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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
s Martyrs,
getheat,ell
255Dynamics of the InfiniteRange Ising SpinGlass Model in a Transverse Field
M. J. Rozenberg1 and D. R. Grempel21Institut LaueLangevin, B.P. 156, 38042 Grenoble, France
2Dpartement de Recherche Fondamentale sur la Matire Condense, SPSMS, CEAGrenoble, 17 rue de38054 Grenoble Cedex 9, France
(Received 10 February 1998)
We use quantum Monte Carlo methods and various analytic approximations to study the infiniteranIsing spinglass 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 longtime behavior of this modand that of the singleimpurity Kondo model. Our predictions for the longtime dynamics of the modeare in good agreement with experimental results on LiHo0.167Y0.833F4. [S00319007(98)072123]
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 infiniterange Ising spinglass 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 spin1y2 operator at theith site of a fully connected lattice of sizeN and the first sum runs over allpairs of sites. The exchange interactionsJij are independent random variables with a Gaussian distributionzero mean and varianceJ kJ2ijl1y2, and G is an applied magnetic field transverse to the easyaxisz. ForG 0 Eq. (1) is the classical SherringtonKirkpatrickspinglass model that has a secondorder phase transiat T0g Jy4. WhenG is finite quantum fluctuations compete with the tendency of the system to develop spinglaorder. As a result a boundaryGsT d appears in theGTplane between spinglass (SG) and paramagnetic (Pphases.
The model of Eq. (1) is relevant for the compoundLiHo0.167Y0.833F4, a sitediluted derivative of the dipolarcoupled 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 indicatessecondorder transition line between SG and PM phasthat ends atT0g 135 mK andHt 0 [5]. Investigationof the longtime 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 00319007y98y81(12)y2550(4)$15.00aers
sel

of
tion
ss
M)
ld
aes
eeldesse
of x 00sv ! 0d across an almost flat line in theGT 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 theGTplane 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 zerotemperature 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 longtime limit, the problem can be mapped to tsingleimpurity Kondo model. The lowenergy 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 lowfrequency response tare in good agreement with the experimental resultsLiHo0.167Y0.833F4 [3].
Bray and Moore [15] have shown that the quantuspinglass problem can be exactly transformed intosinglespin problem with a timedependent selfinteractiQstd 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 selfinteraction 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
nd
e
e
sess
byhe
al
ce
for
rall
icsfluctuating timedependent 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 timeordering operator along the imaginarytime 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 latter is determined by functional minimization of (2) whichgives the selfconsistency 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 imaginarytime 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 selfconsistency is generaattained after about eight iterations except very cloto the quantum critical point. We mapped the spinglass transition line in theGT plane using the wellknown stability criterion [15]1 Jxloc where xloc Rb
0 dt Qstd is the local spin susceptibility. We found asecondorder 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 theGT 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 decays very rapidly for short times,t & t0 , b21. Fort * t0, however, it exhibits a very slow variation which

ity
eis
llyse
al
a
u
e

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 lowenergy 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 longtime 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 ttimeordered exponential under the integral inZloc usinga loworder 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 transverse 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, overestimating 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
sa
itry
oflto
,
nthe
go
r
for
sent
nd
ndhens
ree
are
y
classical SherringtonKirkpatrick 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 lowenergy states represented by thedfunction are responsible for the slow decay observed in tlongtime 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 highenergy 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 singleimpurity Kondo problem that weestablish next. We first perform a Trotter decompositioof the timeordered exponential in Eq. (2) and introducintermediate statesjsl ksj at each imaginary time slicet.The trace is now evaluated using the expression2552e
,

isre.
s
he
ofre].yticthlerby
ialted
ed
gal
ne
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 singleimpurity Kondomodel in the AndersonYuval formalism [18]. In Eq. (8)G and J2Qstd play the roles of the spinflip couplingJ6 and the longrange Isinglike effective interactioin the Kondo model [18]. Still, there is an importandifference between the two problems. In the latter tIsinglike interaction is given and behaves ass2 2 edyt2with e ~ Jz as T ! 0 [18]. When the dynamics ofthe impurity spin is controlled by the strongcouplinfixed point, its timedependent 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 selfconsistesolution of the problem at lowT can be accurately fittedby the finitetemperature generalization of Anderson aYuvals Isinglike interaction [18] usingt0 , G21c as ashorttime cutoff. This analogy between our problem athe Kondo model provides a simple way to estimate tenergy scale associated with the lowenergy 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

nt

gy
ts
,
B
B
.
tt.
,
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 lowlying excitations represented by thedfunction 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...