Table Of Content

Pis’ma v ZhETF Full replica symmetry breaking in p-spin-glass-like systems T. I. Schelkacheva1, and N. M. Chtchelkatchev1 6 − 1InstituteforHighPressurePhysics,RussianAcademyofSciences, 142190, Troitsk,Moscow,Russia 2 L.D.LandauInstitute forTheoretical Physics,RussianAcademyofSciences,142432, MoscowRegion,Chernogolovka, Russia 3UralFederalUniversity,620002Ekaterinburg, Russia 7 4DepartmentofTheoretical Physics,MoscowInstitute ofPhysicsandTechnology, 141700Moscow,Russia 1 0 5All-RussiaResearchInstituteofAutomatics,22Suschevskaya, Moscow127055, Russia 2 6InstituteofMetallurgy,UralDivisionofRussianAcademyofSciences, Yekaterinburg620016, Russia n a J 1 It isshown that continuously changing theeffectivenumberof interacting particles in p-spin-glass-likemodel allows to describe the transition from the full replica symmetry breaking glass solution to stable first replica ] h symmetry breaking glass solution in the case of non-reflective symmetry diagonal operators used instead of c Isingspins. Asanexample,axialquadrupolemomentsinplaceofIsingspinsareconsideredandtheboundary me valuepc1 ∼=2.5 is found. - tIntroduction The basis of understanding glasses is netic state to the glass state (like, e.g., in SK model). a tthe Sherrington– Kirkpatrick(SK)model[1]: the Ising The second class of models can be called 1RSB-models s .model with random links. A stable solution for SK (p-spin model, Potts models). In this case there is a t amodel was obtained by Parisi [2, 3] with a full replica finite range of temperatures where stable 1RSB glass m symmetry breaking (FRSB) scheme. Later it was re- solution occurs. What is important that this 1RSB so- -alised that replica symmetry is not abstract and aca- lution mostly appears abruptly. d ndemic question but it corresponds to formation of the 1RSB-models and especially p-spin glasses in re- ospecific hierarchy of basins in the energy landscape of cent years attract much interest in connection with c the glass forming system. the fact that there is a close relationship between [ A natural generalization of the SK model with pair static replica approachand dynamic consideration. For 1 interaction of spins is a model with p-spin interac- example the Random First Order Transition theory v 0tions [3, 4]. Unlike of the SK model, p-spin model for structural glasses is inspired by the p-spin glass 3has a stable first replica symmetry breaking (RSB) so- model [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. 2 lution that arises abruptly. A very low-temperature Itshouldbealsonotedthatthetwoclassesofmodels 0 boundary of the 1RSB stability region is given by so mentioned above distinguish essentially by their energy 0 .called Gardner transition temperature intensively dis- landscape [16], which is an important concept in the 1 cussed last time [4, 5] where a valley in configuration dynamics of liquids and glasses. 