Symmetry Breaking in Finite Volume

Spontaneous symmetry breaking is a cooperative phenomenon for systems with inGnitely many degrees of freedom and it plays an essential role in quantum Held theories.Lattice O(N)model is studied within the Hamiltonian approach using an adiabatic approximation.It is shown that the low-lying spectrum of the system in the broken phase can be understood by using the adiabatic^or Born-Oppenheimer approximation,which turns out to become 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-momGntum mode restores the symmetry and gives rise to the rotator spectrumf which has been observed in realistic Monte Carlo simulations.