Spin one p -spin glass: the Gardner transition
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Spin one p -spin glass: the Gardner transition. T.I. Schelkacheva
Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk142190, Moscow Region, Russia
E.E. Tareyeva
Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk142190, Moscow Region, Russia
Abstract.
We examine the phase diagram of the p -spin mean field glass modelin the spin one case, that is when S = 0 , +1 , −
1. For large p the model is solvedexactly. The analysis reveals that the phase diagram is in some way similar tothat of Ising spins. However, as we show, the quadrupolar regular ordering aswell as qudrupolar glass order are present now. The temperature of the Gardner1RSB – FRSB transition is obtained explicitly for large p . The case of higherspins is discussed briefly. pin one p -spin glass: the Gardner transition.
1. Introduction
The p -spin spin-glass model of p randomly interacting Ising spins was introduced[1, 2] as a natural generalization of the Sherrington-Kirkpatrick model [3]. In the p -spin model there exists temperature interval where the first step replica symmetrybreaking (1RSB) solution is stable. In mean field pure p -spin spherical glasses [4] thisinterval extends to zero temperature and the transition from replica symmetric (RS)to 1RSB solution is jumpwise in the glass order parameter. It is not so the case whenthe model contains terms with different values of p [5]. The papers on 1RSB spinglasses make now the basis for so called equilibrium approaches to real glasses (see forreviews [6, 7, 8]).In discontinuous mean field Ising spin p -spin glass of [1, 2] the mentioned intervalof 1RSB stability is finite and increases with increasing p . It is zero for p = 2 [3].The low-temperature boundary of the 1RSB stability region is given by the so calledGardner transition temperature [2]. Recently it was shown [9, 10, 11] that the Gardnertransition plays an important role in the equilibrium theory of structural glasses at highpressures near jamming transition. Jamming transition takes place at pressures higherthan the Gardner pressure. One can say that at high pressure it is just non-spherical p -spin glasses that are the prototypes of glassy behavior. At very low temperaturesin the case of spin glasses or at very high pressure in the case of structural glasses theslow dynamics is determined by the state landscape of FRSB [12].The detailed work by Rizzo [13] and our own experience in the investigation ofarbitrary operator p-operator spin-glass-like models [14, 15, 16, 17, 18] as well as ofPotts spin-glass models [19, 20] bring us to the conclusion that Gardner transitionis the phenomenon which is common for a large class of models, although explicitanalytical result, as far as we know, was obtained in only one case [2]. So, in thepresent paper we investigate this problem in the case of the spin one p -spin glass.The p -spin spin glass with S = 0 , ± p -spin interaction. The investigation of these two problems presents agoal of the present paper.The paper is organized as follows. In sect.2.1 the model is described and themain equations obtained by replica approach are obtained. In sect.2.2 the case ofinfinite and that of large p are investigated. Replica symmetric solution as well as thefirst step replica symmetry breaking solution are considered. The stability of 1RSBsolution is examined for large p . The low-temperature instability (Gardner transition)point is obtained explicitly in analytical way. In sect.2.3 for large p the existence ofquadrupole orientational glass is demonstrated. In sect.3 the analogous results forhigher spin models are derived and discussed.
