Solvable self-dual impurity models

P u b l i s h e d f o r SISSA b y S p r i n g e r

Received: May 21, 2019 A ccepted: July 7, 2019 Published: July 25, 2019

Solvable self-dual impurity models

C. A dam ,“ K . Oles,b J . M . Queiruga,c,d T . Rom anczukiewiczb and A. W ereszczynskib

aDepartamento de Fisica de Particulas, Universidad de Santiago de Compostela, and Instituto Galego de Fisica de Altas Enerxias (IGFAE),

E-15782 Santiago de Compostela, Spain bInstitute of Physics, Jagiellonian University,

Lojasiewicza 11, Kraków, Poland

cInstitute fo r Theoretical Physics, Karlsruhe Institute of Technology (K IT ), 76131 Karlsruhe, Germany

dInstitute fo r Nuclear Physics, Karlsruhe Institute o f Technology (K IT ), Hermann-von-Helmholtz-Platz 1, D-763ĄĄ Eggenstein-Leopoldshafen, Germany E-mail: adam@fpaxp1.usc.es, katarzyna.slawinska@uj.edu.pl,

queimanu@hotmail.com, eviltrom@gmail.com, andwereszczynski@gmail.com

Ab s t r a c t: We find a fam ily of (half) self-dual im purity m odels such th a t th e self-dual (B PS) sector is exactly solvable, for any sp atial d istrib u tio n of th e im purity, b o th in th e topologically triv ial case and for kink (or antikink) configurations. T his allows us to derive th e m etric on th e corresponding one-dim ensional m oduli space in an an alytical form . Also th e generalized tra n sla tio n a l sym m etry is found in an exact form. This sym m etry provides a m otion on m oduli space w hich transfo rm s one B P S solution into another. Finally, we analyse exactly how v ib ration al p roperties (spectral m odes) of th e B P S solutions depend on th e actual position on m oduli space.

T hese results are o b tained b o th for th e nontrivial topological sector (kinks or antikinks) as well as for th e topologically triv ial sector, w here th e m otion on m oduli space represents a kink-antikink an n ih ilatio n process.

Ke y w o r d s: N o n p e rtu rb a tiv e Effects, Solitons M onopoles and In stan to n s, Effective Field Theories

ArXiv ePr i n t: 1905.06080

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C o n te n ts

1 I n tr o d u c tio n 1

2 S t a tic B P S s o lu tio n s 3

3 M e tr ic o n m o d u li sp a c e a n d g e o d e s ic flow 6

4 S p e c tr a l p r o b le m 8

5 N o n -lo c a liz e d im p u r itie s — s te p - f u n c tio n e x a m p le s 12

6 S e lf-d u a l k in k -a n tik in k s o lu tio n s 13

6.1 A nnihilation of solitons on th e m oduli space 13

6.2 A nnihilation and th e spectral s tru c tu re 17

7 S u m m a r y 20

1 In tr o d u c tio n

T he recently discovered Bogom olny-Prasad-Som m erfeld (B P S )-im pu rity m odels (or self­

dual im purity m odels — we will use b o th nam es interchangeably) [1, 2] possess features which m ake th em a very a ttra c tiv e th eoretical tool for th e stu d y of dynam ical aspects of topological solitons. In co n trast to th e usual (1+ 1) dim ensional solitonic m odels (w ith or w ith ou t im purity ), th ey possess a nontrivial m oduli space, i.e., a space of energetically equivalent sta tic solutions which are solutions of a p e rtin en t Bogom olny equatio n and, therefore, s a tu ra te th e corresponding topological bound. Of course, as all configurations have exactly th e sam e energy, th ere is no sta tic force betw een th e sta tic B P S soliton and th e im purity. Im p o rtan tly , however, th e form of th e solution changes at different positions on th e m oduli space, i.e., for B P S solitons w ith different distances from th e im purity.

T here is, in fact, a tran sfo rm atio n called a generalized tra n sla tio n which transform s one B P S solution into an oth er. T his tra n sfo rm atio n is a sym m etry of th e Bogom olny equation b u t not of th e full action (th e full E uler-L agrange (EL) equation ). As a consequence, th e low energy dynam ics of th e B P S -im purity solution follows a (nontrivial) geodesic flow on m oduli space. As a consequence of th is nontrivial sym m etry, th e spectral stru c tu re of th e B P S solution depends on its position on m oduli space. In o th er words, a B P S solution v ib rates differently, depending on its distance from th e im purity. T his allows us to analyze interactions betw een th e B P S soliton and th e excited m odes in a very clear set-up. N ote th a t, in th e usual solitonic m odels in (1+ 1) dim ensions, th e m oduli space is trivial. It consists of tra n sla tio n a lly equivalent B P S solutions (one-soliton solutions) w ith a fixed spectral stru c tu re which does not depend on its position on m oduli space. O n th e o th er

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hand, th e sta n d a rd ways to couple im purities (see for exam ple [3]- [14]) destroy th e self­

dual (B PS) sector completely. Hence, solitons are a ttra c te d or repelled by th e im purity, im plying a preferred distance betw een th e soliton and th e im purity. As a result, no m oduli space survives.

As a consequence, th e B P S -im purity m odels in one sp atial dim ension reveal striking sim ilarities to higher dim ensional B P S (self-dual) theories, like th e A belian Higgs m odel at critical coupling, or th e self-dual t ’H ooft-Polyakov m onopoles. For exam ple, B P S vortex solutions at different positions on th e corresponding m oduli space have different shapes and different sp ectral stru c tu re (th e spectral s tru c tu re has been calculated explicitlty only for cylindrical vortices [15]- [17]). Therefore, th e self-dual im pu rity m odels in (1+1) dim ensions (ra th e r th a n th e usual solitonic m odels w ithout im purities) m ay serve as a lab o ra to ry for higher dim ensional self-dual (B PS) theories, especially as far as dynam ical properties are concerned. Indeed, th ey reproduce th e usual geodesic flow on m oduli space which describes th e low speed dynam ics of B P S solitons. Moreover, th ey allow for an analy tical stu d y of effects beyond th e geodesic appro xim atio n like, e.g., an in teractio n of B P S solitons w ith in tern al (oscillating) modes, which can be crucial for a b e tte r u n d erstan d in g of th e full dynam ics as well as q u a n tu m prop erties (see for exam ple th e role of th e vib ratio n al m odes in th e semiclassical q u an tizatio n of th e Skyrm e m odel [18]- [19]).

It this work, we present a ty p e of B P S -im p u rity m odels which, while keeping all th e m ain properties of th e B P S -im p u rity theories unchanged, are analytically solvable in th e B P S regim e for any im purity. Furtherm ore, th e m etric on th e m oduli space and, th e re ­ fore, th e geodesic flow, as well as th e effective p o ten tial in th e spectral problem can be d eterm ined in an exact m anner. This will allow us to u n d e rsta n d in an an alytical way how th e m otion on th e m oduli space (generalized tra n sla tio n ) is reflected on th e level of th e spectral properties of B P S solitons. Thus, we provide th e sim plest (solvable and (1+1) dim ensional) exam ple of a m odel w ith a nontrivial m oduli space.

