Fermion transfer in the ${\varphi }^4$ model with a half-BPS preserving impurity

We study a fermion field coupled to a scalar via a Yukawa term. The scalar field is the ${\varphi }^4$ model with an impurity that preserves half of the Bogomol’nyi-Prasad-Sommerfield (BPS) property. We analyze the spectrum of the defects of the model and collisions between them both close to the BPS regime and not. As the fermion binds to these defects, it may be transferred from one to the other, which we quantify via overlaps, known as Bogoliubov coefficients. BPS collisions are less likely to transfer the fermion between defects and can be adiabatic for nonrelativistic velocities, especially for small coupling constants. Moreover, closer to the BPS limit only a small fraction of the fermion number is radiated away. In contrast, non-BPS collisions lead to more radiation in the fermion field and excitation of the fermion to higher bound states, and the result is more sensitive to the parameters.

Figure 1

BPS solutions of the scalar field. The full line is the lump solution, The dashed line is an antikink to the left and a lump and the dotted line is the antikink-on-impurity solution. This figure is a reproduction of results in [45].

Figure 2

Spectrum of the fermion field coupled to (a) the lump and (b) to the BPS antikink with $\lambda =-1.0$. We set $g=2.0$ (the supersymmetric case). The spectrum is identical to the scalar field spectrum in [45], as expected.

Figure 3

Upper graphs: Evolution of the scalar field during a collision between (a) an antikink and a lump and (b) an antikink and a kink-on-impurity. These two graphs are the reproduced results in Figs. 14 and 24 in [45] with different parameters. Lower graphs: Evolution of the fermion field during the background collision between (c) the antikink and the lump and (d) the antikink and the kink-on-impurity. Parameters are $g=2.0$, $v=0.3$ and $\lambda =-1.0$.

Figure 4

(a) Snapshots (solid) of the scalar field configuration during an adiabatic evolution of an antikink-lump collision with $v=0.02$ and $g=2.0$. Superimposed in the curves we have the static BPS antikink solution (dotted) and the two curves are indistinguishable. (b) Same as before for the fermion density. The dotted curves are now the density of the fermion eigenstates coupled to the corresponding static BPS antikinks.

Figure 5

Evolution of the Bogoliubov coefficients as a function of the final time $t$. (a) Corresponds to the antikink-lump collision and (b) to the collision between antikink and kink-on-impurity. Parameters are $v=0.3$, $g=2.0$ and $\lambda =-1.0$.

Figure 6

Bogoliubov coefficients versus $g$ for an antikink-lump collision with different values of $v$. We take $\lambda =-1.0$.

Figure 7

Bogoliubov coefficients versus $g$ for the collision between an antikink and the kink-on-impurity, with different values of $v$. We take $\lambda =-1.0$.