Full replica symmetry breaking in p-spin-glass-like systems
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Full replica symmetry breaking in p-spin-glass-like systems
T. I. Schelkacheva , and N. M. Chtchelkatchev − Institute for High Pressure Physics, Russian Academy of Sciences, 142190, Troitsk, Moscow, Russia L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 142432, Moscow Region, Chernogolovka, Russia Ural Federal University, 620002 Ekaterinburg, Russia Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Moscow, Russia All-Russia Research Institute of Automatics, 22 Suschevskaya, Moscow 127055, Russia Institute of Metallurgy, Ural Division of Russian Academy of Sciences, Yekaterinburg 620016, Russia
It is shown that continuously changing the effective number of interacting particles in p-spin-glass-like modelallows to describe the transition from the full replica symmetry breaking glass solution to stable first replicasymmetry breaking glass solution in the case of non-reflective symmetry diagonal operators used instead ofIsing spins. As an example, axial quadrupole moments in place of Ising spins are considered and the boundaryvalue p c ∼ = 2 . Introduction
The basis of understanding glasses isthe Sherrington – Kirkpatrick (SK) model [1]: the Isingmodel with random links. A stable solution for SKmodel was obtained by Parisi [2, 3] with a full replicasymmetry breaking (FRSB) scheme. Later it was re-alised that replica symmetry is not abstract and aca-demic question but it corresponds to formation of thespecific hierarchy of basins in the energy landscape ofthe glass forming system.A natural generalization of the SK model with pairinteraction of spins is a model with p -spin interac-tions [3, 4]. Unlike of the SK model, p -spin modelhas a stable first replica symmetry breaking (RSB) so-lution that arises abruptly. A very low-temperatureboundary of the 1RSB stability region is given by socalled Gardner transition temperature intensively dis-cussed last time [4, 5] where a valley in configurationspace transforms to a multitude of separated basins.Now there is a reborn of interest to spin modelsshowing glassy behaviour [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].It turned out that these models can qualitatively explainphysics of “real” glasses [15]. On the other hand, thereis limited number of analytically solvable glassy modelsand each such model is interesting itself. Here we pro-pose analytical solution of p-spin-glass-like system anddiscuss physical applications of this model.For a long time there was a conjecture that there aremore or less two classes of models, depending on how thereplica symmetry breaking appears [18]. In one class ofmodels full replica symmetry breaking (FRSB) occurscontinuously at the transition point from the paramag- netic state to the glass state (like, e.g., in SK model).The second class of models can be called 1RSB-models( p -spin model, Potts models). In this case there is afinite range of temperatures where stable 1RSB glasssolution occurs. What is important that this 1RSB so-lution mostly appears abruptly.1RSB-models and especially p -spin glasses in re-cent years attract much interest in connection withthe fact that there is a close relationship betweenstatic replica approach and dynamic consideration. Forexample the Random First Order Transition theoryfor structural glasses is inspired by the p -spin glassmodel [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].It should be also noted that the two classes of modelsmentioned above distinguish essentially by their energylandscape [16], which is an important concept in thedynamics of liquids and glasses.In the context of SK-like and 1RSB-models it ispossible to develop very advanced theoretical toolsthat can be reused in other contexts. These rela-tively simple models based on well-designed solutionsallow to explore qualitatively an extensive range of is- c1 p c2 pcontinuous1RSB transition discontinuous1RSB transitionstable 1RSB full RSB we investigate: Fig.1 We have got FRSB glass solution of a p-spin-likemodel when the effective number of interacting particles2 < p < p c . We considered general diagonal operatorsˆ U with Tr ˆ U k +1 = 0, k = 1 , , ... instead of