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PHYSICAL REVIEW B 70, 024422 (2004)<strong>Effects</strong> <strong>of</strong> <strong>dissipation</strong> on disordered quantum spin modelsL. F. Cugliandolo, 1,2 D. R. Grempel, 3 G. Lozano, 4,5 and H. Lozza 51 Laboratoire de Physique Théorique de l’École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France2 Laboratoire de Physique Théorique et Hautes Énergies, Jussieu, 1er étage, Tour 16, 4 Place Jussieu, 75252 Paris Cedex 05, France3 CEA-Saclay, DSM/DRECAM/SPCSI, 91191 Gif-sur-Yvette Cedex, France4 Department <strong>of</strong> Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, United Kingdom5 Departamento de Física, FCEyN, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina(Received 10 February 2004; revised manuscript received 10 May 2004; published 30 July 2004)We study the effects <strong>of</strong> the coupling to an Ohmic quantum environment on the static and dynamicalproperties <strong>of</strong> a class <strong>of</strong> disordered spin models in a transverse magnetic field using a method <strong>of</strong> direct spinsummation. We discuss the influence <strong>of</strong> the environment on various features <strong>of</strong> the phase diagram <strong>of</strong> themodels as well as on the stability <strong>of</strong> the possible phases.DOI: 10.1103/PhysRevB.70.024422PACS number(s): 75.10.Jm, 75.10.Nr, 75.40.GbI. INTRODUCTIONThe coupling <strong>of</strong> quantum two-level systems (TLS) to adissipative environment has decisive effects on their dynamicalproperties. The case <strong>of</strong> dilute systems, in which interactionsbetween the TLS can be neglected, has been extensivelyinvestigated in the literature 1,2 and is now wellunderstood. The physics that emerges in the cases in whichinteractions between the TLS may not be neglected is muchless understood.In this paper we study the effects <strong>of</strong> a dissipative environmenton the equilibrium and dynamical properties <strong>of</strong> quantumglassy systems. Spin-glass phases in which quantumfluctuations play an important role were found in a number<strong>of</strong> experimental systems 3–5 and the importance <strong>of</strong> interactionson the properties <strong>of</strong> tunneling defects in structuralglasses was recently demonstrated. 6Mean-field 7 and finite dimensional 8 quantum spin-glassmodels have been studied intensively in the last few years.The effects <strong>of</strong> a coupling to the environment have been discussedfor some <strong>of</strong> these models such as the quantum sphericalp-spin glass model, 9–11 the SUN random Heisenbergmodel in the limit N→, 12 and the quantum random walkproblem. 13 In all these cases the relevant degrees <strong>of</strong> freedomare continuous, a fact that makes analytical treatments possible.Here we analyze the more realistic case <strong>of</strong> systems <strong>of</strong>interacting S=1/2 spins coupled to a bath <strong>of</strong> harmonic oscillators.We consider the case in which the interactions involvep-uplets <strong>of</strong> spins and are <strong>of</strong> infinite range. For p=2 we recovera model for metallic spin glasses studied previously. 14For p3 the model exhibits richer behavior, 15–18 includingthe possibility <strong>of</strong> having first order transition lines.We use a method <strong>of</strong> direct spin summation (DSS) firstused in the context <strong>of</strong> disordered spin models by Goldschmidtand Lai. 19 In this method the disorder-averaged freeenergydensity is computed using the replica method to averageover the random quenched interactions. Next, a Trotterdecomposition is performed in order to express the partitionfunction <strong>of</strong> the resulting single-site self-consistent problemas a sum over different contributions, each coming from apossible spin history = ±1. The continuous imaginarytimevariable 0 is discretized on a grid t =t/M,t=0,¯ ,M −1, and the partition function is computed bynumerically performing the exact sum over the 2 M possiblediscrete spin histories, t t = ±1. Physical results areobtained repeating this procedure for various values <strong>of</strong> Mand extrapolating to M →.The main conclusion <strong>of</strong> this work is that the coupling tothe enviromement reduces the strength <strong>of</strong> the quantum fluctuationsthus favoring the appearance <strong>of</strong> the spin-glass phase.Whereas for p=2 the phase transition is always second order,for p3 quantum fluctuations drive the transition first orderbelow a tricritical temperature T . We find that T decreaseswith the strength <strong>of</strong> the coupling to the bath. For p3 adynamic transition precedes the equilibrium phase transition.The coupling to the bath also stabilizes the dynamic glassyphase.The organization <strong>of</strong> the paper is as follows. In Sec. II weintroduce the coupled p-spin-bath model and the formalismthat we use to solve it. In Sec. III we discuss the numericalmethod and present our results. Section IV contains our conclusions.II. THE MODELWe are interested in disordered Ising spin models in atransverse field described by Hamiltonians <strong>of</strong> the typeNH s = H L − ˆ ix1i=1withH L =−Ni 1 ¯i pJ i1 ,...,i pˆ z i1¯ ˆ z ip.Here, ˆ x, ˆ y, ˆ z are the standard Pauli matrices, J i1¯i pdenotesa quenched random exchange between groups <strong>of</strong> pspins, is a transverse magnetic field, and N is the totalnumber <strong>of</strong> spins. The sum runs over all possible p-uplets <strong>of</strong>spins. The model is then fully connected and mean-field in20163-1829/2004/70(2)/024422(12)/$22.50 70 024422-1©2004 The American Physical Society

CUGLIANDOLO, GREMPEL, LOZANO, AND LOZZA PHYSICAL REVIEW B 70, 024422 (2004)character. The transverse field introduces quantum fluctuationsin what would otherwise be a purely classical model.To completely define the model we must choose p and thedistribution PJ <strong>of</strong> random interactions. We consider thecase in which the random independent variables J i1 ,...,i pareGaussian with zero mean and variance p!J 2 /2N p−1 . The scaling<strong>of</strong> the variance with N is chosen so as to ensure a goodthermodynamic limit.We study the thermodynamics and some aspects <strong>of</strong> thenonequilibrium dynamics <strong>of</strong> model 1 coupled to a quantumenvironment represented by a set <strong>of</strong> Ñ independent harmonicoscillators. We assume that the oscillators are in thermalequilibrium and that each <strong>of</strong> the spins in the system iscoupled to a different subset <strong>of</strong> these.The bosonic Hamiltonian <strong>of</strong> the isolated reservoir isÑH b = l=11pˆ l2 +2m ll=1Ñ12 m l l 2 xˆl2 . 3The coordinates xˆl and the momenta pˆ l satisfy canonicalcommutation relations. For simplicity we consider a bilinearcouplingNH sb =−i=1ш iz c il xˆll=1that involves only the oscillator coordinates. The Hamiltonianfor the coupled system is then given byH = H s + H b + H sb + H ct ,where we added a counter termÑH ct = l=11N2m l l2i=145c il ˆ iz2, 6whose effect is to eliminate a possible mass normalizationinduced by the coupling to the bath. 2The partition function <strong>of</strong> the combined system for a particularrealization <strong>of</strong> the bondsZ =Tre −H involves a sum over all states <strong>of</strong> the system and <strong>of</strong> the bath.The trace over the variables <strong>of</strong> the bath can be performedexplicitly using standard techniques. 2,20,21 The result <strong>of</strong> tracingout these variables can be expressed in terms <strong>of</strong> the spectralfunction <strong>of</strong> the bath7I =2 max − with the friction constant, max an ultraviolet cut<strong>of</strong>f andx the Heaviside theta function.This problem can be mapped onto a classical Ising modelusing the Totter-Suzuki formalism. 15,19,22 This amounts towriting the path integral for the partition function as a sumover spin and oscillator variables evaluated on a discreteimaginary-time grid on the points t =/Mt labeled by theindex t=0,...,M −1. Periodic boundary conditions are imposedon the discretized time axis. To recover the correctrepresentation <strong>of</strong> the trace the limit M → should be ultimatelytaken. The finite M expression yields a sequence <strong>of</strong>M approximants to the asymptotic M → formula.The Mth approximant <strong>of</strong> the “reduced” partition functionobtained after integrating out the bath isM−1Z =Tr ti exp M t=0whereM−1 N+ t=0i=1C t−t = 2 maxd 0Ni 1 ¯i pt tJ i1¯i p i1¯ ipM−1 NA + B t i t+1 i − 1− t i ti Ct−t ,t,t=0 i=1A = 1 2 ln sinh Mcosh M,B = 1 2 ln coth M,9101112 cosht − t/M − 1/2sinh2 /2M. sinh/213The trace represents the sum over all 2 NM distinct classicalIsing spin configurations, t i = ±1, for each spin, i=1,...,N,evaluated at each time-slice, t=0,...,M −1.The disordered averaged free-energy is calculated usingthe replica trick 23 Z n −1F¯=−ln Z = − lim . 14n→0 nAfter some standard manipulations, and up to some irrelevantfactors, we obtainÑI ij 2 l=1cil c jlm l l − l = ij I.We chose to study the effect <strong>of</strong> an Ohmic bath parameterizedas8withnZ n = M−1a,b=1 t,t=0 DQ atbt D atbt exp− NP,Q,15024422-2