It should be underlined th a t, alth ough th e assum ed form of th e m odel requires a quite specific coupling betw een th e im p u rity (which, on th e o th er hand, can have any spatial d istrib u tio n ) and th e pre-potential, th e sam e qu alitativ e features are shared by all oth er self-dual im p u rity m odels. T he coupling considered here is chosen to render analytical co m p utatio n s possible. In addition, th is coupling enlarges th e acceptable class of im purities, which now do not have to be L² integrable. Instead, our im p urity can approach a co n stan t value or even diverge a t spatial infinity. T his opens a new avenue of applications of th e self­

dual im pu rity m odels for a b e tte r u n d erstan d in g of th e dynam ics of an nihilatio n processes in kink-antikink scattering.

Indeed, we show th a t such a non-localized im purity can lead to a th eo ry whose B PS sector consists of topologically triv ial solutions. T he corresponding moduli space describes the annihilation (creation) of an asym ptotically infinitely sep arated soliton-antisoliton pair.

As th ey are solutions of a Bogom olny equation, th ere is no s ta tic force betw een th e kink and th e antikink. Again, th e m oduli space m etric as well as th e effective p o ten tial of th e sp ectral problem can be found analytically. This m ay provide a chance to analytically u n d e rsta n d th e role of in ternal m odes in th e ann ihilatio n of solitons.

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2 S ta tic B P S so lu tio n s

It has been shown recently th a t for a given (1+1) dim ensional scalar field th eo ry w ith topological solitons (and therefore w ith a p o ten tia l w ith a t least two vacua), th ere are infinitely m any couplings to an im pu rity a = a (x ) such th a t half of th e B P S sector of th e original m odel keeps th e B P S p ro p erty [20]. T h a t is to say, after th e coupling w ith th e im purity, eith er th e kink or th e antik ink (b u t never b oth) is still a solution of a first order differential equation, th e so-called Bogom olny equation. In th e lim it a ^ 0 all these extensions reproduce th e original scalar theory.

In th e present work we w ant to consider th e following B P S -im p u rity m odel

(2 . 1)

which in th e lim it a = 0 gives th e usual scalar m odel w ith a p o ten tial U(fi) = W 2(fi) (w ith two vacua fi+ > fi- ). N ote th a t th is B P S -im p urity th eo ry differs in some details from th e previously analyzed exam ples [1, 2], b u t th e crucial properties, discussed briefly above, rem ain unchanged.

In order to find a Bogom olny equation, we rew rite th e sta tic energy as

w here th e bound is s a tu ra te d if th e following Bogom olny equ atio n holds

fi^x + V ^W + ^ [ 2 a W = 0. (2.3)

Here Q is th e topological charge Q = (fi(x = to ) — fi(x = —t o ) ) /( 0 + — fi- ). One m ay easily verify th a t th e Bogom olny eq u atio n implies th e full eq u atio n of m otion. N ote also th a t, as always happens for th e B P S -im p u rity models, we have only one Bogom olny equ ation which m eans th a t only one soliton (eith er th e kink or th e antikink) belongs to th e B P S sector, while its charge conjugate p a rtn e r only solves th e full E L equations and does not s a tu ra te th e topological bound. F u rtherm o re, th ere is no restriction on th e sp atial d istrib u tio n of th e im purity.

A peculiarity of th is p a rticu la r class of B P S -im purity m odels consists in th e fact th a t th e Bogom olny equ atio n can be brought into th e n o-im purity form. Namely, consider a coo rdin ate tran sfo rm atio n y = y ( x ) such th a t

dy f^x

— = 1 + a (x ) ^ y(x) = x + a ( x ' ) d x ' = x + A^a(x) (2.4) d x

w here Aa(x) is a coord in ate shift due to th e im purity. T hen, th e Bogom olny equatio n is ju s t th e Bogom olny eq u ation of th e no-im purity lim it (in th e new coord in ate y)

fi^y + ^ 2 W = 0 (2.5)

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

/

-TO L2 2 J

/

T O 2

/

-CO J Ó —

(2.2)

which enjoys th e triv ial tra n sla tio n a l sym m etry y ^ y + y0, y 0 € R. Now, we can derive th e form of th e generalized tra n sla tio n analogous to [2], which acts on th e B P S solution in term s of th e x variable. In th e present case, it is a pure co o rd inate tran sfo rm atio n x ^ X (x ,x 0), w here x and X are related by

As we see, th e coord in ate tra n sfo rm atio n depends on th e p a rticu la r form of th e im purity, i.e., on A a (x). T he tran sfo rm atio n (2.6) is an exact, global version of th e previously iden­

tified infinitesim al generalized tra n sla tio n s (which, however, in [²] are not pure coordinate tran sfo rm atio n s). Obviously, th e Bogom olny equ ation in th e new variable (2.5) is solvable by an integration, which ensures th e solvability of th is self-dual im pu rity theory.

T he B P S sector still possesses th e two co n stan t topologically triv ial solutions (lum ps) 0 = 4>±, which correspond to th e vacua of th e non-im purity m odel. As th ey s a tu ra te th e topological bound, th eir energy m ust be 0. Obviously, th e generalized tra n sla tio n m aps th e lum p solution into itself, as expected on general grounds, see [2]. As we will see below, despite th e ir sim plicity these lum p sta te s are not trivial and do co n trib u te to th e dynam ics and spectral properties of th e solutions.

Let us observe th a t th e s ta tic energy w ritte n in th e variable y is not com pletely equiv­

alent to th e no-im purity case. Indeed,

w here <r(y) = a ( x( y ) ) is th e im p urity expressed in th e y coordinate. N ote th a t the energy density is p ro p ortio nal to th e Bogom olny equ ation squared. It is therefore a triv ia l observation th a t any solution of th e Bogom olny equ ation <fiy + \/2 W = 0 obeys th e full equatio n of m otion. T he energy functional (2.7) can no t be w ritte n as E = J ' ^ dy (1 + <r(y)) ( 2 + W 2) i.e., w ith th e cross term o m itted, because th is term is m ul­

tiplied by th e im p u rity function and is not a pure b o u n d ary term . Observe also th a t in th e usual, a = 0 case, th e full sta tic equation of m otion can be once integ rated to a con stan t pressure equ atio n 1 + W 2 = P w ith P € R+ being a pressure. In o th er words, any static solution is a solution of th is co n stan t pressure generalization of th e Bogom olny equation.

