Nonlinear susceptibility of a quantum spin glass under uniform transverse and random longitudinal magnetic fields
S. G. Magalhaes, C. V. Morais, F. M. Zimmer, M. J. Lazo, F. D. Nobre
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Nonlinear susceptibility of a quantum spin glass under uniform transverse andrandom longitudinal magnetic fields
S. G. Magalhaes , C. V. Morais , F. M. Zimmer , M. J. Lazo , F. D. Nobre Instituto de Fisica, Universidade Federal do Rio Grande do Sul, 91501-970 Porto Alegre, RS, Brazil ∗ Instituto de F´ısica e Matem´atica, Universidade Federal de Pelotas, 96010-900 Pelotas, RS, Brazil Departamento de Fisica, Universidade Federal de Santa Maria, 97105-900 Santa Maria, RS, Brazil Programa de P´os-Gradua¸c˜ao em F´ısica - Instituto de Matem´atica, Estat´ıstica e F´ısica,Universidade Federal do Rio Grande, 96.201-900, Rio Grande, RS, Brazil and Centro Brasileiro de Pesquisas F´ısicas and National Institute of Science and Technology for Complex Systems,Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil (Dated: April 3, 2018)The interplay between quantum fluctuations and disorder is investigated in a quantum spin-glassmodel, in the presence of a uniform transverse field Γ, as well as of a longitudinal random field h i , which follows a Gaussian distribution characterized by a width proportional to ∆. The interac-tions are infinite-ranged, and the model is studied through the replica formalism, within a one-stepreplica-symmetry-breaking procedure; in addition, the dependence of the Almeida-Thouless eigen-value λ AT (replicon) on the applied fields is analyzed. This study is motivated by experimentalinvestigations on the LiHo x Y − x F compound, where the application of a transverse magnetic fieldyields rather intriguing effects, particularly related to the behavior of the nonlinear magnetic suscep-tibility χ , which have led to a considerable experimental and theoretical debate. We have analyzedtwo physically distinct situations, namely, ∆ and Γ considered as independent, as well as these twoquantities related, as proposed recently by some authors. In both cases, a spin-glass phase transitionis found at a temperature T f , with such phase being characterized by a nontrivial ergodicity break-ing; moreover, T f decreases by increasing Γ towards a quantum critical point at zero temperature.The situation where ∆ and Γ are related [∆ ≡ ∆(Γ)] appears to reproduce better the experimentalobservations on the LiHo x Y − x F compound, with the theoretical results coinciding qualitativelywith measurements of the nonlinear susceptibility χ . In this later case, by increasing Γ gradually, χ becomes progressively rounded, presenting a maximum at a temperature T ∗ ( T ∗ > T f ), withboth the amplitude of the maximum and the value of T ∗ decreasing gradually. Moreover, we alsoshow that the random field is the main responsible for the smearing of the nonlinear susceptibility,acting significantly inside the paramagnetic phase, leading to two regimes delimited by the temper-ature T ∗ , one for T f < T < T ∗ , and another one for T > T ∗ . It is argued that the conventionalparamagnetic state corresponds to T > T ∗ , whereas the temperature region T f < T < T ∗ may becharacterized by a rather unusual dynamics, possibly including Griffiths singularities.Keywords: Spin Glasses, Critical Properties, Non-Linear Susceptibility, Replica-Symmetry Break-ing. PACS numbers: 75.10.Nr, 75.50.Lk, 05.70.Jk, 64.60.F-
I. INTRODUCTION
Nature is quantum in its essence, although classicaltheories may be employed under certain conditions. Instatistical mechanics, the temperature range becomescrucial for the use of classical or quantum approaches.Typical examples appear in magnetism, where the use ofclassical models is justified when the temperature rangesare high enough, when compared to some reference tem-perature. In many magnetic systems the quantum effectsbecome relevant, and should be taken into account, likethose within the realm of quantum magnetism .In what concerns spin glasses (SGs), models basedon Ising variables have been able to describe fairlywell, at least qualitatively, a wide variety of experimen-tal behavior, even for sufficiently low temperatures .Some of these results have been obtained at mean-fieldlevel, based on the infinite-range-interaction Sherrington-Kirkpatrick (SK) model , either by means of the replica- symmetric (RS), or replica-symmetry-breaking (RSB),solutions . Although this may seem paradoxical, due tothe fact that Ising SG Hamiltonians are not formulated interms of quantum operators, it is understood since theirbinary variables capture an essential ingredient of manyphysical systems, for which strong anisotropy fields arepresent, leading to two significant states associated withthe spin operators. However, in some compounds, thequantum fluctuations controlled by a given parameter(e.g., magnetic field, and/or doping) depress the tran-sition temperature T f , changing radically the physicalproperties of the system . In some cases, a field trans-verse to such spin operators appears to be relevant, andso, the simpler Ising SG Hamiltonian should be replacedby a quantum type of Hamiltonian.The Ising dipolar-coupled ferromagnet LiHoF is awell known system in which quantum fluctuations be-come important by applying a transverse magnetic field H t , which induces quantum tunnelling through the bar-rier separating the two degenerate ground states of Ho ions . Moreover, disorder can be introduced, by replac-ing the magnetic Ho ions by nonmagnetic Y ones.Therefore, the resulting LiHo x Y − x F compound is con-sidered as an ideal ground for investigating the interplaybetween quantum fluctuations and disorder in Ising spinssystems .In these physical systems, coefficients of the expansionof the magnetization m , in powers of a small externallongitudinal field H l , are quantities of great interest , m = χ H l − χ H l − χ H l − · · · , (1)corresponding to the linear susceptibility, χ = ∂m∂H l (cid:12)(cid:12)(cid:12)(cid:12) H l → , (2)and nonlinear susceptibilities, χ = − ∂ m∂H l (cid:12)(cid:12)(cid:12)(cid:12) H l → ; χ = − ∂ m∂H l (cid:12)(cid:12)(cid:12)(cid:12) H l → . (3)Since measurements of χ (and higher-order susceptibil-ities) may become a hard task, very frequently in theliterature one refers to χ as the nonlinear susceptibility.Moreover, χ is directly related to the SG susceptibil-ity , χ = β (cid:18) χ SG − (cid:19) , (4)with the latter representing an important theoreticaltool, being defined as χ SG = βN X i,j (cid:2) ( h S i S j i − h S i ih S j i ) (cid:3) av , (5)where h .. i and [ .. ] av denote, respectively, thermal averagesand an average over the disorder.In fact, the interplay between quantum fluctuationsand disorder stands for the physical origin of the intrigu-ing behavior found in the magnetic susceptibility χ ofLiHo x Y − x F , which has been the object of a consid-erable experimental and theoretical debate. In the ab-sence of H t , the LiHo . Y . F compound displays asharp peak in χ at the temperature T f , which resem-bles a conventional second-order SG phase transition .Surprisingly, the sharp peak of χ becomes increasinglyrounded when the transverse field H t is applied and en-hanced, so that the resulting smooth curve presents amaximum located at a temperature T ∗ , with T ∗ > T f .Such behavior was initially interpreted as a changing inthe nature of the transition, from second order at T f tofirst order at T ∗ . More recently, J¨onsson and col-laborators investigated the behavior of χ for dopings x = 0 .
