We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori\u27s line [tanh(βJ)=2p−1 for the ±J distribution]. One scaling axis is along the line, and the other is along the temperature direction. This scenario is generic for any random Ising model with a Nishimori line, in any number of dimensions, if the transitions are second order. The renormalization-group fixed point located inside Nishimori\u27s manifold is expected to control multicriticality for a wider class of models

Harris, A. Brooks

Le Doussal, Pierre

English

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University of PennsylvaniaScholarlyCommonsDepartment of Physics Papers Department of Physics8-1-1988Location of the Ising Spin-Glass Multicritical Pointon Nishimori's LinePierre Le DoussalA. Brooks HarrisUniversity of Pennsylvania, harris@sas.upenn.eduFollow this and additional works at: http://repository.upenn.edu/physics_papersPart of the Physics CommonsThis paper is posted at ScholarlyCommons. http://repository.upenn.edu/physics_papers/480For more information, please contact repository@pobox.upenn.edu.Recommended CitationLe Doussal, P., & Harris, A. (1988). Location of the Ising Spin-Glass Multicritical Point on Nishimori's Line. Physical Review Letters, 61(5), 625-628. http://dx.doi.org/10.1103/PhysRevLett.61.625Location of the Ising Spin-Glass Multicritical Point on Nishimori's LineAbstractWe present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses shouldbe located on a particular line of the phase diagram known as Nishimori's line [tanh(βJ)=2p−1 for the ±Jdistribution]. One scaling axis is along the line, and the other is along the temperature direction. This scenariois generic for any random Ising model with a Nishimori line, in any number of dimensions, if the transitionsare second order. The renormalization-group fixed point located inside Nishimori's manifold is expected tocontrol multicriticality for a wider class of models.DisciplinesPhysicsThis journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/480VOLUME 61, NUMBER 5 PHYSICAL REVIEW LETTERS 1 AUCUST 1988Location of the Ising Spin-Glass Multicritical Point on Nishimori s LinePierre Le Doussal 'Center for Theoretical Physics, Yale University, New Haven, Connecticut 06511andA. Brooks HarrisDepartment of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104(Received 4 May 1988)We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori's line[tanh(PJ) =2p —I for the ~ J distribution]. One scaling axis is along the line, and the other is alongthe temperature direction. This scenario is generic for any random Ising model with a Nishimori line, inany number of dimensions, if the transitions are second order. The renormalization-group fixed point lo-cated inside Nishimori's manifold is expected to control multicriticality for a wider class of models.PACS numbers: 75.50.Kj, 05.50.+q, 64.60.KwThe phase diagram of the Ising spin-glass in zero mag-netic field, described by the nearest-neighbor Hamiltoni-an H=p&;1) J1S;S1, has been determined for asym-metric distributions of couplings P(J;1) by the methodsof Migdal-Kadanoff renormalization group ' (RG),Monte Carlo RG, domain-wall RG, and series expan-sions. It is now believed that (i) in three dimensionsthere are three phases, paramagnetic (P), (anti) ferro-magnetic (F), and spin-glass (SG), which meet at a mul-ticritical point M; (ii) in two dimensions there is nospin-glass phase, but there is still a multicritical point Mdelimiting two critical behaviors on the F-P boundary.The phase diagram is sketched in Fig. 1, for the J dis-tribution:P(J) =pB(J—Jp)+ (1 —p) 8(J+Jp) .On the other hand, Nishimori has shown that theinternal energy [(H)] can be computed exactly when thedistribution of couplings P(J) satisfies the conditionP( —J) =exp( —2PJ)P(J)P( t) =(1 —t)(P—(t)/(1+t), (2)where t =tanh(pJ). In the phase diagram, (2) definesthe Nishimori line: tanh(pJp) =2p —1 for the + Jp dis-tribution (1) and pJ =Jp for the Gaussian distribution. 