Question

I am studying the spontaneous symmetry breaking in the mean-field Ising model and it's clear to me the necessity of taking first the thermodynamic limit and then the zero-field limit to see the phase transition, otherwise we only see zero magnetisation even under Tc. I was wondering if there was a phase space approach underlying this result and if the spontaneous symmetry breaking implied and ergodicity breaking as well, as I imagine that the ensemble average is anyway zero, but this is not what we observe physically. Can someone give me a hint?

Answer 1

In the mean-field Ising model, spontaneous symmetry breaking leads to non-zero magnetization below Tc, which implies ergodicity breaking in the phase space approach.

Step-by-step explanation:

When studying the mean-field Ising model for spontaneous symmetry breaking and phase transitions, it is crucial to take first the thermodynamic limit and then the zero-field limit to witness the symmetry breaking resulting in non-zero magnetization below the critical temperature, Tc.

If these limits are not taken in the proper order, the magnetization appears to be zero which contradicts physical observations. In terms of phase space approach, spontaneous symmetry breaking indicates that below Tc, the phase space is divided into distinct regions corresponding to different spontaneous magnetizations.

This indeed suggests an ergodicity breaking since the system can no longer explore all microstates within the phase space due to the emergence of discrete magnetized states.

Consequently, ensemble averages yield zero magnetization, but this is not consistent with the macroscopic reality where non-zero magnetization exits.