T his allows, for exam ple, for th e co n struction of solitons in a box (or on a circle S 1 for a periodic p re-p otential). Here, on th e contrary, solutions of th e co n stan t pressure equation do not solve th e equ atio n of m otion unless P = 0. U nfortunately, we do not know any generalisation of th e Bogom olny equation (2.5) (or (2.3)) which would be equivalent to th e sta tic equatio n of m otion. A related question is w h eth er it is possible to define th e self-dual im p urity m odel on a circle (which for topologically nontrivial solutions, of course, requires a periodic im pu rity). N ote th a t for th e original non-im purity m odel on S 1, solitonic solu­

tions do not s a tu ra te th e p e rtin en t topological bound as th ey obey th e Bogom olny type equation w ith a non-zero pressure (here related to th e radius of S 1). Hence, strictly speak­

ing, th ey are not of th e B P S type. O n th e o th er hand, if p u t on a circle, th ey obviously enjoy th e usual tra n sla tio n sym m etry (w ith th e periodic bo u n d ary condition), form ing a triv ia l m oduli space.

x + A a (x) = x + x 0 + A a (x). (2 .6)

(2.7)

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f ~ ( 1 \ 2

E = J dy (1 + d(y )) I <fty + W ) + b o u n d ary term

Figure 1. Profile of the BPS antikink for the 4>A B PS-im purity model (2.9) w ith a = —2 for different values of the moduli space param eter x 0.

As an exam ple, let us consider <fi4 theory, th a t is, we take W = (1 — 0 2)/a /2 - F u rth e r­

m ore, we assum e th e previously considered, exponentially localized form of th e im purity a ( x ) = a / cosh2 x, w here a € R m easures th e stre n g th of th e im p u rity [2] (see also [14]).

T hen, th e Bogom olny equation (2.5) form ally has an an tikin k solution in th e y coordinate, which is th e antikink of th e pure 0 4 th eory

^(y)BPS = — ta n h (y + yo). (2.8)

Hence, th e B P S an tikink in th e self-dual im pu rity m odel is

0 (x)bps = — ta n h (x + A a (x) + xo) (2.9) w here th e coo rdinate shift reads

A a (x) = a (ta n h x + 1 ). (2.10)

Its value changes from A a (x = —to ) = 0 to A a (x = to ) = 2 a . T he m oduli space m ay be param etrized by x 0 € R. In figure 1 we present profiles of th e B P S an tikink w ith a = —2 for several values of th e m oduli space p a ra m ete r x 0 which represents different positions of th e soliton relatively to th e im purity. T he im pact of th e im p u rity on th e soliton is visible only when th e soliton is in th e vicinity of th e im purity. O therw ise, th e profile ten d s to th e usual 4>4 an tik in k solution.

N ote th a t for such a stro n g negative im purity (i.e., for a = —2), th e position of th e zero of th e field (i.e., th e position a such th a t 0 BPS(a) = 0) cann ot be used globally as a coo rdin ate on m oduli space. In fact, for a < —1 th e field is not a m onotonically decreasing function. T his leads to th e ap pearan ce of th re e zeros for a certain range of x 0. Hence, we explicitly see th e problem existing in th e previously discussed self-dual im purity models, w here th e generalised tra n sla tio n was known only in an infinitesim al form. T here, th e m oduli space m etric was calculated num erically, and th e zero of th e field was used as a

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m oduli space coord in ate because of its num erical simplicity. As a consequence, for to o negative a, th is coord in ate on m oduli space was not globally defined.

It is im p o rta n t to observe th a t th e Bogom olny equ atio n (2.5) possesses an additio n al solution

theory, namely, 0(y) ^ 1 /0 (y ). Obviously, th is solution is not a regular solution on th e full R due to a singularity a t y = — y0. Accordingly, it is not relevant for a co nstru ction of th e an tikin k in th e presence of th e localized im purity. However, as we will see in section 6, th is class of solutions of th e Bogom olny equatio n is very im p o rta n t for B P S solutions w ith vanishing to ta l topological charge. We rem ark th a t th e coth solution has been considered in th e 0 4 m odel w ith a bo u n d ary [21].

3 M e tr ic on m o d u li sp a ce an d g e o d e sic flow

H aving solved th e B P S (self-dual) sector completely, we now analyze some aspects of its dynam ics, again in an analy tical way. N ote th a t b o th sta tic and dynam ical p roperties of th e non-B PS solitons d rastically differ from th e ir B P S co u n terp arts. T his is obviously related to th e fact th a t th ere is no m oduli space for non-B PS s ta tic solitons. Here we are interested in an analytical description of th e B P S sector and do not consider non-B PS solutions. T he full L agrangian is

w here th e topological (boundary) term has been su b tra c te d . T hen, th e dynam ical Euler- Lagrange equatio n reads

In a low-velocity collision of th e B P S soliton on th e im purity, th e system undergoes a sequence of B P S states, which leads to a dom ain wall th a t does not get stuck on the im p urity [2 , 22]. Hence, it generates a flow on m oduli space [23]. To find an analytical form of th is flow, we assum e th a t th e position of th e soliton on th e m oduli space is a dynam ical, tim e dep endent q u a n tity x 0(t), i.e.,

and insert it into th e Lagrangian. As a result, we get an effective m odel which has a nontrivial kinetic p a rt only. T he p o ten tial p a rt reduces to a c o n stan t due to th e B PS n a tu re of our im pu rity theory. Here,

whose existence is related to an o th er sym m etry of th e Bogom olny equatio n of th e pure 4>4

0 tt — <fixx — + 2 W W ^ (1 + ct)2 = 0. (3.2)

0 (x ,t) = 0BPs(x,Xo(t)), (3.3)

(3.4)

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0(y) = ~ A = - coth(y + yo)

9 { V )b p s

(2.11)

L = j d x 2 — 2 { ^ x + V ^W + V 2 a W ^ — J dxV2W<fix (3.1)

Leff = 1 M (xo)X2

Figure 2. M etric on the moduli space for the 4>A B PS-im purity model (3.6) for different values of the im purity strength a.

w here th e effective m ass (m etric on m oduli space) is

M = J ^ b p s (x + A a (x) + x 0( t ) )^ . (3.5) T he obvious solution of th e effective problem is

. dx

M (xo) = 4--- . J -IX cosh4(x + a ( ta n h x + 1) + x0(t))

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■ U\ Im (xo = —ro)

xo(t) = M (xo)

w here vin is th e velocity of th e incom ing antikink a t m inus infinity, which we assum e to be th e initial sta te . N ote th a t, since th e solution exists for any x⁰ € R, M ( x 0) is a positive definite function. Hence, th e m etric is globally well defined on th e whole m oduli space.

For 04 th eo ry w ith th e exponentially localized im purity, th e m etric is given by th e expression

(3.6) T he integral is finite, g reater th a n 0 for any value of a and any x0 € R. We plot it in figure 2 for several values of th e stre n g th a of th e im purity. For a > 0 th e m etric is always below th e asy m pto tic m ass, approaching arb itra rily small positive num bers if a ^ ro. For negative im purities, a € ( a crit, 0) th e m etric is above th e asym p totic m ass w ith one local extrem um . However, if th e stre n g th of th e im pu rity is below a crit, th re e local ex trem a show up. T his stru c tu re is related to th e p a rticu la r shape of th e im purity ra th e r th a n to a generic feature of th e m odel like, for exam ple, th e existence of th re e zeros in th e profile for some values of x⁰ if a < - ¹.