2When more parameters are added (2) generally defines amanifold. Surprisingly, the internal energy was found tobe regular along the line although this line crosses theboundary of the ferromagnetic phase at point N (Fig. 1).It was thus natural to ask whether M must coincide withThis is true for all infinite-dimensional models(Sherrington-Kirkpatrick or diluted models, ' Bethelattice" ) and is in good agreement with approximatedeterminations of phase diagrams in 2D and 3D. Letus emphasize that there was, up to now, no compellingreason why this should be true in general, since the regu-larity of the internal energy at point N along the lineIi kTJ0FerroPARA IIIIIlIIIIIIIrIrrrrFERRO.--" M=NrrrrI)(IIIihPureIP/S GSPiN GLASSSG ~P1/2FIG. 1. "Generic scenario" proposed here for the phase dia-gram of the ~ J spin-glass in d=3 [in the (p, T) plane], alsosupported by numerical studies (Refs. 2-5). In d=2 it is thesame picture, without the SG phase. Nishimori's line isdashed. The RG flow on corresponding hierarchical latticesand its fixed points are indicated.could be due to some accidentally vanishing critical am-plitude (no result is available for the free energy).In this Letter we give two independent arguments infavor of an exact location of the multicritical point M onNishimori's line (M =N), which also allow us to identifyprecisely the "nongeneric" cases where the situationcould be more complicated. These cases are clearly not625VOLUME 61, NUMBER 5 PHYSICAL REVIEW LETTERS 1 AUGUST 1988Z2k+ l (l ~J ) Z2k+2(li J ) (3)supported by present numerical or analytical results onIsing spin-glasses. They are (i) first-order P-SG or P-Ftransitions, (ii) the existence of more than one multicriti-cal point in Fig. 1, or (iii) some very unlikely instancesdiscussed below. Moreover, independently of M =N, onecan give an exact characterization of the scaling axis atN Ex.plicit proofs can be found in Ref. 8.The first argument relies on the exact equality of thespin-glass susceptibility Z2 and the ferromagnetic suscep-tibility Z~ on Nishimori s line. Here we set Xk(ij)=[(S;S~) ] and the susceptibilities Lk are defined byXk =+~1k(i,j ) (thermal averages are denoted by angu-lar brackets and configurational averages by squarebrackets). In Ref. 8 it is proven that, when the distribu-tion P(J) satisfies Nishimori s condition (2), one has forgeneral integer k,[note that (3) in the particular case k =0 was obtainedin Ref. 11 for a random-clock model].As is the case of all results in connection withNishimori s line, (3) is a simple consequence of the in-variance of the Hamiltonian H under the local gaugetransformation J;1 e;eJJ& and S; e;S; and holds fora system of any size I. t is rigorous for a system with anygiven (disorder independent) subset of spins frozen toS;=+1. From (3) we obtain, as an intermediate (new)result that the line cannot enter the spin glass-phase,where one expects':1Iin Z2k+ ~ (i,j ) =0, lim 22k+2(i,j )WD.Since for the present purpose one needs only to consid-er the disordered (replica symmetric' ) phase, we givehere a diH'erent derivation of (3) than in Ref. 8, based onthe n-replica Hamiltonian H„:exp( pH„)=— exp p g J;~S SJ' = (au+a~ij a 1 ijwith ak = [c"t"],where c =cosh(pJ). Nishimori's condi-tion (2) implies that a2k+~ =a2k+2, for any integer k andany n. As a consequence, it is straightforward to checkthat on Nishimori's line the replica Hamiltonian (4) has,for any n, an additional gauge invariance: S =U,S =U U; (a & 1), where the U; act as local Ising vari-ables. The exact equality between 12k+] (i,j) andX2k+2(i,j ) then follows immediately from this additionalinvariance, since one hasZ~(i,j)= lim(S; S~ )H = lim(U U; UJ )H =X2(i,j),n 0 " n 0and similar relations hold for any integer k.Summing (3) over j, we