EFFECTS OF DISSIPATION ON DISORDERED … PHYSICAL REVIEW B 70, 024422 (2004)n M−1P,Q = atbta,b=1 M 2 Qatbt − 2 J 24M 2Qatbt •pt,t=0+ C t−t ab 1−Q atbt − lnTr at e H eff, 16nH eff = a,b=1 t,t=0M−1atbtM 2n M−1 at bt + A + B at at+1 ,a=1 t=017where the bullet is used to distinguish the ordinary powerfrom the matrix power. In the thermodynamic limit, N→,the integrals in Z¯n can be evaluated with the saddle pointmethod at the expense <strong>of</strong> interchanging the order <strong>of</strong> the N→ and n→0 limits. The disordered-averaged free-energyper spin is thenP 0 ,Q 0 f¯= lim , 18n→0 nwhere 0 and Q 0 satisfy PQ,QQ 0 , 0=0, PQ,Q 0 , 0=0. 19Hereafter we omit subscripts in the saddle-point values Q 0and 0 . The disorder-averaged entropy per spin is easily obtainedfrom the disorder-averaged free-energy density and − s¯k = f¯+ 1 n M−1B2 J 2n 2M 2 Qatbt •pa,b=1 t,t=0− ab 1−Q atbt C t−t +−sinh2/Mcosh2/M 1 n a=1nQ at+1at.20Another physical observable <strong>of</strong> interest is the magnetic susceptibility = M, 21 hh=0where M=N −1 N i=1 z i is the total disorder-averaged magnetizationand h a longitudinal external magnetic field. Interms <strong>of</strong> Q atbt the susceptibility is given byn = M 2 M−1a,b=1 t,t=0Q atbt .22The right-hand sides <strong>of</strong> Eqs. (20) and (22) should be evaluatedat the saddle point.The matrix elements Q atbt are the order parameters <strong>of</strong> themodelQ atbt = 1 atN i bti .23i=1Because <strong>of</strong> the translational invariance in the Trotter timedirection the diagonal terms in the replica indices depend onthe time difference onlyQ atat = q d t − t.24Notice that due to the periodic boundary condition q d t=q d M −t. In addition, as q d 0=1, only t−t=1,2,¯ ,intM 2 need to be considered. The <strong>of</strong>f-diagonalelements in the replica indices, Q atbt with ab, are t and tindependent as shown by Bray and Moore. 24In order to determine the different phases <strong>of</strong> the model,we consider the following Ansätze.(1) Paramagnetic phase. The matrices Q and are takento be diagonal in replica spaceQ atbt = q d t − t ab , atbt = d t − t ab . 25Using Eqs. (16)–(18) the disorder-averaged free-energyper spin can be expressed asf¯= 2 J 2 M−14M 2p −1 q p d t − t − 2 J 2 M−14M + tt− lnTr t e H pmeffC t−t tt26withH pm eff = 1 M−1M−1M 2 d t − t t t + A + B t t+1 .ttt=027Here, q d t−t and d t−t are obtained self-consistentlyfrom the extremum condition by summing over all 2 M spinconfigurations t =±1:q d t − t = Tr t e H pmeff t t Tr t e H pm, 28eff d t − t = 2 J 2 pq p−14 d t − t + M 2 C t−t . 29(2) Equilibrium spin-glass phase. In order to characterizethis phase we use a one-step replica symmetry breaking(RSB) AnsatzQ atbt = q d t − t − q ea ab + q ea ab , atbt = d t − t − ea ab + ea ab ,where ab is a block-diagonal matrix in replica space= 1 if a and b belong to the same m m ab diagonal block,0 otherwise.303132024422-3

CUGLIANDOLO, GREMPEL, LOZANO, AND LOZZA PHYSICAL REVIEW B 70, 024422 (2004)The parameter m will be referred below as the breakpoint.Using Eqs. (16)–(18) the disordered-averaged free-energydensity becomesf¯= 2 J 2 M−14M 2 p −1 q p d t − t + m −1 2 J 24 p −1q eaptt ea = 2 J 2 pq p−14 ea .39As it has been discussed in a number <strong>of</strong> papers on classical 25and quantum 15–17,26 spin-glass models, two choices for thedetermination <strong>of</strong> the parameter m lead to physically differentresults. The use <strong>of</strong> the extremum condition, that correspondsto taking the value <strong>of</strong> m at which the disorder-averaged freeenergyis stationary, leads towithM−1− 2 J 24M − 1 m ln dTr t e H esgeff m + C t−t tt33m = I −1m2 2 J 2 pp −1q p4ea +ln dX Tr t e H esgeff m,40whereesg = 1 M−1M 2 d t − t − ea t t − M−1eaM + 2 eaMx ttH effttM−1+ A + B t t+1 tand the integration measured dx 2e −x2 2 .3435Here and in what follows all integrals over x extend from− to .As in the paramagnetic phase, the order parameters q d t−t, q ea , d t−t and ea are determined self-consistentlyfrom the extremum conditions that involve a sum over all 2 Mspin configurations, t =±1:q d t − t =q ea = d Tr t e H esgeff m−1 Tr t e H esgeff t t , d Tr t e H esgeff m36 d Tr t e H esgeff m−2 Tr t e H effesg t t /M 2, 37 d Tr t e H esgeff m d t − t = 2 J 2 pq p−14 d t − t + M 2 C t−t ,38I = dX Tr t e H esgeff m lnTr t e H esgeff . 41 dX Tr t e H esgeff mWith this choice one describes the equilibrium properties <strong>of</strong>the model.This Ansatz yields the exact solution 17 to the sphericalversion <strong>of</strong> the p3 model. The p=2 spherical model issolved by a simpler replica symmetric form.We do not expect the one-step RSB Ansatz to be stableeverywhere in the phase diagram in the case <strong>of</strong> the discretespin models that we investigate here. The stability <strong>of</strong> theone-step Ansatz for the quantum models can be tested byextending to the quantum case the classical analysis <strong>of</strong> deAlmeida and Thouless. 