Ref.20might be a transient behavior due to the small sizes,which re?ected the“coupling”behavior expected at L

20

cessible by the present computational capability L?20, being only comparable to the crossover length for the spin-chirality coupling/decoupling to occur,is still not large enough to clear see this behavior.We do expect, however,that the correlation lengths for larger lattices L>20would eventually exhibit a clear spin-chirality decoupling behavior.

We note that such a coupling/decoupling behavior of the SG correlation length in smaller/larger lattices was indeed observed recently in the2D Heisenberg SG.24For the2D Heisenberg SG with the Gaussian coupling,Kawa-mura and Yonehara calculated the dimensionless SG cor-relation lengthξSG/L up to the size L=40,and found thatξSG/L for the smaller sizes L=10,16,20crossed almost at a common temperature T/J?0.022,dis-guising the occurrence of a?nite-temperature SG transi-tion:See the inset of Fig.6(a)of Ref.24,while the data for the larger sizes L=30and40data eventually came down,no longer making a crossing at T/J?0.022.The asymptotic non-ordering behavior observed for L>20is consistent with a zero-temperature SG transition,which has been well established in2D.23,24Meanwhile,the CG correlation length exceeds the SG correlation length at around T/J?0.022,which might naturally explain the reason whyξSG/L for smaller sizes L<～20exhibited a crossing behavior.Anyway,this observation in2D gives us a warning that one should be careful in interpreting the crossing-like behavior ofξ/L observed for smaller sizes as an unambiguous evidence of a true SG phase transition.

E.Finite-size scaling of the order parameter Matsubara,Shirakura,Endoh and Takahashi made a ?nite-size scaling analysis of the SG order parameter q(2)SG for the3D±J Heisenberg SG,and claimed that the qual-ity of the scaling was much better when ones assumed a nonzero SG transition temperature T SG/J=0.18than a zero SG transition temperature T SG/J=0.39Their conclusion is in apparent contrast to that of our present work based on a similar scaling analysis in Sec.V C.We note that the quality of the?nite-size scaling is some-times sensitive to the range of lattice sizes and the range of temperatures used in the?t.

As already noticed in Sec.V C,this point could be particularly serious in the present model.In the spin-chirality coupling/decoupling scheme,the SG correlation and the CG correlation are trivially coupled at shorter length scale L<～L×≈20,so that even the SG order parameter q(2)SG would be scaled for smaller sizes with as-suming a simultaneous SG and CG transition,with ap-parent(not true)SG pseudo-exponentsνe?SG?νCG and 1+ηe?SG?(1+ηCG)/3.Indeed,as was shown in Fig.8, our present data,particularly those of L<～16,turned out to be scaled reasonably well by assuming a simultaneous SG and CG transition at T/J=0.19,which we inter-preted as a pre-asymptotic pseudo-critical behavior real-ized in the short-scale coupling regime.Furthermore,the relation betweenηCG andηe?SG mentioned above roughly holds at short length scale;(1+ηCG)/3～0.60versus 1+ηe?SG～0.88at T/J=0.19.At longer length scales, however,the spin is eventually decoupled from the chiral-ity.Then,if ones continues to put T SG=T CG in the?t of q(2)SG,the good data collapse obtained for smaller sizes would eventually deteriorate for larger sizes.Indeed,as shown in Fig.8,our L=20data of q(2)

SG

showed such a deviation expected for larger sizes.

Within the spin-chirality coupling/decoupling scheme, in order to see the true asymptotic critical behavior of the SG transition occurring at T SG(

By contrast,we have observed that,if we use the data points only of larger lattices L>～16and only at tem-peratures below T CG,the data were scaled reasonably well even with assuming T SG=0:See Fig.9.Therefore, we believe that there still exists a good possibility that the SG order occurs only at T=0as has widely been believed in the community,although it is also quite pos-sible that it occurs at a low but nonzero temperature, 0

As discussed in some detail above,any of the re-cent works claiming the simultaneous spin and chiral transition in the3D Heisenberg SG appears not con-clusive.As far as the authors are aware,all of these observations are consistent with the spin-chirality cou-pling/decoupling scheme with the crossover length scale of20lattice spacings and the crossover time scale of 105?106MCS.Rather,we believe that some of the ob-servations reported in the present paper give a strong numerical support that the SG transition indeed occurs at a temperature below the CG transition temperature, i.e.,T SG

