Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields

0.36 MB·

by  S. G. Magalhaes

Nonlinear susceptibility of a quantum spin glass under uniform transverse and
random longitudinal magnetic fields
S. G. Magalhaes1, C. V. Morais2, F. M. Zimmer3, M. J. Lazo4, F. D. Nobre5
1Instituto de Fisica, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, RS, Brazil∗
3Instituto de F´ısica e Matema´tica, Universidade Federal de Pelotas, 96010-900 Pelotas, RS, Brazil
2Departamento de Fisica, Universidade Federal de Santa Maria, 97105-900 Santa Maria, RS, Brazil
4Programa de Po´s-Gradua¸c˜ao em F´ısica - Instituto de Matema´tica, Estat´ıstica e F´ısica,
7 Universidade Federal do Rio Grande, 96.201-900, Rio Grande, RS, Brazil and
1 5 Centro Brasileiro de Pesquisas F´ısicas and National Institute of Science and Technology for Complex Systems,
0 Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil
2 (Dated: January 17, 2017)

n
Theinterplaybetweenquantumfluctuationsanddisorderisinvestigatedinaquantumspin-glass
a
model, in the presence of a uniform transverse field Γ, as well as of a longitudinal random field
J
hi, which follows a Gaussian distribution characterized bya width proportional to ∆. Theinterac-
6
tionsareinfinite-ranged,andthemodelisstudied throughthereplica formalism, within aone-step
1
replica-symmetry-breaking procedure; in addition, the dependence of the Almeida-Thouless eigen-

value λAT (replicon) on the applied fields is analyzed. This study is motivated by experimental
]
h investigationsontheLiHoxY1−xF4 compound,wheretheapplication ofatransversemagneticfield
c yieldsratherintriguingeffects,particularlyrelatedtothebehaviorofthenonlinearmagneticsuscep-
e tibilityχ3,whichhaveledtoaconsiderableexperimentalandtheoreticaldebate. Wehaveanalyzed
m twophysically distinct situations, namely,∆ andΓ considered asindependent,as well asthesetwo
- quantitiesrelated,asproposedrecentlybysomeauthors. Inbothcases,aspin-glassphasetransition
t isfoundatatemperatureT ,withsuchphasebeingcharacterized byanontrivialergodicity break-
a f
ing; moreover, T decreases by increasing Γ towards a quantum critical point at zero temperature.
t f
s Thesituation where∆ and Γare related [∆≡∆(Γ)]appearstoreproducebettertheexperimental
.
t observations on the LiHoxY1−xF4 compound, with the theoretical results coinciding qualitatively
a
m with measurements of the nonlinear susceptibility χ3. In this later case, by incre∗asin∗g Γ gradually,
χ3 becomes progressively rounded, presenting a maximum at a temperature T (T > Tf), with
- both the amplitude of the maximum and the value of T∗ decreasing gradually. Moreover, we also
d
show that the random field is themain responsible for thesmearing of thenonlinear susceptibility,
n
actingsignificantly insidetheparamagnetic phase,leadingtotworegimes delimitedbythetemper-
o ature T∗, one for T < T < T∗, and another one for T > T∗. It is argued that the conventional
c f
∗ ∗
[ paramagnetic state corresponds to T >T , whereas the temperature region Tf <T < T may be
characterized by a rather unusualdynamics, possibly including Griffiths singularities.

1
v Keywords: Spin Glasses, Critical Properties, Non-Linear Susceptibility, Replica-Symmetry Break-
3 ing.
7
3 PACSnumbers: 75.10.Nr,75.50.Lk,05.70.Jk,64.60.F-
4
0
1. I. INTRODUCTION symmetric (RS), or replica-symmetry-breaking (RSB),
0 solutions7. Although this may seem paradoxical, due to
7 thefactthatIsingSGHamiltoniansarenotformulatedin
Nature is quantum in its essence, although classical
1 terms of quantum operators, it is understood since their
theories may be employed under certain conditions. In
:
v statistical mechanics, the temperature range becomes binary variables capture an essential ingredient of many
i crucial for the use of classical or quantum approaches. physical systems, for which strong anisotropy fields are
X
present, leading to two significant states associated with
Typicalexamplesappearinmagnetism,wherethe useof
r the spin operators. However, in some compounds, the
a classicalmodels isjustified whenthe temperatureranges
quantum fluctuations controlled by a given parameter
are high enough, when compared to some reference tem-
(e.g., magnetic field, and/or doping) depress the tran-
perature. Inmanymagneticsystemsthequantumeffects
sition temperature T , changing radically the physical
become relevant, and should be taken into account, like f
those within the realm of quantum magnetism1. properties of the system8. In some cases, a field trans-
verse to such spin operators appears to be relevant, and
In what concerns spin glasses (SGs), models based
so, the simpler Ising SG Hamiltonian should be replaced
on Ising variables have been able to describe fairly
by a quantum type of Hamiltonian.
well, at least qualitatively, a wide variety of experimen-
tal behavior, even for sufficiently low temperatures2–5. The Ising dipolar-coupled ferromagnet LiHoF is a
4
Some of these results have been obtained at mean-field well known system in which quantum fluctuations be-
level,basedontheinfinite-range-interactionSherrington- come important by applying a transverse magnetic field
Kirkpatrick(SK)model6,either bymeansofthe replica- H , which induces quantum tunnelling through the bar-
t
2
rierseparatingthetwodegenerategroundstatesofHo3+ of any nature. In contrast, Ancona-Torres and collabo-
ions9. Moreover, disorder can be introduced, by replac- rators17 performed measurements for doping x = 0.167,
ing the magnetic Ho3+ ions by nonmagnetic Y3+ ones. notonlyofχ ,butalsoofχ ,aswellasofthe acsuscep-
3 5
Therefore,the resulting LiHo Y F compoundis con- tibility, reasserting T∗ as the SG critical temperature.
x 1−x 4
sideredas an idealgroundfor investigatingthe interplay On the theoretical side, the debate on this particular
betweenquantumfluctuationsanddisorderinIsingspins issue has also been intense (see, for instance, Refs.18,19).
