An investigation of the Casimir energy for a fermion coupled to the sine-Gordon soliton with parity decomposition

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, i.e. the spectrum of the states and the phase shifts, 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 the 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 show considerable sensitivity to the fine details of the shape of the background field 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.

Scattering of fermionic isodoublets on the sine-Gordon kink

“…In addition, the kink's field distorts the energy levels of the fermionic vacuum; bound states can arise and continuum states can change compared to a free fermion. These lead to a change in the zero-point fermion energy, and consequently to the Casimir effect, in the presence of the kink [28].…”

Fermions on wobbling kinks: normal versus quasinormal modes

“…The system composed of a fermion interacting with background φ 4 kink is analytically solvable and exhibits bound states as well as scattering states [6][7][8]. The properties of fermion-kink systems, such as the fermion energy spectrum and Casimir energy, have been studied for many different models [9][10][11][12][13][14]. In these systems, it is also possible to consider the back-reaction of the fermion on the soliton, which modifies the kink profile, especially for strong coupling constants [15][16][17][18].…”

Fermion bound states in geometrically deformed backgrounds

“…There are many motivations to study the interaction of fermion fields with bosonic backgrounds since it may create or affect other interesting physical phenomena like the Casimir effect [12,13], the Bose-Einstein condensation [14], and the localization of fermions in braneworld scenarios [15][16][17][18]. Another motivation is the current in-terest in the study of miniaturized samples of magnetic materials [11,[19][20][21][22] and the recent investigation [10].…”