a Xi :1305.4251 1 [hep- h] 18 May 2013
epl d a
De o ma ion me hod o gene alized Abelian Higgs-Che n-Simons
models
L. Losano1,2,3, J. M. C. Malbouisson1,4, D. Rubie a-Ga cia5and C. dos San os1
1Cen o de F´ısica e Depa amen o de F´ısica e As onomia, Faculdade de Ciˆencias da Uni e sidade do Po o, 4169-
007 Po o, Po ugal
2Depa amen o de F´ısica, Uni e sidade Fede al da Pa a´ıba, 58051-900 Jo˜ao Pessoa, PB, B azil
3Depa amen o de F´ısica, Uni e sidade Fede al de Campina G ande, 58109-970 Campina G ande, PB, B azil
4Ins i u o de F´ısica, Uni e sidade Fede al da Bahia, 40210-340 Sal ado , BA, B azil
5Depa amen o de F´ısica, Uni e sidad de O iedo, 33007 O iedo, As u ias, Spain
PACS 11.27.+d – Ex ended classical solu ions
PACS 11.15.Yc – Che n-Simons gauge heo y
Abs ac – We p esen an ex ension o he de o ma ion me hod applied o sel -dual solu ions o
gene alized Abelian Higgs-Che n-Simons models. S a ing om a model de ined by a po en ial
V(|φ|) and a non-canonical kine ic e m ω(|φ|)|Dµφ|2whose analy ical domain wall solu ions a e
known, we show ha his me hod allows o ob ain an uncoun able numbe o new analy ical
solu ions o new models de ined by o he unc ions
e
Vand eω. We p esen some examples o
de o ma ion unc ions leading o new amilies o models and hei associa ed analy ic solu ions.
In oduc ion. – Topological de ec s play impo -
an ole in se e al impo an a eas such as high en-
e gy physics [1], cosmology [2] and condensed ma e
physics [3]. Such de ec s eme ge as classical solu ions o
nonlinea ield heo ies which possess degene a ed acua.
Typical examples a e domain walls desc ibed by kink so-
lu ions o he φ4model, Ginzbu g-Landau o ices and
monopoles.
Usually, domain walls a e solu ions connec ing wo dis-
inc acua o scala ield heo ies in one-space dimension,
o in hei inse ions in highe dimensions, while o ices
eme ge as solu ions o models ha couple cha ged ma e
ields wi h gauge ields li ing in a (a leas ) 3-dimensional
space- ime, and monopoles lie in a 4-D space- ime.
In a (2+1)-dimensional space- ime, minimal coupling
be ween cha ged-ma e and gauge ields can be imple-
men ed by he Che n-Simons (CS) ac ion. Al hough he
CS ield can no be concei ed as a ee ield, i s coupling
wi h ma e ields imposes cons ain s in he dynamics
which ha e e y ele an consequences, in bo h, classi-
cal and quan um heo ies, wi h ei he ela i is ic o non-
ela i is ic kine ics. In he non- ela i is ic (NR) ame-
wo k, pa icles coupled h ough he CS ield ca y bo h
elec ic cha ge and magne ic lux, and possess ac ional
s a is ics [4]. Addi ionally, he NR scala CS model con-
s i u es a seminal example o a Galilean-in a ian gauge-
ield heo y [5]. Also, o a c i ical s eng h o a qua ic
sel -in e ac ion o he scala ield, which es o es he scale
in a iance [6], his model p o ides a ield- heo e ical de-
sc ip ion o he Aha ono -Bohm (AB) sca e ing [7]; con-
side ing he Lo en z co a ian ield heo y, ela i is ic co -
ec ions o he AB sca e ing a e ob ained [8].
Sel -dual soli on solu ions ha e been ound in he el-
a i is ic, U(1)-in a ian , Abelian, Higgs-Che n-Simons
(HCS) gauge- heo y whe e he symme y-b eaking po en-
ial o he Higgs ield is U(ϕ)∼ |ϕ|2(|ϕ|2− 2)2[9]; o -
ex and domain-wall solu ions ha e been ob ained o his
model [10]. This model was gene alized by conside ing
a non-canonical kine ic e m o he complex scala ield,
W(|ϕ|)|Dµϕ|2, p o iding sel -dual o ex [11] and domain-
wall [12] solu ions. Models wi h noncanonical kine ic
e ms (k- ields) ind also applica ions in s ong-in e ac ion
physics [13] and in cosmology [14].
Due o he nonlinea i y, he e is no gene al in eg a-
ion me hod o sol e analy ically he equa ions o mo-
ion o non-linea ield heo ies; only o a small se o
models, solu ions o he equa ions o mo ion can be di-
ec ly de e mined. Howe e , o scala ields in (1+1)-
dimensions, s a ing om a nonlinea model wi h known
solu ions, in ini ely many new models and hei co e-
p-1
L. Losano e al.
sponding s a ic solu ions can be ound using he de o ma-
ion me hod [15]. This me hod wo ks as ollows. Choos-
ing a de o ma ion unc ion (φ), he model de ined by
he de o med po en ial e
V(φ) = V[( (φ))]/[ ′(φ)]2, whe e
′means he de i a i e o , possesses s a ic solu ions
gi en by e
φ(x) = −1(φ(x)), whe e φ(x) is a solu ion o he
s a ic equa ion o mo ion o he o iginal model wi h po-
en ial V(φ). This p ocedu e has been applied o gene a e
de ec solu ions o many models ha ing polynomial in e -
ac ions [16] and new amilies o sine-Go don and mul i-
sine-Go don models [17]. Also, an o bi -based ex ension
o his me hod has been applied o models in ol ing wo
in e ac ing scala ields [18].
