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VOLUME 85, NUMBER 4 PHYSICAL REVIEW LETTERS 24 JULY 2000 Mean Field Theory of a Quantum Heisenberg Spin Glass Antoine Georges, 1 Olivier Parcollet, 1,2 and Subir Sachdev 3 1 CNRS-Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond, 75005 Paris, France 2 Serin Physics Laboratory, Rutgers University, Piscataway, New Jersey 08854 3 Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120 (Received 16 September 1999) A full mean-field solution of a quantum Heisenberg spin-glass model is presented in a large-N limit. A spin-glass transition is found for all values of the spin S. The quantum critical regime associated with the quantum transition at S 0 and the various regimes in the spin-glass phase at high spin are analyzed. The specific heat is shown to vanish linearly with temperature. In the spin-glass phase, intrigu- ing connections between the equilibrium properties of the quantum problem and the out-of-equilibrium dynamics of classical models are pointed out. PACS numbers: 75.10.Nr, 64.60.Cn The interplay between quantum effects and disorder in spin glasses have been a subject of great recent interest [1]. On the experimental side, the strength of quantum fluctua- tions can be continuously tuned by varying, e.g., an applied transverse magnetic field [2]. Progress on the theoretical side has followed two different routes. From the higher di- mensional end, mean-field solutions and effective Landau theories have been obtained [3,4] for quantum Ising and rotor spin glasses, with a special focus on the vicinity of the quantum-critical point where the glass transition tem- perature is driven to zero. In low dimensions [5–7], it has been shown that the low-T physics is controlled by rare events (Griffiths-McCoy effects) at strong disorder fixed points. However, no established mean-field theory of the ex- perimentally important case of quantum Heisenberg spin glasses, with full SU2 symmetry, is yet available. Unlike the rotor / Ising models above, each site has nontrivial Berry phases which impose the spin commutation relations, and this is expected to place these models in a different uni- versality class [8]. Bray and Moore [9] pioneered the study of a model of Heisenberg spins on a fully connected (Sherrington-Kirkpatrick) lattice of N sites. In this Let- ter, we report a full solution of this model, both in the paramagnetic and the glassy phase, when the spin sym- metry group is extended from SU2 to SUN and the large-N limit is taken. The Hamiltonian is H 1 p N N X i ,j J ij S i ? S j , (1) where the J ij are independent, quenched random variables with distribution: PJ ij ~ e 2J 2 ij 2J 2 . In an imaginary time path-integral formalism, the model is mapped onto a self-consistent single site problem with the action [8,9] S eff S B 1 J 2 2N Z b 0 dt dt 0 Q ab t2t 0 S a t ? S b t 0 , (2) with b 1k B T , and the retarded interaction Q ab t2 t 0 obeys the self-consistency condition Q ab t2t 0 1N 2 S a t S b t 0 S eff . (3) Here, a, b 1,..., n denote the replica indices (the limit n ! 0 has to be taken later), and S B is the Berry phase in the spin coherent state path integral. For N 2 the prob- lem remains of considerable difficulty even in this mean- field limit. In [9], as well as in most subsequent work [10], the static approximation was used in which the t de- pendence of Q ab t is neglected; this may be appropriate in some regimes but prevents a study of the quantum equi- librium dynamics, in particular, in the quantum-critical regime. This imaginary time dynamics has, however, been studied recently in a Monte Carlo simulation with spin S 12 by Grempel and Rozenberg [11], but their study was limited to the paramagnetic phase. In our large-N limit, the problem is exactly solvable and, as explained be- low, this limit provides a good description of the physics of the N 2 mean-field model, as far as the latter is known. We find that in the paramagnetic phase, at low S (where the quantum fluctuations are the strongest), the quantum- critical regime is a gapless quantum paramagnet already studied in [8,12] and radically different from the paramag- net obtained in the classical regime (at large S), in which a local moment behavior persists down to the glass transi- tion. In the spin-glass phase, various regimes are obtained as a function of temperature T . The thermodynamic prop- erties and the dynamical response functions are analyzed below. Most notably, the low-T specific heat is found to have a linear T dependence, a behavior commonly ob- served experimentally in spin glasses but not often realized in mean-field classical models. Furthermore, the equi- librium dynamics of the quantum case reveals intriguing connections with some known features of the out-of equi- librium dynamics of classical glassy models, an observa- tion already made in [13] in a different context. To handle the large-N limit, we use a Schwinger boson representation of the SUN spin operators: S ab b y a b b 2 Sd ab , corresponding to fully symmetric repre- sentations (one line of NS boxes in the language of Young tableaux) where the number of bosons is constrained by 840 0031-9007 00 85(4) 840(4)$15.00 © 2000 The American Physical Society

