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

TL;DR

This paper rigorously explores the relationship between long-range order and spontaneous symmetry breaking in quantum lattice systems, revealing the role of low-energy excited states and the formation of a superposed ground state.

Contribution

It extends previous theories by providing a comprehensive and rigorous analysis of LRO and SSB, introducing the concept of a 'tower' of states in finite quantum systems with continuous symmetry.

Findings

Ground states with LRO but no SSB are linked to a tower of low-energy states.

Superpositions of these low-energy states form physically realistic ground states.

The theory applies to various quantum many-body systems with continuous symmetry.

Abstract

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.