VII.SUMMARY

In summary,we studied the equilibrium properties of the three-dimensional isotropic Heisenberg spin glass by means of extensive MC simulations.We presented evidence of a?nite-temperature CG transition with-out accompanying the conventional SG order through the observation of various physical quantities including the order parameters,equilibrium static and dynamic correlation functions,Binder parameters and overlap-

黄自艺术歌曲钢琴伴奏及艺术成就

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奈达翻译理论简介 （一）奈达其人尤金?奈达（EugeneA.Nida）1914年出生于美国俄克勒荷马州，当代著名语言学家、翻译家和翻译理论家。也是西方语言学派翻译理论的主要代表，被誉为西方“现代翻译理论之父”。尤金是当代翻译理论的主要奠基人，其理论核心是功能对等。 尤金先后访问过90个国家和地区，并著书立说，单独或合作出版了40多部书，比较著名的有《翻译科学探索》、《语言与文化———翻译中的语境》等，他还发表论文250余篇，是世界译坛的一位长青学者。他还参与过《圣经》的翻译工作。他与塔伯合著的《翻译理论与实践》对翻译界影响颇深。此书说明了中国与西方译界人士思维方式的巨大差别：前者是静的，崇尚“信、达、雅”，讲究“神似”，追求“化境”；后者是动的，将语言学、符号学、交际理论运用到翻译研究当中，提倡“动态对等”，注重读者反应。中国译论多概括，可操作性不强；西方译论较具体，往往从点出发。他在该书中提到了动态对等，详细地描述了翻译过程的三个阶段：分析、转换和重组，对于翻译实践的作用是不言而喻的。 （二）奈达对翻译的定义 按照奈达的定义：“所谓翻译，是指从语义到文体（风格）在译语中用最切近而又最自然的对等语再现 源语的信息。”其中，“对等”是核心，“最切近”和“最自然”都是为寻找对等语服务的。奈达从社会语言学和语言交际功能的观点出发，认为必须以读者的反应作为衡量译作是否正确的重要标准。翻译要想达到预期的交际目的，必须使译文从信息内容、说话方式、文章风格、语言文化到社会因素等方面尽可能多地反映出原文的面貌。他试图运用乔姆斯基的语言学理论建立起一套新的研究方法。他根据转换生成语法，特别是其中有关核心句的原理，提出在语言的深层结构里进行传译的设想。 奈达提出了词的4种语义单位的概念，即词具有表述事物、事件、抽象概念和关系等功能。这4种语义单位是“核心”，语言的表层结构就是以“核心”为基础构建的，如果能将语法结构归纳到核心层次，翻译过程就可最大限度地避免对源语的曲解。按照4种语义单位的关系，奈达将英语句子归结为7个核心句：（1）Johnranquickly.（2）JohnhitBill.（3）JohngaveBillaball.（4）Johnisinthehouse.（5）Johnissick.（6）Johnisaboy.（7）Johnismyfather. （三）奈达翻译理论的经历阶段 奈达翻译理论的发展经历过三个阶段，分别是描写语言阶段、交际理论阶段和和社会符号学阶段。 第一个阶段始于1943年发表《英语句法概要》，止于1959年发表《从圣经翻译看翻译原则》。这一阶段是奈达翻译思想及学术活动的初期。 第二阶段始于1959年发表的《从圣经翻译看翻译原则》，止于1969年出版的《翻译理论与实践》。主要著作有《翻译科学探索》、《信息与使命》。在这10年中，奈达确立了自己在整个西方翻译理论界的权威地位。1964年出版的《翻译科学探索》标志着其翻译思想发展过程中一个最重要的里程碑。第三阶段始于70年代，奈达通过不断修正和发展自己翻译理论创建了新的理论模式———社会符号学模式。奈达在继承原有理论有用成分的基础上，将语言看成一种符号现象，并结合所在社会环境进行解释。在《从一种语言到另一种语言》一书中，奈达强调了形式的重要性，认为形式也具有意义，指出语言的修辞特征在语言交际及翻译中的重要作用，并且用“功能对等”取代了“动态对等”的提法，是含义更加明确。 三、对奈达翻译理论的评价 （一）贡献 奈达是一位硕果累累的翻译理论家。可以说，在两千年的西方翻译思想发展史上，奈达的研究成果之丰是名列前茅的。他的研究范围从翻译史、翻译原则、翻译过程和翻译方法到翻译教学和翻译的组织工作，从口译到笔译，从人工翻译到机器翻译，从语义学到人类文化学，几乎无所不包，从而丰富并拓展了西方的翻译研究领地。 奈达的理论贡献，主要在于他帮助创造了一种用新姿态对待不同语言和文化的气氛，以增进人类相互之间的语言交流和了解。他坚持认为：任何能用一种语言表达的东西都能够用另一种语言来表达；在语言之间、文化之间能通过寻找翻译对等语，以适当方式重组原文形式和语义结构来进行交际。因此也说明，某

