We present a mathematically simple procedure to explain spontaneous symmetry breaking in quantum systems. The procedure is applicable to a wide range of models and can be easily used to explain the existence of a symmetry broken state in crystals, antiferromagnets, and even superconductors. It has the advantage that it automatically brings to the fore the main players in spontaneous symmetry breaking: the symmetry-breaking field, the thermodynamic limit, and the global excitations of a “thin” spectrum.
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Notice that ferromagnetism is explicitly not included in this list. The ferromagnet has a large number of possible exact ground states that are precisely degenerate and all have a finite magnetization. The singling out of one of these eigenstates is more like classical symmetry breaking than like the quantum symmetry breaking discussed here. Quantum symmetry breaking causes a state that is not an eigenstate of the Hamiltonian to be realized, and thus goes much further than singling out only one particular eigenstate.
This argument holds only for the lowest eigenstates. As can be seen from Eq. (10), these are the only states contributing to the symmetry-broken ground state, as all other states are exponentially suppressed.
Note that the wavefunction in Eq. (12) is not properly normalized due only to the non-normalizability of the free-particle wavefunction in quantum mechanics. This non-normalizability can be cured in the usual way by introducing a finite space and periodic boundary conditions. By letting the size of the space tend to infinity, the properties of the free particle in infinite space are recovered.