27 When the lowest eigenvalue <strong>of</strong> thestability matrix (also called replicon) vanishes, the one-stepRSB Ansatz is marginally stable. When the replicon is negative,this Ansatz is unstable.By evaluation <strong>of</strong> the replicon for the values <strong>of</strong> the orderparameters and the breakpoint obtained from the extremumconditions we found that the one-step RSB Ansatz is unstablein the full spin-glass phase when p=2 [Sherrington-Kirkpatrick (SK)] model indicating the need to break thereplica symmetry further.In the case <strong>of</strong> the p3 classical spin model, the one-stepRSB Ansatz is unstable below a temperature T g T s asshown by Gardner 28 in the classical case. Thus, the solutionfor the classical Ising p spin model also requires full RSB atvery low temperatures. T g depends on the parameter p and,as expected, it tends to T s =J when p→2 + and it vanisheswhen p→. We thus expect to find a Gardner line <strong>of</strong> instabilityalso when quantum fluctuations are taken into account.A careful study <strong>of</strong> the dependence <strong>of</strong> this line on and thecoupling to the bath requires solving the quantum problem atrather low temperatures. This is done in Sec. III where wecompute the location <strong>of</strong> the Gardner instability line. As seenin Fig. 8, the region where the one-step RSB static Ansatz isunstable is quite small. Outside this region the one-step RSBAnsatz is exact and can be used to study the properties <strong>of</strong> the024422-4

EFFECTS OF DISSIPATION ON DISORDERED … PHYSICAL REVIEW B 70, 024422 (2004)p3 quantum S=1/2 model. Elsewhere, and for p=2, weshall regard this solution as a suitable approximation to thecorrect solution.(3) Dynamic spin-glass phase. The marginality conditionleads to a different equation for m. With this condition onerequires that the saddle-point is only marginally stable, i.e.,the matrix <strong>of</strong> quadratic fluctuations has a zero replicon eigenvalue(and one does not impose the condition <strong>of</strong> extremeon m). It has been checked by comparison to the real timedynamics, 9 that this condition yields the location <strong>of</strong> thefreezing transition <strong>of</strong> the spherical quantum p-spin modelwith p3 coupled to the oscillator reservoir at the initialtime t=0. 10 Here we use it as an indication <strong>of</strong> where such adynamic transition line should be located for the discretequantum spin systems.Adapting the calculation <strong>of</strong> de Almeida and Thouless 27 tothe quantum problem under study we find that the repliconeigenvalue is given bywithP =1−kq p−2 ea t,The factors r, u, and t are R = P −2Q + RQ =−kq p−2 ea u,R =−kq p−2 ea r.4243r = a b c d = dTr te H dsgeff m−4 Tr teH effdsgt dTr te H dsgeff m dTr te H dsgeff m−3 Tru = 1 teH effM 2 at b a d =tt = 1 M 4 at bt a b =ttdsgt t /M 4 t /M 2 Tr te H eff dTr te H dsgeff m dTr te H dsgeff m−2 Tr teH effdsgtt dTr te H dsgeff m, 44dsgtt t t /M 2 2 t t /M2 , 45. 46Here, we have definedk 2 J 2pp −1 472and H dsg eff is the Hamiltonian <strong>of</strong> Eq. 34. Finally, the values<strong>of</strong> q d t−t and q ea are fixed by the extremal conditions.III. RESULTSFIG. 1. Disorder-averaged free-energy density, f¯, <strong>of</strong> the quantumS=1/2 p=3 model at temperature T=0.3 as a function <strong>of</strong> thetransverse magnetic field . The coupling to the bath is =1.0. Thethree phases <strong>of</strong> the model are represented: a physical paramagnetlabeled PM1, an unphysical paramagnet that one discard on physicalgrounds labeled PM2 and the spin glass. The values <strong>of</strong> f¯ obtainedfor finite M are shown with thin lines [M =8 (bottom), M=9 (middle), and M =10 (top)]. The result <strong>of</strong> the extrapolation toM → is displayed with bold lines.In this section we describe the outcome <strong>of</strong> solving theequations we derived in the previous section and we discusshow the coupling to the Ohmic bath <strong>of</strong> harmonic oscillatorsmodifies the behavior <strong>of</strong> the spin model.A. Numerical methodThe free-energy density and derived magnitudes dependon the parameter M that in practice takes finite values. Severalstrategies were proposed to study the limit M →. Usadeland Schmitz 29 noted that M should be such that024422-5

