Exactly solvable spin-glass models with ferromagnetic couplings: the spherical multi-p-spin model in a self-induced field 3 1 0 Andrea Crisanti1,2, Luca Leuzzi1,3 2 1 Dipartimento di Fisica, Universita`di Roma “La Sapienza”, P.le Aldo Moro 5, 00185 n Roma, Italy a J 2CNR-ISC, Viadei Taurini 19, 00185 Rome, Italy 8 3 CNR-IPCF, UOSRoma, P.le Aldo Moro 5, 00185 Rome, Italy ] n n - s i d Abstract . t a WereportsomeresultsonthequencheddisorderedSphericalmulti-p-SpinModel m in presence of ferromagnetic couplings. In particular, we present the phase di- - agrams of some representative cases that schematically describe, in the mean- d fieldapproximation,thebehaviorofmostknowntransitionsinglassymaterials, n including dynamic arrestinsuper-cooledliquids, amorphous-amorphoustransi- o c tions and spin-glass transitions. A simplified notation is introduced in order to [ computesystemspropertiesintermsofaneffective,self-induced,fieldencoding 1 the whole ferromagnetic information. v 5 7 1. Introduction 4 1 In the very extended framework of complex systems, spin glasses have be- . 1 come the source of ideas and techniques now representing a valuable theoret- 0 ical background in diverse fields, with applications far beyond the physics of 3 amorphous materials (both magnetic and structural). These systems are char- 1 : acterizedbyastrongdependencefromthe details,sostrongthattheirbehavior v cannot be rebuilt starting from the analysis of a single cell constituent. Their i X analysis cannot be carried out without considering the collective behavior of r the whole system. One of the common features is the occurrence of a large a number of stable and metastable states or, in other words,a largechoice in the possible realizations of the system. This goes along with a rather slow evolu- tion through many, detail-dependent intermediate states, looking for a global equilibrium state (or optimal solution). Mean-field models have largely helped incomprehending manyof the mechanisms yielding suchcomplicatedstructure andalsohaveproducednewtheoriesorcombinedamongeachotheroldconcepts pertaining to other fields such as, e.g., the spontaneous breaking of the replica symmetry and the ultrametric structure of states. Among mean-field models, Preprint submitted toElsevier December 11, 2013 spherical models - i.e., continuous dynamic variables with a global constraint [1]- are analytically solvable even in the most complicated cases. Multi-p-spin spherical models have been shown to yield low temperature amorphous phases that, depending on the dominant interaction terms, can bothbedescribedbydiscontinuousandcontinuousReplicaSymmetryBreaking (RSB) Ansa¨tze. In particular, (i) one step replica symmetry breaking (1RSB) phaseswerestudied,becauseoftheirrelevanceforthestructuralglasstransition [2, 3, 4, 5], (ii) two step RSB phases [6, 7] are found that are thermodynami- cally stable and whose dynamics models secondary relaxation in glass-forming liquids (see, e.g., [8] and for a thorough overview [9] and references therein) and study the singularities in the phase diagrams predicted by the mode cou- pling theory [10, 11, 12], (iii) the Full RSB phase represents spin-glasses in the propersenseand,moregenerally,thefrozenphaseinrandommanifoldproblems [13,14,15,16,17]. ThepossibilityoftheexistenceofFullRSBinsphericalmod- elswasfirstpointedoutbyNieuwenhuizen[18]onthebasisofthesimilaritybe- tweenthereplicafreeenergymulti-spinmodelsandthe relevantpartofthefree energy of the Sherrington-Kirkpatrick model. In Refs. [19, 20] thermodynamic stable Full RSB phases have been actually computed and analyzed. Spherical models,thus,alsoprovideamuchsimplerrealizationofthis Ansatz thaninthe spin-glass mean-field prototype model, i.e., the Sherrington-Kirkpatrick model [21]. Further including ordered interaction terms representing attractive ferro- magnetic couplings between spins, one can use these models to study diverse problems, such as disordered systems along the Nishimori line [22, 23], or the states following problem[24, 25, 26], else the random pinning with a system at a very high temperature, or in presence of external random constraints, as, e.g., in porous media [27, 28, 29]. Spherical models with competing disordered