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[tex] \hat{H}=-J \sum_{\langle i,j \rangle} \vec{S}_i \cdot \vec{S}_j[/tex]

where ##J>0## and summation is between nearest neighbours. Hamiltonian is perfectly rotational symmetric. However, the ground state “spontaneously” chooses a particular orientation ##|\psi\rangle=|\uparrow...\uparrow...\uparrow \rangle## and hence is not invariant under the symmetry (rotation).

I do not understand part

**the ground state “spontaneously” chooses a particular orientation.**As far as I understand ground state is the state with minimal energy. So state ##|\psi_2\rangle=|\downarrow...\downarrow...\downarrow \rangle## also corresponds to same energy. And state ##|\psi_3\rangle=|←...←...← \rangle## coresponds to same energy. Why then is the part

**the ground state “spontaneously” chooses a particular orientation**so important?