One step replica symmetry breaking and overlaps between two temperatures
OONE STEP REPLICA SYMMETRY BREAKING ANDOVERLAPS BETWEEN TWO TEMPERATURES
BERNARD DERRIDA , AND PETER MOTTISHAW Abstract.
We obtain an exact analytic expression for the average distribu-tion, in the thermodynamic limit, of overlaps between two copies of the samerandom energy model (REM) at different temperatures. We quantify the non-self averaging effects and provide an exact approach to the computation of thefluctuations in the distribution of overlaps in the thermodynamic limit. Weshow that the overlap probabilities satisfy recurrence relations that generaliseGhirlanda-Guerra identities to two temperatures.We also analyse the two temperature REM using the replica method. Thereplica expressions for the overlap probabilities satisfy the same recurrencerelations as the exact form. We show how a generalisation of Parisi’s replicasymmetry breaking ansatz is consistent with our replica expressions. A crucialaspect to this generalisation is that we must allow for fluctuations in the replicablock sizes even in the thermodynamic limit. This contrasts with the singletemperature case where the extremal condition leads to a fixed block size inthe thermodynamic limit. Finally, we analyse the fluctuations of the blocksizes in our generalised Parisi ansatz and show that in general they may havea negative variance. Introduction
Since replica symmetry breaking (RSB) was invented by Parisi, 40 years ago [1],it has been used in many different contexts and the subtle physical meaning of thescheme he used has been elucidated [2, 3, 4] (for reviews see [5] or [6]). Here wewould like to provide a simple example to explore how the Parisi scheme could beextended to calculate correlations between different temperatures.In the replica approach, a central role is played by the overlaps which representthe correlations between pure states. For a system of N Ising spins with the inter-actions sampled from some disorder distribution (as in the Sherrington-Kirkpatrickmodel [7], for example), the overlap between a configuration C and a configuration C (cid:48) is defined by q ( C , C (cid:48) ) = 1 N N (cid:88) i =1 σ C i σ C (cid:48) i (1)where σ C i = ± i in configuration C . The distribution P ( q ) of this overlap at a single inverse temperature β for a particular sample isthen given by [2] P ( q ) = (cid:88) C , C (cid:48) e − βE ( C ) Z ( β ) e − βE ( C (cid:48) ) Z ( β ) δ ( q − q ( C , C (cid:48) )) , (2) Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France Laboratoire de Physique de l’Ecole Normale Sup´erieure, ENS, Universit´e PSL,CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France SUPA, School of Physics and Astronomy, University of Edinburgh, Peter GuthrieTait Road, Edinburgh EH9 3FD, United Kingdom
E-mail addresses : [email protected], [email protected] . a r X i v : . [ c ond - m a t . d i s - nn ] D ec ONE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES where Z ( β ) = (cid:80) C e − βE ( C ) is the partition function at inverse temperature β and E ( C ) is the energy of configuration C for the particular sample. In a disorderedsystem the energies are quenched random variables and P ( q ) is itself a randomquantity, sample dependent in the sense that it depends on the energies E ( C ).One of the achievements of Parisi’s theory of spin glasses was to predict that P ( q )remains sample dependent even in the thermodynamic limit, and to allow the cal-culation of various averages and moments which characterize its sample to samplefluctuations [3, 4, 8, 9].The notion of overlap distribution can be generalized when the two configurationsare at different temperatures P β,β (cid:48) ( q ) = (cid:88) C , C (cid:48) e − βE ( C ) Z ( β ) e − β (cid:48) E ( C (cid:48) ) Z ( β (cid:48) ) δ ( q − q ( C , C (cid:48) )) . (3)This clearly reduces to (2) when β = β (cid:48) . These multiple temperature overlapshave mostly been studied in the context of temperature chaos, in order to see howthe random free energy landscapes are correlated at different temperatures (see forexample [10]). Several spin glass models exhibit temperature chaos, meaning thatthe overlap between different temperatures vanishes in the thermodynamic limit (inwhich case, the question of the fluctuations of P β,β (cid:48) ( q ) becomes superfluous). Oneway to predict temperature chaos is to show that P β,β (cid:48) ( q ) vanishes exponentiallywith the system size when β (cid:54) = β (cid:48) and q > E ( C ) are 2 N independent random variables distributedaccording to a Gaussian distribution of width proportional to N . The overlap canthen only take two values q ( C , C (cid:48) ) = δ C , C (cid:48) . Therefore P ( q ) consists of two delta function peaks [5, 17], P ( q ) = (1 − Y ) δ ( q ) + Y δ ( q −
1) (4)where Y is the probability, at equilibrium, of finding two copies of the same samplein the same configuration. Y = (cid:88) C (cid:18) e − βE ( C ) (cid:80) C e − βE ( C ) (cid:19) = Z (2 β ) Z ( β ) . (5)In the large N limit, Y vanishes in the high temperature phase ( β < β c ), while inthe low temperature phase ( β > β c ) it takes non zero values with sample to samplefluctuations.A direct calculation [8, 9] as well as a replica calculation [4] lead to (cid:104) Y (cid:105) = 1 − µ ; (cid:104) Y (cid:105) − (cid:104) Y (cid:105) = µ − µ (cid:104) Y (cid:105) − (cid:104) Y (cid:105) NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 3 where (cid:104) . (cid:105) denotes the disorder average i.e. the average over the random energies E ( C ) and µ = β c β . (7)The quantity Y can be generalized to the probabilities Y k of finding k copies ofthe same sample in the same configuration Y k = (cid:88) C (cid:18) e − βE ( C ) (cid:80) C e − βE ( C ) (cid:19) k = Z ( kβ ) Z ( β ) k . (8)As for Y , the large N limits of the disorder averages of these overlaps are known[3, 4, 18, 19] (cid:104) Y k (cid:105) = Γ( k − µ )Γ(1 − µ ) Γ( k ) . (9)Since µ = 1 − (cid:104) Y (cid:105) , equation (9) implies( kβ − β c ) (cid:104) Y k (cid:105) = kβ (cid:104) Y k +1 (cid:105) (10)which can be seen as simple cases of the Ghirlanda-Guerra identities [20, 21, 22, 23].At two different temperatures, the overlap distribution (3) for the REM is stilla sum of two delta functions P β,β (cid:48) ( q ) = (1 − Y , ) δ ( q ) + Y , δ ( q −
1) (11)where the random variable Y k,k (cid:48) = (cid:88) C (cid:18) e − βE ( C ) Z ( β ) (cid:19) k (cid:32) e − β (cid:48) E ( C ) Z ( β (cid:48) ) (cid:33) k (cid:48) = Z ( kβ + k (cid:48) β (cid:48) ) Z ( β ) k Z ( β (cid:48) ) k (cid:48) (12)is the probability that k copies of a given sample at temperature β and k (cid:48) attemperature β (cid:48) are all in the same configuration.In section 2 and in the Appendix A we will derive the following exact expressionsof the sample averages of these generalized overlaps (cid:104) Y k,k (cid:48) (cid:105) = ββ c k ) 1Γ( k (cid:48) ) (cid:90) ∞ dv v k (cid:48) − Ψ (cid:16) k + k (cid:48) β (cid:48) β ; v (cid:17)(cid:0) − ψ ( v ) (cid:1) (13)where the functions ψ ( v ) and Ψ( v ) are given by ψ ( v ) = (cid:90) ∞ du ( e − u − vu β (cid:48) β − u − − βcβ (14)and Ψ( z ; v ) = (cid:90) ∞ du e − u − vu β (cid:48) β u z − − βcβ (15)These two functions are generalisations of the Gamma function. Like the Gammafunction which satisfies Γ( z +1) = z Γ( z ), they obey some recursion relations (whichcan be obtained from (15) via integrations by parts) leading to the following rela-tions ( kβ + k (cid:48) β (cid:48) − β c ) (cid:104) Y k,k (cid:48) (cid:105) = kβ (cid:104) Y k +1 ,k (cid:48) (cid:105) + k (cid:48) β (cid:48) (cid:104) Y k,k (cid:48) +1 (cid:105) (16)which generalize (10).We will also show that (cid:104) (cid:0) Y , (cid:1) (cid:105) = ββ c (cid:90) ∞ dv v Ψ (cid:16) β (cid:48) β ; v (cid:17)(cid:0) − ψ ( v ) (cid:1) + Ψ (cid:16) β (cid:48) β ; v (cid:17) (cid:0) − ψ ( v ) (cid:1) (17)By varying β one can draw, using the exact expressions (13,17), the variance (cid:104) (cid:0) Y , (cid:1) (cid:105) − (cid:104) Y , (cid:105) versus the average (cid:104) Y , (cid:105) as in Figure 1. Clearly the relation (6) ONE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES (which is a direct consequence of Parisi’s ansatz) is no longer satisfied when thetwo temperatures are different ( β (cid:54) = β (cid:48) ). . . . . . . . . . . . h Y , i h ( Y , ) i − h Y , i Variance vs Mean of Y , β = β simulation β = 2 β simulation β = β theory β = 2 β theory Figure 1.