andorderednon-linearcouplingsalsodescribemode-lockinglasermodels,where sphericalspins are used to represent both realand imaginaryparts of the com- plex amplitude of photonic modes [30, 31, 32]. In particular, they can be used to address the problem of random lasers [33, 34], whose statistical mechanics description involves interactions between modes that are both non-linear and partially quenched disordered [35, 36]. In the latter case, we notice that the global spherical constraint on continuous variables is not implemented to ap- proximate discrete spin variables or ease the computation of the properties of continuous spins of fixed magnitude (like XY or Heisenberg spins), but it rep- resents the total amount of energy that an external pumping laser beam forces into the random laser to activate its modes. We will show in this work that adding purely ferromagnetic terms to the quencheddisorderedones (a particularcase ofwhich is to havequencheddisor- der with non-zeroaverage)can be simply encoded into adding an effective field to the purely disorderedsystem. The paper is organizedas follow: in Sec. 2 we introduce the model and present a formal solution in the framework of Parisi ReplicaSymmetryBreakingTheoryforthe generalcase;inSec. 3wespecialize the analysis to the s+p case in an uniform external field; in Secs. 4 and 5 we study the behavior in presence of ferromagnetic couplings of two qualitatively 2 different models both yielding RS and 1RSB phases: the 3+4 and the 2+3 models; in Sec. 6 we consider Replica Symmetry Breaking phases with contin- uous breakings and in Sec. 7 we show an explicit case in which these phases appear, even in presence of competing ferromagnetic interactions. Eventually, in Sec. 8 we show the termperature vs. degree of order phase diagrams for the 2+pandthe3+4models,wherethedegreeororderisyieldedbyacombination of ferromagnetic interaction magnitudes. A word of caution. When the phase is described by a step-like order parameter function, as the 1RSB phase, or it possesses a step-like part, the transition between different phases can differ if oneconsidersthe staticordynamicpropertiesofthe model[2,3,37,38]. When they are distinct one speaks of the static and dynamic transitions. In the main text we shall consider only the static transitions. The changes associated with the dynamic transition will be briefly discussed in Appendix B. 2. The Model We consider the general model system described by the spin-Hamiltonian J(k) = J(p) σ σ 0 σ σ (1) H − i1···ip i1··· ip − Nk−1 i1··· ik Xp≥2i1<X···<ip kX≥1 i1<X···<ik with both quenched, independently distributed, Gaussianp-spininteractions of zero mean and variance 2 p!J2 J(p) = p , (2) (cid:20)(cid:16) i1···ip(cid:17) (cid:21) 2Np−1 and uniform k-spin interactions J(k), with the k = 1 term representing the 0 interaction with an external uniform field. The scaling of the interaction with the system size N ensures the correct thermodynamic limit N . The → ∞ spins are real continuous variables ranging from to + , subjected to the −∞ ∞ global spherical constraint σ2 = N that limits the fluctuations and makes i i thepartitionfunctionwelldPefined. Thedynamicsofthecasewithasinglep>2 term and k =2 was treated in Ref. [39]. The model can also be seen as a spherical multiple-spin interaction Spin Glass model with random couplings of non-zero average. The formulation (1) is,however,moregeneralsinceitgivesmorefreedominchoosingtheinteractions in the disordered and ordered part of the Hamiltonian. To stress this point we havedeliberatelyuseddifferentindexes,namelypandk,forthe disorderedand ordered interactions. Inthepresentstudyweshallconsiderthesub-classofmodelswhereonlytwo terms,onewithsandonewithp>sinteractions,areretainedinthedisordered part. These models have been called spherical s+p models [19, 20, 40, 41, 6]. The hallmark of these models are different phase diagrams depending on the values of s and p. Representative values of s and p will be discussed when needed. 