0 7space transforms to a multitude of separated basins. In the context of SK-like and 1RSB-models it is 1 possible to develop very advanced theoretical tools Now there is a reborn of interest to spin models : vshowingglassybehaviour[5,6,7,8,9,10,11,12,13,14]. that can be reused in other contexts. These rela- XiItturnedoutthatthesemodelscanqualitativelyexplain tively simple models based on well-designed solutions allow to explore qualitatively an extensive range of is- rphysics of “real” glasses [15]. On the other hand, there a is limited number of analyticallysolvable glassymodels and each such model is interesting itself. Here we pro- stable 1RSB pose analytical solution of p-spin-glass-like system and we investigate: continuous discontinuous discuss physical applications of this model. fullRSB 1RSBtransition 1RSBtransition Foralongtimetherewasaconjecturethatthereare 2 pc1 pc2 p moreorlesstwoclassesofmodels,dependingonhowthe Fig.1 Wehave got FRSB glass solution of a p-spin-like modelwhentheeffectivenumberofinteractingparticles replica symmetry breaking appears [18]. In one class of models full replica symmetry breaking (FRSB) occurs 2<p<pc1. We considered general diagonal operators Uˆ withTrUˆ2k+1 6=0,k=1,2,...insteadofIsingspins. continuously at the transition point from the paramag- 1 2 T. I. Schelkacheva, and N. M. Chtchelkatchev sues, ranging from magnetic to structural glass transi- crossoverfrom the full replica symmetry breaking glass tions [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. solution to stable first replica symmetry breaking glass solution (in our case of non-reflective symmetry diag- Therefore, a natural generalizations of these basic onal operators used instead of Ising spins). For illus- models leads to a successful description of the various trating example we take operators of axial quadrupole types of glasses, such as orientation glasses and clus- moments in place of Ising spins. For this model we find ter glasses. Replacing the simple Ising spins by more complex operators Uˆ dramatically expands the range the boundary value pc1 ∼=2.5. of solvable tasks. The operator Uˆ can be, for example, the axial quadrupole moment (quadrupole glass The model The staring point is the p-spin-glass-like in molecular solid hydrogen at different pressures) or Hamiltonian the role of Uˆ can be played by certain combinations H = J Uˆ Uˆ ...Uˆ , (1) of the cubic harmonics (orientational glass of the clus- − i1...ip i1 i2 ip ters, for example, in C60 in a wide pressure range), i1≤Xi2...≤ip see [18, 19, 20, 21, 22, 23, 24, 25, 26]. where the quenched interactions J are distributed i1...ip It was shown recently that many statements re- with Gaussian probability: lated to the models based on Ising spins can be applied to p-spin-like models that use instead of spins diago- P(J )= √Np−1 exp (Ji1...ip)2Np−1 . (2) nal operators Uˆ when there is “reflection symmetry”: i1...ip √p!πJ − p!J2 (cid:20) (cid:21) TrUˆ2k+1 =0, k =1,2.... For such models we can build Here arbitrarynon-reflective symmetry diagonalopera- a stable 1RSB solution for p > 2 in a wide region of tors Uˆ are located on the lattice sites i instead of Ising temperatures. It was shown that the point p = 2 is spins, N is the number of sites. We remind that it special for such models [26, 27, 28]. implies that TrUˆ2k+1 = 0, k= 1,2,.... Here we inves- For p-spin-like model we previously have got signif- 6 tigate how orderparametersofthis model developwith icantly different solutions when diagonal operators Uˆ the continuous parameterp and the specific type of the havebroken reflective symmetry,thenTrUˆ2k+1 =0like operators Uˆ. 6 (e.g., for quadrupole operators). And for these models