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I. MODEL AND OUTLINE OF THE RESULTS
The numerous recent studies of quantum fluctuations in spin glasses [14], have focused either on infinite-range models of Ising and rotor models [13, 15] or models in low dimensions which flow to strong disorder fixed points [16, 17, 18]. Here we shall continue the study of infinite range models, but will consider a model of Heisenberg spins :inthiscase,thepathintegralforeachspinhasaimportantBerryphasetermwhichimposesthespincommutation relations. As we will see, this leads to a great deal of new physics[19] and non-trivial dynamic spin correlations even in the spin glass state. More specifically, we present a complete solution of the quantum Heisenberg spin glass on afullyconnectedlatticeof sites with strong Gaussian disorder, both in the paramagnetic and the glassy phase, when the spin symmetry groupN is extended from SU(2) to SU(N)andthelarge-N limit is taken. In the limit of large connectivity, (dynamical) mean-field techniques apply and the model can be reduced to the study of a self- consistent single-site problem, which is however still highly non-trivial because of quantum effects. The large-N limit is instrumental in allowing for an explicit solution. Nevertheless some of our results regarding the quantum critical regime have been extended beyond the large-N limit. In a recent publication [20], we summarized the main results of the present study. Here, we provide detailed derivations and new results, such as a full discussion of the paramagnetic phases and a discussion beyond large-N. The model considered in this paper is defined by the Hamiltonian: 1 H = J S⃗ S⃗ , (1) √ ij i · j N i 1 J2 /(2J2) P (Jij)= e− ij (2) J√2π As already pointed out by Bray and Moore [21], after using the replica trick to average over the disorder [22], the mean-field (infinite dimensional) limit maps the model onto a self-consistent single site model with the action (in imaginary time τ,withβ the inverse temperature) : 2 β J ab a b Seff = SB dτdτ ′ Q (τ τ ′)−→S (τ) −→S (τ ′)(3) − 2N − · "0 and the self-consistency condition ab 1 a b Q (τ τ ′)= 2 −→S (τ) −→S (τ ′) (4) − N · Seff # $ where a, b =1, ,n denote the replica indices (the limit n 0hastobetakenlater)andSB is the Berry phase of the spin [19].··· Due to their time-dependence, the solution→ of these mean-field equations remains a very difficult problem for N =2,evenintheparamagneticphase.Thus,inRef.21,aswellasinmostsubsequentwork[23], the static approximation was used, neglecting the τ dependence of Qab(τ). This approximation may be reasonable in some regimes but prevents a study of the quantum− equilibrium dynamics, and is particularly inappropriate in the quantum-critical regime. However, this imaginary time dynamics has been explicitly studied in a Quantum Monte Carlo simulation in the paramagnetic phase with spin S =1/2byGrempelandRozenberg[24].Recently, we introduced a large-N solution of the mean field problem [20], in which the problem is exactly solvable and, as explained below, the solution provides a good description of the physics of the N =2meanfieldmodel,totheextent the latter is understood. More specifically, in the following we will consider two different types of spin representations for the SU(N)spins: a) Bosonic representations : the spin operator S is represented using Schwinger bosons b by S = b† b Sδ , αβ α β − αβ with the constraint α bα† bα = SN (0 S). In the language of Young tableaux, these representations are described by one line of length SN.Theyareanaturalgeneralisationofan≤ SU(2) spin of size S. % b) Fermionic representations : the spin operator S is represented using Abrikosov fermions f by S = f† f q δ , αβ α β − 0 αβ with the constraint f† f = q N (0 q 1). In the language of Young tableaux, these representations are α α α 0 ≤ 0 ≤ described by one column of length q0N.NotethatforSU(2), only S =0andS =1/2canberepresentedin this manner. % 3 In the following, we refer to the model with bosonic (resp. fermionic) representations as the bosonic (resp. fermionic) model. In the fermionic model, quantum fluctuations are so strong in large-N that the spin glass ordering is destroyed [19], contrary to the bosonic model, where a spin glass phase exists, as explained below. The two models have different theoretical interest : if one wants to concentrate on the quantum critical regime, above the spin glass ordering temperature, one can use the fermionic model (as e.g. in Ref. 25). However, since we are interested in the spin glass phase itself, we will now focus on the bosonic model. Nevertheless, our results on the paramagnet will be valid for both cases with only slight modifications explicitly quoted below. In the N limit, the mean field self-consistent model (3) reduces to an integral equation for the Green’s function →∞ab a b of the boson Gb (τ) Tb (τ)b† (0) where the bar denotes the average over disorder and the brackets the thermal average [19] : ≡−⟨ ⟩ 1 ab a ab (G− ) (iν )=iν δ + λ δ Σ (iν )(5a) b n n ab ab − b n 2 Σab(τ)= J 2 Gab(τ) Gab( τ)(5b) b b b − aa G (τ =0−)= S (5c) b − & ' Similarly for the fermionic model, we have : 1 ab a ab (G− ) (iω )=iω δ + λ δ Σ (iω )(6a) f n n ab ab − f n 2 Σab(τ)= J 2 Gab(τ) Gab( τ)(6b) f − f f − aa Gf (τ =0−)=q0 & ' (6c) In these equations, ν (resp. ω)arethebosonic(resp.fermionic)Matsubarafrequenciesandtheinversionshouldbe taken with respect to the replica indices a, b.NotethatourconventionsforthesignoftheGreenfunctionsinthis paper slightly differ from those of Ref. 19. Note that, although the equations are written in term of G,thephysical quantity is the local spin susceptibility χ (τ)= S(τ)S(0) which is given in the large-N limit by loc ⟨ ⟩ χ (τ)=Gaa(τ)Gaa( τ)(7) loc b b − In many instances below (where we mostly focus on the bosonic case), we shall drop the index b in Gb. From both analytical and numerical analysis of these integral equations, we have constructed the phase diagram displayed on Figure 1, as a function of the size of the spin S and the temperature T .Letusgivehereabriefoverview of the main features of this phase diagram, which will be studied in great detail in the rest of this paper. As is evident from Fig 1, it is useful to divide the discussion into models with S large and S small. Both regimes are accessible in the large N limit, where S is effectively a continuous parameter taking all positive values. For the physical case N =2,wewillpresentevidencelaterthatatleastS =1/2isinthesmallS regime for the infinite-range model; moreover, we can expect that at least some of the consequences of increased quantum fluctuations in a realistic model with finite-range interactions are mimicked by taking small S values in the large N theory of the infinite-range model. For large S,thegroundstatemustclearlybeaspinglass(Fig1).However,evenforverylargeS,itisnecessary to consider quantum effects in understanding the low T excitations and thermodynamics, and these have not been previously described. In this paper (Sections III C 2 and III D) we will show that the local spin susceptibility has a low energy density of states which increases linearly with energy. At the same time, the specific heat also has a linear dependence upon temperature. These results hold for temperatures T 6 5 4 3 2 1 PARAMAGNET SPIN GLASS 0 012345 FIG. 1: Phase diagram of the mean field