If we are considering a system, like a paramagnet, above its critical temperature $T_c$ we can statistically describe it using a canonical ensemble: $$ \langle A \rangle = Tr\left[ \rho A \right]; \quad \rho = \frac{e^{-\beta H}}{Tr\; e^{-\beta H}}. $$ Let us further suppose that system obeys rotational $SO(3)$ symmetry. Now, the expectation value (and correlation functions in general) should also obey this symmetry. This can be easily seen if we denote arbitrary rotation operator by $U$: $$ [\rho, U] = 0 \implies \langle U A U^{\dagger}\rangle = Tr\left[ \rho U A U^{\dagger} \right] = Tr\left[ U^{\dagger} \rho U A \right] = Tr\left[ \rho A \right] = \langle A \rangle , $$ where we used the cyclic property of trace and the fact that density matrix commutes with rotation operator.
However, if we lower the temperature below the critical temperature $T_c$, the system transitions in the ferromagnetic phase, and the $SO(3)$ symmetry is broken. Even though the Hamiltonian still possesses the $SO(3)$ symmetry, the procedure we conducted above cannot be repeated, since we know that it is not true for magnetization (for example) $$ \langle U M U^{\dagger}\rangle \neq \langle M\rangle. $$ This means that the density matrix in the ferromagnetic (ordered) state is not determined by $$ \rho = \frac{e^{-\beta H}}{Tr\; e^{-\beta H}} $$ How do we then do the statistical physics in the ordered (like ferromagnet) phase? What density matrix should we use? Does that mean that straightforward canonical ensemble does not apply?
From this, we can conclude that ergodicity does not hold anymore.