165 and 0 . performed measurements for doping x = 0 . χ , but also of χ , as well as of the ac suscep-tibility, reasserting T ∗ as the SG critical temperature.On the theoretical side, the debate on this particularissue has also been intense (see, for instance, Refs. ).The suggestion that an effective longitudinal random field(RF) h i can be induced from the interplay of a transverseapplied field H t , with the off-diagonal terms of the dipo-lar interactions in LiHo x Y − x F , represents a very inter-esting hint to clarify these controversies concerning themeaning of T ∗ . According to the droplet picture forSGs, the rounded behavior of χ in the presence of thefield-induced RF h i is interpreted as a suppression of theSG transition , similar to what a uniform field does inthat picture . On the other hand, Tabei and collabo-rators working within Parisi’s mean-field theory , usingan effective Hamiltonian defined in terms of the field-induced RF h i and a transverse field Γ [where Γ = Γ( H t )represents some monotonically increasing function of H t ],reproduced quite well the experimental behavior of χ ,with an increasingly rounded peak at T ∗ when H t is en-hanced.It should be remarked that the results described aboveare based on a particular approach of the quantum SKmodel proposed by Kim and collaborators . In thisapproach, the SK model is analyzed in the presence ofa transverse field Γ, within the static approximation.Mostly important, a region inside the SG phase wasfound where the RS approximation is stable. Actually,the RSB solution exists only for sufficiently low valuesof Γ, at temperatures lower than the SG transition tem-perature. The main consequence of this scenario is thatthe sharp peak of χ , which signals the SG phase tran-sition temperature, does not coincide with the onset ofRSB. Nevertheless, this result is also highly controversial,since other works indicate precisely the opposite, i.e., theRS approximation is unstable throughout the whole SGphase (see, for instance, Refs. ) except, possibly, atthe zero-temperature Quantum Critical Point (QCP) .Consequently, when Γ enhances, the RSB transition tem-perature T f decreases, so that, for finite temperatures,the critical behavior appears in χ as χ ∝ [(Γ − Γ f ( T )) / Γ f ( T )] − δ ′ , (6)where Γ f ( T ) denotes the critical value of Γ for a giventemperature, from which its corresponding value at thezero-temperature QCP is obtained as lim T → Γ f ( T ) =Γ c .In the classical case, it is well known that χ SG is in-versely proportional to the Almeida-Thouless eigenvalue λ AT , the so-called replicon . Therefore, the divergingbehavior of χ at the SG transition, χ ∝ [( T − T f ) /T f ] − γ , (7)is a direct consequence of λ AT = 0 at T f , occurring to-gether with the onset of RSB. Similarly, in the quan-tum case, one expects that the divergence of χ at Γ f ( T )should coincide with the onset of RSB.Indeed, the presence of a RF can produce deep changesin the scenario described previously. For instance, in theclassical SK model , the RF induces the RS order param-eter q , which becomes finite at any temperature . Asa consequence, q versus temperature presents a smoothbehavior, being no more appropriate for identifying a SGtransition in the SK model. Nevertheless, such a transi-tion may still be related with the onset of RSB, signaledby λ AT = 0 . In spite of this, the derivative of q with re-spect to the temperature increases as one approaches T f from above; such an increase is the ultimate responsiblefor the rounded maximum in χ at the temperature T ∗ ,which does not coincide with the SG transition tempera-ture T f ( T ∗ > T f ). In fact, the maximum value of χ at T ∗ reflects the effects of the RF inside the paramagneticphase, instead of the non-trivial ergodicity breaking ofthe SG phase transition . Therefore, one can raise thequestion of whether such scenario for χ , found in theclassical SK model, is robust in the corresponding quan-tum model, when the transverse field is considered, i.e.,Γ = 0, and no longer independent from the RF, as pro-posed by Tabei and collaborators .The purpose of the present work is to study the suscep-tibility χ , using the so-called fermionic Ising SG modelin the presence of a longitudinal RF h i and a transversefield Γ. In this model, the spin operators are writtenin terms of fermionic occupation and destruction op-erators , whereas the spin-spin couplings { J ij } andrandom fields { h i } follow Gaussian distributions. Thegrand-canonical potential is obtained in the functionalintegral formalism, and the disorder is treated using thereplica method; moreover, the SG order parameters areobtained in the static approximation and investi-gated within the one-step RSB scheme . It should beremarked that the fermionic Ising SG model is defined onthe Fock space, where there are four possible states persite: one state with no fermions, two states with a singlefermion, and one state with two fermions, leading to twononmagnetic states. In particular, one can consider twocases: the 4 S model that allows the four possible statesper site and the 2 S model, which restricts the spin op-erators to act on a space where the nonmagnetic statesare forbidden. In the present work we will consider thelater model, by imposing a restriction to remove the con-tribution of these nonmagnetic states, i.e., taking intoaccount only the sites occupied by one fermion in thepartition-function trace .In order to deal appropriately with the experimentalbehavior of χ in the LiHo x Y − x F compound, we focusour calculations on the 2S model by proposing a relation-ship between ∆ (the width of the distribution of randomfields h i ) and Γ, following the approach introduced byTabei and collaborators . The main characteristic ob-served experimentally in χ concerns the peak for small H t (classical limit) being replaced by a rounded maxi-mum which becomes increasingly rounded for large H t (quantum limit). Besides the progressive smearing of χ ,the amplitude of its maximum also decreases as H t in-creases. Therefore, the effects of the RF triggered by H t , as suggested by Tabei and collaborators , shouldprovoke simultaneously both effects, i.e., the smearing ofthe peak and the decrease of the maximum amplitudevalue of χ . For the present fermionic Ising SG model,we have tested a relationship involving ∆ and Γ, partic-ularly in the power-like form, ∆ /J ∝ (Γ /J ) B ′ , where J represents the width of the Gaussian distribution for thecouplings { J ij } . Considering the interval for the expo-nent, 1 . < B ′ < .