bosonic model. There is a spin glass phase below the spin glass temperature c eq Tsg,whichisdeterminedwiththemarginalitycondition(SeeSectionIII). Tsg is the spin glass temperature as determined with the stationarity criterion. II. PARAMAGNETIC PHASE Contrary to the classical case, the paramagnetic phase of quantum spin-glass models is non trivial in mean field theory. An early discussion of these solutions has been given in Ref. 19, but we present here a much more complete description, and compare our results to the N =2case,whennumericalresultsareavailable.Sinceinthissection ab we look for paramagnetic solutions, we will consider only replica diagonal solutions of (5) : G δab.Twotypes of paramagnetic solutions have been found, that we will now consider successively : the spin-liquid∝solutions and the local moment solutions. ASpin-liquidsolutions 1Thelarge-N limit Alow-frequency,long-timeanalysisoftheintegralequations(5)revealsthat,undertheconditionthatλ Σ(i0+) vanishes at low temperature, a solution can be found which displays a power law decay of the Green’s function− at long time [19]: G(τ) 1/√τ.Thesesolutionsdisplayasingularityinthecomplexplaneoffrequencies,z,atz =0, with an amplitude which∼ can be parameterized by an angle θ as: Ae iπ/4 iθ G(z) − − for z 0, Imz>0(8) ∼ √z → (Values of z on the imaginary frequency axis at the Matsubara frequencies will be denoted by νn,whileontherealaxis will be denoted, ω.) Thus, these solutions display a slow local spin dynamics : Imχloc(ω) sgn ω for ω 0,T =0. β ∝ → Moreover, the local susceptibility χ (T ) χ (τ) dτ diverges as χ (T ) lnT/J at low temperature. More loc ≡ 0 loc loc ∼ ( 5 precisely, one can find the thermal scaling function characterizing the τ ,T 0limit,asexplainedinaprevious paper [25] : →∞ → π/β ω χ (τ,β) + ... Jχ′′ (ω,T) tanh (9) loc ∝ sin πτ/β loc ∝ 2T ) * Note that in the paramagnetic phase of quantum models, the local susceptibility χloc(T )(whichistheresponsetoa local magnetic field) differs from the uniform susceptibility χ(T )(responsetoaconstantmagneticfield),contraryto classical spin glass models where χ = χloc [22] : this is a consequence of the commutation relations of the spin, as can be seen for example in the high-temperature expansion in the SU(2) model. In this large-N limit, it can be shown that χ(T ) χloc(T )forT 0andnumericalcomputationsindeedsuggestthatχ(T ) const. [25]. Remarkably,≪ the parameter→θ which characterizes the spectral asymmetry [26] of the spectral∼ density at low frequency can be explicitly related to the size S of the spin (which involves a priori an integral of the spectral density over all frequencies). This is very similar to a kind of Friedel sum rule applying to this problem, and indeed the derivation follows a very similar route, based on the existence of a Luttinger-Ward functional. (Interestingly enough, the “boundary term” which usually vanishes in such derivations contributes here a finite value). This derivation is presented in detail in Appendix A, where the following relation between θ and S is established: 1 θ sin 2θ + S in the bosonic model + = 2 (10) π 4 ⎧ 1 ⎪ q in the fermionic model ⎨ 2 − 0 This relation has important consequences for the⎩⎪ physical properties of the spin-liquid solutions. First, we note that + + the spectral density must obey the positivity conditions: ImGf (ω + i0 ) < 0andsgn(ω)ImGb(ω + i0 ) < 0. Hence, in the fermionic case, θ must obey π θ π .Itiseasilycheckedfrom(10)thatθ precisely describes this range − 4 ≤ ≤ 4 of parameters as q0 is varied from q0 =0toq0 =1,andthattheθ(q0)relationisunique.Thissuggeststhatthe spin-liquid solution is an acceptable low-temperature solutions for the whole range of q0 in the fermionic case. In 0.06 0.05 0.04 0.03 0.02 0.01 0 0.25 0.3 0.35 0.4 0.45 0.5 FIG. 2: S as a function of θ for the bosonic model. The solid line is given by relation (10) and the points were obtained previously from a numerical solution of the saddle-point equation at zero temperature [19]. contrast, in the bosonic case, the plot in Fig.2 shows that (10) actually defines two values of θ (in the allowed range π 3π 4 θ 4 )foragivenspinS as long as S ∂ sin(π/4 θ) S =ln − (12) ∂q0 sin(θ + π/4) The entropy is then obtained by integration over the size of the spin, with the physically obvious boundary conditions (q =0)= (q =1)=0.Theresultingvalueoftheentropyasafunctionofq is plotted in Fig.3. S 0 S 0 0 0.5 0.4 0.3 S(q0) 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 q0 FIG. 3: Entropy as a function of the size of the spin (q0)inthefermionicmodel. Finally, we comment on the physical nature of the spin-liquid paramagnetic solutions found in this section. These solutions correspond to a partial screening of the local moment at each site, due to the interaction with the other spins. As a result the local susceptibility diverges logarithmically (much slower than a Curie law), but an extensive entropy is still present at T =0,indicatingadegeneratestate.Fromalocalpointofview,thephysicsissomewhat similar to an overscreened Kondo system, but here the gapless bath which quenches the spin is not external but self- consistently generated by the other spins. We suspect that the physics of this phase has to do with the degeneracy of the (large-N generalization) of the “triplet” state in which two spins are bound whenever a strong ferromagnetic bond Jij is encountered. In Section II A 2, we show that this spin-liquid regime is not a peculiarity of the large-N limit but indeed survives in the mean-field description of the quantum critical regime of a SU(2) quantum Heisenberg spin-glass. It would be very valuable to gain a more direct understanding of this gapless spin-liquid regime from astudyoftheproblemforafixedconfigurationofbonds,beforeaveragingoverdisorder.Thiscouldbeachieved numerically and is left for future studies. 7 2Beyondthelarge-N limit This subsection will show how recent renormalization group analysis of related models[11, 28, 29] imply that the above spin-liquid solution applies to all orders in 1/N .Inparticular,thelargeN solution with Imχloc sgn ω for small ω acquires no corrections to its functional form: the only changes are to the non-universal proportionality∝ constant. All the discussion below will be in a paramagnetic phase where it is sufficient to consider only a single replica, and so we will drop replica indices in this subsection. We begin by rewriting (3) in the following form[11] β Seff = SB γ0 dτ−→S (τ) −→φ (τ)(13) − · "0 2 ϵ where γ0 is a coupling constant and −→φ is an annealed Gaussian random field with −→φ (τ) −→φ (0) =1/ τ − .Itis reasonable to expect that the spin correlations in the quantum ensemble defined by⟨ (13) decay· with⟩ the| power-law| −→S (τ) −→S (0) 1/ τ σ,andweareinterestedindeterminingthevalueoftheexponentσ.Asimpleextension[19]of the⟨ solution· discussed⟩∼ | | above implies that in the large N limit σ = ϵ.Herewewillarguethatthisequalityisinfact exact for all N.Nowusingtheself-consistencycondition(4)weobtainσ = ϵ =1,whichthenimpliesImχloc sgn ω. The field-theoretic renormalization group analysis of (13) was discussed in Ref. 29, and we will highlight∝ the main results. The key observation is that renormalization of the theory (13) requires only a single wave-function renormalization factor Z,andthatthereisnoindependentrenormalizationofthecouplingconstantγ0.Thisresult was established