Replica symmetry breaking

In the EA model the spin-spin interaction is only of the nearest-neighbor type. The Sherrington-Kirpatrick (SK) model [79] is the infinite-range version of the EA model. It is most useful as a basis for mean-field calculations. One such solution is the replica symmetry breaking theory of Parisi [80-82]. 

By a synthesis of the partition function for a supercooled liquid at some hctive temperature in the inherent structure formalism [ 1,4] with the configurational entropy obtained by restricting mrepiica replicas to be in the same state [161], Mohanty has uncovered a relationship between Parisi s replica symmetry-breaking parameter mrepiica(T), and the Narayanaswamy-Moynihan non-linear parameter x, a parameter that provides a metric on the deviation of the glassforming system from equilibrium [162] ... 

Here, Cv h(T) and Svlh(T) are the vibrational contributions to the heat capacity and the entropy, respectively. Note that the slope of the replica symmetry-breaking parameter with respect to temperature is not unity as predicted by one-step replica symmetry breaking. Rather, the slope is governed by three factors the Narayanaswamy-Moynihan nonlinearity parameter x, the Kauzmann temperature, and the ratio of the Kauzmann temperature to the glass transition temperature. 

Figure 3. Illustration of first step replica symmetry breaking. The limits n —> 0 and m Xc have to be taken with 0 < Xc < 1.

For non-asjrmptotic values of p and L the non-linear stationarity equations equations could be solved numerically [20] using a standard iterative method [25]. We fotmd that for a given set of parameters there is a chmn length (which depends on the strength of the disorder) such that for 0 < T < Tc there is only a replica sjunmetric solution. This is the case when the variational p irameters satisfy Xc = 1 and sq = Si. For L > Lc there is still a replica symmetric solution but we also find an additional replica symmetry breaking solution. So in this regime we find an additional solution such that 0 < Xc < 1... 

It should be emphasized that the subtle dependence on the volume of the system is a direct consequence of replica symmetry breaking. In fact, as shown in Fig. 4 the replica symmetric solution does not correctly describe the size of the polymer chain, since it fails to capture the dominance of localized tail states. 

We also gave a physical interpretation of the 1-step replica-symmetry-breaking solution, and elucidated the connection with the statistics of localized tail states. Our coucusions support the heuristic arrguments of Cates ajid Ball, but it starts with the microscopic model. 

Fig. 18. Magnetic phase diagram for the SK model (EA model for Ising spins with infinite-range couplings). J and / denote the width and mean of the exchange distribution. P = paramagnet FM = ferromagnet SG = spin glass. F is a ferromagnetic phase viith replica symmetry breaking, i.e. irreversibility ( mixed phase ) and is separated from FM by an AT line.

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