The phi(4) model with the BPS preserving defect

JHEP03(2019)131 Published for SISSA by Springer Received:December 18, 2018 Accepted:March 17, 2019 Published:March 22, 2019 The φ4model with the BPS preserving defect C. Adam,aT. Romanczukiewiczband A. Wereszczynskib aDepartamento de F´ısica de Part´ıculas, Universidad de Santiago de Compostela and Instituto Galego de F´ısica de Altas Enerxias (IGFAE), Campus Sur, E-15782 Santiago de Compostela, Spain bInstitute of Physics, Jagiellonian University, Lojasiewicza 11, Krak´ow, Poland E-mail: [email protected],[email protected], [email protected] Abstract: The φ4model is coupled to an impurity in a way that preserves one-half of the BPS property. This means that the antikink-impurity bound state is still a BPS solution, i.e., a zero-pressure solution saturating the topological energy bound. The kink-impurity bound state, on the other hand, does not saturate the bound, in general. We found that, although the impurity breaks translational invariance, it is, in some sense, restored in the BPS sector where the energy of the antikink-impurity solution does not depend on their mutual distance. This is reflected in the existence of a generalised translational symmetry and a zero mode. We also investigate scattering processes. In particular, we compare the antikinkimpurity interaction close to the BPS regime, which presents a rather smooth, elastic like nature, with other scattering processes. However, even in this case, after exciting a sufficiently large linear mode on the incoming antikink, we can depart from the close-to-BPS regime. This results, for example, in a backward scattering. Keywords: Field Theories in Lower Dimensions, Solitons Monopoles and Instantons, Integrable Field Theories, Space-Time Symmetries ArXiv ePrint: 1812.04007 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP03(2019)131 JHEP03(2019)131 Contents 1 Introduction 1 2 The BPS φ4impurity model 3 2.1 The BPS soliton-impurity models 3 2.2 Kink-form preserving impurity 5 2.3 An exact example 6 3 Static solutions 7 3.1 The BPS sector — the antikink-impurity state 7 3.2 The generalised translational symmetry 9 3.3 Analytical description of the antikink-impurity state 11 3.3.1 Weak/strong impurity expansion 11 3.3.2 The composite impurity picture 12 3.4 Topologically trivial lumps 15 3.5 Non BPS kink-impurity state 17 4 Spectral structure 19 4.1 BPS antikink-impurity solution 20 4.2 The topologically trivial lump 21 4.3 The kink-impurity solution 23 5 Scattering processes 23 5.1 Moduli space approximation 24 5.2 Antikink-impurity scattering — numerical results 25 5.3 Kink-impurity scattering 32 5.4 Antikink-(kink-on-impurity) scattering 35 5.5 Kink-(antikink-on-impurity) scattering 40 6 Summary 42 1 Introduction Scalar field theories in (1+1) dimensions with double-vacuum potentials are known to support topological solitons, called kinks or antikinks, depending on the value of the topological charge. Here the topological charge is proportional to the difference of the asymptotic values of the scalar field at plus and minus infinity. Further, static solitons are zero-pressure configurations and satisfy the so-called Bogomolny equation [1] which, in contrast to the full Euler-Lagrange (EL) equation, is a first order equation. This guarantees that the pertinent energy bound is saturated, leading to the topological stability of the solitons. A model which possesses these properties is usually referred to as a Bogomol’nyi-Prasad- Sommerfield (BPS) theory. – 1 – JHEP03(2019)131 In fact, it has been shown recently that all scalar soliton models in (1+1) dimensions with at least two vacua enjoy the BPS property. This comprises both the inclusion of an arbitrary number of scalar fields and any dependence of the energy density on the first spatial derivative [2] as well as on higher derivatives [3–6]. In other words, the BPS property in (1+1) dimensions is a completely generic feature which does not imply any specific requirements on translationally invariant models. This differs completely from higher-dimensional generalizations, where only very particular models enjoy the BPS property — for example the O(3) σ-model and the Abelian Higgs model in (2+1) dimensions, the t’Hooft-Polyakov monopole (see [7] for a review) and the BPS Skyrme model in (3+1) dimensions [8], as well as the self-dual sector of the SU(2) Yang-Mills theory in (4+0) dimensions. In some sense, this drastic difference limits the application of (1+1) dimensional theories for the modeling of dynamical processes of higher dimensional solitons. In addition, the BPS sector in (1+1) dimensions contains only one-particle states, i.e., a kink or antikink separately. For models supporting compactons, i.e., solitons which differ from the vacuum only on a finite segment, one may easily construct a multi-particle state by a simple collection of non-overlapping compactons. However, by construction, all constituents do not interact with each other and can in fact be treated independently. In higher dimensions, the BPS sector is significantly more complicated and usually allows for multi-soliton configurations. One way to destroy the BPS property is to break the translational invariance by adding an impurity located in space [3,9,10]. However, quite interestingly, among the infinitely many couplings to defects (see for example [9,10]), there is a specific way of introducing an impurity where one-half of the BPS-ness is preserved [3]. This means that only one topological sector is a BPS sector saturating the corresponding topological bound. Hence, we have a rather unique chance to investigate, within the same model, both BPS and non-BPS solitons, which can allow us to analyze the impact of the BPS property on the dynamics of solitons. Such a BPS preserving coupling of defects exists for any 1+1 dimensional scalar soliton model without any restriction on the potential. Also the specific spatial distribution of the impurity is arbitrary, except for the finiteness of the L2norm, which physically means that the impurity is sufficiently well localized. This large freedom may give the chance for an experimental realization of this class of models, especially if one extends the impurity to a periodical lattice of BPS preserving impurities. This can be compared with the integrability preserving defect [11,12]. It is not surprising that the integrability puts much more constraints on the defect, as there are infinitely many currents which must be conserved. Therefore, it strongly restricts its form. To study properties of solitons in a model with the BPS preserving defect, we choose the φ4theory, which together with the sine-Gordon model, is perhaps one of the most studied examples of solitonic field theories in (1+1) dimensions. It is a prototypical field theory with solitons (kinks), which found applications in various physical systems from condensed matter [13] and biophysics to cosmology. It has been widely studied, especially in the context of time dependent solutions describing a collision of kinks, annihilation and creation processes as well as an interaction with radiation. Unlike the sine-Gordon model, where the collisions between defects are perfectly elastic, the φ4model reveals a very interesting resonant structure. The presence of this structure is related to the – 2 – JHEP03(2019)131 existence of the internal mode of the kink which can store the energy and give it back to the translational degrees of freedom in a resonant way [14–16]. Later, the effect with some improvement was discussed in [17,18] where a typographic error was corrected and the role of the shape mode was questioned. Similar structures were later observed in many other models, including the nonlinear Schr¨odinger equation [19,20], double sine-Gordone model [21], multicomponent models [22–26] or collisions with a dynamical boundary [27] or impurity [28]. A different mechanism was found in the case of non-symmetric kinks when the bound modes were formed between the kinks [29]. Recently, it was shown that also quasi-normal modes can be responsible for the creation of such a structure [30]. The main aim of the present paper is to better understand the role of the BPS property in the dynamics of solitonic models. Specifically, we begin with a systematic description of the static spectrum of the model in the topologically trivial sector (lump) as well as in the topologically nontrivial sectors (Q=±1) where kink and antikink exist. As only one of the solitons is a BPS state, we carefully examine their asymmetry as a function of a parameter which measures the strength of the impurity. Next, we