CUGLIANDOLO, GREMPEL, LOZANO, AND LOZZA PHYSICAL REVIEW B 70, 024422 (2004)FIG. 2. The critical averaged free-energy density, f¯, at inversetemperature =3.3, as a function <strong>of</strong> the inverse number <strong>of</strong>imaginary-time slices, 1/M. The coupling to the bath is =1.0. Thedashed and solid lines are the results <strong>of</strong> fits linear in 1/M and in1/M 2 , respectively. Circles: results <strong>of</strong> DSS for M =8,9,10,11,12,and 13. Stars: results <strong>of</strong> DSS for M =14,15, and 16.FIG. 4. Upper panel: Static phase diagram for the p=2 model asobtained using the DSS technique for finite number <strong>of</strong> time slices Mand extrapolating the data to M →. The three lines correspond to=0,0.5,1, from bottom to top. Lower panel: Static phase diagramfor the p=3 model obtained using the same numerical method. Thecontinuous line (dashed line) indicates a second order (first order)phase transition. The critical lines continue below the lowest value<strong>of</strong> T for which we trust the algorithm, T0.25, to reach a quantumcritical point at T=0.FIG. 3. Upper panel: The diagonal order parameter q d for thep=3 model as a function <strong>of</strong> /=t/M where t is the Trotterindex, t=0,1,¯ ,M. The temperature is T=0.3 and the transversemagnetic field is =0.8. The coupling to the bath is =1. The sixupper curves are the solutions for M =8,9,10,11,12,13, from topto bottom. The symbols are the actual data and the lines representthe results <strong>of</strong> the spline interpolation. The lowest curve is the result<strong>of</strong> the extrapolation to M →. Lower panel: The limiting curvelim M→ q d M for three couplings to the environment: =0 (bottomcurve), =0.5 (middle curve), and =1 (top curve), The other parametersare the same as in the upper panel. (See text for the details<strong>of</strong> the method <strong>of</strong> extrapolation used.)/M 1. For low temperatures this criterion becomesquickly impractical since one cannot perform the completesum over states for such large values <strong>of</strong> M. As an alternative,these authors proposed to use a Monte Carlo procedure toestimate the sum over configurations when M is large. 29,30In this paper we adopt another method that has been previouslyused to study the isolated quantum SK model 19 andS=1/2 p-spin models in a transverse field. 15 Physical quantitiesare computed by DSS for values <strong>of</strong> M in the range 8M 13. The results thus obtained are fitted to series <strong>of</strong>powers <strong>of</strong> 1/M that allows to perform the extrapolation toM →. In almost all cases the expected 31 1/M 2 law isverified.As an example, consider the free-energy density <strong>of</strong> thedifferent phases <strong>of</strong> the p=3 model with =1 (that will bediscussed in detail later) displayed in Fig. 1. The four curvescorrespond to three values <strong>of</strong> M, M =8,9,10, and to the result<strong>of</strong> the extrapolation to M →, respectively. Figure 2 showsthe M dependence <strong>of</strong> the free energy at the value <strong>of</strong> thetransverse field at which the curves for the paramagnetic andspin-glass phases cross. The circles represent the data forM =8, 9, 10, 11, 12, and 13. The dashed and solid lines arethe results <strong>of</strong> fits linear in 1/M and in 1/M 2 , respectively.024422-6

EFFECTS OF DISSIPATION ON DISORDERED … PHYSICAL REVIEW B 70, 024422 (2004)FIG. 5. Free energy density, f¯, entropy, s¯, andsusceptibility, , as functions <strong>of</strong> the transversefield, , for the p=3 model at T=0.5T s * forthree values <strong>of</strong> the coupling to the bath, =0,0.5,1. The continuous (dashed) line correspondsto the paramagnetic (glassy) phase. Theentropy and susceptibility are continuous at thetransition indicating a second order phasetransition.The difference between the asymptotic values obtained usingthese two extrapolations is <strong>of</strong> the order <strong>of</strong> 10%. The lastthree points, represented by stars, are the result <strong>of</strong> the DSSfor M =14,15,16. It can be seen that they fall nicely on the1/M 2 extrapolation curve obtained for the smaller values <strong>of</strong>M thus supporting the results <strong>of</strong> the 1/M 2 extrapolations thatwe shall use hereafter.The same method can be used to determine the diagonalorder parameter, q d . Since the latter is only known on theimaginary-time grid 0,/M ,2/M , ¯ , and this dependson M, we first perform a spline interpolation <strong>of</strong> thedata for each value <strong>of</strong> M and use the interpolated curves asimput for the polynomial extrapolation described earlier.The method is illustrated in the upper panel <strong>of</strong> Fig. 3where we show the diagonal order parameter q d as a function<strong>of</strong> at constant temperature and transverse magneticfield for the p=3 model. Data for six values <strong>of</strong> M, M=8,¯ ,13, are represented by the symbols and the lines goingthrough them represent the spline interpolations. Thelowest curve is the result <strong>of</strong> the extrapolation to M →.024422-7