The variance of Y , versus its average when β = β (cid:48) and β = 2 β (cid:48) . The curves are obtained by varying β (cid:48) between β c and ∞ .The lines represent the expression (6) in the case β = β (cid:48) and (13) when β = 2 β (cid:48) . The points are the results of Monte Carlo simulations in thesetwo cases. In the rest of the paper we will show in section 2 how the expressions (13,14,15,17)can be derived directly. We will also give the generalisation of these expressionswhen the overlaps are weighted by the partition functions to some power (in thereplica language when the number of replicas is non-zero). Then in section 3 wewill show what needs to be done in order to calculate these overlaps in a replicaapproach and in section 4 we will propose a scheme which generalises Parisi’s ansatzand is compatible with our exact results. Finally, in section 5 we will explore thenature of the fluctuations in block size that we observe in this generalisation ofParisi’s ansatz to two temperatures.2.
The direct calculation of the overlaps
In this section, after recalling the definition of the REM, we explain how theexpressions of the overlaps such as (13,14,15) can be derived. In the REM, a sampleis determined by the choice of 2 N random energies E ( C ) chosen independently froma Gaussian distribution P ( E ) = 1 √ N πJ exp (cid:20) − E N J (cid:21) (18)It is known that, in the thermodynamic limit ( N → ∞ ), there is a phase transition[13, 14] at an inverse temperature β c = 2 √ log 2 J (19)and that in the frozen phase β > β c (which is the only phase with non-zero overlaps)the partition function is dominated by energies close to the ground state whichitself has fluctuations of order 1 around a characteristic energy E = − JN √ log 2 + NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 5 J log N/ (4 √ log 2). As only energies at a distance of order 1 (i.e (cid:28) N ) contribute tothe partition function one can replace the REM by a Poisson REM [24] which has,in the frozen phase and for large system sizes, the same properties as the originalREM [13]. In this Poisson REM (PREM), the values of the energies, for a givensample, are the points generated by a Poisson process [18] on the real line withintensity ρ ( E ) = C exp[ β c ( E − E )] with C = 1 J √ π . (20)One way to think of it is to slice the real axis into infinitesimal energy intervals( E, E + dE ) indexed by ν , and to say that there is an energy E ν in the interval ν with probability p ν = ρ ( E ) dE . In other words the partition function is given by Z ( β ) = ∞ (cid:88) ν = −∞ y ν exp[ − βE ν ] (21)where the y ν are independent binary random variables such that y ν = 1 with prob-ability p ν and y ν = 0 with probability 1 − p ν (because the intervals are infinitesimal,there is no interval ν occupied by more than one energy).The details of the calculation leading to the expressions (13,14,15) are given inAppendix A.One can generalize (13) to obtain (see (76) of Appendix A) the average overdisorder of Y k ,k ; k (cid:48) ,k (cid:48) defined by Y k ,k ; k (cid:48) ,k (cid:48) = (cid:80) ν (cid:54) = ν (cid:48) y ν y ν (cid:48) e − ( βk + β (cid:48) k (cid:48) ) E ν − ( βk + β (cid:48) k (cid:48) ) E ν (cid:48) Z ( β ) k + k Z ( β (cid:48) ) k (cid:48) + k (cid:48) In particular this allows one to obtain (17) from (76) as one has from the definition(12) ( Y , ) = Y , + Y , , . In the replica approach, as we will see, it is often convenient to deal with weightedoverlaps defined as (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = (cid:68) Z ( kβ + k (cid:48) β (cid:48) ) Z ( β ) n − k Z ( β (cid:48) ) n (cid:48) − k (cid:48) (cid:69) (cid:104) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:105) (22)(see (12)). These averages can be performed (see Appendix A) to get expressions(83,82) (valid for n < n (cid:48) <
0) which generalize (13) and (16) (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = ( − r ) Γ( − n ) Γ( − n (cid:48) )Γ( k − n ) Γ( k (cid:48) − n (cid:48) ) (cid:82) ∞ dv v k (cid:48) − n (cid:48) − Ψ (cid:16) k + k (cid:48) β (cid:48) β ; v (cid:17) ( − ψ ( v )) r − (cid:82) ∞ dv v − n (cid:48) − ( − ψ ( v )) r (23)where r = nβ + n (cid:48) β (cid:48) β c (24)and( kβ + k (cid:48) β (cid:48) − β c ) (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = ( k − n ) β (cid:104) Y k +1 ,k (cid:48) (cid:105) n,n (cid:48) + ( k (cid:48) − n (cid:48) ) β (cid:48) (cid:104) Y k,k (cid:48) +1 (cid:105) n,n (cid:48) . (25) Remark . The n, n (cid:48) → limit As explained in Appendix A (see (85)) the expression (23) reduces to (13) in thelimit n → − and n (cid:48) → − . Very much like the integral representation of theGamma function the expression (23) would take a different form for n and/or n (cid:48) >
0, and so we would need to use these alternative expressions to verify that thelimits n → + and n (cid:48) → + lead also to (13). ONE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES
Remark . The β (cid:48) = β case In the particular case where β = β (cid:48) , one can perform the integrals in (14,15) as in(79,80) and obtain more explicit expressions of the overlaps as in (81). In particularone gets (see (81)) that (cid:104) Y k,k (cid:48) (cid:105) = 1Γ( k + k (cid:48) ) Γ( k + k (cid:48) − β c β )Γ(1 − β c β ) (26)which agrees with (9) as Y k,k (cid:48) = Y k + k (cid:48) when β = β (cid:48) (see the definitions (9,12)).Similarly (23) becomes when β = β (cid:48) (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = Γ(1 − n − n (cid:48) )Γ( k + k (cid:48) − n − n (cid:48) ) Γ( k + k (cid:48) − β c β )Γ(1 − β c β ) (27)As in this single temperature case Y k,k (cid:48) = Y k + k (cid:48) , (16) reduces to equation (10).Therefore once (cid:104) Y (cid:105) is known, all the other (cid:104) Y k (cid:105) can be determined by the relations(10). Clearly for β (cid:48) (cid:54) = β this is not the case. However, (16) or (25) would allow todetermine all the (cid:104) Y k,k (cid:48) (cid:105) from the knowledge of all the (cid:104) Y k, (cid:105) .In the rest of the paper we will see how expressions (13) or (23) can be interpretedin terms of replica symmetry breaking.3. The replica method
In this section we apply the replica method to the REM. To illustrate the ap-proach we first recall the computation of the free energy and overlap probability (cid:10) Y k (cid:11) n for a single temperature. It is well known that a single step in Parisi’s replicasymmetry breaking scheme [1] gives the correct low temperature solution [14, 17].Here we will start with a slightly more general approach that allows for fluctuationsin the block sizes. In the single temperature case the need to allow fluctuations inthe block sizes has been discussed in [25, 26] and used in [24, 27] to compute finitesize corrections in the REM.In the two temperature case block size fluctuations have also been discussed inthe context of temperature chaos in spin glasses (see [28] and [29] Appendix Gfor a detailed discussion), but computing the full overlap distribution between twotemperatures by averaging over these block fluctuations has proved challenging. Inthis section we outline a replica symmetry breaking scheme for the two temperaturecase and show that it satisfies the same recursion (16) as the exact solution.3.1. The REM at a single temperature.
To implement the replica method, thefirst step is to calculate the integer moments of the partition function, (cid:10) Z ( β ) n (cid:11) ,then one assumes that the expression is valid for non-integer n and finally onemakes use of (cid:10) log Z (cid:11) = lim n → log (cid:10) Z ( β ) n (cid:11) n (28)to obtain the disorder average of the free energy.3.1.1. Integer moments of the partition function.