diagrammatically in Ref. 29, and we will not reproduce the argument here. So if we renormalize ϵ/2 the spin by −→S = √Z−→S R,thenthecouplingconstantrenormalizationissimplyγ0 = µ γ/√Z,whereµ is a renormalization scale. The renormalization constant is in general a complicated function of γ,andwasdeterminedto two-loop order in Ref. 29: 2γ2 γ4 Z =1 + + ... (14) − ϵ ϵ in a minimal subtraction scheme. However, even though Z is not known exactly, the exponent σ can be determined exactly. Standard field-theoretical technology shows that the above renormalizations imply the β-function 1 ϵγ 1 ∂ lnZ − β(γ)= 1 . (15) − 2 − 2 ∂ lnγ ) * Furthermore, the exponent σ is given by the value of ∂ lnZ σ(γ)=β(γ) (16) ∂γ at the fixed point γ = γ∗ where β(γ)vanishes.Comparing(15)and(16)weseethat β(γ)= (ϵ σ(γ))γ/2. (17) − − Clearly, a zero of the β function must have σ = ϵ,andthisestablishestherequiredresult. We note that similar examples of a critical exponent being valid to all orders (in spite of a non-trivial β-function) can be found for other models in the statistical mechanics of disordered systems (see e.g [30, 31]). B“Localmoment”solutions In a mean field model, one usually expects to find locally stable (while possibly unstable to ordering) paramagnetic solutions of the mean-field equations down to zero temperature. Hence, the absence of solutions of the spin-liquid type for S>Smax suggests that a different kind of paramagnetic solution should exist for those values of the spin. Indeed, we have found that the integral equations (5) have another class of paramagnetic solutions in the bosonic case. These solutions actually exist for all values of the spin S and down to zero temperature. Hence, they coexist at low S with the spin-liquid solutions in some range of temperature. Their physical nature is very different from the previous spin-liquid solutions, and as discussed below they are not very physical solutions when considered at 8 low temperature. They are characterized by a Green’s function which does not decay at long times and obeys the J2 βS3τ asymptotic behavior: G (τ) S e− .Incontrasttothespin-liquidcase,λ diverges for T 0inthisregime: b ≃− − → J 2S3 λ (18) ∼ T Finding numerically these solutions of (5) requires some care. We have used an algorithm in which we solve (5) β in imaginary time for G(τ), for a fixed value of r = 0 G(τ)dτ,andthenadjustthenumberofparticlestoS by a dichotomy on r.Thelocalsusceptibilityχloc(τ)obtainedinthismannerisdisplayedonFigure4.Athightemperature, ( 2 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 FIG. 4: χloc(τ) extracted from a numerical solution of saddle point equation (5) in imaginary time, for S =1, J =1. The solid curve is low temperature (Jβ =10), the dashed curve is high temperature (Jβ = .1); for intermediate temperatures, the curves interpolate between the two. S(S+1) we find χloc(T )= T as expected since the spin is essentially free. At low temperature, we find another Curie law, with a reduction of the Curie constant due to quantum fluctuations : S2 χ (T )= for T 0(19) loc T → Hence, for these solutions, the effect of the interactions with the other spins is not strong enough to result in a qualitatively different screening regime, resulting merely in a reduction of the Curie constant. This is analogous to an underscreened Kondo regime. These solutions are of a similar type than those found in in a quantum Monte-Carlo simulation of the SU(2) model [24]. There also, a reduction of the Curie constant from S(S +1)/3toS2/3wasclearlyobserved.Thus,contrary to one of the conclusions of Ref. 24, the large-N limit correctly reproduces the paramagnetic local moment solution found for the physical N =2case.Moreover,anumericalsolutionofthelarge-Nintegralequations(5)forreal frequencies can also be obtained both at high and low temperature. The results are presented in Figure 5 for both ρ (ω) 1/πImG (ω)andχ′′ (ω) 1/πImχ (ω). At high temperature, ρ is centered around λ T ln((S +1)/S) b ≡− b loc ≡ loc b ∼ and χloc′′ (ω)isasimplepeak.Atlowtemperature,wefindinχloc′′ (ω)adelta-functionpeakatzerofrequencywith weight S2,whichisassociatedwithlongitudinalrelaxation(seeRef.24foradiscussionforN =2)andtwopeaks, centered around λ 1/T (Cf Eq. (18)) with a constant width, which are associated with transverse relaxation. Again, these results± are∝ very similar to the conclusions reached in Ref. 24 from a fit of the imaginary-time data for N =2,usingamodelχ′′ (ω). The only difference is that the central peak is not broadened by thermal fluctuations in the large-N limit. Finally, since these solutions describe the formation of a local moment at low temperature, and that the onset of the quenching temperature (at which the reduction of the Curie constant sets in) turns out to be lower than the temperature where spin-glass ordering occurs (as shown in the next section), we consider these solutions as a mean- field artefact of the spin glass ordering. Static limits of such solutions are actually known to occur in the classical SK model. We also note that the internal energy of these solutions have an unphysical divergence as T 0. → 9 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0 0 −10 0 10 20 −15− 10 −5 0 5 10 15 0.3 0.15 0.2 0.1 0.1 0.05 0 0 −10 0 10 20 −20− 10 0 10 20 FIG. 5: Spectral densities for Gb (left) and for the local susceptibility χloc (right), for high temperature (top) and a very low temperature (bottom). These results are extracted from a numerical solution of saddle point equation (5) in real frequencies. We close this section by noting that, for small values of the spin S,aratherintricatepatternemergesforthestability and coexistence of the two kinds of paramagnetic solutions described above. We have studied this numerically in some detail but do not report this here, since most of these phenomena occur below the spin-glass ordering temperature anyway. The important features have been displayed on Fig.1. We emphasize again that the spin-liquid solutions are the relevant solutions describing the whole quantum-critical regime. Even though they are unstable at T =0for S>Smax they remain consistent finite temperature solutions in a much wider range of values of S for T III. THE SPIN-GLASS PHASE In this section, we investigate spin-glass ordering in this model. The first observation that we make (Sec.III A) is that the spin-glass susceptibility (i.e the response to a spin-glass ordering field) is actually of order 1/N in the large-N limit. This does not preclude a spin-glass phase, but means that the transition is not associated associated with a linear instability. Indeed, we shall find explicit solutions in the ordered phase in the bosonic case, while the fermionic case does not have a spin-glass phase at N = (but does order as soon as 1/N corrections are considered). ∞ ASpin-glasssusceptibility Here, we derive an exact expression for the spin-glass susceptibility, valid for arbitrary N in this mean-field model. The derivation is most conveniently performed using replicas. We note that in the presence of spin-glass order, the a b correlation function −→S (τ) −→S (τ ′) acquires a non-zero value for a = b,whichishoweverstatic,i.eindependent · ̸ # $ 10 of τ τ ′ (see e.g Ref. 8). This crucial point is due to the fact that different replicas are independent of one another before− averaging, so that for a = b: ̸ a b a b a b −→S (τ) −→S (τ ′) = −→S (τ) −→S (τ ′) = −→S (0) −→S (0) (20) · · · # $ # $ # $ # $ # $ In the following, we shall denote by qab the (normalized) off-diagonal correlation function which is an order parameter for the spin-glass phase: 1 a b qab −→S (τ) −→S (τ ′) (21) ≡ N 2 · a=b !