Dynamic spin-glass behavior in a disorder-free, two-component model of quantum frustrated magnets

6 Pages·0.22 MB·English

by  Frederic Mila

Dynamic spin-glass behavior in a disorder-free, two-component model of quantum
frustrated magnets
Fr´ed´eric Mila(a) and David Dean(b)
(a) Institut de Physique Th´eorique, Universit´e de Lausanne, CH-1015 Lausanne, Switzerland
2 (b) Laboratoire de Physique Quantique, Universit´e Paul Sabatier, 31062 Toulouse, France
0
0
2
Motivatedbytheobservationofaspin-glasstransitioninalmostdisorder-freeKagomeantiferromag-
n
nets, and by the specific form of the effective low-energy model of the S=1/2, trimerized Kagome
a
antiferromagnet, we investigate the possibility to obtain a spin-glass behavior in two-component,
J
disorder-freemodels. Weconcentrateonatoy-model,amodifiedAshkin-Tellermodelinamagnetic
3
fieldthatcouplesonlytoonespeciesofspins,forwhichweprovethatadynamicspin-glassbehavior
2
occurs. Thedynamicsof themagnetization is closely related tothat of theunderlyingIsing model

in zero field in which spins and pseudo-spins are intimately coupled. The spin-glass like history
]
l dependenceof themagnetization is aconsequence of theageing of theunderlying Ising model.
e
-
r
t PACS Nos : 75.10.Jm, 75.10.Nr, 75.50.Lk
s
.
t
a
m I. INTRODUCTION ing periodic Josephson arrays in a transverse magnetic
field, but these models are not directly related to the
-
d The experimental investigation of quantum magnets Kagome AF.
n is undergoing rapid progress with the synthesis of new In parallel, a lot of progress has been made in the un-
o
and better controlled samples. Of special interest are derstanding of the spectrum of the S=1/2 AF Heisen-
c
[ frustrated magnets, for which very unusual behavior has berg model on various frustrated lattices, in particular
been reported [1]. In particular, a behavior reminiscent the Kagome and the pyrochlore ones [1]. So far, it is
2 ofspinglasseshasbeenreportedinanumberofKagome well established that frustration can have two effects: It
v
6 antiferromagnets(AF)[1,2]. Whiletheroleofdisorderin canopenagaptotripletexcitations[5],like forthe non-
0 thesephenomenacannotbeexcludedyet,thepersistence frustrated spin 1 chain [6], but it can also lead to a pro-
4 ofthis behaviorinprogressivelycleaner samplescalls for liferationoflow-lyingsingletsinsidethis gap,likeforthe
9 explanations in terms of disorder-free models. S=1/2 Kagome antiferromagnet [7]. These singlets can
0 The possibilityofa spin-glassbehaviorin the spin1/2 beinterpretedasRVB(ResonatingValenceBond)states
1 Heisenberg model on the Kagome lattice without any [8–10], and they could lead to a power-law behavior of
0
disorder was first suggested in the pioneering work of the low-temperature specific heat [11–13]. Experimental
/
t Chandra et al [3] in 1993. Since the classical Heisenberg systems known so far have a larger spin however (3/2,
a
m model on the Kagome lattice does not exhibit a spin- 5/2,...) [1], and there is room for new physics in these
glass behavior, the main problem one is facing is how cases since the presence of a singlet-triplet gap is un-
-
d to include the quantum aspects of the problem into a likely given the rather small value already reported for
n description in terms of classical variables for which one S=1/2. Possibleimplicationsofthisstrangespectrumre-
o can use standard techniques to study the dynamics. In garding in particular a possible spin-glass behavior have
c Ref. [3], the authors assumed that quantum fluctuations notbeendiscussedyet,mainlyduetothelackofmethods
:
v select coplanar configurations, which led them to con- to attack this problem.
i centrateontheanisotropicXYversionofthemodel. For In this paper, we continue the quest initiated in Ref.
