Long-Range Order, “Tower” of States, and Symmetry Breaking in Lattice Quantum Systems
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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 Néel order, and lattice boson systems which exhibits 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.
KeywordsLong-range order Spontaneous symmetry breaking “Tower of states” Quantum spin systems
It is a pleasure to thank Hosho Katsura for his essential contribution in the derivation of (4.43), which considerably simplified the proof, and for useful comments. I also thank Tohru Koma and Haruki Watanabe for indispensable discussions and comments which made the present work possible, and Masaki Oshikawa and Masafumi Udagawa for useful discussions. The present work was supported by JSPS Grants-in-Aid for Scientific Research no. 16H02211.
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