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

In a quantum many-body system where the Hamiltonian and the order operator do not commute, it often happens that the unique ground state of a finite system exhibits long-range order (LRO) but does not show spontaneous symmetry breaking (SSB). Typical examples include antiferromagnetic quantum spin systems with Neel order, and lattice boson systems which exhibit Bose-Einstein condensation. By extending and improving previous results by Horsch and von der Linden and by Koma and Tasaki, we here develop a fully rigorous and almost complete theory about the relation between LRO and SSB in the ground state of a finite system with continuous symmetry. We show that a ground state with LRO but without SSB is inevitably accompanied by a series of energy eigenstates, known as the "tower" of states, which have extremely low excitation energies. More importantly, 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 require any knowledge about the previous works on the subject.