Indeed, e.g., for a rectan g u lar im p urity a (x ) = a ( Q ( x + 1 ) - Q ( x —1)) [6]- [12], w here 0(x) is th e Heaviside step function, th e m etric can be derived in a closed form . T he shift func­

tio n is A a (x) = a ((x + 1)0(x + 1) — (x — 1)0(x — 1)) which seems to be a m onotonously

(3.7)

4 S p e c tr a l p ro b lem

T he next step of th e analysis of dynam ical properties of th e model, which goes beyond th e geodesic approxim ation, considers small p e rtu rb a tio n s ab o u t th e B P S solutions. As th e B P S solutions are exactly known, we have th e unique o p p o rtu n ity to get an an alytical insight into th e stru c tu re of th e oscillating m odes as a function of th e position on th e m oduli space. For this purpose we consider a small p e rtu rb a tio n of th e B P S solution, 4>(x,t) = 0 b p s(x ) + cos(w t)n(x). T hen, th e m ode (Schrodinger) equ atio n is

— Vxx + V (x)n = (4.1)

w here th e effective p o ten tial reads

V (x) = 2(1 + a ) 2( W | + W W ^ ) — V 2 a x W ^ , (4.2) and W, W<£, W ^ are evaluated for th e B P S solution 0 BPS. This can also be w ritte n as

V (x) = 2(1 + a f O ^ W W ^ ) — V 2 a x W ^ = (1 + a ) 2d ^ W 2 — ^ 2 m Ą W . (4.3) As th e im pu rity is spatially well-localized a t th e origin, asym pto tically (i.e., for x 0 ^ ± ro ) th e B SP antikink solution consists of an infinitely sep arated free soliton and a lum p. T here­

fore, th e corresponding sp ectral problem also decouples. As a consequence, th e spectru m of bound sta te s is a sum of b ound states of th e free antik ink (in th e original no-im purity m odel) and th e lum p. Observe th a t, although th e lum p solutions are spatially constan t, th ey have a nontrivial effective p o ten tial in th e spectral problem . Namely,

Viump(x) = 2 ( 1 + a ) 2 W 2Iump — V 2 a x W ^ lumP (4.4) w here W ^ denotes W^ com puted on a lum p solution, which coincides w ith one of th e vacua of th e original, n on-im purity model. T hus, W^Iump is ju s t a m odel dep en den t nu­

m erical co n stant. Interestingly, th e p o ten tial in th e spectral problem of th e lum p solutions takes th e form of a super-sym m etric q u an tu m m echanical p o ten tial

Viump(x) = U 2 (x) — (4.5)

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growing function for a < —1 and, therefore, th e field possesses th re e topological zeros for some values of th e m oduli space p aram eter. Now, th e m etric reads

M (x0) = 3 [4 + T + a 1 ) ( 2 c o s h 2 (x 0 — 1) — 1) 3 [ ¹ + a cosh²(x⁰ — ¹)

— ta n h <x0 + T + 2 a ) (2cosh2(xo + 1 + 2a ) — 1)1 1 + a cosh2 (xo + 1 + 2a j V 0 ' ,

w here we assum e th a t a = —1. This m etric has no local m inim a for a < 0 b u t for sufficiently negative a it reveals a p lateau . T he plateau, for asym ptotically large, negative x 0, is located at §. O n th e o th er hand, th e B P S antikink solutions possess a very sim ilar s tru c tu re as in th e case of th e exponential im purity. Namely, th ere are th re e topological zeros for some values of x0, for any a < —1.

w here U (x) = \/2 W ^ lump(1 + a (x )). L et us assum e th a t we have two sym m etric vacua (as in theory). Then, V±mp = U2 ± Ux and U = V/2|W ^lump|(1 + a ). We can use th e sta n d a rd techniques of supersym m etric q u a n tu m m echanics to o b tain some inform ation a b o u t th e sp ectral problem [24]. F irst, we define two first order differential o perators

d d

A+ = - - d + U, A- = - d + U. (4.6)

dx dx

T he m ode equ atio n can be w ritte n as

Hon = A±ATn(x) = w²n (4.7)

for lum ps. H± are known as p a rtn e r H am iltonians. From th e last form ula it follows th a t th e (possible) zero m ode for th e lum ps verify th e first order equations

AonO = 0. (4.8)

We have two possibilities of th e zero m ode stru c tu re for lum ps depending of th e form of a. Namely, no norm alizable zero m odes or only one zero m ode (for one of th e lum p solutions b u t never for b o th ). A n o th er pro p erty of th e p a rtn e r H am iltonians is th a t, regardless of th e form of a, if th ere are excited b ound states, th e sp ectra of b o th lum ps coincide (except for th e zero m ode). It is also interestin g to note th a t, in th e context of SUSY QM, H- and H + are th e bosonic and ferm ionic ham iltonians respectively. Therefore, th e m odes for th e 0- lum p correspond to bosonic states in SUSY QM and th e m odes for th e <p+ lum p correspond to ferm ionic sta te s in SUSY QM. So we can define a W itte n index in our system .

T he zero m ode equations can be easily solved

n± = N e^± f dxU (x) = N e±2(x+^ y(x)dx) (⁴.⁹) N orm alizability of th e zero m odes requires th a t th e integral a d x a t infinity grows a t least like x. B u t th is implies th a t a at infinity m ust be a nonzero co n stan t, in con tradictio n w ith our assum ption of localization. As a consequence, in th e background of a lum p there are no zero m odes for localized im purities.

W hen th e B P S an tikink moves on th e m oduli space, approaching th e im pu rity (x0 ^ 0), th e sp ectral stru c tu re changes due to a nontrivial deform ation of th e solution. In some sense, we observe a nonlinear superposition of th e m odes of th e asy m pto tic states. In p articu lar, oscillating m odes (originating b o th in th e free soliton and th e lum p) move and can even be pushed into th e continuous spectrum . T his leads to th e recently discovered spectral wall effect [25].

T he sim plest version of such a deform ation of a m ode while moving on th e m oduli space can be studied in th e case of th e zero m ode. In fact, th is m ode, which is related to th e generalized tra n sla tio n , rem ains a zero m ode on th e whole m oduli space and is given by no = i+ y d<tedPxs(x). Since th e B P S solutions are explicitly known, we can find th e zero m ode in a closed form . Moreover, for spatially well-localized im purities, we can conclude th a t asym ptotically th e zero m ode is confined to th e asy m p totic (free) soliton. Hence, th e zero m ode in th is asym p to tical regim e acts nontrivially only on th e outgoing /inco m ing soliton.