As shown in Appendix B theinteger moments of the partition function for the REM are given by (cid:10) Z ( β ) n (cid:11) = (cid:88) r ≥ r ! (cid:88) µ ≥ · · · (cid:88) µ r ≥ C n,r ( { µ i } ) (cid:10) Z ( βµ ) (cid:11) (cid:10) Z ( βµ ) (cid:11) · · · (cid:10) Z ( βµ r ) (cid:11) (29)(see (91) in appendix B) where C n,r ( { µ i } ) = n ! µ ! µ ! · · · µ r ! δ (cid:34) r (cid:88) i =1 µ i = n (cid:35) (30) NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 7 and the Kronecker delta δ [ (cid:80) ri =1 µ i = n ] ensures that the { µ i } sum to n . As (cid:10) Z ( β ) (cid:11) = (cid:90) ∞−∞ ρ ( E ) e − βE dE = e N f ( β ) (31)with f ( β ) = log 2 + ( βJ ) . (32)one can rewrite (29) as (cid:104) Z ( β ) n (cid:105) = (cid:88) r ≥ r ! (cid:88) µ ≥ · · · (cid:88) µ r ≥ C n,r ( { µ i } ) e NA ( r, { µ i } ) (33)where A ( r, { µ i } ) = r (cid:88) i =1 f ( µ i β ) = r log 2 + ( βJ ) r (cid:88) i =1 µ i . (34)One can interpret a given term of the sum (29) or (33) as n replicas distributedover r distinct configurations with µ i replicas in configuration i . Before using thereplica method to compute the free energy from these expressions we compute theoverlap probability for integer n .An expression for the overlap probability Y k , defined in (8), can be obtainedfrom the ratio of integer moments (cid:104) Y k (cid:105) n = (cid:104) Y k Z ( β ) n (cid:105)(cid:104) Z ( β ) n (cid:105) = (cid:10) Z ( kβ ) Z ( β ) n − k (cid:11) (cid:104) Z ( β ) n (cid:105) . (35)The denominator in the rightmost expression is the single temperature moment in(33). If n and k are positive integers with n > k then the numerator is a twotemperature moment of the partition function that is computed (see (96,97) inAppendix B) and one gets (cid:10) Y k (cid:11) n = (cid:28) r µ ( µ − · · · ( µ − k + 1) n ( n − · · · ( n − k + 1) (cid:29) { µ i } (36)where the average (cid:104) . (cid:105) { µ i } means that for any function F ( { µ i } ), (cid:104) F ( { µ i } ) (cid:105) { µ i } = (cid:88) r ≥ (cid:88) { µ i ≥ } F ( { µ i } ) W r ( { µ i } ) (cid:88) r ≥ (cid:88) { µ i ≥ } W r ( { µ i } ) . (37)with (see (30,33)) W r ( { µ i } ) = C n,r ( { µ i } ) r ! e NA ( r, { µ i } ) . The thermodynamic limit and the extremal condition.
In the thermodynamiclimit (cid:104) Z ( β ) n (cid:105) in equation (33) should be dominated by terms which maximize A ( r, { µ i } ) in equation (34). At high temperatures the maximum corresponds to all n replicas being in different configurations. Thus r = n , µ i = 1 for all i . Then (cid:104) Z ( β ) n (cid:105) (cid:39) e N n f ( β ) (38)which gives (see (28)) (cid:104) log Z (cid:105) = N f ( β ) = N (cid:20) log 2 + ( βJ ) (cid:21) . (39)The entropy of this solution is (cid:10) S (cid:11) = N [ f ( β ) − βf (cid:48) ( β )] = N (cid:20) log 2 − ( βJ ) (cid:21) . (40) ONE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES
There is a critical inverse temperature β c where this entropy vanishes. It is thesolution of f ( β c ) − β c f (cid:48) ( β c ) = 0 (41)and therefore given by (19). When β < β c the entropy in (40) is positive and (39)is indeed the right free energy [13]. On the other hand at low temperatures, when β > β c , the entropy is negative and one must look for a different solution.To do so we proceed as Parisi did in his original papers [1, 2, 30, 31] on replicasymmetry breaking. To identify the terms that dominate the sum in (33) in thethermodynamic limit ( N → ∞ ) in the low temperature phase we make the followingthree assumptions:(1) We expect all the dominant terms to have a large N behaviour of the formexp N A ( r, { µ i } ) with the same value of A and the same value of r .(2) The dominant terms in the n → A ( r, { µ i } ) and not the maximum. This seems an unreasonable assumption,but gives the correct result when replica symmetry is broken. One argu-ment to support this assumption is that when the number of independentparameters we are maximising over is negative the maximum becomes aminimum [5]. In (34) there are r − µ i (due to theconstraint (cid:80) ri =1 µ i = n ) and, as we will see in (44) below, r − n < β c β .(3) We allow n, r, µ i to become real parameters when we compute the minimumof A ( r, { µ i } ).As for Parisi’s original ansatz, it is clear that these assumptions, as such, have norigorous justification. However, the free energy obtained using these assumptionshas been verified for a number of spin glass models by a rigorous mathematicalanalysis (for reviews see [32, 33]). They also lead to the correct free energy ofthe REM in the low temperature phase [14]. In addition, there is a representationof the free energy as a contour integral in the complex plane [24, 27], where theminimum of A ( r, { µ i } ) and the non-integer values of r and the µ i appear naturallyas a consequence of taking the saddle point along the contour.The minimum of A ( r, { µ i } ) in (34) with respect to the { µ i } subject to the con-straint (cid:80) ri =1 µ i = n can be found using a Lagrange multiplier and it correspondsto all µ i taking the same value. The constraint then gives immediately µ i = nr . (42)for all i . Then (34) gives A ( r, { µ i } ) = rf ( nβr ) and taking the extremal value withrespect to r one gets f (cid:18) βnr (cid:19) − βnr f (cid:48) (cid:18) βnr (cid:19) = 0 . (43)Comparison with (41) gives r = n ββ c , so that µ = β c β (44)so that for large N we can approximate (29) as (cid:10) Z ( β ) n (cid:11) ∼ exp (cid:26) N n (cid:20) µ log 2 + β µ (cid:21)(cid:27) = exp (cid:26) N n β β c (cid:27) , (45)where we have defined µ = β c β . The free energy (28) in the frozen phase is therefore (cid:104) log Z (cid:105) = N β β c which is known to be the correct expression [13] NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 9
The extremal condition (42) tells us that µ i = µ and r = nµ for the dominantterms. So the µ i do not fluctuate and we can immediately write (cid:104) Y k (cid:105) n = r µ ( µ − · · · ( µ − k + 1) n ( n − · · · ( n − k + 1) = Γ( k − µ ) Γ(1 − n )Γ(1 − µ ) Γ( k − n ) (46)which in the n → (cid:104) Y (cid:105) =1 − µ . We are now going to see that in the two temperature case this simple laststep is not possible because fluctuations of µ i remain even in the thermodynamiclimit.3.2. The REM at two temperatures.
Integer moments of the partition function.