Ising spins. T. I. Schelkacheva, and N. M. Chtchelkatchev sues, ranging from magnetic to structural glass transi-tions [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].Therefore, a natural generalizations of these basicmodels leads to a successful description of the varioustypes of glasses, such as orientation glasses and clus-ter glasses. Replacing the simple Ising spins by morecomplex operators ˆ U dramatically expands the rangeof solvable tasks. The operator ˆ U can be, for exam-ple, the axial quadrupole moment (quadrupole glassin molecular solid hydrogen at different pressures) orthe role of ˆ U can be played by certain combinationsof the cubic harmonics (orientational glass of the clus-ters, for example, in C in a wide pressure range),see [18, 19, 20, 21, 22, 23, 24, 25, 26].It was shown recently that many statements re-lated to the models based on Ising spins can be appliedto p -spin-like models that use instead of spins diago-nal operators ˆ U when there is “reflection symmetry” :Tr ˆ U k +1 = 0, k = 1 , ... . For such models we can builda stable 1RSB solution for p > p = 2 isspecial for such models [26, 27, 28].For p-spin-like model we previously have got signif-icantly different solutions when diagonal operators ˆ U have broken reflective symmetry , then Tr ˆ U k +1 = 0 like(e.g., for quadrupole operators). And for these models1RSB glass solution behavior have been recently investi-gated near the glass transition at different (continuous) p [26, 29]. It turned out that there is a finite region ofinstability of 1RSB solutions for 2 ≤ p < p c where p c is determined by the specific form of ˆ U . 1RSB solutionis stable for p > p c . Wherein the transition from para-phase to 1RSB glass is continuous for p c < p < p c .When p > p c [26, 29] 1RSB glass occurs abruptly justas in the conventional p -spin model. We should notethat p c is not universal, but it depends on the particu-lar type of ˆ U . We have built [30, 31] FBSB solution forthese models with a pair interaction p = 2.In this letter we investigate in detail the generalised p -spin glass forming models in the region 2 ≤ p < p c where instability of 1RSB glass solution is expected.We build a solution with full replica symmetry break-ing. The very existence of the domain with full replicasymmetry breaking is a surprising result especially com-pared with the traditional p -spin model of Ising spins.Continuously changing the effective number of interact-ing particles in p -spin model allows us to describe the crossover from the full replica symmetry breaking glasssolution to stable first replica symmetry breaking glasssolution (in our case of non-reflective symmetry diag-onal operators used instead of Ising spins). For illus-trating example we take operators of axial quadrupolemoments in place of Ising spins. For this model we findthe boundary value p c ∼ = 2 . The model
The staring point is the p-spin-glass-likeHamiltonian H = − X i ≤ i ... ≤ i p J i ...i p ˆ U i ˆ U i ... ˆ U i p , (1)where the quenched interactions J i ...i p are distributedwith Gaussian probability: P ( J i ...i p ) = √ N p − √ p ! πJ exp (cid:20) − ( J i ...i p ) N p − p ! J (cid:21) . (2)Here arbitrary non-reflective symmetry diagonal opera-tors ˆ U are located on the lattice sites i instead of Isingspins, N is the number of sites. We remind that itimplies that Tr ˆ U k +1 = 0, k= 1 , , . . . . Here we inves-tigate how order parameters of this model develop withthe continuous parameter p and the specific type of theoperators ˆ U .We write down in standard way the disorder aver-aged free energy using the replica approach [3]. We as-sume like it was done in [30, 31] that the order param-eter deviations δq αβ are small from replica symmetryorder parameter q RS . To describe the RSB-solution nearthe bifurcation point temperature T , where RS “trans-forms” into RSB, we expand the expression for the freeenergy up to the fourth order of δq αβ . We write belowin (3) the deviation ∆ F ( p ) of the free energy from itsRS-part. It is important to note that the expression forFree energy have not been written before for arbitrary p . We should also emphasize that it includes the termswith odd number of identical replica indices [33, 34, 35]unlike