Coupled fermion-kink system in Jackiw-Rebbi model

In this paper we study Jackiw-Rebbi model, in which a massless fermion is coupled to the kink of $\lambda \phi^4$ theory through a Yukawa interaction. In the original Jackiw-Rebbi model the soliton is prescribed. However, we are interested in the back-reaction of the fermion on the soliton besides the effect of the soliton on the fermion. Also, as a particular example, we consider a minimal supersymmetric kink model, $\mathcal{N}=1$, in ($1+1$) dimensions. In this case, the bosonic self-coupling, $\lambda$, and the Yukawa coupling between fermion and soliton, $g$, have specific relation, $g=\sqrt{\lambda/2}$. As the set of coupled equations of motion of the system is not analytically solvable, we use a numerical method to solve it self-consistently. We obtain the bound energy spectrum, bound states of the system and the corresponding shape of the soliton using a relaxation method, except for the zero mode fermionic state and threshold energies which are analytically solvable. With the aid of these results we are able to show how the soliton is affected in general and supersymmetric cases. The results we obtain are consistent with the ones in the literature, considering the soliton as background.


Introduction
Solitons named by Zabusky and Kruskal in 1965 [1], first appeared as a solution for KdV equation [2]. They play important roles in diverse areas of physics, biology and engineering [3,4,5]. In one spatial dimension kinks, and in higher spatial dimensions vortices, monopoles, instantons and domail walls, are amongst the most important ones in this category. They are topological configurations which appear in different areas of physics such as high energy, atomic and condensed matter physics [6,7,8,9,10]. These topologically nontrivial configurations cannot be deformed continuously into a trivial vacuum configuration.
Models including coupled fermionic and bosonic fields are crucial in many branches of physics, specially when the bosonic field has the form of a soliton. As solitons can be viewed as extended particles with finite mass, finite energy at rest, the systems consisting of coupled fermionic and solitonic fields can be a good candidate to describe extended objects such as hadrons. Since 1958, with Skyrme's pioneering works [11,12,13,14,15], many physicists have tried to explain the hadrons and their strong interactions nonperturbatively using phenomenological nonlinear field theories [16,17,18].
When a fermion interacts with a soliton, an interesting phenomenon occurs which is the assignment of fractional fermion number to the solitonic state. In a theory where all the fields carry integer quantum numbers, the emergence of fractional quantum numbers has attracted a lot of interest. Much of the work in this area has been inspired by the Jackiw and Rebbi's pioneering work [19]. The presence of a soliton distorts the vacuum state and consequently can induce nonzero vacuum polarization and Casimir energy. Imposing a boundary condition or a background field like soliton can alter the fermionic spectrum in the system which results in Casimir energy. These phenomena have been widely discussed in the literature for different models and different dimensions [20,21,22,23].
The issue of quantum corrections to the classical soliton mass in (1 + 1) dimensional theories, specially in the context of minimal supersymmetric solitons, N = 1, have received a great deal of attention since 1970's [24,25]. Supersymmetric soliton models are amongst the most important coupled fermion-soliton systems. BPS saturation in N = 1 supersymmetric solitons at quantum level has a long and controversial history [26,27,28,29]. BPS solitons can emerge in the supersymmetric models in which superalgebras are centrally extended, having mass equal to the central charge. Centrally extended supersymmetric algebras admit a special class of massive representations which preserve some fraction of the supersymmetry of the vacuum and consequently form multiplets, shortened or BPS supermultiplets, which are smaller than a generic massive representation. In supersymmetric theories with solitons, topological quantum numbers appear as central extensions of the supersymmetric algebra. Furthermore, in a certain class of N = 2 supersymmetric field theories some nonperturbative effects have been calculated using dualities between field theories with pointlike particles and field theories with solitons [31,30].
In most of the models investigated in the literature consisting of coupled fermion-soliton systems, the soliton is considered as a background field. The main reason is that solving the nonlinear system treating both fields as dynamical (not prescribed) is in general extremely difficult to do analytically. In principle, the soliton in these systems can have infinitely different shapes. Based on Jackiw and Rebbi's work [19], the back-reaction of the fermion on soliton is small in the weak coupling regime and can be treated perturbatively. Therefore, in the weak coupling region considering soliton to be a prescribed field is a good approximation, although out of this region including strong coupling limit it fails considerably. In this paper we investigate a minimal supersymmetric kink model, N = 1, self-consistently and exactly within our numerical precision. As we solve the system self-consistently, it is possible to analyze not only the effect of the soliton on the fermion field but also the backreaction of the fermion field on the soliton. We study the system in both weak and strong coupling regimes. We show that our results are consistent with the ones discussed in literature in the weak coupling regime. This paper is organized in four sections: In section 2 we briefly introduce the minimal supersymmetric kink model in (1 + 1) dimensions and the formulation of the problem. In this section we write the lagrangian describing our model, in components, and the resultant equations of motion. In section 3 we obtain the bound states and bound energies of the system. We find the fermionic zero energy bound state and threshold bound energies analytically. To obtain other fermionic bound energies, bound states and the shape of the soliton, we use a numerical method called relaxation method. In the same section we show and compare the results in weak and strong coupling regimes. At the end of the section we show the classical soliton mass in the intervals including both weak and strong coupling regions. We also obtain the back-reaction of the fermion on the soliton. Based on the result, we compare the back-reaction in weak and strong coupling regions. Finally, section 4 is devoted to summarize and discuss our results.

