Symmetry Breaking in Finite Volume — a Study in the O(N) Model**The project supported by National Natural Science Foundation of China under Grant No. 19705001, Climbing-Up Program of China and the Natural Science Fund from Peking University

Abstract

Spontaneous symmetry breaking plays a crucial role in quantum field theories and it occurs only for systems with infinite degrees of freedom. For a system in a large but finite volume, although the symmetry is not broken spontaneously, an adiabatic approximation can be applied to study the would-be broken symmetry. Lattice O(N) model is studied within the Hamiltonian approach. It is shown that the low-lying spectrum of the system can be explained using an adiabatic, or Born–Oppenheimer approximation, which turns out to become 1 an expansion in the inverse power of volume. In the infinite volume limit, the symmetry is broken while in the finite volume the slow rotation of the zero-momentum mode restores the symmetry and gives rise to the rotator spectrum, which has been observed in real Monte Carlo simulations.