Complete spectral analysis of the Jackiw-Rebbi model, including its zero mode

Abstract

In this paper we present a complete and exact spectral analysis of the (1+1)-dimensional model that Jackiw and Rebbi considered to show that the half-integral fermion numbers are possible due to the presence of an isolated self charge conjugate zero mode. The model possesses the charge and particle conjugation symmetries. These symmetries mandate the reflection symmetry of the spectrum about the line E = 0. We obtain the bound state energies and wave functions of the fermion in this model using two different methods, analytically and exactly, for every arbitrary choice of the parameters of the kink, i.e. its value at spatial infinity (θ0) and its scale of variations (µ). Then, we plot the bound state energies of the fermion as a function of θ0. This graph enables us to consider a process of building up the kink from the trivial vacuum. We can then determine the origin and evolution of the bound state energy levels during this process. We see that the model has a dynamical mass generation process at the first quantized level and the zero-energy fermionic mode responsible for the fractional fermion number, is always present during the construction of the kink and its origin is very peculiar, indeed. We also observe that, as expected, none of the energy levels crosses each other. Moreover, we obtain analytically the continuum scattering wave functions of the fermion and then calculate the phase shifts of these wave functions. Using the information contained in the graphs of the phase shifts and the bound states, we show that our phase shifts are consistent with the weak and strong forms of the Levinson theorem . Finally, using the weak form of the Levinson theorem, we confirm that the number of the zero-energy fermionic modes is exactly one. * Electronic address: f charmchi@sbu.ac.ir † Electronic address: ss-gousheh@sbu.ac.ir

Fermionic bound states in distinct kinklike backgrounds

“…Similar investigations have been carried out in Refs. [16][17][18][19][20][21]. It is worth mentioning that both potentials (2.7) and (2.8) reproduce the φ 4 model in the case p = n = 1.…”

“…For n = 1, the system is analytically solvable and it is easy to show that there are three fermionic bound states with energies E = 0 and E = ± √ 3 and two threshold ones with energies E = ±2 (see [20,21]). Besides analytical results E = 0 and E = ±2, our numerical method results in the bound energies E = ±1.7316.…”

Coupled fermion–kink system in Jackiw–Rebbi model

“…the kink of λ φ 4 theory. The solutions of some similar systems in this limit have been studied in detail in [25,36]. In this limit, the solutions for this equation are…”

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

“…However, it is well known that the phase shifts of the upper and lower components are generically not equal, and this is indeed the case when the Fermi field is coupled to a topologically nontrivial configuration. We have devised a simple prescription to define a unique phase shift for the Dirac spinor [100] and have tested its validity using both the strong and the weak forms of the Levinson theorem [62,100,101]. We shall use this prescription in this paper, and this will be yet another test.…”

“…We have devised a prescription to define a unique phase shift for the Dirac spinors in these cases: We define the phase shift to be the average of those two phase shifts [100]. We have checked [62,100,101] the correctness of this prescription in several different cases using both the weak [102] and the strong [100] forms of the Levinson theorem. In this paper we also check the consistency of our results with the Levinson theorem for the phase shifts of the parity eigenstates and also the scattering states.…”