CUGLIANDOLO, GREMPEL, LOZANO, AND LOZZA PHYSICAL REVIEW B 70, 024422 (2004)FIG. 6. Free energy, f¯, entropy, s¯, and susceptibility,, as function <strong>of</strong> the transverse field, ,for the p=3 model at T=0.3T s * for three values<strong>of</strong> the coupling to the bath, =0,0.5,1. The continuous(dashed) line corresponds to the paramagnetic(glassy) phase. The entropy and susceptibilityare discontinuous at the transition indicating afirst order phase transition.The method <strong>of</strong> Goldschmidt and Lai 19 just described issimple to implement and very efficient but it is limited torelatively high temperatures as the extrapolation becomesless and less reliable as the temperature decreases.B. Static phasesLet us first discuss the effect <strong>of</strong> the environment on thestatic phase diagram <strong>of</strong> the p=2 (SK) and p3 quantum S=1/2 spin models. The spin-glass disorder-averaged freeenergydensities are obtained using the one-step RSB Ansatzdiscussed in Sec. II. We discuss the limits <strong>of</strong> validity <strong>of</strong> thisAnsatz later.As in other disordered quantum spin models 15–17,22 twoparamagnetic solutions coexist. As in the spherical p-spinmodel coupled to a bath the one labeled PM2 in Fig. 1 can bediscarded since its entropy becomes negative at sufficientlylow temperatures. Thus, we do not discuss it further in thispaper.The critical line T s , s separating the paramagnetic (PM)and spin-glass (SG) phases is determined by the values <strong>of</strong> thepairs T, where the physical paramagnetic (called PM1 in024422-8

EFFECTS OF DISSIPATION ON DISORDERED … PHYSICAL REVIEW B 70, 024422 (2004)FIG. 7. Dependence <strong>of</strong> m andq ea on the critical temperatureT s , for the p=3 model. Threevalues <strong>of</strong> are considered, =0,0.5,1. For increasing values<strong>of</strong> the interval in which m=1increases and, hence, the regionwhere a thermodynamic first ordertransition occurs decreases.Fig. 1) and spin-glass free-energy densities cross.We show in the upper and lower panels <strong>of</strong> Fig. 4 the staticcritical line in the T, plane separating a high T, high PM from a low T, low SG for the p=2 and p=3 models,respectively. The three curves in each figure correspond to=0 and two nonzero couplings =0.5,1 from bottom totop.For both models, the classical transition temperature,T s class , corresponding to s →0, remains unchanged by thecoupling to the quantum heat reservoir. This value is T s =Jfor when p=2 (Ref. 23) and it coincides with the one givenby Gross and Mézard, T s 0.67, for the classical problemwith p=3. 32For the three values <strong>of</strong> , the static critical transversefield, s T, is a decreasing function <strong>of</strong> T, which is consistentwith the fact that quantum fluctuations tend to destroy theglassy phase. We also see from the figures that the couplingto a quantum thermal bath favors the formation <strong>of</strong> the glassyphase: the coupling to the environment effectively reducesthe strength <strong>of</strong> the quantum fluctuations that tend to destroyit. For any value <strong>of</strong> the temperature that satisfies TT s class theextent <strong>of</strong> the spin-glass phase is larger for stronger couplingsto the bath. Moreover, we observe that the effect <strong>of</strong> the bathis stronger for lower temperatures.When p=2 the transition is always continuous andsecond-order thermodynamically. For p=3 instead, as in thespherical case 16,17,10 and the isolated quantum S=1/2model, 15 an interesting change from a second-order to a firstordertransition appears. We demonstrate these statements bydisplaying in Figs. 5 and 6 the behavior <strong>of</strong> the free-energydensity, entropy, and susceptibility <strong>of</strong> the p=3 S=1/2 spinmodel as a function <strong>of</strong> the transverse field for T=0.5T s*and T=0.3T s * .At sufficiently high temperatures, TT s * , one finds a spinglasssolution for increasing transverse fields until the breakpointm reaches the value m=1. The values T, where m=1 coincides with the ones obtained by analyzing the crossing<strong>of</strong> the free-energy densities <strong>of</strong> the paramagnetic and spinglasssolutions. Thus, for the chosen temperature TT s * thisis the critical transverse field. Even if the Edwards-Andersonparameter, q ea , and the diagonal element, q d , are nonzeroat this point in parameter space, one can check, as shown inFig. 