An Investigation of the Casimir Energy for a Fermion Coupled to the Sine-Gordon Soliton with Parity Decomposition

TL;DR

This paper calculates the Casimir energy for a fermion coupled to a sine-Gordon soliton, analyzing parity channels separately, and finds that the most stable configuration involves a soliton with winding number one.

Contribution

It introduces a method to compute parity-decomposed Casimir energies and examines their behavior in fermion-soliton systems, revealing new insights into stability and spectral properties.

Findings

Distinct deformations occur at zero-energy fermionic bound state crossings.

Parity-decomposed Casimir energies reveal properties not visible in total energy.

The ground state favors a soliton with winding number one.

Abstract

We consider a fermion chirally coupled to a prescribed pseudoscalar field in the form of the soliton of the sine-Gordon model and calculate and investigate the Casimir energy and all of the relevant quantities for each parity channel, separately. We present and use a simple prescription to construct the simultaneous eigenstates of the Hamiltonian and parity in the continua from the scattering states. We also use a prescription we had introduced earlier to calculate unique expressions for the phase shifts and check their consistency with both the weak and strong forms of the Levinson theorem. In the graphs of the total and parity decomposed Casimir energies as a function of the parameters of the pseudoscalar field distinctive deformations appear whenever a fermionic bound state energy level with definite parity crosses the line of zero energy. However, the latter graphs reveal some properties of the system which cannot be seen from the graph of the total Casimir energy. Finally we consider a system consisting of a valence fermion in the ground state and find that the most energetically favorable configuration is the one with a soliton of winding number one, and this conclusion does not hold for each parity, separately.