systems10–12. Thesuggestionthataneffectivelongitudinalrandomfield
Inthese physicalsystems,coefficientsoftheexpansion (RF)h canbeinducedfromtheinterplayofatransverse
i
of the magnetization m, in powers of a small external applied field H , with the off-diagonalterms of the dipo-
t
longitudinalfieldH ,arequantitiesofgreatinterest2,3,13, larinteractionsinLiHo Y F ,representsaveryinter-
l x 1−x 4
esting hint to clarify these controversies concerning the
m=χ1Hl−χ3Hl3−χ5Hl5−··· , (1) meaningofT∗20–24. Accordingto the dropletpicture for
SGs, the rounded behavior of χ in the presence of the
3
corresponding to the linear susceptibility, field-inducedRFh is interpretedasasuppressionofthe
i
SGtransition20,21,similartowhatauniformfielddoesin
∂m
χ = , (2) that picture25,26. Onthe other hand, Tabei andcollabo-
1
∂Hl(cid:12)Hl→0 rators22workingwithinParisi’smean-fieldtheory7,using
(cid:12)
(cid:12) an effective Hamiltonian defined in terms of the field-
and nonlinear susceptibilities,(cid:12) inducedRFh andatransversefieldΓ[whereΓ=Γ(H )
i t
representssomemonotonicallyincreasingfunctionofH ],
1 ∂3m 1 ∂5m t
χ = ; χ = . (3) reproduced quite well the experimental behavior of χ ,
3 −3! ∂Hl3(cid:12)Hl→0 5 −5! ∂Hl5(cid:12)Hl→0 with an increasingly rounded peak at T∗ when Ht is en3-
(cid:12) (cid:12)
(cid:12) (cid:12) hanced.
Since measuremen(cid:12)ts of χ5 (and higher-orde(cid:12)r susceptibil- Itshouldberemarkedthattheresultsdescribedabove
ities) may become a hard task, very frequently in the
are based on a particular approach of the quantum SK
literatureone refersto χ3 asthe nonlinearsusceptibility. model proposed by Kim and collaborators27. In this
Moreover, χ is directly related to the SG susceptibil-
3 approach, the SK model is analyzed in the presence of
ity2,3,
a transverse field Γ, within the static approximation.
Mostly important, a region inside the SG phase was
2
χ =β2 χ , (4) found where the RS approximation is stable. Actually,
3 SG
− 3
(cid:18) (cid:19) the RSB solution exists only for sufficiently low values
of Γ, at temperatures lower than the SG transition tem-
with the latter representing an important theoretical
perature. The main consequence of this scenario is that
tool, being defined as
the sharp peak of χ , which signals the SG phase tran-
3
β sition temperature, does not coincide with the onset of
χ = ( S S S S )2 , (5)
SG N h i ji−h iih ji av RSB.Nevertheless,thisresultisalsohighlycontroversial,
Xi,j (cid:2) (cid:3) sinceotherworksindicatepreciselytheopposite,i.e.,the
RS approximation is unstable throughout the whole SG
where .. and[..] denote,respectively,thermalaverages
h i av phase (see, for instance, Refs.28,29) except, possibly, at
and an averageover the disorder.
the zero-temperature Quantum Critical Point (QCP)30.
In fact, the interplay between quantum fluctuations
Consequently,whenΓenhances,theRSBtransitiontem-
anddisorderstandsforthe physicaloriginoftheintrigu-
perature T decreases, so that, for finite temperatures,
ing behavior found in the magnetic susceptibility χ of f
3 the critical behavior appears in χ as
LiHo Y F , which has been the object of a consid- 3
x 1−x 4
erable experimental and theoretical debate. In the ab- χ [(Γ Γ (T))/Γ (T)]−δ′ , (6)
3 f f
sence of H , the LiHo Y F compound displays a ∝ −
t 0.167 0.833 4
sharp peak in χ at the temperature T , which resem-
3 f where Γ (T) denotes the critical value of Γ for a given
f
bles a conventional second-order SG phase transition14.
temperature, from which its corresponding value at the
Surprisingly, the sharp peak of χ becomes increasingly
3 zero-temperature QCP is obtained as lim Γ (T) =
T→0 f
rounded when the transverse field Ht is applied and en- Γ0.
c
hanced, so that the resulting smooth curve presents a
In the classical case, it is well known that χ is in-
SG
maximum located at a temperature T∗, with T∗ > T .
f versely proportionalto the Almeida-Thouless eigenvalue
Such behavior was initially interpreted as a changing in λ , the so-called replicon2,31. Therefore, the diverging
AT
the nature of the transition, from second order at T to
f behavior of χ at the SG transition,
3
first order at T∗14,15. More recently, J¨onsson and col-
laborators16 investigated the behavior of χ3 for dopings χ3 [(T Tf)/Tf]−γ , (7)
x = 0.165 and 0.0045, obtaining the same roundings of ∝ −
thepeakinbothcases;theseauthorsunderstoodthisbe- is a direct consequence of λ = 0 at T , occurring to-
AT f
haviorasanevidenceofabsenceofaSGphasetransition gether with the onset of RSB. Similarly, in the quan-
3
tumcase,oneexpectsthatthedivergenceofχ atΓ (T) (quantumlimit). Besidestheprogressivesmearingofχ ,
3 f 3
should coincide with the onset of RSB. the amplitude of its maximum also decreases as H in-
t
Indeed,thepresenceofaRFcanproducedeepchanges creases. Therefore, the effects of the RF triggered by
in the scenariodescribedpreviously. For instance, in the Ht, as suggested by Tabei and collaborators22, should
classicalSKmodel6,theRFinducestheRSorderparam- provokesimultaneouslyboth effects,i.e., the smearingof
eter q, which becomes finite at any temperature32,33. As the peak and the decrease of the maximum amplitude
a consequence, q versus temperature presents a smooth value of χ3. For the present fermionic Ising SG model,
behavior,beingnomoreappropriateforidentifyingaSG we have tested a relationship involving ∆ and Γ, partic-
transition in the SK model. Nevertheless, such a transi- ularly in the power-like form, ∆/J (Γ/J)B′, where J
∝
tion may still be related with the onset of RSB, signaled representsthe width of the Gaussiandistribution for the
sbpyeλctATto=th0e34t.emInpsepraitteuoreftihnicsr,etahseesdaesrivoanteivaepopfroqawchitehsrTef- ncoeunpt,lin1.g8s <{JiBj}′.<C2o.n5s,idweerinhgavtehebeiennteravballeftoor othbetaeinxpχo3-
from above; such an increase is the ultimate responsible as a function of temperature and Γ resembling qualita-
for the rounded maximum in χ3 at the temperature T∗, tively the experimental behavior for χ3 described above.
whichdoesnotcoincidewiththe SGtransitiontempera- As already mentioned, the transverse field Γ used in the
ture Tf (T∗ >Tf). In fact, the maximum value of χ3 at effective model to describe LiHoxY1−xF4 is expected to
T∗ reflects the effects of the RF inside the paramagnetic be related to the experimental applied field Ht11,12; in
phase, instead of the non-trivial ergodicity breaking of fact, at least for low Ht, Γ∝Ht2 (see, e.g., Ref.42).
the SG phase transition35. Therefore, one can raise the The paper is structured as follows: in Section II we
question of whether such scenario for χ , found in the define the model and find its grand-canonical poten-
3
classicalSK model, is robust in the corresponding quan- tial within the one-step RSB scheme; in Section III we
tum model, when the transverse field is considered, i.e., presentadetaileddiscussionoftheorderparameters,the
Γ = 0, and no longer independent from the RF, as pro- susceptibility χ3, and some phase diagrams. Finally, the
po6sed by Tabei and collaborators22. last section is reserved to conclusions.