3 2.1. The partition sum and replicas The static properties of the model are obtained from the free energy com- putedforfixedinteractionsandthenaveragedoverthedisorder. Thisquenched free energy can be computed using the replica trick: one first computes the annealed free energy density Φ(n) of n non-interacting identical replicas of the system by rising the partition sum Z = Tr e−βH to the n’th power and aver- σ aging it over the disorder: 1 Φ(n)= lim ln[Zn]. (3) −N→∞βNn The quenched free energy density Φ is then obtained from the continuation of Φ(n) to non-integer values of n down to n=0: 1 Φ= lim lim ([Zn] 1)= lim Φ(n). (4) −N→∞ n→0βNn − n→0 In the last equality we assumed that the thermodynamic limit N and → ∞ the replica limit n 0 can be exchanged. The calculation of [Zn] is rather → standard, so we report the main steps, just in order to introduce our notation. The interested reader can find more details in Refs. [42, 5]. By introducing the collective variables 1 1 q = σaσb, m = σa (5) ab N i i a N i Xi Xi where a,b = 1,...,n are replica indexes, with q = 1 from the spherical con- aa straint, the leading contribution to [Zn] for N can be written as →∞ [Zn] [q,λ,m,y]e−NG[q,λ,m,y], N (6) ∼Z D →∞ where [q,λ,m,y] dq λ dm dy denotes integrations D ∝ a<b ab a≤b ab a a a a over all (free) variablesQand Q Q Q 1 1 G[q,λ,m,y] = g(q ) κ(m )+ λ q + y m ab a ab ab a a −2 − 2 Xab Xa Xab Xa 1 1 + lnDet( λ)+ y (λ−1) y . (7) a ab b 2 − 2 Xab We have introduced the short-hand notation: µ p g(q)= p qp, µ = β2J2 (8) p p 2 p Xp≥2 β κ(m)= b mk, b = J(k) (9) k p k! 0 kX≥1 4 In the thermodynamic limit N the integrals can be evaluated using the → ∞ saddle-point approximation,leading to 1 βΦ= lim ExtrG[q,m] (10) n→0n with 1 1 G[q,m]= g(q ) κ(m ) Trln(q m m ) (11) ab a ab a b −2 − − 2 − Xab Xa ThefunctionalG[q,m]mustbeevaluatedatitsstationarypointthat,asn 0, → gives the maximum with respect to variationsof q and the minimum for vari- ab ations in m . The variablesλ and y havebeen eliminated via the stationary a ab a point equations. In the expression (10) we have not included a constant term βΦ that comes from neglected O(N) terms. This fixes the zero temperature 0 value of energy and entropy, but it is not relevant for the study of the phase diagram. Since we are interested into the limit n 0, the expression of G[q,m] can → be simplified further by noticing that Trln(q m m )=Trlnq m (q−1) m +O(n2) (12) ab a b a ab b − − Xab so we arrive at the final expression 1 1 1 G[q,m]= g(q ) κ(m ) Trlnq+ m (q−1) m +O(n2). ab a a ab b −2 − − 2 2 Xab Xa Xab (13) By imposing stationarity of G[q,m] with respect to variations with respect to m and q (a=b) we obtain the stationary point equations: a ab 6 b(m )= (q−1) m (14) a ab b Xb Λ(q )+(q−1) (q−1) m (q−1) m =0, a=b (15) ab ab ac c bc c − 6 Xc Xc dg(q) dκ(m) Λ(q) ; b(m) . (16) ≡ dq ≡ dm) To solve these equations we observe that eq. (14) can be inverted to give m = q b(m ). (17) a ab b Xb If we retain only the k = 1 term in κ(m), then b(m) = b . The equation 1 becomes m =b q and m does not depend on the replica index a. This a 1 b ab a remains true in thPe generalcase because there are no explicit replica symmetry 5 breaking fields. The stationary point equation for the magnetization m m a ≡ then becomes n m=b(m) q a=1,...,n (18) ab ∀ Xb=1 and the stationary point equation for q ab Λ(q )+(q−1) +b(m)2 =0 a=b. (19) ab ab 6 Note that if we consider the value of b(m) as given, that is b(m)=b, Eq. (19) reducesto thatofthe modelin anexternaluniformconstantfieldh=Tb. This a rather important technical point because we can split-up the resolution of the stationary point equation into two steps. First we solve eq. (19) assuming b(m)=b as fixed. Next we look for m solution of eq. (18) such that b(m)=b. In the following we will generically refer to b as “field”. 2.2. Parisi Parametrization: Replica Symmetry Breaking To solvethe self-consistentstationarypointequation(19)anassumptionon the structure of the overlap matrix q must be done. As the n replicas of the ab realsystemareidentical,onemayreasonablyassumethatthesolutionshouldbe symmetric under the exchange of any pair of replicas. In the high temperature (or field) case this holds true, and the solution is of the form q =δ +(1 δ )q . (20) ab ab ab 0 − This form of q is known as the Replica Symmetric (RS) solution. ab As the temperature (and field) decrease the symmetry under replica ex- change is spontaneously broken, and the overlap matrix becomes