analyze spectral properties of the static solutions which means the linear perturbation theory. Finally, we consider scattering processes between these objects. However, as one may choose the form of the impurity arbitrarily, we also want to investigate which properties are generic, i.e., independent on a particular choice of the defect. Here, the main finding is that a moduli space for the topologically nontrivial BPS soliton exists. As a consequence, there is a generalized translation symmetry which is reflected by the existence of a zero mode. Furthermore, we want to disentangle dynamical features which are inherited from the original pure φ4model from those which are due to the (specific) BPS preserving impurity. The last motivation comes from the fact that, in many aspects, our model is very similar to the Abelian Higgs model with the (one-half) BPS preserving defect introduced in [31]. In fact, the coupling to the impurity is given by exactly the same two terms. In the first one, the impurity couples to a topological quantity (the topological charge density and the magnetic field, respectively) while in the second one it multiplies the square root of the potential. As numerical studies in (1+1) dimensions are much simpler, our paper may also be viewed as a guideline for numerics for this vortex model [32,33]. In addition, we prove the existence of the zero mode (moduli space), which has not yet been completely achieved for the vortex model. 2 The BPS φ4impurity model 2.1 The BPS soliton-impurity models Here we briefly summarize some recent findings [3] on BPS soliton-impurity models. Let us begin with a general Lagrange density L=1 2φ2 t−1 2φ2 x−U−2σ√U−√2σφx−σ2(2.1) where φ(t, x) is a real scalar field in one spatial dimension, σ(x) is a space-located impurity and U(φ) is a potential with at least two isolated global minima φv +> φv −. For static – 3 – JHEP03(2019)131 configurations the energy reads E=Z∞ −∞ dx 1 2φ2 x+U(φ)+2σ√U+√2σφx+Z∞ −∞ dxσ2(2.2) where the last term, obviously, does not contribute to the field equations but sets the zero of the energy scale. It leads to the following Euler-Lagrange equation φxx −Uφ−σUφ √U+√2σx= 0 (2.3) where Uφdenotes the derivative w.r.t. the target space variable. Now, using the standard completing of the square trick, we can compute a lower bound on the energy E=Z∞ −∞ dx 1 √2φx+ (σ+√U)2 −√2Z∞ −∞ dxφx√U≥ −√2Zφ(+∞) φ(−∞) dφ√U(2.4) =−Q√2Zφv + φv − dφ√U(2.5) which is saturated by solutions of the following first order equation (the so-called Bogomolny equation) 1 √2φx+σ+√U= 0.(2.6) This is equivalent to the zero pressure condition T11 = 0. Here the topological charge Q=φ(+∞)−φ(−∞) φv +−φv − (2.7) is +1 for the kink and −1 for the antikink. Obviously, the topological charge is just a spatial integral of the temporal component of the topological current which is jµ= (φv +−φv −)−1µν∂νφ. Note that, contrary to the usual solitonic models, there is a fixed sign in front of the potential term in the Bogomolny equation. Therefore, only the antikink (or the kink if we change some signs in the model) is a BPS solution in such an impurity extended system, while the kink is a genuine non-BPS solution which generally does not saturate the bound. Furthermore, the energy bound suggests an asymmetry in the energies of kink and antikink. We also remark that, although the energy can be negative, it is bounded from below in each topological sector. Hence, the model is well defined and has a true vacuum. Let us remark that in the case of the BPS preserving impurity model a natural object is the prepotential W(also called superpotential), where U=W2(φ). The energy then reads E=Z∞ −∞ dx 1 2φ2 x+W2(φ)+2σW +√2σφx+Z∞ −∞ dxσ2.(2.8) Although the BPS soliton-impurity model seems to contain quite nontrivial couplings between the scalar field and the impurity via a potential as well as a derivative term, it may be written in a sort of effective potential way. Indeed, the term which is linear in φx – 4 – JHEP03(2019)131 can be replaced, up to a total derivative, by a new potential-like term. Then we get an equivalent formulation of the model E=Z∞ −∞ dx 1 2φ2 x+U(φ)+2σ√U−√2σxφ+Z∞ −∞ dxσ2(2.9) or simply E=Z∞ −∞ dx 1 2φ2 x+Ueff(φ, σ)(2.10) where the effective potential is Ueff(φ, σ) = U(φ)+2σ√U−√2σxφ+σ2.(2.11) Observe that, since the effective potential explicitly depends on the spatial coordinates, we cannot apply the former trick to derive the pertinent Bogomolny equation. We believe that this formulation might open a way for an experimental realization of this solitonimpurity model. 2.2 Kink-form preserving impurity It is an important consequence of our construction that the particular space distribution of the impurity is completely arbitrary provided it is L2integrable. Hence, a specific form can be dictated by a physical system we would like to model. However, there exists a special class of impurities which are selected by a mathematical property of the model, i.e., the original kink (or antikink) profile remains unchanged. This means that besides the nice and useful BPS property of the antikink, which allows for a reduction of the second order field equation to the first order Bogomolny equation, we get an exact solution of the non-BPS kink, which, in general, would require to solve the full static EL equation. Suppose that the scalar field in the model without impurity solves the equation φxx =Uφ⇒φx=±√2U . (2.12) The solution remains unchanged in the BPS soliton-impurity model if the impurity satisfies the following equation −σUφ+√2Uσx= 0 ,(2.13) which, using the original no-impurity equation (2.12), can be rewritten as −σφxx ±φxσx= 0 .(2.14) In the first case with a minus sign it reduces to −(σφx)0= 0 ⇒σ=α φx .(2.15) In the second case we can divide the equation by σ2and obtain −σφxx −φxσx σ2=−φx σ0 = 0 ⇒σ=αφx.(2.16) As we will see later, the parameter αcontrols the qualitative type of kink-impurity interaction (repulsive or attractive) in the non-BPS sector as well as its strength. – 5 – JHEP03(2019)131 2.3 An exact example Let us now consider the soliton-impurity model in the case of the φ4potential U=1 2(1 −φ2)2(2.17) where we suppressed the coupling constant. If the impurity is neglected we find the usual kink K0and antikink ¯ K0solutions φ=±tanh x(2.18) which saturate the energy bound E= 4/3 and have Q= 1 and Q=−1 topological charge, respectively. There are also two vacuum solutions φ=φv ±=±1 saturating the energy bound in the trivial sector. Now we add the impurity in the way described above. In our example the prepotential is chosen to be W(φ) = (1 −φ2)/√2. Hence, it is not a positive definite function. Nonetheless, it reproduces the φ4potential and therefore it is perfectly fine for our purposes. Observe that a different choice, for example W(φ) = |1−φ2|/√2, can modify our findings. From now on, the energy reads E=Z∞ −∞ dx 1 2φ2 t+1 2φ2 x+1 2(1 −φ2)2+√2σ(1 −φ2) + √2σφx+Z∞ −∞ dxσ2.(2.19) Then, the EL equation is −φtt +φxx + 2φ(1 −φ2)+2√2σφ +√2σx= 0.(2.20) The corresponding Bogomolny equation is 1 √2φx+σ+1 √2(1 −φ2) = 0.(2.21) Again, this equation implies the full static Euler-Lagrange equation. Thus, its solutions are solutions of the full variational problem. The energy bound is E≥ −4 3Q(2.22) and is saturated for the BPS solution. To specify a concrete example, we choose the impurity in the kink-preserving form which guarantees that the non-BPS kink solution is given in an exact way σ=α cosh2x.(2.23) Observe that this impurity is localized exponentially and centered at x= 0. Further, αis a real parameter. Finally, the Euler-Lagrange equation is −φtt +φxx + 2φ(1 −φ2) 1 + α√2 cosh2x(1 −φ2)!−2√2αtanh x cosh2x= 0.(2.24) – 6 – JHEP03(2019)131 3 Static solutions We shall find that we will be able to identify certain simple structures within general static solutions, so here we want to introduce some notation for these substructures. A kink and antikink are denoted by Kand ¯ K, respectively, and a kink bound to the impurity is denoted by K0. Further, topologically trivial lumps with asymptotic values ±1 are denoted by Σ±. 3.1 The BPS sector — the antikink-impurity state We start with the BPS sector defined by the Bogomolny equation 1 √2φx=−1 √2(1 −φ2)−α cosh2x.