Thepurposeofthepresentworkistostudythesuscep-
tibility χ , using the so-called fermionic Ising SG model
3 II. MODEL
in the presence of a longitudinal RF h and a transverse
i
field Γ. In this model, the spin operators are written
in terms of fermionic occupation and destruction op- The model is defined by the Hamiltonian
erators36,37, whereas the spin-spin couplings J and
{ ij} N N
rgarnanddom-cafineolndisca{lhpi}otefonltlioawl iGsaoubstsaiainneddiisntritbhuetifounnsc.tioTnhael H =− JijSˆizSˆjz− hiSˆiz −2Γ Sˆix, (8)
integral formalism, and the disorder is treated using the (Xi,j) Xi=1 Xi=1
replica method; moreover, the SG order parameters are
where the summation applies to all distinct pairs
obtained in the static approximation38,41 and investi- (i,j)
of spin operators, whereas the couplings J and mag-
gated within the one-step RSB scheme7. It should be P { ij}
neticfields h arequenchedrandomvariables,following
remarkedthatthefermionicIsingSGmodelisdefinedon { i}
independent Gaussian distributions,
the Fock space, where there are four possible states per
site: one state with no fermions,two states with a single 1/2
N N
fermion,andone state with twofermions, leadingto two P(J )= exp (J J /N)2 ,
ij 32πJ2 −32J2 ij − 0
nonmagnetic states. In particular, one can consider two (cid:20) (cid:21) (cid:20) (cid:21)
(9)
cases: the 4S model that allows the four possible states
and
per site and the 2S model, which restricts the spin op-
erators to act on a space where the nonmagnetic states 1/2
1 1
are forbidden. In the present work we will consider the P(h )= exp h2 . (10)
i 32π∆2 −32∆2 i
latermodel,byimposingarestrictiontoremovethecon- (cid:20) (cid:21) (cid:20) (cid:21)
tribution of these nonmagnetic states, i.e., taking into
In order to obtain susceptibilities [cf. Eqs. (2) and (3)],
account only the sites occupied by one fermion in the
oneintroducesalongitudinaluniformfieldH ,byadding
partition-function trace39,40. l
an extra term N H Sˆz in the Hamiltonian above.
In order to deal appropriately with the experimental − i=1 l i
Moreover,the spin operators in Eq. (8) are defined as
behavior of χ3 in the LiHoxY1−xF4 compound, we focus P
ourcalculationsonthe2Smodelbyproposingarelation- 1 1
Sˆz = [nˆ nˆ ] ; Sˆx = [cˆ† cˆ +cˆ† cˆ ] , (11)
shipbetween∆(the width ofthe distributionofrandom i 2 i↑− i↓ i 2 i↑ i↓ i↓ i↑
fields h ) and Γ, following the approach introduced by
i
Tabei and collaborators22. The main characteristic ob- where nˆ = cˆ† cˆ and nˆ = cˆ† cˆ , with cˆ† denoting a
i↑ i↑ i↑ i↓ i↓ i↓ i↑
served experimentally in χ concerns the peak for small creationoperatorfor a fermionwith spinup atsite i, cˆ
3 i↓
H (classical limit) being replaced by a rounded maxi- anannihilationoperatorforafermionwithspindownat
t
mum which becomes increasingly rounded for large H sitei,andsoon. Inthisfermionicproblem,thepartition
t
4
function is expressed by using the Lagrangian path in- and the Fourier representation may be used to express
tegral formalism in terms of anticommuting Grassmann
fields (φ and φ∗)37. The restriction in the 2S-model is 1 2π
Λ y = dx e−yα D[φ∗,φ ]exp[H ] .
imposedby meansofaKroneckerdeltafunction, insuch { } 2π α α α eff
α Z0 Z
a way to take into account only those sites occupied by Y
(17)
onefermion(n +n =1)inthepartition-function39,40.
i↑ i↓ Above, one has an “effective Hamiltonian” in replica
Therefore, adopting an integral representation for this
space,
delta function, onecanexpressthe partitionfunction for
both 2S and 4S models in the following form,
β2∆2 βJ
H = Aα +4 SzSz+ 0 m Sz
eff 0Γ 2 α γ 2 α α
Z y =es−22Nβµ D(φ∗φ) 1 2πdxje−yjeA{y} , Xα " Xα,γ Xα
{ } 2π
Z Yj Z0 + β2J2 q SzSz +2 q SzSz ,
(12)  αα α α αγ α γ
where Xα (Xα,γ)
  (18)
A y = βdτ φ∗ (τ) ∂ + yj φ (τ) with
{ }  jσ ∂τ β jσ (13)
−HZ0 φ∗jσ(τ)Xj,,φσjσ(τ) (cid:20). (cid:21) Aα0Γ =Xω ϕ†α(ω)(iω+yα+βΓσx)ϕα(ω), (19)
1
(cid:0) (cid:1)(cid:9) Sz = ϕ (ω)σzϕ (ω),
In the equations above, β = 1/T (T being the tem- α 2 α α
ω
perature), y = ix for the 2S-model, or y = βµ for X
j j j
the 4S-model, s = 2,4 denotes the number of states wheretheMatsubara’sfrequenciesareω = π, 3π, ,
± ± ···
per site allowed in each model, respectively, and µ is σx and σz denote the Pauli matrices, and ϕ†(ω) =
α
the chemical potential. Moreover, φ and φ∗ repre-
jσ jσ φ∗ (ω) φ∗ (ω) .
sentGrassmannfields atsite j andspinstate σ, whereas ↑α ↓α
H φ∗jσ(τ),φjσ(τ) standsforaneffectiveHamiltonianat (cid:16) Moreover, the(cid:17)functional integrals over qαγ, qαα and
a given value of the integration variable τ. mαinEq.(15)havebeenevaluatedthroughthesteepest-
(cid:0) (cid:1) descent method, yielding
Now,weusethereplicamethod, sothatstandardpro-
cedures lead to the grand-canonical potential per parti-
m = S ; q = SzSz ; q = (Sz)2 , (20)
cle25, α h αi αγ h α γi αα h α i
with .. representingathermalaverageovertheeffective
1 1 Z y n 1 h i
J,h Hamiltonian of Eq. (18).