Supersymmetric kink model
We consider the minimal supersymmetry, N = 1, in a two dimensional field theory. Adding a real Grassmann variable θ α , one can promote the two dimensional space x µ = {t, x} to superspace, taking α = 1, 2. The coordinate transformations that add supersymmetry to the Poincaré invariance are where γ µ are the Dirac matrices,θ = θγ 0 and α parametrizes the supersymmetry transformations. The superfield Φ(x, θ) is a function defined in superspace and can be expanded in a Taylor series in θ as φ(x), ψ(x) and F (x) being component fields of the superfield. The supersymmetric action is defined as The spinoral derivatives are defined as In terms of component fields components, the supersymmetric lagrangian ends up with the form where the subscript φ shows the derivative with respect to φ and F is an auxilary field. Using the Euler-Lagrange equations and choosing the bosonic potential to be the kink potential in which g = λ 2 and φ is considered static. It is interesting to notice that the supersymmetry relates the bosonic self-coupling λ and the Yukawa coupling g, although they can be different in general [19,24,32]. In order to guarantee a well-defined energy for the soliton, we have where prime denotes the differentiation with respect to x. Defining χ ≡ φ/φ 0 and ψ = e −iEt ψ 1 + i ψ 3 ψ 2 + i ψ 4 , these equations become in which the representation for the Dirac matrices is chosen as γ 0 = σ 1 , γ 1 = iσ 3 and γ 5 = σ 2 .
Due to the symmetry in the equation system, there are only two independent degrees of freedom in ψ. Therefore, we can solve only the real part of the spinor field, components ψ 1 and ψ 2 . Also, we rescale all the quantities to dimensionless ones as ψ → √ mψ, χ → χ, E → mE, λ → m 2 λ and x → x/m 1 .
As can be seen in the lagragian (2.7), the fermion field interacts nonlinearly with the pseudoscalar field. The system cannot be solved analytically without imposing the soliton to be a background field. Thus, we solve this coupled set of differential equations self-consistently and find the fermionic bound states and bound energies as well as the shape of the soliton using a numerical method. The shape of the static soliton in the model we considered is not prescribed and is determined by the equations of motion. With this, besides the effect of the soliton on the fermion, we are able to obtain at classical level the effect of the fermion on the soliton (the back-reaction) in the whole range of the coupling constant λ, within our numerical restrictions. The main advantage is to help us understand the system in both strong and weak coupling regimes.
In the weak coupling limit, λ → 0 (φ 0 → ∞), the last equation in (2.9) decouples from the others and has analytical solution, i.e. kink of λ φ 4 theory. The solutions of some analogous systems in this limit have been studied in detail in [19,32]. In this limit, the solutions for this equation are and there are five fermionic bound states with energies 0, ± 3 2 (m) and ± √ 2 (m). The E = ± √ 2 (m) states are threshold ones as the bound state goes to a constant at infinity. In this paper we consider the positive sign in eq. (2.10).
One can calculate the classical soliton mass using the expression for the kink of λ φ 4 theory. We solve the set of coupled equations (2.9) self-consistently and check the results with the ones in the weak coupling limit.
In the weak coupling limit the system has charge conjugation and particle conjugation symmetries. In this system, the charge conjugation operator is γ 1 , which is also the particle conjugation operator. However, as the coupling increases the λ 2ψ ψ term in second equation of (2.8) cannot be neglected anymore and breaks these symmetries. As a result, the energy spectrum is not symmetric around E = 0 in general.