obtain that on Nishimori'sline the ferromagnetic susceptibility X~ and the spin-glasssusceptibility Z2 are equal. The consequences are the fol-lowing. In all cases where the P-F boundary is unambi-guously defined by X~ ~, X2/X~ 0 and the P-SGboundary by Z2 ~, Z~/Z& 0, this result implies thatthese two boundaries must meet at point N onNishimore's line (e.g. , M=N), where ferromagnetic andspin-glass fluctuations simultaneously become critical(X~ =Xq ~). Moreover one has rt~ =rt2 at M=N,where these exponents are defined by the asymptoticlarge-separation behavior Xk(i,j ) Ak ~i —j ~-This has been obtained to order e=6 —d by Chen andLubensky' but our result indicates exact equality ofboth exponents and amplitudes for Z2ky~ and X2k+2 (forany k).The argument for M =N relies on the assumption thatonly one of the Z; (i =1,2) is dominant on either the P-For P-SG boundary (it obviously also requires second-order transitions). This is true in the d ~ limit for Is-ing spin-glasses and we have checked that this remainstrue within a systematic 1/d expansion. ' The argument626gS SJ'+a2 g S S~SJSJt'+. . .),a a(Pis certainly valid whenever mean-field exponents are ex-act (since then only one Z; diverges on each boundary)which is expected for d &6. Moreover, as discussedbelow, the 6 —e expansion reveals no bifurcation in thisscenario near d =6. Although it makes sense physically,this argument alone becomes less conclusive in low di-mensions (d =2,3) since the two susceptibilities X~ andX2 could diverge similarly on the P-F boundary. Thesecond argument, that we now introduce, clarifies the sit-uation and allows us to identify possible alternativescenarios.This second argument is based on the fact thatNishimori's condition (2) exhibits a remarkable propertyof scale invariance which does not seem to have been no-ticed previously: Namely, the distribution of disorderP(J) satisfies on the line a symmetry property which ispreserved by appropriate exact renormalization schemes.Physically, the symmetry (2) is reproduced at all scalesin the system: For instance, the distribution of (S;SJ ) forany (i,j) satisfies (2) (in the t variable ). As a result,Nishimori's line (or manifold) plays a central role inthe structure of the RG jhow, as is the case for the line ofsymmetric disorder p = —,' which has similar properties:They both contain RG fixed points, and this will now beprecisely formulated.These properties are most simply illustrated for spin-glasses on hierarchical lattices, where the Migdal-KadanoA' RG recursion relations are exact. ' Theselattices are constructed iteratively by the replacement ofeach bond at each step by a given association of b bondsin series and 1=b ' in parallel. For two bonds inparallel the recursion relation is k'=K~+K2, and ist'=t~tz for two bonds in series [with K=PJ andt =tanh(PJ) ]. Couplings remain nearest neighbor andVOLUME 61, NUMBER 5 PHYSICAL REVIEW LETTERS 1 AUGUST 1988uncorrelated by renormalization, but their (common)distribution is changed accordingly. Finding the phasesand critical exponents is then equivalent to studying thetransformation P(K) P'(K') and its fixed distribu-tions, starting from an initial distribution Po(K) [wechoose (1) to be specific ].It is explicitly shown in Ref. 8 that, for any hierarchi-cal lattice, if the initial distribution Po(K) satisfiesNishimori's condition (2) then all the iterates P'(K')satisfy the same condition (2) [one easily checks that (2)is invariant under addition both in parallel and in series].The immediate consequence is that, starting from the in-itial distribution Pp(K) corresponding to point N of Fig.1, one flows towards a fixed distribution P*(K) whichcannot be any of the four following fixed distributions:pure critical fixed point (K=K ) or low-temperature(K ~ ~) critical fixed distribution —since they do notsatisfy (2); ferromagnetic (K=+~) or