βΩ= lnZ y = lim hh { } ii − ,
J,h
−Nhh { }ii −N n−→0 n Herein, the problem will be analyzed within one-step
(14) RSB Parisi’s scheme7, in which q =p, and the replica
αα
where .. denoteaveragesoverthequenchedrandom
hh iiJ,h matrix elements are parametrized as
variables. Thereplicatedpartitionfunction Z y n
J,h
hh { } ii
becomes
q if I(α/a)=I(γ/a)
q = 1 (21)
αγ
∞ ∞ n (q0 if I(α/a)=I(γ/a)
Z y n J,h =es−22Nβµ dqαγ dqαα 6
hh { } ii N
Z−∞(Yα,γ) Z−∞αY=1 where I(x) gives the smallest integer greater than, or
∞ n equal to x.
dm exp[NβΩ (q ,q ,m )] The parametrization given by Eq. (21) allows to
α n αγ αα α
×
Z−∞α=1 perform the sums over replica indexes and then, the
Y
(15) quadratictermsinEq.(18)canbelinearizedthroughthe
introductionofnewauxiliaryfields. Fromthis point,the
where α (α = 1,2, ,n) stands for a replica in- integrals over the Grassmann variables in Eq. (17) can
···
dex, (α,γ) denotes distinct pairs of replicas, and = be performed and the sum over Matsubara’s frequencies
(βJ N/2π)n(n+1)/2. Assuming the static approxNima- can be obtained, like in Ref.40. Therefore, the resulting
tion38,41, one obtains grand-canonicalpotential is obtained from Eq. (14),
p
(βJ)2 βJ
βΩ (q ,q ,m )= β2J2 q2 βΩ= [(x 1)q2 xq2+p2]+ 0m2 ln2
n αγ αα α − αγ 2 − 1 − 0 2 −
β2J2 βJ (Xα,γ) (16) (s−2)βµ 1 Dzln Dv[K(z,v)]x ,
q2 0 m2 +lnΛ y , − 2 − x
− 2 αα− 2 α { } Z (cid:26)Z (cid:27) (22)
α α
X X
5
where 1
Γ/J=0.00
(s 2) 1 ΓΓ//JJ==01..5000
K(z,v)= − cosh(βµ)+ Dξcosh[ Ξ(z,v,ξ)], p
2 0.8
Ξ(z,v,ξ)=[βh(z,v,ξ)]2+(βΓ)Z2, p 0.5 q1 (∆a/)J=0.0
h(z,v,ξ)=βJ[ 2q0+(∆/J)2z+ 2(q1 q0)v 0.6 0 q0
− 0 0.6 1.2
+ 2(pp−q1)ξ], p 0.4 δ=q1-q0
(23)
p
and Dx dx e−x2/2/√2π (x=z, v or ξ). 0.2
≡
The parameters q , q , x, p, and m are obtained
0 1
through extremization of the grand-canonical potential 0
inEq.(22),andresultsfor2Sand4Smodelsareobtained 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
T/J
by considering s = 2 and s = 4, respectively. Moreover,
the RS solution is recovered for q0 = q1 = q, and x = 0. 1 ∆/J=0.00
In this way, the linear susceptibility of Eq. (2) becomes 1 q ∆/J=0.25
1 ∆/J=0.50
χ =β[p q +x(q q )]7.
1 1 1 0 0.8
− −
As usual, the RSB parameters, the magnetization m, 0.5 (b)
and the quadrupolar parameter p, form a set of coupled q Γ/J=0.0
equations, to be solved simultaneously. Particularly, the 0.6 0
0
parameterp is quite dependent onΓ, andinfact, for the 0 0.6 1.2
δ=q -q
2S model, p 1 only as Γ 0; it should be mentioned 0.4 1 0
→ →
that p plays an important role in the nonlinear suscep-
tibility χ . This aspect represents a crucial difference of
3
0.2
thepresentinvestigationwithrespecttopreviousone,by
Kimandcollaborators[cf. Ref.27],wheretheparameters
q1,q0,andx(orevenqintheRSsolution)donotdepend 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
on p. In the present work, for the above one-step RSB
T/J
solution,χ willbeobtainedbynumericalderivatives;for
3
the RSsolution,ananalyticalformforχ3 ispresentedin FIG. 1: The one-step RSB parameter δ ≡ q1 −q0 is pre-
Appendix A. As mentioned before, in order to deal with sented versus the dimensionless temperature T/J, for ∆=0
the LiHo Y F compound, we will restrict ourselves andtypicalvaluesofΓ/J [panel(a)],aswellasforΓ=0and
x 1−x 4
to the 2S model, considering J0 =0. typical values ∆/J [panel (b)]. The parameters q1, q0, and p
arealsoexhibitedversustemperatureintherespectiveinsets;
one notices that the quadrupolar parameter p becomes rele-
vantonlyinthecasesΓ>0,forwhichitdecreasesbylowering
III. RESULTS
the temperature. Due to theusual numerical difficulties, the
low-temperature results [typically (T/J)<0.05)] correspond
Hence, considering the 2S model, in this section we to smooth extrapolations from higher-temperaturedata.
analyze the behavior of the nonlinear susceptibility χ ,
3
either by varying the temperature (for fixed typical val-
ues of ∆/J and Γ/J), or by considering joint variations verifiedinFig.1(b)byincreasingthe widthofthe distri-
in some of these parameters. Since χ3 is directly related butionofrandomfields∆(Γ=0). IntheHamiltonianof
with the order parameters that appear in the thermo- Eq.(8)oneseesthatthelimitΓ=0correspondstoasim-
dynamic potential of Eq. (22), we first discuss the SG ple,diagonalizable,quantumIsingSGmodel,whereonly
order parameters q1 and q0, as well as the quadrupolar the spin components Sˆiz are present, leading to a trivial
parameter p. Moreover,the onset of RSB is signalled by quadrupolar parameter p = 1 (for all temperatures), as
δ = q1 q0 > 0, which locates the freezing temperature shown in the inset of Fig. 1(a). This particular case,
−
Tf; it should be mentioned that for Γ = 0 and ∆ = 0, for which the SG parameters are exhibited in Fig. 1(b),
one has that T =√2J. yields results qualitatively similar to those found in the
f
In Fig. 1 we exhibit the one-step RSB parameter previous study of the classical SK model in the presence
δ q q versus the dimensionless temperature T/J, ofaGaussianrandomfield,carriedinRef.35. Onenotices
1 0
≡ −
for typical choices of Γ/J and ∆/J. The corresponding that for (∆/J)> 0 the RS order parameter q = q = q
0 1
parameters q , q , and p are also presented versus T/J isinduced[cf. theinsetofFig.1(b)],presentingasmooth
1 0
in the respective insets. From Fig. 1(a) one notices that behavior versus temperature; consequently, the freezing
the freezing temperature gets lowered for increasing val- temperature T can only be found by means of the RSB
f
ues of the transverse field Γ, up to the zero-temperature scheme, with the SG transition coinciding with the on-
QCP locatedat Γ0 =2√2J (∆=0)40; a similar effect is set of the parameter δ. However, for Γ > 0, the spin
c
6
500 500
Γ/J=0.0 T/J=0.2
Γ/J=1.0 T/J=1.0
400 (∆a=)0 103 0.24((T-Tf)/Tf)-1 400 (∆b=)0 103 0.4((Γ - Γf)/Γf)-1
102 102
101 101
300 300
3Jχ3 10010-3 10-2 10-1 3χJ 3 10010-3 10-2 10-1
200 (T - Tf)/Tf 200 (Γ - Γf)/Γf
100 100
0 0
1.2 1.4 1.6 1.2 1.6 2 2.4 2.8 3.2 3.6 4
T/J Γ/J
FIG. 2: Plots of the dimensionless nonlinear susceptibility [computed from Eq. (3)] are exhibited for ∆ = 0: (a) J3χ3
versus T/J for two different values of Γ/J; (b) J3χ3 versus Γ/J for two different temperatures. In all cases one notices sharp
divergencesofχ3,signalingevidentphasetransitions. Thecorrespondingcriticalexponentsareestimatedthroughlog-logplots
(shown in the respective insets), where in each case, the fittingproposal is represented bya dashed-dotted line (see text).