2. Spin one p-spin glass model
Let us consider the p -spin model with the Hamiltonian H = − X i ≤ i ... ≤ i p J i ...i p ˆ S i ˆ S i ... ˆ S i p , (1) pin one p -spin glass: the Gardner transition. S now is the diagonal spin one operator ( S = 0 , +1 , − N is the number oflattice sites, i = 1 , , ...N , and p is the number of interacting particles. J i ...i p areindependent random variables with Gaussian distribution P ( J i ...i p ) = √ N p − √ p ! πJ exp (cid:20) − ( J i ...i p ) N p − p ! J (cid:21) . (2)Using replica approach we can write the free energy averaged over disorder in thefollowing form that we write here because it is instructive to compare our case withthe free energy of the random p -spin model in the case of Ising spins [1, 2]: FN kT = lim n → α n max " − t X α (cid:18) y α (cid:19) p + t X α µ α y α − t X α = β ( q αβ ) p + t X ( αβ ) q αβ λ αβ − ln Tr exp ˆ θ (3)with ˆ θ = t X ( αβ ) λ αβ ˆ S α ˆ S β + t X α µ α Q α . ˆ Q is the quadrupole-moment operator,( ˆ S α ) = 2 + ˆ Q α S = 1, y α = < ˆ Q α > is the regular quadrupole order parameter and q αβ isthe spin glass order parameter; λ, µ are Lagrange multipliers, t = J/kT .Below we will use another form which follows explicitly from our papers where thedetailed calculations for the case of interaction of p arbitrary diagonal operators ˆ U aregiven (see, e.g.[14]). Let us perform the first stage of the replica symmetry breaking(1RSB) ( n replicas are divided into n/m groups with m replicas in each group)and obtain the expression for the free energy. Glass order parameters are denoted by q αβ = q if α and β are from different groups and q αβ = q if α and β belong to thesame group. So F = − N kT (cid:26) m t ( p − q p − m )( p − t ( q ) p − t ( p − w p m Z dz G ln Z ds G h Tr exp (cid:16) ˆ θ (cid:17)i m (cid:27) . (4)Hereˆ θ = zt r pq p − S + st s p [( q ) ( p − − q p − ]2 ˆ S + t p [ w p − − ( q ) ( p − ]4 ˆ S , (5) Z dz G = Z ∞−∞ dz √ π exp (cid:18) − z (cid:19) . (6) pin one p -spin glass: the Gardner transition. F yield equations for the order parameters. We getglass order parameters q and q , the auxiliary order parameter w , the regular orderparameter x and the parameter m . Auxiliary order parameter w arises from thefact that ˆ S in Eq. (1) are not Ising spins. q = Z dz G R ds G h Tr exp ˆ θ i ( m − h Tr ˆ S exp ˆ θ iR ds G h Tr exp ˆ θ i m , (7) q = Z dz G R ds G h Tr exp ˆ θ i ( m − h Tr ˆ S exp ˆ θ i R ds G h Tr exp ˆ θ i m ,w = Z dz G R ds G h Tr exp ˆ θ i ( m − h Tr ˆ S exp ˆ θ iR ds G h Tr exp ˆ θ i m , (8) x = Z dz G R ds G h Tr exp ˆ θ i ( m − h Tr ˆ S exp ˆ θ iR ds G h Tr exp ˆ θ i m . (9)Similarly, we obtain the equation for the order parameter m .The corresponding expressions for the RS approximation can be easily obtainedfrom the preceding formulas (4)-(9) when q = q . We have q RS = Z dz G Tr h ˆ S exp (cid:16) ˆ θ RS (cid:17)i Tr h exp (cid:16) ˆ θ RS (cid:17)i , (10) w RS = Z dz G Tr h ˆ S exp (cid:16) ˆ θ RS (cid:17)i Tr h exp (cid:16) ˆ θ RS (cid:17)i , (11) x RS = Z dz G Tr h ˆ S exp (cid:16) ˆ θ RS (cid:17)i Tr h exp (cid:16) ˆ θ RS (cid:17)i , (12)Here ˆ θ RS = zt r p q RS ( p − S + t p [ w RS ( p − − q RS ( p − ]4 ˆ S . (13) p solutions. Gardner transition temperature. In the case of p → ∞ the problem can be solved exactly [21, 22]. Consideration ofsuch a limiting case makes it possible to describe many properties of the model forfinite values of p .It is easy to see that order parameters come in ˆ θ and F in the form of a powerfunction q p and w p . Herewith 0 ≤ q ≤ ≤ w ≤ p → ∞ . Let’s pretendthat 0 ≤ q < ≤ w <
1. Then q p = w p = 0 and we get directly q RS = 0, pin one p -spin glass: the Gardner transition. x RS = < ˆ S > = 0 and w RS = 2 / Q = 3 ˆ S − − , , < ˆ Q > = 3 w RS − F/ ( N kT ) = − ln 3.There is another solution in the replica symmetric consideration. When 0 ≤ q < w = 1 we have q p = 0 , w p = 1. It turns out from Eq. (10) - Eq. (13) that q RS = 0, x RS = 0 and w RS = 1. Then average values < ˆ Q > = 3 w RS − < ˆ S > = 0.The phase is not ordered in spins, but there is a quadrupole ordering.It is important to note that the contribution to ordering is given only by thestates S = +1 , −
