PHYSICAL REVIEW D 101, 021701(R) (2020) Rapid Communications Kinks bounded by fermions Ilya Perapechka Department of Theoretical Physics and Astrophysics, BSU, Minsk 220004, Belarus Yakov Shnir BLTP, JINR, Dubna 141980, Moscow Region, Russia Department of Theoretical Physics, Tomsk State Pedagogical University, Tomsk 634061, Russia (Received 24 October 2019; published 28 January 2020) We present and study a new mechanism of interaction between solitons based on the exchange interaction mediated by the localized fermion states. As particular examples, we consider solutions of simple (1 þ 1)-dimensional scalar field theories with self-interaction potentials, including the sine-Gordon model and the polynomial ϕ4 and ϕ6 models, coupled to the Dirac fermions with a backreaction. We discover that there is an additional fermion exchange interaction between the solitons, and it leads to the formation of static multisoliton bound states. Furthermore, we argue that similar mechanisms of formation of stable coupled multisoliton configurations can be observed for a wide class of physical systems. DOI: 10.1103/PhysRevD.101.021701 I. INTRODUCTION zero mode is generated via supersymmetry transformations of the boson field of the static soliton. The breaking of One of the most interesting features of the topological supersymmetry of the configurations, in agreement with the solitons, like kinks, vortices, or monopoles (see, e.g., [1,2]), index theorem, shows a spectral flow of the eigenvalues of the is the remarkable relation between the topological charge of Dirac operator with some number of normalizable bounded the configuration and the number of fermionic zero modes modes crossing zero. localized on a soliton. The fundamental Atiyah-Patodi- The typical assumption in most such considerations is that Singer index theorem requires one normalizable fermionic the spinor field does not backreact on the soliton [5–7,24]; zero mode per unit topological charge [3]. Moreover, apart moreover, only the fermion zero modes were considered in from zero modes, most configurations support the existence most cases. A completely different approach to the problem of a tower of localized fermionic modes with nonzero energy. was proposed in our previous works [25–27], where we The fermion zero modes of solitons have been studied reconsidered this problem consistently. We found that the for many decades; these states localized on the vortices backreaction of the localized fermions significantly modifies were discussed first in Ref. [4]. There has been substantial both the spectral flow and the structure of the coupled interest in the study of these localized states in various configurations. In particular, it results in deformations of dimensions; examples are fermion modes of the kinks [5–8], the solitons, which to a certain extent resemble the excitations monopoles [9,10], sphalerons [11,12], and skyrmions of the internal modes of the scalar configurations [27–29]. [13–16]. The existence of localized fermions leads to many In contrast to previous studies, our focus here is not to interesting and unusual phenomena such as fermion number consider localized fermion states but rather to investigate fractionization [6,17], monopole catalysis [9,10], and string the pattern of interactions between the solitons in the superconductivity in cosmology [18] or in chiral super- presence of the additional exchange interaction mediated conductors [19]. by localized fermions with finite energy. Our aim is to study Fermion zero modes naturally appear in supersymmetric 1 the collective fermion modes in the system of kinks in theories; several simple examples are the N ¼ chiral 1 1 1 1 ( þ )-dimensional scalar field theories, including the scalar superfield in þ dimensions [20] or supersym- ϕ4 ϕ6 metric extensions of the O(3) nonlinear sigma model [21] sine-Gordon model and the polynomial and models, and baby-Skyrme model [22,23]. In such a case, the fermion coupled to the Dirac fermions with a backreaction. Notably, solitons in all these models may support the fermion zero mode, which does not affect the kink for any values of the Yukawa coupling. Furthermore, a single ϕ4 kink or a kink Published by the American Physical Society under the terms of in the sine-Gordon model [30] may bound other localized the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to fermion modes with finite energy; the number of these the author(s) and the published article’s title, journal citation, modes extracted from the positive and negative continuum and DOI. Funded by SCOAP3. increases as the Yukawa coupling becomes stronger. 