500 500
∆/J=0.000 ∆/J=0.000
∆/J=0.001 ∆/J=0.001
(a) ∆/J=0.005 (b) ∆/J=0.005
∆/J=0.010 ∆/J=0.010
400 Γ=0.00 400 T/J=1.00
103 103
102 γ 102 δ’
300 300
101 101
3Jχ3 100 3Jχ3 100
200 10-3 10-2 10-1 200 10-3 10-2 10-1
(T - T*)/T* (Γ - Γ*)/Γ*
100 100
0 0
1.4 1.41 1.42 1.43 1.44 1.45 1.46 1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58
T/J Γ/J
FIG. 3: The behavior of the dimensionless nonlinear susceptibility (in two typical cases exhibited in Fig. 2) is presented for
increasingvaluesof∆/J: (a)J3χ3 versusT/J,for(Γ/J)=0.0;(b)J3χ3 versusΓ/J,for(T/J)=1.0. Ineachcaseonenotices
∗ ∗
a rounded peak for (∆/J) > 0, with its maximum value located at a temperature T [panel (a)], or at a transverse field Γ
[panel (b)], such that its height decreases for increasing values of ∆/J. The log-log plots in the respective insets show that
the divergences of Eq. (7), leading to the exponent γ [inset of (a)], or in Eq. (6), leading to the exponent δ′ [inset of (b)], are
fulfilled only for (∆/J)=0.
components Sˆx become important, so that one expects T is lowered by increasing values of Γ/J. The criti-
i f
a nontrivial behavior for the quadrupolar parameter p; cal exponents associated with the divergences of χ may
3
indeed, p should decrease for increasing values of Γ (at be obtained by log-log plots, as shown in the insets of
a fixed temperature), whereas for a fixed Γ, it decreases Fig. 2. In the inset of Fig. 2(a) we have verified that the
by lowering the temperature, as shown in the inset of behavior of Eq. (7) (represented by the dashed-dotted
Fig. 1(a). line) fits well the region 0.001 < (T T )/T < 0.1,
f f
−
with the same critical exponent, γ = 1, for both val-
Inthepresentproblem,clearphasetransitionsmaybe ues Γ/J = 0.0 (full line) and Γ/J = 1.0 (dotted line),
verified only for ∆ = 0, like those exhibited in Fig. 2. suggesting that the transverse field Γ should not change
From the χ3 plots of Fig. 2(a) one confirms two impor- the universality class of the exponent γ. It is important
tant features shown in Fig. 1(a), concerning the behav- to mention that this estimate coincides with the well-
ior of the order parameters q0, q1 and δ: (i) The freez- known value found for the SK model2. In Fig. 2(b) one
ing temperature Tf, signaled by the divergence of χ3 sees divergences of χ3 at given values of Γ [defined as
in Fig. 2(a), coincides with the onset of RSB indicated Γ (T) in Eq. (6)], for two typical fixed temperatures;
f
by the parameter δ of Fig. 1(a); (ii) The temperature
7
1.2
T/J=1.35; (Γ*=0.42J) 120 T=1.405J (Γf=0.15J, Γ*=0.24J)
1 (∆a/)J=0.25 T/J=1.00; TT(Γ//JJf===011...532100J;; ,(( ΓΓΓ***===011...604813JJJ))) 100 (b) TTT===111...433050JJJ (((ΓΓΓfff===000...14697649J,JJ Γ,, *ΓΓ=**==0.008..226092JJJ)))
T/J=0.60; (Γf=1.39J, Γ*=1.95J) T=1.20J (Γf=0.93J, Γ*=1.108J)
T/J=0.20; (Γf=1.91J, Γ*=2.40J) T=1.00J (Γf=1.30J, Γ*=1.495J)
0.8 80 T=0.60J (Γf=1.794J, Γ*=2.02J)
T=0.20J (Γf=2.22J, Γ*=2.479J)
3χJ3 0.6 3χJ3 60 8
6
0.4 ← 40 ↑ 4
↑ ↑ 2
↑
0.2 20 0
↓
↓ ↓ 1↓ 1.5 ↓ 2 ↓ 2.5 3
0 0
0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3
Γ/J Γ/J
FIG.4: (a)ThedimensionlessnonlinearsusceptibilityisrepresentedversusΓ/J,fortypicalfixedtemperaturesandanonzero
width for the random fields [(∆/J) = 0.25]. The divergences that occur for ∆ = 0 at Γ (T) [following Eq. (6)], signalled by
f
∗
arrows in some cases, get smoothened due to the random fields, so that their corresponding maxima [located at Γ (T)] are
∗
shifted towards higher values of the transverse field, i.e., Γ (T) > Γ (T). (b) The behavior of the dimensionless nonlinear
f
susceptibility is shown versus Γ/J, for typical fixed temperatures, by considering a particular relation involving ∆ and Γ
[(∆/J) = 0.02(Γ/J)2]; the inset represents an amplification of the region for higher values of Γ/J. In all cases, the maxima
[locatedatΓ∗(T)]appearshiftedwithrespecttotheonsetofRSB[locatedatΓ (T),signalledbyarrowsinsomecases]towards
f
higher values of thetransverse field, i.e., Γ∗(T)>Γ (T).
f
like in Fig. 2(a), these divergences coincide with the on- 0.001 < (Γ Γ∗)/Γ∗ < 0.1 cannot be fitted by Eq. (6)
set of RSB indicated by the parameter δ. One notices with a critic−al exponent δ′ = 1.0, as shown in the inset
that, as one approaches zero temperature [cf., e.g., the of Fig. 3(b).