In the two temperature case ourstarting point is the following expression for the moments (see Appendix B) (cid:68) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:69) = (cid:88) r ≥ r ! (cid:88) { µ i ≥ } (cid:88) { µ (cid:48) i ≥ } δ [ µ i + µ (cid:48) i ≥ × C n,r ( { µ i } ) C n (cid:48) ,r (cid:0) { µ (cid:48) j } (cid:1) e NA ( r, { µ i ,µ (cid:48) i } ) (47)where the sum on { µ i } , { µ (cid:48) i } is over all non-negative integers, the C n,r ( { µ i } ) and C n (cid:48) ,r ( { µ (cid:48) i } ) are combinatorial factors defined in (30) and A (cid:0) r, { µ i , µ (cid:48) i } (cid:1) = r (cid:88) i =1 f ( µ i β + µ (cid:48) i β (cid:48) ) = r log 2 + J r (cid:88) i =1 ( µ i β + µ (cid:48) i β (cid:48) ) . (48)As in (33) each term in the sum (47) corresponds to a different grouping of the n + n (cid:48) replicas: in configuration i there are µ i replicas at inverse temperature β and µ (cid:48) i replicas at inverse temperature β (cid:48) . There is an additional constraint associatedwith each configuration i that µ i + µ (cid:48) i ≥
1; in other words we need at least onereplica, which can be from either n or n (cid:48) , in each configuration.We are interested in how the single temperature overlap calculation leadingto (46) generalises to the two temperature case. We start with the weighted formof Y k,k (cid:48) defined in (22) (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = (cid:68) Y k,k (cid:48) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:69) (cid:104) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:105) = (cid:68) Z ( βk + β (cid:48) k (cid:48) ) Z ( β ) n − k Z ( β (cid:48) ) n (cid:48) − k (cid:48) (cid:69) (cid:104) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:105) (49)The denominator in the rightmost expression is the two temperature moment in(47). The numerator is a three temperature moment of the partition function (95)computed in appendix B. By a direct generalisation of the derivation (96,97) of (36)one gets (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = (cid:28) r µ ( µ − · · · ( µ − k + 1) n ( n − · · · ( n − k + 1) µ (cid:48) ( µ (cid:48) − · · · ( µ (cid:48) − k (cid:48) + 1) n (cid:48) ( n (cid:48) − · · · ( n (cid:48) − k (cid:48) + 1) (cid:29) { µ i ,µ (cid:48) i } (50)where the average (cid:104) . (cid:105) { µ i ,µ (cid:48) i } means that for any function F ( { µ i , µ (cid:48) i } ), (cid:104) F ( { µ i , µ (cid:48) i } ) (cid:105) { µ i ,µ (cid:48) i } = (cid:88) r ≥ (cid:88) { µ i ≥ } (cid:88) { µ (cid:48) i ≥ } F ( { µ i , µ (cid:48) i } ) W r ( { µ i , µ (cid:48) i } ) (cid:88) r ≥ (cid:88) { µ i ≥ } (cid:88) { µ (cid:48) i ≥ } W r ( { µ i , µ (cid:48) i } ) (51)with (see (30,48)) W r ( { µ i , µ (cid:48) i } ) = C n,r ( { µ i } ) C n (cid:48) ,r ( { µ (cid:48) i } ) r ! e NA ( r, { µ i ,µ (cid:48) i } ) r (cid:89) i =1 θ [ µ i + µ (cid:48) i ≥ . Here θ [ µ i + µ (cid:48) i ≥
1] is one if µ i + µ (cid:48) i ≥ The thermodynamic limit and the extremal condition.
Let us focus on thelow temperature phase and take β > β c and β (cid:48) > β c when replica symmetry isbroken. We proceed as we did in the single temperature case, by making a similarset of three assumptions on how to take the thermodynamic limit as in the singletemperature case. As before we look for the minimum of A ( r, { µ i , µ (cid:48) i } ) in (48) andthe only difference is that we now have the additional parameters n (cid:48) and { µ (cid:48) i } .Using Lagrange multipliers we find that the minimum corresponds to βµ i + β (cid:48) µ (cid:48) i being independent of i . Summing on i and using the constraints (cid:80) ri =1 µ i = n and (cid:80) ri =1 µ (cid:48) i = n (cid:48) we find that βµ i + β (cid:48) µ (cid:48) i = βn + β (cid:48) n (cid:48) r for all i. (52)The value of r that gives the minimum of A ( r, { µ i , µ (cid:48) i } ) = rf ( nβ + n (cid:48) β (cid:48) r ) is then givenby f (cid:18) βn + β (cid:48) n (cid:48) r (cid:19) − βn + β (cid:48) n (cid:48) r f (cid:48) (cid:18) βn + β (cid:48) n (cid:48) r (cid:19) = 0 . (53)so that from (41) r = βn + β (cid:48) n (cid:48) β c . (54)Together with equation (52) this gives βµ i + β (cid:48) µ (cid:48) i = β c , (55)which constrains the fluctuations of µ i and µ (cid:48) i , but, unlike the single temperaturecase, (55) does not eliminate them completely.One can however, without any further assumption, recover (25) from (50). Usingthe replica form (50) one can see that( k − n ) β (cid:104) Y k +1 ,k (cid:48) (cid:105) n,n (cid:48) + ( k (cid:48) − n (cid:48) ) β (cid:48) (cid:104) Y k,k (cid:48) +1 (cid:105) n,n (cid:48) = (cid:68) ( βk + β (cid:48) k (cid:48) − βµ − β (cid:48) µ (cid:48) ) × r µ ( µ − · · · ( µ − k + 1) n ( n − · · · ( n − k + 1) µ (cid:48) ( µ (cid:48) − · · · ( µ (cid:48) − k (cid:48) + 1) n (cid:48) ( n (cid:48) − · · · ( n (cid:48) − k (cid:48) + 1) (cid:29) µ i ,µ (cid:48) i . (56)If we take the large N limit of the right hand side we expect (see(55)) that theextremal condition βµ + β (cid:48) µ (cid:48) = β c should apply. Then (56) simplifies to give( k − n ) β (cid:104) Y k +1 ,k (cid:48) (cid:105) n,n (cid:48) + ( k (cid:48) − n (cid:48) ) β (cid:48) (cid:104) Y k,k (cid:48) +1 (cid:105) n,n (cid:48) = ( kβ + k (cid:48) β (cid:48) − β c ) (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) (57)the same recursion relation (25) as in the direct calculation. This gives at leastsome confidence in the assumptions that we have made in developing the replicaapproach so far for the two temperature problem.Ideally we would like to go a step further and recover the exact solution (23)directly from (50) using the replica approach. The challenge is to find a way tocompute the average over { µ i , µ (cid:48) i } in (50) subject to the constraints (54) and (55)that is valid when n, n (cid:48) are no-longer integers. We have not found an approachthat is sufficiently convincing to merit inclusion here. The problem essentially hasto do with the r and the { µ i , µ (cid:48) i } becoming non-integer. We will see however insection 5 that by matching with the exact expressions of section 2 one can obtainthe generating function of the { µ i , µ (cid:48) i } . NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 11 Parisi overlap matrices
For the REM the replica method in section 3 could be implemented withoutexplicitly using the replica overlap matrices. However, for more complex problemssuch as the Sherrington-Kirkpatrick model [7] the saddle point equations are ex-pressed in terms of replica overlap matrices and so it is useful to see what theylook like in the case of the REM. In this section we describe the structure of thesematrices in the single and two temperature case of the REM. The Parisi ansatz isused in the single temperature case and we show how it can be generalised to twotemperatures in the case of the REM.We could have approached this by applying the replica method to the large p limitof the p-spin models introduced in [14] as was done in [17] for the single temperaturecase. However, the corresponding two temperature calculation is rather long andis not essential to understanding how to generalise the Parisi ansatz.4.1. Single temperature case.
In the single temperature case (cid:104) Z ( β ) n (cid:105) is ex-pressed in equation (33) as a sum over the parameters r and µ , µ , . . . µ r with theconstraint (cid:80) ri =1 µ i = n . We can use these parameters to define an n × n replicaoverlap matrix Q ( { µ i } ). We divide the n replicas into r groups of sizes µ , µ , . . . µ r .The overlap matrix is then defined as Q a,b ( { µ i } ) = (cid:40) , if replicas a, b are in the same group ,0 , otherwise. (58)This means that, up to a permutation of the replica indices, the n × n matrix Q ( { µ i } ) consists of r blocks of size µ × µ , µ × µ , . . . , µ r × µ r , along the diag-onal where the matrix elements take value unity and they are zero elsewhere. Forexample if n = 6 , r = 3 , µ = 2 , µ = 3 , µ = 1 Q ( { µ i } ) =
00 0
00 0
00 0 0 0 0 . (59)(Here we have taken Q a,a = 1 for simplicity). In terms of this overlap matrix, onecan rewrite (34) as A ( r, { µ i } ) = r log 2 + ( βJ ) n (cid:88) a =1 n (cid:88) b =1 Q a,b ( { µ i } ) . (60)where we have used that (cid:80) na =1 (cid:80) nb =1 Q a,b ( { µ i } ) = (cid:80) ri =1 µ i .The thermodynamic limit gives us the extremal condition (42) which indicatesthat the dominant form of Q ( { µ i } ) has all the µ i equal. This fixed block structuregives the one step RSB form of the overlap matrices introduced by Parisi [1] tosolve mean field spin glass models such as the Sherrington-Kirkpatrick model [7].It should be noted that any permuation of the replica indices will also give a Q ( { µ i } )that satisfies the extremal condition and we should sum over all these saddle pointswhen computing physical properties such as P ( q ) (see [34]).4.2. Two temperature case.
