Spectral Walls

During defect-antidefect scattering, bound modes frequently disappear into the continuous spectrum before the defects themselves collide. This leads to a structural, nonperturbative change in the spectrum of small excitations. Sometimes the effect can be seen as a hard wall from which the defect can bounce off. We show the existence of these spectral walls and study their properties in the $\phi^4$ model with BPS preserving impurity, where the static force between the antikink and the impurity vanishes. We conclude that such spectral walls should surround all solitons possessing internal modes.

A detailed understanding of the interaction of topological solitons in non-integrable models is still a very difficult task.For example, in the φ 4 model, which is the prototypical soliton model in (1+1) dimensions, the kink-antikink collisions reveal a chaotic structure which is typically associated with the existence of an internal mode which may be exited during the scattering process [1][2][3][4][5].This mode can store energy, binding the solitons for a while.After some time the mode can transfer its energy to the translational degrees of freedom.This energy exchange has many properties of a resonant phenomenon and eventually leads to an intriguing fractallike pattern of multiple bounce windows as a function of the initial velocity.Such a mechanism was observed in other solitonic models, as well, including soliton-impurity collisions [6,7], multi-component fields [8,9] and quasinormal modes [10].A bound oscillational mode can even be excited when there are no such modes on the original solitons (asymptotic states).This may happen, for example, in a model with two vacua with two different mass parameters [11].Solitons in one alignment generate a potential well which hosts oscillational modes, giving rise to a resonance structure.In the opposite alignment of the solitons the interaction potential has a hill in the center, excluding bound modes.As a result, no resonant structure emerges.
The effective model proposed in [1] and later studied and modified by many others initially confirmed the above results quite well.The the original work, however, had a typographical error which was later inherited in further studies.It was not until very recently that the error was corrected [12,13].Unfortunately, the corrected picture did not result in a correct quantitative description of multi-bounce windows characteristic for the soliton collisions.As pointed out in [2], one of the reasons was that all analytical attempts assumed that the field can be written as a simple superposition of the kink, antikink and mode profiles.The dynamical variables in this effective model are the relative position, a, of the colliding defects and the amplitude of the bound mode A. However, this picture is not correct, in general, because of several problems: • before the solitons collide they are highly deformed by mutual interactions (static force), • none of the intermediate state is static, therefore the eigen-problem for the bound modes is badly defined, • at the moment of the collision a = 0 the solitons vanish, which is identified as a zero vector problem, • when solitons momentarily vanish, there cannot be bound states.Actually, the bound modes vanish even earlier (see, e.g., Fig. 1, upper panel, for the φ 4 model).
Thus, the computed spectral structure does not correspond to the real dynamics of the process.Even beyond the effective model, the mixing of the the kink-mode interaction with the (static) forces between solitons which deform the soliton profiles and also strongly influence the spectral properties, i.e., the mode structure, renders any analytical treatment very difficult.
This mixing problem could be avoided for a theory with static multi-soliton solutions of the Bogomolny-Prasad-Sommerfield (BPS) type.This means that individual solitonic constituents of the static multi-soliton configuration do not interact with each other, like, e.g., the vortices in the Abelian Higgs model at critical coupling.These vortices can be put in arbitrary positions, leading to a finite-dimensional moduli space.Although each point on the moduli space corresponds to an energetically equivalent state (no static inter-soliton forces), their spectral structures differ.The low energy scattering of such solitons can be described as a geodesic motion on the moduli space.In a next step, a mode on a scattered soliton can be excited.In this way, one could disentangle the soliton-mode interaction from the inter-soliton forces.
Unfortunately, in soliton models in (1+1) dimensions only one-soliton solutions belong to the BPS sector.The corresponding moduli space is trivial and given by translations of the kink center.In particular, the spectral structure remains unchanged along this very simple moduli space.
Very recently, however, it has been observed that this arXiv:1903.12100v1[hep-th] 28 Mar 2019 situation may change when an impurity is added.In particular, there exist BPS-impurity models in (1+1) dimensions [14,15] whose moduli space resembles the higher-dimensional BPS counterparts.Namely, the spectral structure of the soliton-impurity solution depends on its position on the moduli space.This gives us the unique opportunity to disentangle the above-mentioned mixing between the kink-mode interaction and the inter-kink force (and deformations due to it).The aim of this letter is to analyze the interaction of the excited mode on the BPS soliton in the BPS-impurity φ 4 theory.We discover a universal phenomenon, a spectral wall, which denotes a spatially localised region, defined by the point where an oscillation mode enters the continuous spectrum, and at which a soliton interacts nontrivially.