case (T/J) = 0.2], the divergence at Γ (T) approaches
f In Fig. 4 we represent the dimensionless nonlinear
the one that occurs at the QCP, Γ0 = 2√2J30. In
c susceptibility χ3 versus Γ/J, for typical fixed tempera-
the inset of Fig. 2(b), the critical behavior described by
tures, in two cases: (a) Γ/J and ∆/J as independent
Eq. (6) (represented by the dashed-dotted line) was ful-
quantities [Fig. 4(a)]; (b) Imposing a relation involv-
filled in both cases, showing a good agreement in the
ing ∆ and Γ [Fig. 4(b)]. In Fig. 4(a) we consider a
region 0.001<(Γ Γ (T))/Γ (T)<0.1, with the same
f f fixed value for the width of the Gaussian random fields
exponentδ′ =1for−thetwovaluesoftemperaturesinves-
[(∆/J) = 0.25], showing that the sharp SG transitions
tigated, (T/J)=0.2 (full line) and (T/J)=1.0 (dashed
occurringfor∆=0,signaledbydivergencesofχ atthe
3
line). Hence, similarly to the results of Fig. 2(a) con-
correspondingcriticalvalues Γ (T)[accordingto Eq.(6)
f
cerning the critical exponent γ of Eq. (7), the present
andshownbyarrowsinsomecurves],changeintosmooth
estimates of δ′ suggest that the temperature should not curveswith maximaatΓ∗(T),shifted to highervaluesof
change the universality class of this later exponent. Γ, i.e., Γ∗(T)>Γ (T). Following the proposalof Ref.22,
f
In agreement with the previous study of the SK fordealingproperlywiththeexperimentalbehaviorofχ
3
model in the presence of a Gaussian random field35, the inthe LiHo Y F compound,weanalyzedthe present
x 1−x 4
smoothening of χ is verified in Fig. 3 for the cases system by imposing a relation involving ∆ and Γ, i.e.,
3
(∆/J) > 0. For instance, Fig. 3(a) displays χ versus ∆ ∆(Γ). According to the experimentalinvestigations
3
≡
T/J, for increasing values of ∆/J, in the case Γ = 0, of Ref.14, such a relation should satisfy certain require-
showing that the divergent peak of the nonlinear sus- ments,e.g.,∆shouldincreasemonotonicallywithΓ,and
ceptibility is replaced by a broad maximum at a tem- one should get ∆ = 0 for Γ = 0. The simplest proposal
perature T∗. One observes that such a peak becomes obeying these conditions comes to be a power function,
smoother, decreasing its height for increasing values of (∆/J) = A(Γ/J)B, where A and B are fitting param-
∆/J. Particularly,in the inset of Fig. 3(a) one sees that eters. Herein, these parameters were computed by ad-
the temperature range 0.001 < (T T∗)/T∗ < 0.1 no justing our results to those of the experiments of Ref.14,
−
longer can be fitted by Eq. (7) with the critical expo- leading to the optimal values A = 0.02 and B = 2. In
nent γ = 1.0, in the cases (∆/J) > 0. In a similar way, Fig.4(b)weexhibitthedimensionlessnonlinearsuscepti-
Fig. 3(b) shows χ versus Γ/J, for (T/J) = 1.0, consid- bility,versusΓ/J,fortypicalfixedtemperatures,bycon-
3
ering the same values for ∆/J of Fig. 3(a); again, the sideringthisparticularrelationinvolving∆andΓ. Inall
peak of nonlinear susceptibility gets flattened due to the cases,themaxima[locatedatΓ∗(T)]appearshiftedwith
presence of an applied random field, now displaying a respect to the onset of RSB [located at Γ (T)] towards
f
maximum at Γ∗. Consequently, in such cases the region highervalues ofthe transversefield,i.e., Γ∗(T)>Γ (T).
f
8
0.6
∆/J = 0.1
1.6 Γ/J=1.0 1.6 Γ=0 ∆∆//JJ==00..01 Γ/J=1.0 ∆/J = A (Γ/J)B
(a) 1.2 χ3 (b)
0.8
1.2 0.4
0.4 b
χ3 (∆/J=0.1) 0 ↓ λAT b = λAT
1.2 ↑ 1.6
0.8
b
0.2
0.4
b (∆/J=0.1)
λAT b
0 ↓ λAT (∆/J=0.1) ↓ b = λAT (∆=0) 0 ↓ ↓ λAT
1 1.2 1.4 1.6 1.8 1 1.2 1.4 1.6 1.8
T/J T/J
FIG. 5: The softening of the nonlinear susceptibility is illustrated by means of the denominator of q2, i.e., q2 ∝ b−1,
b=1−2(βJ)2I0(Γ)[cf. Eq.(24)],whichappearsintheexpressionofχ3calculatedinAppendixA,withintheRSapproximation.
The Almeida-Thouless eigenvalue λAT, associated with the onset of RSB and defining the SG critical temperature Tf is also
shown, for comparison. (a) Results for (Γ/J) = 1.0 are exhibited for two typical values of ∆/J, namely, (∆/J) = 0.0 and
(∆/J) = 0.1, in the case where Γ and ∆ are independent; similar results are presented in the inset for (Γ/J) = 0. The full
lines represent the cases (∆/J) = 0.0, showing that λAT and the denominator b coincide, becoming zero at the temperature
T . The cases (∆/J) = 0.1 show that b is always positive, presenting a smooth minimum around a temperature T∗, leading
f
to the rounding of χ3, whereas λAT becomes zero at a lower temperature Tf. (b) Results for (Γ/J) = 1.0 and (∆/J) = 0.1
areshown,bycomparingthecasewherethesetwoquantitiesareconsideredasindependent(dashedlines),withtheonewhere
they follow the relation proposed in in in Fig. 4(b) [(∆/J) = 0.02(Γ/J)2] (dotted lines). In all cases, the arrows locate the
freezing temperature T .
