VOLUME 85, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 18 SEPTEMBER 2000

From Second to First Order Transitions in a Disordered Quantum Magnet

Leticia F. Cugliandolo,1 Daniel R. Grempel,2,* and Constantino A. da Silva Santos3

1Laboratoire de Physique Théorique de l’École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, Franceand Laboratoire de Physique Théorique et Hautes Énergies, Jussieu, 5ème étage, Tour 24, 4 Place Jussieu, 75005 Paris, France

2CEA-Service de Physique de l’État Condensé, CEA-Saclay, 91191 Gif-sur-Yvette CEDEX, France3Laboratoire de Physique Théorique et Modèles Statistiques, Bâtiment 100, Université Paris-Sud, Orsay, F-91405, France

(Received 17 March 2000)

We study the spin-glass transition in a disordered quantum model. In a region of the phase dia-gram quantum effects are small and the phase transition is second order, as in the classical case. Inanother region, quantum fluctuations drive the transition to first order. Across the first order line thesusceptibility is discontinuous and shows hysteresis. Our findings qualitatively reproduce observationson LiHoxY12xF4. We also discuss a marginally stable spin-glass state and derive some results previouslyobtained from the real-time dynamics of the model coupled to a bath.

PACS numbers: 75.10.Nr, 05.70.Fh, 42.50.Lc

The study of quantum effects on the properties of spinglasses is a subject of great experimental and theoreticalinterest. Spin-glass phases have been identified in systemssuch as mixed hydrogen-bonded ferroelectrics [1], thedipolar magnet LiHoxY12xF4 [2,3], or Sr-doped La2CuO4[4] where quantum mechanics plays a fundamental role.An important question is whether quantum spin glassesare qualitatively different from their classical counterpartsat low temperature. There is growing experimental evi-dence that the answer to this question is affirmative bothin and out of equilibrium [2]. The thermodynamics ofseveral models of disordered magnetic systems has beeninvestigated with various techniques. Mean-field-likemodels have been solved using the replica formalism inimaginary time [5–13] and the Ising model in a transversefield has also been studied in finite dimensions [14–16].It is generally found that, in terms of a suitably definedquantum parameter G, a boundary Gc�T � in the G-T planeseparates spin-glass (SG) and paramagnetic (PM) phases.The transition line ends at a quantum critical point at�T � 0, Gc�0�� above which the system is paramagneticat all temperatures. In the case of the quantum sphericalp-spin model, the real-time dynamics of the systemcoupled to a phonon bath was also investigated [17].In this case, a boundary Gd�T � was found across whichthere is a dynamic phase transition from a PM state withequilibrium dynamics to a SG with nonstationary, aging,dynamics.

In this paper we investigate in detail the equilibriumproperties of this model. We find that a tricritical point�T�, G�� divides the line Gc�T � in two parts. For T $ T�,the SG transition is of second order and the behavior ofthe quantum system is qualitatively similar to that of theclassical one. However, for T , T� quantum fluctuationsdrive the transition first order. The magnetic susceptibil-ity is discontinuous and shows hysteresis across the firstorder line. These findings reproduce qualitatively the ob-served behavior of LiHoxY12xF4 in a transverse magnetic

0031-9007�00�85(12)�2589(4)$15.00

field [2,3]. The equations describing this system are non-linear and there is a multiplicity of solutions in parts ofthe phase diagram. We found that the usual criteria usedto choose between them have to be reinterpreted in or-der to get physically meaningful solutions in the regionT , T�. We also discuss the properties of solutions ob-tained through the use of the marginality condition, an ap-proach recently applied to the study of quantum problems[10,18]. Classically [19], the results of this approach areclosely related to corresponding ones from the analysis ofthe real-time dynamics of the system. We explicitly showthat this holds true in our quantum problem. This enablesus to identify the dynamical transition line Gd�T � and toderive certain properties of the nonequilibrium dynamicsusing the replica calculation.

The Hamiltonian of the quantum p-spin spherical modelis

H�P, s, J� �1

2M

NXi�1

P2i 2

NXi1,···,ip

Ji1···ip si1 · · · sip , (1)

where si is a scalar spin variable and the conju-gated momenta Pi satisfy the commutation relations�Pi , sj� � 2ihdij. A Lagrange multiplier z enforces thespherical constraint 1�N

PNi�1�s2

i � � 1. The interactionsJi1···ip are chosen from a Gaussian distribution with zeromean and variance �J2

i1···ip� J � J 2p!��2Np21�. The

model has glassy properties for all p $ 2. The Hamilto-nian (1) is a nonlinear generalization of the quantum-rotorspin-glass models discussed in [12]. It also describes aquantum particle moving in an N- (eventually infinite)dimensional space in the presence of a random potential.Finally, its partition function is formally identical to thatof a classical chain of “length” L � bh embedded in anN-dimensional random environment.