5, we have been able to obtain χ as a function of temperature and Γ resembling qualita-tively the experimental behavior for χ described above.As already mentioned, the transverse field Γ used in theeffective model to describe LiHo x Y − x F is expected tobe related to the experimental applied field H t ; infact, at least for low H t , Γ ∝ H t (see, e.g., Ref. ).The paper is structured as follows: in Section II wedefine the model and find its grand-canonical poten-tial within the one-step RSB scheme; in Section III wepresent a detailed discussion of the order parameters, thesusceptibility χ , and some phase diagrams. Finally, thelast section is reserved to conclusions. II. MODEL
The model is defined by the Hamiltonian H = − X ( i,j ) J ij ˆ S zi ˆ S zj − N X i =1 h i ˆ S zi − N X i =1 ˆ S xi , (8)where the summation P ( i,j ) applies to all distinct pairsof spin operators, whereas the couplings { J ij } and mag-netic fields { h i } are quenched random variables, followingindependent Gaussian distributions, P ( J ij ) = (cid:20) N πJ (cid:21) / exp (cid:20) − N J ( J ij − J /N ) (cid:21) , (9)and P ( h i ) = (cid:20) π ∆ (cid:21) / exp (cid:20) − h i (cid:21) . (10)In order to obtain susceptibilities [cf. Eqs. (2) and (3)],one introduces a longitudinal uniform field H l , by addingan extra term − P Ni =1 H l ˆ S zi in the Hamiltonian above.Moreover, the spin operators in Eq. (8) are defined asˆ S zi = 12 [ˆ n i ↑ − ˆ n i ↓ ] ; ˆ S xi = 12 [ˆ c † i ↑ ˆ c i ↓ + ˆ c † i ↓ ˆ c i ↑ ] , (11)where ˆ n i ↑ = ˆ c † i ↑ ˆ c i ↑ and ˆ n i ↓ = ˆ c † i ↓ ˆ c i ↓ , with ˆ c † i ↑ denoting acreation operator for a fermion with spin up at site i , ˆ c i ↓ an annihilation operator for a fermion with spin down atsite i , and so on. In this fermionic problem, the partitionfunction is expressed by using the Lagrangian path in-tegral formalism in terms of anticommuting Grassmannfields ( φ and φ ∗ ) . The restriction in the 2S-model isimposed by means of a Kronecker delta function, in sucha way to take into account only those sites occupied byone fermion ( n i ↑ + n i ↓ = 1) in the partition-function .Therefore, adopting an integral representation for thisdelta function, one can express the partition function forboth 2 S and 4 S models in the following form, Z { y } = e s − Nβµ Z D ( φ ∗ φ ) Y j π Z π dx j e − y j e A { y } , (12)where A { y } = Z β dτ X j,σ φ ∗ jσ ( τ ) (cid:20) ∂∂τ + y j β (cid:21) φ jσ ( τ ) − H (cid:0) φ ∗ jσ ( τ ) , φ jσ ( τ ) (cid:1)(cid:9) . (13)In the equations above, β = 1 /T ( T being the tem-perature), y j = ix j for the 2 S -model, or y j = βµ forthe 4 S -model, s = 2 , µ isthe chemical potential. Moreover, φ jσ and φ ∗ jσ repre-sent Grassmann fields at site j and spin state σ , whereas H (cid:0) φ ∗ jσ ( τ ) , φ jσ ( τ ) (cid:1) stands for an effective Hamiltonian ata given value of the integration variable τ .Now, we use the replica method, so that standard pro-cedures lead to the grand-canonical potential per parti-cle , β Ω = − N hh ln Z { y }ii J,h = − N lim n −→ hh Z { y } n ii J,h − n , (14)where hh .. ii J,h denote averages over the quenched randomvariables. The replicated partition function hh Z { y } n ii J,h becomes hh Z { y } n ii J,h = e s − Nβµ N Z ∞−∞ Y ( α,γ ) dq αγ Z ∞−∞ n Y α =1 dq αα × Z ∞−∞ n Y α =1 dm α exp [ N β Ω n ( q αγ , q αα , m α )](15)where α ( α = 1 , , · · · , n ) stands for a replica in-dex, ( α, γ ) denotes distinct pairs of replicas, and N =( βJ p N/ π ) n ( n +1) / . Assuming the static approxima-tion , one obtains β Ω n ( q αγ , q αα , m α ) = − β J X ( α,γ ) q αγ − β J X α q αα − βJ X α m α + ln Λ { y } , (16) and the Fourier representation may be used to expressΛ { y } = Y α π Z π dx α e − y α Z D [ φ ∗ α , φ α ] exp[ H eff ] . (17)Above, one has an “effective Hamiltonian” in replicaspace, H eff = X α A α + 4 " β ∆ X α,γ S zα S zγ + βJ X α m α S zα + β J X α q αα S zα S zα + 2 X ( α,γ ) q αγ S zα S zγ , (18)with A α = X ω ϕ † α ( ω )( iω + y α + β Γ σ x ) ϕ α ( ω ) ,S zα = 12 X ω ϕ α ( ω ) σ z ϕ α ( ω ) , (19)where the Matsubara’s frequencies are ω = ± π, ± π, · · · , σ x and σ z denote the Pauli matrices, and ϕ † α ( ω ) = (cid:16) φ ∗↑ α ( ω ) φ ∗↓ α ( ω ) (cid:17) .Moreover, the functional integrals over q αγ , q αα and m α in Eq. (15) have been evaluated through the steepest-descent method, yielding m α = h S α i ; q αγ = h S zα S zγ i ; q αα = h ( S zα ) i , (20)with h .. i representing a thermal average over the effectiveHamiltonian of Eq. (18).Herein, the problem will be analyzed within one-stepRSB Parisi’s scheme , in which q αα = p , and the replicamatrix elements are parametrized as q αγ = ( q if I ( α/a ) = I ( γ/a ) q if I ( α/a ) = I ( γ/a ) (21)where I ( x ) gives the smallest integer greater than, orequal to x .The parametrization given by Eq. (21) allows toperform the sums over replica indexes and then, thequadratic terms in Eq. (18) can be linearized through theintroduction of new auxiliary fields. From this point, theintegrals over the Grassmann variables in Eq. (17) canbe performed and the sum over Matsubara’s frequenciescan be obtained, like in Ref. . Therefore, the resultinggrand-canonical potential is obtained from Eq. (14), β Ω = ( βJ ) x − q − xq + p ] + βJ m − ln 2 − ( s − βµ − x Z Dz ln (cid:26)Z Dv [ K ( z, v )] x (cid:27) , (22)where K ( z, v ) = ( s − βµ ) + Z Dξ cosh[ p Ξ( z, v, ξ )] , Ξ( z, v, ξ ) = [ βh ( z, v, ξ )] + ( β Γ) ,h ( z, v, ξ ) = βJ [ p q + (∆ /J ) z + p q − q ) v + p p − q ) ξ ] , (23)and Dx ≡ dx e − x / / √ π ( x = z, v or ξ ).The parameters q , q , x , p , and m are obtainedthrough extremization of the grand-canonical potentialin Eq. (22), and results for 2S and 4S models are obtainedby considering s = 2 and s = 4, respectively. Moreover,the RS solution is recovered for q = q = q , and x = 0.In this way, the linear susceptibility of Eq. (2) becomes χ = β [ p − q + x ( q − q )] .As usual, the RSB parameters, the magnetization m ,and the quadrupolar parameter p , form a set of coupledequations, to be solved simultaneously. Particularly, theparameter p is quite dependent on Γ, and in fact, for the2S model, p → →