X
this model,they suggestedthe presenceatfinite temper- [3] for spin-glass behavior in the disorder-free, quan-
ar atureof3-spinorder,andofaglassytransitionrelatedto tum Heisenberg model on the Kagome lattice. However,
the binding of non-abelian defects that are expected to buildingontherecentresultsobtainedonthelow-energy
be the natural point defects associated with this type of spectrumof the model, we proposeanother approximate
order. For details, the reader should consult Ref. [3]. To way of including quantum fluctuations into a classical
proveordisprovethis scenarioturned outto be difficult, description. The startingpointisthe effective modelob-
and whether this indeed provides an explanation of the tained in Ref. [14] for the spin 1/2 Heisenberg model on
spin-glassbehavior of Kagomeantiferromagnetswithout the trimerized Kagome lattice (see Fig. (1)). This is
introducing any disorder remains unsettled. Since then, a modified version of the spin 1/2 Heisenberg model on
clearevidenceofglassinesshasbeenreportedbyChandra the Kagome lattice in which the exchange integrals take
etal[4]foranotherclassofdisorder-freemodels describ- two different values J and J′ accordingto the patternof
1
Fig. (1). This is actually the relevant description of the the spins are polarized, and the pseudo-spins will adapt
Kagome layers in SrCr Ga O since the presence to keep the energy as low as possible. Hence the result-
9p 12−9p 19
ofatriangularlayerbetweenpairsofKagomelayerslead ing magnetization at low temperature is expected to be
to two types of bonds with precisely the pattern of Fig. significant. However,in a zero-field cooled (ZFC) exper-
(1). Then,since the groundstateofatriangleis fourfold iment, the system is first quenched without a magnetic
degenerateandcanbedescribedbytwospin1/2degrees field,anditwilluse bothdegreesoffreedomto minimize
of freedom, the total spin~σ and the chirality pseudospin the energy. When the magnetic field is switched on, in
~τ, it was shown in Ref. [14] that the low-energyeffective order to polarize, the system will have to overcome the
Hamiltonian in the limit J′ J can be written intimatemixingofspinsandpseudo-spinstogetasignif-
≪
icantpolarization. Thisislikelytotakeaverylongtime,
H˜ =(J′/9) H˜iσjH˜iτj, H˜iσj = ~σi.~σj, andtheapparentsusceptibilitywillbemuchsmallerthan
<Xi,j> <Xi,j> the one measured in a FC experiment.
H˜τ =(1 2(α τ−+α2 τ+))(1 2(β τ−+β2τ+)) (1) The analysis of the model of Eq. (1) with the spin
ij − ij i ij i − ij j ij j and the pseudo-spintreated as classicalHeisenberg vari-
where<i,j >denotespairsofnearestneighbors. InH˜τ, ables is a considerable task, and before starting such an
ij
endeavor, one would like to know whether the scenario
α and β are complex parameters that take the values
ij ij
outlined in the previous paragraph can indeed lead to a
1, exp(2πi/3) or exp(4πi/3) depending on the bond (for
spin glass behavior. In fact, other models, like the fully
details, see Ref. [14]).
frustratedXYmodel[15]orsomevectormodels[16],can
be describedintermsofspinandchiralityvariables,and
no spin-glass behavior was ever reported for these mod-
els. Consequently the rest of the paper is devoted to a
detailedanalysisofatoymodeltotestwhetherthepres-
ence of two local degrees of freedom can indeed lead to
a spin-glass behavior. The simplest model of this kind is
a modified Ashkin-Teller model defined by the Hamilto-
nian:
=J S S T T h S (2)
i j i j i
H −
hXi,ji Xi
In this Hamiltonian, S and T are Ising variables that
FIG.1. Trimerized Kagome lattice showing the difference i i
describe the spin and the pseudo-spin respectively. Note
betweenstrongbonds(blacktriangles)andweakbonds(white
that the magnetic field h is coupled only to the spin de-
triangles).
gree of freedom. Since frustration has already been in-
In the case of the trimerized Kagome lattice, quan- cluded in the model as an extra degree offreedom, there
tum fluctuations thus introduce an extra local degree of is no need to work on a frustrated lattice any more, and
freedom in the Hamiltonian. For the general case, the for simplicity we study this model on a square lattice.