1. The free energy is: F RS / ( N kT ) = − J / (2 kT ) − ln 2 . (14)It is identical to the result for the case of Ising spin ˆ S = (+1 , −
1) [1, 2].All these RS states as well as the transitions between them are described in detailsin [21, 22].Let us consider now the 1RSB case in the limit p → ∞ . Let us emphasize thatnow it is the paraphase with the nonzero quadrupolar ordering that bifurcates.Inaccordance with the Parisi approach [23] we have q < q . Hence we immediatelyobtain the order parameters from Eq. (7) - Eq. (9): q = 0, q = 1, x = < ˆ S > = 0and w = 1. So we have got a glass phase with a nonzero spin glass order parameter q = 1. The quadrupole ordering is preserved < ˆ Q > = 3 w − S = +1 , −
1. A value of S = 0 does not make a contribution.The expression for the free energy is given by (Eq. (4)) and has the form: F / ( N kT ) = − m J / (2 kT ) − (1 /m ) ln 2 . (15)It coincides with that which was obtained for the Ising spins [1, 2]. The expressionfor m can be obtained as the extremum condition for F / ( N kT ): m J / (2 kT ) = ln 2 . (16)When m = 1 free energies Eq. (14) and Eq. (15) become equal. From Eq. (16) wehave kT c /J = 1 / (2 √ ln 2). Since m J/ (2 kT ) is independent of temperature. F is independent of temperature too below T c . This is exactly the same results as forthe problem with Ising spins [1, 2]. This 1RSB solution was obtained in [22]. Theproblem of its stability was not considered in that paper. Let us investigate it now.We can break the replica symmetry in our model Eq. (1) once more and obtainthe corresponding expressions for the free energy and the order parameters. Thebifurcation condition λ ( = 0 determining the temperature T = T G (the Gardnertemperature) of instability follows from the condition that a nontrivial small solutionfor the 2RSB glass order parameter appears (see [15]). We have: λ ( = 1 − t p ( p − q ) ( p − × Z dz G R ds G h Tr exp (cid:16) ˆ θ (cid:17)i m ( Tr [ ˆ S exp ( ˆ θ )] Tr [ exp ( ˆ θ )] − (cid:20) Tr [ ˆ S exp ( ˆ θ )] Tr [ exp ( ˆ θ )] (cid:21) ) . R ds G h Tr exp (cid:16) ˆ θ (cid:17)i m (17)Eq. (17) depends only on 1RSB-solution. It has been shown that 1RSB solution isstable when λ ( > pin one p -spin glass: the Gardner transition. p → ∞ the 1RSB glass solution is stable, because we have(2 kT ) λ ( > T >
0. At large but finitevalues of p the condition λ ( > T G = 0.Let us calculate the Gardner transition temperature T G explicitly from therequirement λ ( = 0. In our case ˆ θ RSB = st p p andΨ( s ) ≡ Tr exp ˆ θ RSB = 1 + 2 cosh st r p . (18)We can rewrite the equation for λ in the form: λ = 1 − t p ( p − R ds G Ψ( s ) m − (3 + Ψ( s )) R ds G Ψ( s ) m (19)Calculating the transition point we keep in mind that at large p the values of T and m are small. So we obtain after deriving the asymptotics of integrals and summingthe series the equation for the limit of stability of 1RSB phase:1 = p / t √ π p (cid:18) − π √ (cid:19) , (20)so, that kT G /J = 0 . √ π p / p . (21)Let us note that the p -dependence is of the same form as in the case of Ising spins. p solution. The existence of the quadrupole glass. The main difference of Ising and spin one cases is the presence of a quadrupole orderingin the latter one. We can define the quadrupole glass (orientational) order parameterby Eq. (7) and Eq. (8) replacing ˆ S to ˆ Q and keeping in mind the zero limit of interactionof quadrupoles . The function ˆ θ is not changed at the shutdown of quadrupole-quadrupole interaction. q Q = Z dz G R ds G h Tr exp ˆ θ i ( m − h Tr ˆ Q exp ˆ θ iR ds G h Tr exp ˆ θ i m , (22) q Q = Z dz G R ds G h Tr exp ˆ θ i ( m − h Tr ˆ Q exp ˆ θ i R ds G h Tr exp ˆ θ i m . (23)In the limit of infinite p we obtain quadrupole glass order parameters q Q = 1 and q Q = 1. But we can not say that, along with the spin glass we obtain a quadrupoleglass, because ( < ˆ Q > RSB ) = q Q = q Q = 1.To clarify the question of the presence or absence of quadrupole glass let usconsider now the case of large but finite values of p . It is suitable to write the equationsfor < ˆ Q > RSB and q Q as follows: < ˆ Q > RSB = 1 − R ds G Ψ( s ) m − R ds G Ψ( s ) m pin one p -spin glass: the Gardner transition. q Q = 1 − R ds G Ψ( s ) m − R ds G Ψ( s ) m + 9 R ds G Ψ( s ) m − R ds G Ψ( s ) m and we have q Q − < ˆ Q > RSB = 9 " R ds