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In a quantum many-body system where the Hamiltonian and the order operator donot commute, it often happens that the unique ground state of a finite systemexhibits long-range order (LRO) but does not show spontaneous symmetry breaking(SSB). Typical examples include antiferromagnetic quantum spin systems withNeel order, and lattice boson systems which exhibit Bose-Einstein condensation.By extending and improving previous results by Horsch and von der Linden and byKoma and Tasaki, we here develop a fully rigorous and almost complete theoryabout the relation between LRO and SSB in the ground state of a finite systemwith continuous symmetry. We show that a ground state with LRO but without SSBis inevitably accompanied by a series of energy eigenstates, known as the"tower" of states, which have extremely low excitation energies. Moreimportantly, we also prove that one gets a physically realistic "ground state"by taking a superposition of these low energy excited states.  The present paper is written in a self-contained manner, and does not requireany knowledge about the previous works on the subject.

Long-range order, "tower" of states, and symmetry breaking in lattice quantum systems