In the two temperature case (cid:68) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:69) isexpressed in equation (47) as a sum over the parameters r ; µ , µ , . . . µ r and µ (cid:48) , µ (cid:48) , . . . µ (cid:48) r with the constraints (cid:80) ri =1 µ i = n and (cid:80) ri =1 µ (cid:48) i = n (cid:48) . We can usethese parameters to define three different replica overlap matrices. We divide the n replicas into r groups of size µ , µ , . . . µ r and the n (cid:48) replicas into r groups of size µ (cid:48) , µ (cid:48) , . . . µ (cid:48) r . We then have the single temperature n × n replica overlap matrix Q ( { µ i } ) defined in equation (58) and the equivalent n (cid:48) × n (cid:48) matrix Q (cid:48) ( { µ (cid:48) i } ) atinverse temperature β (cid:48) . We can also define an n × n (cid:48) overlap matrix R ( { µ i , µ (cid:48) i } )between the inverse temperature β and the inverse temperature β (cid:48) by R a,b (cid:48) ( { µ i , µ (cid:48) i } ) = (cid:40) , if replicas a, b (cid:48) are in the same group;0 , otherwise. (61)So all the matrix elements of the rectangular matrix R ( { µ i , µ (cid:48) i } ) are zero except r blocks of sizes µ × µ (cid:48) , µ × µ (cid:48) , . . . , µ r × µ (cid:48) r , along the diagonal where they takevalue unity. It has the property that (cid:80) na =1 (cid:80) n (cid:48) b (cid:48) =1 R a,b (cid:48) ( { µ i , µ (cid:48) i } ) = (cid:80) ri =1 µ i µ (cid:48) i sothat we can write (48) as A (cid:0) r, { µ i , µ (cid:48) i } (cid:1) = r log 2 + ( βJ ) n (cid:88) a =1 n (cid:88) b =1 Q a,b ( { µ i } )+ ββ (cid:48) J n (cid:88) a =1 n (cid:48) (cid:88) b (cid:48) =1 R a,b (cid:48) ( { µ i , µ (cid:48) i } ) + ( β (cid:48) J ) n (cid:48) (cid:88) a (cid:48) =1 n (cid:48) (cid:88) b (cid:48) =1 Q (cid:48) a (cid:48) ,b (cid:48) ( { µ (cid:48) i } ) . (62)As an example of the overall matrix, for n = 6 , n (cid:48) = 9 , r = 3 , µ = 2 , µ =3 , µ = 1 , µ (cid:48) = 4 , µ (cid:48) = 1 , µ (cid:48) = 3, one has (cid:18) Q RR T Q (cid:48) (cid:19) = (63)This type of two temperature order parameter has already been discussed in thecontext of spin models in a number of works on temperature chaos (see [10] for areview).4.3. When the numbers n and n (cid:48) of replicas become non integer. In thethermodynamic limit, in the case of a single temperature, we have seen in section3.1.2 that the number r of blocks is fixed and that all the µ i are equal to the value µ = β c β (see (44)). Therefore the matrix Q in (59) takes precisely the form firstproposed by Parisi [1] with blocks of equal sizes along the diagonal.In the case of two temperatures, we have seen in section 3.2.2 that the number r of blocks is still fixed (see (54)) and that there is a constraint βµ i + β (cid:48) µ (cid:48) i = β c (see(55)) for each pair µ i , µ (cid:48) i . The simplest assumption would be to take µ i = µ and µ (cid:48) i = µ (cid:48) independent of i . This choice is not consistent with the exact expressions(13) of (cid:104) Y k,k (cid:48) (cid:105) presented in section 2 and therefore µ i and µ (cid:48) i fluctuate subject tothe constraint (55).However, to obtain the exact results (13) or (23), using the replica method wemust sum over all the saddle points that satisfy the constraints (54) and (55). Inthe single temperature case this was fairly straightforward (see [34]) because all the NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 13 saddle points are related by a simple permutation of the replica indices. In the twotemperature case this is no longer true, as discussed in detail in [12], and it is notclear how to sum over the saddle points.5.
The fluctuations in block sizes µ i , µ (cid:48) i In this section we analyse the fluctuations of the block sizes µ i , µ (cid:48) i in the ther-modynamic limit. We first obtain the mean and the variance of the µ i and µ (cid:48) i . Wewill then obtain the moment generating function for the distribution of these blocksizes from the exact expression (23) for (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) . One outcome of our results isthat the µ i and the µ (cid:48) i do fluctuate even in the equal temperature case. Howeverextracting the distribution of P ( µ i , µ (cid:48) i ) from this generating function is not an easytask and can be interpreted as a signed measure, i.e. a measure with negative prob-abilities. Also not all properties of the distribution of these block sizes have physicalimplications: for example, we will see that in the limit n → − and n (cid:48) → − thesedistributions depend on the ratio n (cid:48) /n although all physical properties have a limitindependent of this ratio.In this section n, n (cid:48) are negative real numbers because our analysis is based onthe exact expression (23) which is only valid for this range of values.5.1. The first moments of µ i . As Y , = 1 (see (12)) and as r does not fluctuate(see (54)) one can show from (50) that (cid:104) µ i (cid:105) { µ i ,µ (cid:48) i } = nβ c nβ + n (cid:48) β (cid:48) (cid:104) µ i (cid:105) { µ i ,µ (cid:48) i } = nβ c (1 − (1 − n ) (cid:104) Y , (cid:105) n,n (cid:48) ) nβ + n (cid:48) β (cid:48) = nβ c ( β c − n (cid:48) β (cid:48) (cid:104) Y , (cid:105) n,n (cid:48) ) β ( nβ + n (cid:48) β (cid:48) ) (64)where we have used the relation between (cid:104) Y , (cid:105) n,n (cid:48) and (cid:104) Y , (cid:105) n,n (cid:48) (cid:104) Y , (cid:105) n,n (cid:48) = β − β c + n (cid:48) β (cid:48) (cid:104) Y , (cid:105) n,n (cid:48) (1 − n ) β which follows from (25) and the fact that Y , = 1.From these expressions (64) one can notice first that the limit of the first moment (cid:104) µ i (cid:105) , when n → − and n (cid:48) → − , depends on the ratio n (cid:48) /n . This means that notall the properties of the µ i have a physical meaning, since one expects all physicalproperties to be independent of this ratio when n and n (cid:48) vanish.One can also notice that the variance of µ i is in general non-zero. Depending on n, n (cid:48) , β, β (cid:48) , this variance may change its sign, implying that the distribution of µ i is not really a probability distribution. For example when β = β (cid:48) , one has (27) (cid:104) Y , (cid:105) n,n (cid:48) = (cid:104) Y , (cid:105) n,n (cid:48) = 1 − β c β − n − n (cid:48) and (cid:104) µ i (cid:105) { µ i ,µ (cid:48) i } − (cid:104) µ i (cid:105) { µ i ,µ (cid:48) i } = nn (cid:48) β c ( β ( n + n (cid:48) ) − β c ) β ( n + n (cid:48) ) ( n + n (cid:48) − µ (cid:48) i by using either the symmetry n, β ↔ n (cid:48) β (cid:48) or