(3.1) The equation is invariant under the antisymmetric transformation φ(x)→ −φ(−x). Therefore, in the simplest case, it supports odd-symmetric solutions. The results are plotted in figure 1. For α= 0 we obviously re-obtain the usual antikink solution of the pure φ4theory, φ=−tanh x. As αincreases, the antikink-impurity solution initially goes beyond the ±1 values and then, in the vicinity of the location of the impurity, i.e., at x= 0, it exhibits a steeper and steeper descent. Solutions exist for arbitrarily large values of α. More interesting features happen for α < 0. Then, the BPS antikink is confined within the ±1 strip. For a sufficiently small α, the originally decreasing profile starts to grow close to the location of the impurity, which in the odd-symmetric case corresponds to φ= 0. The critical value of the parameter αbelow which φx(x= 0) takes a positive value is αcr =−1/√2. As αfurther decreases, this inner structure begins to look like a hidden pair of kink and antikink. Hence, the full BPS charge -1 solutions look like a composite three particle object, consisting of a left antikink, a central kink located at the impurity and a right antikink. For α→ −√2, the central kink approaches the tanh xprofile, while the two antikinks get repelled to plus/minus infinity. Of course, as the impurity is exponentially localized, they take the −tanh(x±x0) form, where x0→ ∞. Hence, the Bogomolny equation supports Q=−1 solutions for all α > −√2, with the energy E= 4/3. However, as we approach α=−√2, the just described inner structure emerges. The impurity hosts a hidden antikink-kink pair which together with the original antikink leads to a sort of three body bound state ¯ KK ¯ K, which is further bound to the impurity. In the limit α=−√2 both antikinks are sent to infinity and the kink tanh xremains located at the impurity. Indeed, in this limit the Bogomolny equation is 1 √2φx=−1 √2(1 −φ2) + √2 cosh2x(3.2) and has φ(x) = tanh xas its solution. Its energy is E=−4/3. However, the two antikinks contribute with 2×4/3, so the total energy agrees with the BPS energy in the Q=−1 sector. To summarize, the Bogomolny equation supports Q=−1 BPS solitons for α > −√2, which smoothly (as described above), tends to an isolated BPS Q= 1 kink for α=−√2. Probably, we can even say that if α=−√2, then both Q= 1 and Q=−1 are BPS – 7 – JHEP03(2019)131 −1.5 −1 −0.5 0 0.5 1 1.5 −10 −50 5 10 15 φBP S x α= 1.0 α= 0.5 α= 0 α=−0.5 α=−1 α=−1.3 α=−√2 + 10−2 α=−√2 + 10−3 α=−√2 + 10−4 α=−√2 + 10−5 α=−√2 + 10−6 Figure 1. Solutions of the Bogomolny equation for different values of αcentered at a= 0. −1.5 −1 −0.5 0 0.5 1 1.5 −10 −5 0 5 10 a) α=−1 −5 0 5 10 b) α= +1 φBP S x x x0=−5 x0=−4 x0=−3 x0=−2 x0=−1 x0= 0 x0= 1 x0= 2 x0= 3 x0= 4 x0= 5 Figure 2. Solutions of the Bogomolny equation for different positions of the topological zeros for α=±1. solutions. Thus, the model for this particular impurity is fully BPS, not half-BPS as occurs for all other impurities. However, remember that the Q=−1 solution contains two pure φ4antikinks located at plus/minus infinity. Note also that the usual energy degeneracy of the BPS solitons is lifted. In the case α < −√2 there are no finite energy BPS solutions. It turns out that the Bogomolny equation supports also non odd-symmetric solutions. Since they are BPS solutions, their energy is exactly the same as the energy of the oddsymmetric BPS antikink (and of a free φ4antikink). In the limiting case, they represent a configuration consisting of a well separated antikink (of the pure φ4theory) and an impurity-induced topologically trivial lump, while all other intermediate configurations are also possible. In other words, we can say that in the BPS sector the antikink and the topologically trivial lump admit a nonlinear superposition which changes the shape of the BPS solution while the energy is kept constant and all the time saturates the bound. In figure 2we show the solutions of the BPS equations with the condition φ(x0) = 0 for α=±1, where x0can be identified with the position of the antikink. For large values of |x0|the solutions look like an antikink centered at x0and a lump solution attached to the – 8 – JHEP03(2019)131 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −10 −5 0 5 10 φv ± x α=−√2 + 10−6 α=−1.4 α=−1.2 α=−1.0 α=−0.8 α=−0.6 α=−0.4 α=−0.2 α= 0.0 α= 0.2 α= 0.4 α= 0.6 α= 0.8 α= 1.0 α= 1.2 Figure 7. Topologically trivial lumps Σ+and Σ−for different values of α. Interestingly, the well pronounced three soliton hidden structure of the Q=−1 BPS solution occurs in the strong coupling regime, i.e., where αtakes the smallest possible value. In other words, when the impurity is strong then the Q=−1 solution reveals a structure of three weakly coupled constituents. On the other hand, when the impurity is weak, i.e., for α→0 (which for example allows for a perturbative treatment of the impurity), then the hidden solitons are in some sense strongly coupled and lose their identity. All this suggests a sort of weak-strong correspondence between the strength of the impurity and the strength of the interaction between the hidden solitons. 3.4 Topologically trivial lumps It is a straightforward consequence of the nonlinear superposition of antikink and topologically trivial lump in the BPS sector that, after sending the antikink to infinity, we remain with the pure lump solution located on the impurity. As the infinitely separated antikink has the form −tanh(x−x0) (with x0→ ∞), its energy is E= 4/3. Thus, the energy of the lump localised on the impurity is exactly 0. This means that the topologically trivial lump also saturates the energy bound and, in the above sense, is a BPS solution. In figure 7 we plot the topologically trivial lumps Σ+and Σ−for different values of α. Note that the lump is not symmetric under the x→ −xtransformation. The symmetry center of the lump is shifted from x= 0 where the impurity is located. First of all, let us observe that in the limit when α→ −√2, the topologically trivial lump, similarly as it happens for the solution in the Q=−1 BPS sector, clearly exhibits a hidden structure. In this limiting case, it consists of a kink located at the origin and an antikink sent to spatial infinity. Again, this picture, to some extent, is visible also for bigger values of α. To verify this conjecture, we assume the following decomposition of the lump φK0¯ K0 lump =−1 + tanh(x−xk)−tanh(x−xa) (3.25) where xkand xaare the positions of the pure φ4kink and the antikink, respectively. This trial function clearly takes into account the non-symmetric shape of the lump. We determine the optimal values of the fit parameters xkand xaby minimizing R(φnum −φK0¯ K0 lump )2, – 15 – JHEP03(2019)131 i.e., by a “least square” fitting procedure. In figure 8(left) we plot smin =xa−xk. In figure 5 (lower panel, solid blue curve) we also plot the energy of the two soliton approximation. As the energy of the BPS lump is always 0, the energy scale to which our approximation should be compared is the mass of the antikink. Hence, we get an accuracy better than 3.75%. It is clearly seen that the kink-antikink interpretation of the lump works extremely well for α→ −√2. When αgrows, then the kink and antikink quickly approach each other and lose their identity. However, our simple two particle approximation reproduces the qualitative features of the true solution reasonably well, even for bigger α, which supports our identification of the lump as a mixed state of kink and antikink. Of course, for higher α, the discrepancy is always visible. This means that the true lump solution is not a simple superposition of these solitons but that a mutual, nonlinear interaction modifies the state. Therefore, for αoutside a certain region close to −√2, the lump as a superposition of a kink and an antikink as well as their position inside the lump is again an interpretation which, however, works quite well. In particular, we may appreciate in figure 8that the distance between kink and antikink is zero for α= 0 and approaches infinity for α→ −√2, as it must be. In order to get an analytic insight into the separation distance between the hidden kink and antikink in the lump solution we apply a very simplified trial function where the position of the kink is always fixed at the origin φ=−1 + tanh x−tanh(x−s) (3.26) From the numerics we know that is it a good approximation only when α→ −√2. Once αgrows to 0 the kink also moves. But we may consider the above trial function as a crude approximation to the distance — not to the actual positions of the constituent solitons or the shapes of the solutions, especially as the position of the hidden kink xkis bounded from above and varies much less than its antikink counterpart. Then, again we may apply the standard energy minimization which gives α=−2√2es−12scosh s+ 9 sinh s+ sinh(3s) −5−4s−4(−1+2s)e2s+e4s(3.27) while the integration of the BPS equation leads to α=−√2(scoth s−s−1).