Mean Field Theory of a Quantum Heisenberg Spin Glass

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VOLUME 85, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 24 JULY 2000

840

Mean Field Theory of a Quantum Heisenberg Spin Glass

Antoine Georges,1 Olivier Parcollet,1,2 and Subir Sachdev3

1CNRS-Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond, 75005 Paris, France2Serin Physics Laboratory, Rutgers University, Piscataway, New Jersey 08854

3Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120(Received 16 September 1999)

A full mean-field solution of a quantum Heisenberg spin-glass model is presented in a large-N limit.A spin-glass transition is found for all values of the spin S. The quantum critical regime associatedwith the quantum transition at S � 0 and the various regimes in the spin-glass phase at high spin areanalyzed. The specific heat is shown to vanish linearly with temperature. In the spin-glass phase, intrigu-ing connections between the equilibrium properties of the quantum problem and the out-of-equilibriumdynamics of classical models are pointed out.

PACS numbers: 75.10.Nr, 64.60.Cn

The interplay between quantum effects and disorder inspin glasses have been a subject of great recent interest [1].On the experimental side, the strength of quantum fluctua-tions can be continuously tuned by varying, e.g., an appliedtransverse magnetic field [2]. Progress on the theoreticalside has followed two different routes. From the higher di-mensional end, mean-field solutions and effective Landautheories have been obtained [3,4] for quantum Ising androtor spin glasses, with a special focus on the vicinity ofthe quantum-critical point where the glass transition tem-perature is driven to zero. In low dimensions [5–7], it hasbeen shown that the low-T physics is controlled by rareevents (Griffiths-McCoy effects) at strong disorder fixedpoints.

However, no established mean-field theory of the ex-perimentally important case of quantum Heisenberg spinglasses, with full SU�2� symmetry, is yet available. Unlikethe rotor/Ising models above, each site has nontrivial Berryphases which impose the spin commutation relations, andthis is expected to place these models in a different uni-versality class [8]. Bray and Moore [9] pioneered thestudy of a model of Heisenberg spins on a fully connected(Sherrington-Kirkpatrick) lattice of N sites. In this Let-ter, we report a full solution of this model, both in theparamagnetic and the glassy phase, when the spin sym-metry group is extended from SU�2� to SU�N� and thelarge-N limit is taken. The Hamiltonian is

H �1

pN N

Xi,j

Jij�Si ? �Sj , (1)

where the Jij are independent, quenched random variableswith distribution: P�Jij� ~ e2J2

ij��2J2�. In an imaginarytime path-integral formalism, the model is mapped ontoa self-consistent single site problem with the action [8,9]

Seff � SB 1J2

2N

Z b

0dt dt0 Qab�t 2 t0� �Sa�t� ? �Sb�t0� ,

(2)

with b � 1�kBT , and the retarded interaction Qab�t 2

t0� obeys the self-consistency condition

0031-9007�00�85(4)�840(4)$15.00

Qab�t 2 t0� � �1�N2� � �Sa�t� �Sb�t0��Seff . (3)

