Dynamics of the Infinite-Range Ising Spin-Glass Model in a Transverse Field

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  • VOLUME 81, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 21 SEPTEMBER1998

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    255Dynamics of the Infinite-Range Ising Spin-Glass Model in a Transverse Field

    M. J. Rozenberg1 and D. R. Grempel21Institut Laue-Langevin, B.P. 156, 38042 Grenoble, France

    2Dpartement de Recherche Fondamentale sur la Matire Condense, SPSMS, CEA-Grenoble, 17 rue de38054 Grenoble Cedex 9, France

    (Received 10 February 1998)

    We use quantum Monte Carlo methods and various analytic approximations to study the infinite-ranIsing spin-glass model in a transverse field in the disordered phase. We focus on the behavior offrequency dependent susceptibility of the system above and below the critical field. We establish that the quantum critical point, there exists an equivalence between the long-time behavior of this modand that of the single-impurity Kondo model. Our predictions for the long-time dynamics of the modeare in good agreement with experimental results on LiHo0.167Y0.833F4. [S0031-9007(98)07212-3]

    PACS numbers: 75.10.Jm, 75.10.Nr, 75.40.Gbiedhsedl

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    The physics of frustrated quantum spin systems isfascinating and rapidly growing area of condensed mattphysics [1]. One of the most widely investigated systemis the infinite-range Ising spin-glass model in a transverfield, a model that combines simplicity and experimentaaccessibility [2,3]. Its Hamiltonian reads

    H 21pN

    Xi,j

    JijSzi S

    zj 2 G

    Xi

    Sxi , (1)

    where Smi , m x, z are components of a three dimen

    sional spin-1y2 operator at theith site of a fully con-nected lattice of sizeN and the first sum runs over allpairs of sites. The exchange interactionsJij are inde-pendent random variables with a Gaussian distributionzero mean and varianceJ kJ2ijl1y2, and G is an ap-plied magnetic field transverse to the easy-axisz. ForG 0 Eq. (1) is the classical Sherrington-Kirkpatrickspin-glass model that has a second-order phase transiat T0g Jy4. WhenG is finite quantum fluctuations com-pete with the tendency of the system to develop spin-glaorder. As a result a boundaryGsT d appears in theG-Tplane between spin-glass (SG) and paramagnetic (Pphases.

    The model of Eq. (1) is relevant for the compoundLiHo0.167Y0.833F4, a site-diluted derivative of the dipolar-coupled Ising ferromagnet LiHoF4 [2]. An externalmagnetic fieldHt perpendicular to the easy axis splitsthe doubly degenerate ground state of the Ho31 ion.This splitting is proportional toH2t and plays the roleof G in Eq. (1) [2,4]. Experimentally, LiHo0.167Y0.833F4is paramagnetic at all temperatures above a critical fieHct 12 kOe [2,3]. Below Hct and for T * 25 mKthe behavior of the nonlinear susceptibility indicatessecond-order transition line between SG and PM phasthat ends atT0g 135 mK andHt 0 [5]. Investigationof the long-time dynamics of this system above this linhas revealed the existence of a fast crossover in the fidependence of the absorption at very low frequenci[2,3]. This crossover is characterized by a steep increa0 0031-9007y98y81(12)y2550(4)$15.00aers

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    of x 00sv ! 0d across an almost flat line in theG-T planeat G Gc and up toT , T0g .

    While the phase diagram of the model has been studtheoretically using a variety of methods [811], mucless is known about its dynamics that has been discusonly for large GyJ [12] and near the quantum criticapoint [13].

    In this paper we use a recently developed quantMonte Carlo method (QMC) [14] to numerically find thparamagnetic solutions of the model throughout theG-Tplane and also obtain analytic expressions that we dein several limiting cases. We find that the behaviorx 00svd above and below the critical field is qualitativeldifferent. ForG . Gc the zero-temperature spectrum omagnetic excitations has a gapD [13] that vanishes asG ! Gc 0.76J. At finite but low temperatures,T ,D, the gap edge develops a tail of exponentially smweight. On the other hand, for smallG and low T ,we find a narrow feature aroundv 0 whose intensitydecreases rapidly with increasing field or temperaturespectral weight is transferred to higher frequencies. Wfurther demonstrate that at the quantum critical point,the long-time limit, the problem can be mapped to tsingle-impurity Kondo model. The low-energy propertieof the system in the neighborhood of this point acharacterized by a new energy scale,T0 0.08J. Atfinite temperature there is a crossover between the regijust mentioned that is essentially controlled byG up toT , T0g . We finally present detailed predictions for thG and T dependence of the low-frequency response tare in good agreement with the experimental resultsLiHo0.167Y0.833F4 [3].