2470-0010=2020=101(2)=021701(5) 021701-1 Published by the American Physical Society ILYA PERAPECHKA and YAKOV SHNIR PHYS. REV. D 101, 021701 (2020) x The novel aspect is that an exponentially localized fermion ΦSG ¼ 4 arctan e ; Φϕ4 ¼ tanhðxÞ; bound state may appear in the system of kinks, just in the rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ tanh x same way as the collective scalar bound states become Φϕ6 ¼ : ð3Þ trapped by the kink-antikink potential in the ϕ6 model [31]. 2 In this paper, we investigate multisoliton-fermion sys- The spectrum of linearized perturbations of the kinks tems with a backreaction numerically and elucidate the always includes a translational zero mode ΦðxÞ → mechanism for the appearance of the collective fermionic Φðx − x0Þ and the continuum modes (e.g., see [2,28]). modes. Our computations reveal sequences of new ϕ4 bounded collective fermion states in various systems of A peculiarity of the model is that, apart from these modes, Φ 4 solitons. Furthermore, we show that these localized modes there is also an internal bosonic mode of the kink ϕ . give rise to additional interaction between the solitons. The coupled pair of the first-order differential equations in (2) over the background provided by the kinks (3) can be transformed into two decoupled Schrödinger-type equa- II. THE MODEL tions for the spinor components u and v [5]: The (1 þ 1)-dimensional field theory we are interested in − Φ 2 − Φ ϵ2 ; is defined by the following Lagrangian: uxx þððm þ g Þ g xÞu ¼ u 2 2 −vxx þððm þ gΦÞ þ gΦxÞv ¼ ϵ v: ð4Þ 1 μ μ L ¼ ∂μϕ∂ ϕ þ ψ¯ ½iγ ∂μ − m − gϕψ − UðϕÞ; ð1Þ 2 We can easily see that the ϕ4 potential supports exponen- tially localized fermion zero mode ϵ0 ¼ 0, up to some ϕ where Uð Þ is a potential of the self-interacting real scalar modifications of the coupling term [30] or the potential; ϕ ψ field , is a two-component Dirac spinor, and m and g are such a mode exists for all types of the configurations above. the bare mass of the fermions and the Yukawa coupling The zero eigenvalue does not depend on the Yukawa 1 γ γ σ constant, respectively. The matrices μ are 0 ¼ 1 and coupling g, and there is no level crossing spectral flow γ σ σ ψ¯ ψ †γ0 1 ¼ i 3, where i are the Pauli matrices, and ¼ . in one spatial dimension. In the special case of the N ¼ 1 The sine-Gordon (SG) model corresponds to the periodic supersymmetric generalization of the model (1) [20], this potential UðϕÞ¼1 − cos ϕ with an infinite number of mode is generated via supersymmetric transformation of 4 degenerate vacua ϕ0 ¼ 2πn, n ∈ Z, the ϕ model corre- the translational mode of the static kink. 1 2 2 4 sponds to the quartic potential UðϕÞ¼2 ð1 − ϕ Þ with One of the essential features of the symmetric ϕ kink 6 two vacua ϕ0 ∈ f−1; 1g, and the ϕ theory is defined by is that, for large values of the Yukawa coupling, other UðϕÞ¼1 ϕ2ð1 − ϕ2Þ2 with triple degenerated vacuum localized fermionic states with nonzero energy eigenvalues 2 ε − – ϕ0 ∈ f−1; 0; 1g. j j < jg mj appear in the spectrum [5 8,27,30]. Using the standard parametrization for a two-component The consideration of the fermion modes bounded to a spinor soliton usually invokes a simplifying assumption that the backreaction of the localized fermions is negligible [5–7,24]. However, coupling to the higher localized modes may − ϵ uðxÞ ψ ¼ e i t ; significantly distort the ϕ4 kink; its profile deforms as a v x ð Þ fermion occupies an energy level [27]. Furthermore, since such exponentially localized fermion modes may occur in we obtain the following coupled system of dynamical multisoliton systems, localized fermions could mediate the equations: exchange interaction between the solitons. In particular, they may appear as a collective state trapped by the kink-antikink ϕ 2 − 0 0; xx þ guv U ¼ pair [7,30], in a way analogous to the appearance of the 6 ux þðm þ gϕÞu ¼ ϵv; collective bosonic modes in the ϕ model [31]. −vx þðm þ gϕÞv ¼ ϵu: ð2Þ III. NUMERICAL RESULTS RThis system is supplemented by the normalization condition To find a numerical solution of the complete system of ∞ 2 2 1 −∞ dxðu þ v Þ¼ , which we impose as a constraint on integral-differential equation (2) with the normalization the system (2). condition on the spinor field, we used the eighth-order For all previously mentioned potentials, in the decoupled finite-difference method. The system of equations is dis- 0 limit g ¼ , the model (1) supports spatially localized static cretized on a uniform grid with the usual size of 5000 topological solitons, the kinks: points. For the sake of simplicity, we considered the fermions with zero bare mass m ¼ 0. The emerging system 1Here we are using rescaled dimensionless variables. of nonlinear algebraic equations is solved using a modified 021701-2 KINKS BOUNDED BY FERMIONS PHYS. REV. D 101, 021701 (2020) Newton method. The underlying linear system is solved with the Intel MKL PARDISO sparse direct solver.