4 - Physics of replica symmetry breaking

Summary

In this chapter the physical interpretation of the formal RSB solution will be proposed, and some new concepts and quantities will be introduced. The crucial concept that is needed to understand physics behind the RSB structures is that of the pure states.

The pure states

Consider again a simple example of the ferromagnetic system. Here, spontaneous symmetry breaking takes place below the critical temperature T_c, and at each site the non-zero spin magnetizations 〈σ_i〉 = ±m appear. As we have already discussed in Section 2.2, in the thermodynamic limit the two ground states with the global magnetizations 〈σ_i〉 = +m and 〈σ_i〉 = –m are separated by an infinite energy barrier. Therefore, once the system has happened to be in one of these states, it will never be able (during any finite time) to jump into the other one. In this sense, the observable state is not the Gibbs one (which is obtained by summing over all the states), but one of these two states with non-zero global magnetizations. To distinguish them from the Gibbs state they could be called the ‘pure states’. More formally, the pure states could also be defined by the property that all the connected correlation functions in these states, such as 〈σ_iσ_j〉_c ≡ 〈σ_iσ_j〉 – 〈σ_i〉〈σ_j〉, tend towards zero at large distances.

In the previous chapter we obtained a special type of spin-glass ground-state solution.