f
ThemostimportantnoveltyofFig.4(b)[tobecontrasted [either in Eq. (A1), or in Eq. (A13)], which may present
with the results of Fig. 4(a)], concerns the fact that the a divergence at finite temperatures. Moreover, one can
amplitudeofthemaximumofχ decreasesforincreasing show that for ∆ = 0, the so-called “dangerous” eigen-
3
values of Γ/J, and consequently, for decreasing temper- value31 in Eq. (A10) is equal to the denominator of q ,
2
atures. i.e., λ =1 2(βJ)2I (Γ), even for Γ>0. The mecha-
AT 0
−
Recent studies in the compound LiHo Y F sug- nism behind the flattening of the χ peak at T∗ is illus-
x 1−x 4 3
gested that the transverse field Γ introduced in the tratedinFig.5,whereweplotthequantitybofEq.(24),
Hamiltonian of Eq. (8) should be related to the experi- λ , and χ , versus the dimensionless temperature, for
AT 3
mental applied field in a real system, H 11,12; in fact, at typicalchoices ofΓ/J and∆/J. Results for Γ and∆ in-
t
least for low H , Γ H2 (see, e.g., Ref.42). Hence, con- dependent are presentedin Fig. 5(a); the full lines [cases
t ∝ t
sidering a new dimensionless variable, Ht (Ht Γ/J), (∆/J)=0.0]showthatthequantitiesbandλAT become
wehaveverifiedthatthesamequalitativebehav≡iorshown zerotogether,being associatedwiththe divergenceofχ3
in both Figs. 4(a) and 4(b) occur in representaptions of [accordingtoEq.(7)], signallingthe SGphase-transition
the dimensionless nonlinear susceptibility χ3 versus Ht. temperature Tf. However, the results for (∆/J) = 0.1
Consequently,Fig.4(b)preservestheagreementwithex- show that such a small value for the width of the RFs
perimental observations, showing that besides the pro- distribution yields b>0, which presents a smooth mini-
gressive smearing of χ , the amplitude of its maximum mumaroundatemperatureT∗,beingdirectlyassociated
3
also decreasesas the realfield Ht increases,as suggested withtheroundingbehaviorofχ3;ontheotherhand,one
by Tabei and collaborators22. has λAT = 0 at a temperature Tf, with Tf < T∗. In
In Appendix A we have calculated χ3 analytically, Fig. 5(b) we present the denominatorb and λAT, for the
within the RS approximation, for both Γ > 0 [cf. caseswhereΓand∆areindependent(dashedlines),and
Eqs. (A1) and (A2)] and Γ = 0 [cf. Eq. (A13)]. In wherethesequantitiesarerelatedthroughthepowerlaw
thesecalculations,animportantquantityemerged,given (∆/J) = 0.02(Γ/J)2. In this later case, since one has
in Eq. (A2) for Γ>0, as (∆/J)>0 for any (Γ/J)>0, the denominator b will al-
waysdisplayaminimumvaluearoundatemperatureT∗,
q = 2(βJ)2I0(Γ) ; b=1 2(βJ)2I (Γ) . (24) higher than Tf. Particularly, by means of this relation,
2 b − 0 highervaluesofΓimplyonhighervaluesof∆,increasing
the values of b at the minima, resulting in a decrease in
Notice thatthe denominatorbmaybecome zero,leading the amplitude of the maxima of χ .
3
to a divergence in the nonlinear susceptibility; it should
bementionedthatq istheonlyquantityappearinginχ In Fig. 6 we present phase diagrams T/J versus Γ/J
2 3
9
1.6 1.6
0.2
(a) (b)
1.4 ∆/J=0.25 1.4 T* ∆/J 0.1
1.2 T* 1.2 0
Tf 0 1 2 3
1 1 Γ/J
PM2
T/J 0.8 T/J 0.8
Tf PM1 PM2
0.6 0.6 PM1
RSB SG RSB SG
0.4 0.4
0.2 0.2
0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
Γ/J Γ/J
FIG. 6: Phase diagrams T/J versusΓ/J showing thefrontiers separating theSGand paramagnetic phases (full lines), which
represent the behavior of the temperature Tf for increasing values of Γ, located by the onset of RSB, i.e., λAT = 0. The
∗
dashed lines correspond to the temperature T , associated with the maximum of the nonlinear susceptibility χ3, and herein
interpreted as a crossover between two distinct regions of the paramagnetic phase (PM1 and PM2). (a) Phase diagram for
(∆/J)=0.25, in thecase where Γ and ∆are independent. (b)Phase diagram for which Γ and ∆ follow therelation proposed
in Fig. 4(b) [(∆/J)=0.02(Γ/J)2], whose parabolic behavior is presented in theinset. Due to the usual numerical difficulties,
thelow-temperature results [typically (T/J)<0.05)] correspond tosmooth extrapolations from higher-temperature data.
showing a decrease in the temperature T for increas- Γ(i.e.,theclassicalregime),hasalsofoundatemperature
f
ing values of Γ (full lines). These lines delimit the SG T∗, associated with the rounded maximum of χ , with
3
phase and were identified with the onset of RSB, by set- T∗ >T . In this case, T∗ was interpreted as an effect of
f
ting λ =0; throughout the whole SG phases ones has theRFsactinginsidetheparamagneticphase,insteadof
AT
λ <0. The temperature T∗ (dashed lines), associated some type of non-trivial ergodicity breaking. Herein, we
AT
withthemaximumofthenonlinearsusceptibilityχ ,sig- claim that the temperature T∗, although it may be also
3
nalsacrossoverbetweentworegionsoftheparamagnetic affected by the transverse field Γ, should be interpreted
phase (PM1 and PM2), as will be discussed next. The in a similar manner. Hence, along the line signaled by
phase diagramshown in Fig. 6(a) corresponds to a fixed T , the growth of Γ produces an enhancement of quan-
f
valueof∆[(∆/J)=0.25],andwasobtainedbyconsider- tum fluctuations, which become increasingly dominant
ing Γ and ∆ as independent quantities. In this case, one as compared with thermal fluctuations, driving the non-
notices that the two lines (full and dashed lines) remain trivial ergodicity breaking of the SG phase transition to
essentially parallel to one another, up to zero tempera- aQCP.FortemperaturesintheregionT <T <T∗,the
f
ture, where the full line reaches a QCP, which appears enhancement of quantum fluctuations by Γ along with
to be shifted towards lower values of Γ, when compared the spinfluctuationsdue to the RFs insidethe paramag-
with the QCP for ∆ = 0, Γ0 = 2√2J 2.828J40. The netic phase create two distinct scenarios, more precisely
c ≈
case shown in Fig. 6(b) corresponds to Γ and ∆ follow- concerning the PM1 region, as discussed next: (a) For
ing the relation (∆/J) = 0.02(Γ/J)2 (see inset), so that fixed ∆ [e.g., Fig. 6(a)], one has Γ and ∆ independent,
for Γ = 0, one has ∆ = 0, giving T∗ = T . By increas- so that the smearing of the nonlinear susceptibility is
f
ing values of Γ, the width of RFs also increases, leading caused only by the RFs, leading to the effect that the
to a rounded peak in the nonlinear susceptibility, yield- full and dashed lines remain essentially parallel to one
ingT∗ >T ,andconsequently,the regionPM1emerges. another,up to zerotemperature; (b) The phase diagram
f
DuetothejointincreaseofbothΓand∆,asshowninthe of Fig. 6(b), for which ∆ and Γ are related through the
inset, the region PM1 gets enlarged up to zero tempera- parabolicbehaviorshowninthe inset, the appearanceof
ture,whereonegetsaQCP,shiftedtowardslowervalues T∗ occurs for Γ > 0 (i.e., ∆ > 0). Hence, the region
of Γ as compared with the QCP for ∆ = 0, similarly to PM1 starts very narrow for low Γ, and gets enlarged for
the one occurring in Fig. 6(a). increasing values of Γ, showing that the rounding of χ
3
is dominated by the enhancement of the RFs, leading to
ItshouldbeemphasizedthatthetemperatureT∗plays
spinsfluctuationsduetotheRFsinsidetheparamagnetic
aroledifferentfromT ,inthe sensethatitdoesnotcor-
f phase.
respond to a phase transition, but rather to a crossover
between two distinct regions of the paramagnetic phase.