(3.28) Again, both expressions reproduce the asymptotic behavior correctly. Furthermore, they coincide quite well — see figure 8(right panel). As the one parameter trial function does not reproduce the proper shapes for higher α, the derived formulas could be less precise than in the Q=−1 sector, especially when α→0. In the limit α→0 we can linearize the BPS equation and obtain for φ=−1 + √2αη ηx+1 cosh2(x)−2η= 0.(3.29) This equation has an exact solution: η(x)=2e2x1 1 + e2x+ 2x−log 1 + e2x+Ce−2x.(3.30) – 16 – JHEP03(2019)131 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −1.5−1−0.5 0 0.5 x α stop sfit 0 1 2 3 4 5 −1.4−1.2−1−0.8−0.6−0.4−0.2 0 s α s0 smin Figure 8. Left: the distance between the hidden kink and antikink in the lump (blue line), together with the positions of the topological zeros (red line). Right: the distance sbetween the antikinks and the central kink in the Q= 0 BPS solution in the fixed kink approximation, as a function of the impurity parameter α. Blue line — the energy minimization. Red line — the integration of the BPS equation. This time, the part coming from the homogeneous part has to vanish (C= 0), otherwise the solution would not be normalized. This is due to the fact that the lump is bound to the impurity and there is no translational freedom, as it happened in the case of the BPS antikink. This approximation works perfectly where it should, that is |α|<0.2. Clearly, the profile of the lump is not symmetric, and is not centered around x= 0. Obviously, these features are not captured by the symmetric K¯ Kapproximation in the limit of α→0 φ(x)≈ −1−√2 2 α cosh2x+O(α2).(3.31) 3.5 Non BPS kink-impurity state By construction, i.e., by the assumed form of the impurity, there is an exact non-BPS kink-impurity solution Kof the full second order static equation φK= tanh x(3.32) for any value of α. Although the profile of the kink is αindependent, its energy varies and reads EK=4 31+2√2α+α2.(3.33) Moreover, as we already remarked, for one isolated value of the parameter, α=−√2, this kink-impurity configuration is also a proper BPS solution, i.e., it saturates the bound and solves the Bogomolny equation. The energy of this BPS kink is E(α=−√2) = −4/3 and obviously saturates the topological bound for Q= +1. The non-BPS kink-impurity solution φKappears to be stable for α∈[−√2,0] while it develops an instability for α > 0. This means that there is a repulsive interaction between the kink and the impurity which expels the soliton to infinity. This can be easily proven if we compare the energy of the kink-impurity bound state (3.33) and the energy of the infinitely separated topologically trivial lump with E= 0 and free kink E= 4/3 EK−0 + 4 3=4α 32√2 + α(3.34) – 17 – JHEP03(2019)131 −3 −2 −1 0 1 2 3 −10 −5 0 5 a) α= 1 −10 −5 0 5 10 b) α=−1 φ(x) x x Σ+ Σ− K ¯ K tanh x K Figure 9. Kink-impurity K, antikink-impurity ¯ Kand topologically trivial lumps Σ±for the impurity model with α= 1 (left) and α=−1 (right). The dotted purple line corresponds to a static singular solution. which is positive for α > 0. Hence, for the bound state it is energetically favorable to decay into its infinitely separated constituents. As an example, in figure 9we show the kink, antikink and lump solutions for α= 1 and α=−1. The antikink (red curve) is a stable, BPS solution in both cases. For α=−1 a nontrivial inner structure (which grows to a kink-antikink pair as α→ −√2) is already visible. The kink solution, φK= tanh xis stable for α=−1 and unstable for α= 1 where it decays into the vacuum impurity solution and an infinitely separated kink of the pure φ4theory. To visualize this fact, we included a snapshot of the decay (kink running to the left) which was generated by introducing a small perturbation to the unstable kink solution (we remark that in figure 9the lump Σ+which remains close to x= 0 as a result of this decay cannot be distinguished from the pure lump solution Σ+). Note that the second order equation has even more static solutions satisfying the appropriate boundary conditions (for Q=±1) but developing a singularity at some point. For α=0 these singular solutions Kare given analytically in the form φK(x)=±coth(x−x0). When considered on a full line, such solutions have infinite energy, but they can play an important role in models with restricted domains [27]. In figure 9, the numerically found singular solutions are shown with dotted purple lines. Let us summarize the spectrum of static solutions in the BPS φ4-impurity model as a function of the strength of the impurity (the parameter α). In figure 10 we show the energies of the kink φK= tanh xand the BPS antikink as a function of the parameter α. The BPS antikink-impurity solution always saturates the energy bound E= 4/3, which obviously enforces the topological stability of the solution. Strictly speaking, there are infinitely many solutions representing the antikink at an arbitrary distance from the lump (always located on the impurity). Hence, we dynamically recover the translational invariance of the solitons in the pure φ4model. Here, the change of the position of the antikink modifies the shape of the solution but leaves the energy unchanged. As we will see below, this corresponds to the existence of a zero mode. – 18 – JHEP03(2019)131 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 −1.5−1−0.5 0 0.5 1 1.5 −√2 E α anti-K kink lump Figure 10. Energies of the kink-impurity, antikink-impurity and topologically trivial lump solutions as functions of α. The stable kink-impurity solution exists for α∈[−√2,0] and has energy smaller than the BPS antikink-impurity state. Its energy is, on the other hand, bigger than the bound, except for the very special case, α=−√2, when it becomes a BPS solution with energy E=−4/3. Note that in this limit the energies of the kink and antikink exactly cancel. As a consequence, such a pair has the energy of the topologically trivial lump. In other words, the energy needed for the creation of a kink-antikink pair drops to 0 as we approach α=−√2. This obviously coincides with our finding that for α=−√2 the lump is a composition of the kink and antikink. Moreover, since the constituents are exponentially localized and their distance grows, the separation energy also quickly drops to 0. Surprisingly, for α=−√2 + 1 the kink-impurity bound state and the topologically trivial lumps have exactly the same energies E= 0 which may be interpreted as an enhancement of the degeneracy of the vacuum, which contains not only topologically trivial lumps but also the kink-impurity solution. For α < −√2+1 the kink-impurity is the lowest energy state, and we have a unique and topologically nontrivial vacuum. Nonetheless, we will find that these facts do not have any impact on the dynamics of solitons in our model. 4 Spectral structure In order to investigate the spectral structure of the model, we have to perturb the solutions in each topologically distinct sector. We introduce a small perturbation around the static kink-impurity, antikink-impurity and topologically trivial lump, φ=φstatic +Aeiωtη(x) + c.c., where the perturbation obeys −η00 + (6φ2 static −2−2√2σ)η=ω2η(4.1) It is a well documented fact that the spectral structure of the static solutions plays an important and sometimes even crucial role not only in the analysis of stability and relaxation, but also in processes of kink-antikink and kink-impurity scatterings [14,29,30]. – 19 – JHEP03(2019)131 0 1 2 3 4 a) BPS ¯ K 0 1 2 3 4 a) BPS ¯ K -1.5 -1 -0.5 0 0.5 1 1.5 b) BPS Lump -1.5 -1 -0.5 0 0.5 1 1.5 b) BPS Lump -1 0 1 2 3 4 5 -1.5 -1 -0.5 0 0.5 1 1.5 c) non-BPS K -1 0 1 2 3 4 5 -1.5 -1 -0.5 0 0.5 1 1.5 c) non-BPS K ω2 n= 0 n= 1 n= 2 n= 3 n= 4 ω2 αα ω2 α ω2 α Figure 11. Spectral structure of the antikink (left), the lump (right) and the kink (bottom). 