paramagneticfixed point (K=O), since N is on the critical surface,which is stable under renormalization. We thus obtainthe rigorous result that N corresponds to a nontrivial RGfixed point. Two cases are now possible in principle: (a)P (K) has two (or more) unstable directions and N is amulticritical point. (b) P (K) has one unstable direc-tion and is a "random critical fixed point" attracting awhole portion of the P-F boundary.The fixed distribution P (K) is computed in Ref. 8,numerically for low-dimensional hierarchical lattices,and analytically in the limit i ~. In all cases it isfound that (i) P has exactly two unstable directionsand is thus multicritical and (ii) there is only one mul-ticritical point in the phase diagram of Fig. 1 andM =N. This is presumably a generic feature:P[J}=exp( PH(J, I—S; =+1}) )@[J}, (5)where H is the Hamiltonian and N[J} an arbitrarygauge invariant -function of the couplings. Condition (5)is the correct generalization of Nishimori's condition (2)to arbitrary multispin couplings. The invariance of (5)has important implications for the field-theoreticalrenormalization-group formulation, as we now discuss.Note that the invariance of (5) [and of (2) for hierarchi-cal lattices] again implies that Nishimori's line cannotenter the spin-glass phase.In the 6 —e expansions, both P-SG and P-F transitionscan be described by the following Landau-Ginzburg en-ergy density F(M, Q ):Nishimori s subspace (2) is repulsive in one direction,but attractive in all others, so that the fixed distributionP*(K) attracts the whole multicritical manifold [a codi-mension 2 manifold in the space of distributions P(K), towhich point M of Fig. 1 belongs] or a large portion of it[no dependence of P*(K) on the starting distribution Powas found]. Thus we reject case (b) above, not only forhierarchical lattices, but more generally since it seemshighly nongeneric [it would imply that an infinite set ofequalities, like (3) for large Ii —j I, hold on a wholeportion of the P-F boundary].On Bravais lattices the exact real-space RG generatesmany-spin couplings with statistical correlations. How-ever, as shown in Ref. 8, there still exists a symmetry ofthe joint distribution of al/ interactions P [J},satisfied onNishimori's line (2), which is invariant by summationover any subset of spins. If we denote by J the set of allinteractions J = [J;,,J;,kl, . . .} the following form for P [J}is RG invariant:F =g[(VM.)'+rMM.'}+g [(~g.,)'+p~g.'t +wig. &M.M&}+w Z Q.pgp, g.,+a a&P z(P( y (6)where the fields M, and Q,~ are conjugate to S andS Si', respectively. This expression can be derived ex-actly from H„by use of the Stratonovich transformation,and we find' that, for a Gaussian distribution P(J),Nishimori's line is defined by rM =rg, w~ =w. One cancheck that the one-loop formulas for the RG flow ob-tained in Ref. 13 leave this line invariant and give themulticritical fixed point rM =rg =0 on the line. In (6) itcan be seen as an exact symmetry under the interchangeof M, and Q~. (a & 1) corresponding to the additionalinvariance of the replicated Hamiltonian H„noticedabove. This symmetry should hold to all orders, and nobifurcation occurs near six dimensions.The scaling axis at the multicritical point N can befound exactly in general, from powerful topological con-straints, first derived in a weaker version by Nishimori,and renormalization arguments. Let Pp(K) be any dis-tribution satisfying Nishimori s condition (2). It is prov-en in Ref. 8, that for any P(K) obtained from P~(K) byany odd change of variable [e.g., P(K) =P&(p(K))xI dpldK I with p( —K) = —p(K) ], one has I [(S,)]p I~ [I (S;) I ]p . Thus, if Plv(K) is inside the paramagnet-ic phase, P(K) cannot be ferromagnetic. On hierarchi-cal lattices the following arguments can be maderigorous. At the multicritical fixed point P*(K) one hasthat (i) the eigendirection of largest eigenvalue