G Ψ( s ) m − R ds G Ψ( s ) m − (cid:18) R ds G Ψ( s ) m − R ds G Ψ( s ) m (cid:19) . (24)Now we can proceed as when obtaining the Gardner temperature. At large p theintegrals R ds G Ψ( s ) η with η > ∗ p while those for η < / √ πp . So, the considered difference is q Q − < ˆ Q > RSB = 9 " Σ √ πp p − (cid:18) Σ √ πp p (cid:19) > i standing for converging sums that can be easily evaluated.This means that for large but finite p the quadrupolar orientational glass is presentalong with the spin glass in spin one system.Such a phenomenon was first encountered in generalized SK [3] model for spinone case investigated in RS approximation in the paper Ref. [24]. After performinghigh-temperature series expansion of the RS equations Eq. (10) - Eq. (13), one easilymakes sure that average value of quadrupole < ˆ Q > RS = 3 w RS − S produces anon-zero average value of the quadrupole moment, which gradually increases withdecreasing temperature to a critical temperature T c . At temperatures below T c , thesystem continues to have quadrupole ordering. At the point T c appears spin glassand quadrupole glass, too, i. e., the order parameter defined in the replica symmetricconsideration by the relation q Q RS = Z dz G Tr h ˆ Q exp (cid:16) ˆ θ RS (cid:17)i Tr h exp (cid:16) ˆ θ RS (cid:17)i , (26)ceases to be equal to [ < ˆ Q > RS ] . We define the parameters of the quadrupole glassin the limit off quadrupole interaction.Low-temperature asymptotic behavior of the order parameters may be obtainedanalytically: < ˆ Q > RS = 1 + O ( e − t ) , q S RS = 1 − √ π t − + O ( e − t ) . (27)The most interesting of these results is the fact that < ˆ Q > RS = 1 and q S RS = 1 at T = 0. This means that at zero temperature, all spins take only the values +1 and-1. The state S = 0 is absent. It may be used to determine the number of metastablestates < N S > = e . N at zero temperature. It is known result for the SK model[25]. So that our spin one glass is quite similar to the Ising spin glass at T = 0.
3. Higher spins p -spin glass Consider now the case of Hamiltonian Eq. (1) with larger spin values j = 2 , , ... . Wewill use normalized operators ˆ S = ˆ j z /j that is much more convenient for calculationsand does not change the symmetry of the problem. So when j = 1 we have pin one p -spin glass: the Gardner transition. , , −
1) as before. For j = 2 we use ˆ j z = (2 , , , − , − j = 3 we use ˆ j z = (3 , , , , − , − , −
3) and so on.As is known, the quadrupole moment is ˆ Q ∼ [3ˆ j z − j ( j +1)] in the space j = const .We will use normalized expression for quadrupole moment [3ˆ j z − j ( j + 1)] /j foruniformity of computing that does not change the symmetry of the problem.We operate on completely similar to the previous case of spin one. First ofall, we get completely disordered paraphase. We have not glass q SRS = 0. Thereis no ordering of the spins. We obtain 0 < w RS < w RS = 2 / j = 1, w RS = 1 / j = 2, w RS = 4 / j = 3, w RS = 5 /
12 for j = 4. Foraverage valule < ˆ Q j > = 3 < (ˆ j z /j ) > − ( j + 1) /j = 3 w RS − ( j + 1) /j we get < ˆ Q j =1 > = < ˆ Q j =2 > = < ˆ Q j =3 > = ... = 0. Hence, we have no quadrupoleordering. Free energy is F/ ( N kT ) = − ln(2 j + 1), so F/ ( N kT ) = − ln(3) for j = 1, F/ ( N kT ) = − ln(5) for j = 2, and so on. We obtain from q RS < θ RS = t pw RS( p − ˆ S where ˆ S = ˆ j z /j . So we get from w RS < p → ∞ . Forarbitrarily large but finite values of p such a phase takes place only at T → ∞ (or t → R ds G [Tr exp( s p p t ˆ S )] m only the termswith the largest absolute values of the normalized operator ˆ S = ˆ j z /j do contribute.We obtain two low-temperature phases: disordered phase of spin values ( q RS = 0, x RS = < ˆ j z /j > = 0, w RS = 1) and 1RSB spin glass phase ( q s = 0, q s = 1, x s = < ˆ j z /j > = 0 and w = 1). The first of these phases (no glass) for j ≥ < ˆ Q j =1 > = 1, < ˆ Q j =2 > = 3 / < ˆ Q j =3 > = 5 / q Q = q Q = [ < ˆ Q j =1 > ] = 1 for j = 1, q Q = q Q = [ < ˆ Q j =2 > ] = [3 / for j = 2,.. This is a consequenceof the fact that only the maximum values of the operator ˆ Q significantly contributeto Tr ˆ Qexp (ˆ θ ) under the integral when p → ∞ . So there is no quadrupole glassalong with the spin glass in the limit p → ∞ limit.