0; it should be mentionedthat p plays an important role in the nonlinear suscep-tibility χ . This aspect represents a crucial difference ofthe present investigation with respect to previous one, byKim and collaborators [cf. Ref. ], where the parameters q , q , and x (or even q in the RS solution) do not dependon p . In the present work, for the above one-step RSBsolution, χ will be obtained by numerical derivatives; forthe RS solution, an analytical form for χ is presented inAppendix A. As mentioned before, in order to deal withthe LiHo x Y − x F compound, we will restrict ourselvesto the 2S model, considering J = 0. III. RESULTS
Hence, considering the 2S model, in this section weanalyze the behavior of the nonlinear susceptibility χ ,either by varying the temperature (for fixed typical val-ues of ∆ /J and Γ /J ), or by considering joint variationsin some of these parameters. Since χ is directly relatedwith the order parameters that appear in the thermo-dynamic potential of Eq. (22), we first discuss the SGorder parameters q and q , as well as the quadrupolarparameter p . Moreover, the onset of RSB is signalled by δ = q − q >
0, which locates the freezing temperature T f ; it should be mentioned that for Γ = 0 and ∆ = 0,one has that T f = √ J .In Fig. 1 we exhibit the one-step RSB parameter δ ≡ q − q versus the dimensionless temperature T /J ,for typical choices of Γ /J and ∆ /J . The correspondingparameters q , q , and p are also presented versus T /J in the respective insets. From Fig. 1(a) one notices thatthe freezing temperature gets lowered for increasing val-ues of the transverse field Γ, up to the zero-temperatureQCP located at Γ c = 2 √ J (∆ = 0) ; a similar effect is δ =q -q (a) ∆ /J=0.0 Γ /J=0.00 Γ /J=0.50 Γ /J=1.00 0 0.5 1 0 0.6 1.2q q p 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6T/J δ =q -q (b) Γ /J=0.0 ∆ /J=0.00 ∆ /J=0.25 ∆ /J=0.50 0 0.5 1 0 0.6 1.2q q FIG. 1: The one-step RSB parameter δ ≡ q − q is pre-sented versus the dimensionless temperature T /J , for ∆ = 0and typical values of Γ /J [panel (a)], as well as for Γ = 0 andtypical values ∆ /J [panel (b)]. The parameters q , q , and p are also exhibited versus temperature in the respective insets;one notices that the quadrupolar parameter p becomes rele-vant only in the cases Γ >
0, for which it decreases by loweringthe temperature. Due to the usual numerical difficulties, thelow-temperature results [typically (
T /J ) < . verified in Fig. 1(b) by increasing the width of the distri-bution of random fields ∆ (Γ = 0). In the Hamiltonian ofEq. (8) one sees that the limit Γ = 0 corresponds to a sim-ple, diagonalizable, quantum Ising SG model, where onlythe spin components ˆ S zi are present, leading to a trivialquadrupolar parameter p = 1 (for all temperatures), asshown in the inset of Fig. 1(a). This particular case,for which the SG parameters are exhibited in Fig. 1(b),yields results qualitatively similar to those found in theprevious study of the classical SK model in the presenceof a Gaussian random field, carried in Ref. . One noticesthat for (∆ /J ) > q = q = q is induced [cf. the inset of Fig. 1(b)], presenting a smoothbehavior versus temperature; consequently, the freezingtemperature T f can only be found by means of the RSBscheme, with the SG transition coinciding with the on-set of the parameter δ . However, for Γ >
0, the spin J χ T/J(a) ∆ =0 Γ /J=0.0 Γ /J=1.010 -3 -2 -1 (T - T f )/T f f )/T f ) -1 J χ Γ /J(b) ∆ =0 T/J=0.2T/J=1.010 -3 -2 -1 ( Γ - Γ f )/ Γ f Γ - Γ f )/ Γ f ) -1 FIG. 2: Plots of the dimensionless nonlinear susceptibility [computed from Eq. (3)] are exhibited for ∆ = 0: (a) J χ versus T /J for two different values of Γ /J ; (b) J χ versus Γ /J for two different temperatures. In all cases one notices sharpdivergences of χ , signaling evident phase transitions. The corresponding critical exponents are estimated through log-log plots(shown in the respective insets), where in each case, the fitting proposal is represented by a dashed-dotted line (see text). J χ T/J(a) Γ =0.00 ∆ /J=0.000 ∆ /J=0.001 ∆ /J=0.005 ∆ /J=0.01010 -3 -2 -1 (T - T * )/T * γ J χ Γ /J(b)T/J=1.00 ∆ /J=0.000 ∆ /J=0.001 ∆ /J=0.005 ∆ /J=0.01010 -3 -2 -1 ( Γ - Γ * )/ Γ * δ ’ FIG. 3: The behavior of the dimensionless nonlinear susceptibility (in two typical cases exhibited in Fig. 2) is presented forincreasing values of ∆ /J : (a) J χ versus T /J , for (Γ /J ) = 0 .
0; (b) J χ versus Γ /J , for ( T /J ) = 1 .
0. In each case one noticesa 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 thatthe divergences of Eq. (7), leading to the exponent γ [inset of (a)], or in Eq. (6), leading to the exponent δ ′ [inset of (b)], arefulfilled only for (∆ /J ) = 0. components ˆ S xi become important, so that one expectsa nontrivial behavior for the quadrupolar parameter p ;indeed, p should decrease for increasing values of Γ (ata fixed temperature), whereas for a fixed Γ, it decreasesby lowering the temperature, as shown in the inset ofFig. 1(a).In the present problem, clear phase transitions may beverified only for ∆ = 0, like those exhibited in Fig. 2.From the χ plots of Fig. 2(a) one confirms two impor-tant features shown in Fig. 1(a), concerning the behav-ior of the order parameters q , q and δ : (i) The freez-ing temperature T f , signaled by the divergence of χ in Fig. 2(a), coincides with the onset of RSB indicatedby the parameter δ of Fig. 1(a); (ii) The temperature T f is lowered by increasing values of Γ /J . The criti-cal exponents associated with the divergences of χ maybe obtained by log-log plots, as shown in the insets ofFig. 