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Symmetry Breaking and Finite Size Effects in Quantum Many-Body Systems
We consider a quantum many-body system on a lattice with a continuous
symmetry which exhibits a spontaneous symmetry breaking in its infinite volume
ground states, but in which the order operator does not commute with the
Hamiltonian. A typical example is the Heisenberg antiferromagnet with a Neel
order. In the corresponding finite system, the symmetry breaking is usually
"obscured" by "quantum fluctuation" and one gets a symmetric ground state with
a long range order. In such a situation, we prove that there exist ever
increasing numbers of low-lying eigenstates whose excitation energies are
bounded by a constant times 1/N, where N denotes the number of sites. By
forming linear combinations of these low-lying states and the (finite-volume)
ground state, and by taking infinite volume limits, we construct infinite
volume ground states with explicit symmetry breaking. Our general theorems do
not only shed light on the nature ofsymmetry breaking in quantum many-body
systems, but provide indispensable information for numerical approaches to
these systems. We also discuss applications of our general results to a variety
of examples. The present paper is intended to be accessible to the readers
without background in mathematical approaches to quantum many-body systems.Comment: LaTeX, 58 pages, no figures. Notes about Bose-Einstein condenstaion
are added after the publicatio