Quantum tunneling for the sine-Gordon potential: Energy band structure and Bogomolny-Fateyev relation

Abstract

A path-integral calculation of the energy band structure for the sine-Gordon potential with the help of the instanton method is presented. The periodic pseudoparticle configuration obtained recently is seen to be responsible for tunneling at a finite energy which leads to the level splitting of excited states. The lowest band structure due to the tunneling of vacuum instantons is recovered from our result in the low energy limit. The above result is obtained by considering a half period of the classical solution as a kinklike configuration (with nontrivial topological charge). On the other hand if a full period of the solution is treated as a nontopological (bouncelike) pseudoparticle configuration, the imaginary parts of energy eigenvalues can be calculated. We then obtain the Bogomolny-Fateyev relation which is well known in perturbation theory at large order. The agreement of our results with those of WKB calculations also resolves a controversy concerning the method: In fact this method yields the same correct result as the method of complex paths.