5, that the entropy and susceptibility do not show ajump. Thus, for TT s * the transition is discontinuous [due tothe jump in q ea and q d ] but <strong>of</strong> second order thermodynamically.The situation is different at lower temperatures. In Fig. 6we show the free energy, entropy and susceptibility <strong>of</strong> thep=3 S=1/2 model for T=0.3T s * . In this case, the point inwhich the free-energy <strong>of</strong> the paramagnetic and spin-glasssolution cross corresponds to m1 and as shown in the figurethis leads to a discontinuity <strong>of</strong> the entropy and susceptibility.In this case, the transition is discontinuous and firstorderthermodynamically.In Fig. 7 we show the dependence <strong>of</strong> q ea and m on thecritical temperature T s for three values <strong>of</strong> the coupling to thebath. The model is again the p=3 quantum S=1/2 spin glass.As already mentioned we observe that for all temperaturesq ea is different from zero, leading to a discontinuous phasetransition. m equals one for T s T s * but m1 for T s T s * .The figure also shows that T s * decreases with increasing couplingto the bath . Again, this result is reminiscent <strong>of</strong> whatwas found in the spherical case. 10C. Stability <strong>of</strong> the one-step static solutionIn order to study the stability <strong>of</strong> the one-step solution weevaluated the replicon eigenvalue R on the values <strong>of</strong> theorder parameters and m obtained from the static solution, andwe searched for the parameters T, such that R vanishes.In the classical limit this yields Gardner’s classical criticaltemperature that takes a rather low value, T G =00.25. 28Since we expect to find a decreasing value <strong>of</strong> the instabilitytemperature with the strength <strong>of</strong> the transverse field, we needto control the numerical algorithm for T0.25. Even if thismight seem, at first sight, impossible, we managed to obtainsensible results keeping reachable values <strong>of</strong> M, M 13,since the small values <strong>of</strong> the transverse field compensate thelarge value <strong>of</strong> in the condition /M 1.First, we analyzed the p=2 case that corresponds to theSK model in a transverse field. In the absence <strong>of</strong> the environmentwe found that the one-step RSB solution is notstable in the full spin-glass phase supporting the idea that thesolution to the statics <strong>of</strong> this model needs a full RSB scheme,024422-9

CUGLIANDOLO, GREMPEL, LOZANO, AND LOZZA PHYSICAL REVIEW B 70, 024422 (2004)FIG. 8. Comparison betweenthe static critical line T s , s andGardner’s instability line T G , G for the p=3 model with =0.just as in its classical limit and in contrast to recent claims inthe literature. 33In Fig. 8 we compare the static critical line T s , s asfound from the one-step RSB Ansatz, with Gardner’s line <strong>of</strong>instability for the p=3 model. We see that the region wherethe one-step RSB static Ansatz is not stable is quite small.Since we only trust the extrapolation from low values <strong>of</strong> Mto M → above temperatures <strong>of</strong> the order <strong>of</strong> T0.1, we donot explicitly extrapolate the instability line to lower temperatures.Nevertheless, the existing data suggest that in thezero temperature limit the static critical transverse field, s ,and Gardner’s critical field, G , do not coincide s T s=0 G T G =0.D. The dynamic transitionAs already explained in Sect. II the value <strong>of</strong> m found bysetting the replicon eigenvalue to zero leads to differentequations that encode some information about the nonequilibriumrelaxation dynamics <strong>of</strong> the system. Using this prescriptionwe obtained, for the p3 models a different criticalline that lies above the static transition. This result issimilar to those found in a series <strong>of</strong> other classical 25 andquantum 15–17 problems. In Fig. 9 we compare the static andmarginal critical lines for the p=10 quantum S=1/2 model.We chose a larger value <strong>of</strong> p to make the difference betweenthe two lines easier to visualize. The glassy static region issmaller than the glassy region determined by the marginalitycondition. When approaching the glassy phase from any directionin parameter space, the dynamic transition, associatedto