Casimir Energy for a Coupled Fermion-Soliton System

Abstract

In this paper we compute the Casimir energy for a coupled fermion-pseudoscalar field system. In the model considered in this paper the pseudoscalar field is \textit{static} and \textit{prescribed} with two adjustable parameters. These parameters determine the values of the field at infinity ($\pm {\theta }_0$) and its scale of variation ($\mu$). One can build up a field configuration with arbitrary topological charge by changing ${\theta }_0$, and interpolate between the extreme adiabatic and non-adiabatic regimes by changing $\mu$. This system is exactly solvable and therefore we compute the Casimir energy exactly and unambiguously by using an energy density subtraction scheme. We show that in general the Casimir energy goes to zero in the extreme adiabatic limit, and in the extreme non-adiabatic limit when the asymptotic values of the pseudoscalar field properly correspond to a configuration with an arbitrary topological charge. Moreover, in general the Casimir energy is always positive and on the average an increasing function of ${\theta }_0$ and always has local maxima when there is a zero mode, showing that these configurations are energetically unfavorable. We also compute and display the energy densities associated with the spectral deficiencies in both of the continua, and those of the bound states. We show that the energy densities associated with the distortion of the spectrum of the states with $E>0$ and $E<0$ are mirror images of each other. We also compute and display the Casimir energy density. Finally we compute the energy of a system consisting of a soliton and a valance electron and show that the Casimir energy of the system is comparable with the binding energy.

11 pages, 6 figures

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• An Investigation of the Casimir Energy for a Fermion Coupled to the Sine-Gordon Soliton with Parity Decomposition
• Majorana zero mode-soliton duality and in-gap and BIC bound states in modified Toda model coupled to fermion
• Massive fermion between two parallel chiral plates
• Vacuum Polarization and Casimir Energy of a Dirac Field Induced by a Scalar Potential in One Spatial Dimension