a non-trivial functionofthereplicaindexes. InthisregimetheRSassumptionisnotvalidand a more complex structure arises. Following the parameterizationintroducedby Parisi[43,44],theoverlapmatrixq forRstepsofreplicapermutationsymme- ab trybreaking–calledRSBsolution–isdividedalongthediagonalintosuccessive blocks of decreasing size p , with p =n and p =1, and elements given by: u 0 R+1 q =q =q , u=0,...,R+1 (21) ab a∩b u with 1 = q q q > q . In this notation u = a b denotes the R+1 R 1 0 ≥ ≥ ··· ≥ ∩ overlap between the replicas a and b, and means that a and b belong to the same box of size p but to two distinct boxes of size p <p . u u+1 u The case R =0 gives back the RS solution, while the limiting case R →∞ producesthesolutioncalledFullReplicaSymmetryBreaking(FRSBor -RSB) ∞ solution[44,45]. Inthislimitq q 0foru=1,...,R,andthematrixq u u−1 ab − → is described by a continuous, non-decreasingfunction q(x), where, in the Parisi parameterization, x varies between 0 and 1. Solutions with a finite value of R are called R-RSB solutions [45, 5, 40, 41, 6]. These solutions can be described by a step-like function q(x). Mixed-type solutions, with both discontinuous R- RSB-typeandcontinuousFRSB-typepartsforsomexinterval,arealsopossible [19, 20]. 6 Insertingtheform(21)intothefreeenergyfunctionalG[q,m],eq. (13),with m =m, one obtains a R 2 G[q,m] = g(1) (p p )g(q ) ln(1 q ) u u+1 u R n − − − − − uX=0 R 1 qˆ q m2 u 0 ln − 2κ(m) (22) − p qˆ − qˆ − uX=1 u u+1 1 where qˆ is the Replica Fourier Transform (RFT) of q [46, 47]: u ab R+1 qˆ = p (q q ). (23) u v v v−1 − vX=u The freeenergyfunctionalcanbeconvenientlyexpressedbyintroducingthe auxiliary function R x(q)=p + (p p )θ(q q ) (24) 0 u+1 u u − − uX=0 which gives the fraction of pair of replicas with overlap q q. In terms of ab ≤ x(q) the functional G[q,m] takes the form 2 1 qR dq m2 G[q,m]= dqx(q)Λ(q) ln(1 q )+ 2κ(m) (25) R n −Z −Z χ(q) − − χ(0) − 0 0 where 1 χ(q)= dq′x(q′). (26) Z q Note that χ(q ) = qˆ and, moreover, χ(q) = χ(q ) = χ(0) for 0 q q u u+1 0 0 ≤ ≤ since x(q)=0 for q [0,q ]. 0 ∈ The stationary point equations are obtained from the first variation of the free energy functional G[q,m] with respect to x(q) and m: 2 1 m δG[q,m]= dqF(q) δx(q) 2 b(m) δm (27) n −Z − (cid:20) − χ(0)(cid:21) 0 where q dq′ m2 F(q)=Λ(q) + (28) −Z χ(q′)2 χ(0)2 0 and R R δx(q)= [θ(q q ) θ(q q )] δp (p p )δ(q q )δq . (29) u−1 u u u+1 u u u − − − − − − uX=1 uX=0 Stationarity of G[q,m] with respect to variations of m, q and p gives: u u m=χ(q )b(m) (30) 0 7 F(q )=0, u=0,...,R (31) u qu dqF(q)=0, u=1,...,R. (32) Z qu−1 ThefunctionF(q)iscontinuous,thuseqs. (31)and(32)requirethatbetween any two successive pairs (q ,q ) there must be at least two extrema of F(q). u−1 u Denoting these by q∗, the extrema condition F′(q∗)=0 implies that 1 1 dqx(q)= (33) Zq∗ Λ′(q∗) p where the prime denotes the derivative with respect to the argument q. The function x(q) is a non-decreasing function of q, and the left hand side of this equation is, thus, a concave function. The solutions to this equation, thus, depend from the convexity properties of 1/ Λ′(q): in the region where it is concave a continuum of solution can be foupnd, while where it is convex only discrete solutions exist. In the first case we deal with a continuous solution of the FRSB-type, while in the second case with a R-RSB-type solution. If 1/ Λ′(q) changes concavity for different intervals of q, we have a mixed-type solpution. In the above argument the presence of the ordered part of the Hamiltonian doesnotplay anyrole, onceencodedintob(m). Inthiswayonecandecouplethe computationstudyingthebehaviorofamodelina(self-induced)“external”field apart from the relationship between the field and the magnetizations induced by the ferromagnetic couplings. Thevalueofthefieldonlyentersinsettingthevalueofq ,thelowestpossible 0 valueofq(x). Asaconsequence,thevaluebofb(m)fine-tunestherange[q ,q ] 0 R where solutions of the stationary point equations must be searched. Since q is 0 anincreasingfunctionofbthepresenceofaneffectivefieldbcanonlyreducethe “complexity”ofthe solutionfound in absence ofit. In particular,by increasing the value of b we can eventually force q = q , that is a transition to the RS 0 R solution. For