Here, a, b � 1, . . . , n denote the replica indices (the limitn ! 0 has to be taken later), and SB is the Berry phase inthe spin coherent state path integral. For N � 2 the prob-lem remains of considerable difficulty even in this mean-field limit. In [9], as well as in most subsequent work[10], the static approximation was used in which the t de-pendence of Qab�t� is neglected; this may be appropriatein some regimes but prevents a study of the quantum equi-librium dynamics, in particular, in the quantum-criticalregime. This imaginary time dynamics has, however, beenstudied recently in a Monte Carlo simulation with spinS � 1�2 by Grempel and Rozenberg [11], but their studywas limited to the paramagnetic phase. In our large-Nlimit, the problem is exactly solvable and, as explained be-low, this limit provides a good description of the physics ofthe N � 2 mean-field model, as far as the latter is known.We find that in the paramagnetic phase, at low S (wherethe quantum fluctuations are the strongest), the quantum-critical regime is a gapless quantum paramagnet alreadystudied in [8,12] and radically different from the paramag-net obtained in the classical regime (at large S), in whicha local moment behavior persists down to the glass transi-tion. In the spin-glass phase, various regimes are obtainedas a function of temperature T . The thermodynamic prop-erties and the dynamical response functions are analyzedbelow. Most notably, the low-T specific heat is found tohave a linear T dependence, a behavior commonly ob-served experimentally in spin glasses but not often realizedin mean-field classical models. Furthermore, the equi-librium dynamics of the quantum case reveals intriguingconnections with some known features of the out-of equi-librium dynamics of classical glassy models, an observa-tion already made in [13] in a different context.

To handle the large-N limit, we use a Schwingerboson representation of the SU�N� spin operators: Sab �by

abb 2 Sdab, corresponding to fully symmetric repre-sentations (one line of NS boxes in the language of Youngtableaux) where the number of bosons is constrained by

© 2000 The American Physical Society

VOLUME 85, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 24 JULY 2000

Pa by

aba � NS. In the SU�2� case, S coincides with theusual definition of spin. Fermionic representations canalso be considered but they actually do not lead to a spin-glass phase at any temperature in the N � ` limit [8]. Inthe large-N limit, the self-consistent single-site problemreduces to a nonlinear integral equation for the replicatedboson Green’s function: Gab�t� � 2

Pa�Tba

a�t� 3

byba �0���N [8]:

�G21�ab�inn� � inndab 1 ladab 2 Sab�inn� , (4)

Sab�t� � J2≥Gab�t�

¥2Gab�2t� , (5)

Gaa�t � 02� � 2S . (6)

Here, nn are the bosonic Matsubara frequencies, and G21

stands for the inverse in replica space. The (disorder-averaged) local spin correlation function is related to

Gab�t� by xloc�t� � � �Si�0� ? �Si�t�� � Gaa�t�Gaa�2t�.The resulting phase diagram, obtained by both analyticaland numerical studies of these equations, is displayed inFig. 1, as a function of S and T�J. Spin-glass orderingis found at any value of S. The critical temperatureincreases as JS2 at large S (see below) and vanishes in thelimit S ! 0, as found earlier in [10]. The point S � 0,T � 0 is the quantum critical point of this model. Severalcrossovers are found within the spin-glass phase, whichwill be described later.

We first describe the paramagnetic phase and the as-sociated crossovers. In this phase, the Green’s function isreplica diagonal Gab�t� � G�t�dab and thus Eqs. (4)–(6)reduce to a single nonlinear integral equation. We empha-size that, as in any mean-field theory, paramagnetic solu-tions of the mean-field equations can be found even belowthe critical T where an instability to ordering occurs. Athigh T , we have nearly free spins with an almost con-stant correlation function xloc�t� � S�S 1 1� and a Curielocal susceptibility xloc �

Rb0 xloc�t� dt � S�S 1 1��T .

FIG. 1. Mean-field phase diagram and crossovers of thelarge-N quantum Heisenberg spin glass (the various regimesare discussed in the text).

For large values of S, these solutions smoothly evolve, asT is reduced, into solutions which still behave locally aslocal moments, but with a Curie constant reduced by quan-tum fluctuations: xloc � S2�T . This partial quenchingoccurs at a temperature of order JS2 at large S, of thesame order but smaller than the glass transition tempera-ture. These solutions actually have unphysical low-T prop-erties, such as a divergent internal energy U � 2J2S4�2Tand a negative entropy (~ 2J2S4�4T2). These featuresare well known in classical mean-field models and sim-ply signal the tendency to spin-glass ordering. At smallervalues of S (Fig. 1), a crossover to a different kind of para-magnetic solution is found below T � J , where we enterthe quantum-critical regime. In this gapless quantum para-magnet (spin liquid), investigated previously in [8,12],the local response displays a scaling form for v, T ø J ,Jx