4. Conclusions
The phase diagram of the p -spin mean field glass model in the spin one case isexamined in details for large values of p . Some new facts, as compared with [21, 22],are established. It is shown that 1RSB phase is unstable at low temperatures andthe low-temperature boundary of stability, the Gardner temperature, is calculatedexplicitly in analytical way. An interesting feature of spin glass phase, as we show, isthe existence of quadrupole orientational glass along with the spin glass in additionto known regular quadrupole ordering. The case of higher spins is discussed, too.It is interesting that the contribution to the values of order parameters comesfrom all the values of ˆ S only at high temperatures. At low temperatures, the maincontribution is made by the maximum values of the spin operators. It is possiblethat for other operators and models of cluster glass [26, 27] some components or some pin one p -spin glass: the Gardner transition.
5. Acknowledgments
This work was supported in part by the Russian Foundation for Basic Research ( No.14-02-00451). [1] Gross D.J., Mezard M. 1984 Nuc.Phys.
B240 , 431[2] Gardner E 1985 Nuc. Phys.
B257 , 747[3] Sherrington D and Kirkpatrick S 1975 Phys. Rev. Lett. , 4384[4] Kirkpatrick T. and Thirumalai D. 1987 Phys. Rev. Lett. , 2091[5] Crisanti A., Leuzzi L. 2004 Phys. Rev. Lett. , 217203[6] Parisi G. and Zamponi F. 2010 Rev.Mod.Phys. , 789 .[7] Wolynes P.G. and Lubchenko V., Structural Glasses and Supercooled Liquids: Theory,Experiment, and Applications (Wiley, com, 2012).[8] Berthier L. and Biroli G. 2011 Rev. Mod. Phys. , 587[9] Kourchan J., Parisi G., Urbani P. and Zamponi F. 2013 J. Phys. Chem.B , 12979[10] Charbonneau P., Kim Y., Parisi G., Rainone C., Seoane B., Zamponi F. 2015 Phys. Rev. E ,012316[11] Urbani P., Biroli G. 2015 Phys. Rev. B , 032135[14] Schelkacheva T I and Chtchelkatchev N M 2011 J. Phys. A: Math. Theor. , 445004[15] Schelkacheva T I, Tareyeva E E and Chtchelkatchev N M 2010 Phys. Rev. B , 134208[16] Schelkacheva T I, Tareyeva E E and Chtchelkatchev N M 2009 Phys. Rev. E , 021105[17] Tareyeva E. E., Schelkacheva T. I. and Chtchelkatchev N. M. 2014 J. Phys. A: Math. Theor. 47,075002.[18] Schelkacheva T.I., Tareyeva E.E., and Chtchelkatchev N.M. 2014 Phys. Rev. E 89, 042149[19] Gribova N.V., Ryzhov V.N., Tareyeva E.E. 2003 Phys.Rev.E, , 067103-1[20] Gribova N V, Schelkacheva T I and Tareyeva E E 2010 J. Phys. A: Math. Theor. , 495006[21] Mottishaw P. 1986 Europhys. Lett., , 409[22] de Araujo J.M., da Costa F.A. , Nobre F.D., 2000 J. Phys. A: Math. and Gen. , 1987[23] Parisi G 1980 J. Phys. A L115[24] Luchinskaya E. A., Tareyeva E E 1992 Theor. Math. Phys, , 185[25] Tanaka F., Edwards S. F., 1980 J. Phys. F., , 2769[26] Chtchelkatchev N.M., Ryzhov V.N., Schelkacheva T.I., and Tareyeva E.E., 2004, Phys. Lett. A , 244[27] Ryzhov V.N., Tareyeva E.E., Schelkacheva T.I., and Chtchelkatchev N.M., 2004, Theor. Math.Phys.141