The equilibrium properties of the model are obtainedusing a replicated imaginary-time path integral formalism[5]. In the large N limit, the saddle-point evaluation of thepartition function allows us to define the order parameter

© 2000 The American Physical Society 2589

VOLUME 85, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 18 SEPTEMBER 2000

Qab�t 2 t0� � 1�NPN

i�1�T sai �t�sb

i �t0�� where a, b � 1, . . . , n denote the replica indices and T denotes the imaginary-time ordering operator. In terms of Qab the free energy per spin reads

F � limn!0

12n

(2

1b

Xvk

"Tr ln

√Q�vk�

bh

!2 n

√�Mv2

k 1 z�qd�vk�

h2 1

!#2 nz 2

J2

2h

Xab

Z b h

0dt Q

pab�t�

), (2)

where b � 1��kBT � is the inverse temperature, vk �2pk�b are the Matsubara frequencies, Qab�vk� �Rb h

0 dt Qab�t�eivkt , and qd�vk� � Qaa�vk�. From hereon we take J as the unit of energy, h�J as the unit of time,and work with dimensionless quantities. Quantum fluc-tuations are controlled by the parameter G � h2��JM�.The classical limit [20] is recovered when G ! 0. Theequilibrium solutions are determined by requiring thatQab�vk�, parametrized according to different Ansätze bean extremum of F. In the following, we concentrate onthe case p $ 3. The phenomenology of the p � 2 case[12,13] is not as rich [21].

For sufficiently high T and/or G, thermal and/or quan-tum fluctuations destroy the SG phase and the systemis in the PM phase. The free energy is extremal forQab�t� � dabqd�t� whose Fourier transform satisfies

qd�vk� � �v2k�G 1 z 2 S�vk��21, (3)

with S�vk� � p�2Rb

0 dt qp21d �t�eivkt and z is deter-

mined from qd�0� � 1. The above equation is nonlinearand may have several solutions. Some of them may bespurious. We solved Eq. (3) numerically for p � 3. ForT . T� 1�6 there is only one solution, irrespective ofthe value of G. For T , T�, three solutions coexist in afinite region of the T -G plane (not including the G � 0axis). One of them is unstable and can be discarded fromthe start. We discuss below how to choose the physicalsolution between the remaining two. We have shown thata one-step replica symmetry breaking Ansatz solves theproblem exactly in the SG phase [22], as in the classi-cal case [20]. Within this Ansatz, Qab is parametrized asQab�t� � q0

d�t�dab 1 qEAeab , where eab � 1 if a and bbelong to the same diagonal block of size m 3 m and zerootherwise, and q0

d�t� � qd�t� 2 qEA. The diagonal part,qd�t�, the breaking point, m, and the Edwards-Andersonorder parameter, qEA, are determined by extremizing F.We find

q0d�vk� � v2

k�G 1 z0 2 �S0�vk� 2 S0�0���21, (4)

where S0�t� � p�2�qp21d �t� 2 q

p21EA �, z0 � p�

2bmqp21EA �1 1 xp��xp , and

m � Txp

q2�� p�1 1 xp��q2p�2

EA . (5)

The parameter xp , solution of a transcendental equation,depends on p only and x3 � 1.817. Solutions of Eq. (4)exist only for G # Gmax�T �. Above this value, quan-tum fluctuations destroy the SG phase. The conditionqd�0� � 1 yields an equation for the breaking point of the

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form m � mT �G�. This function has two real branches inthe interval 0 # G # Gmax�T �. Physical values of m areon the branch that verifies mT �0� � mclass�T �, the clas-sical breaking point. For T $ T� (but lower than theclassical transition temperature), mmax � m�Gmax� � 1, itslargest possible value. For T , T�, instead, mmax , 1.In both cases, qEA is finite at Gmax�T �. We found thatlimT!0 mmax�T � � 0, implying that replica symmetry isrestored at the quantum critical point as in [10]. Theseconclusions also follow from an approximate analytical so-lution for arbitrary p $ 3 [22].