̸ # $ We consider the stability of the paramagnetic phase to this type of ordering, and introduce an ordering field Hab conjugate to qab.Thishastwoeffects: i) It adds to the effective action (3) an explicit term: 1 δS = dτ dτ ′ H S⃗ (τ) S⃗ (τ ′)(22) N ab a · b a=b " " !̸ The normalization of Hab has been chosen in such a way that the change in the total free-energy is of order N. ii) It modifies the value of the self-consistent field Qab.Thechangeoftheoff-diagonalcomponent,tolinearorder, ab is imposed by the self-consistency condition (4) to be: δQ = δqab,withδqab the induced order parameter. One can then perform an expansion of the off-diagonal correlation function up to linear order in both Hab and δQab.Thisyields: 1 δq = χ2 H + J 2δQab (23) ab N loc ab & ' ab in which χloc is the local susceptibility of the paramagnetic phase. Since δQ = δqab,thisfinallyyieldsthesuscepti- bility to spin-glass ordering: δq 1 χ2 χ ab = loc (24) sg ≡ H N 1 (Jχ )2/N ab − loc This formula has two important consequences. The susceptibility to spin-glass ordering is of order 1/N and any spin-glass instability at N = must be associated with a non-linear effect of higher order (i.e come from terms of ∞ higher than quadratic order in qab in the free energy). Furthermore, (24) shows that for finite N,a(linear)instability into a spin-glass phase will occur when Jχloc(T )=√N.Hencethefermionicmodel,forwhichχloc diverges at low-T, will have a spin-glass instability for arbitrary large but finite N.Moreprecisely,sincethelow-temperaturebehavior J Jχloc ln T has been shown above to hold for all N for the paramagnetic solution of the fermionic case, we conclude ∼ f √N from (24) that the spin-glass transition temperature depends on N in that case as : T Je− . c ∼ BSpinglasssolutionsandthe“replicon”problem We now turn to the explicit construction of solutions of the integral equations (5) with spin-glass ordering, in the bosonic case. The same reasoning as above shows that the Green’s function Gab(τ)doesnotdependonτ for a = b. Thus the most general Ansatz for the Green’s function Gab can be written as : ̸ G(τ) g (a = b) Gab(τ)= − 1 (25) / gab (a = b) 0 − ̸ where G(τ)isafunctionoftheimaginarytime,gab aconstantmatrixandg1 aconstant.Bydefinition,g1 is fixed so that G is regular at T =0,i.e. G(τ) 0asτ .Inthefollowingdiscussion,wewillrestrictourselvesto solutions0 given by a Parisi Ansatz for g (a→ replica symmetry→∞ breaking scheme). In the n 0limit(wheren is the ab → number of0 replica), this matrix becomes0 a function g(u)ofacontinuousvariableu with 0 u 1[22]. ≤ ≤ 11 Our equations (5) involve the Green function G,whereasthephysicalquantityisthesusceptibilityχab(τ)= ab ab 2 G (τ)G ( τ). The order parameter qab widely introduced in the spin glass literature [22] is given here by qab = gab, or in the limit− n 0: → q(u)=g(u)2 (26) 2 The Edwards-Anderson parameter is qEA = q(1) = g(1) [8, 22]. Since at zero temperature, in the long time limit, aa aa we find limτ G (τ)G ( τ)=qEA we find g1 = g(1), by definition of g1.Moreprecisely,welookforsolutions →∞ − in which these two definitions of qEA coincide but it is not really an assumption in our computation : if this relation was violated, we would simply find for G anonvanishinglimitforτ . Among the various possible replica symmetry breaking schemes [22],→∞ we will now focus on one-step solutions, since we have not found any other, either two-steps0 or with continuous replica symmetry breaking. In this case, the function g(u)ispiecewiseconstant:g(u)=g for 0 Σ(τ) J 2g3 (a = b) Σab(τ)= − (27) 2 3 / J gab (a = b) 0 − ̸ with Σisgivenby(31b).UsingthestandardformulastoinverttheParisimatricesinthelimitn 0[32],wefind: → 1 0 G− (iνn)=iνn + λ Σ(iνn)(28) − 2 2 2 2 3 J g G(iνn =0) =1+J βxg G(iνn =0) (29) 0 0 At this stage, it is useful to introduce here0 a new parameter Θdefined0 by : Θ G(iν )= (30) n −Jg We then eliminate λ in (28) and G(0) in (29) and obtain0 finally a closed set of equations for G and g : 1 Jg 0 G(iν ) − = iν Σ(iν ) Σ(0) 0 (31a) n n − Θ − n − 1 2 1 2 0 0 0 Σ(τ)=J 2 G2(τ)G( τ) 2gG(τ)G( τ) gG2(τ)+2g2G(τ)+g2G( τ) (31b) − − − − − 1 2 0 0 0 0 0 0 0 0 G(τ =0−)= (S g)(31c) − − 0 1 1 βx = Θ (31d) Jg2 Θ − ) * The crucial observation at this point is that these saddle-point equations possess a one parameter family of solutions, parametrised by Θor equivalently by the breakpoint x.Thisphenomenonalreadyoccurinothermodelswhichhave a one-step replica symmetry breaking solution [33]. The determination of the breakpoint turns out to be the most difficult question of this analysis. Two possible criteria are : 1. To minimize the free energy (x)asafunctionofx,aswouldberequiredbythethermodynamics.Inthe following, we will refer to thisF as the equilibrium criterion.Thiscriterionhasbeenusedinapreviousattempt to understand this spin glass phase [23]. 2. To impose a vanishing lowest eigenvalue of the fluctuation matrix in the replica space. We will refer to this as the marginality or replicon criterion. Although it is not really justified up to now, we will argue that it is the correct choice. This problem is not due to the quantum aspect of our model : it already appears similarly in some classical spin glass models, in the p-spin model for example. In this classical model, the study of the dynamics shows the existence of a dynamical transition T dyn above the static spin glass temperature T eq given by the static solution of the mean field 12 model. It turns out that, in this classical model, the replicon criterion has been proven to give the same transition temperature T c = T dyn.Moreover,ithasbeenshown[34,35]thatthesamephenomenonoccursinsomequantum version of the p-spin model. Thus, using this condition, it is possible in some sense to mimic the dynamics by simply solving a static problem, although this is not fully understood at present. In the present model, the two criteria give a coherent solution but with totally different spectra of equilibrium fluctuations : the equilibrium criterion leads to a gap in χ′′(ω)whereastherepliconcriterionistheonlyonewhich give a gapless χ′′(ω)(asimilarobservationwasmadeinRef.33inaonedimensionalquantummodelwithdisorder). We believe that in this quantum Heisenberg spin glass the replicon criterion provides us with the correct physical solution (Tc), contrary to the equilibrium solution, which gives the static transition temperature (Teq). However this claim cannot be proved in the present context : in particular, the static solution does give a full solution of (5). A study of the true Hamiltonian dynamics in real time and finite temperature of this quantum problem is necessary for a deeper understanding of this question, but this is beyond the scope of this paper. Let us now examine the two criteria separately in more details. 1Therepliconcriterion To apply the replicon criterion, we need to study the fluctuations of the free energy in the replica space around the one-step solution. In the large-N limit, the free energy is given by the expression : 2 β 1 3J 2 [Gab,λ]= Tr ln iν + λ Σab(iν ) + dτ Gab(τ)Gab( τ) λS (32) F β n − n 4 − − n ab "0 ! 1 2 ! 