statistical mechanics - Spontaneous symmetry breaking at a finite temperature $T$: How is the state dscribed as a function of $T$?

Consider the equilibrium state of a statistical system with infinite DOF at a finite temperature $T$. For example, a Heisenberg ferromagnet with Hamiltonian

$$H=-J\sum\limits_{i,j}\textbf{s}_i\cdot \textbf{s}_j\tag{1}$$ One can see that $\textbf{s}_i\cdot\textbf{s}_j$ is a scalar and therefore invariant under rotation.

However, if the temperature $T$ is greater than a critical value $T_c$, then the equilibrium state respects the symmetry of the Hamiltonian. And if $T

I want to make this picture mathematically precise. Can we describe the equilibrium state, in general, at temperature $T$ irrespective of whether $T>T_c$ or $T

How can I mathematically describe the equilibrium configuration so that I can explicitly see (like in Eqn. (1)) it breaks rotational invariance (symmetry of the Hamiltonian) for $TT_c$?