Stable PM solutions exist throughout the T -G plane.The free energies of the different states must thus be com-pared in order to construct a phase diagram. Figures 1(a)and 1(b) show the G dependence of the PM and SG freeenergies for the case p � 3 computed from Eq. (2) for twotemperatures, above and below T�. Solid lines and sym-bols represent the PM and SG solutions, respectively. Thecurves end at the point where the corresponding solutiondisappears. It may be seen that for T . T� the free ener-gies of the two states merge precisely at Gc�T� � Gmax�T �.The SG solution does not extend beyond the transitionpoint but the PM solution does. Below the critical point,FSG . FPM meaning that the SG solution maximizes thefree energy. Stability arguments do not exclude the PMsolution below Gc as a metastable state. However, this

-1.5

-1.0

-0.5

0.0

0.5

Γc = Γ

max

SG PM

(a) β=4

F

0 1 2 3 4

-4

-2

0

SGPM1

PM2

(b) β=20

F

Γ

2.5 3.0 3.5

0.2

0.3

Γmax

Γc

F

Γ

FIG. 1. Free energies of the different PM (solid lines) and SG(symbols) phases above (a) and below (b) T�. The inset inpanel (b) shows in detail the crossing of the free energies at thecritical point for T , T�.

VOLUME 85, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 18 SEPTEMBER 2000

solution has unphysical properties and can be discarded.Indeed, it can be shown that the ground-state energy andsusceptibility of Hamiltonian (1) are finite, whereas bothquantities diverge for the PM solution when T ! 0. Thisis the usual situation encountered in replica theories ofclassical spin glasses. As in the classical case [20], qEA isdiscontinuous at the transition. The latter is neverthelessof second order because m � 1 at Gc and, therefore, theeffective number of degrees of freedom participating in thetransition �1 2 m�qEA ! 0 at Gc. There is no latent heatand the linear susceptibility is continuous.

The situation is more involved below T� as there are twoPM solutions labeled PM1 and PM2 in Fig. 1(b). Naively,one would choose the solution with the lowest free energy,i.e., PM2. This solution disappears at a finite value of G

(not shown in the figure) and cannot be reached startingfrom G � `. Its free energy never intersects that of the SGphase. PM2 is, in fact, the continuation to low temperaturesof PM discussed in the previous paragraph and exhibits thesame unphysical behavior. Thus, PM1 has to be choseneven if its free energy is higher [23]. The free energiesof the SG and PM1 states cross at Gc , Gmax as shownin the inset in Fig. 1(b). In the low temperature phase,FSG , FPM, the opposite of what we found for T . T�

[24]. The SG and PM solutions extend beyond the pointwhere they cross. There is a region of phase coexistenceand hysteresis is expected.

Since now qEA and m are discontinuous at Gc the ther-modynamic transition is first order with latent heat anddiscontinuous susceptibility (see below). The phase dia-gram resulting from this analysis is represented in Fig. 2(thin lines). The flat section is the first order line. We havecomputed qEA and the susceptibility, x �

Rb0 dt�qd�t� 2

�1 2 m�qEA�, as functions of G for the p � 3 model. Theresults are displayed in Fig. 3. The susceptibility has acusp at Gc for T . T� and a discontinuity for T , T�.The dotted lines correspond to the regions of metastability.Their end points give the amplitude of the ideal hystere-sis cycle. The importance of quantum fluctuations can beappreciated from the fact that half way from the transition

0.0 0.2 0.4 0.60

1

2

3

m < 1

m=1

T*

SG

PMentertexthere

Γ

T

FIG. 2. Static (thin lines) and dynamic (thick lines) phase dia-rgrams of the p-spin model for p � 3. Solid and dashed linesrepresent second and first order transitions, respectively.

the order parameter is already reduced by a factor of 2. Itcan be shown analytically [22] that the spectrum of mag-netic excitations at T � 0 is gaped both in the PM and SGphases for all G fi 0 (see below, however). Consequently,the latent heat vanishes exponentially as T ! 0. Since italso vanishes at T�, it must have a maximum at some in-termediate temperature.

First order transitions were also found in the fermionicSK-like spin-glass model [9] and a p-spin model in a trans-verse field [11]. In contrast, the SG transitions of theHeisenberg EA model and of the Ising model in a trans-verse field are known to be second order [8,14–16]. Earlyexperiments on LiHoxY12xF4 suggested that the secondorder SG transition might become first order at low tem-peratures [2] as a discontinuity in the ac susceptibility isfound at a critical transverse field with no divergence of thenonlinear susceptibility. More recently, hysteresis has beenobserved in this system [3], giving further support to thisidea. While the model that we study here is not intendedto describe microscopically this compound, it captures itsphenomenology.