00loc�v� ~ tanh�v�2T �, and the local susceptibility di-

verges only logarithmically Jxloc ~ ln�J�T �. In contrastto the local-moment solutions, this paramagnet has finiteresidual low-temperature entropy [14], so that the quench-ing of the entropy as T is decreased takes place muchmore gradually at low S, when quantum fluctuations arestrong, than at large S in the classical regime. It can beshown analytically [14] that these solutions of the mean-field equations exist down to T � 0 only for very lowvalues of S, smaller than Sc � 0.05. For larger spins, alocal-moment-like solution is retrieved as T is loweredbelow a temperature of order JS2 (again below the actualglass transition). However, the spin-liquid solutions arethe relevant ones in the quantum-critical regime at finitetemperature JS2 , T , J for an extended range of spinvalues which extend up to S � 1. The detailed analysis ofthe coexistence between these two kinds of paramagneticsolutions at low S is rather intricate and will be presentedelsewhere [14].

In the quantum Monte Carlo results of [11] for the para-magnetic phase of the S � 1�2, SU�2� model, the samereduction of the Curie constant from S�S 1 1� to S2 wasobserved. Furthermore, the relaxation function x 00�v��v

evolves from a single peak of width JS centered at v � 0to a three peak structure in the low-T local moment regime.The central peak of weight S2 corresponds to the resid-ual local moment while two side peaks at an energy scaleJ2S3�T correspond to transverse relaxation [11]. All thesefeatures are captured by our solution in the large-N limit,the only qualitative difference being that no thermal broad-ening of the central peak is found in this limit. Furthermore[15], numerical results not reported in [11] reveal that, in alimited intermediate T range of the SU�2� S � 1�2 model,spin liquid solutions similar to those found here in thequantum-critical regime are observed. Although a logarith-mic regime is not directly visible in the T dependence ofthe local susceptibility because of this limited range, quan-tum criticality is directly apparent in a nonmonotonic Tdependence of the local spin correlation function xloc�t�.

We now turn to the analysis of the spin-glass phase. Wefirst note that the spin-glass transition is not signaled by

841

VOLUME 85, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 24 JULY 2000

the divergence of the spin-glass susceptibility (which isactually of order 1�N) [14,16]. In the ordered phase, theboson Green’s function can be parametrized as follows:

Gab�t� � � eG�t� 2 gdab 2 gab�1 2 dab� , (7)

where gab is a constant n 3 n matrix and g1 a constant,fixed so that eG is regular at T � 0, i.e., eG�t ! `� � 0.The usual spin-glass order parameter [17] is qab � g2

ab .We have searched for replica-symmetric broken solutionswith a general Parisi ansatz for gab and found only single-step replica symmetry breaking solutions (as in [10]). TheParisi function g�x� associated with gab is thus piecewiseconstant: g�x� � 0 for x , xc, g�x� � g�1� �

pqEA �

g for x . xc, where qEA is the Edwards-Anderson orderparameter; this also implies that g � g. For the followingdiscussion, it is convenient to define the parameter Q �2J eG�in � 0��g. Using standard inversion formulas for aParisi matrix [18], the full set of mean-field equations read

� eG�inn�21 � inn 2 Jg�Q 2 �eS�inn� 2 eS�0� , (8)

eS�t� � J2� eG2�t� eG�2t� 2 2g eG�t� eG�2t�

2 g eG2�t� 1 2g2 eG�t� 1 g2 eG�2t� , (9)

eG�t � 02� � g 2 S , (10)

bxc � �1�Q 2 Q��Jg2. (11)

However, these equations do not determine Q (or equiva-lently the breakpoint xc) as also happens in a classicalspin-glass model with a single step of replica symmetrybreaking: there is a continuous family of solutions pa-rametrized by Q, which has to be determined by indepen-dent considerations. Two possibilities have appeared inprevious work: (i) Determine Q by minimizing the freeenergy, as a function of Q, or (ii) impose a vanishinglowest eigenvalue of the fluctuation matrix in the replicaspace (the “replicon” mode). Criterion (i) is certainly thenatural one from the point of view of equilibrium ther-modynamics. However, studies of out-of-equilibrium dy-namics of classical spin glasses have revealed [19] thatthese lowest free-energy solutions can never be reachedand that the system “freezes” at a dynamical temperatureTc

sg, given precisely by the onset of solutions satisfyingthe replicon criterion (ii). In our quantum problem, bothchoices give sensible solutions, but with entirely differ-ent spectra of equilibrium dynamical fluctuations: (i) leadsto a gap in x