the line <strong>of</strong> marginal stability, occurs before the static one.As on the critical static line, the curve determined with themarginal stability criterion is made <strong>of</strong> two pieces, on one <strong>of</strong>them the transition is <strong>of</strong> second-order (indicated with a solidline in Fig. 9) and on the other the transition is <strong>of</strong> first-order(indicated with a dashed line on the same figure). The firstordernature <strong>of</strong> the dynamic transition is displayed by, forinstance, a jump in the asymptotic value <strong>of</strong> the averagedinternal energy. The marginal tricritical point occurs athigher temperature than the static one.The external noise also has a strong effect on the dynamiccritical line. The stronger the coupling to the environment(larger value <strong>of</strong> ), the larger the spin-glass region in thephase diagram. This is also shown in Fig. 9 where a couple<strong>of</strong> curves, corresponding to =0 and =0.5 are drawn (seethe caption in the figure for more details).Finally, let us mention that there is an empirical relationbetween the value <strong>of</strong> the parameter m as found from themarginality condition and how the fluctuation-<strong>dissipation</strong>theorem is modified in the real time nonequilibrium relax-FIG. 9. Comparison between the static andmarginal critical lines for the p=10 model. Thesolid lines represent second order transitions andthe dashed lines first order ones. The set <strong>of</strong> curveswith T s class 0.6 (thin lines) are the static transitionlines for =0 (below) and =1 (above); theset <strong>of</strong> curves with T s class 0.82 (bold lines) correspondto the dynamic transition for =0 (below)and =1 (above).024422-10

EFFECTS OF DISSIPATION ON DISORDERED … PHYSICAL REVIEW B 70, 024422 (2004)ation <strong>of</strong> the quantum model. 9,10 Using this relation and interpretingthen the parameter m/T as an effective temperature 34we find that the modification <strong>of</strong> the fluctuation-<strong>dissipation</strong>theorem, and hence, T eff , depend on the strength <strong>of</strong> the couplingto the bath.IV. CONCLUSIONSIn this paper we studied the effect <strong>of</strong> an Ohmic quantumbath on the statics and dynamics <strong>of</strong> quantum disordered S=1/2 spin models <strong>of</strong> mean-field type. We found that thecoupling to the environment favors the appearance <strong>of</strong> thespin-glass phase reducing the strength <strong>of</strong> the quantum fluctuationsthat tend to destabilize it. As in the case <strong>of</strong> thespherical model 10,11 the phase transition is always secondorder for p=2. For p3 there exists a tricritical temperatureT below which quantum fluctuations drive the transitionfirst order. T decreases with the strength <strong>of</strong> the coupling tothe bath. For p3 a dynamic transition precedes the equilibriumphase transition. The coupling to the bath also stabilizesthe dynamic glassy phase.The physical origin <strong>of</strong> these effects is very simple: frictionand spin-spin interactions separately counteract the transversefield tending to suppress quantum fluctuations. Whenthe two effects are simultaneously present they reinforceeach other.It would be interesting to check if the same tendency toordering appears in macroscopic spin models in finite dimensions.One could attempt to study this problem in the context<strong>of</strong> frustrated spin magnets or the much studied, numericallyand analytically, quantum S=1/2 spin chain with and withoutdisorder. This problem is <strong>of</strong> interest for the possibleimplementation <strong>of</strong> quantum computers where the interaction<strong>of</strong> the system with its environment needs to be controlled.The effect <strong>of</strong> an environment on the properties <strong>of</strong> Griffithphases has also been the focus <strong>of</strong> a hot debate. 35 We expectto report on these problems in the near future.ACKNOWLEDGMENTSThe authors acknowledge financial support from theEcos-Sud travel grant, the ACI project “Optimisation algorithmsand quantum disordered systems.” L.F.C. is researchassociate at ICTP Trieste and acknowledges financial supportfrom the J. S. Guggenheim Foundation. G.S.L. was supportedby EPSRC Grants Nos. GR/N19359 and GR/R70309.This research was supported in part by the National ScienceFoundation under Grant No. PHYS99-07949. G.L. acknowledgesfinancial support from CONICET under PICT 03–11609. The authors thank F. Ritort for very useful discussions.1 A. 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