4.1 BPS antikink-impurity solution We start with the BPS antikink-impurity solution. Unfortunately, we do not know the analytic form of the profiles of these solutions. However, because they are of the BPS type and obey first order equations, they are very easy to find numerically. We have found the eigen-frequencies of the bound vibrational modes of the BPS solution centered at x= 0 using the linearized equation (4.1) substituting φwith the numerically found antikink-impurity solution φ¯ K. The results are gathered in figure 11 (top left). Obviously, for α= 0 the defect has exactly the same spectral structure as the pure φ4kink: a single mode with ω2= 3 and a translational zero mode ω= 0. The first main result is that this zero mode exists for any acceptable value of the parameter α. Hence, even if the impurity is coupled, the model recovers a sort of translational symmetry. This we have already understood as a nonlinear superposition of the antikink and the trivial lump which leaves the energy unchanged. Next, for α=−√2 we know the exact BPS solution which now represents the kink. It has a single even mode with a frequency ω=√3. Surprisingly, although the frequency of the mode coincides with the frequency of the pure φ4kink, the potential generated by the kink trapped by the impurity is a P¨oschl-Teller potential with the same depth as in the sine-Gordon model. When αhas a slightly larger value α=−√2 + the BPS antikink-impurity solution looks like a widely separated ¯ KK ¯ Kconfiguration. Both antikinks have the same structure as the free defects in the pure φ4model. Namely, one translational mode ω= 0 and one oscillating bound mode ω=√3. The whole configuration in the limit of α→ −√2 has two modes — ω= 0 (twice degenerated) and ω=√3 (three times degenerated). However, for – 20 – JHEP03(2019)131 nonzero the mutual interaction lifts the degeneracy and the modes split, which is clearly visible in figure 11. The modes with the frequency ω=√3 split into three modes, one even (with the lowest frequency) and two odd modes. The mode with ω= 0 splits into two modes. The odd mode has a frequency which increases as αincreases and becomes the single vibrational mode. It exists for α < 0.39 where it reaches the mass threshold. The second mode still has zero frequency and reflects the BPS-ness property of the solution. The excitation of this mode does not cost any energy, therefore the defect can be moved away from the impurity. The existence of this zero mode indicates the existence of a symmetry transformation generalising the translational symmetry of the model without impurity. We demonstrated the existence of this generalised translational symmetry in section 3.2. 4.2 The topologically trivial lump The spectral structure of the trivial lump possesses at least one oscillating mode for α∈[−√2,0). It reaches the mass threshold for α= 0, that is, in the limit of the pure φ4 theory, while it tends to a zero mode for α=−√2. For the same value α=−√2, two other oscillating modes merge to a single mode of φ4theory with ω3= 3. This is related to the fact that in this limit the topologically trivial lump can be viewed as a kink-antikink bound state where each of its constituents looks like a soliton of the pure φ4model. Hence again, the number of degeneracies reflects the number of hidden constituents while their mutual interaction (for α > −√2) leads to a splitting of the frequencies. It is interesting to note that the hidden structure of the lump, which first clearly appears at α=−1/√2 and gets more visible as we approach α=−√2, amounts to an appearance of a oscillating kink-antikink bound state in the spectrum of perturbations of the lump. As we clearly showed in a previous section, the lump can be described with very good accuracy as a pair of two solitons with a separation distance governed by the impurity. This picture results in the identification of ω0as an oscillation frequency of the kinkankikink pair. Precisely speaking, the kink is relatively strongly trapped by the impurity while the antikink performs oscillations whose period grows as we tend to the limiting α. The mechanism of this behavior is simple. The negative impurity always attracts the kink while, at least in the BPS sector, it does not interact with the antikink. Hence, the kink is more and more frozen on the impurity, as we approach α=−√2, while the antikink oscillates via an interaction with the kink. In the linear approximation, such a state exists for sufficiently negative α→ −√2. However, since in this limit the separation energy of the pair also vanishes, in realistic processes there will always be a critical α > −√2 for which the antikink will be released from the oscillating pair while the kink remains confined to the impurity. Such a behavior will be visible in many scattering processes. We remark that the vanishing of ω0as αtends to −√2 leads to a very high stability of the oscillating topologically trivial lump (wobbling kink-antikink pair). The wobbling state can radiate but only by higher harmonics, which strongly suppresses its decay [29,35,36]. Note also that this oscillation mode approaches the mass threshold for α= 0. Moreover, for α≥0 there are no oscillating modes at all. One should underline that we found a smooth transition between an oscillating excitation of the lump and an oscillating kink-antikink pair trapped on the impurity as we – 21 – JHEP03(2019)131 0 1 2 3 4 a) α=−1.0 0 1 2 3 4 a) α=−1.0 b) α=−0.5 b) α=−0.5 0 1 2 3 4 -10 -5 0 5 10 c) α= 0.3 0 1 2 3 4 -10 -5 0 5 10 c) α= 0.3 -10 -5 0 5 10 d) α= 1.0 -10 -5 0 5 10 d) α= 1.0 ω2 ω2 ω0 ω1 ω2 ω3 ω4 ω0 ω1 ω2 ω3 ω4 ω2 x0 ω2 x0x0 x0 Figure 12. Spectral structure of the BPS solution for different values of α, as a function of the topological zero of the antikink. decrease α. In other words, the K¯ Koscillating state originates in the first oscillating mode. One may even say that the exited lump is just the oscillating kink-antikink pair, especially in the limit when α→ −√2. Therefore, it can be interpreted as a linear effect. However, one should remember that the ground state above which the perturbation is performed, i.e., the lump, is a highly nonlinear solution. Since the interaction energy between the lump and the antikink is exactly zero, the position of the static antikink can be arbitrary. As the position changes so does the spectral structure. When the antikink is very far from the impurity, i.e., |x0|  1, then the influence of the impurity is almost negligible. The spectrum consists of separated modes of the free φ4soliton (ω=√3 and ω= 0) and the modes of the α-dependent lump. As the defect approaches the lump, the modes starts to interact and for x0= 0 the structure becomes the one described in the previous subsection. The frequencies of the normalizable eigenmodes are shown in figure 12 for four different values of α. For large distances, the number of modes is the sum of all the modes of the antikink and the lump. For example in figure 12 a) for α=−1 there are five modes for x0=−10, two of them are from the antikink and three from the lump (see figure 11b)). As the distance becomes smaller one of the modes enters the continuous spectrum, and for x0= 0 there are only four modes (see figure 11 a)). For α= 1 figure 12 d) there are two modes: one oscillational mode of the antikink and one translational mode. For x0<2 the oscillational mode becomes a narrow quasi-normal mode (QNM) and for x0= 0 its frequency becomes ω= 2.478 + 0.096i. Such a QNM can also play an important role in the dynamics of the system [30]. Note that for α < 0.38 the mode always stays below the threshold as in the figure 11 c). Note, that in [37] it was conjectured that a similar disappearance of the shape modes of a colliding kink and antikink is responsible for the failure of the collective coordinate method applied to such a – 22 – JHEP03(2019)131 collision. On the other hand, this distance-dependence of the spectral structure is crucial for the existence of the resonance bounce structure in the φ6model [29]. 4.3 The kink-impurity solution Due to the very special form of the kink solution, we can analytically study its spectral structure −η00 + 4−6+2√2α cosh2x!η=ω2η . (4.2) This is a P¨oschl-Teller potential with a depth equal to V0= 6 + 2√2α. The bound modes of the potential are known ω2 n= 4 −(λ−n)2,0≤n≤λ , λ(λ+ 1) = 6 + 2√2α . (4.3) Obviously, for α= 0 we recover the spectral structure of the kink in the usual φ4theory. For example, the first bound mode has the frequency ω2 0= 0. This mode is the translational zero mode of the kink, reflecting the translational symmetry of the model. The behavior of this mode in the impurity extended model strongly depends on the value of the parameter α. For negative α, the mode transforms into an oscillating mode with frequency growing to ω2= 3 at α=−√2. This means that the pure φ4kink is trapped by the impurity, forming a true kink-impurity bound state. Small perturbations oscillate around the static solution with frequency ω0. For positive α, the zero mode transforms into a negative frequency mode, ω2<0, which means that the static kink solution centered at x= 0 is not a stable solution. Any small perturbation violating the symmetry results in the ejection of the kink by the impurity. As we see, the spectral structure or, more precisely, the sign of ω2 0, fully confirms the previously found structure of solitonic solutions. In the pure φ4theory, the static kink solution has a second (shape) mode which is responsible for oscillations with ω1=√3. This mode exists in the impurity model, as well. As the depth of the potential changes, so