    Bray and Moore [15] have shown that the quantuspin-glass problem can be exactly transformed intosingle-spin problem with a time-dependent self-interactiQstd determined by the feedback effects of its couplinto the rest of the spins. As we have shown elsewhfor a related problem [14], much progress can be maby eliminating the self-interaction in favor of an auxiliar 1998 The American Physical Society

  • VOLUME 81, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 21 SEPTEMBER1998

    ofytic

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    icsfluctuating time-dependent fieldhstd coupled to the spins.The free energy per siteF in the paramagnetic phase canthen be written as

    bF minQstd

    (J2

    4

    Z b0

    Z b0

    dt dt0Q2st 2 t0d 2 ln ZlocfQg

    ),

    ZlocfQg Z

    D h exp"

    212

    Z b0

    Z b0

    dt dt0Q21st, t0d

    3 hstdhst0d

    #

    3 TrT exp"Z b

    0dtfJhstdSzstd 1 GSxstdg

    #,

    (2)

    whereT is the time-ordering operator along the imaginary-time axis0 # t # b. Zloc can be thought of as theaverage partition function of a spin in an effective magnetic field $heff Jhstdez 1 Gex whosez component isa random Gaussian function with varianceQstd. The lat-ter is determined by functional minimization of (2) whichgives the self-consistency condition [15]

    Qstd kT SzstdSzs0dlhstd , (3)where the average is taken with respect to the probabildensity associated toZloc. We solved Eqs. (2) and(3) iteratively using the QMC technique that we havdescribed elsewhere [14]. The imaginary-time axisdiscretized in up toL 128 time slices with Dt byL # 0.5. An iteration consists of at least 20 000 QMCsteps per time slice and self-consistency is generaattained after about eight iterations except very cloto the quantum critical point. We mapped the spinglass transition line in theG-T plane using the wellknown stability criterion [15]1 Jxloc where xloc Rb

    0 dt Qstd is the local spin susceptibility. We found asecond-order transition line ending at a quantum criticpoint at T 0 in agreement with previous work [810]. Going down in temperature toT , 1022J andextrapolating the results toT 0 we determined a precisevalue for the critical fieldGcyJ 0.76 6 0.01, whichlies in between previous estimates [10,13].

    We have studied the dynamical properties of the parmagnetic state throughout theG-T plane even belowTg,where it is unstable. Indeed, the analysis of the evoltion of the paramagnetic solution for smallG providesinsight on the physics of this problem as the states blow and aboveTg are continuously connected. In Fig. 1we show the correlation function for several values ofGandT . For G . Gc (panel a),Qstd decays exponentiallywith a time constantt0 G21 b that depends onlyweakly onT . This behavior is characteristic of the existence of a gapD , G in the excitation spectrum of thesystem. ForG , Gc andT J (panel c),Qstd also de-cays very rapidly for short times,t & t0 , b21. Fort * t0, however, it exhibits a very slow variation which-

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    FIG. 1. Qstd as a function oftyb. The crosses correspondto QMC data. The error bars are smaller than the sizethe crosses. The solid lines are obtained using the analexpressions discussed in the text.

    indicates the presence of excitations in the low-energy eof the spectrum,v T . With increasing temperaturet0increases and reaches a valueO sJ21d when T , J. Itthen is no longer possible to distinguish two different timscales. The caseG Gc at low temperature (panel b) isintermediate between the other two and the long-time bhavior becomes a power law,Qstd ~ t22 as T ! 0, asanticipated by Miller and Huse [13] using internal consitency arguments [16]. The solid lines in Fig. 1 are thresults of various analytic approximations that we discunext. We begin by consideringGyJ 1. In this case,the effective fields appearing in Eq. (2) are dominatedtheir x component. We may thus evaluate the trace of ttime-ordered exponential under the integral inZloc usinga low-order cumulant expansion. To the lowest nontriviorder we find

    x 00svd sgnsvd2GJmx

    qs2GJmxd2 2 sv2 2 G2d2 , (4)

    where mx 1y2 tanhsbGy2d is the zeroth order trans-verse magnetization. Equation (4) predicts the existenof a gapD

    pG2 2 2GJmx in the excitation spectrum.

    This gap has a very weak temperature dependenceT G and vanishes atGcyJ 1 at T 0, overesti-mating the critical field. It can be shown [17] that, foT D, the gap edge develops a tail that carries a smweightO se2DyT d.

    This procedure breaks down atGyJ , 1 and a differentapproach must be taken in order to describe the physat low fields. At G 0, the problem reduces to the2551

  • VOLUME 81, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 21 SEPTEMBER1998

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    classical Sherrington-Kirkpatrick model andQstd ; 1y4at all temperatures. We thus expect that forGyJ 1 theeffective field $heffsvnd will be dominated by itsv 0component. Indeed, setting for the momenthsvd 0 forv fi 0 in Eq. (3) and performing the functional averagsolely over static fieldshstd ; h0, we find

    Qsvnd

    bJ2h20y4G2 1 J2h20

    h0

    dn,0 1

    Gmsh0d cosux

    v2n 1 G2 1 J2h20

    h0

    ,

    (5)

    where msh0d 1y2 tanhsbp

    G2 1 J2h20y2d andcosux Gy

    pG2 1 J2h20 . The average is per-

    formed with respect to the probability distributionP sh0d ~ exps 2 bh20y2xlocd coshsb

    pG2 1 J2h20y2d.