00loc�v�, while (ii) is found to be the unique

choice leading to a gapless spectrum. A similar observa-tion was made in the work of Giamarchi and Le Doussal[13] in their study of a one-dimensional quantum modelwith disorder. In the present context, it seems naturalto expect local gapless modes in the ordered phase of aquantum spin glass with continuous spin symmetry, andthese various considerations lead us to adopt (ii). Diago-nalizing the fluctuation matrix in replica space, we veri-fied stability and obtained the lowest eigenvalue e1 �3bJ2g2�1 2 3Q2�. The replicon criterion thus leads toQ � 1�

p3. The same value is also selected by imposing

842

that eG has a gapless spectral weight. In contrast, crite-rion (i) leads to 2 lnQ 1 1��4Q2� 1 1�2 2 3Q2�4 � 0,or Q � 0.44 . . . , and a gapped solution, which is alsostable because e1 . 0. We also note that the previous com-putation shows that the replica symmetric solution Q � 1is unstable in the spin-glass phase. Moreover, it can beshown that it leads to unphysical negative spectral weightat large S. Hence, a correct description of the low-energyexcitations of the quantum model requires replica symme-try breaking at any finite T in the spin-glass phase, al-though the replica symmetry is restored at T � 0 wherexc � 0 [from (11)].

Once Q is determined, a full numerical solution ofthe above equations can be performed. In particular, the“equilibrium” spin-glass temperature T

eqsg obtained from

criterion (i) is lower than the “dynamical” transition tem-perature Tc

sg obtained from criterion (ii) (see Fig. 1): thisis not obvious a priori, but is certainly required in ourinterpretation. Further analytical insight can be obtainedin the limit of large S. This limit can actually be takenin two distinct ways, revealing two crossovers within thespin-glass phase displayed in Fig. 1. If we take S ! `

while keeping T�JS2 fixed (i.e., staying close to the criti-cal temperature), all nonzero Matsubara frequencies canbe neglected (the static approximation is accurate). In thislimit, we find, in particular, Tc

sg 2JS2�33�2. Alterna-tively, keeping T � T�JS and v � v�JS fixed, we ac-cess the “semiclassical” regime of the spin-glass phase.In this limit, the Green’s function obeys a scaling formeG�v, T � � f�v���JS�, where f turns out to be indepen-dent of T and satisfies

f�v�21 � v 2 1�Q 2 3Q 2 f�v� 2 f��2v� .(12)

Eliminating f��2v� leads to a quartic equation for f�v�on which all the above properties can be checked moreexplicitly. A plot of the (gapless) relaxation function in the

FIG. 2. Relaxation function x 00�v��v in the large-S limit, ob-tained from (12).

VOLUME 85, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 24 JULY 2000

FIG. 3. Specific heat C�T� and internal energy U�T � (inset) vstemperature T , from a numerical solution of Eqs. (8)– (11) forS � 5.

spin-glass phase x 00�v��v obtained from (12) is displayedin Fig. 2.

Finally, we briefly describe the thermodynamic proper-ties, focusing on the T dependence of the specific heat.Numerical results for this quantity for intermediate spinare displayed on Fig. 3. They have been obtained fromthe T derivative of the internal energy U � 2J2�2 3Rb

0 Gab�t�2Gab�2t�2 dt, where G is a numerical solutionof Eqs. (8)–(11). Furthermore, a large-S, low-T expan-sion of U�T � can be done analytically and leads to [14]:U�T � � U�0� 1 aST4 1 bT2 1 . . . where a and b arepositive numerical coefficients. Hence, in the quantumregime defined by T , J

pS (see Fig. 1), the specific heat

depends linearly on temperature. Moreover, this behavioractually holds numerically for intermediate values of thespin as displayed in Fig. 3.