3 4 Under infinitesimal variation δg for a = b,thevariationofthefreeenergyis(uptosecondorder) ab ̸ δ = Mab,cdδgabδgcd. (33) F a>b !c>d Strictly speaking, as this is a quantum problem, we have to simultaneously consider the variation of the diagonal component, δG(τ)in(25)inastudyofthefluctuations.Inaspinglassphase,thereisindeedacouplingbetween δgab and δG(τ)whichmodifiesthefluctuationeigenvalues.Fortunatelyhowever,asweshowinAppendixB,this coupling does0 not modify the eigenvalue e1 and our main result (35) below, and so we will neglect δG(τ)here.The diagonalization0 of the n(n 1)/2 n(n 1)/2matrixM is briefly explained in Appendix B and gives three eigenvalues − × − 2 2 2 0 e1 =3βJ g (1 3Θ ) 2 2 − e = 3βJ g Θ2 3+3βJg2Θ(1 + Θ) (34) 2 Θ2 − e =6βJ2g2 3βJg2Θ 1 3 & − ' Afirstconsequenceofthisanalysisisthatreplicasymmetricsolutionsareunstable,sincefromEq.(31d)they& ' correspond to θ =1andthene1 < 0. Hence, these solutions will not be considered in the following discussion. AfullsolutionofEqs.(34)isrequiredtoshowthepositivityofe2,e3,butweimmediatelyseethate1 =0for 1 ΘR = (35) √3 Quite remarkably, we will see below in Section IIIC2 and Appendix B2 that precisely the same value of Θis selected by a criterion which is seemingly entirely independent. We will study the dynamic spectral functions in the spin-glass phase, as defined by G(τ), and show that their associated spectral densities are non-zero as ω 0onlyfor the value of Θin (35). So marginal stability in replica space appears to be connected to a gapless quantum| |→ excitation spectrum. We may intuitively understand0 this as due to the availability of many low energy states when the system first freezes, but a better understand should emerge from a real-time analysis. 2Theequilibriumcriterion To apply the equilibrium criterion, we start from the expression of the free energy and solve for x : F d (x) F =0 (36) dx 13 The computation of the total derivative (36) reduces to ∂ (x) because the saddle-point equations (5) for finite xF |G,λ n are equivalent to ! ∂ ∂F F = =0 (37) ∂Gab ∂λ as can be checked by an explicit calculation. Computing the logarithm in (32) (using Appendix II of Ref. 32) and taking the derivative leads to : 3 2 4 2 g J g 2 ln JgG(iνn =0))= (38) 4 − (βx) − −βxG(iνn =0) & 0 and finally to a equation for Θ: 0 1 1 3Θ2 2lnΘ+ + =0 (39) 4Θ2 2 − 4 This equation has two solutions : the replica symmetric one Θ= 1 (unstable, as explained above), and a non trivial one Θ=Θeq 0.4421 ....Contrarytotheprevioussolution,wewillseeinSectionIIIC2thatImG has a gap for this value of Θ≈. 0 CThephasediagram Once Θhas been determined, the equations (31) can be solved either numerically (both in imaginary time and in real frequency) or analytically in the S limit. In the following, we will mainly restrict ourselves to Θ=Θ since →∞ R we believe that it is the correct solution. However all calculations have been redone for Θ=Θeq with related results. 1Numericalsolution First, the critical temperature, is obtained from the numerical solution of Eqs.(31) in imaginary time : the spin glass order parameter q(T )=g2(T )andthebreakpointx(T )aredisplayedinFigure6asafunctionofthetemperature. x increases linearly with T from 0 at T =0(thereisnoreplicasymmetrybreakingatzerotemperature)andTsg is 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 FIG. 6: The Edwards-Anderson parameter qEA and the breakpoint x as a function of the temperature T for a fixed value of the size of the spin S =1(J =1). The transition to the paramagnet is given by the condition x =1. determined by the condition x(Tsg)=1,sincewemusthave0 x 1bydefinition[22].Hencethereisadiscontinuity in q at the transition, but we will show below that the transition≤ ≤ is second order. A careful numerical study shows that the transition is always driven by x =1forallvaluesofS and produces the critical temperature displayed 14 c on Figure 1. The computation is similar for the critical temperature Tsg given by the “replicon” criterion and for eq c the critical temperature Tsg given by the equilibrium criterion. Moreover, we find that Tsg,thedynamic transition eq temperature, is higher than Tsg ,thestatic transition temperature : this is required by our physical interpretation of the two solutions but it was not obvious a priori from the integral equations solved. 2Thelarge-S limit and spectral densities Further analytical insight into the spin glass phase itself can be obtained by considering various large-S limits, which differ by the manner in which temperatures and frequencies are scaled with S (See Fig 1). In a first simple large-S limit, we take T large enough so that T/JS2 is of order unity. This is the simple classical limit in which we can neglect all non-zero Matsubara frequencies, and (1) reduces to the classical problem in which S⃗ are commuting vectors of length S.Theequations(5)areanalyticallysolvableandwecanobtainaclosedform expression for the critical temperature at which spin-glass order vanishes: 2 T c JS2. (40) sg ∼ 3√3 Asecond,moresophisticatedlimit,validatlowertemperatures(wellwithinthespin-glassphase)iswhenwe examine ω and T of order JS.Itisthereforeusefultodefinethevariablesω = ω/(JS)andT = T/(JS), which remain of order unity at large S.Withthisscaling,theintegralequations(31)reducetoindependentquarticequations for each frequency. More precisely, if we make the following Ansatz for the Green’s function 1 1 G(ω,T)= g (ω, T)+ g (ω,T)+..., (41) JS 1 JS2 2 then to leading order in 1/S,(31)reduceto0 ∞ g(T )=S ρ (ω)dω − 1 "−∞ 1 1 g (ω)− = ω 3Θ 2g (ω) g ( ω)(42) 1 − Θ − − 1 − 1 − where ρ = Img /π as usual. Eliminating the frequency ω,wefindaquarticequationforg (ω). We do not 1 − 1 − 1 explicitly display the far more complicated equation for the subleading term g2. The solution of the quartic equation for Θ=ΘR is presented on Figure 7 together with a numerical solution of the full integral equation for S =5.Fromthesolutionofthequarticequationwefindthatρ1 vanishes linearly frequency at low frequencies; indeed, we find the analytic expansion 1 (1 + i) (2 3i)√3 2 g1(ω)= ω + − ω (43) −√3 − 2 4 at low frequencies. We expect that the full Green’s function in (41) also has a similar low frequency expansion, although this has not been proved. It is not difficult to show that the linear low frequency spectral density holds at higher orders in the 1/S expansion; moreover, our numerical results, shown in Fig 7, also clearly indicate a linear behavior at small ω.AtdominantorderinthepresentlargeS theory, the spin susceptibility is given by : ω 2 χ′′(ω)= π dxρ (x)ρ (x ω)+gπ (ρ (ω) ρ ( ω)) + πg βωδ(ω)(44) − 1 1 − 1 − 1 − "0 and this is also shown in Figure 7. It is important to realize that the deceptively simple structure in (43) relies on the special value Θ=ΘR =1/√3 determined by the entirely different replicon argument in Section III B 1. For arbitrary values of Θwe either find no physically sensible solution of the large S quartic equation (this is the case at the replica symmetric value Θ= 1 where the spectral density does not satisfy the required positivity criteria) or a solution with a spectral gap. In the latter case, the solution for g is real for small real ω,andthereisanonsetintheimaginarypart (ω ω )1/2 above 1 ∼ − c some critical frequency ωc.ThesolutionforΘ=Θeq is of the second type: it has a finite energy gap, but does not violate any spectral positivity criteria. This subsection has so far identified two distinct large S regimes. In the regime T JS2 we have purely classical behavior (the non-zero Matsubara frequencies can be neglected for static properties)∼ and a phase transition at a 15 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 −0.05 −50 −25 0 25 50 −40 −20 0 20 40 FIG. 7: a) ρ (ω) at T 0 for S =5and Θ=Θ :thesolidlineisthenumericalsolutionfortheintegralequation 1 ≈ R (31), the dashed line is the solution of the quartic equation (41) for g .b)χ′′(ω)/ω at T 0 from the quartic equation. 