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

For exam ple, in th e case of th e <fi4 m odel and for an a rb itra ry im purity, th e effective po ten tial takes th e following form

V (x) = 2 (1 + a ) ( —1 + 3 ta n h 2(x + A a (x) + x 0)) + 2 a x ta n h (x + A a (x) + x 0). (4.10)

Now it is perfectly visible th a t th e m otion on th e m oduli space, i.e., th e change of th e p a ra m ete r x 0, is reflected on th e level of th e effective p o ten tial, and, as a consequence, in th e spectral p roperties of th e B P S solutions in a ra th e r nontrivial m anner. To be specific, for th e exponentially localized im p u rity we find

w here t corresponds to th e 0 lump = ± 1 lum ps, respectively. In figure 3 we show how th e effective p o ten tial gets deform ed w hen th e B P S antikin k approaches th e im purity. In some sense, we observe a ‘collision’ of two asym pto tically independent spectral problem s. In th e u p p er panel (th e im p u rity w ith a = 2), in th e initial s ta te there is only one oscillating m ode localized on th e free 4>4 antikink. T he im p urity does not host any b ound sta te . W hen th e soliton approaches th e im purity, th e <fi4 p o ten tial well becomes to o narrow and shallow, and th e m ode disap pears into th e continuous spectrum . In th e b o tto m panel ( a = - 2 ) , th e im pu rity (lum p) possesses th re e oscillating m odes. W hen th e B P S soliton m eets th e im purity, th e p o ten tial wells merge. In th is case, all oscillating m odes survive for any position on th e m oduli space (see figure 4) .

T he dependence of th e m ode stru c tu re on th e position on th e m oduli space is strongly sensitive to th e details of th e assum ed im pu rity (and th e field th eo retical pre-po ten tial).

As is clearly visible in figure 4, th e num ber of m odes can be c o n stan t du rin g th e collision.

However, it is not a rare phenom enon th a t one m ode enters th e continuous spectrum . It can even reap p ear again when th e soliton is sufficiently close to th e im purity. In fact, th e richness of possibilities is ra th e r unlim ited. N ote th a t a m ode which disap pears into th e continuous spectrum m ust finally re tu rn to th e discrete spectrum , because of th e spatial localization of th e im purity.

W henever a bound m ode enters th e continuous spectrum , a sp ectral wall is expected.

T he fam ily of solvable self-dual im purity m odels considered here may, therefore, be ideal for fu rth e r studies of th is phenom enon in a very clear num erical and m ath em atical set-up.

N ote th a t for spatially localised im purities, th ere always exists a p air of sp ectral walls, i.e., on b o th sides of th e im purity.

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

a \ 2 4 a ta n h x klump(x) = 4 I 1 + ~2 ) T T“2

cosh x cosh x (4.12)

while

V (x) = 2 ( n --- a ^ l ( —1 + 3 ta n h2(x + a (ta n h x + 1) + xo)) cosh2 x

4 a ta n h x

---2— ta n h (x + a (ta n h x + 1) + x0)

cosh² x (4.11)

F ig u r e 3. Potential V(x) in the spectral problem for the f>4 B PS-im purity model (4.11) . L eft: a = 2 and x 0 = —10, —2, —1, 4. R ight: a = —2 and x 0 = —10,1, 2, 4.

Figure 4. Number of discrete modes (zero mode and oscillating modes) in the <fA B PS-im purity model (4.11) as a function of the im purity strength a and the position on the moduli space x 0.

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

5 N o n -lo c a liz e d im p u r itie s — s te p -fu n c tio n e x a m p le s

It is a n o th er ad vantage of th e present m odel th a t it allows for im purities which do not ap proach zero a t infinity. Thus, it does not have to be a localized, L 2 integrable im purity.

T his results in a divergent shift function A a (x). T his case is possible because of th e fact th a t th e im purity is always m ultiplied by th e p rep o ten tial W , ensuring th e d isap pearance of th is term if th e field takes its vacuum values (th e zeros of W ). Physically, th is m eans th a t th e im p u rity changes th e asy m p to tic value of th e W 2(1 + a ) 2 term , i.e., th e asym p totic m ass of th e soliton. For instance, let us consider a very sim ple im p u rity in th e following form, a = aQ(x). Then,

A a = a x d ( x ) (5.1)

w here 0(x) is th e Heaviside step function. N ote th a t a m ay be even divergent a t infinity.

Here, as an exam ple one can consider a = a x 2 which leads to A a = a x 3/3 .

Interestingly, such a non-sym m etric im p urity (5.1) , which can affect th e asym ptotic m ass of th e field (at infinity), can com pensate a non-sym m etric (p re)p o ten tial. Here an exam ple m ay be given by 0 6 theory, w here W = 0(1 — 0 2)/a /2 . In th e no-im purity lim it th e m ass of th e field (sm all p e rtu rb a tio n s aro un d a vacuum ) is different a t 0 = 0 (m = 1) and 0 = ± 1 (m 2 = 4). A fter coupling th e m odel to an im p urity in th e self-dual m anner, th ere are two B P S solutions: th e antikink 1 —> 0

j 1 — ta n h (x + xo + A CT (x))

b p s = y ---2--- (5-2)

and th e kink 1 0

1 — ta n h (x + xo + A a (x))

0 K b p s = — Y--- 2 ---• ( ) T he two o th er sta tic solitons are not B P S solutions. Then, for th e B P S an tikin k and kink th e m ass of small p e rtu rb a tio n s a t 0 = + 1 and a t 0 = 0 are equal if A = 0(x). For th e non-B PS solitons, th e m asses of th e field a t plus and m inus infinity are different. T his m ay affect th e resonant, fractal-like stru c tu re s in a kink-antikink collision.

If we assum e th e step function im p urity [13], th e n th e m etric can be com puted in an exact form. For th e 0 4 model, it reads

M (xo) = a ( ta n h x o — 1 ta n h 3 xo J + 2 ( 1 + — 1— ) (5.4)

1 + a 3 3 1 + a

for a > —1. As rem arked before, th e asy m p to tic m asses of th e solitons a t plus and m inus infinity (on th e m oduli space) are not th e same. Namely, M ( x o = to) = | and M ( x o = —to) = 3 i + a . One can check th a t for any a > —1 th e m etric is globally well- defined on th e whole m oduli space i.e., M (x o) > 0 for all x o € R. For a > 0 ( a < 0) th e m etric function is a strictly growing (decreasing) function, see figure 5. Similarly, for 0 6 B P S solitons and th e step function im purity we get

M ( x o) ^ 1---- - (4 + a + 2 a ta n h x o + a ta n h 2 x o) (5.5) 32(1 ~+ a )

which m onotonically in terp o lates betw een M ( x o = to) = | and M (x o = —to) = 8(1+a ) .

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

F ig u r e 5. M etric on the moduli space in the 04 B PS-im purity model w ith non-localized, step function im purity (5.4) for different values of the im purity strength a.

Here, as th e effective p o ten tial does not have to ten d to th e sam e asym pto tic value at

±to, th e sp ectral wall m ay exist only on one side of th e im purity, w ith ou t any p a rtn e r.