derivation is no longer valid since J′/J is not a small This model is of course a very dramatic simplification
parameter, but the low-energy spectrum found in ex- sincethe Heisenbergspinsandpseudo-spinsarereplaced
actdiagonalizationsofsmallclusters[7]isstillconsistent by Ising variables. Still, as we shall see, the physics of
withsuchadescription: Theexponentiallylargenumber thismodelisveryrich,andtoalargeextentconfirmsthe
of low-lying singlets suggest that there must be another simple picture of the previous paragraph. Shortcomings
quantumnumber inadditiontothe spinto classifythese thatmightbeovercomebygoingtothemorephysicalde-
eigenstates. This is precisely the role played by the chi- scription in terms of Heisenberg spins will be discussed
rality pseudo-spin in the trimerized Kagome lattice. A in the last section.
detailed comparison of both spectra has actually shown
that they are very similar indeed [10].
II. MONTE-CARLO RESULTS
We would therefore like to propose a new approach
to the spin-glassbehaviorofdisorder-freequantummag-
nets in terms of models with two-local degrees of free- Let us first discuss the equilibrium properties of this
dom. Our basic motivation for concentrating on such model. If h = 0, this model is equivalent to the antifer-
models is that they are a priori good candidates to ex- romagneticIsingmodelafterthelocalgaugetransforma-
hibitspin-glassbehaviorevenwithoutanydisorderinthe tion σi SiTi, and all states have an additional degen-
Hamiltonian. The basic argument is the following. In a eracy of≡2Nsites since σi = 1 (resp. 1) can be achieved
−
field-cooled (FC) experiment, in which one quenches the with (Si,Ti) = (1,1) and ( 1, 1) (resp. (1, 1) and
− − −
sample from high temperature in a magnetic field, the ( 1,1)). In fact, for any value of h, the partition func-
−
systemwillautomaticallychooseconfigurationsinwhich tioncanbefactorizedasZ =ZIsingZS whereZIsing isthe
2
partition function of the Ising model on a square lattice, curves depend on t and t (this dependence should of
m w
andZ isthepartitionfunctionofparamagneticspinsin course vanish if t was infinite) but only weakly in a
S m
a magnetic field. Accordingly, the free energy per site f large parameter range, as we shall explain below.
is given by: To gain some insight into the origin of this behavior,
it is very useful to study the time evolution of the mag-
f =f k T ln(2cosh(βh)) (3)
Ising− B netizationatfixedtemperature fordifferentvaluesoftw.
with β = 1/k T. The magnetization per site is defined Typical results are given in Fig. (3). The most salient
B
as usually by m = ∂f/∂h, and since f does not features are: i) A muchfaster increase at shorttimes for
Ising
depend on h, we ob−tain m = tanh(βh). So if the sys- the FCexperiment(tw =0)thanforthe ZFCones;ii)A
tem has reachedits equilibrium state, the magnetization similarityofthe shape ofthe ZFCcurves. Infact, avery
smoothly saturates as the temperature goes to zero. good scaling can be obtained if we plot the magnetiza-
However, as we shall see below, this equilibrium state tion as a function of tm/tw (see inset of Fig. (3)). Such
mightbe verydifficultto reachdepending onthe history a scaling is typical of ageing phenomena [17].
of the system. To be specific, let us consider the follow-
ing protocol: The quench takes place at t = 0, the field 0.5
0.05
is switched on after a waiting time t , and the magne-
w
tization is measured after a measuring time tm elapsed 0.4 T/J =.6 m
after the field has been switched on. The time elapsed
between the quench and the measurement is denoted by
0.3 0
t. In a FC experiment, t = 0 and t = t , while in a 0 t /t 0.5
w m m m w
ZFC one, t >0 and t=t +t .
w w m
0.2 ZFC
FC
0.4
0.1
Equilibrium
m=tanh(βh)
0.3
0
0 1000 2000 3000 4000
t
m
0.2 Tg
FIG. 3. Time dependence of the magnetization at given
T
FC, t =500 c temperature for various measurement conditions (FC and
m
0.1 ZFCwithtw =500,1000,2000,3000). Inset: PlotoftheZFC
ZFC, t =500, t =500 data as a function of tm/tw.