does the frequency of the mode and for the special value of α=−√2 (or λ= 1) this mode disappears at the mass threshold. Interestingly enough, for this value of αthe spectral structure is identical to the spectral structure of the kink in the sine-Gordon (sG) model, but because of the mass threshold differences (mφ4= 2 and msG = 1) the mode corresponding to the translational mode of the sG kink (ω= 0) in our model oscillates with the frequency of the vibrational mode of the φ4kink ω=√3. We remind that in this limit the kink becomes a BPS solution and the model is fully BPS. 5 Scattering processes Having identified the structure of static solutions (asymptotic states structure) as well as their spectral properties, we can investigate scattering processes. As the first case, we consider the scattering of an incoming antikink (which at t→ −∞ is just the antikink solution of the pure φ4model) with the topologically trivial lump Σ−located on the impurity. As we know, such an antikink-impurity bound state lives in the BPS sector. – 23 – JHEP03(2019)131 5.1 Moduli space approximation We start with a theoretical description of the scattering of a slow-moving antikink on the impurity within the framework of the moduli space approximation, using the generalised translation symmetry of section 3.2 as a collective coordinate. The existence of this symmetry implies that there exists a one-parameter family of antikink BPS solutions. Let us denote this family as φ(x;a) where ais a parameter which uniquely defines the BPS solution. In our further examples we will assume that ais the position of the topological zero x0, which is a proper parameter for α > −1/√2 (see section 3.1). However, in principle, a can be any other parameter which uniquely identifies a solution, like, for example, a field value at a certain point. Let us add a perturbation ψto the static BPS solution φ(x;a) in such a way that φ(x; ˜a) = φ(x;a) + (˜a−a)ψ(x) (5.1) is still a solution to the BPS equation for a different parameter ˜a. Assuming that, for a small change of the parameter ˜a=a+da, the perturbation function ψda is also small, we can linearize the BPS equation and obtain an equation for ψ ψx= 2φψ. (5.2) This linearised BPS equation implies that ψis a zero mode, i.e., it obeys the mode equation (4.1) for ω= 0, as may be checked easily. The above equation can be formally solved ψ=Ce2Rφ dx ,(5.3) where Cis an arbitrary constant. If we use the definition (5.1) for ψ, however, then this constant has a definite value for each particular choice of the parameter a. For example, when ais the position of a topological zero x0, the transformation x0→˜x0=x0+dx0 is the translation of the topological zero by a small value dx0. Note that the field value does not change (the transformation changes zero to zero, φ(˜x0; ˜x0) = φ(x0;x0) = 0) so the total derivative vanishes dφ(x0;x0) = φx(x0;x0)dx +φx0(x0;x0)dx0= 0,(5.4) from which it follows that ψ(x0) = φx0(x0;x0) = −φx(x0;x0) since dx0=dx. Because φ(x0;x0) = 0 from the definition of x0, we get from the BPS equation C=ψ(x0) = φx0(x0;x0) = −φx(x0;x0) = 1 + √2σ(x0) (5.5) and finally ψ(x) = h1 + √2σ(x0)iexp 2Zx x0 φ(x0)dx0(5.6) For σ≡0, φ(x) = −tanh(x−x0), Rφ=−log cosh(x−x0) and eventually ψ= cosh−2(x− x0), which is exactly the translational mode of the antikink. Next, we want to study the antikink-impurity scattering in the moduli space approximation, where the only modulus (collective coordinate) is given by the parameter a. That is to say, we limit ourselves to a slow evolution of the nearly BPS state, φ≈φ¯ K(x;a(t)) (5.7) – 24 – JHEP03(2019)131 −20 −15 −10 −5 0 5 10 15 20 a) A= 0.20, α = 1.0 φ(0, T ) −20 −15 −10 −5 0 5 10 15 20 b) A= 0.40, α = 1.0 φ(0, T ) −20 −15 −10 −5 0 5 10 15 20 0 50 100 150 200 c) A= 0.70, α = 1.0 d) A= 2.00, α =−0.3 e) A= 4.80, α =−0.3 0 50 100 150 200 f) A= 6.00, α =−0.3 x −1−0.5 0 0.5 1 x −1−0.5 0 0.5 1 x time t time t Figure 18. Different scenarios of perturbed antikink-impurity ( ¯ K∗+ Σ, left panel) and antikinkperturbed impurity ( ¯ K+ Σ∗, right panel) collisions. The parameter values are a) A= 0.2, α= 1, b) A= 0.4, α= 1, c) A= 0.7, α= 1, d) A= 2, α=−0.3, e) A= 4.8, α=−0.3 and f) A= 6, α=−0.3. is more appropriate to consider it as some oscillon-type state bound to the impurity. The capturing-bouncing process seems to be also chaotic. Looking from a wider perspective, a plausible explanation of the richer structure of the antikink-lump scattering with one object being excited is that the initial state is not a solution of the BPS sector. To summarize all the possible results of the collisions, it is helpful to study the final state at a specific point, say x= 0 at certain time. We choose T= 100 + v/|x0|. This gives some information about the final state of the whole system, figure 19. However, the information can also be misleading, especially near α→0 when all final states have similar values φf≡φ(0, T)≈0 or when a few different results can have the same value at the center of the collision. One hint that can help in identifying those regions is looking at dislocations of constant phase lines. In many cases, the final state continuously depend on parameters until the structure of results suddenly changes. This can be seen for example in the figure 19 near α≈ −1.4 and v≈0.8. Below we summarize the possible results of the antikink-impurity collisions. The last column is the color we use to mark the final state in the phase diagram (for short we use R-red, O-orange, G-gray, B-blue, I-indigo): ¯ K+ Σ−−→      Σ++¯ Ksmooth passage φf≈1 RO ¯ K+K0+¯ Klump decay φf≈0 G ¯ K+ Σ−bounce φf≈0 G α→ −√2 (5.16) – 31 – JHEP03(2019)131 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1−0.5 0 0.5 φ(0, T ) v α −1−0.5 0 0.5 1 −1−0.5 0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1−0.5 0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ¯ K+K0+¯ K Σ++¯ K ¯ K+ Σ−Σ++¯ K/ ¯ K+K0+¯ K v α ¯ K+K0+¯ K Σ++¯ K ¯ K+ Σ−Σ++¯ K/ ¯ K+K0+¯ K Figure 19. Different scenarios of antikink-impurity ( ¯ K+ Σ−) collisions. The left plot shows the field value at center at time Tafter the collision. The right plot shows the products of the collisions. Line filled regions correspond to chaotic behaviour. Note that the last possibility happens when αis very close to the critical value −√2. For this value the field admits similar values φ(0, T)≈0 for all possibilities. Only a more careful analysis of the whole scattering process gives the appropriate identification. Those regions were marked on the right panel of figure 19 with different colors. There is yet another region, between smooth passage and lump decay, where the system behaves chaotically (the boundary has some fractal properties). Such regions we mark with line patterns. 5.3 Kink-impurity scattering As the next process, we consider the scattering of an incoming kink (from the right) on the topologically trivial lump Σ−located on the impurity. The main possibilities occurring at v= 0.2 are presented in figure 20 and, for some other velocities and small α, in figure 21. For positive α, the impurity always repels the kink as there is no kink-impurity bound state. Thus, the incoming kink is elastically reflected back — see figure 20a) with α= 1. Of course, for sufficiently large initial velocities the kink should have enough energy to go through the impurity. In the limiting case, we expect that the field should go through a static configuration representing the kink at x= 0. On the one hand, this is the lowest energy state satisfying the condition that φ(0) = 0 in the Q= 1 topological sector but, on the other hand, for positive values of α, the lowest energetic state is when the kink is far away from the impurity. Therefore the static kink configuration is a saddle point. The kink with energy below the energy of the static unstable kink is unable to penetrate the potential barrier generated by the impurity. Therefore the limiting case gives an estimate for the critical velocity when the kink can go through to the other side. Using the energetic argument we find that 4 3 1 p1−v2 cr =4 31+2√2α+α2.