the fact that the sum βµ i + β (cid:48) µ (cid:48) i = β c does not fluctuate (55). The generating function of µ i and µ (cid:48) i . We are now going to obtain theexact expression of the generating function (cid:68) x µ y µ (cid:48) (cid:69) { µ i ,µ (cid:48) i } where the average (cid:104)·(cid:105) { µ i ,µ (cid:48) i } is defined in equation (51). Taking the Taylor expansions of x µ , y µ (cid:48) about x = 1 , y = 1 the generating function can be written as (cid:68) x µ y µ (cid:48) (cid:69) { µ i ,µ (cid:48) i } = (cid:88) k ≥ (cid:88) k (cid:48) ≥ ( x − k k ! ( y − k k (cid:48) ! × (cid:104) µ ( µ − · · · ( µ − k + 1) µ (cid:48) ( µ (cid:48) − · · · ( µ (cid:48) − k (cid:48) + 1) (cid:105) { µ i ,µ (cid:48) i } . (65)In the thermodynamic limit we can express the average on { µ i , µ (cid:48) i } on the righthand side in terms of (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) using equation (50). This gives (cid:104) µ ( µ − · · · ( µ − k + 1) µ (cid:48) ( µ (cid:48) − · · · ( µ (cid:48) − k (cid:48) + 1) (cid:105) { µ i ,µ (cid:48) i } = 1 r n (cid:48) ( n (cid:48) − · · · ( n (cid:48) − k (cid:48) + 1) n ( n − · · · ( n − k + 1) (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) (66)where we have used the fact that r does not fluctuate in the thermodynamic limit(see (24)). Using the exact expression (23) for (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) we obtain (cid:104) µ ( µ − · · · ( µ − k + 1) µ (cid:48) ( µ (cid:48) − · · · ( µ (cid:48) − k (cid:48) + 1) (cid:105) { µ i ,µ (cid:48) i } = ( − k + k (cid:48) (cid:82) ∞ dv v k (cid:48) − n (cid:48) − Ψ (cid:16) k + k (cid:48) β (cid:48) β ; v (cid:17) ( − ψ ( v )) r − (cid:82) ∞ dv v − n (cid:48) − ( − ψ ( v )) r (67)Finally, substituting into equation (65) and summing on k, k (cid:48) we obtain (cid:68) x µ y µ (cid:48) (cid:69) { µ i ,µ (cid:48) i } = x βcβ (cid:82) ∞ dv v − n (cid:48) − (cid:16) − ψ (cid:16) vyx − β (cid:48) β (cid:17)(cid:17) ( − ψ ( v )) r − (cid:82) ∞ dv v − n (cid:48) − ( − ψ ( v )) r . (68)where r is given by (24). (Note that, as mentioned at the beginning of this sectionthe above expression (65) is valid for n < n (cid:48) < Remark . One recovers (55)Making the substitution x = z β , y = z β (cid:48) in (68) gives (cid:68) z βµ + β (cid:48) µ (cid:48) (cid:69) { µ i ,µ (cid:48) i } = z β c . (69)This confirms the fact that the sum βµ + β (cid:48) µ (cid:48) does not fluctuate and takes thevalue β c , as expected from (55). Remark . The n, n (cid:48) → − limit One can show using the asymptotics (85) that, in the limit n → − and n (cid:48) → − ,the generating function (68) becomes (cid:68) x µ y µ (cid:48) (cid:69) { µ i ,µ (cid:48) i } = βnx βcβ + β (cid:48) n (cid:48) y βcβ (cid:48) βn + β (cid:48) n (cid:48) + ββ (cid:48) nn (cid:48) ( y βcβ (cid:48) − x βcβ ) β c ( βn + β (cid:48) n (cid:48) ) log Γ (cid:16) − β c β (cid:48) (cid:17) Γ (cid:16) − β c β (cid:17) + βnn (cid:48) βn + β (cid:48) n (cid:48) x βcβ (cid:90) ∞ log v ddv ψ ( yx − β (cid:48) β v ) ψ ( v ) dv (70)+ o ( n, n (cid:48) )To leading order one finds that the distribution of µ and µ (cid:48) consists of two deltafunctions NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 15 P ( µ , µ (cid:48) ) = (cid:20) nβnβ + n (cid:48) β (cid:48) δ (cid:18) µ − β c β (cid:19) + n (cid:48) β (cid:48) nβ + n (cid:48) β (cid:48) δ ( µ ) (cid:21) β (cid:48) δ ( βµ + β (cid:48) µ (cid:48) − β c )Clearly this expression does not contain any information on the overlaps (cid:104) Y k,k (cid:48) (cid:105) . Infact the generating function of these overlaps only appears in the first order term in(70). Note also that as for the variance of µ , the n → − , n (cid:48) → − limit dependson the ratio n (cid:48) /n .5.3. Trying to describe P ( µ , µ (cid:48) ) . As the variance of the µ i can become negative,it is clear from the very beginning that it is not possible to find a meaningfuldistribution of the block sizes compatible with the generating function (68). Wemade a number of attempts which became rather complicated and we don’t thinkit is of much interest to mention them here. Let us however discuss briefly onecase for which we could get a rather simple picture, the equal temperature case,i.e. when β = β (cid:48) .In this case we have an explicit expression (79) of the function ψ ( v ). Then (68)becomes (cid:68) x µ y µ (cid:48) (cid:69) { µ i ,µ (cid:48) i } = (cid:82) ∞ v − n (cid:48) − (1 + v ) n + n (cid:48) − βcβ ( x + vy ) βcβ dv (cid:82) ∞ v − n (cid:48) − (1 + v ) n + n (cid:48) dv After a simple change of variable v = (1 − t ) /t this becomes (cid:68) x µ y µ (cid:48) (cid:69) { µ i ,µ (cid:48) i } = Γ( − n − n (cid:48) )Γ( − n ) Γ( − n (cid:48) ) (cid:90) dt t − − n (1 − t ) − − n (cid:48) (cid:16) tx + (1 − t ) y (cid:17) βcβ (71)In order to give an interpretation to (71) let us consider a random variable s ,sum of m i.i.d. random variables τ i which take the value τ i = 1 with probability t and τ i = 0 with probability 1 − t . The distribution of s is a binomial distributionand one has (cid:104) z s (cid:105) = ( zt + 1 − t ) m . (72)Let us further consider that the parameter t is itself randomly distributed accordingto some distribution ρ ( t ) so that the distribution of s becomes a superposition ofbinomial distributions. Then the generating function of s becomes (cid:104) z s (cid:105) = (cid:90) ρ ( t )( zt + 1 − t ) m dt . This is exactly the form we have in (71) (by taking x = z and y = 1) if one choosesfor ρ ( t ) ρ ( t ) = Γ( − n − n (cid:48) )Γ( − n ) Γ( − n (cid:48) ) t − − n (1 − t ) − − n (cid:48) (remember that here n and n (cid:48) are negative).Therefore the distribution of µ can be thought as a superposition of binomialdistributions. The only odd aspect is that s is a sum of m = β c β binary variables,that is s is a sum of a non-integer number of random variables! Remark . A signed measure
If one takes a non-integer m in (72) one gets by expanding in powers of z (cid:104) z s (cid:105) = (cid:88) p =0 (1 − t ) m − p t p m ( m − · · · ( m − p + 1) p ! z p which one can interpret, for t < , as the probability P ( s ) of s being a signedmeasure concentrated on positive integers. Expanding in powers of 1 /z leads, for < t <
1, to a different signed measure. Combining these two representations by cutting the integral (71) into two parts ( t < and t > ) leads to a signed measureconcentrated on the points ( µ i = p, µ (cid:48) i = β c β − p ) and ( µ i = β c β − p, µ (cid:48) i = p ) for allpositive integers p ≥
0. 6.