w m
0
0 1 2 3
III. ANALYTICAL APPROACH: AGEING AND
T/J
PERSISTENCE
FIG.2. Temperature dependenceof the magnetization for
different measurement conditions. TounderstandthebehavioroftheZFCandFCexperi-
mentsletusfirstconsiderthedynamicsoftheunderlying
Tomimicsuchexperiments,wehaveperformedMonte Ising model in the absence of a magnetic field. After a
Carlo simulations of the model of Eq. (2). The elemen- quench below the Curie temperature Tc, the model has
tary step consists in flipping either a spin or a pseudo- a spontaneous staggered magnetization in the variables
spinaccordingto Glauber prescription. The site andthe σi in terms of statics. However, in the absence of a field
variable (spin or pseudo spin) are chosen randomly, and thatbreaksthesymmetrybetweenthetwogroundstates,
the time unit corresponds to a number of steps equal to no global magnetization develops for an infinite system
the number of degrees of freedom, i.e. twice the number andthe system staysout of equilibrium. Domains of the
of sites. In all numerical experiments reported below, twophasesformandcoarsenwithacharacteristiclength
the starting configuration is completely random, corre- scale √t [18] as the evolution is via diffusion of the do-
sponding to an infinite temperature, and when it is not mainwallsandcoalescenceofdomains. Theergodictime
switched off, the magnetic field h is equal to 0.2 in units terg for such a system scales with the linear size L of the
of J. Typical results for the magnetization as a function systemlikeL2,andthusaslongast terg,thesystemis
≪
of temperature are shown in Fig. (2) for a system of out of equilibrium and has no global magnetization. We
size400 400. Belowa temperatureT whichissmaller have checked that for T = .6J and L =400, the ergodic
g
×
thantheCurietemperatureT 2.27J oftheunderlying time is ofthe orderof4000. Since this regimeis the only
c
≃
Ising model, there is a clear difference between the ZFC one accessible for very large samples, we will limit our
and FC measurements: The ZFC magnetization drops discussion to that regime.
quiteabruptly,asintypicalspin-glasses. Notethatthese
3
Another point worth noticing about the dynamics of To understandthe magnetizationprocesswe therefore
theIsingmodelisthat,aftertheshortinitialstagewhere onlyhavetoconsiderc (t ,t )fortheIsingmodel. After
0 m w
spins with 3 or 4 parallel neighbors have disappeared, the firstfew steps, domainscanbe identified, and0-sites
flips take place only at sites with two neighbors up and areontheboundaryofthedomains. Onasquarelattice,
two neighbors down. For such sites, the effective field they correspond to the diagonal portions of the bound-
coming from the coupling to the neighbors is equal to ary. Now,afteratimet thecharacteristicsizeofthedo-
w
zero, and we call them 0-sites. Since there is no energy mainsisoforder√t . Consequentlythetotalnumberof
w
cost to flip the spin σ of such a site, the probability to domains is of order1/t , and the total length of domain
i w
flipaccordingtoGlauberdynamicsis1/2. Sotheflipping walls is proportional to 1/√t . When t t we re-
w m w
≪
rate shouldbe half the concentrationof 0-sites. We have mark two important points: (i) the domain walls appear
checked numerically that this is indeed the case after a locally flat, the effects of surface tension and hence cur-
few sweeps. For T =.6J, this is true as soon as t>40. vature are negligible and hence the domain walls diffuse
Let us now turn to an analysis of the problem in the (aszeromodesaretheonlyonesgeneratingdynamicsby
presence of a magnetic field. Let us forget the first few this stage),a givenpointona domain wallwill therefore
steps and concentrate on times where flips take place at diffuse a length √t ; (ii) one canneglect the interaction
m
0-sites. If the magnetic field is small compared to J, a between domain walls, that is to say the coalescence of
condition usually fulfilled in experiments, the dynamics domains. The total number of spins which flip between
of the Ising spins σ will be essentially unaffected, and t and t +t is therefore proportional to √t total
i m m w m
×
the magnetic field will just favor configurations where length of domain walls. We therefore find that at short
the spins S are parallel to the field [19]. Now let us times t (compared to t ) one has
i m w
supposethata0-siteremainssoforawhile. Theequation
that governs the appearance of a magnetization for the tm
c (t ,t )=Const. (7)
S spins can be easily deduced. If we denote by p (resp. 0 m w rt
+ w
p )theprobabilityforS tobeup(resp. down),Glauber
−
where the constant should be independent of t . This
dynamics implies that p satisfies the equation w
+
provides an excellent fit of the data at short times (see
dp eβh e−βh Fig. (4). WehavecheckedthatConst. 0.05atT =.6J
+ = p p (4) ≃
dt e−βh+eβh − − e−βh+eβh + is indeed independent of tw. This argument provides a
verysimpleexplanationofthefactthatthelargert ,the
w
The magnetization, which is related to p and p by
+ − slower the initial increase in the magnetization.