Despite being formulated over two decades ago [9], acomplete understanding of the quantum Heisenberg spinglass at the mean-field level has proven elusive. Here, wehave obtained a complete solution in a large-N limit, andpresented evidence that global aspects of the phase dia-gram pertain also to the physical SU�N � 2� case. Wedescribed crossovers in the vicinity of a quantum criticalpoint accessed by varying the spin S, but we can expectthat some features and intermediate temperature regimeswill survive when it is accessed by varying other parame-ters in the Hamiltonian, including doping with metallic car-riers as in Kondo lattice models [12,20]. We have alsodescribed the T ! 0 thermodynamics and spectral func-tions within the spin-glass phase, which is something notpreviously analyzed in any mean-field quantum spin-glassmodel: we found a specific heat linear in temperature, anda dynamical susceptibility x 00�v��v ! const as v ! 0.

We thank G. Biroli, L. Cugliandolo, T. Giamarchi,D. Grempel, P. Le Doussal, G. Kotliar, M. Rozenberg,

and A. Sengupta for useful discussions. Laboratoire dePhysique Théorique de l’ENS is UMR 8549 associéeau CNRS et à l’Ecole Normale Supérieure. S. S. wassupported by NSF Grant No. DMR 96-23181. A. G. andO. P. also acknowledge support of a NATO collaborativeresearch grant.

Note added.—It has been recently proven by one of us[21] that the behavior Jx

00loc�v� ~ const found above in the

quantum critical regime also holds for SU�2�.

[1] For a recent review and references, see, e.g., R. N. Bhatt,in Spin Glasses and Random Fields, edited by A. P. Young(World Scientific, Singapore, 1998).

[2] J. Brooke et al., Science 284, 779 (1999).[3] J. Miller and D. A. Huse, Phys. Rev. Lett. 70, 3147 (1993).[4] N. Read, S. Sachdev, and J. Ye, Phys. Rev. B 52, 384

(1995).[5] D. S. Fisher, Phys. Rev. B 51, 6411 (1995).[6] C. Pich et al., Phys. Rev. Lett. 81, 5916 (1998).[7] O. Motrunich et al., cond-mat/9906322.[8] S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993).

This paper demonstrated a spin-liquid ground state atN � ` for the fermionic representations of SU�N�, andincorrectly conjectured that a similar saddle point was theground state for a finite range of S . 0 in the bosonicrepresentations: the spin-glass state is always the globallypreferred solution at T � 0 for the latter representations.

[9] A. Bray and M. Moore, J. Phys. C 13, L655 (1980).[10] T. K. Kopec, Phys. Rev. B 52, 9590 (1995). In this pa-

per the static approximation was used to determine Tsg�S�,with results similar to Fig. 1. However, it was apparentlynot noticed that (4)– (6) allow a one-parameter family ofsolutions and the minimization criterion (i) was used with-out questioning its validity.

[11] D. Grempel and M. Rozenberg, Phys. Rev. Lett. 80, 389(1998).

[12] O. Parcollet and A. Georges, Phys. Rev. B 59, 5341 (1999);in this paper only fermionic spin representations are con-sidered, but the results apply also to the bosonic case.

[13] T. Giamarchi and P. Le Doussal, Phys. Rev. B 53, 15 206(1996).

[14] A. Georges, O. Parcollet, and S. Sachdev (to be published).[15] D. Grempel (private communication).[16] This is also clear from the manner in which the spin-glass

order vanishes, with xc ! 1 and g finite [10].[17] M. Mézard, G. Parisi, and M. Virasoro, Spin Glass Theory

and Beyond (World Scientific, Singapore, 1987).[18] M. Mézard and G. Parisi, J. Phys. (Paris) I 1, 809 (1991).[19] L. F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. 71, 173

(1993).[20] A. M. Sengupta and A. Georges, Phys. Rev. B 52, 10 295

(1995); S. Sachdev, N. Read, and R. Oppermann, Phys.Rev. B 52, 10 286 (1995).

[21] S. Sachdev, C. Buragohain, and M. Vojta, Science 286,2479 (1999); M. Vojta, C. Buragohain, and S. Sachdev,Phys. Rev. B 61, 15 152 (2000).

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