奈达翻译理论动态功能对等的新认识

To Equivalence and Beyond: Reflections on the Significance of Eugene A. Nida for Bible Translating1 Kenneth A. Cherney, Jr. It’s been said, and it may be true, that there are two kinds of people—those who divide people into two kinds and those who don’t. Similarly, there are two approaches to Bible translation—approaches that divide translations into two kinds and those that refuse. The parade example of the former is Jerome’s claim that a translator’s options are finally only two: “word-for-word” or “sense-for-sense.”2 Regardless of whether he intended to, Jerome set the entire conversation about Bible translating on a course from which it would not deviate for more than fifteen hundred years; and some observers in the field of translation studies have come to view Jerome’s “either/or” as an unhelpful rut from which the field has begun to extricate itself only recently and with difficulty. Another familiar dichotomy is the distinction between “formal correspondence” translating on one hand and “dynamic equivalence” (more properly “functional equivalence,” on which see below) on the other. The distinction arose via the work of the most influential figure in the modern history of Bible translating: Eugene Albert Nida (1914-2011). It is impossible to imagine the current state of the field of translation studies, and especially Bible translating, without Nida. Not only is he the unquestioned pioneer of modern, so-called “meaning-based” translating;3 he may be more responsible than any other individual for putting Bibles in the hands of people around the world that they can read and understand. 1 This article includes material from the author’s doctoral thesis (still in progress), “Allusion as Translation Problem: Portuguese Versions of Second Isaiah as Test Case” (Stellenbosch University, Drs. Christo Van der Merwe and Hendrik Bosman, promoters). 2 Jerome, “Letter to Pammachius,” in Lawrence Venuti, ed., The Translation Studies Reader, 2nd ed. (NY and London: Routledge, 2004), p. 23. 3 Nigel Statham, "Nida and 'Functional Equivalence': The Evolution of a Concept, Some Problems, and Some Possible Ways Forward," Bible Translator 56, no. 1 (2005), p. 39.