II. THE BPS-IMPURITY MODEL
The BPS-impurity model is defined by the following Lagrangian where φ(t, x) is a scalar field and W = W (φ) a prepotential such that the potential of the original model without impurity is U (φ) = W 2 .Finally, σ = σ(x) is a spatially localized (in principle arbitrary) impurity.The model is a half-BPS theory in the sense that half of the solitons of the no-impurity field theory remain BPS solitons, that is, saturate a pertinent topological bound and obey the corresponding Bogomolny (first order static differential) equation.Furthermore, the BPS sector also contains topologically trivial solutions (lumps) which are the counterparts of the vacuum solutions φ = φ v = const.
(such that U (φ v ) = 0) of the model without impurity.
The moduli space M is now a one-dimensional space which can be parametrized, e.g., by a point a ∈ R which measures the distance between the static BPS soliton and the impurity (no static force between them).A move on M represents a generalized translation which nontrivially transforms one BPS solitonic solution into another.
Of course, all points on the moduli give solutions with exactly the same energy.
In the present work, we analyze the BPS-impurity version of the φ 4 model.
Hence, we assume Furthermore, we choose an impurity of the form σ = α/ cosh 2 x, located at x = 0, where α is a real parameter which measures the strength of the impurity.
As announced above, one advantage of the BPSimpurity model is the nontrivial dependence of the spectral structure on the position on the moduli space, i.e., on the distance of the BPS soliton (here the antikink) from the impurity.Concretely, we choose the position a of the topological 0, i.e., the point where the antikink vanishes, which is a typical measure for the (anti-)kink position.
(a) Spectral structure of the K K superposition in φ 4 (i.e., spectrum of linear perturbations about φ 0 = tanh(x − a) − tanh(x + a) + 1).(b) Spectral structure of the static BPS K solution in the BPS-impurity model for two values of α.
Remember that the static BPS solution φ 0 (x; a) exists for any a and obeys the Bogomolny equation Therefore, in the BPS-impurity model the static solution and the spectral structure are well-defined for any position of the soliton.The modes can be found by a small perturbation around the static BPS solution φ = φ 0 (x; a) + Aη(x, t; a)e iωt where the mode amplitude A is assumed to be small.Further, W and its derivatives are calculated for φ = φ 0 .In this letter, we focus on the two simplest cases with one discrete mode (and one zero mode corresponding to the generalized translation invariance of the BPS solution).
In the first example, the mode exists for all points on the moduli space and its frequency grows while we approach the impurity, see Fig. 1 (lower panel, dotted curve).In the second example, we choose the impurity such that, at a certain point on the moduli space (which corresponds to a certain distance between the BPS antikink and the impurity) the mode enters the continuous spectrum, becoming a quasinormal mode, see Fig. 1 (lower panel, solid curve).Here it happens for a = ±1.68.Obviously, there is a fundamental qualitative difference in these two cases.While for α = 0.3 we expect a smooth evolution (captured by an effective model), for α = 3 we expect some nontrivial and novel effects.Indeed, as we will see below, this drastic change in the spectrum of the discrete modes will lead to the appearance of a spectral wall, i.e., a spatially well-localized region (barrier) at which the BPS antikink with the pertinent mode excited may be trapped, even though the unexcited BPS antikink goes through this point without any interference (no energy loss due to the geodesic motion on moduli space).
Of course, the spectral structure can reveal even more complicated patterns with a bigger number of modes entering the continuous spectrum at different points.This may lead to more involved structures and new effects which, however, we leave for future investigations.

III. EFFECTIVE MODEL
A standard construction of an effective model is to express the field as a superposition of known profiles such as kinks and their bound modes [1], leaving the positions and mode excitations as the only dynamical variables of the collective coordinate model.As explained in the introduction, this approach has important problems, in general.In the BPS model, this approach corresponds to considering a field in the form φ(x, t) = φ 0 (x; a(t)) + A(t)η(x; a(t)). ( This is a consistent procedure in the BPS model, because at all times the static solution and the spectral structure are well defined.Assuming slow, almost adiabatic evolution we can expect that the system will evolve through the static states with additional small perturbations.Inserting the above expression into the lagrangian, we get at quadratic order in a and where M (a) is the effective mass of the BPS soliton or a metric on moduli space and, therefore, is well-defined for any position on this space.However, the two other integrals can be divergent as a approaches a critical separation a cr , simply because the mode becomes non-normalizable at ω = 2 when it enters the continuum (Fig. 2).Obviously, at this point the effective model breaks down and some new effects can be expected.So, while the effective model breaks down also in the BPS case for some parameter values, it does so in a controlled way.The effective model itself contains all the relevant information about its range of validity and the points in parameter space where its break-down occurs.Note that the existence of singular points in the lagrangian does not necessarily mean that the effective model breaks down.For example, the effective mass M (a) vanishes at a = 0 for α = −1/ √ 2, which is a reflection of a bifurcation of the zero of the antikink φ 0 .This just indicates that the parameter a is no longer a good collective coordinate, but this can be easily cured by another choice of parameter.However, the problem of the vanishing mode is deeper.There are two different sets of independent variables on both sides of a cr and there is no obvious way to match effective models in both regions.Moreover, the bound mode entering the continuous spectrum becomes a quasi-normal mode.In [10] it was shown that such modes can also be responsible for creating a resonance structure.But constructing the effective model with a quasinormal mode is not as straightforward as in the case of bound modes.