We discuss next the consequences of the use of themarginality condition [19] to determine the breaking point.In this approach it is not required that F be an extremumwith respect to m but that the SG phase be marginallystable. This implies that the “replicon” eigenvalue L mustvanish throughout the low-temperature phase. The calcu-lation of L [22] is analogous to the classical one [20]. Theresult is

L � b2��qd�0� 2 bqEA�2 2 b2�2p� p 2 1�qp22EA . (6)

The value of m follows from the equation L � 0, whichcombined with the equation dF�dqEA � 0 yields

m � T � p 2 2�q

2�� p� p 2 1�� q2p�2EA . (7)

0 1 2 30.0

0.5

1.0 (b) β=12

q EA

Γ

1.5

2.0

2.5

3.0(a)

β=4

β=12

χ

FIG. 3. Magnetic susceptibility (a) and Edwards-Andersonorder parameter (b) of the p � 3 model.

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VOLUME 85, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 18 SEPTEMBER 2000

This expression is equivalent to Eq. (5) with the substitu-tion xp ! � p 2 2�. Interestingly enough, the results ob-tained using the marginality condition coincide with thecorresponding ones from the real-time dynamic calcula-tion taking the limits of long times and weak coupling tothe bath (in this order) [17]. Equation (7) is identical tothe equation found for the fluctuation-dissipation theorem(FDT) violation parameter, X, in the real-time dynamicalcalculation. Since the static and dynamical equations forqEA are also identical, m � X. The coincidence betweenthe values of X and m has been noticed several times forclassical models. This is the first explicit evidence of itsvalidity in a quantum problem. X is related to the effec-tive temperature [25] of the system, Teff � X21T , whereT is the temperature of the environment. Values of X fi 1signal FDT violations. The fact that bX � bm ! constwhen T ! 0 shows that a nontrivial Teff is generated evenin this limit. We have also shown analytically [22] that theinternal energy, computed from U � ≠�bF��≠b at con-stant m, coincides with the energy per spin as obtainedfrom dynamics. The dynamic transition line Gd�T � is thusthe boundary of the region in the T -G plane where themarginally stable SG exists. Below this line, the dynam-ics of the system becomes nonstationary and FDT viola-tions set in. The dynamic phase diagram for p � 3 isshown in Fig. 2 (thick lines). As in the equilibrium case,m is discontinuous across the dashed line. Gd lies alwaysabove Gc suggesting that the equilibrium state can neverbe reached dynamically starting from an initial state in thePM phase. The two lines come extremely close to eachother for T � T�. Within the accuracy of our calcula-tions we cannot assert whether they precisely touch at T�,an intriguing possibility. For T , T�, m varies contin-uously along Gd�T � and vanishes at the quantum criticalpoint. This has a consequence of potential interest for ex-periment: FDT violations are predicted to appear sud-denly as Gd is crossed coming from the high G region forT , T�. These could be detected by comparing indepen-dent measurements of the dynamic susceptibility and thenoise fluctuation spectrum. The stationary part of the time-dependent susceptibility in the SG phase can be calculatedby analytic continuation of q0

d�vk�. The excitation spec-trum of the marginal SG state is gapless [22]. In the limitT , v ! 0, x 00�v� is linear,

limv!0

x 00�v��v � 1�p

G �2q�22p�EA ��� p� p 2 1���3�4, (8)

as in the Heisenberg spin glass [10]. However, in ourmodel a gapless spectrum is not a consequence of Gold-stone’s theorem since there is no continuous symmetry.

In conclusion, we have studied a quantum disorderedmodel whose low-temperature behavior is qualitatively dif-ferent from that of its classical counterpart. This featureis likely to be more general as suggested by existing ex-

2592

perimental evidence. We have also shown that partial in-formation about the dynamics of the ordered phase can beobtained from a purely static calculation.

C. A. d. S. S. acknowledges financial support fromthe Portuguese Research Council, FCT, BPD/16303/98.L. F. C. and D. R. G. thank ECOS-Sud for a travel grantand D. R. G. thanks the Newton Institute for hospitality.We thank T. F. Rosenbaum for communicating to us theresults of Ref. [3] prior to publication. We also thank G.Biroli, L. Ioffe, J. Kurchan, G. Lozano, and M. Rozenbergfor discussions.

*On leave from CEA-Grenoble, SPSMS, F-38054 GrenobleCedex 9, France.

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[22] L. F. Cugliandolo, D. R. Grempel, and C. A. da Silva Santos(unpublished).

[23] A similar situation arises in the Ising SG model in a trans-verse field at low temperatures. M. J. Rozenberg and D. R.Grempel (unpublished).

[24] This has also been found in an exactly solvable classicalmodel, P. Mottishaw, Europhys. Lett. 1, 409 (1986). It canbe interpreted as being due to the competition between therequirement of maximization (minimization) with respectto Qafib (Qaa).

[25] L. F. Cugliandolo et al., Phys. Rev. E 55, 3898 (1997).