We write down in standard way the disorder aver- 1RSBglasssolutionbehaviorhavebeenrecentlyinvesti- aged free energy using the replica approach [3]. We as- gatednear the glasstransitionat different (continuous) sume like it was done in [30, 31] that the order param- p [26, 29]. It turned out that there is a finite region of eter deviations δqαβ are small from replica symmetry instability of 1RSB solutions for 2 p <p where p is determined by the specific form o≤f Uˆ. 1RcS1B solutiocn1 orderparameterqRS. TodescribetheRSB-solutionnear the bifurcationpointtemperatureT ,where RS“trans- 0 is stable for p>p . Wherein the transition frompara- c1 forms” into RSB, we expand the expressionfor the free phase to 1RSB glass is continuous for pc1 < p < pc2. energy up to the fourth order of δqαβ. We write below When p > p [26, 29] 1RSB glass occurs abruptly just c2 in (3) the deviation ∆F(p) of the free energy from its as in the conventional p-spin model. We should note RS-part. Itis importantto note thatthe expressionfor that p is notuniversal,but it depends on the particu- c2 Free energy have not been written before for arbitrary lar type of Uˆ. We have built [30, 31] FBSB solution for p. We should also emphasize that it includes the terms these models with a pair interaction p=2. with odd number ofidentical replicaindices [33,34, 35] In this letter we investigate in detail the generalised unlike Ising-spin SK-models. p-spin glass forming models in the region 2 p < p All coefficients in (3) depend only on RS-solution ≤ c1 where instability of 1RSB glass solution is expected. at T . They are given in Supplementary Material. The 0 We build a solution with full replica symmetry break- primeon ′meansthatonlythesuperscriptsbelonging ing. The very existence of the domain with full replica tothesameδq arenecessarilydifferent. Thecoefficients P symmetrybreakingisasurprisingresultespeciallycom- of the second and third orders, λ ,L,B ,..B , (RS)repl 3 4 pared with the traditional p-spin model of Ising spins. havebeenobtainedearlier[18,26,27],butwealsowrite Continuouslychangingthe effectivenumberofinteract- theminSupplementaryMaterialforreadability. Thisis ing particles in p-spin model allows us to describe the the only overlap with our previous publications. FRSB in p-spin-glass-like systems... 3 ∆NFk(Tp) =nlim0n1(t42p(p2−1)qR(pS−2) λ(p)(RS)repl ′ δqαβ 2−t24L(p) ′δqαβδqαδ−t6 B2(p) ′δqαβδqαγδqβδ+ → α,β α,β,δ (cid:20) α,β,γ,δ (cid:2) (cid:3)X (cid:0) (cid:1) X X ′ ′ ′ ′ B (p) δqαβδqαγδqαδ+B (p) δqαβδqβγδqγα+B (p) δqαβ 2δqαγ +B (p) δqαβ 3 + 2′ 3 3′ 4 α,β,γ,δ α,β,γ α,β,γ α,β (cid:21) X X X (cid:0) (cid:1) X (cid:0) (cid:1) ′ ′ ′ ′ t8 D (p) δqαβ 4+D (p) δqαβ 3δqαγ +D (p) (δqαβ)2 δqαδ 2+D (p) δqαβ 2δqαγδqγβ+ 2 31 32 33 (cid:20) α,β α,β,γ α,β,δ α,β,γ X (cid:0) (cid:1) X (cid:0) (cid:1) X (cid:0) (cid:1) X (cid:0) (cid:1) ′ ′ ′ D (p) δqαβ 2δqαγδqαδ+D (p) δqαβ 2δqαγδqβδ+D (p) δqαβ 2δqαγδqγδ+ 42 43 45 α,β,γ,δ α,β,γ,δ α,β,γ,δ X (cid:0) (cid:1) X (cid:0) (cid:1) X (cid:0) (cid:1) ′ ′ ′ D (p) δqαβδqαγδqαδδqβγ +D (p) δqαβδqβγδqγδδqδα+D (p) δqαβδqαγδqαδqαµ+ 46 47 53 α,β,γ,δ α,β,γ,δ α,β,γ,δ,µ X X X ′ ′ D (p) δqαβδqαγδqαδqβµ+D (p) δqαβδqαγδqγδqδµ , (3) 54 55 α,β,γ,δ,µ α,β,γ,δ,µ (cid:21)) X X The order parameters and the replicon mode λ (re- whicharenotformallydescribedbytheParisirulescan