2. In the inset of Fig. 2(a) we have verified that thebehavior of Eq. (7) (represented by the dashed-dottedline) fits well the region 0 . < ( T − T f ) /T f < . γ = 1, for both val-ues Γ /J = 0 . /J = 1 . γ . It is importantto mention that this estimate coincides with the well-known value found for the SK model . In Fig. 2(b) onesees divergences of χ at given values of Γ [defined asΓ f ( T ) in Eq. (6)], for two typical fixed temperatures; J χ Γ /J(a) ∆ /J=0.25 ↑ ↑ ← T/J=1.35; ( Γ *=0.42J)T/J=1.30; ( Γ *=0.68J)T/J=1.20; ( Γ *=1.01J)T/J=1.00; ( Γ f =0.51J, Γ *=1.43J)T/J=0.60; ( Γ f =1.39J, Γ *=1.95J)T/J=0.20; ( Γ f =1.91J, Γ *=2.40J) 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 J χ Γ /J(b) ↑↑ ↓ ↓ ↓ ↓ ↓ ↓ T=1.405J ( Γ f =0.15J, Γ *=0.24J)T=1.40J ( Γ f =0.194J, Γ *=0.29J)T=1.35J ( Γ f =0.479J, Γ *=0.62J)T=1.30J ( Γ f =0.66J, Γ *=0.820J)T=1.20J ( Γ f =0.93J, Γ *=1.108J)T=1.00J ( Γ f =1.30J, Γ *=1.495J)T=0.60J ( Γ f =1.794J, Γ *=2.02J)T=0.20J ( Γ f =2.22J, Γ *=2.479J) 0 2 4 6 8 1 1.5 2 2.5 3 FIG. 4: (a) The dimensionless nonlinear susceptibility is represented versus Γ /J , for typical fixed temperatures and a nonzerowidth for the random fields [(∆ /J ) = 0 . f ( T ) [following Eq. (6)], signalled byarrows in some cases, get smoothened due to the random fields, so that their corresponding maxima [located at Γ ∗ ( T )] areshifted towards higher values of the transverse field, i.e., Γ ∗ ( T ) > Γ f ( T ). (b) The behavior of the dimensionless nonlinearsusceptibility is shown versus Γ /J , for typical fixed temperatures, by considering a particular relation involving ∆ and Γ[(∆ /J ) = 0 . /J ) ]; the inset represents an amplification of the region for higher values of Γ /J . In all cases, the maxima[located at Γ ∗ ( T )] appear shifted with respect to the onset of RSB [located at Γ f ( T ), signalled by arrows in some cases] towardshigher values of the transverse field, i.e., Γ ∗ ( T ) > Γ f ( T ). like in Fig. 2(a), these divergences coincide with the on-set of RSB indicated by the parameter δ . One noticesthat, as one approaches zero temperature [cf., e.g., thecase ( T /J ) = 0 . f ( T ) approachesthe one that occurs at the QCP, Γ c = 2 √ J . Inthe inset of Fig. 2(b), the critical behavior described byEq. (6) (represented by the dashed-dotted line) was ful-filled in both cases, showing a good agreement in theregion 0 . < (Γ − Γ f ( T )) / Γ f ( T ) < .
1, with the sameexponent δ ′ = 1 for the two values of temperatures inves-tigated, ( T /J ) = 0 . T /J ) = 1 . γ of Eq. (7), the presentestimates of δ ′ suggest that the temperature should notchange the universality class of this later exponent.In agreement with the previous study of the SKmodel in the presence of a Gaussian random field , thesmoothening of χ is verified in Fig. 3 for the cases(∆ /J ) >
0. For instance, Fig. 3(a) displays χ versus T /J , for increasing values of ∆ /J , in the case Γ = 0,showing that the divergent peak of the nonlinear sus-ceptibility is replaced by a broad maximum at a tem-perature T ∗ . One observes that such a peak becomessmoother, decreasing its height for increasing values of∆ /J . Particularly, in the inset of Fig. 3(a) one sees thatthe temperature range 0 . < ( T − T ∗ ) /T ∗ < . γ = 1 .
0, in the cases (∆ /J ) >
0. In a similar way,Fig. 3(b) shows χ versus Γ /J , for ( T /J ) = 1 .
0, consid-ering the same values for ∆ /J of Fig. 3(a); again, thepeak of nonlinear susceptibility gets flattened due to thepresence of an applied random field, now displaying amaximum at Γ ∗ . Consequently, in such cases the region 0 . < (Γ − Γ ∗ ) / Γ ∗ < . δ ′ = 1 .
0, as shown in the insetof Fig. 3(b).In Fig. 4 we represent the dimensionless nonlinearsusceptibility χ versus Γ /J , for typical fixed tempera-tures, in two cases: (a) Γ /J and ∆ /J as independentquantities [Fig. 4(a)]; (b) Imposing a relation involv-ing ∆ and Γ [Fig. 4(b)]. In Fig. 4(a) we consider afixed value for the width of the Gaussian random fields[(∆ /J ) = 0 . χ at thecorresponding critical values Γ f ( T ) [according to Eq. (6)and shown by arrows in some curves], change into smoothcurves with maxima at Γ ∗ ( T ), shifted to higher values ofΓ, i.e., Γ ∗ ( T ) > Γ f ( T ). Following the proposal of Ref. ,for dealing properly with the experimental behavior of χ in the LiHo x Y − x F compound, we analyzed the presentsystem by imposing a relation involving ∆ and Γ, i.e.,∆ ≡ ∆(Γ). According to the experimental investigationsof Ref. , such a relation should satisfy certain require-ments, e.g., ∆ should increase monotonically with Γ, andone should get ∆ = 0 for Γ = 0. The simplest proposalobeying these conditions comes to be a power function,(∆ /J ) = A (Γ /J ) B , where A and B are fitting param-eters. Herein, these parameters were computed by ad-justing our results to those of the experiments of Ref. ,leading to the optimal values A = 0 .
02 and B = 2. InFig. 4(b) we exhibit the dimensionless nonlinear suscepti-bility, versus Γ /J , for typical fixed temperatures, by con-sidering this particular relation involving ∆ and Γ. In allcases, the maxima [located at Γ ∗ ( T )] appear shifted withrespect to the onset of RSB [located at Γ f ( T )] towardshigher values of the transverse field, i.e., Γ ∗ ( T ) > Γ f ( T ). Γ =0 b = λ AT b λ AT χ ↓ ↑ ∆ /J=0.0 ∆ /J=0.1 Γ /J=1.0(a) b = λ AT ( ∆ =0)b ( ∆ /J=0.1) λ AT ( ∆ /J=0.1) χ ( ∆ /J=0.1) ↓ ↓ Γ /J=1.0(b) b λ AT b λ AT ↓ ↓ ∆ /J = 0.1 ∆ /J = A ( Γ /J) B FIG. 5: The softening of the nonlinear susceptibility is illustrated by means of the denominator of q , i.e., q ∝ b − , b = 1 − βJ ) I (Γ) [cf. Eq. (24)], which appears in the expression of χ calculated in Appendix A, within the RS approximation.The Almeida-Thouless eigenvalue λ AT , associated with the onset of RSB and defining the SG critical temperature T f is alsoshown, for comparison. (a) Results for (Γ /J ) = 1 . /J , namely, (∆ /J ) = 0 . /J ) = 0 .
1, in the case where Γ and ∆ are independent; similar results are presented in the inset for (Γ /J ) = 0. The fulllines represent the cases (∆ /J ) = 0 .
