10 - Ward–Takahashi Identities and Goldstone modes

Summary

In the previous chapter we have explicitly computed components of the (bare) propagators in the replicon sector and found that we had a spectrum of masses bounded below by zero modes. In this chapter we show that these massless modes are indeed Goldstone modes related to the spontaneous breaking of a continuous symmetry of the Lagrangian, and thus are to remain massless under loop corrections. We have already analyzed a case of spontaneous symmetry breaking in the different context of the TAP free energy in Chapter 7. Here, we focus on the invariance properties of the spin glass Lagrangian and exhibit the continuous symmetry spontaneously broken below the AT line, which entails corresponding Ward–Takahashi Identities. We start in Section 10.1 by describing the problem in the simpler setting of a system with a two-component order parameter. In Section 10.2 we generalize to the spin glass case, by defining the symmetry which undergoes spontaneous breaking. In Section 10.4 we give a derivation of the Ward–Takahashi Identities, expressing the fact that the massless transverse modes are indeed Goldstone modes. Details of what follows can be found in De Dominicis, Kondor and Temesvari (1998).

The Legendre Transform and invariance properties

To explain the general problem we would like to address, it is convenient to consider first the case of a simpler field theory (at least for what concerns the notation).