sponsible for RSB stability) we find as follows: be reduced to the standard form as well [30, 31]. The equation for the order parameter q(x) follows λ(p)(RS)repl =1−t2p(p2−1)qR(pS−2)× fromThthereefsotaret,iownearciatnyicsoandsiimtioilnarδwqδ(axy)∆finFd(pt)he=b0r.anching 2 Tr Uˆ2expθˆRS TrUêxpθˆ 2 condition (appearance of FRSB glass solution), which dzG RS ; (4) were derived in detail for p = 2 (see [30, 31] and Refs. Z  (cid:16)TrexpθˆRS (cid:17) −" TrexpθˆRS #  therein): λ(p)(RS)repl |T0= 0, which produces the temperature T (p). 0   Forclaritywewrite δ ∆F(p)=0uptotheterms 2 δq(x) Tr Uêxp θˆ of the second order. We get RS q(p) = dzG ; (5) RS Z  Trhhexp(cid:16)(cid:16)θˆRS(cid:17)(cid:17)ii − t220p(p2−1)qR(pS−2)d λ(RdSt)repl |t0∆tq(x)− w(p)RS =Z dzGTrThrUê2xepxpθˆ(cid:16)RθˆSRS(cid:17)i; (6) So if the operators(cid:2)Uˆ do no(cid:3)tt40Lhahqvie+th..e.=re0fl.ect(iv7e) h (cid:16) (cid:17)i symmetry (therefore L = 0 in that case) we need where dzG = −∞∞ √d2zπ exp −z22 , t = J/kT = an additional branching c6ondition, hqi ≡ 01q(x)dx = t0+∆tR; α,β labelRn replicas an(cid:16)d (cid:17) 0+o(∆t)2 [31], insuring the appearance of non-trivial R new solutions. θˆ(p)RS =zt pqRS2(p−1) Uˆ +t2p[wRS(p−1)4−qRS(p−1)]Uˆ2. Integralequation, δqδ(x)∆F =0,thatdeterminesthe r functionq(x), asusuallycanbesimplifiedusingthedif- ferential operator Oˆ = 1 d 1 d , where q = dq(x): FRSB Below we find answer of two questions: 1) q′dxq′dx ′ dx if FRSB realises for given model (we derive simple cri- terium), 2)weprovidenecessarytoolsforcalculationof t6 B B x +t8 [ 2D +4xD ] xq(x) 4 3 33 47 FRSB order parameter q(x). { − } (− (cid:20)− − To describe the FRSB function q(x) of the variable 1 xweincludeintheconsiderationthefourth-orderterms dyq(y) +[ 4D2+2xD33]q(x) =0. (8) intheexpansionof∆F. Weusethestandardformalized Zx (cid:21) − ) Parisialgebrarules [2,3]to writethe freeenergyasthe ThecoefficientsaregiveninSupplementaryMaterialfor functionalofq(x)andsotoconstructFRSB.Theterms p 2. ≥ 4 T. I. Schelkacheva, and N. M. Chtchelkatchev 3 3.0 a) a) 2 2.5 xD 1 T0 2.0 0 1.5 -1 1.0 -2 0.5 -3 0.0 1.0 1.5 2.0 2.5 3.0 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 p p 100 0.9 b) b) 0.8 qp B3 0 B4/ 0.7 0.6 0.5 -100 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 0.4 p 1.0 1.5 2.0 2.5 3.0 Fig. 2 a).The dependence of the denominator xD = p (cid:2)−D2+D33x−D47x2(cid:3) |x˜ of the function dqd(xx) |x˜ on Fig. 3 a) Dependence of branching temperature T0 of the effective number of interacting quadrupoles p ob- glass solution on the effective number of interacting tainedfromEq.(9). b)Thedependenceofthefunction quadrupoles p. b) Dependence of x˜(p) = B4(p)/B3(p) qp = dqd(xx) |x˜ on the effective number of interacting on theeffective numberof interacting quadrupoles p. quadrupoles p obtained from Eq. (9). what one usually has in the case of Ising-like opera- Since q(x) can only be a non-decreasing function of tors [2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28]. xweshouldconsiderhowthesignofq′ = dqd(xx) depends We have done earlier calculations of 1RSB solu- on the parameter p. We obtain from Eq. (8): tion for various values of p that could change continuously [26]. It has been found that for 2 < p < p x c1 B +4[xq(x) dyq(y)]D +2D q(x) q′(p)= 3 − 0 47 33 . (pc1 = 2.5) the solution is