6 S elf-d u a l k in k -a n tik in k so lu tio n s

6 .1 A n n ih ila tio n o f s o lito n s o n t h e m o d u li sp a c e

T here is a class of non-localized im purities for which th e Bogom olny equ atio n su p po rts topologically triv ial solutions only. Surprisingly, such solutions result in a m oduli space w ith a nontrivial m etric. Consider, for exam ple, 0⁴ th eory w ith th e following im purity

gj (x) = ² ta n h x — ¹ (⁶.¹)

w here j is a positive integer num ber. T he shift function is now A j(x ) = | ln co sh x — x, which leads to th e following solution of th e Bogom olny equation

, , , , / j , \ coshj x — a

ffl(x) = — ta n h - ln cosh x + x0 = --- ;--- (6.2)

' ^{\ 2} ) coshj x + a V '

w here a = e-2x0. In spite of th e fact th a t we used a form al antikink B P S solution in th e y coordinate (tre ate d as a solution of (2.5)), this solution is topologically trivial. Indeed, th e B P S solution (6.2) has a m axim um for x = 0, w here it reaches — t a n h x 0, and th en it approaches th e same value a t ± t o , i.e., 0 (x = ± to ) = —1. T reated as an elem ent of th e m oduli space, th e B P S solution for x0 ^ to (a ^ 0) approaches th e lum p solution

^iump = —1. W hen x0 decreases, th e solutions develop a hill centered a t x = 0. By fu rth e r decreasing x 0, th e hill grows until it develops a p late au a t 0 ^ + 1 . For x0 ^ —to (a ^ to ), th e p late au expands and th e solution represents a kink-antikink p air w ith th e separation distan ce growing to infinity. In o th er words, th e m otion on m oduli space, from x0 = —to

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

F ig u r e 6. BPS solutions for different values of the moduli param eter X for th e annihilating im purity with j = 1.

to x⁰ = to , is equivalent to th e scatterin g of an initially infinitely sep arated pair of a kink and an antikink. A t th e final stage, th e solitons com pletely an n ih ilate and we have th e lum p solution 0 - mp only. Observe th a t, as all B P S solutions have th e sam e energy, there is no sta tic force betw een th e co n stitu en ts, kink and antikink. However, th is th re e body (kink-im purity-antikink) system is not com pletely free, as th e solitons m ust be a t equal distan ce from th e im purity.

Surprisingly, it tu rn s o u t th a t xo does not cover th e whole m oduli space for th is im pu­

rity, and th a t th e full m oduli space of topologically trivial solutions is even larger. Indeed, one can easily verify th a t th e B P S solution (6.2) rem ains a solution of th e Bogom olny equ ation if th e p a ra m ete r a takes negative values. If a < — 1 th e solution diverges a t x sing such th a t coshj x sing + a = 0. Therefore, these are not physically acceptable solutions.

O n th e o th er hand, for a € ( —1,0), th e solutions are globally well-defined solutions of th e Bogom olny equ atio n and, as a consequence, m ust co n trib u te to th e m oduli space. We plot th is new, negative p a rt of th e B P S solutions (to geth er w ith th e first, positive p a rt) in figure 6 in a new m oduli space coord inate X € ( —to , to ), defined as

a = —1 + ejX . (6.3)

T his new m oduli space coord in ate b e tte r describes th e zero of th e field, i.e., th e position of th e soliton in th e large sep aratio n lim it. However, in some form ulas we will still keep a due to its simplicity. For a = 0 (X = 0) we s ta rt w ith th e <fi = —1 lum p, which was th e end of th e first, positive a fam ily of th e B P S solutions. Once a (or X ) takes negative values, th e field develops a dip located at x = 0. As a ten d s to -1 (X tend s to —to ) th e dip approaches an a rb itra ry large (negative) value.

It is interestin g to notice th a t, while th e positive p a rt of th e B P S solutions emerges from a regular solution of th e original (no-im purity) theory, th e negative p a rt stem s from th e singular solutions of th e no-im purity m odel. Indeed, th e Bogom olny equation (2.5)

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

possesses th e following solution

0(y) = — co th y (6.4)

which is singular a t y = 0, see our discussion below eq. (2.11) . However, in th e case of th e im purity Oj, th e variable y is expressed via x as y = | ln co sh x + x 0, w here x⁰ is a co n stan t. Hence, y is bounded from below by ymin = Xo and th e solution does not reach th e singularity if X⁰ > 0. As a consequence, we get

,. . , / j , _ \ coshj x + e-2x0 . .

0 (x ) = — coth - l n c o s h x + x0 = --- :--- — , (6.5)

\ 2 J coshj x — e-2x0

th a t is, a solution (6.2) w ith a = — e-2x0. T he fact th a t th e m oduli space needs to be com plem ented by th e originally (no im purity theory) divergent solutions is a new and ra th e r unexpected result, which m ay have some im pact on th e co n stru ctio n of th e m oduli space of o th er self-dual im p urity models, also in higher dim ensions [20].

To sum m arize, th e full m otion on th e m oduli space describes an an nih ilation of th e initially infinitely sep arate kink-antikink pair. T he process does not stop a t th e 0 = —1 lum p b u t goes th ro u g h th is point and ends on a well form ed by solutions corresponding to th e singular solutions of th e no-im purity m odel. N ote th a t such a well is observed in kink-antikink collisions in th e pure 04 theory. In fact, du ring th e collision, th e solitons form a sort of well w ith a shape th a t m ay be approxim ated by

0(x) = — ta n h (x — X ) + ta n h (x + X ) — ¹ (⁶.⁶) w ith X > —1. A t th e m inim um of th is well, th e field takes th e value —3, at m ost. In this regime, our negative ty p e of solutions reproduces this behavior. Therefore, th e appearance of th e negative ty p e of solutions should be considered as an advantage of th e im pu rity model. Of course, in co n tra st to th e pure 0⁴ theory, in our m odel th ere is no m echanism preventing th e well from developing an arb itra rily large d epth.

T he low speed dynam ics of th is an nih ilation process is described by th e geodesic ap ­ proxim ation. T he m oduli space m etric for th e corresponding topologically triv ial B PS sector is

„ t cosh2j x d x ,

M j (a) = 4 / — -- --- - I (6.7)

■J-™ (coshj x + a) which for j = 1 has th e following exact form

4 / 4 i Oj2 a __1

M j = l(a ) = — 3 (1 — a2)3 ( “ 4 — 1 0 a 2 —6 —6a a rc tan (6'8) In figure 7, we plot th e m etric M ( X ) corresponding to th e collective coord inate X , related to M (a ) via M ( X ) = ( d a / d X )2M (a ) = j 2(a + 1)2M (a ). In b o th coordinates, th e m etric is a m onotonic function which sm oothly interpolates betw een infinity at X = —to (a = —1) and a finite value a t X = to (a = to ). Specifically, for asym ptotically large a th e m etric ten d s to zero in a power like m anner, M j= 1 (a) = ( 4 /3 ) a -2 + O ( a - 4 ). T hus, th e zero of the m etric is approached in infinite tim e a a e^const. In th e X coord inate th e m etric ten d s to a nonzero value M j = 1(X = to ) = 4 / 3 also in infinity tim e

X a t (6.9)

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

F ig u r e 7. M etric on the moduli space for the annihilating im purity with j = 1. The horizontal line denotes the asym ptotic values M (X = to ). A logarithm ic like scale is used for M , because M grows very fast for negative X .

For a ^ —1 th e m etric asym ptotically reads M j = i(a ) = (1 + a) 7/2 + O((1 + a) 5/2).