1 ≈ critical temperature in (40) where the spin-glass order vanishes. At lower temperatures, T JS,wearewellwithin the spin-glass phase, and the semiclassical dynamics is described by the solution of a quartic∼ equation defined by (42). As we noted in Fig 1, there is a third “quantum” regime at even lower temperatures, T J√S,andthisbecomes evident in a study of the thermodynamic properties presented in the following section. ∼ DThermodynamics We now turn to the internal energy U and the specific heat C.BycomputingtheaverageoftheHamiltonianin the N limit, we find that U is given by : →∞ 2 β J 2 U(T )= dτ Gab(τ)Gab( τ) (45) − 2 − "0 3 4 Using (25) and the one-step replica symmetry breaking Ansatz in the spin glass phase, we find : J 2 β G(τ)2G( τ)2dτ in the paramagnetic phase − 2 − U(T )=⎧ "0 (46) 2 β 2 ⎪ J 2 2 J ⎨⎪ G(τ) g G( τ) g dτ β(x 1)g4 in the spin glass phase − 2 − − − − 2 − "0 ⎪ & ' & ' Anumericalcomputationoftheinternalenergy⎩⎪ 0 0 U(T )andthespecificheatC(T )isdisplayedinFigure8for Θ=ΘR.TheconditionforthephasetransitionbetweenthetwophasesisthatthebreakpointintheParisifunction reaches its limiting value x =1.Inthislimittheequationsdeterminingtheparametersinthespinglassphase,(31), transform continuously to those for the paramagnet. As the equations are believed to have a unique solution, this implies that there is no discontinuity in the internal energy at the transition, indicating its second-order nature. This is confirmed by the numerical solution displayed on Figure 8. Moreover, in the large-S limit defined above, we can perform a low temperature expansion of the internal energy. Inserting (41) and (42) into (45), and keeping the dominant terms at large S we find J 1 1 U(T )= +3Θ g2 J 2g2 G(iν )G( iν )+2G2(iν ) + ... − 2 Θ − β n − n n ν ) * !n 5 6 JS2 1 1 0 0 0 = +3Θ + JS +3Θ T g (iν ) JST g (iν )g ( iν )+2g2(iν ) + (JS0)(47) − 2 Θ Θ 1 n − 1 n 1 − n 1 n O ) * ) * νn νn ! ! 3 4 16 3 0 −10 2 −20 −30 −40 −50 0 20 40 60 80 100 1 0 0 20 40 60 80 100 FIG. 8: The specific heat C(T ) and the internal energy U(T ) vs. the temperature T ,fromanumericalsolutionof Eqs. (31) for S =5and Θ=ΘR. where νn = νn/(JS). Clearly, this result indicates that the leading term in U(T )isatemperature-independent constant of order JS2,followedbyatermoforderJS whose coefficient is a function only of (T/JS). Evaluation of the latter function at low T for Θ=ΘR yields a curious accident: the gapless structure of the spectral functions suggests that the low T expansion should depend only on even powers of T/JS,butitisnotdifficulttoshowusing (43) that the coefficient of the term of order S(T/JS)2 vanishes. The first non-vanishing, T -dependent term among those shown explicitly in (47) turns out to be order JS(T/JS)4.ToobtainthetruelowT behavior we need to expand (47) to one higher-order in 1/S,andthisrequiresuseofthesecondterm,g2,in(41).Wedonotexpectany cancellation of the term of order (T/JS)2 at this point, and so the low T expansion for U looks like U(T )=U(0) + aS(T/JS)4 + b(T/JS)2 + ... (48) Rather than numerically evaluating the values a and b,wewillbesatisfiedbythefullnumericalsolutionof(31), followed by the evaluation of (46). The results are shown in Fig 8 and are consistent with (48). The structure of the expansion in (48) suggests that these results are valid for T IV. CONCLUSION We believe that the results of this paper provide a reasonably complete understanding of the infinite-range quantum Heisenberg spin glass. While there have been a large number of previous studies of quantum spin glasses of Ising spins and rotors (including models with (p>2)-spin interactions), none of these models contain quantum Berry phases in their effective actions, as is the case with the Heisenberg model. They have strong consequences: the spin-liquid solution of Section II A and its spectral density (9) are novel properties of the Heisenberg model. There is an intricate 17 interplay in stability between this spin-liquid state and the state with spin-glass order at low T which we have also described. At sufficiently low T ,thespin-glassorderalwaysappears,andwehavealsodescribedthethermodynamic properties of this state. An important issue not resolved in our analysis is the origin of the marginal stability criterion in the fluctuation eigenvalues in replica space. We imposed this criterion in a rather ad hoc manner, and found that it was the unique case under which the quantum excitation spectrum was gapless. Ultimately, the selection criterion for the spin glass state has to be a dynamic one, and this requires an analysis of the approach to equilibrium in real-time dynamics. Such an analysis was not carried out here, and is an important direction for future research. Another interesting open problem is to extend the study of (1) to cases where Jij has a non-zero average value. This will allow for ground states with other types of magnetic order, ferromagnetic and antiferromagnetic, and their competition with the spin glass state should be of some experimental interest. Interesting transitions in the paramagnetic states from the spin liquid state discussed also appear possible. We have already mentioned a recent study [25] of the quenching of the spin liquid state by mobile charge carriers into a disordered Fermi liquid. Combining this with models just mentioned, with a non-zero average Jij,shouldlead to results of direct physical interest in the heavy fermion and cuprate series of compounds. ACKNOWLEDGMENTS We thank G. Biroli, L. Cugliandolo, D. Grempel, P. Le Doussal, M. Rozenberg for useful discussions. S.S. was supported by US NSF Grant No DMR 96–23181. O.P is supported by the Center of Material Theory, Rutgers University, NJ,USA. APPENDIX A: COMPUTATION OF THE SPECTRAL ASYMMETRY This appendix is devoted to the derivation of Eqs (10). We will consider hereafter the fermionic case (the bosonic one is very similar). At zero temperature, the number of particles is given by ∞ dω + ∞ dω + ∞ dω + q = i GF (ω)eiω0 = iP ∂ ln GF (ω)eiω0 iP GF (ω)∂ ΣF (ω)eiω0 (A1) 0 2π f 2π ω f − 2π f ω f "−∞ "−∞ "−∞ where GF is the Green function with Feynman prescription on the real axis, and the symmetric principal part is defined by η ∞ − ∞ P =lim + (A2) η 0 η "−∞ → "−∞ " Using the relation between GF and the retarded Green function GR,wefindforthefirstterm: R R ∞ dω + arg G (0−) arg G ( ) ∞ dω + iP ∂ ln GF (ω)eiω0 = f − f −∞ + iP ∂ ln GR(ω)eiω0 (A3) 2π ω f π 2π ω f "−∞ "−∞ The arguments can be extracted from the low-energy and the high-energy behavior of the Green function, which R R leads to arg Gf (0−)= 3π/4 θ and arg Gf ( )= π respectively. The integral on the right of (A3) can be easily evaluated : we close− the contour− of integration,−∞ avoiding− the singularity at ω =0andusetheanalyticityofthe retarded Green function in the upper half plane, in which it has no