0, showing that λ AT and the denominator b coincide, becoming zero at the temperature T f . The cases (∆ /J ) = 0 . b is always positive, presenting a smooth minimum around a temperature T ∗ , leadingto the rounding of χ , whereas λ AT becomes zero at a lower temperature T f . (b) Results for (Γ /J ) = 1 . /J ) = 0 . /J ) = 0 . /J ) ] (dotted lines). In all cases, the arrows locate thefreezing temperature T f . The most important novelty of Fig. 4(b) [to be contrastedwith the results of Fig. 4(a)], concerns the fact that theamplitude of the maximum of χ decreases for increasingvalues of Γ /J , and consequently, for decreasing temper-atures.Recent studies in the compound LiHo x Y − x F sug-gested that the transverse field Γ introduced in theHamiltonian of Eq. (8) should be related to the experi-mental applied field in a real system, H t ; in fact, atleast for low H t , Γ ∝ H t (see, e.g., Ref. ). Hence, con-sidering a new dimensionless variable, H t ( H t ≡ p Γ /J ),we have verified that the same qualitative behavior shownin both Figs. 4(a) and 4(b) occur in representations ofthe dimensionless nonlinear susceptibility χ versus H t .Consequently, Fig. 4(b) preserves the agreement with ex-perimental observations, showing that besides the pro-gressive smearing of χ , the amplitude of its maximumalso decreases as the real field H t increases, as suggestedby Tabei and collaborators .In Appendix A we have calculated χ analytically,within the RS approximation, for both Γ > >
0, as q = 2( βJ ) I (Γ) b ; b = 1 − βJ ) I (Γ) . (24)Notice that the denominator b may become zero, leadingto a divergence in the nonlinear susceptibility; it shouldbe mentioned that q is the only quantity appearing in χ [either in Eq. (A1), or in Eq. (A13)], which may presenta divergence at finite temperatures. Moreover, one canshow that for ∆ = 0, the so-called “dangerous” eigen-value in Eq. (A10) is equal to the denominator of q ,i.e., λ AT = 1 − βJ ) I (Γ), even for Γ >
0. The mecha-nism behind the flattening of the χ peak at T ∗ is illus-trated in Fig. 5, where we plot the quantity b of Eq. (24), λ AT , and χ , versus the dimensionless temperature, fortypical choices of Γ /J and ∆ /J . Results for Γ and ∆ in-dependent are presented in Fig. 5(a); the full lines [cases(∆ /J ) = 0 .
0] show that the quantities b and λ AT becomezero together, being associated with the divergence of χ [according to Eq. (7)], signalling the SG phase-transitiontemperature T f . However, the results for (∆ /J ) = 0 . b >
0, which presents a smooth mini-mum around a temperature T ∗ , being directly associatedwith the rounding behavior of χ ; on the other hand, onehas λ AT = 0 at a temperature T f , with T f < T ∗ . InFig. 5(b) we present the denominator b and λ AT , for thecases where Γ and ∆ are independent (dashed lines), andwhere these quantities are related through the power law(∆ /J ) = 0 . /J ) . In this later case, since one has(∆ /J ) > /J ) >
0, the denominator b will al-ways display a minimum value around a temperature T ∗ ,higher than T f . Particularly, by means of this relation,higher values of Γ imply on higher values of ∆, increasingthe values of b at the minima, resulting in a decrease inthe amplitude of the maxima of χ .In Fig. 6 we present phase diagrams T /J versus Γ /J T / J Γ /J(a) ∆ /J=0.25T f T* PM2PM1RSB SG 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5 T / J Γ /J(b) T f T* PM2PM1RSB SG 0 0.1 0.2 0 1 2 3 ∆ / J Γ /J FIG. 6: Phase diagrams
T /J versus Γ /J showing the frontiers separating the SG and paramagnetic phases (full lines), whichrepresent the behavior of the temperature T f for increasing values of Γ, located by the onset of RSB, i.e., λ AT = 0. Thedashed lines correspond to the temperature T ∗ , associated with the maximum of the nonlinear susceptibility χ , and hereininterpreted as a crossover between two distinct regions of the paramagnetic phase (PM1 and PM2). (a) Phase diagram for(∆ /J ) = 0 .
25, in the case where Γ and ∆ are independent. (b) Phase diagram for which Γ and ∆ follow the relation proposedin Fig. 4(b) [(∆ /J ) = 0 . /J ) ], whose parabolic behavior is presented in the inset. Due to the usual numerical difficulties,the low-temperature results [typically ( T /J ) < . showing a decrease in the temperature T f for increas-ing values of Γ (full lines). These lines delimit the SGphase and were identified with the onset of RSB, by set-ting λ AT = 0; throughout the whole SG phases ones has λ AT <
0. The temperature T ∗ (dashed lines), associatedwith the maximum of the nonlinear susceptibility χ , sig-nals a crossover between two regions of the paramagneticphase (PM1 and PM2), as will be discussed next. Thephase diagram shown in Fig. 6(a) corresponds to a fixedvalue of ∆ [(∆ /J ) = 0 . c = 2 √ J ≈ . J . Thecase shown in Fig. 6(b) corresponds to Γ and ∆ follow-ing the relation (∆ /J ) = 0 . /J ) (see inset), so thatfor Γ = 0, one has ∆ = 0, giving T ∗ = T f . By increas-ing values of Γ, the width of RFs also increases, leadingto a rounded peak in the nonlinear susceptibility, yield-ing T ∗ > T f , and consequently, the region PM1 emerges.Due to the joint increase of both Γ and ∆, as shown in theinset, the region PM1 gets enlarged up to zero tempera-ture, where one gets a QCP, shifted towards lower valuesof Γ as compared with the QCP for ∆ = 0, similarly tothe one occurring in Fig. 6(a).It should be emphasized that the temperature T ∗ playsa role different from T f , in the sense that it does not cor-respond to a phase transition, but rather to a crossoverbetween two distinct regions of the paramagnetic phase.The previous analysis of the SK model in the presenceof a Gaussian random field Ref. , which should corre-spond herein to the region of high temperatures and low Γ (i.e., the classical regime), has also found a temperature T ∗ , associated with the rounded maximum of χ , with T ∗ > T f . In this case, T ∗ was interpreted as an effect ofthe RFs acting inside the paramagnetic phase, instead ofsome type of non-trivial ergodicity breaking. Herein, weclaim that the temperature T ∗ , although it may be alsoaffected by the transverse field Γ, should be interpretedin a similar manner. Hence, along the line signaled by T f , the growth of Γ produces an enhancement of quan-tum fluctuations, which become increasingly dominantas compared with thermal fluctuations, driving the non-trivial ergodicity breaking of the SG phase transition toa QCP. For temperatures in the region T f < T < T ∗ , theenhancement of quantum fluctuations by Γ along withthe spin fluctuations due to the RFs inside the paramag-netic phase create two distinct scenarios, more preciselyconcerning the PM1 region, as discussed next: (a) Forfixed ∆ [e.g., Fig. 6(a)], one has Γ and ∆ independent,so that the smearing of the nonlinear susceptibility iscaused only by the RFs, leading to the effect that thefull and dashed lines remain essentially parallel to oneanother, up to zero temperature; (b) The phase diagramof Fig. 6(b), for which ∆ and Γ are related through theparabolic behavior shown in the inset, the appearance of T ∗ occurs for Γ > > χ is dominated by the enhancement of the RFs, leading tospins fluctuations due to the RFs inside the paramagneticphase.0 IV. CONCLUSIONS