(5.17) Once αbecomes negative, the kink can go through it. For small negative α, such a transition does not significantly change the velocity of the incoming kink as is visible in figure 20b) where α=−0.1. If we make αsmaller, then the attractive force of the impurity – 32 – JHEP03(2019)131 -20 -10 0 10 20 a) v= 0.200, α = 1.00000 φ(0, T ) b) v= 0.200, α =−0.10000 φ(0, T ) -20 -10 0 10 20 0 50 100 150 c) v= 0.200, α =−0.20000 0 50 100 150 d) v= 0.200, α =−1.41000 x −1−0.5 0 0.5 1−1−0.5 0 0.5 1 x time t time t Figure 20. Different scenarios of kink-impurity (Σ−+K) collisions for v= 0.2. The parameter values are a) α= 1, b) α=−0.1, c) α=−0.2 and d) α=−1.41. -20 -10 0 10 20 a) v= 0.270, α =−1.40000 φ(0, T ) b) v= 0.360, α =−1.40000 φ(0, T ) -20 -10 0 10 20 c) v= 0.370, α =−1.40000 d) v= 0.710, α =−1.40000 -20 -10 0 10 20 0 50 100 150 e) v= 0.720, α =−1.40000 0 50 100 150 f) v= 0.760, α =−1.40000 x −1−0.5 0 0.5 1−1−0.5 0 0.5 1 x x time t time t Figure 21. Chaotic behaviour of the oscillon after the kink-impurity (Σ−+K) collisions for α=−1.4. The velocity values are a) v= 0.27, b) v= 0.36, c) v= 0.37, d) v= 0.71, e) v= 0.72 and f) v= 0.76. overcomes the kinetic energy of the kink, which is now trapped by the impurity forming an excited bound state, as presented in figure 20c) (α=−0.2). It oscillates and, slowly radiating energy, decays into the static kink-impurity solution. When α→ −√2, the inner structure of the lump starts to play a role. Then, the capture process is a bit more involved. – 33 – JHEP03(2019)131 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −1.4−1.2−1−0.8−0.6−0.4−0.2 0 0.2 0.4 αcr vf α ¯ K+ Σ− ¯ K+ Σ− Σ++K vi= 0.3 Figure 22. Velocities of free defects propagating after impurity-defect type collisions. The incoming kink annihilates with the antikink from the hidden kink-antikink pair of the lump. In the shown example they form an oscillon which goes towards the impurity. This subprocess is inherited from the pure φ4model, as both solitons are quite far away from the impurity (remember it is exponentially localized). The remaining kink behaves as a spectator and is initially unaffected but later is perturbed (excitation of the oscillating mode) by the remains of the annihilation process. Note that in contrast to the previous case, we find a lot of radiation emitted during the kink-impurity scattering. This is, of course, related to the fact that the static kink-impurity solution is not a BPS configuration. To summarize the defect-impurity collisions, we measured the final velocity of defects after the collision for initial velocity vi= 0.3 as a function of α. The red curve in plot 22 shows the final velocity of the antikink in the antikink-lump collision. For a large range of values α > −0.7, the final velocity is almost constant and almost equal to the initial velocity. This means that the energy exchanged between the defect and the impurity is negligible. This proves that the collisions of the BPS solutions with the generalised translational symmetry are almost elastic. In the range −1.25 < α < −0.7 the scattering is less elastic and more energy is transferred to the lump. Below α < −1.25 the lump is destroyed in the collision and a second antikink is ejected from the impurity, leaving a kink tightly bound to the impurity. This also proves that in some sense, for values of αclose to −√2 the lump has a structure of a kink-antikink pair stabilized by the impurity. The second observation is that in the case of kink-impurity scattering the kink which is reflected from the impurity also loses very little of its initial kinetic energy. However, just after the transition is possible (marked by dotted line and αcr in the figure 22) the kink indeed can go through the impurity but loses more and more energy as the strength of the impurity increases. For α < −0.17 the kink is captured by the impurity. The energy is then radiated out from the internal modes of the configuration described in the previous section. – 34 – JHEP03(2019)131 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1−0.5 0 0.5 φ(0, T ) v α −1−0.5 0 0.5 1 −1−0.5 0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1−0.5 0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 K+ Σ+ Σ−+KK0 v α K+ Σ+ Σ−+KK0 Figure 23. Different scenarios of kink-impurity (Σ−+K) collisions. The left plot shows the field value at center at time Tafter the collision. The right plot shows the products of the collisions. Line filled regions correspond to chaotic behaviour. In figure 23 we have shown the final states of φ(0, T) (left) and the identification of the final state (right) Σ−+K−→      Σ−+Kbounce φf≈ −1 I K+ Σ+passage φf≈1 RO K0capture φf≈0 G (5.18) The last reaction, during which the kink is captured by the impurity, can produce a lot of radiation in order to get rid of the excess of energy. Sometimes this radiation takes the form of an oscillon. The oscillon can be recaptured by the impurity and adopted eventually as one of the internal modes of the kink, or it can pass through the impurity or can be torn apart into a kink-antikink pair. All of these processes lead to similar final values at the center of collisions, φ(0, T )≈0. However, the excitation of the captured kink, or the kink with a recaptured oscillon, lead to different excitation amplitudes, which can be seen as discontinuities of the constant phase lines in the lower left corner of the left plot in figure 23 or near the boundary with the passage region. 5.4 Antikink-(kink-on-impurity) scattering Now we analyze the scattering of the incoming antikink with the kink-impurity bound state. We remind that such a bound state is stable only for negative values of α. In figure 24 we present the main qualitatively different scenarios of the scattering at the velocity of the incoming antikink v= 0.2. For positive α, the kink-impurity is an unstable configuration. The kink can be ejected both towards positive 24a) and negative x-values 24b). The incoming antikink can then either go through the impurity as in the previous case or can capture the kink from the impurity — see figure 24b) with α= 0.3. The mechanism is straightforward. The antikink attracts the kink which is simultaneously repelled by the impurity. However, the energy stored in the unstable state is high enough to avoid the annihilation. Both the incoming antikink as well as the released kink are scattered to minus infinity. – 35 – JHEP03(2019)131 -20 -10 0 10 20 a) v= 0.250, α = 0.20000 φ(0, T ) b) v= 0.200, α = 0.30000 φ(0, T ) c) v= 0.200, α =−0.10000 -20 -10 0 10 20 d) v= 0.200, α =−1.00000 e) v= 0.200, α =−1.05000 f) v= 0.200, α =−1.07000 -20 -10 0 10 20 0 50 100 150 g) v= 0.200, α =−1.30000 0 50 100 150 h) v= 0.630, α = 0.30000 0 50 100 150 i) v= 0.890, α =−0.30000 x −1−0.5 0 0.5 1−1−0.5 0 0.5 1 xx time t time t time t Figure 24. Different scenarios of antikink-(kink-on-impurity) ( ¯ K+K0) collisions, mostly for velocity v= 0.2. The other velocity values are a) v= 0.25, h) v= 0.63 and i) v= 0.89. Further, the parameter values are a) α= 0.2, b) α= 0.3, c) α=−0.1, d) α=−1.0, e) α=−1.05, f) α=−1.07, g) α=−1.3, h) α= 0.3 and i) α=−0.3. For negative α, the incoming antikink is attracted by the kink and both get annihilated by forming an oscillon, as shown in figure 24c) with α=−0.1. As the impurity is weak, the behavior is similar to the pure φ4theory. However, the oscillon is trapped by the impurity and performs a sort of coherent wobbling around the impurity. In figure 25 we analyse this particular case in detail. Figure 25a) shows the power spectrum of the field measured at x=−2 for the time range t∈[1000,2000]. There are certain peaks in the spectrum. The highest peak corresponding to the basic frequency ω≈1.6 is very different from the frequency obtained in the linear theory ω0= 1.96 corresponding to the only eigen-mode of the lump. The power spectrum resembles the typical power spectra of oscillons created via defect collisions in the pure φ4model (figure 25b)). Therefore we can identify this object as a trapped oscillon. Oscillons are famous for their long and slow decay. To increase the rate of the decay we added an additional damping term γφtto the equation for times in the range 200 <t<1000. After that time we calculated again the power spectrum. As the damping term increases, the spectrum becomes more and more similar to the spectrum predicted by the linear approximation. After a small damping one can see that a single frequency – 36 – JHEP03(2019)131 1×10−5 0.0001 0.001 0.01 0.1 1ω0 a) 1×10−5 0.0001 0.001 0.01 0.1 1ω0 a) 0.0001 0.001 0.01 0.1 1 0.5 1 1.5 2 2.5 3 3.5 4 b) 0.0001 0.001 0.01 0.1 1 0.5 1 1.5 2 2.5 3 3.5 4 b) 1×10−5 0.0001 0.001 0.01 0.1 1ω0 c) 1×10−5 0.0001 0.001 0.01 0.1 1ω0 c) 1×10−5 0.0001 0.001 0.01 0.1 1 d) 1×10−5 0.0001 0.001 0.01 0.1 1 d) 1×10−6 1×10−5 0.0001 0.001 0.01 0.1 0.5 1 1.5 2 2.5 3 3.5 4 e) 1×10−6 1×10−5 0.0001 0.001 0.01 0.1 0.5 1 1.5 2 2.5 3 3.5 4 e) |F[φ(x0, t)](ω)||F[φ(x0, t)](ω)| |F[φ(x0, t)](ω)| ω |F[φ(x0, t)](ω)| ω ωω Figure 25. (a) Power spectrum of the field after the collision ¯ K+K0for α=−0.1, v= 0.2 measured at x=x0=−2 compared with (b) the power spectrum of the oscillon α= 0 and (c-e) in the damped evolution with increasing damping term. is