“功能对等”翻译理论奈达翻译理论体系的核心

［摘要］传统的只围绕直译与意译之争，而奈达从《圣经》翻译提出功能对等即读者同等反应。“功能对等”翻译理论是奈达翻译理论体系的核心,是从新的视角提出的新的翻译方法，它既有深厚的理论基础,也有丰富的实践基础，对翻译理论的进一步完善是一大贡献。 ［关键词］功能对等；奈达翻译；英语论文范文 尤金·A·奈达博士是西方语言学翻译理论学派的代表人物之一。在他的学术生涯中，从事过语言学、语义学、人类学、通讯工程学等方面的研究，还从事过《圣经》的翻译工作，精通多国文字，调查过100多种语言。经过五十多年的翻译实践与理论研究，取得了丰硕的成果。至今他已发表了40多部专着、250余篇论文。“自八十年代初奈达的理论介绍入中国以来，到现在已经成为当代西方理论中被介绍的最早、最多、影响最大的理论。他把信息论与符号学引进了翻译理论，提出了‘动态对等’的翻译标准；把现代语言学的最新研究成果应用到翻译理论中来；在翻译史上第一个把社会效益（读者反应）原则纳入翻译标准之中。尤其是他的动态对等理论，一举打破中国传统译论中静态分析翻译标准的局面，提出了开放式的翻译理论原则，为我们建立新的理论模式找到了正确的方向。奈达在中国译界占据非常重要的地位。”“奈达的理论贡献，主要在于他帮助创造了一种新姿态对待不同语言和文化的气氛，以增进人类相互之间的语言交流和了解。”[1] 翻译作为一项独立的学科，首先应回答的问题就是：什么是翻译？传统翻译理论侧重语言的表现形式，人们往往醉心于处理语言的特殊现象，如诗的格律、诗韵、咬文嚼字、句子排比和特殊语法结构等等。现代翻译理论侧重读者对译文的反应以及两种反应（原文与原作读者、译文与译作读者）之间的对比。奈达指出：“所谓翻译，就是指从语义到文体在译语中用最贴切而又最自然的对等语再现原语的信息”，奈达在《翻译理论于实践》一书中解释道，所谓最切近的自然对等，是指意义和语体而言。但在《从一种语言到另一种语言》中，奈达又把对等解释为是指功能而言。语言的“功能”是指语言在使用中所能发挥的言语作用；不同语言的表达形式必然不同，不是语音语法不同就是表达习惯不同，然而他们却可以具有彼此相同或相似的功能。奈达所强调的是“对等”“、信息”“、意义”和“风格”，奈达从语义学和信息论出发，强调翻译的交际功能，正如他自己所说“：翻译就是交际”，目的是要寻求原语和接受语的“对等”。他所说的“信息”包括“意义”和“风格”，着重于交际层面。他实质上要打破的是传统的翻译标准。他把翻译看成是“语际交际”，也就是在用交际学的观点来看问题。交际至少应当是三方的事情：信息源点———信息内容———信息受者，也就是说话者———语言———听话者。奈达注重译文的接受者，即读者，而且都对读者进行了分类。奈达根据读者的阅读能力和兴趣把读者分为四类：儿童读者、初等文化水平读者、普通成人读者和专家。他曾说过，一些优秀的译者，常常设想有一位典型的译文读者代表就坐在写字台的对面听他们口述译文，或者正在阅读闪现在电脑显示屏上的译文。这样，就好像有人正在听着或读着译文，翻译也就不仅仅是寻求词汇和句法的对应过程。运用这种方法，译者就可能更自觉地意识到“翻译就是翻译意思”的道理。 可译性与不可译性是翻译界长期争论的一个问题。奈达对不同的之间的交流提出了新的观点。他认为每种语言都有自己的特点，一种语言所表达的任何东西都可以用另一种语言来表达。尽管不同民族之间难以达到“绝对的”交流，但是可以进行“有效的”交流，因为人类的思维过程、生产经历、社会反应等有许多共性。他这种思想主要基于他对上帝的信仰和对《圣经》的翻译。在他看来，上帝的福音即是真理，可以译成不同的语言，也可以为不同国家的人所理解。因此，他提出“最贴近、最自然的对等”。 奈达把翻译分为两种类型：形式对等翻译和动态功能对等。翻译形式对等是以原语为中心，尽量再现原文形式和内容。功能对等注重读者反映，以最贴近、最自然的对等语再现原文信息，使译文读者能够达到和原文读者一样的理解和欣赏原文的程度。奈达的形式对等要求严格地再现原语的形式，其实也就是“逐字翻译”或“死译”。奈达本人也不主张形式对等的翻译，他认为严格遵守形式无疑会破坏内容。 奈达的“功能对等”理论的提出是对译学研究的一个重大贡献。首先，他提出了一个新的翻译评价标准。他指出：翻译准确与否取决于普通读者正确理解原文的程度，也就是把译文读者反应与原文读者反应进行对照，看两者是否达到最大限度的对等。其次，他提出的“最贴切、最自然的对等”标准也不同于传统的“忠实”

The-theory-and-practice-of-translation-奈达的翻译理论与实践

The theory and practice of translation Eugene A. Nida and Charles R. Taber 1974 Contents 1.A new concept of translation 2.The nature of translating 3.Grammatical analysis 4.Referential meaning 5.Connotative meaning 6.Transfer 7.Restructuring 8.\ 9.Testing the translation Chapter One The old focus and the new focus The older focus in translating was the form of the message, and the translator too particular delight in being able to reproduce stylistic specialties, ., rhythms, rhymes, plays on words, chiasmus, parallelism, and usual grammatical structures. The new focus, however, has shifted from teh form of the message to the response of the receptor. Therefore, what one must determine is the response of the receptor to te translated message, this response must be compared with the way in which the original receptors presumably reacted to the message when it was given in its original setting. Chapter Two Translating consists in reproducing in the receptor language the closest natural equivalent of the source-language message, first in terms of meaning and secondly in terms of style. But this relatively simple statement requires careful evaluation of several seemingly contradictory elements. Reproducing the message Translating must aim primarily at “reproducing the message.” To do anything else is essentially false to one’s task as a translator. But to reproduce the message one must make a good many grammatical and lexical adjustments. Equivalence rather than identity 。 The translator must strive for the equivalence rather than identity. In a sense, this is just another way of emphasizing the reproduction of the message rather than the conversation of the form of the utterance, but it reinforces the need for radical alteration of a phrase, which may be quiet meaningless.