We have investigated a quantum spin-glass model inthe presence of a uniform transverse field Γ, as well asof a longitudinal random field h i , the later following aGaussian distribution characterized by a width propor-tional to ∆. The model was considered in the limitof infinite-range interactions and studied through thereplica formalism, within a one-step replica-symmetry-breaking procedure. The spin-glass critical frontier, sig-naled by the temperature T f , was identified with the on-set of replica-symmetry breaking, calculated through theAlmeida-Thouless eigenvalue (replicon) λ AT , i.e., by set-ting λ AT = 0. In this approach, the whole spin-glassphase becomes characterized by λ AT <
0, and conse-quently, it was treated through replica-symmetry break-ing. Such analysis was motivated by experimental inves-tigations on the LiHo x Y − x F compound. In this sys-tem, the application of a transverse magnetic field yieldsrather intriguing effects, particularly related to the be-havior of the nonlinear magnetic susceptibility χ , whichhave led to a considerable experimental and theoreticaldebate.We have analyzed two physically distinct situations,namely, ∆ and Γ considered as independent, as wellas these two quantities related, as proposed recentlyby some authors (see, e.g., Ref. ). In both cases, wehave found a spin-glass critical frontier, given by T f ≡ T f (Γ , ∆), with such phase being characterized by a non-trivial ergodicity breaking. In the first case, for ∆ fixed,we have found that T f (Γ , ∆) decreases by increasing Γtowards a quantum critical point at zero temperature,whereas in the second, we have found a similar behaviorfor this critical frontier, with ∆ changing according tovariations in Γ. In this later case, we have taken into ac-count previous experimental investigations which sug-gest that a relation of the type ∆ ≡ ∆(Γ) should satisfycertain requirements, e.g., ∆ should increase monoton-ically with Γ, and one should get ∆ = 0 for Γ = 0.Although such a relation may not be unique, the sim-plest proposal following such conditions appears to be apower function, (∆ /J ) = A (Γ /J ) B . In the present work,the parameters A and B were computed by adjusting ourresults to those of the experiments of Ref. , leading tothe the optimal values A = 0 .
02 and B = 2.We have shown that the present approach, consider-ing the relation (∆ /J ) = 0 . /J ) , was able to repro-duce adequately the experimental observations on theLiHo x Y − x F compound, with theoretical results coin-ciding qualitatively with measurements of the nonlinearsusceptibility χ . As a consequence, by increasing Γgradually, our results indicate that χ becomes progres-sively rounded, presenting a maximum at a temperature T ∗ ( T ∗ > T f ); moreover, both amplitude of the maxi-mum and the value of T ∗ diminish, by enhancing Γ.From the analysis where ∆ and Γ are considered asindependent, we have concluded that the random field isthe main responsible for the smearing of the nonlinear susceptibility. Hence, the random field acts significantlyinside the paramagnetic phase, leading to two regimesdelimited by the temperature T ∗ , one for T f < T < T ∗ (called herein as PM1), and another one for T > T ∗ (denominated as PM2). In the paramagnetic regime for T > T ∗ one should have weak correlations and conse-quently, the usual paramagnetic type of behavior. How-ever, close to T ∗ , and particularly for temperatures inthe range T f < T < T ∗ , one expects a rather nontrivialbehavior in real systems, as happens with experimentsin the compound LiHo x Y − x F , resulting in very con-troversial interpretations . Hence, as alreadyargued in the analysis of the SK model in the presence ofGaussian random field , the line PM1–PM2 may notcharacterize a real phase transition, in the sense of adiverging χ , but the region PM1 should be certainlycharacterized by a rather nontrivial dynamics. As onepossibility, one should have a growth of free-energy bar-riers in this region, leading to a slow dynamics, whereasonly below T f the nontrivial ergodicity breaking appears,typical of RSB in SG systems. Also, one could haveGriffiths singularities along PM1, which are found cur-rently in disordered magnetic systems, like site-dilutedferromagnets , ferromagnet in a random field , classicalIsing spin glasses , and also claimed to occur in quan-tum spin glasses . Whether such curious propertiesmay appear throughout the region PM1 in the presentproblem, represents a matter for further investigation.In fact, recent experiments in the above compound for x = 0 .
045 strongly suggest this picture : these authorsclaim an “unreachable” transition due to an ultra-slowdynamics (of the order 10 times slower than the ones ofconventional spin-glass materials) and argue that such adynamics should be caused by a Griffiths phase betweenthe paramagnetic and spin-glass phases.Next, we discuss some contributions of the presentwork, as compared to previous theoretical approaches inthis problem. (i) The analysis of Ref. did not takeinto account the random field, which in our view, rep-resents a key ingredient for an appropriate descriptionof the properties of LiHo x Y − x F . Moreover, as it wasshown herein, the RSB SG parameters, together with themagnetization m , and the quadrupolar parameter p , allform a set of coupled equations, to be solved simulta-neously. The approach of Ref. considered p as inde-pendent from the remaining parameters; this could bedirectly related with the curious result concerning a partof the SG phase characterized by stability of the replica-symmetric solution, along which these authors find therounded maximum of χ . (ii) The study of Ref. hasconsidered an effective Hamiltonian characterized by anextra two-body interacting term (as compared with theHamiltonian used herein), coupling spin operators in the x and z directions. Moreover, these authors have sug-gested a relation ∆ ≡ ∆(Γ), which due to the Hamilto-nian employed, turned out to be slightly different fromours, e.g., (∆ /J ) = A (Γ /J ) B , with an exponent B < χ corroborate those of Ref. ; however, we under-stand that the present analysis, characterized by a singletwo-body interacting term, − P ( i,j ) J ij ˆ S zi ˆ S zj , leads to amuch simpler analysis to the problem, when compared tothe one carried in this previous work.To conclude, we have considered a model able to re-produce theoretically many properties observed in ex-periments on the LiHo x Y − x F compound, particularlythose related to the nonlinear susceptibility χ . Thepresent theoretical proposal appears to be simpler thanprevious ones, and consequently, its results should beeasier to compare with further experimental investiga-tions. Obviously, the observation of a clear spin-glassstate, characterized by a nontrivial ergodicity breakingat a temperature T f (below the temperature T ∗ whereone observes rounded effects on the nonlinear suscepti-bility) represents a challenge for experiments. Acknowledgments
We acknowledge the partial financial support fromCNPq, FAPERGS, FAPERJ, and CAPES (Brazilianfunding agencies).