Conclusion
In this paper we have analysed the distribution of overlaps (11) between twocopies of the same REM at two temperatures. A direct calculation was used toobtain exact expressions (23) for the two temperature overlaps (12) in the thermo-dynamic limit. Generalising this approach allows us to quantify (17) the non-self-averaging effects illustrated in Figure 1.An alternative approach using the replica method enables us to obtain expres-sions for the two temperature overlaps in terms of replicas (50). In the thermody-namic limit the exact and replica expressions satisfy the same Ghirlanda-Guerratype recurrence relation, (25) and (57), giving confidence that the replica expres-sions are valid. We also proposed a way to generalise the Parisi ansatz (63), inthe one step RSB form, to the two temperature case which is consistent with thereplica expressions for the overlaps. In contrast to the single temperature case wefind that the block sizes at the two different temperatures fluctuate even in the ther-modynamic limit subject the constraint (55). We characterised these fluctuationsin terms of a moment generating function for the block sizes (68). It is well knownthat in the single temperature case the strange properties of Parisi’s RSB ansatz(non-integer block sizes and number of blocks) lead to a perfectly good physicaldescription in terms of overlaps [3]. In the two temperature case the distribution of µ i and µ (cid:48) i analysed in section 5 also has strange properties, but it remains an openquestion as to which of these properties have a clear physical interpretations.It would be interesting to extend both the exact and replica approaches to thegeneralised random energy model [35], directed polymer in a random medium [36]and other models where exact methods are likely to be tractable. In contrast to thesingle temperature case, the multi-temperature overlaps should be different in theREM and in the directed polymer problem on a tree because the lowest energies ofthe directed polymer can be thought of as a decorated Poisson process and it hasbeen proved that the multi-temperature overlaps depend on the decoration [37].One could also look at spin models where one step RSB occurs to see if the twotemperature ansatz with fluctuating block sizes is applicable. An obvious startingpoint would be the p-spin spherical model proposed in [38]. In order to addressthese spin problems, where exact expressions for the two temperature overlaps arenot currently available, it would be essential to develop a systematic approach tosumming over the fluctuations in block sizes in the replica expressions. Appendix A. Direct calculation of the overlaps (13,23)
To begin with, it is easier to think that the energies can take only a discrete setof values E ν indexed by ν and that the partition function at inverse temperature β is given by Z ( β ) = (cid:88) ν y ν e − βE ν where y ν = (cid:26) p ν − p ν . So a given sample is specified by the value of all these binary random variables y ν . Then the probability of finding k copies at temperature β and k (cid:48) copies at NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 17 temperature β (cid:48) in the same configuration is given by Y k,k (cid:48) = (cid:80) ν y ν e − ( βk + β (cid:48) k (cid:48) ) E ν Z ( β ) k Z ( β (cid:48) ) k (cid:48) These Y k,k (cid:48) are random quantities as they depend on the realization of the y ν ’s.Using the identity Z − k = Γ( k ) − (cid:82) ∞ dt e − tZ t k − one gets Y k,k (cid:48) = (cid:88) ν y ν e − ( βk + β (cid:48) k (cid:48) ) E ν (cid:90) ∞ t k − dt Γ( k ) (cid:90) ∞ t (cid:48) k (cid:48) − dt (cid:48) Γ( k (cid:48) ) exp (cid:34) − (cid:88) ν (cid:48) y ν (cid:48) (cid:16) te − βE ν (cid:48) + t (cid:48) e − β (cid:48) E ν (cid:48) (cid:17)(cid:35) Averaging over the y ν ’s leads to (cid:104) Y k,k (cid:48) (cid:105) = (cid:90) ∞ t k − dt Γ( k ) (cid:90) ∞ t (cid:48) k (cid:48) − dt (cid:48) Γ( k (cid:48) ) (cid:88) ν p ν e − ( βk + β (cid:48) k (cid:48) ) E ν exp (cid:104) − te − βE ν − t (cid:48) e − β (cid:48) E ν (cid:105) × (cid:89) ν (cid:48) (cid:54) = ν (cid:16) − p ν (cid:48) + p ν (cid:48) exp (cid:104) − te − βE ν (cid:48) − t (cid:48) e − β (cid:48) E ν (cid:48) (cid:105) (cid:17) Now if we go to the continuum limit, by saying that each energy interval (
E, E + dE ) is either occupied by an energy level or empty and if we choose as in (20) p ν = C (cid:48) e β c E dE with C (cid:48) = Ce − β c E one gets (cid:104) Y k,k (cid:48) (cid:105) = (cid:90) ∞ t k − dt Γ( k ) (cid:90) ∞ t (cid:48) k (cid:48) − dt (cid:48) Γ( k (cid:48) ) W ( k, k (cid:48) ; t, t (cid:48) ) e w ( t,t (cid:48) ) where w ( t, t (cid:48) ) = C (cid:48) (cid:90) e β c E dE (cid:16) exp (cid:104) − te − βE − t (cid:48) e − β (cid:48) E (cid:105) − (cid:17) and W ( k, k (cid:48) ; t, t (cid:48) ) = C (cid:48) (cid:90) e β c E dE e − ( βk + β (cid:48) k (cid:48) ) E exp (cid:104) − te − βE − t (cid:48) e − β (cid:48) E (cid:105) Then these expressions can be simplified by noticing that w ( t, t (cid:48) ) = C (cid:48) β t βcβ ψ (cid:32) t (cid:48) t β (cid:48) β (cid:33) and W ( k, k (cid:48) ; t, t (cid:48) ) = C (cid:48) β t βcβ − k − β (cid:48) β k (cid:48) Ψ (cid:32) k + k (cid:48) β (cid:48) β ; t (cid:48) t β (cid:48) β (cid:33) where ψ ( v ) = (cid:90) ∞ du ( e − u − vu β (cid:48) β − u − − βcβ (73)and Ψ( z ; v ) = (cid:90) ∞ du e − u − vu β (cid:48) β u z − − βcβ (74)This leads to (cid:104) Y k,k (cid:48) (cid:105) = ββ c k ) 1Γ( k (cid:48) ) (cid:90) ∞ dv v k (cid:48) − Ψ (cid:16) k + k (cid:48) β (cid:48) β ; v (cid:17)(cid:0) − ψ ( v ) (cid:1) (75) Remark
A.1 . To generalize (75) one can define the probability of finding k copiesat inverse temperature β and k (cid:48) copies of the same system at inverse temperature β (cid:48) in the same configuration, and similarly k and k (cid:48) in a different configurationand so on i.e. Y k ,k (cid:48) ; ··· k p ,k (cid:48) p = (cid:80) ν ··· ν p e − ( βk + β (cid:48) k (cid:48) ) E ν −··· ( βk p + β (cid:48) k (cid:48) p ) E νp Z ( β ) k + ··· k p Z ( β (cid:48) ) k (cid:48) + ··· k (cid:48) p where, in the sum, the configurations ν (cid:54) = ν (cid:54) = · · · ν p are all different. By astraightforward extension of the above calculation one gets (cid:104) Y k ,k (cid:48) ; ··· k p ,k (cid:48) p (cid:105) = ββ c Γ( p ) 1Γ( k + · · · k p ) 1Γ( k (cid:48) + · · · k (cid:48) p ) × (cid:90) ∞ dv v k (cid:48) + ··· k (cid:48) p − Ψ (cid:16) k + k (cid:48) β (cid:48) β ; v (cid:17) · · · Ψ (cid:16) k p + k (cid:48) p β (cid:48) β ; v (cid:17)(cid:0) − ψ ( v ) (cid:1) p (76) Remark
A.2 . In the way the above formulae are written, β and β (cid:48) seem to playasymmetric roles. One can however check from the definitions (73,74) of ψ and Ψthat ψ β,β (cid:48) ( v ) = ββ (cid:48) v βcβ (cid:48) ψ β (cid:48) ,β ( v − ββ (cid:48) ) (77)Ψ β,β (cid:48) ( z ; v ) = ββ (cid:48) v βcβ (cid:48) − ββ (cid:48) z Ψ β (cid:48) ,β (cid:18) ββ (cid:48) z ; v − ββ (cid:48) (cid:19) (78)and using these relations one can easily prove that the expressions (75) and (76)are left unchanged by the symmetry (cid:16) β, β (cid:48) , { k , · · · k p } , { k (cid:48) , · · · k (cid:48) p } (cid:17) ←→ (cid:16) β (cid:48) , β, { k (cid:48) , · · · k (cid:48) p } , { k , · · · k p } (cid:17) Remark