larger value of b only the RS solution exists. If terms besides the k = 1 term (the uniform external field) are present in the orderedpart of the Hamiltonian, cf. Eq. (9), what we just said is only part ofthegame. Inthiscase,indeed, b(m)isafunctionofm,andwemustconsider the possible solutions to Eq. (30) such that b(m)=b (34) to unfold the complete solution. The unfolding depends on the form of b(m). Therefore, starting from the same phase diagram expressed as function of b, different phase diagrams can be produced in the coupling constants, depending ontheactualb(m). AnexplicitinstanceofcompletephasediagramsinT,J(s,p) andJ(s,p) canbefoundinRef. [26]andthe cases2+p(p 4)and3+4willbe 0 ≥ reported in Section 8. In the forthcoming part of the paper we shall, instead, address the fate of the different type of solutions as the value of the effective 8 field is varied. To illustrate the results, we shall study s+p models in presence of an external uniform field b described by the stationary point equations (18), (19). Stability of the stationary point requires that the quadratic form Λ′(q )(δq )2+Tr q−1δq 2, (35) ab ab − Xab (cid:0) (cid:1) mustbe positive (semi)definite, where δq =δq is the fluctuation ofq from ab ba ab the stationary point value. 3. The Spherical s+p model in an uniform external field The Sphericals+pmodelisthe particularmodelobtainedfromthe general Hamiltonian (1) in which one retains only two terms with random s-spin and p-spin interactions and a (k = 1) uniform external field in Eq. (1). Without loosing in generality, we assume s<p from now on. For this model we have: µ µ g(q) = sqs+ pqp, Λ(q)=µ qs−1+µ qp−1, (36) s p s p κ(m) = bm The phase diagrams in the plane (µ ,µ ), i.e. for b = βh = 0, are well p s known [20]. Depending on the value of s and p different type of solutions can be found, of both FRSB and R-RSB type and mixed. Since we are interested on the effect of the external field b on these different phases, it can be useful to use the following parametrization Λ(q)=µ (rqs−1+qp−1) (37) p where µ sJ2 r = s = s, 0 r < (38) µ pJ2 ≤ ∞ p p gives the relative strength of the s and p interaction terms, and we use µ and p b as free parameterfor givenr. The temperature, when needed, is computed as T/J = p/(2µ ). p p p 3.1. The RS Solution All models, regardlessof the value of s and p, for large enough temperature (i.e., small enough µ and µ ) present a RS phase. The equation for the RS s p phase are obtained inserting x(q)=θ(q q ) (39) 0 − into equations (30), (31) and (32), or into the functional (25), then making it stationary with respect to m and q . In either cases one ends up with: 0 m=(1 q )2b (40) 0 − 9 and q Λ(q )= 0 b2 (41) 0 (1 q )2 − 0 − The RS phase remains stable as long as the relevant eigenvalue Λ of the fluc- 1 tuations remains positive: 1 Λ = Λ′(q )+ 0. (42) 1 − 0 (1 q )2 ≥ 0 − Using the stationary point equation (41) one obtains the equivalent condition Λ(q ) q Λ′(q )+b2 0. (43) 0 0 0 − ≥ The equal sign defines the critical line on which the RS phase ends. With the help of the parameterization (37) the parametric equation of the critical line reads: 1 1 µ =  p (1−q0)2r(s−1)q0s−2+(p−1)q0p−2  µs =rµp 0≤q0 ≤1 (44) 1 r(s 2)qs−1+(p 2)qp−1  b2 = (1−q0)2r(s−−1)q00s−2+(p−−1)q00p−2 Depending on the values of µ and µ , the curve may show points where p s dµ p =0, (45) db (cid:12) (cid:12)µs/µp (cid:12) (cid:12) thatis dT/dh=0 inthe (h,T)plane. When present,one ofsuchcriticalpoints occurs where the transition between the RS phase and the RSB one changes from continuous to discontinuous. A discontinuity of finite height appears in q(x) at the transition and the critical line (44) stops there. By using the parametricform(44), the points wheredµ /db=0 correspond p to the value of q solution of 0 2Λ′(q )+(q 1)Λ′′(q )=0. (46) 0 0 0 − Thelargestsolution0 q <1ofthisequation,whenitexists,givesthecritical c ≤ point where the line (44) ends, and q gets restricted to q q 1. For r =0 0 c 0 ≤ ≤ we recover q = 1 2/p , while it is q = 1 2/s in the opposite limit r . c c − − →∞ This is the critical value of the pure p-spin, or s-spin, spherical model. Beyond this point one must resort to a 1RSB Ansatz in order to obtain the expression for the transition line. 10