Therefore, th e infinity of th e m etric is approached again in infinite tim e (1 + a) a t -4/3 or

X a ln t (6.10)

Finally, th e point X = 0 (a = 0) w here th e positive and negative ty p e of solutions are connected is a com pletely regular point of th e m etric. Here M j = 1(a = 0) = 8. All th is m eans th a t th e singular b ranch of B P S solutions (w ith a < —1) is dynam ically separated from th e regular b ranch (formed by b o th positive and negative types of solutions sm oothly joined a t a = 0). Indeed, th e a ^ —1 b o u n d ary is approached for t ^ to .

An analogous behavior is found for th e o th er im purities a j . For exam ple, for j = 2 th e m oduli space m etric is

M j= 2(a) = 1 ( —3 + 4 a (4 ^+ a) + 3(1 + 62a) a rc ta n h , ) . (6.11) j w 6 V a(1 + a ) 3 (1 + a ) 7/2a 3/2 V 1 + a j

Again, th e m etric has a finite value a t a = 0 (or X = 0), M j= 2(a = 0) = 16/3, w here th e positive p a rt sm oothly joins th e negative p a rt of th e self-dual solutions.

To some ex ten t, th e ob tain ed m oduli space m etric reveals some sim ilarities w ith th e m etric for th e C P 1 a-m odel in (2+ 1) dim ensions [26]. In th e sim plest set-up, for two or m ore solitons sittin g on to p of each other, th e m oduli p a ra m ete r is given by th e height of th e soliton A, while its position is kept fixed. Hence, th e com plex field which is th e prim ary field of th e m odel reads u = An (x + iy)n , w here n is th e p e rtin en t topological charge carried by th e soliton. W hen A is small, th e soliton is small and broad, while for large A th e soliton gets narrow and tall. T h en th e m etric is ju s t M (A) = 2Cn A- 4 , w here Cn is a c o n stan t [27].

T he zero of th e m etric, which corresponds to th e soliton collapsing to an infinitely th in and peaked configuration, is approached in a finite tim e as A ~ 1 /( tcrit — t). At th is point, th e configuration possesses a singularity in th e energy density, and th e geodesic approxim ation

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

as well as th e full num erics break down [27]. In our case, th e zero as well as th e infinity of th e m etric are approached in infinite tim e. Hence, we have a globally well defined geodesic dynam ic.

From a m ore general perspective, th is is a unique exam ple of a solitonic m odel which has a B P S sector w ith soliton-antisoliton solutions. Typically, if a Bogom olny equation su p p o rts m ultisoliton configurations, th e co n stitu en ts carry eith er positive or negative topological charge, b u t never b o th . This is exactly w hat h appens in th e A belian Higgs m odel a t critical coupling, w here a B P S solution is provided by a ratio n al function of an eith er holom orphic or antiholom orphic variable. T he sam e occurs for self-dual instanton s, w here in sta n to n -a n tiin sta n to n solutions do not belong to th e B P S sector. S tatic soliton- an tisoliton solutions are ra th e r exam ples of sphalerons, i.e., un stab le solutions which are not m inim a b u t m o u ntain pass solutions of th e energy functional. Here, th e C P N m odel w ith N > 1 m ay serve as an exam ple, see [28]. S tatic soliton-antisoliton configurations do not obey a first order Bogom olny ty p e equation and do not s a tu ra te any topological energy bound and, therefore, are not B P S solutions. Observe th a t a B P S sector containing solitons w ith positive and negative topological charge (coexisting holom orphic and anti- holom orphic m aps) has been discovered very recently for o th er B P S -im p urity theories in 2+ 1 dim ensions. Namely, in th e B P S -im p urity C P 1 m odel [20] and late r in th e im p u rity A belian Higgs m odel [31]. Furth erm ore, m ixed holom orphic and antiholom orphic m aps are B P S solutions of a m odel of th e so-called m agnetic Skyrm ions at th e critical cou­

pling [29]- [30]. In fact, a closed relation betw een this th eo ry and self-dual im p u rity m odels has been recently pointed ou t in [20] and [30].

N ote also th a t b o th th e kink and th e an tikin k co n stitu en ts, if considered separately, are not solutions of th e Bogom olny equ ation and, therefore, are not contained in th e B P S sector.

Only th eir b ound sta te , altho u gh w ith zero binding energy, solves th e Bogom olny equation.

We rem ark th a t a collective coordinate analysis w ith th e relative distan ce betw een kink and antik in k as one collective coo rd inate was applied to th e kink-antikink in teractio n in th e usual, no im purity, scalar field th eory in (1+ 1) dim ension. However, such a kink-antikink configuration is not a B P S solution, and th e collective coordin ate does not belong to a m oduli space. It is, therefore, not g u aran teed th a t th is coord inate covers th e dynam ics of th is process on th e whole R, even approxim ately. T his is one of th e reasons why th ere is no correct an alytical description of th e annihilatio n process in, e.g., th e <fi4 m odel [32]- [33].

6 .2 A n n ih ila tio n a n d t h e s p e c tr a l s tr u c tu r e

A nother im p o rta n t observation is th a t this choice of th e im pu rity leads to th e Poschl-Teller p o ten tial in th e sp ectral problem of th e lum p solutions. Indeed, we find th a t U = j ta n h x and

V ±mP = j 2 — j ( j T 1)cosh2^ • (6.12)

Hence, we triv ially find th e spectral stru c tu re of th e lum p solutions. For 0 lump = —1 (which does belong to th e m oduli space of th e B P S solution) th e p o ten tial is V - . Thus, th ere are j discrete bound sta te s w ith energies E n = n ( 2 j — n ), n = 0 , 1 , . . . , j — 1, which includes a norm alizable zero m ode n - = (cosh x ) - j . T he <filump = 1 leads to th e V + potential, which

J H E P 0 7 ( 2 0 1 9 ) 1 5 0

possesses j — 1 discrete m odes w ith energies coinciding w ith th e negative lum p, except, of course, th e zero m ode which is absent. T he continuous sp ectru m sta rts above E = j 2 in b o th cases. T he existence of th e zero m ode for th e fixed im purity can be, a t first glance, a surprising feature. However, in th is set-up th e lum p solution is not an isolated topolog­

ically triv ial solution b u t it belongs to the m oduli space. So, it is precisely th e zero m ode responsible for th e generalized tra n sla tio n which provides th e m otion on th e m oduli space.

T he an n ihilatio n process is p articu larly tra n s p a re n t on th e level of th e scatterin g of th e p o ten tials in th e sp ectral problem . In fact, th e full expression of th e spectral p o ten tial reads

. . j 2 , 2 ( /c o s h j x — a \ 2\ j coshj x — a

V (x) = ^ - ta n h 2 x —1 + 3 ---=--- --- ---=---. (6.13) 2 \ \ coshj x + a / J cosh2 x coshj x + a

At one b o u n d ary of th e m oduli space, w hen a ^ to (X ^ to), th e p o ten tial tend s to th ree sep arate p o tentials which are joined sm oothly, consisting of th e a t th e origin and two 0 4 ty p e (an ti)kin k p o ten tial wells approaching th e sp atial infinities, see figure 8 a) where a = 108 is p lo tted . (O f course, th ey q u a n tita tiv e ly reproduce th e 0 4 effective poten tials only for j = 2.) T his regim e corresponds to th e w ell-separated kink-antikink pair. As X decreases, th e soliton p o ten tial wells approach th e central region and merge, form ing th e V —mp for a = 0 (or X = 0), figure 8 e). It is clear th a t th e zero m ode, initially confined to th e incom ing solitons, gets tra p p e d on th e lum p solution 0 = —1. For negative a (or X ) th e effective p o ten tial looks like a well located a t x = 0, which becomes deeper and narrow er as we approach th e second bo u n d ary of th e m oduli space a t a ^ —1 (X ^ —to).