The previous analysis of the SK model in the presence
of a Gaussian random field Ref.35, which should corre-
spond herein to the regionof high temperatures and low
10
IV. CONCLUSIONS susceptibility. Hence, the random field acts significantly
inside the paramagnetic phase, leading to two regimes
delimited by the temperature T∗, one for T < T < T∗
We have investigated a quantum spin-glass model in f
(called herein as PM1), and another one for T > T∗
the presence of a uniform transverse field Γ, as well as
(denominated as PM2). In the paramagnetic regime for
of a longitudinal random field h , the later following a
i T > T∗ one should have weak correlations and conse-
Gaussian distribution characterized by a width propor-
quently, the usual paramagnetic type of behavior. How-
tional to ∆. The model was considered in the limit
ever, close to T∗, and particularly for temperatures in
of infinite-range interactions and studied through the
the range T < T < T∗, one expects a rather nontrivial
replica formalism, within a one-step replica-symmetry- f
behavior in real systems, as happens with experiments
breaking procedure. The spin-glass critical frontier, sig-
in the compound LiHo Y F , resulting in very con-
naledby the temperature T ,wasidentified with the on- x 1−x 4
f troversial interpretations11,16,17,20–24. Hence, as already
setofreplica-symmetrybreaking,calculatedthroughthe
arguedintheanalysisoftheSKmodelinthepresenceof
Almeida-Thouless eigenvalue (replicon) λ , i.e., by set-
AT Gaussian random field35, the line PM1–PM2 may not
ting λ = 0. In this approach, the whole spin-glass
AT
characterize a real phase transition, in the sense of a
phase becomes characterized by λ < 0, and conse-
AT
diverging χ , but the region PM1 should be certainly
quently, it was treated through replica-symmetry break- 3
characterized by a rather nontrivial dynamics. As one
ing. Such analysis was motivated by experimental inves-
possibility, one should have a growth of free-energy bar-
tigations on the LiHo Y F compound. In this sys-
x 1−x 4
riers in this region, leading to a slow dynamics, whereas
tem, the applicationofa transversemagnetic field yields
onlybelowT thenontrivialergodicitybreakingappears,
rather intriguing effects, particularly related to the be- f
typical of RSB in SG systems. Also, one could have
haviorofthe nonlinearmagneticsusceptibilityχ ,which
3
Griffiths singularities along PM1, which are found cur-
have led to a considerable experimental and theoretical
rently in disordered magnetic systems, like site-diluted
debate.
ferromagnets43,ferromagnetinarandomfield44,classical
We have analyzed two physically distinct situations,
Ising spin glasses45, and also claimed to occur in quan-
namely, ∆ and Γ considered as independent, as well
tum spin glasses46–48. Whether such curious properties
as these two quantities related, as proposed recently
may appear throughout the region PM1 in the present
by some authors (see, e.g., Ref.22). In both cases, we
problem, represents a matter for further investigation.
have found a spin-glass critical frontier, given by T
f
≡ In fact, recent experiments in the above compound for
T (Γ,∆), with such phase being characterizedby a non-
f x = 0.045 strongly suggest this picture49: these authors
trivial ergodicity breaking. In the first case, for ∆ fixed,
claim an “unreachable” transition due to an ultra-slow
we have found that T (Γ,∆) decreases by increasing Γ
f dynamics (of the order107 times slowerthanthe ones of
towards a quantum critical point at zero temperature,
conventionalspin-glass materials) and argue that such a
whereas in the second, we have found a similar behavior
dynamics should be caused by a Griffiths phase between
for this critical frontier, with ∆ changing according to
the paramagnetic and spin-glass phases.
variationsin Γ. In this later case,we have takeninto ac-
count previous experimental investigations14 which sug- Next, we discuss some contributions of the present
gest that a relation of the type ∆ ∆(Γ) should satisfy work, as compared to previous theoretical approaches in
≡
certain requirements, e.g., ∆ should increase monoton- this problem. (i) The analysis of Ref.27 did not take
ically with Γ, and one should get ∆ = 0 for Γ = 0. into account the random field, which in our view, rep-
Although such a relation may not be unique, the sim- resents a key ingredient for an appropriate description
plest proposal following such conditions appears to be a of the properties of LiHo Y F . Moreover, as it was
x 1−x 4
powerfunction, (∆/J)=A(Γ/J)B. Inthe presentwork, shownherein,theRSBSGparameters,togetherwiththe
theparametersAandB werecomputedbyadjustingour magnetization m, and the quadrupolar parameter p, all
results to those of the experiments of Ref.14, leading to form a set of coupled equations, to be solved simulta-
the the optimal values A=0.02 and B =2. neously. The approach of Ref.27 considered p as inde-
We have shown that the present approach, consider- pendent from the remaining parameters; this could be
ing the relation (∆/J) = 0.02(Γ/J)2, was able to repro- directlyrelatedwiththe curiousresultconcerninga part
duce adequately the experimental observations on the of the SG phase characterizedby stability of the replica-
LiHoxY1−xF4 compound, with theoretical results coin- symmetric solution, along which these authors find the
ciding qualitatively with measurements of the nonlinear rounded maximum of χ . (ii) The study of Ref.22 has
3
susceptibility χ3. As a consequence, by increasing Γ considered an effective Hamiltonian characterized by an
gradually, our results indicate that χ3 becomes progres- extra two-body interacting term (as compared with the
sively rounded, presenting a maximum at a temperature Hamiltonianusedherein),coupling spinoperatorsin the
T∗ (T∗ > Tf); moreover, both amplitude of the maxi- x and z directions. Moreover, these authors have sug-
mum and the value of T∗ diminish, by enhancing Γ. gested a relation ∆ ∆(Γ), which due to the Hamilto-
≡
From the analysis where ∆ and Γ are considered as nian employed, turned out to be slightly different from
independent, we have concludedthat the randomfield is ours, e.g., (∆/J) = A(Γ/J)B, with an exponent B < 1.
the main responsible for the smearing of the nonlinear Theresultsobtainedhereinforthenonlinearsusceptibil-