qualitatively similar to one t24[−D2R+D33x−D47x2] which occurs in the case of pair interaction (p = 2). (9) This behavior is natural for continuity reason. At high This expression is one of the central results of our pa- temperaturesT >T (we remindthatT is the bifurca- 0 0 per. Depending on the signof q we can conclude if the ′ tion temperature) there is a stable non-trivial solution system falls into FRSB (positive q ) state or not. ′ for the RS order parameter q . But q =0 for any fi- RS RS Ifweconfineourselvestothetermsofthethirdorder nite temperature T >T unlike conventio6 nal Ising-spin 0 over δq in ∆F and (8) then we obtain t6{B4−B3x}+ model. For T < T0 we have Almeida-Thoulles repli- ...=0. So the significantly depending on x partof q(x) conmode λ <0andRS-solutionbecomes unsta- (RS)repl is concentrated in the neighborhood of ble. Similarly, when T < T we have λ < 0 0 (1RSB)repl determining the condition when the 1RSB glass is un- x˜(p)=B (p)/B (p). (10) 4 3 stable [19, 24, 26, 27]. Soweinvestigatedthestabilityof1RSBsolutionand Only in the case ofoperatorsUˆ with TrUˆ(2k+1) =0 for foundthat1RSBsolutionisunstablewhenp<2.5. The 6 k =1,2,.. we get x˜=B /B =0. For Ising-like opera- 4 3 1RSB solutionbecomes stable when p>2.5, see Fig.1. torswithTrUˆ(2k+1) =0wege6tp=2(see [26,27,29]), The instability region of 1RSB solution is an area in B =0 and sox˜=0 in accordancewith usualstandard 4 which there may be FRSB solution. case of the Parisi theory [2, 3]. Using Eqs. (9)-(10), we can describe FRSB solution forT <T nearT for2<p<p . Wealsowanttofind 0 0 c1 Example As an illustrating example, we consider the value of pc1 just numerically solving the equations quadrupole glass in the space, J = 1. Operator Eqs. (9)-(10). We intend to compare this result of pc1 Uˆ = Qˆ is the axial quadrupole moment and it takes top =2.5wehadpreviouslyreceived,consideringthe c1 values ( 2,1,1). So there is no reflection symmetry. stability of the 1RSB solution [26]. − Behavior of this model is significantly different from For pair interaction, p = 2, we earlier obtained [31] FRSB in p-spin-glass-like systems... 5 that T0 = 1.37 and x˜ = 0.43 for FRSB. Wherein 9. S.Franz,H.Jacquin,G.Parisi, P.Urbani,F.Zamponi, q = q(x) is a continuous increasing function of x in J. Chem. Phys. 138, 12A540 (2013). narrow neighborhood, e.g., ∆x = 0.016 near x˜ for 10. F. Caltagirone, U. Ferrari, L. Leuzzi, G. Parisi, F. (T T ) = 0.2. For other values of x, not in the Ricci-Tersenghi,T.Rizzo,Phys.Rev.Lett.108,085702 0 − − neighborhood of x˜, q(x) is very close to q . (2012). 1RSB So in our case 2 < p <= 2.5 we can with good ac- 11. T. Rizzo, Euro Phys.Lett. 106, 56003 (2014). ∼ curacy put q(x˜) = q(x˜) = q(0) and [x˜q(x˜) 12. T. Rizzo, Phys.Rev.E 87, 022135 (2013). 1RSB 1RSB x˜dyq(y)]=0 in (9). We also have got that q(x˜) −is 13. G. Parisi and F. Zamponi, Rev. Mod. Phys. 82, 789 0 1RSB a small quantity and so we can neglect 2D q(x˜) com- (2010). RparedwithB . Forothervaluesofxnotver3y3closetox˜ 14. L. Berthier and G. Biroli, Rev. Mod. Phys. 83, 58 3 (2011). inexactlythesamewayasitwasobtainedforp=2[31] we find that q(x) is very close to q . 