奈达和纽马克翻译理论对比初探

[摘要] 尤金·奈达和彼得·纽马克的翻译理论在我国产生很大的影响。他们在对翻译的认识及处理内容与形式关系方面有共识亦有差异。他们孜孜不倦发展自身理论的精神值得中国翻译理论界学习。 [关键词] 翻译; 动态对等; 语义翻译和交际翻译; 关联翻译法 Abstract :Both Nida and Newmark are outstanding western theorist in the field of translation. They have many differences as well as similarities in terms of the nature of translation and the relationship between the form and content. Their constant effort to develop their theories deserve our respect. Key words :translation ; dynamic equivalence ; semantic translation ; communicative translation ; a correlative approach to translation 尤金·奈达( Eugene A1Nida) 和彼得·纽马克是西方译界颇具影响的两位翻译理论家, 他们在翻译理论方面有诸多共通之处, 同时又各具特色。 一、对翻译的认识 对翻译性质的认识, 理论界的讨论由来已久。奈达和纽马克都对翻译是科学还是艺术的问题的认识经历了一个变化的过程。 奈达对翻译的认识经历了一个从倾向于把翻译看作科学到把翻译看作艺术的转化过程。在奈达翻译理论发展的第二个阶段即交际理论阶段, 他认为, 翻译是科学, 是对翻译过程的科学的描写。同时他也承认, 对翻译的描写可在三个功能层次上进行: 科学、技巧和艺术。在奈达逐渐向第三个阶段, 即社会符号学和社会语言学阶段过渡的过程中, 他越来越倾向于把翻译看作是艺术。他认为翻译归根到底是艺术, 翻译家是天生的。同时, 他把原来提出的“翻译是科学”改为“翻译研究是科学”。到了上世纪90 年代, 奈达又提出, 翻译基本上是一种技艺。他认为: 翻译既是艺术, 也是科学, 也是技艺。 纽马克对翻译的认识也经历了一定的变化。最初, 他认为, 翻译既是科学又是艺术, 也是技巧。后来他又认为翻译部分是科学, 部分是技巧, 部分是艺术, 部分是个人品位。他对翻译性质的阐释是基于对语言的二元划分。他把语言分为标准语言和非标准语言。说翻译是科学, 因为标准语言通常只有一种正确译法, 有规律可循, 体现了翻译是科学的一面。如科技术语。非标准语言往往有许多正确译法, 怎么挑选合适的译法要靠译者自身的眼光和能力, 体现了翻译是艺术和品位的性质。但译文也必须得到科学的检验, 以避免明显的内容和用词错误, 同时要行文自然, 符合语言环境要求。纽马克虽然认为翻译是科学, 但他不承认翻译作为一门科学的存在。因为他认为目前的翻译理论缺乏统一全面的体系, 根本不存在翻译的科学, 现在没有, 将来也不会有。 二、理论核心 奈达和纽马克都是在各自翻译实践的基础上, 为了解决自己实践中的实际问题, 提出了相应的翻译理论。实践中要解决的问题不同, 翻译理论也就各成一派。但毕竟每种实践都要有一定的规律存在, 因此两位的理论又有着不可忽视的相似。 奈达提出了著名的“动态对等”。他对翻译所下的定义: 所谓翻译, 是在译语中用最切近而又最自然的对等语再现源语的信息, 首先是意义, 其次是文体。这一定义明确指出翻译的本质和任务是用译语再现源语信息, 翻译的方法用最切近而又最自然的对等语。同时这一定义也提出了翻译的四个标准: 1 (1) 传达信息; 2 (2) 传达原作的精神风貌;