    Using Eqs. (2) and (5) we can deduce the region of validity of this ansatz from the estimatesjhsvndyhs0dj , 32GT2yJ3 for T J, and,Gys2pnT dwhen T J. Within this approximation, the imaginarypart of the response on the real axis is given by

    x 00svdpv

    bJ2h20y4

    G2 1 J2h20

    h0

    dsvd

    1G2

    2v2tanhsbjvjy2dP s

    pv2 2 G2dp

    v2 2 G2. (6)

    In this regime the relaxation functionx 00svdyv splitsinto two contributions, an elastic peak atv 0 and acontinuum starting atv G. The fraction of spectralweight contained in each of these two contributionsdetermined by the transverse field and the temperatuEvaluating the coefficient of thed function in Eq. (6) wefind that the relative intensity of the central peak variebetween1 2 s8GTyJ2d2 for T J, andJ2ys4G2 1 J2dfor T J. The low-energy states represented by thedfunction are responsible for the slow decay observed in tlong-time behavior ofQstd at low T in Fig. 1c. The factthat the central peak has a zero width is a shortcomingthe approximation leading to Eq. (5) as it does not captuthe slow relaxational processes which broaden it [14Nevertheless, the excellent agreement between the analand the numerical results of Fig. 1c indicates that the widof the central peak must be, in any case, much smalthan the temperature. The high-energy states describedthe inelastic part of the response control the exponentdecay observed for short times. The decay rate predicby Eq. (6) ist210 , J2by8 for T J, andt210 , J forT J in agreement with the numerical results.

    The approximations discussed above cannot be usnearG Gc. However, one can still gain insight on thedynamics in the critical region by exploiting an interestinanalogy between the model (1) at the quantum criticpoint and the single-impurity Kondo problem that weestablish next. We first perform a Trotter decompositioof the time-ordered exponential in Eq. (2) and introducintermediate statesjsl ksj at each imaginary time slicet.The trace is now evaluated using the expression2552e

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    ksjeDt $heffstd? $Sjs0l eDtJhstds

    3

    dss0 1 dss0

    GDt

    21 O sDt2d

    (7)

    valid in the limit when the width of the time sliceDt !0. The partition function can then be expressed in termof a sum over histories, each of them defined byparticular sequence of the eigenvaluesszstd 61y2 ofthe intermediate states. Going over to the continuum limand performing the Gaussian integral over the auxiliafieldshstd, we obtain

    Zloc Z

    Dsz exp

    J2

    2

    Z b0

    Z b0

    dt dt0 szstd

    3 Qst 2 t0dszst0d 1 lnsGy2d

    3 snumber of spin flipsd

    , (8)

    where by number of spin flips we mean the numbertimes that the functionszstd changes sign in the interva0 # t # b. This expression has a form analogousthat of the partition function of the single-impurity Kondomodel in the Anderson-Yuval formalism [18]. In Eq. (8)G and J2Qstd play the roles of the spin-flip couplingJ6 and the long-range Ising-like effective interactioin the Kondo model [18]. Still, there is an importandifference between the two problems. In the latter tIsing-like interaction is given and behaves ass2 2 edyt2with e ~ Jz as T ! 0 [18]. When the dynamics ofthe impurity spin is controlled by the strong-couplinfixed point, its time-dependent correlation function is als~ t22 at long times. In contrast, in our problem,Qstdis a priori unknown. However, one realizes that if, fosomeG, the asymptotic behavior ofQstd is ,t22, thenthe two problems become equivalent at low energiesthat value of the field andT 0 by virtue of (8) andthe identifications just made. This is indeed the caat Gc where, as shown in Fig. 1b, the self-consistesolution of the problem at lowT can be accurately fittedby the finite-temperature generalization of Anderson aYuvals Ising-like interaction [18] usingt0 , G21c as ashort-time cutoff. This analogy between our problem athe Kondo model provides a simple way to estimate tenergy scale associated with the low-energy excitatioat the quantum critical point. It is well known [19]that the local susceptibility of the Kondo impurity isgiven by T0xlocsT d fsTyT0d, wheref is a universalfunction with fs0d 0.0796 [19] and T0 is the Kondoscale. Since in our casexloc J21 at the quantumcritical point, it follows thatT0 0.0796J. We expectthe quantum critical region to extend up to a temperatuTqc of this order. Our detailed numerical results for thtemperature dependence of the local susceptibilityconsistent withTqc , T0y4 [17].

    Finally, we would like to discuss the experimentallobserved crossover dynamics in LiHo0.167Y0.833F4 in the

  • VOLUME 81, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 21 SEPTEMBER1998

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    oe,light of the results presented here. By measuring thesponse of the system at 1.5 Hz, the experiment [3]probing the intensity of the low-lying excitations represented by thed-function peak in Eq. (6). In Fig. 2 wecompare the intensity of the latter (upper panel) and texperimental data of Ref. [3] (lower panel) at various temperatures. To make the connection with the experimewe plot the theoretical results as a function ofsGyGcd1y2since the splitting of...