Ising-spin SK-models.All coefficients in (3) depend only on RS-solutionat T . They are given in Supplementary Material. Theprime on P ′ means that only the superscripts belongingto the same δq are necessarily different. The coefficientsof the second and third orders, λ ( RS ) repl , L, B , ..B ,have been obtained earlier [18, 26, 27], but we also writethem in Supplementary Material for readability. This isthe only overlap with our previous publications. RSB in p-spin-glass-like systems. . . F ( p ) N kT = lim n → n ( t p ( p − q ( p − RS (cid:2) λ ( p ) ( RS ) repl (cid:3) X α,β ′ (cid:0) δq αβ (cid:1) − t L ( p ) X α,β,δ ′ δq αβ δq αδ − t (cid:20) B ( p ) X α,β,γ,δ ′ δq αβ δq αγ δq βδ + B ′ ( p ) X α,β,γ,δ ′ δq αβ δq αγ δq αδ + B ( p ) X α,β,γ ′ δq αβ δq βγ δq γα + B ′ ( p ) X α,β,γ ′ (cid:0) δq αβ (cid:1) δq αγ + B ( p ) X α,β ′ (cid:0) δq αβ (cid:1) (cid:21) + t (cid:20) D ( p ) X α,β ′ (cid:0) δq αβ (cid:1) + D ( p ) X α,β,γ ′ (cid:0) δq αβ (cid:1) δq αγ + D ( p ) X α,β,δ ′ ( δq αβ ) (cid:0) δq αδ (cid:1) + D ( p ) X α,β,γ ′ (cid:0) δq αβ (cid:1) δq αγ δq γβ + D ( p ) X α,β,γ,δ ′ (cid:0) δq αβ (cid:1) δq αγ δq αδ + D ( p ) X α,β,γ,δ ′ (cid:0) δq αβ (cid:1) δq αγ δq βδ + D ( p ) X α,β,γ,δ ′ (cid:0) δq αβ (cid:1) δq αγ δq γδ + D ( p ) X α,β,γ,δ ′ δq αβ δq αγ δq αδ δq βγ + D ( p ) X α,β,γ,δ ′ δq αβ δq βγ δq γδ δq δα + D ( p ) X α,β,γ,δ,µ ′ δq αβ δq αγ δq αδ q αµ + D ( p ) X α,β,γ,δ,µ ′ δq αβ δq αγ δq αδ q βµ + D ( p ) X α,β,γ,δ,µ ′ δq αβ δq αγ δq γδ q δµ (cid:21)) , (3)The order parameters and the replicon mode λ (re-sponsible for RSB stability) we find as follows: λ ( p ) ( RS ) repl = 1 − t p ( p − q ( p − RS × Z dz G Tr (cid:16) ˆ U exp ˆ θ RS (cid:17) Tr exp ˆ θ RS − " Tr ˆ U exp ˆ θ RS Tr exp ˆ θ RS ; (4) q ( p ) RS = Z dz G Tr h ˆ U exp (cid:16) ˆ θ RS (cid:17)i Tr h exp (cid:16) ˆ θ RS (cid:17)i ; (5) w ( p ) RS = Z dz G Tr h ˆ U exp (cid:16) ˆ θ RS (cid:17)i Tr h exp (cid:16) ˆ θ RS (cid:17)i ; (6)where R dz G = R ∞−∞ dz √ π exp (cid:16) − z (cid:17) , t = J/kT = t + ∆ t ; α, β label n replicas andˆ θ ( p ) RS = zt r pq RS ( p − U + t p [ w RS ( p − − q RS ( p − ]4 ˆ U . FRSB
Below we find answer of two questions: 1)if FRSB realises for given model (we derive simple cri-terium), 2) we provide necessary tools for calculation ofFRSB order parameter q ( x ).To describe the FRSB function q ( x ) of the variable x we include in the consideration the fourth-order termsin the expansion of ∆ F . We use the standard formalizedParisi algebra rules [2, 3] to write the free energy as thefunctional of q ( x ) and so to construct FRSB. The terms which are not formally described by the Parisi rules canbe reduced to the standard form as well [30, 31].The equation for the order parameter q ( x ) followsfrom the stationarity condition δδq ( x ) ∆ F ( p ) = 0.Therefore, we can is a similar way find the branchingcondition (appearance of FRSB glass solution), whichwere derived in detail for p = 2 (see [30, 31] and Refs.therein): λ ( p ) ( RS ) repl | T = 0, which produces the tem-perature T ( p ).For clarity we write δδq ( x ) ∆ F ( p ) = 0 up to the termsof the second order. We get − t p ( p − q ( p − RS d (cid:2) λ ( RS ) repl (cid:3) dt | t ∆ tq ( x ) − t L h q i + ... = 0 . (7)So if the operators ˆ U do not have the reflectivesymmetry (therefore L = 0 in that case) we needan additional branching condition, h q i ≡ R q ( x ) dx =0 + o (∆ t ) [31], insuring the appearance of non-trivialnew solutions.Integral equation, δδq ( x ) ∆ F = 0, that determines thefunction q ( x ), as usually can be simplified using the dif-ferential operator ˆ O = q ′ ddx q ′ ddx , where q ′ = dq ( x ) dx : t { B − B x } + t ( [ − D + 4 xD ] (cid:20) − xq ( x ) − Z x dyq ( y ) (cid:21) + [ − D + 2 xD ] q ( x ) ) = 0 . (8)The coefficients are given in Supplementary Material for p ≥ T. I. Schelkacheva, and N. M. Chtchelkatchev x D pa) q p p b) Fig. 2 a).The dependence of the denominator x D = (cid:2) − D + D x − D x (cid:3) | ˜ x of the function dq ( x ) dx | ˜ x onthe effective number of interacting quadrupoles p ob-tained from Eq. (9). b) The dependence of the function q p = dq ( x ) dx | ˜ x on the effective number of interactingquadrupoles p obtained from Eq. (9). Since q ( x ) can only be a non-decreasing