p =(1+m)/2 and p =(1 m)/2, is thus given by
+ −
−
0.35
m=tanh(βh)(1 exp( t/τ)) (5)
− − FC
0.3
where the relaxation time τ is equal to 1 in the chosen
m=tanh(βh)(1−(τ/t)θ), τ=2.6, θ=.2
time units. This is a very short time, especially consid- 0.25
ering the very small flipping rate which is achieved af-
ter a few steps (already below 10−2 after 100 sweeps for 0.2
m
T =.6J). Underthesecircumstances,thesitesthathave
0.15
been 0-sites in the presence of the magnetic field should
ZFC t =50
have enough time to reach equilibrium, and on average 0.1 w
their magnetization should be equal to tanh(βh). So if m=tanh(βh)(1−(τ/t)θ), τ=62, θ=.22
0.05
we call c0(tm,tw) the proportion of sites that have been m=Const. (t/t)1/2, Const.=.05
m w
0-sitesbetweentw andtw+tm thissimpleargumentleads 0
0 1000 2000 3000 4000
to the prediction that
t
m(t ,t )=tanh(βh)c (t ,t ) (6)
m w 0 m w
FIG. 4. Fits of the FC and ZFC data at T/J = 0.6 with
Notethatc (t ,t )dependsonlyonthedynamicsofthe thehelp of Eqs. (7-9).
0 m w
underlying Ising model and not on the magnetic field.
To check this prediction, we have calculated c (t ,t ) In the case where t = 0, the above argument can-
0 m w w
for different t corresponding to our ZFC numerical ex- not be applied as it is because of the initial steps, where
w
periments. The agreement is very good - the curves are flipsoccuratsiteswith3and4parallelneighborsaswell,
indistinguishable from the the ZFC calculations of m on andbecausedomainsarenotpresentrightaway. Stillthe
thescaleofFig. (3)-andwehavecheckedthatitremains samekindofreasoningsuggestthateveryspinwhichhas
soaslongast isnottoosmallsothatthemagnetization flipped has an average magnetization tanh(βh). There-
w
is indeed controlled by 0-sites. This analysis shows that fore one has m(t,0)=tanh(βh)(1 p(t,0)) where p(t,0)
−
the ageing behavior of the model of Eq. (1) is indeed istheprobabilitythatagivenspinhasnotflippedbefore
closely related to the dynamics of the underlying Ising time t. The quantity p(t,0) has received much atten-
model. tion in the literature [22] and is called the probability of
4
persistence. Extensivenumericalsimulationshaveshown by T (t )/J 8/ln(t ). So, unlike the standard ther-
g m m
thatforlargetimesp(t) 1/tθ,whereθisthepersistence modynamic sp∼in glass transition, the freezing tempera-
∼
exponentwhichhasbeenmeasuredasθ 0.2[20]forthe ture depends ont . This estimate of T agreeswith our
m g
≃
twodimensionalIsingmodel. Onethereforededucesthat Monte Carloresults. If one takesa measurementtime of
t = 500, as for the curves shown in Fig. (2), one finds
m
m(t,0)=tanh(βh)(1 (τ/t)θ) (8) thatT /J 1.287,closetothe value ofT showninFig.