IV. THE SPECTRAL WALL
We have collided the BPS antikink initially separated by a(0) = 10 with the impurity with α = 0.3 and α = 3.0.In both cases the pure, unexcited solitons smoothly travel through the impurity, as expected from the geodesic flow on the moduli space.However, when the internal mode of the antikink is excited, we have found that the soliton can bounce back for small velocities v = ȧ(0).However, the position of the turning point in the case of α = 0.3 is changing smoothly both with the amplitude of excitation and with the velocity reaching the origin.
The situation in the α = 3 case is different.Increasing the amplitude of the excitation for fixed velocity, the soliton is slowing down at the position of the spectral wall a cr = −1.68.If the excitation is large enough, the antikink bounces from the wall.Decreasing the amplitude slightly we have found another intriguing effect.The soliton can go through the wall but is reflected from the second (symmetric) wall behind the impurity.Sometimes even a few internal reflections can be observed.When the perturbation of the antikink is radiated out, it can finally pass one of the walls.Effectively there is a small window in the amplitude range, inside which the soliton can bounce back or go through the impurity after a series of internal reflections.This suggests that the energy stored in the mode even after a long time is still somehow attached to the kink, and it takes some time to radiate it out.Indeed, after entering the continuous spectrum, the normal mode attached to the antikink turns into a quasinormal mode whose frequency and width increases and admits the highest value ω = 3.72 + 0.11i for a = 0 (determined using Prone's method).The existence of such a mode can prevent the immediate emission of energy.We also have found that the condition for the bounce, for small velocity, obeys a linear scaling law A ≈ 1.70v.
We have compared the results of the numerical simulation of the full PDE problem with predictions from the collective coordinate effective model.In the α = 0.3 case, when the effective model is applicable for all separations, we have found a very good agreement.For example, for initial velocity v = 0.01 the critical excitation A(0), separating bounce from passage in both cases agrees with an accuracy of about 1%: A full = 0.0186 vs.A eff = 0.0188.In the α = 3 case, the effective model works very well until the the antikink gets close to the wall.In the full model for v = 0.05, we have found a critical excitation A full = 0.0847 and A eff = 0.0878 which is slightly less accurate, but the integrals calculated near the wall have larger numerical errors.
Higher, relativistic velocities require a higher mode excitation.However, in nonlinear models the frequency of the highly excited mode depends on the amplitude, and usually is lower than the eigenfrequency found from the linearization.The frequency shift means that the highly excited mode enters the continuous spectrum for smaller values of a.This effect was also observed numerically, when A cr > 0.2 the wall shifted visibly.This may be important in the more general, non-BPS case, because the colliding objects can attract each other, therefore there should exist some minimal A cr below which the full collision would take place even with zero initial velocity.Note that in the pure φ 4 model the resonant structure can be observed for relativistic velocities 0.18 < v < 0.26 measured in the center of mass.Such high velocity collisions mean that both the inter-soliton interaction and the mode excitation are very large.
A similar wall was also found when the system undergoes a transition from three to two oscillating bound modes (α < −0.35).

V. SUMMARY
First of all, we expect that the spectral wall will be a generic phenomenon for the interaction of solitons in non-integrable theories if a discrete mode undergoes a transition to the continuous spectrum.Of course, it can be less visible than in the BPS-impurity model, as other effects may interfere.In the case of solitons with long range tails, e.g., it is practically impossible to find an unperturbed initial state, therefore all collisions in such systems are collisions in the excited systems [16].Furthermore, in more realistic physical systems solitons are always excited due to quantum or thermal fluctuations.
The spectral wall should be especially easy to find in BPS theories (if the mode transition occurs) like the Abelian Higgs model in (2+1) dimensions at critical coupling.In other words, our work provides new insight into the dynamics of BPS solitons beyond the geodesic approximation, where the spectral wall effect may play a significant role.
Moreover, we show that the disappearance of the bound modes may be responsible for the failure to construct a precise collective coordinate model for kinkantikink collision processes in theories like the φ 4 model.

FIG. 2 .
FIG. 2. Regular (a) and singular (b) terms in the lagrangian of the effective model.The singularities in (b) occur at the spectral wall acr = 1.68.

FIG. 3 .
FIG. 3. Comparison between the smooth evolution for α = 0.3 and v = 0.01 (a) and a meandering kink trapped between spectral walls for α = 3 and v = 0.05 (b) for different excitations of the mode.Dashed lines correspond to the positions of the spectral walls.