zeros nor poles. We find finally : 1 θ ∞ dω + q = iP GF (ω)∂ ΣF (ω)eiω0 (A4) 0 2 − π − 2π f ω f "−∞ The problem is now reduced to the computation of the integral in (A4) as function of θ,whichturnouttobethe most difficult point. An analogous computation was performed in the overscreened regime of a large-N description of Kondo effects [26, 27], but it turns out to be more complex here. Proceeding along the lines of Ref. 26, 27, we 2 2 note the existence of Luttinger-Ward functional ΦLW = dt G (t)G ( t)whichhastwoproperties: firstwehave R R − Σ (ω)=δΦLW /δG (ω); second ΦLW is invariant in the transformation G(ω) G(ω + ϵ). From this, we could ( → 18 naively think that the integral of (A4) vanishes. However, it is not possible to find a regularization for the integral for which we could use the invariance of the Luttinger-Ward functional and a more careful analysis shows that : ∞ dω sin 2θ iP GF (ω)∂ ΣF (ω)= (A5) 2π f ω f 4 "−∞ To obtain this result, we introduce the following parametrisation of the singularity at ω =0: C + for ω>0 √ω ρ(ω) ⎧ (A6) ∼ C ⎪ − for ω<0 ⎨ ω | | ⎪ The principle of the computation is very simple :⎩⎪ we7 compute explicitly the integral with a regulator η>0andthen perform the limit η 0. Going to the real axis, we find : → F ρ(ω1)ρ(ω2)ρ(ω3) Σ (ω)= dω1dω2dω3 + (A7) − ω1>0 ω1<0 ω1 + ω2 ω3 ω i0 sgn ω1 "ω2>0 ou ω2<0 − − − ω3<0 ω3>0 + Using the notations a = a + iϵa, b = b + iϵb, ϵa/b = 0 and ψη(x)=Θ(x η)(a are b real and Θis the Heaviside function), we obtain (using the definition of the principal± part (A2)) | |− ∞ dz φη(a, b)=P (A8) (z a)2 (z b) "−∞ − − 1 (η + b)(η a) 1 1 1 = ln − + iπψ (b)sgn ϵ iπψ (a)sgn ϵ + + (a b)2 (η b)(η + a) η b − η a a b η a η + a ) 8 8 * ) * − 8 − 8 − − 8 F 8 Using the spectral representation8 for G and (A7),8 we find : 3 = dωk ρ(ω0)ρ(ω1)ρ(ω2)ρ(ω3)φη(ω1 + ω2 ω3 iϵ1sgn ω1,ω0 iϵ0sgn ω0)(A9) I − ∆1 ∆2 − − − " ∪ k9=0 with an explicit integration over ω with (A8). In this expression, the integration domains are defined as : ω0 < 0 ω0 > 0 ω0 > 0 ω0 < 0 ω1 > 0 ω1 < 0 ω1 > 0 ω1 < 0 ∆ = ⎧ ⎫ ⎧ ⎫ ∆ = ⎧ ⎫ ⎧ ⎫ (A10) 1 ⎪ ω > 0⎪ ⎪ ω < 0⎪ 2 ⎪ ω > 0⎪ ⎪ ω < 0⎪ ⎪ 2 ⎪ ⎪ 2 ⎪ ⎪ 2 ⎪ ⎪ 2 ⎪ ⎨ ⎬ = ⎨ ⎬ ⎨ ⎬ = ⎨ ⎬ ω3 < 0 ω3 > 0 ω3 < 0 ω3 > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Since φ ( a, b)= φ (a, b⎩⎪), a simple⎭⎪ change⎩⎪ of variable⎭⎪ leads to : ⎩⎪ ⎭⎪ ⎩⎪ ⎭⎪ η − − − η = ρ(ω1)ρ(ω2)ˇρ(ω3)ˇρ(ω0) ρˇ(ω1)ˇρ(ω2)ρ(ω3)ρ(ω0) φη(x1 + x2 + x3 iϵ1, x0 + iϵ0) I − xi>0> − − − " 1 2 + ρ(ω1)ρ(ω2)ˇρ(ω3)ρ(ω0) ρˇ(ω1)ˇρ(ω2)ρ(ω3)ˇρ(ω0) φη(x1 + x2 + x3 iϵ1,x0 iϵ0) (A11) − − − ? 1 2 withρ ˇ(ω)=ρ( ω). To take the limit η 0, we use the new variables xi = ηui and the behavior of ρ(x)forx 0, parametrised according− to (A6). The first→ integral in (A11) vanishes at dominant order in η (this term is proportional→ 2 2 2 2 to C+C C C+ =0),butthesecondintegralgives: − − − 3 3 C+C C C+ = − − − φη=1(u1 + u2 + u3 iϵ1,u0 iϵ0)(A12) I √u u u u − − "ui>0 0 1 2 3 19 3 3 Using x = u0, y = u1 + u2 + u3 and polar coordinates in √ui,wefind =2π(C+C C C+) 2 with I − − − I ∞ ∞ dx 1 1+x 1 y = √ydy ln − + iπ (ψ (y) ψ (x)) + I2 √x (x y + iϵ)2 1 x 1+y 1 − 1 "0 "0 @ − ) 8 − 8 * 8 8 1 1 1 8 8 + (A13) 8 8 y x iϵ 1 y + iϵ 1+y iϵ − − ) − 1 − 1 *A After an integration by parts on y and using 1 dx 1+x π2 ln = (A14) x 1 x 4 "0 8 8 8 − 8 8 8 we find 8 8 3 ∞ dω F F π 3 3 iP Gf (ω)∂ω Σf (ω)= (C+C C C+)(A15) 2π 2 − − − "−∞ Finally, a analogous computation can be performed in the bosonic case, leading in both cases to : ∞ dω sin 2θ iP GF (ω)∂ ΣF (ω)= (A16) 2π ω 4 "−∞ (in this expression, π θ π). These expressions have been shown to agree perfectly with numerical computations in imaginary time for− the≤ fermionic≤ case and on the real axis at zero temperature in the bosonic case. Let us note finally that we can guess the result if we admit a priori that the integral is given by an homogeneous polynomial of degree 4 : due to the particle-hole symmetry (in the fermionic case : f f†,theresultcanbeexpressed 4 4 3 3 ↔ as a function of C+ C and C+C C C+.Thefirsttermisrejectedsinceitleadstoasingularityatθ = π/4. − − − − − ± The proportionality coefficient is fixed by imposing θ = π/4forq0 =0. APPENDIX B: THE MARGINALITY CRITERION 1 Diagonalization of the fluctuation matrix First, we diagonalize the fluctuation matrix M in the replica space defined by Eq. (33). A priori M is a n(n − 1)/2 n(n 1)/2matrix.However,wehavetakenforgab the simple one step replica symmetry breaking Ansatz on g , i.e.× the−n n matrix splits into n/m n/m blocks : g = g if a/m = b/m ,0otherwise.ThusM splits ab × × ab ⌊ ⌋ ⌊ ⌋ into n/m identical m(m 1)/2 m(m 1)/2blocks(Mab,cd does not vanish if and only if all indices are in the same block a/m = b/m =− c/m ×= d/m− ). Hence the diagonalization is to be performed only on one block (we set 1 a,⌊ b, c, d⌋ m⌊), which⌋ ⌊ elements⌋ are⌊ given⌋ by (with a, b, c, d distinct replica indices) : ≤ ≤ Θ 2 M = A 3βJ2g2 1 3β2J 2g2 g2 + g + (B1) ab,ab ≡ − βJg @ ) 1 2 *A Θ M = B 9β3J 4g5 2g + (B2) ab,ac ≡− βJg ) * M = C 18β3J 4g6 (B3) ab,cd ≡− This matrix has already been diagonalized in Ref. 36 and its eigenvalues are given by : e = A 2B + C (B4) 1 − e = A +2(m 2)B +(m 2)(m 3)C/2(B5) 2 − − − e = A +(m 4)B (m 3)C (B6) 3 − − − Moreover, the degeneracies of e1, e2 and e3 are n(m 3)/2, n/m and n(m 1)/m respectively. Using (B1) and (B4), we find finally the result of the text (34). − − The above calculation has entirely ignored perturbations, δG(τ)inthediagonalelementsof(25).Includingthese greatly complicates the analysis, but a simple observation will suffice for our purposes. Our main attention is on the 0 20 cross-coupling between δG(τ)andtheδgab.Asimpleconsequenceoftheblock-diagonalstructureofthegab in the mean-field solution is that this cross-coupling has the form 0 β δ dτδG(τ)δg . (B7) F∼ ab a>b, a/m = b/m "0 ⌊ !⌋ ⌊ ⌋ 0 Now we can expand δgab in terms of the eigenvectors associated with (B4), which were computed in Ref. 36. The key observation is that after the sum over a, b in (B7), the cross-terms corresponding to all the eigenvectors associated with e1 vanish. Consequently these eigenvectors remain eigenvectors even upon including δG(τ), and the eigenvalue e1 remains unchanged. A similar argument shows that the eigenvalue e3 also remains unchanged, and only the eigenvectors associated with e2 are modified non-trivially by the coupling to δG(τ). 