− g ∼ g
(2).
Here τ is a microscopic time scale related to the flipping Another original aspect of our data with respect to
rate of the spins. This provides and excellent fit of the standard spin glasses is the minimum of the ZFC mag-
data except at very short times (see Fig. 4). netization at a temperature far below T . To under-
g
In the case of tw nonzero, and for t tw, one may stand this, one may make an approximate decomposi-
≫
adopt a coarse grained point of view and cut the sys- tion of the magnetization in terms of the activated com-
tem up into units of size √tw. These blocks may be re- ponent ma within the domains and the component gen-
garded as effective spins of the sign of the average block erated by the domain walls m . We find m (t ,t ) =
d a m w
staggered magnetization. This is related to block per- tanh(βh)(1 exp( α(h)t /τ ))(solvingtheGlauberdy-
m a
sistence introduced in [21]. Moreover, the characteristic namicsexpl−icitly f−ora single spinwith its four neighbors
time scale τ(tw) for these effective spins is proportional fixed parallel shows that α=1+cosh(2βh)+O(1/τ2)).
a
to the time it takes to reverse the staggered magnetiza-
Asm isgeneratedwithindomainsandhenceinthebulk
a
tion of a given block, since this is done by diffusion one
of the system (that is to say not at domain interfaces),
hasτ(t ) t . Onemaynowapplythereasoningofthe
w w it is very weakly dependent on t . Another type of ac-
∼ w
case tw =0 to obtain tivated process may occur at domain interfaces where
a spin with three neighbors antiparallel and one parallel
m(t,t )=tanh(βh)(1 (τ(t )/t)θ) (9)
w − w flips. Theenergybarrierforthisis4J. Thisgivesanother
characteristic time τ∗ with a corresponding time depen-
for t t . This again provides a reasonable fit of the a
data ≫(seewFig. (4)), and we have checked that τ(tw) is dasenint Fteimg.p(e2r)atounreefiTnd∗(stTm∗)//JJ =∼04./64ln3(6tmw)h.icFhocrortrmes=po5n0d0s
proportional to t .
w to the minimum seen on the ZFC curve. For T T∗,
To summarize, the very slow onset of magnetization ≪
the system behaves as if J is infinite on the experimen-
in ZFC experiments follows scaling laws typical of age-
tal time scale and only domain wall diffusion occurs, the
ing, and this is the origin of the spin-glass behavior.
only energy involved being the external field energy h.
This ageing is a consequence of the very slow march to-
Here the dynamics is well described by Eq. (6), hence
wardsequilibriumoftheunderlyingmodelinwhichspins
the measured magnetization m(t ,t ) increases as T is
andpseudo-spinsarecoupledintoaneffectiveIsingspins m w
lowered below T∗.
since the physical field, which couples only to the spin
Boththeseeffectsaredynamicinnatureandshowthat
and not to the pseudo-spin, does not act as a symmetry
the behavior we have observed is typical of the out of
breaking field for that model.
equilibriumdynamicsobservedinspin-glasses. Thiscon-
clusion is in fact consistent with another aspect of our
data,namely the sensitivity of the results to the dynam-
IV. DISCUSSION
ics used inthe simulation. To performMonte Carlosim-
ulations, we have made the assumption that spins and
While the difference between FC and ZFC measure-
pseudo-spins flip independently of each other. However
mentsofthemagnetizationisverytypicalofspin-glasses,
if one allows simultaneous flips as well, the behavior will
a bona fide spin-glass has many other characteristics
be very different: These moves will allow the magnetiza-
which are not shared by the present model.
tion to developinside the Ising domains, and the system
First of all, for an infinite system the freezing tran-
willpolarizeafter onlya few sweeps. Inother words,the
sition of a standard spin glass is believed to be a ther-
breaking of ergodicity is related to the dynamics.