奈达翻译理论

Eugene Nida的翻译理论（转） 摘要：本文介绍了奈达的翻译思想，对其功能对等理论进行了分析，对奈达理论在翻译界的重要地位和重大影响进行了简单的评述。然后从五个方面探讨了当今翻译界对奈达思想的争论，介绍了争论中提出的新概念，并进行了一定的分析。最后，文章探讨了奈达思想形成的原因和在翻译实践中可以得到的启示。关键词：奈达功能对等争论启示一、奈达的翻译思想尤金·奈达（EugeneA.Nida）是公认的现代翻译理论奠基人之一。他供职于美国圣经协会，从事圣经翻译和翻译理论研究，其理论在中国甚至在全世界都有着重要的地位。奈达把翻译看作是一种艺术，力图把语言学应用于翻译研究。他认为，对翻译的研究应该看作是比较语言学的一个重要的分支；这种研究应以语义为核心包括翻译涉及的各个方面，即我们需要在动态对等的层次上进行这种比较。由于奈达把翻译和语言学密切联系，他把翻译的过程分成了四个阶段：分析（analysis），转换（transfer），重组（restructuring）和检验（test）。则翻译的具体过程就经过了下图的步骤。在这一过程中，奈达实际上是透过对原文的表层结构的分析，理解深层结构并将其转换重组到译文中。因此，在翻译中所要达到的效果是要让译文读者得到一个自然的译本。他曾说：“最好的译文读起来应不像翻译。”所以，他的翻译中另一个非常突出的特点就是“读者反应论”。他认为，翻译正确与否必须以译文的服务对象为衡量标准，并取决于一般读者能在何种程度上正确地理解译文。由此，他提出了他的“动态对等理论”。奈达在《翻译科学探素》(1964)一书中指出，“在动态对等翻译中，译者所关注的并不是源语信息和译语信息的对应关系，而是一种动态关系；即译语接受者和译语信息之间的关系应该与源语接受者和原文信息之间的关系基本相同”。他认为，所谓翻译，是指从语义到文体在译语中的最贴近而又最自然的对等语再现源语的信息（郭建中，2000）。所谓自然，是指使用译语中的表达方式，也就是说在翻译中使用归化，而不是异化。然而由于动态对等引起不少误解，认为翻译只要内容不要形式，在奈达的《从一种语言到另一种语言：论圣经翻译中的功能对等》一书中，他把“动态对等”的名称改为“功能对等”，并指出信息不仅包括思想内容，也包括语言形式。功能对等的翻译，要求“不但是信息内容的对等，而且，尽可能地要求形式对等”。二、奈达理论的评析毫无疑问，奈达是当代最著名也是最重要的翻译家之一，他的翻译理论，特别是他的功能对等理论在全世界的翻译界产生了重大的影响。约翰·比克曼（JohnBeekman）和约翰·卡洛（JohnCallow）的《翻译圣经》中说：“传译了原文意义和原文动态的翻译，称之为忠实的翻译。”而所谓“传译原文的动态”，就是指译文应使用目标语自然的语言结构，译文读者理解信息毫不费力，译文和原文一样自然、易懂。同时，他们也认为译者所要传达的是源语表达的信息，而不是源语的表达形式。在米尔德里德·L·拉森（https://www.sodocs.net/doc/297143073.html,rson）的《意义翻译法：语际对等指南》中，当谈到形式与意义时，拉森说得更为直截了当：“翻译基本上是改变形式（achangeofform）……是用接受语（目标语）的形式代替源语的形式。”她还表示，语言的深层结构与表层结构是不同的，我们所要翻译的是深层结构，即意义，而不是表层结构，即语言表达形式。然而正如郭建中(2000)所说：“内容与形式，意译与直译，以译文读者和译文为中心与以原作者和原文为中心……是翻译理论和翻译实践中一个永恒的辩论主题。”虽然奈达在全世界有着重要的影响和极高的地位，但他的理论却也一直引起各家争议。关于奈达理论的争议，主要是围绕以下五点展开的：（一）对等理论的适用范围这是长期以来对奈达翻译理论的争议得最多最激烈的问题。许多的翻译家都认为其理论是不适合文学翻译的。以诗歌翻译为例，林语堂曾经说过：“诗乃最不可译的东西。无论古今中外，最好的诗（而尤其是抒情诗）都是不可译的。”而著名诗人海岸（2005）在他的《诗人译诗，译诗为诗》中也指出：不同文化背景之间的符号系统只能在所指层面达到一定的共享，建立在绝对理解上的诗歌翻译只能是一种难以企及的梦想。特别是英语诗中的音韵节律及一些特殊的修辞手法等均不能完全传译，正如辜正坤在《中西诗比较鉴赏与翻译理论》一书中所言，