15. P.G.WolynesandV.Lubchenko,StructuralGlassesand 1RSB Supercooled Liquids: Theory, Experiment, and Appli- The results of our calculations are presented in Figs. 2-3. At pc1 ∼= 2.5 the function dqd(xx) |x˜ diverges 16. Uca.tiBouncsh(eWnailue,y,J.coPmhy,s2.0C121).5, S955 (2003). and changes its sign when p > p contrary to its con- c1 17. V. Alba, S. Inglis, L. Pollet, Phys. Rev. B 93, 094404 ventional probabilistic interpretation. It follows that (2016). FRSB is impossible for p > 2.5. This result agrees ≈ 18. T. I. Schelkacheva, E. E. Tareyeva, and N. M. with p obtained by us previously for 1RSB solution c1 Chtchelkatchev,Phys.Rev.E 79, 021105 (2009). stability border [26]. 19. T.I. Schelkacheva, E.E. Tareyeva, and N.M. For clarity we show the dependence the denomina- Chtchelkatchev,Phys.Rev.B 82, 134208 (2010). tor x = D +D x D x2 on p in Fig. 2. We D − 2 33 − 47 |x˜ 20. T.I. Schelkacheva, E.E. Tareyeva, and N.M. showinFig.3thedependenceofthebranchingtemper- (cid:2) (cid:3) Chtchelkatchev,Phys.Rev.B 76, 195408 (2007). ature T and the dependence of x˜(p)=B (p)/B (p) on 0 4 3 21. E.A.Lutchinskaia and E.E. Tareyeva, Phys. Rev. B 52, the effective number of interacting quadrupoles p. 366 (1995). 22. N.M. Chtchelkatchev, V.N. Ryzhov, T.I. Schelkacheva, Conclusions For the first time it is shown that con- and E.E. Tareyeva, Phys.Lett. A 329, 244 (2004). tinuously changing of one of the parameters of glass 23. K.Walasek, J. Phys. A 28, L497 (1995). formingmodel withone type ofinteractionallowsmov- 24. E.E.Tareyeva, T.I.Schelkacheva and ing from FRSB glass to stable 1RSB solution. As an N.M.Chtchelkatchev, Theor. Math. Phys. 160, example the generalized p-spin quadrupole glass model 1190 (2009) is considered. 25. E.E. Tareyeva, T.I. Schelkacheva and N.M. Chtchelkatchev,Theor. Math. Phys. 155, 812 (2008). Acknowledgments Thisworkwassupportedinpart 26. E.E.Tareyeva,T.I.Schelkacheva,N.M.Chtchelkatchev, bytheRussianFoundationforBasicResearch(No. 16- J. Phys. A 47, 075002 (2014). 02-00295 ) while numerical simulations were funded by 27. T.I. Schelkacheva, N.M.Chtchelkatchev, J. Phys. A 44, Russian Scientific Foundation (grant No. 14-12-01185). 445004 (2011). 28. T.I. Schelkacheva, E.E.Tareyeva, and N.M.Chtchelkatchev,Phys. Lett. A 358, 222 (2006). 1. D.Sherrington,andS.Kirkpatrick,Phys.Rev.Lett.32, 29. E.E.Tareyeva, T.I.Schelkacheva and 1972 (1975); S. Kirkpatrick, and D. Sherrington, Phys. N.M.Chtchelkatchev, Theor. Math. Phys. 182, Rev. B 17, 4384 (1978). 437 (2015). 2. G. Parisi, J.Phys.A: Math. Theor. 13, L115 (1980). 30. E.E.Tareyeva,T.I.Schelkacheva,N.M.Chtchelkatchev, 3. M.Mezard,G.Parisi,andM.Virasoro,SpinGlassThe- Phys.Lett. A 377, 507 (2013). ory and Beyond (World Scientific, Singapore, 1987). 31. T.I.Schelkacheva,E.E.Tareyeva,N.M.Chtchelkatchev, 4. E. Gardner, Nuc.Phys. B257, 747 (1985). Phys.Rev.E 89, 042149 (2014). 5. P. Charbonneau, Y. Jin, G. Parisi, C. Rainone, B. 32. A. Crisanti, L. Leuzzi, Phys. Rev. Lett. 93, 217203 Seoane, F. Zamponi Phys. Rev.E 92, 012316 (2015). (2004). 6. A.Crisanti,H.Horner,andH.Sommers,Z.Phys.B92, 33. T. Temesvari, C. De Dominicis, I. R. Pimentel: Eur. 257 (1993). Phys.J. B 25, 361 (2002). 7. T. R. Kirkpatrick, D. Thirumalai and P. G. Wolynes, 34. T. Temesvari, Nuclear Physics B, 829, 534 (2010). Phys. Rev.A 40 1045 (1989). 35. A.A.Crisanti, C.DeDominicis, J.Phys.A43,055002 8. T. R.Kirkpatrick, D. Thirumalai, Phys. Rev. Lett. 58, (2010). 2091 (1987), Phys.Rev. B 36, 5388(1987).