modynamic transition, and the freezing temperature is
Another way to check how different the present model
expected to be independent of the protocol. In the
is from standard spin glasses is to study the tempera-
presentcase,inordertoflip,aspininsideadomainmust
turedependence ofthenon-linearsusceptibility,whichis
be able to cross an energy barrier of 8J. The Arrhe-
expectedtodivergeatthetransitioninaspin-glass. Pre-
nius law gives therefore a characteristic flipping time of
liminary results [23] indicate that the non-linear suscep-
τ exp(8J/T). Such activated flips are only observed
a tibility is not singular in the present case. The absence
∼
afteratime ofmeasurementoforderτ . Onemaythere-
a of another thermodynamic singularity below the Curie
fore define a temperature depending on the time scale
temperature T is also clear from Eq. (3).
c
of the measurement T (t ), such that if τ t such
g m a ≫ m However, other characteristic aspects of spin-glasses
activatedspinflipsdonotoccurandhencetheequilibra-
are also shared by the present model. For instance,
tionofthemagnetizationwithindomainsisnotpossible.
the thermo-remnant magnetization m shows a very
TMR
The crossoverbetweenthe tworegimesis thereforegiven
strong dependence on the time t elapsed between the
w
5
quench and the switching off of the magnetic field: The [9] C. Zeng and V.Elser, Phys. Rev.B 51, 8318 (1995).
longer t , the slower the decay. A detailed analysis of [10] M. Mambrini and F. Mila, Eur. Phys.J. B, in press.
w
these results along the same line is in progress. [11] P. Sindzingre et al, Phys. Rev.Lett. 84, 2953 (2000).
Coming back to our original purpose, namely the ex- [12] A. P. Ramirez et al, Phys. Rev.Lett. 84, 2957 (2000).
planationofthelow-temperaturebehaviorofKagomean- [13] B.BernuandG.Misguich,preprint(cond-mat/0003381).
tiferromagnets, the present model has both merits and [14] F. Mila, Phys.Rev. Lett. 81, 2356 (1998).
drawbacks. The very clear difference between FC and [15] J. Villain, J. Phys. C 10, 4793 (1977).
ZFCmagnetizationsis themostinterestingaspectofthe [16] For a review, see H. Kawamura, cond-mat/0111060.
[17] J.P. Bouchaud, L.F. Cugliandolo, J. Kurchan, and M.
results. It shows that the presence of an extra degree
M´ezard, in Spin Glasses and Random Field, edited by
of freedom which is not coupled to the external field
A.P. Young(World Scientific, Singapore, 1998).
can indeed lead to a glassy behavior at low tempera-
[18] A.J. Bray, Adv.Phys. 43, 357 (1994).
tures. However most of the differences with standard
[19] Oneshouldnotethatafieldcoupledtobothspinswould
spin-glasses are problematic in that respect: No experi-
break the symmetry between the phases and allow even
mental indication of a dependence of T on the protocol
g an infinitesystem to reach equilibrium in a finitetime.
was reported, and the non-linear susceptibility is indeed
[20] J. Kockelkoren, A. Lemaˆıtre and H. Chat´e, preprint
enhanced close to T . This is not a final blow however.
g (cond-mat/0004429).
In fact, these differences with standard spin glasses all [21] S. Cueille, C. Sire, J. Phys. A. 30, 23, L791, (1997).
depend on the fact that flipping simultaneously a spin [22] S. N. Majumdar, Curr. Sci., 77, 370, (1999).
and a pseudo-spin leaves the Ising spin unchanged. This [23] M. Ferrero, F. Becca, F. Mila, unpublished.
symmetry will not be present in more realistic models
where spins and pseudo-spins are treated as Heisenberg
variables, while the underlying mechanism for the diffi-
culty that the system will have to magnetize after be-
ing cooled in zero-field is still expected to apply. More
realistic models with two degrees of freedom treated as
Heisenberg spins are therefore good candidates for effec-
tive models to get a spin-glass behavior in disorder free
magnets. Work is in progress along these lines.
We thank M. Ferrero and F. Becca for allowing us to
quote some results of Ref. [23] before publication. This
work was partially supported by the Swiss National Sci-
ence Foundation under grant number 21-63749.00.
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