Bound states and bound energies
We obtain zero energy bound state and threshold energies analytically, although to find the other bound states we have to rely on our numerical method.

Zero energy bound state
For this state the equations of motion are simplified and we are able to obtain the analytical solution of the system in the whole φ 0 interval. Interestingly, the solutions show to be independent of φ 0 . Imposing E = 0, the equation system becomes It turns out that the first two equations can be easily solved as functions of χ, yielding As the normalization of the divergent components should be zero, we can conclude either a 1 = 0 or a 2 = 0. Thus, the term with ψ dependence in the last equation vanishes and we find which corresponds to the trivial kink equation with the known solution (eq. (2.10)). This way one can determine f (x) = e Requiring the wave function to be normalized, we obtain It is important to notice that based on this result, the back-reaction of the fermion on the soliton is zero for the fermionic zero mode. This result would be expected if a system has particle conjugation symmetry. If a system does not possess this symmetry the back-reaction for the fermionic zero mode can be nonzero, in general [33]. Surprisingly, although the system considered here does not respect this symmetry, except in the weak coupling limit, it has the same property which means that this symmetry still has an imprint on the system. This property holds for the lagrangian considered in [19,32] with different fermionic and bosonic couplings.

Threshold states
Threshold or half-bound states are the states appearing in (1 + 1) Quantum Field Theory, where the fermion field goes to a constant at spacial infinity. For these states when x → ∞ the wave function is finite, but does not decay fast enough to be square-integrable [29,34,35].
To find such states in our system we solve the system of equations at x → ±∞. We write ψ The constants c 1 and c 2 can be either zero or nonzero constants in general when the energy is nonzero. If c 1 and c 2 are nonzero constants, this system has solution only in the weak coupling limit, φ 0 → ∞. Solving the system in this limit, it is easy to show that the energy of the threshold states are E = ± √ 2 (m), as expected. Except for the limit φ 0 → 0, the back-reaction of the fermion on the soliton is finite. When the disturbance region is finite, the continuum fermionic state should tend to the plane wave out of this region. Therefore, the threshold fermionic state should go to a nonzero constant to match both bound and continuum regions. This condition holds everywhere except for the limit φ 0 → 0, as can be seen in the last equation of (2.9). It means that, except for this limit, no bound state arises from or joins continua besides those at the weak coupling limit.

