^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

This work deals with the behavior of fermions in the background of kinklike structures in the two-dimensional spacetime. The kinklike structures appear from bosonic scalar field models that engender distinct profiles and interact with the fermion fields via the standard Yukawa coupling. We first consider two models that engender parity symmetry, one leading to the exclusion of fermion bound states, and the other to the inclusion of bound states, when the parameter that controls the bosonic structure varies from zero to unity. We then investigate a third model where the kinklike solution explicitly breaks parity symmetry, leading to fermion bound states that are spatially asymmetric.

Article funded by SCOAP^{3}

The study of fermions in the presence of kinklike structures has been initiated long ago, in the pioneering work by Jackiw and Rebbi [

The interest in the fermion number fractionalization goes beyond its mathematical identification since it presents peculiarities that can be physically realized in condensed matter situations, as shown in Refs. [

The geometrical aspects of the structure is of current interest, since experiments may now be carried out on miniaturized samples in constrained geometries, and the geometry may drastically change the conformational structure of the topological object, as experimentally verified for instance in Ref. [

There are many reasons to study the interaction of fermion fields with bosonic backgrounds since it may create or affect other interesting physical phenomena like the Casimir effect [

To implement the investigation, we organize the work as follows: In Sec.

We are interested in studying models described by the Lagrangian
_{ϕ} is the derivative of some function

We consider the topological structure of the kinklike profile which arises from the bosonic Lagrangian

The equation of motion for the fermion field has the form
_{0},_{1},_{5}) = (_{1}, _{3},_{2}). Moreover, since the scalar field describes a static structure we write the spinor field as
_{±}(^{2}, with

We note that equations (^{∓}^{±}^{(±)} = E^{2}^{(±)}, where ^{±} = ±d/d^{±}^{(±)}=0, and obtain
_{±} are normalization constants; for regularity of the ground state one of them has to be zero. To find the massive bound states, one uses Eqs. (_{n}_{n}(_{n}_{0}(_{0}(^{†}^{†} from the left, using the Eq. (^{2} has non-negative eigenvalues.

Another interesting result is that the threshold energy is taken at the limit _{min}, where the bosonic field approaches the minimum of the scalar potential. Moreover, for the well-behaved solutions of the fermionic bound states we require that in this regime ^{(±)} → _{±} and d^{(±)}/d_{th}_{±}−2_{min}_{∓}=0, and thus one finds _{th}=2_{min}, which is equal to the square root of _{±} for spatial infinities, as expected.

Let us now consider some explicit models. We are interested in studying the fermion field behavior when it evolves in the background of a kinklike structure derived from the bosonic model defined in terms of the potential

Kinklike solutions for the bosonic models (^{2}=1/(1−λ)^{2}, which increases as λ → 1 and is not defined at λ = 1. Model (^{2}=4(1−λ^{2}), which decreases as λ → 1 and is well defined at λ = 1, where it is zero. For λ=0 these sets are equivalent, but they are different as we vary the λ parameter. In particular, the topological sector we choose to work on here approaches the vacuumless solution in the limit λ → 1.

The third model is defined by Eq. (

Considering the Lagrangian (^{1} as well as charge-conjugation symmetry which is representation dependent, and in the representation we have chosen is given by _{3}. Therefore, we expect that the fermionic bound energy spectrum is symmetric around the

Due to the relevance of the sine-Gordon model [_{ϕ}=cos(_{SG}=1−2sech^{2}(_{0} = sech(_{±}(±∞)=^{2}. The potential _{_}(_{0} = 0,_{1} = ±1.87806,_{2} = ±2.48335,_{3} = ±2.83358 and _{4} = ±3.03448. The zero mode can be obtained analytically and, up to a normalization factor, is given by
_{2}(_{2}(e^{–x})=i(Li_{2}(–ie^{–x})–Li_{2}(ie^{–x})). For the other bound states, we solve the set of equations in (

(color online) The ^{(+)} (blue, solid line) and ^{(−)} (red, dashed line) components of the massive bound states and the corresponding eigenenergies of the fermion field coupled to the sine-Gordon soliton (

Let us now look at model (^{2}, which implies that the depth of the well increases with λ. For λ = 0 the expression (

Given the Yukawa coupling between the boson and fermion fields, the Dirac field spectrum must be affected by changes in the behavior of the bosonic structure. For Model I, the fermionic eigenstates are given by equations (_{±} have the form
_{_} is shown in Fig. _{±}(±∞)=4^{2}, where _{±}(0)=±2/(1–λ). Therefore, for _{_} the depth of the well increases but its width reduces as λ→1. This effect is illustrated in Fig.

(color online) The _{_} potential that appears in Eq. (

(color online) The fermionic bound state energy spectrum as a function of λ for Model I. The solid (black) curves correspond to threshold energies.

Unfortunately, we could not find the analytical expression for the bound state wave functions corresponding to an arbitrary λ. Nevertheless, we could observe some characteristics of its behavior. In the vicinity of ^{–x/(1–λ)}+^{–2K(λ)x} in the ground state wave fucntion as λ → 1. The behavior of the fermionic zero mode wave function in these two limits suggests that as λ increases, the normalized wave function becomes taller and narrower, as illustrated in Fig.

(color online) The normalized fermion zero mode in Model I derived from equations (

We now study model (

Once we have chosen the solution (_{_}(

(color online) The _{_} potential that appears in (

(color online) The fermionic bound state energy spectrum as a function of λ for Model II. The solid curves correspond to threshold energies.

As in Model I, we could not find analytically the zero energy solution for the Dirac field for general λ, but we could still get some information about its behavior. In the neighborhood of _{λ=1}(^{−2ϕ∞x} for λ≠1, and it decays as e^{4x}^{–4x} for λ=1. We can integrate the vacuumless solution in order to find the exact form of the ground state at λ=1, which is

(color online) The normalized fermion zero mode in Model II derived from the solution (

We now study how asymmetries within the scalar potential can affect the behavior of the fermionic bound states. We perform the numerical analysis of model (^{2}, respectively. Note that as the scalar field asymptotically approaches

(color online) The _{_} potential of Model III, shown for

In Fig. ^{−2x} and ^{2px}, which implies that in the regime ^{−2x−e}^{−2x} and in the limit

(color online) The fermion bound energy spectrum as a function of

(color online) The normalized fermion zero mode derived from Model III, shown for

In Fig. ^{(+)} and ^{(−)} respect the parity symmetry for

The asymmetry of the normalized zero mode shown in Fig.

(color online) The ^{(+)} and ^{(−)} components of the massive bound states in Model III, shown for

(color online) The mean value

In this work we studied the behavior of the fermion field in the background of three kinklike structures that respond with distinct geometric conformations. The three bosonic structures arise from models described by a single real scalar field recently investigated with distinct motivations, but here we use them to see how the fermion bound energies and states behave in each case. The first two models are controlled by a real parameter, λ, which highlights fascinating characteristics of the models. The third model is different and is controlled by an odd integer parameter,

Model I has the peculiarity of describing a background potential for the fermion field, which deepens and narrows as λ approaches unity, such that the presence of the fermion bound states is reduced when λ increases. Model II has a distinct behavior, and the background potential is now capable of adding new fermion bound states as the parameter λ increases in the interval [0,1]. We find that as λ increases from zero to unity, the number of fermion bound states diminishes in Model I, while it increases without limit in Model II.

While Models I and II obey parity symmetry, Model III engenders another behavior, which is also of current interest. It is controlled by an odd integer ^{6} model, and also Ref. [