is insideNishimori's surface (even subspace) and (ii) the secondrelevant eigendirection is tangent to the critical P-F sur-face, since the crossover exponent is different from 1 inany dimension (for d ( 6 on Bravais lattices). Theabove inequality implies that this second eigendirectionis in the odd subspace [generated by infinitesimal odddeformations of P*(K)]. The tangent space at P* splitsinto two locally RG-invariant subspaces, even and odd.Coming back to the plane of initial parameters of Fig. 1this implies, in the absence of discontinuities, that onescaling axis at M is along Nishimori's line, the other be-ing along the temperature direction, which is the onlyodd direction available. On Bravais lattices, the same627VOLUME 61, NUMBER 5 PHYSICAL REVIEW LETTERS 1 AUt-UST 1988picture should hold, and we have checked' from Ref. 15that this is fully confirmed by the 6 —e expansion.When more parameters are added, all points which areboth in the multicritical manifold and inside Nishimori smanifold are such that the second scaling axis is an odddirection (not necessarily the temperature).In conclusion, irrespective of the lattice and dimen-sion, we expect multicriticality in Ising spin-glasses to bedescribed by a RG fixed point with universal propertiescharacteristic of Nishimori's disorder (r)21, y1 = r)2k+2,and many others ): This should be true for a wide classof models which do not necessarily have a Nishimoriline —such as, for a ~ I, P(J) =p8(J —Jo) + (1 —p)~ 8(J+aJo)—and in the case of model (1) one hasM =N. Second-order P-F and P-SG transitions are cru-cial for this scenario. Note that in the random q-stateclock model of Ref. 11, transitions become first order forq &6, and M=N is indeed found for q=3, although notfor q =~. A similar behavior is observed in the [Z"]models. ' Finally, alternative scenarios where N is arandom critical fixed point, or N is multicritical but dis-tinct from M, are highly unlikely.We thank M. E. Fisher and T. C. Lubensky for help-ful discussions. One of us (A.B.H. ) thanks the NationalScience Foundation for partial support under Grant No,DMR 85/20272. Laboratoire de Physique Theorique isa laboratoire propre du Centre National de la RechercheScientifique associe a 1'Ecole Normale Superieure et a1'Universite Paris-Sud.' On leave from Laboratoire de Physique Theorique, EcoleNormale Superieure, Universite de Paris VII, Paris, France.'For a review see K. Binder and A. P. Young, Rev. Mod.Phys. 58, 801 (1986).2A. P. Young, and B. W. Southern, J. Phys. C 10, 2179(1977); R. Maynard and R. Rammal, J. Phys. (Paris) Lett. 43,L347 (1982).3Y. Ozeki and H. Nishimori, J. Phys. Soc. Jpn. 56, 1568,326S (19S7).4W. L. McMillan, Phys. Rev. B 29, 4026 (1986).5M, E. Fisher and R. R. P. Singh, to be published.6H. Nishimori, Prog. Theor. Phys. 66, 1169 (1981).A. Georges, D. Hansel, P. Le Doussal, and J. P. Bouchaud,J. Phys. (Paris) 46, 1827 (1985).P. Le Doussal and A. Georges, Yale University Report No.YCTP-P1-88, 1988 (to be published).D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35,1792 (1975).'OL. Viana and A. J. Bray, J. Phys. C 18, 3037 (1985).' H. Nishimori and M. J. Stephen, Phys. Rev. B 27, 5644(1983).' Replica-symmetry breaking is unlikely to occur on the line,even in the ferromagnetic phase: It would imply that it alsooccurs on the line p= 2, in the limit n 1; see Ref. 8.' J. H. Chen and T. C. Lubensky, Phys. Rev. B 16, 21G6(1977).' A. B. Harris and P. Le Doussal, to be published.'SA. N. Berker and S. Ostlund, J. Phys. C 12, 4961 (1977).' A. Georges, D. Hansel, P. Le Doussal, and J. M. Maillard,J. Phys. (Paris) 48, 1 (1987).628

Location of the Ising Spin-Glass Multicritical Point on Nishimori\u27s Line

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Location of the Ising Spin-Glass Multicritical Point on Nishimori\u27s Line

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