Numerical Method
The other bound states cannot be found analytically and a numerical method is required. We use a relaxation method that starts with an initial guess and iteratively converges to the solution of the system. We start with the known energy spectrum and bound states in the weak coupling limit [32] and find the solution for the whole region of φ 0 including the strong coupling region.
There are two first order differential equations and one second order equation (eq. (2.9)) that can be transformed to a set of two first order equations as To find the fermion energy eigenvalue we use the equation E = 0, reflecting the fact that energy is spatially constant. Moreover, we fix the translational symmetry of the system by choosing x 0 = 0. Now, there are five coupled first-order differential equations to solve which need five boundary conditions. Among the several boundary conditions available, we choose the following The left graph in figure 1 shows the fermionic bound state energies as a function of the asymptotic value of the bosonic field, φ 0 . As can be seen, the weak coupling limit result is retrieved as φ 0 → ∞, i.e. E = ± 3/2 (m). It is important to note that for φ 0 2 the dynamical graphs and the lines E = ± 3/2 (m) are not easily distinguishable, although for smaller φ 0 the negative and positive energies change drastically from the ones in the weak coupling limit. In the numerical simulations the closer φ 0 is to zero, the more difficult is for the solutions to converge. Thus, the smallest values of φ 0 we are able to obtain are φ 0 = 0.501 for the positive bound energy and φ 0 = 0.564 for the negative one, though based on physical intuition it is possible to guess partially how the energy curves would behave below these values. The positive bound energy curve should not cross the zero energy line as it would configure level crossing [32]. Furthermore, as the negative energy curve becomes closer to the threshold line E = − √ 2 (m), its slope decreases considerably at φ 0 ≈ 0.63, as the right graph of figure 1 shows. It means that it probably does not join the Dirac sea, except for the limit φ 0 → 0, which is consistent with the discussion in the section 3.2. are the result in the weak coupling limit. We show the results for two different low values of φ 0 for positive and negative energy states in order to highlight the effects in the strong coupling region. As can be seen in the graphs, in lower φ 0 case the dynamical and static results become more different which confirms that in the low φ 0 region the system cannot be described by the weak coupling approximation.
To investigate the effect of the fermion on the shape of the soliton, in Figures 4 and 5 we show the bosonic field as a function of x for positive and negative energies, respectively. As before, the solid curves show the result of the dynamical model and the dotted ones the result for model with prescribed soliton. Since the results for positive energy change considerably in low φ 0 region, we show the result for three different values of φ 0 to make it possible to track the transition to the strong coupling limit. For each of the graphs we show χ and its spatial derivative, χ , to illustrate how the soliton changes from the prescribed one. It is important to notice that although the soliton can change drastically in the strong coupling region, the changes are limited to a small region around the origin. This result confirms that the back-reaction of the fermion on the soliton and thus the disturbance region are finite, except for the limit φ 0 → 0, which is also consistent with the discussion in the section 3.2.
Using the expression (2.11), the classical mass of the soliton for both positive and negative energy bound states of the dynamical model as well as for the model with background soliton is shown in figure 6 with solid, dashed and dotted curves, respectively. The weak coupling approximation and the concept of the mass of the soliton based on the expression (2.11) is not valid at low φ 0 as can be seen in this graph, although for φ 0 greater than 1 these three curves coincide within the scale shown in the graph.
As a measure of the effect of the fermion on the soliton we calculate the root mean squared deviation between the static and dynamical soliton, δ RM S . In figure 7, we show this result as a function of φ 0 and energy, for both positive and negative bound states. As expected, the back-reaction of the fermion on the soliton goes to zero when E → ± 3/2 (m), i.e. the weak coupling limit. Interestingly, the right graph in this figure shows that the back-reaction decreases almost linearly with energy. Also, the left graph of this figure confirms that when φ 0 goes to zero the back-reaction increases significantly and cannot be neglected in this region, i.e. the strong coupling region.

Conclusion
In this paper we investigate a minimal supersymmetric soliton model in two dimensions in which a static pseudoscalar field is interacting nonlinearly with a Dirac particle. In this system the bosonic self interaction part of the potential that is responsible for creating a soliton with proper topological characteristics is considered to be the potential in λφ 4 theory. By varying the value of the bosonic field at spacial infinity, φ 0 , from zero to infinity we can span the whole region between the strong coupling limit and the weak coupling limit. In this system we find the zero mode fermionic state and threshold energies analytically, although in order to find other bound states and the corresponding shape of the soliton we need a numerical method. We use a relaxation method to calculate the energy spectrum and the bound states as well as the shape of the soliton. Our calculations show that despite the system not having particle conjugation symmetry except in the weak coupling limit, the back-reaction of the fermion on the soliton for the fermionic zero mode is zero. It means that although this symmetry is broken, it still has an imprint on the system. Therefore, the soliton corresponding to the fermionic zero mode is the known kink of λφ 4 theory for the whole interval of φ 0 . The analytic calculation of threshold states and the fact that they should match with bound and continuum states out of the disturbance region show that except for the limit φ 0 → 0, no bound state joins to or arises from the continua besides the ones existing in the weak coupling limit. The shape of the soliton for positive and negative energy states that we calculate numerically also confirms this reasoning. We also calculate the classical soliton mass and show that in the weak coupling region the dynamical and weak coupling approximation results coincide, though they are completely distinguishable in the strong coupling region. In the end, we calculate the back-reaction of the fermion on the soliton for the positive and negative energy states as a function of φ 0 and E.
The results show that the back-reaction of the fermion on the soliton tends to zero as φ 0 → ∞ for both positive and negative bound energy curves, as expected. In contrast, when φ 0 → 0, the strong coupling limit, the back-reaction increases significantly which means that it cannot be omitted in the strong coupling region.   Figure 5: The bosonic field and its derivative with respect to x corresponding to negative energy for φ 0 = 1 and φ 0 = 0.6. Solid and dashed curves depict the bosonic field and its derivative in the dynamical model and the one in the weak coupling limit, respectively.