A.3 . When β = β (cid:48) , the expressions (73) and (74) become ψ ( v ) = Γ (cid:18) − β c β (cid:19) (1 + v ) βcβ (79)Ψ( v ) = Γ (cid:18) z − β c β (cid:19) (1 + v ) βcβ − z . (80)The integrals in (76) can then be performed and one gets (cid:104) Y k ,k (cid:48) ; ··· k p ,k (cid:48) p (cid:105) = ββ c Γ( p )Γ( k + k (cid:48) + · · · k p + k (cid:48) p ) p (cid:89) i =1 (cid:32) Γ( k i + k (cid:48) i − β c β ) − Γ( − β c β ) (cid:33) (81)which was already known (see for example [24]). Remark
A.4 . It is easy to show, using an integration by parts in (74), that (for z > β c β ) (cid:18) z − β c β (cid:19) Ψ( z ; v ) = Ψ( z + 1 ; v ) + β (cid:48) β v Ψ (cid:18) z + β (cid:48) β ; v (cid:19) and this leads (see (76)) to relationships between the (cid:104) Y k ,k (cid:48) ; ··· k p ,k (cid:48) p (cid:105) ( k β + k (cid:48) β (cid:48) − β c ) (cid:104) Y k ,k (cid:48) ; ··· k p ,k (cid:48) p (cid:105) =( k + · · · k p ) β (cid:104) Y k +1 ,k (cid:48) ; ··· k p ,k (cid:48) p (cid:105) + ( k (cid:48) + · · · k (cid:48) p ) β (cid:48) (cid:104) Y k ,k (cid:48) +1; ··· k p ,k (cid:48) p (cid:105) (82)and similar identities for the pairs k , k (cid:48) , · · · k p , k (cid:48) p . NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 19
Remark
A.5 . In the replica approach, one is usually interested in the limit wherethe number of replicas n →
0. It is however often easier to first think in terms ofa non-zero number of of replicas and to take the n → n and n (cid:48) of replicas ( n and n (cid:48) are a priori arbitrary real numbers) by (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = (cid:68)(cid:80) ν y ν e − ( βk + β (cid:48) k (cid:48) ) E ν Z ( β ) n − k Z ( β (cid:48) ) n (cid:48) − k (cid:48) (cid:69) (cid:104) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:105) It turns out that the expressions have somewhat simpler forms when the numbers n and n (cid:48) take negative values and one gets (cid:104) Y k,k (cid:48) (cid:105) n,n (cid:48) = ( − r ) Γ( − n ) Γ( − n (cid:48) )Γ( k − n ) Γ( k (cid:48) − n (cid:48) ) (cid:82) ∞ dv v k (cid:48) − n (cid:48) − Ψ (cid:16) k + k (cid:48) β (cid:48) β ; v (cid:17) ( − ψ ( v )) r − (cid:82) ∞ dv v − n (cid:48) − ( − ψ ( v )) r (83)where r = nβ + n (cid:48) β (cid:48) β c . (84) Remark
A.6 . Using the following asymptotics of ψ ( v ) which can be derived fromexpression (77) ψ ( v ) (cid:39) Γ (cid:16) − β c β (cid:17) as v → ββ (cid:48) Γ (cid:16) − β c β (cid:48) (cid:17) v βcβ (cid:48) as v → ∞ (85)one can show that (83) reduces to (75) in the limit n → − and n (cid:48) → − . Appendix B. Integer moments of the partition function at multipletemperatures
The replica method starts usually with the calculation of integer moments ofthe partition function. In a two or a multiple temperature case, these are ofthe form (cid:104) Z ( β ) n Z ( β ) n Z ( β ) n · · · (cid:105) where n , n , n . . . are positive integers and β , β , β , . . . are inverse temperatures. In this appendix we obtain the expressions(29) and (47) using a generating function defined for p temperatures as G ( t , t , . . . t p ) = (cid:42) exp (cid:32) − p (cid:88) i =1 t i Z ( β i ) (cid:33)(cid:43) . (86)For the REM (see section 2) the partition function is given by Z ( β ) = N (cid:88) C =1 e − βE ( C ) where the 2 N energies E ( C ) take random values distributed according to P ( E )given in (18). Then, because the E ( C ) are independent, G ( t , · · · t p ) = (cid:34)(cid:90) P ( E ) dE exp (cid:32) − (cid:88) i t i e − β i E (cid:33)(cid:35) N = exp (cid:40) N log (cid:32)(cid:90) P ( E ) dE exp (cid:16) − (cid:88) i t i e − β i E (cid:17)(cid:33)(cid:41) which for large N becomes G ( t , · · · t p ) (cid:39) exp (cid:40)(cid:90) ∞−∞ ρ ( E ) (cid:34) exp (cid:32) − p (cid:88) i =1 t i e − β i E (cid:33) − (cid:35) dE (cid:41) . (87)(By this approximation, we in fact replace the REM by a Poisson REM of density(see (18)) ρ ( E ) = 2 N P ( E ) = 2 N √ N πJ exp (cid:20) − E N J (cid:21) . (88)Doing so the error is exponentially small in the system size N as shown in theappendix of [24]). The exponentials on the right hand side of (87) can be expandedto obtain G ( t , · · · t p ) = ∞ (cid:88) r =0 r ! (cid:34) ∞ (cid:88) µ =0 · · · ∞ (cid:88) µ p =0( µ + ··· + µ p ≥ ( − t ) µ µ ! · · · ( − t p ) µ p µ p ! (cid:104) Z ( β µ + · · · + β p µ p ) (cid:105) (cid:35) r (89)where we use the fact that for the REM (as well as for the Poisson REM) (cid:104) Z ( β ) (cid:105) = (cid:90) ∞−∞ ρ ( E ) e − βE dE. (90)The general expression for integer moments at p temperatures is obtained by equat-ing powers of t i in the expansion of the right hand side of equation (86) with theright hand side of equation (89). Here we give the three moments that are used inthe main text.The single temperature moments are then given by (cid:104) Z ( β ) n (cid:105) = (cid:88) r ≥ r ! (cid:88) { µ i ≥ } C n,r ( { µ i } ) (cid:104) Z ( βµ ) (cid:105) (cid:104) Z ( βµ ) (cid:105) · · · (cid:104) Z ( βµ r ) (cid:105) (91)where we have defined (cid:88) { µ i ≥ } = (cid:88) µ ≥ (cid:88) µ ≥ · · · (cid:88) µ r ≥ (92)and C n,r ( { µ i } ) = n ! µ ! µ ! · · · µ r ! δ (cid:34) r (cid:88) i =1 µ i = n (cid:35) (93)The Kronecker delta δ [ (cid:80) ri =1 µ i = n ] ensures that the µ i always sum to n .Similarly the two temperature moments are given by (cid:68) Z ( β ) n Z ( β (cid:48) ) n (cid:48) (cid:69) = (cid:88) r ≥ r ! (cid:88) { µ i ≥ } (cid:88) { µ (cid:48) i ≥ } θ [ µ i + µ (cid:48) i ≥ C n,r ( { µ i } ) C n (cid:48) ,r ( { µ (cid:48) i } ) × (cid:104) Z ( βµ + β (cid:48) µ (cid:48) ) (cid:105) (cid:104) Z ( βµ + β (cid:48) µ (cid:48) ) (cid:105) · · · (cid:104) Z ( βµ r + β (cid:48) µ (cid:48) r ) (cid:105) (94)where θ [ µ i + µ (cid:48) i ≥
1] is unity if the inequality is satisfied for every i = 1 , , . . . , r and zero otherwise. The three temperature moments are given by (cid:68) Z ( β ) n Z ( β (cid:48) ) n (cid:48) Z ( β (cid:48)(cid:48) ) n (cid:48)(cid:48) (cid:69) = (cid:88) r ≥ r ! (cid:88) { µ i ≥ } (cid:88) { µ (cid:48) i ≥ } (cid:88) { µ (cid:48)(cid:48) i ≥ } θ [ µ i + µ (cid:48) i + µ (cid:48)(cid:48) i ≥ × C n,r ( { µ i } ) C n (cid:48) ,r ( { µ (cid:48) i } ) C n (cid:48)(cid:48) ,r ( { µ (cid:48)(cid:48) i } ) (cid:104) Z ( βµ + β (cid:48) µ (cid:48) + β (cid:48)(cid:48) µ (cid:48)(cid:48) ) (cid:105)× (cid:104) Z ( βµ + β (cid:48) µ (cid:48) + β (cid:48)(cid:48) µ (cid:48)(cid:48) ) (cid:105) · · · (cid:104) Z ( βµ r + β (cid:48) µ (cid:48) r + β (cid:48)(cid:48) µ (cid:48)(cid:48) r ) (cid:105) (95) NE STEP RSB AND OVERLAPS BETWEEN TWO TEMPERATURES 21
As a special case of (94) one has (cid:10) Z ( β ) n − k Z ( kβ ) (cid:11) = (cid:88) r ≥ r ! (cid:88) { µ i ≥ } (cid:88) { µ (cid:48) i ≥ } θ [ µ i + µ (cid:48) i ≥ C n − k,r ( { µ i } ) C ,r ( { µ (cid:48) i } ) × (cid:104) Z ( βµ + kβµ (cid:48) ) (cid:105) (cid:104) Z ( βµ + kβµ (cid:48) ) (cid:105) · · · (cid:104) Z ( βµ r + kβµ (cid:48) r ) (cid:105) (96)In this case n (cid:48) = 1, therefore there is a single µ (cid:48) i = 1 all the others being 0. Becauseof the symmetry between the indices i in the previous formula, one can choose µ (cid:48) = 1 and one gets (cid:10) Z ( β ) n − k Z ( kβ ) (cid:11) = (cid:88) r ≥ r − (cid:88) { µ i ≥ } C n,r ( { µ i } ) ( n − k )! n ! µ !( µ − k )! × (cid:104) Z ( βµ ) (cid:105) (cid:104) Z ( βµ ) (cid:105) · · · (cid:104) Z ( βµ r ) (cid:105) (97)where we take µ − k )! = 0 when µ < k . References [1] G. Parisi. Infinite number of order parameters for spin-glasses.
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