even more dominant than in the undamped case. But the frequency is still much below the eigenfrequency due to the nonlinearities. As the solution is damped even more, the frequency becomes closer to the eigenfrequency of the lump. Therefore we can expect that the trapped oscillon would be absorbed by the lump and assimilated as the mode of the lump. A similar behaviour was analyzed in more detail in [35]. Note that although the oscillon is a long living oscillating object, its decay is a more violent phenomenon than in the case of the wobbling kink-antikink pair in the antikink-lump scattering. For example, the amount of radiation emitted in such a process is bigger. In any case, the intermediate topologically trivial state observed here, i.e., the oscillon, behaves differently if compared to the excitation of the lump (which is just the wobbling kink-antkikink pair for α < 0). This may further support our interpretation of the antikink-impurity scattering as a quasiintegrable process. Here, as there is no static BPS antikink-kink-on-impurity solution, the observed scattering does not exhibit any integrable-like properties and is to a high extent governed by the properties of the scattering in the pure φ4model. For smaller negative α, we observe the appearance of the wobbling state, see figure 24d) and figure 24e) with α=−1.0 and α=−1.05 respectively. However, it is much less stable than in the antikink-impurity collision. It radiates a significant amount of energy and decays quite quickly into the topologically trivial lump. Although the perturbation seems to be quite large, all the oscillations can be easily connected with the eigenfrequencies of the trapped kink and their multiplicities (see figure 26). Again, as αdecreases, one of the constituents, that is, the kink, behaves as a spectator only, with almost no oscillations around its initial position. There is a critical αat which the scattering process reveals another final state. Namely, the initially created topologically trivial state breaks into a kink trapped to the impurity – 37 – JHEP03(2019)131 1·10−5 0.0001 0.001 0.01 0.1 1 10 0 0.5 1 1.5 2 2.5 3 3.5 4 ω0ω1ω2 ˜ω02˜ω03˜ω0 1·10−5 0.0001 0.001 0.01 0.1 1 10 0 0.5 1 1.5 2 2.5 3 3.5 4 ω0ω1ω2 ˜ω02˜ω03˜ω0 |F[φ(x0, t)](ω)| ω |F[φ(x0, t)](ω)| ω Figure 26. Power spectrum of the field after the collision ¯ K+K0for α=−1.05, v= 0.2 measured at x=x0=−2. Eigenfrequencies of the kink ωiand multiplicities of the basic frequency n˜ω0are also marked. and a free antikink reflected to minus infinity — see figure 24f). Here this happens at α=−1.07. After that, the scattering tends to an elastic one once we approach α=−√2. A high velocity antikink can also push the kink from the impurity 24h) and 24i). The antikink can be ejected from the impurity leaving the lump at the center of the collision 24i) or replace the kink 24h) at the impurity. Note that we previously described similar processes for the excited antikink and lump collisions, which are shown in figure 18. The picture gets much more complicated once we vary the initial velocity of the incoming antikink. Again, a chaotic structure emerges, which seems to be inherited from the pure φ4theory. In figure 27 we plot some particular cases for a fixed impurity α=−0.1 while we change the velocity of the incoming antikink. It is clearly seen that there are annihilation, bouncing and simple scattering windows both at smaller and higher velocities. However, the velocity (energy of the incoming antikink) determines whether the remainders of the scattering (as, for example, an oscillon) are bounded to the impurity, reflected or transmitted. It is also visible that the oscillon may be trapped by the impurity figure 27a) or behave as a free object figure 27g), h) and j). Furthermore, it can also be confined to the impurity for a short time, making several oscillation as a whole object around the impurity, and then escape to plus or minus infinity — see figure 27b) and figure 27f), respectively. Because this process is quite complex, and in some cases can be regarded as a threebody collision, also the space of final states is quite complex. ¯ K+K0−→                  Σ+annihilation φf≈1 RO ¯ K+K0bounce φf≈0 G ¯ K+ Σ−+Kbounce/knock φf≈ −1 I Σ−+¯ K+Kpassage/knock φf≈1 RO {¯ K, Σ±, K}ejection φf≈1 RO two stage ¯ K0+KBPS capture φf≈0 G (5.19) – 38 – JHEP03(2019)131 -20 -15 -10 -5 0 5 10 15 20 a) v= 0.41613, α =−0.100 φ(0, T ) b) v= 0.54740, α =−0.100 φ(0, T ) c) v= 0.57419, α =−0.100 -20 -15 -10 -5 0 5 10 15 20 d) v= 0.59677, α =−0.100 e) v= 0.60500, α =−0.100 f) v= 0.61000, α =−0.100 -20 -15 -10 -5 0 5 10 15 20 g) v= 0.63000, α =−0.100 h) v= 0.67000, α =−0.100 i) v= 0.69500, α =−0.100 -20 -15 -10 -5 0 5 10 15 20 0 50 100 150 200 j) v= 0.70500, α =−0.100 0 50 100 150 200 k) v= 0.71500, α =−0.100 0 50 100 150 200 l) v= 0.75000, α =−0.100 x −1−0.5 0 0.5 1−1−0.5 0 0.5 1 x x x time t time t time t Figure 27. Different scenarios of antikink-(kink-on-impurity) ( ¯ K+K0) collisions with fixed impurity α=−0.1 and different initial velocities of the incoming antikink. The velocity values are a) v= 0.41613, b) v= 0.5474, c) v= 0.57419, d) v= 0.59677, e) v= 0.605, f) v= 0.61, g) v= 0.63, h) v= 0.67, i) v= 0.695, j) v= 0.705, k) v= 0.715 and l) v= 0.75. In the right (α > 0) lower (small v) corner of the phase diagram 28 for this process the dominant color is red, indicating that φ(T, 0) ≈1. However, in this region the initial state, a kink on top of the impurity, is unstable. The incoming antikink provides a small disturbance resulting in the early ejection of the kink. The defects then collide outside the impurity, which is similar to the collisions known from the pure φ4model. The products (defects or oscillons) can hit the impurity again after the first stage of collision. This two stage process is very chaotic. – 39 – JHEP03(2019)131 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1−0.5 0 0.5 φ(0, T ) v α −1−0.5 0 0.5 1 −1−0.5 0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −1−0.5 0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ¯ K+K0 Σ+/¯ K+K0 Σ+ Σ++¯ K+K ¯ K+ Σ−+K ¯ K+K0 ¯ K+ Σ−+K/ ¯ K0+K {K, Σ±,¯ K} v α ¯ K+K0 Σ+/¯ K+K0 Σ+ Σ++¯ K+K ¯ K+ Σ−+K ¯ K+K0 ¯ K+ Σ−+K/ ¯ K0+K {K, Σ±,¯ K} Figure 28. Different scenarios of antikink-(kink-on-impurity) ( ¯ K+K0) collisions. The left plot shows the field value at center at time Tafter the collision. The right plot shows the products of the collisions. Line filled regions correspond to chaotic behaviour. For high velocities and a weakly attractive impurity, the kink can be kicked from the impurity and be replaced by the antikink. Capturing of the antikink is an unusual situation and must be connected with the excitation of the internal mode of the antikink. Such excitations also lead to a large sensitivity to the change of parameters of collisions. Another chaotic behaviour can be seen between annihiliation Σ+and bounce processes (¯ K+K0). The dashed green-black line on the left plot in figure 28 corresponds to the critical velocity below which the antikink has not enough kinetic energy to free the kink bound to the impurity. Below that velocity, the antikink can only bounce back or annihilate with the kink. 5.5 Kink-(antikink-on-impurity) scattering Finally, we consider the incoming kink scattering on the antikink-impurity state. As there are infinitely many energetically equivalent antikink-impurity configurations, the main ingredients of the process are the kink-impurity interaction (repulsion for α > 0 and attraction for α < 0) as well as the kink-antikink attraction. In figure 29 we plot several scenarios which can occur during this scattering. Here v= 0.2. First of all, for positive αthe incoming kink gets reflected by the impurity. However, it interacts with the antikink and attracts it. As a consequence, the antikink follows the kink. If αis big (α= 1 then the impurity strongly repels the kink, which leads to a weak kink-antikink interaction. As a result we get an almost elastic reflection of the kink and a slowly moving antikink — see figure 29a). For smaller but positive α, the kink-antikink interaction grows (the repulsive force decreases) and the antikink can eventually hit the kink which leads to an annihilation process where an oscillon is formed — figure 29b). Once αbecomes negative, the impurity attracts the kink, and the solitons annihilate on the other side of the impurity — figure 29c). However, if we further decrease α, i.e., increase the kink-impurity attraction, then the kink gets trapped by the impurity, while the antikink is repelled to minus infinity, as demonstrated in figure 29d) with α=−0.3. For even – 40 –