Appendix A: Nonlinear Susceptibility in the RSSolution
In this appendix we obtain the nonlinear susceptibility χ analytically for the 2S model, within the RS solution.Although in the RS solution, these results allow us toanalyze in detail how the RFs and the transverse field Γaffect the nonlinear susceptibility. Particularly, one hasthat the nonlinear susceptibility of Eq. (3) becomes χ = − ∂ m∂H l (cid:12)(cid:12)(cid:12)(cid:12) H l → = β (cid:2) q + 2( βJ ) V (cid:3) − V − βJ ) V , (A1)where q = 2( βJ ) I (Γ)1 − βJ ) I (Γ) , (A2)with the following definitions I (Γ) = V − V βJ ) V V − βJ ) V , (A3) V = Z Dz (cid:20) C C K − C ( C ) K (cid:21) , (A4) V = Z Dz (cid:20) C K − C C K − (cid:18) C K (cid:19) + 12 C ( C ) K − (cid:18) C K (cid:19) , (A5) V = Z Dz (cid:20) C K − C C K − (cid:18) C K (cid:19) + 2 C ( C ) K (cid:21) . (A6)In the equations above, one has that C n = Z Dξ ∂ n f ( h ) ∂h n ; K = Z Dξ f [ h ( z, ξ ) , Γ] , (A7)with f [ h ( z, ξ ) , Γ] = cosh p h ( z, ξ ) + ( β Γ) , (A8)and h ( z, ξ ) = βJ [ p q + (∆ /J ) z + p p − q ) ξ ] . (A9)In addition, the limit of stability of the RS solution isdelimited by λ AT > , which may be expressed in termsof the above quantities as λ AT = 1 − βJ ) Z Dz " C K − (cid:18) C K (cid:19) . (A10)The particular case Γ = 0 gives h ( z ) = βJ ( p q + (∆ /J ) z , (A11)as well as V = V = 0, whereas V = − Z Dz [sech h ( z ) − h ( z )sech h ( z )] . (A12)As a result, χ becomes χ = β q ] I (0) (A13)where q = [2( βJ ) I (0)] / [1 − βJ ) I (0)] and I (0) = Z Dz [sech h ( z ) − h ( z )sech h ( z )] . (A14)In this case, χ coincides with the one found in Ref. . ∗ Electronic address: [email protected] W. Nolting and A. Ramakanth,
Quantum Theory of Mag- netism (Springer-Verlag, Berlin, 2009). K. Binder and A. P. Young, Rev. Mod. Phys. , 801(1986). K. H. Fischer and J. A. Hertz,
Spin Glasses (CambridgeUniversity Press, Cambridge, UK, 1991). V. Dotsenko,
Introduction to the Replica Theory of Dis-ordered Statistical Systems (Cambridge University Press,Cambridge, UK, 2001). H. Nishimori,
Statistical Physics of Spin Glasses and Infor-mation Processing (Oxford University Press, Oxford, UK,2001). D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. ,1792 (1975). G. Parisi, J. Phys. A , 1101 (1980); ibid. S. Sachdev,
Quantum Phase Transitions , second edition(Cambridge University Press, Cambridge, UK, 2011). D. Bitko, T. F. Rosenbaum, and G. Aeppli, Phys. Rev.Lett. , 940 (1996); P. B. Chakraborty, P. Henelius, H.Kjønsberg, A. W. Sandvik, and S. M. Girvin, Phys. Rev.B , 144411 (2004). W. Wu, B. Ellman, T. F. Rosenbaum, G. Aeppli, and D.H. Reich, Phys. Rev. Lett. M. Gingras, P. Henelius, J. Phys: Conf. Ser. , 012001(2011). J. A. Quilliam, S. Meng, J. B. Kycia, Phys. Rev. B ,184415 (2012) J. Chalupa, Sol. Stat. Comm. , 315 (1977). W. Wu, D. Bitko, T. F. Rosenbaum, and G. Aeppli, Phys.Rev. Lett. L. F. Cugliandolo, D. R. Grempel, C. A. da Silva Santos,Phys. Rev. B , 014403 (2001). P. E. J¨onsson, R. Mathieu, W. Wernsdorfer, A. M.Tkachuk, B. Barbara, Phys. Rev. Lett. , 256403 (2007). C. Ancona-Torres, D. M. Silevitch, G. Aeppli, T. F. Rosen-baum, Phys. Rev. Lett. , 057201 (2008). J. F. Fernandez, Phys. Rev. B , 144436 (2010). J. J. J. Alonso, Phys. Rev. B , 094406 (2015). M. Schechter and N. Laflorencie, Phys. Rev. Lett. ,137204 (2006). M. Schechter and P. C. E. Stamp, Phys. Rev. Lett. ,267208 (2005). S. M. A. Tabei, M. J. P. Gingras, Y.-J. Kao, P. Stasiakand J.-Y. Fortin, Phys. Rev. Lett. , 237203 (2006). S. M. A. Tabei, F. Vernay, M. J. P. Gingras, Phys. Rev. B , 014432 (2008). J. A. Mydosh, Rep. Prog. Phys. , 052501 (2015). D. S. Fisher, D. A. Huse, Phys Rev. Lett , 1601 (1986). A. P. Young, H. G. Katzgraber, Phys Rev. Lett , 207203(2004). D. -H. Kim, J. -J. Kim, Phys. Rev. B , 054432 (2002). Y. Y. Goldschmidt, P-Y. Lai, Phys. Rev. Lett. , 2467(1990). J. Miller, D. A. Huse, Phys. Rev. Lett. , 3147 (1993). N. Read, S. Sachdev and J. Ye, Phys. Rev. B , 384(1995). J. R. L. de Almeida, D. J. Thouless, J. Phys. A , 983(1978). T. Schneider, E. Pytte, Phys. Rev B , 1519 (1977) F. Krzakala, F. Ricci-Tersenghi, L. Zdeborova, Phys. Rev.Lett. , 207208 (2010). R. F. Soares, F. D. Nobre and J. R. L. de Almeida, Phys.Rev. B , 6151 (1994). C. V. Morais, F. M. Zimmer, M. J. Lazo, S. G. Magalh˜aes,F. D. Nobre, Phys. Rev. B , 224206 (2016). A. Theumann, M. V. Gusm˜ao, Phys. Lett. A , 311(1984). R. Oppermann, A. Muller-Groeling, Nuclear Phys. B ,507 (1993). A.J. Bray, M.A. Moore, J. of Phys. C: Sol. State , L655(1980). W. Wiethege and D. Sherrington, J. Phys. C F. M. Zimmer, S. G. Magalhaes, Phys. Rev. B , 012202(2006). D. Thihumalai, Q. Li, and T. R. Kirkpatrick, J. Phys. A , 3339 (1989). W. Wu, B. Ellman, T. F. Rosenbaum, G. Aeppli, D. H.Reich, Phys. Rev. Lett , 2076 (1991). R.B. Griffiths, Phys. Rev. Lett. , 17 (1969). V. Dotsenko, J. Stat. Phys. , 197 (2006); V. Dotsenko,J. Phys. A: Math Gen. , 3397 (1994). M. Randeria, J. P. Sethna, and R.G. Palmer, Phys. Rev.Lett. , 1321 (1985). M. Guo, R. N. Bhatt, D. A. Huse, Phys. Rev. B , 3336(1996). H. Rieger, A. P. Young, Phys. Rev. B , 3328 (1996). A. P. Young, H. Rieger, Phys. Rev. B , 8486 (1996).49