Obviously, th e observed changes in th e form of th e p o ten tial in th e spectral problem are reflected in changes of th e m ode stru c tu re while m oving on th e m oduli space. In th e case of th e a j=1 im p u rity for X = to, th e m ode s tru c tu re of 0 4 theory is reproduced, up to a m ultiplicative factor, as visible in figure 9 left panel. T his is because th e 0 = 1 lum p solution does not have any b ound m odes. Hence we have two zero m odes (tran slatio n s of each soliton) and two discrete m odes. As X decreases, th e zero m ode splits into a zero m ode and a discrete m ode whose frequency rises until it hits th e continuous spectrum . T he asy m ptotic bound m odes also split and en ter th e continuous spectrum . T hen, for negative X , th e p o ten tial well hosts only th e zero m ode. In fact, it is a general feature of th e spectral problem th a t as X approaches —to all bound m odes get expelled into th e continuous spectrum . T he p o ten tial well becomes sim ply to o narrow . Thus, finally, for sufficiently small (negative) X th ere is only th e zero m ode. T he sam e p a tte rn occurs for th e a j= 2 im purity, see figure 9, right panel.

B o th im purities m ay be used for a fu rth e r investigation of th e kink-antikink an nih ila­

tio n process in th e 0 4 m odel and th e possible creatio n of an oscillon. Indeed, this m odel gives us th e rare o p p o rtu n ity to stu d y th e role of in tern al m odes in th e annih ilation process in a set-up w here th ere is no sta tic force betw een kink and antikink. In th e next step, such a sta tic force m ay be switched on by th e ad d itio n of an a rb itra rily small new p o ten tial or im p u rity term , which breaks th e self-duality explicitly. As a result, we could, in a controlled m anner, analyze th e m u tual im pact of th e intersoliton sta tic force and th e excited m odes s tru c tu re on th e scatterin g of solitons in th e topologically triv ial sector. Such a self-duality breaking term will also prevent th e effective p o ten tial from reaching an a rb itra ry depth.

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-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15

x x

Figure 8. The effective potential for the annihilating im purity j = 1 for a = 108,104,1 0 2, 1,0, - 0 .5

Figure 9. Dependence of the spectral structure on the position on the moduli space X for j = 1 (left) and j = 2 (right) annihilating impurities.

Let us underline th a t th e p a rticu la r form of th e im p u rity assum ed here was chosen to provide a solvable spectral problem (for th e lum p solution). T he observed topologically triv ia l m oduli space can be found for m any o th er exam ples. T h e only condition is th a t th e im p urity leads to a shift function such th a t x + A a (x) approaches plus (or m inus) infinity for b o th x ^ —to and x ^ to . We can conjecture th a t w henever an im purity hosts a zero m ode, th e Bogom olny equation su p p o rts topologically trivial solutions only.

As a consequence, th ere is a m oduli space of charge Q = 0 B P S solutions describing an an n ih ila tio n /c re a tio n process of solitons.

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7 S u m m a ry

In th is article, we have presented a large fam ily of solvable (half) self-dual soliton-im purity m odels in (1+1) dim ensions. Like all half self-dual soliton-im purity models, these m odels possess a B P S (self-dual) sector to which half of th e s ta tic solitons belong. Hence, these solitons s a tu ra te th e p e rtin en t topological energy bound and obey a Bogom olny ty p e equa­

tion. In com parison w ith o th er field theories in (1+1) dim ensions, one exceptional p ro perty of these m odels is th a t different solutions of th e Bogom olny equ ation v ib ra te differently.

In o th er words, th e sp ectral stru c tu re depends on th e position on th e m oduli space.

T he first new finding is th a t this p a rticu la r fam ily of theories is com pletely solvable in th e self-dual sector for any sp atial d istrib u tio n of th e im purity. T his leads to exact expressions b o th for topological B P S solitons and lum ps, i.e., topologically triv ial solu­

tions. Furth erm ore, an exact expression for th e finite (i.e., non-infinitesim al) generalised tra n sla tio n s was also found. T his generalised tra n sla tio n is a sym m etry of th e Bogom olny equ atio n (not th e full action) transfo rm ing one self-dual solution into another, energeti­

cally equivalent solution. Hence, it generates th e whole m oduli space of B P S states. As th e generalized tra n sla tio n is known in an exact form, we have an analytical description of th e m oduli space to g eth er w ith its m etric. T his allows for an exact u n d erstan d in g of th e geodesic flow, i.e., th e low energy in teraction (scattering) of th e incom ing B P S soliton on th e im purity. In addition, we have established a ra th e r involved relation betw een th e posi­

tio n on th e m oduli space and th e form of th e effective p o ten tial in th e small p e rtu rb a tio n problem , which defines th e sp ectral stru c tu re (vibration al m odes) of th e solution.

T he second m ain result is th a t th e self-dual im p urity m odels m ay lead to a B P S sector containing topologically triv ial configurations. This generalizes previous findings, where th e B P S sector was po p u lated by kinks (or antikinks), th a t is, by solutions w ith Q = 1 (or Q = —1) topological charge. T his generalization is possible because th e solvable self-dual im p urity m odels allow for a spatially non-localized im purity. In o th er words, it is not necessary to have L² integrable im purities. In fact, o can be a co n stan t or even divergent function a t ± infinity. Specifically, we discovered im purities for which th e m oduli space describes th e annihilation of an initially infinitely separated kink-antikink pair. This kink- an tikin k p air forms a B P S solution for all values of th e m oduli space p aram eter, therefore th ere is no sta tic force betw een th e co n stitu en t solitons. P erh ap s th e m ost striking finding is th a t th e full m oduli space is a sm ooth union of solutions bu ilt o ut of th e regular and singular solutions of th e Bogom olny equation of th e no-im purity m odel. T hey join at a com pletely regular point in th e m oduli space.

It is w orth noting th a t th e m oduli space coordinate, in general, can no t be associated w ith th e position of th e B P S soliton. Especially for th e cases w ith a topologically trivial B P S sector, such an identification is not valid all th e tim e.

Looking from a w ider perspective, th e self-dual im pu rity m odels m ay serve as a very useful tool which allows to decouple th e sta tic forces betw een solitons from th e interactions betw een a soliton and in tern al m odes. For a given process in a given scalar field th eo ry in (¹+¹) dim ensions, we can sw itch off th e s ta tic inter-soliton forces by coupling th e m odel w ith a p e rtin en t im pu rity in th e self-dual m anner. Then, we m ay stu d y th e im pact of

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