Free-Energy Fluctuations and Chaos in the Sherrington-Kirkpatrick Model

Abstract

The sample-to-sample fluctuations ∆FN of the free energy in the Sherrington-Kirkpatrick model are shown rigorously to be related to bond chaos. Via this connection, the fluctuations become analytically accessible by replica methods. The replica calculation for bond chaos shows that the exponent µ governing the growth of the fluctuations with system size N , ∆FN ∼ N µ , is bounded by µ ≤ 1 4 .The sample-to-sample fluctuations of the free energy in the mean-field Ising spin glass [1] are a long standing unsolved problem in spin glass physics. In addition to their intrinsic interest as a finite size effect in spin glasses, they are of fundamental importance for the physics of finite-dimensional spin glasses. It has been shown [2] that the finite-size scaling of the free energy fluctuations ∆F N in the mean-field spin glass is equal to the scaling of the domain wall energy ∆F DW in finite dimensionswhere N is the total system size and L its linear dimension (in the case of a finite dimensional system). This implies the relationship θ = dµ between the domain wall exponent θ and the fluctuation exponent µ. This highly nontrivial equivalence between a mean-field quantity and a finitedimensional quantity provides a strong test for replica field theory which was used in Ref. 2 to derive this result.Chaos is also a very important aspect of spin glasses. Chaos refers to the property that an infinitesimal change of, for instance, the temperature or the bond strengths results in a complete change of the equilibrium state. Chaos was first suggested in the context of the droplet picture and finite dimensional spin glasses [3] but has also been studied in the mean field model [4,5,6].In this paper we derive a new and exact connection between the free energy fluctuations and the seemingly unrelated phenomenon of chaos. Such a connection has been suggested by Bouchaud et al. [7] as part of a heuristic argument to obtain the free energy fluctuations. Our results partly corroborate the argument but we will see that a crucial ingredient seems to be missing from it. In addition to making the heuristic argument precise, our results provide a new way to access the fluctuations analytically. The fluctuations are a subextensive quantity such that their calculation usually requires higher order terms in the loop expansion. These are, however, inaccessible due to the massless modes present throughout the spin glass phase. Here we will show that it is sufficient to calculate chaos to zero loop order (which is possible) to obtain the fluctuations. We demonstrate this explicitly above and at the critical temperature but believe and present evidence that it also works in the low temperature phase.The method we use to derive the connection between fluctuations and chaos is a variation of the interpolating Hamiltonian method. Our approach is inspired by the work of Billoire [8] where a similar method was introduced to study the finite size corrections to the free energy numerically. In this paper, we will set up a general formalism which will b...

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