Quantum Rotor — Topological θ-Term

Berry phase and instanton tunneling in the 1D quantum rotor

θ: g (coupling): N (sites):

The quantum rotor Lagrangian L = (1/2g)(dθ/dt)² + iθQ where Q = (1/2π)∫dθ/dt dt is the winding number (topological charge). At θ=π, Haldane's topological term causes level crossing in integer-spin chains. Shown: energy levels E_m = g(m−θ/2π)² as a function of θ, plus the tunneling amplitude landscape.