function of x we should consider how the sign of q ′ = dq ( x ) dx dependson the parameter p . We obtain from Eq. (8): q ′ ( p ) = B + 4[ xq ( x ) − R x dyq ( y )] D + 2 D q ( x ) t − D + D x − D x ] . (9)This expression is one of the central results of our pa-per. Depending on the sign of q ′ we can conclude if thesystem falls into FRSB (positive q ′ ) state or not.If we confine ourselves to the terms of the third orderover δq in ∆ F and (8) then we obtain t { B − B x } + ... = 0. So the significantly depending on x part of q ( x )is concentrated in the neighborhood of˜ x ( p ) = B ( p ) /B ( p ) . (10)Only in the case of operators ˆ U with Tr ˆ U (2 k +1) = 0 for k = 1 , , .. we get ˜ x = B /B = 0. For Ising-like opera-tors with Tr ˆ U (2 k +1) = 0 we get p = 2 (see [26, 27, 29]), B = 0 and so ˜ x = 0 in accordance with usual standardcase of the Parisi theory [2, 3]. Example
As an illustrating example, we considerquadrupole glass in the space, J = 1. Operatorˆ U = ˆ Q is the axial quadrupole moment and it takesvalues ( − , , T pa) B / B pb) Fig. 3 a) Dependence of branching temperature T ofglass solution on the effective number of interactingquadrupoles p . b) Dependence of ˜ x ( p ) = B ( p ) /B ( p )on the effective number of interacting quadrupoles p . what one usually has in the case of Ising-like opera-tors [2, 3, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28].We have done earlier calculations of 1RSB solu-tion for various values of p that could change contin-uously [26]. It has been found that for 2 < p < p c ( p c = 2 .
5) the solution is qualitatively similar to onewhich occurs in the case of pair interaction ( p = 2).This behavior is natural for continuity reason. At hightemperatures T > T (we remind that T is the bifurca-tion temperature) there is a stable non-trivial solutionfor the RS order parameter q RS . But q RS = 0 for any fi-nite temperature T > T unlike conventional Ising-spinmodel. For T < T we have Almeida-Thoulles repli-con mode λ ( RS ) repl < T < T we have λ ( ) repl < p < .
5. The1RSB solution becomes stable when p > .
5, see Fig. 1.The instability region of 1RSB solution is an area inwhich there may be FRSB solution.Using Eqs. (9)-(10), we can describe FRSB solutionfor
T < T near T for 2 < p < p c . We also want to findthe value of p c just numerically solving the equationsEqs. (9)-(10). We intend to compare this result of p c to p c = 2 . p = 2, we earlier obtained [31] RSB in p-spin-glass-like systems. . . T = 1 .
37 and ˜ x = 0 .
43 for FRSB. Wherein q = q ( x ) is a continuous increasing function of x innarrow neighborhood, e.g., ∆ x = 0 .
016 near ˜ x for( T − T ) = − .
2. For other values of x , not in theneighborhood of ˜ x , q ( x ) is very close to q .So in our case 2 < p < ∼ = 2 . q (˜ x ) = q (˜ x ) = q (0) and [˜ xq (˜ x ) − R ˜ x dyq ( y )] = 0 in (9). We also have got that q (˜ x ) isa small quantity and so we can neglect 2 D q (˜ x ) com-pared with B . For other values of x not very close to ˜ x in exactly the same way as it was obtained for p = 2 [31]we find that q ( x ) is very close to q .The results of our calculations are presented inFigs. 2-3. At p c ∼ = 2 . dq ( x ) dx | ˜ x divergesand changes its sign when p > p c contrary to its con-ventional probabilistic interpretation. It follows thatFRSB is impossible for p > ≈ .
5. This result agreeswith p c obtained by us previously for 1RSB solutionstability border [26].For clarity we show the dependence the denomina-tor x D = (cid:2) − D + D x − D x (cid:3) | ˜ x on p in Fig. 2. Weshow in Fig. 3 the dependence of the branching temper-ature T and the dependence of ˜ x ( p ) = B ( p ) /B ( p ) onthe effective number of interacting quadrupoles p . Conclusions
For the first time it is shown that con-tinuously changing of one of the parameters of glassforming model with one type of interaction allows mov-ing from FRSB glass to stable 1RSB solution. As anexample the generalized p-spin quadrupole glass modelis considered.
Acknowledgments
This work was supported in partby the Russian Foundation for Basic Research ( No. 16-02-00295 ) while numerical simulations were funded byRussian Scientific Foundation (grant No. 14-12-01185).
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