0 0 2Therepliconsolutionisgapless Let us assume that there is no gap in the boson spectral density and more precisely that for small ω : Θ α α α G(ω)= +(a + ib)ω Θ(ω)+(a′ + ib′) ω Θ( ω)+o( ω )(B8) Jg | | − | | where Θis the Heaviside function,0 a, b, a′,b′ are real constants, and α>0. Then from (31b) we obtain for ω>0: Im Σ(ω) = J 2g2Im 2G(ω)+G( ω) + ... (B9) − 1 2 1 2 2 2α Σ(ω)=c + d + i(2b b′)J g ω (B10) 0 0 − 0 where c and d are real constants. The other terms are& subdominant in the' limit ω 0ascanbeseenusingaspectral 0 → Jg representation. We then expand (31a) to second order and obtain at first order λ = c + Θ and for the imaginary part at second order : 2 2 b (2b b′)Θ = b′ (2b′ b)Θ =0 (B11) − − − − which leads to (Θ= 1 is excluded since b>0andb′ < 0) 1 Θ2 = (B12) 3 Thus the value of Θgiven by the replicon condition is the only one that leads to a gapless bosonic spectral density. APPENDIX C: FREE ENERGY OF QUANTUM ROTOR AND ISING SPIN GLASSES Quantum spin glasses of quantum rotors and Ising spins were studied extensively in Ref. 13. However, while the paramagnetic phase and the vicinity of the quantum-critical point were fairly completely described, the T 0ther- modynamics within the spin glass phases were only studied in the replica-symmetric solution. A proper understanding→ of this low T limit requires consideration of replica symmetry breaking, and we will provide that here. We will find, as in the more complex Heisenberg spin model considered in the body of the paper, that the specific heat is linear in T at low T . As we are restricting our attention to mean-field theory, we can neglect the spatial dependence of all degrees of freedom. Further, we will also restrict ourselves to the Ising case, and the generalization to the multi-component rotor case is immediate. As discussed in Ref. 13, the effective action of the quantum Ising spin glass is expressed in terms of the order parameter functional ab a b Q (τ,τ′)= σ (τ)σ (τ ′) (C1) ⟨ ⟩ where σa is the Ising spin in replica a.Theimportantlow-ordertermsinthefreeenergydensityare 1 ∂ ∂ aa κ ab bc ca = dτ + r Q (τ1,τ2) dτ1dτ2dτ3 Q (τ1,τ2)Q (τ2,τ3)Q (τ3,τ1) F κ ∂τ1 ∂τ2 − 3 " a @ A 8τ1=τ2 =τ " abc ! 8 ! u 8 y 4 + dτ Qaa(τ,τ)Qaa(τ,τ)8 dτ dτ Qab(τ ,τ ) . (C2) 2 − 6 1 2 1 2 " a " ab ! ! 3 4 21 Here r is the parameter which tunes across the spin glass transition, and κ, y measure the strength of various non- linearities. The analysis of thermodynamic properties in the spin-glass phase in Ref. 13 was carried out with a vanishing coefficient of the quartic term, y =0:inthiscasetheorderparameterhasreplicasymmetry,anditwas found that the specific heat T 3 as T 0. Here, we will extend the solution to small y =0,andshowthatthe solution with broken replica symmetry∼ has→ a linear specific heat. ̸ Time-translational symmetry requires that the mean-field solution take the form ab 1 ab iν (τ τ ) Q (τ ,τ )= Q (iν )e n 1 − 2 . (C3) 1 2 β n ν !n As in (25), we choose the following Ansatz for Qab: ab D(iνn)+βqEA a = b Q (iνn)= (C4) βqab a = b B ̸ where the off-diagonal terms, q ,aretime-independentandcharacterizedbytheParisifunctionq(u), and q(1) q . ab ≡ EA We have included an additive factor of βqEA in the diagonal term for convenience, and without loss of generality: as in the discussion below (25), we will find that this ensures that at T =0thesolutionforD(τ)vanishesasτ . aa→∞ Also, the diagonal components qaa do not appear in the above, and we are therefore free to choose them as q =0. Here, and in the remainder of this appendix we are assuming that r is sufficiently negative so that the system has a spin glass ground state; for larger r,thegroundstateisaparamagnet[13]withqEA = qab =0whosepropertiesare not addressed here. We now need to insert (C4) into (C2) and find the saddle-point with respect to variations in the functions q(u)and D(iνn). This is, in principle, a straightforward exercise, but the computations are somewhat lengthy. We first identify just the terms that depend upon qab;thesehavetheform R R = R Trq2 2 Trq3 3 q4 + ..., (C5) F − 1 − 3 − 6 ab !ab where R1 = βκ(D(0) + βqEA) 2 R2 = κβ R3 = βy. (C6) We first address the problem of determining the saddle-point of (C5) with respect to variations in q(u). Fortunately, this problem has been completely solved in the classical spin glass literature [37], and we can directly borrow the 2 1/2 results: the function q(u)increaseslinearlyasafunctionofu for 0 2 1/2 R2 (R2 4R1R3) qEA = − − . (C7) 2R3 Combining (C7) with (C5), we obtain the simple result q2 = κD(0)/y (C8) EA − Next, we consider the variation of in (C2) with respect to D(iνn). This is most easily done for νn =0,forwhich we obtain the following saddle-pointF equation ̸ 1 2 2 1 2 (ν + r) κD (iν )+u D(iν′ )+q 2yq D( iν ) κ n − n ⎡β n EA⎤ − EA − n !νn′ 2yqEA ⎣ 2y ⎦ D(iν′ )D( iν iν′ ) D(iν′ )D(iν′′)D( iν iν′ iν′′)=0. (C9) − β n − n − n − 3β2 n n − n − n − n !νn′ ν!n′ ,νn′′ Upon consideration of the saddle point equation for D(0) one initially finds a number of additional term associated with the coupling of D(0) to the qab.However,ourparameterizationin(C4)waschosenjudiciously,andhasthe feature that all these additional terms vanish upon using (C8); so, the result (C9) applies also for νn =0. 22 Let us also note the complete expression for the free energy density, obtained by inserting (C4) and (C9) into (C2): 2 q r y2q5 1 κ u 1 /n = EA + EA + (ν2 + r)D(iν ) D3(iν )+ D(iν )+q F κ 5κ βκ n n − 3β n 2 β n EA ν nu > ν ? !n !n !n 2 yqEA 2yqEA D(iν )D( iν ) D(iν )D(iν′ )D( iν iν′ ) − β n − n − 3β2 n n − n − n ν ν ,ν !n !n n′ y D(iν )D(iν′ )D(iν′′)D( iν iν′ iν′′). (C10) − 6β3 n n n − n − n − n νn!,νn′ ,νn′′ We are now left with the task of solving the saddle-point equations (C8) and (C9) for qEA and D(iνn), and inserting the result in (C10). This is clearly a daunting task, and we will be satisfied in describing the T 0limittofirst order in y.ThisissimilarinspirittothelargeS expansion of Sections III C 2, III D, and we expect→ that higher-order corrections in y will not modify the nature of the low T limit. First, we consider the case y =0.Hereacompleteanalyticalsolutionispossible,andwaspresentedinRef.13.We have at y =0: 1 r q0 = ν EA βκ | n|−κu ν !n ν D0(iν )= | n| n − κ 2 r2 0(T )/n = ν 3 3βκ2 n 2κ2u F ν | | − !n 4π3T 4 = 0(0)/n . (C11) F − 45κ2 We observe that the free energy density behaves as T 4,whilethespecificheat T 3 as T 0. Notice also that 0 ∼ → positivity of qEA requires an upper bound on r,whichwehaveassumedtohold. Before considering explicit corrections in powers of y,wemakeanobservationthatisvalidtoallordersiny.The solution for D(iνn)in(C11),whenanalyticallycontinuedtorealfrequencies,ω,hasanimaginarypartwhichvanishes linearly in ω at small ω.Wenowshowthatthisconclusionholdstoallordersiny;theconstraint(C8)willplaya key role in establishing this result. Let us write D(ω)=D(0) + iD1ω + ... for small ω,whereD(0) and D1 are some real constants. Inserting this in (C9) and evaluating it at T =0forsmallω,wenotethatthelasttwotermsin(C9) have imaginary parts which vanish as ω3 and ω5.Keepingonlytheleadingω dependence of the imaginary part, we obtain the simple expression 2iκD(0)D ω 2yq2 iD ω =0. (C12) − 1 − EA 1 From (C8) we see that this condition is satisfied, and so D1 can be non-zero. Now, we consider explicit first-order corrections in y:wewillseethatthisleadstotermsinthethermodynamics which vanish more slowly as T 0. We can easily use (C8) and (C9) to determine the corrections to D(iν )and → n qEA to linear order in y;however,thesearenotneededhereastheshiftinthefreeenergyduetosuchcorrections will only appear at order y2,becausethefreeenergyisatasaddlepoint.Indeed,toobtainthefreeenergycorrect to first order in y,weneedonlyinsert(C11)into(C10).Itisthenquiteeasytoseethatthefreeenergywillhavea term of order yT 2,andthatthecoefficientofthistermwillbenon-universalanddependentuponthenatureofthe 0 high energy cutoff. The required term comes from the T dependence of qEA,whichfrom(C11)isseentobe πT 2 q0 (T )=q0 (0) . (C13) EA EA − 3κ Upon inserting (C13) and (C11) into (C10) we will now obtain numerous terms in which the above T 2 term multiplies T -independent, cutoff-dependent terms coming from the upper bounds in the summations over the D(iνn). As the relative values of these contributions will depend upon the nature of the cutoff, there is no general reason for them to cancel against each other. 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