The pu pose o his Le e is o ex end he de o ma ion
me hod o gauge- ield models conside ing speci ically he
Abelian HCS heo y, ocusing pa icula ly on he Jackiw-
Lee-Weinbe g (JLK) domain-wall solu ion [10]. In Sec-
ion II, we p esen he gene alized Abelian HCS models
and w i e down he i s -o de equa ions obeyed by he
Bogomol’nyi-P asad-Somme eld (BPS) [19] domain-wall
solu ions. In Sec ion III, he de o ma ion me hod is ex-
ended o domain-wall solu ions o gene alized Abelian
HCS models and some examples a e gi en, illus a ing
he powe o he p ocedu e in gene a ing new models wi h
hei s a ic solu ions. Finally, some ema ks a e made.
BPS domain walls in he gene alized Abelian
HCS model. – We conside he gene alized (2 + 1)-
dimensional Abelian HCS model de ined by he La-
g angian densi y [11]
LS=W(|ϕ|)|Dµϕ|2−U(|ϕ|) + κ
4ǫαβγAαFβγ,(1)
whe e ϕis he complex Higgs ield, Dµ=∂µ+ieAµis
he co a ian de i a i e and Fµν =∂µAν−∂νAµis he
ield s eng h enso o he gauge po en ial Aµ. The sel -
in e ac ion po en ial, U(|ϕ|), is assumed o implemen a
symme y-b eaking mechanism and he non-canonici y o
he kine ic e m is engende ed by he unc ion W(|ϕ|); ak-
ing W ≡ 1, one eco e s he s anda d Abelian HCS model.
No e ha , in he CS e m, ǫαβγ is he ully an isymme -
ic enso and he elec ic and he magne ic CS ields a e
Ei=Fi0=−˙
Ai−∇iA0and B=~
∇× ~
A=∂2A1−∂1A2,
espec i ely.
I is con enien o wo k wi h dimensionless quan i ies.
In (2 + 1) dimensions, he scala ield ϕhas mass dimen-
sion equal o 1/2, he same one we ake o he gauge
ield; his choice ensu es ha he mass dimension o Aα
ag ees wi h he one ob ained i a Maxwell e m we e added
o LS. I ollows ha he elec ic cha ge eand he CS
pa ame e κhas mass dimensions equal o 1/2 and 1,
espec i ely, so ha e2/κ is dimensionless. We can ge
an addi ional simpli ica ion i we abso b he pa ame e s
eand κby ede ining space- ime coo dina es and ields.
Thus, wi h Mbeing a mass scale o he model, we de-
ine ¯xµ=Me2xµ/κ,φ=√κϕ/√Me,Aµ=κAµ/Me,
V=κ2U/M3e4and ω=e2W/κ; he dimensionless La-
g angian densi y is hen gi en by L=κ2LS/M3e4and
he ac ion becomes S=κ
e2Rd3¯xL. Fo a simple no a-
ion, we supp ess he ba o e he space- ime coo dina es
and use, om now on, only dimensionless quan i ies.
Va ia ion o he ac ion leads o he equa ions o mo ion
ωDµDµφ+∂µωDµφ−|Dµφ|2∂ω
∂φ∗+∂V
∂φ∗= 0 ,(2)
1
2ǫαβγFβγ =−Jα,(3)
whe e he cu en densi y, Jα= (ρ,~
j), is gi en by
Jα=iω [φ(Dαφ)∗−φ∗Dαφ].(4)
The ime componen o eq. (3) s a es ha he magne ic
ield is equal o he plana elec ic-cha ge densi y, B=ρ,
which is he CS Gauss law. Also, o s a ic ield con igu-
a ions, we ind
B=ρ= 2A0|φ|2ω(|φ|), Ea=ǫabjb,(5)
which shows ha he elec ic-cu en densi y is pe pen-
dicula o he elec ic ield.
The ene gy-momen um enso is gi en by
Tµν =ω[Dµφ(Dνφ)∗+Dνφ(Dµφ)∗]
−gµν ω|Dαφ|2−V(|φ|)(6)
om which we ob ain he ene gy densi y, ε=T00, and he
p essu e componen s, P1=T11 and P2=T22.
We a e in e es ed in s a ic domain-wall solu ions.
Fi s ly, no e ha he complex phase o he scala ield
φcan be supp essed by a sui able gauge ans o ma ion.
Then, ixing he Coulomb gauge, we can sea ch o solu-
ions o he o m [10,12]
φ=h(x), Aµ=A0(x), A1= 0, A2=A(x),(7)
whe e h(x) and A(x) a e eal unc ions and xdeno es he
x1-coo dina e. This ansa z co esponds o domain-walls
(ac ually lines in he plane) pa allel o he x2-axis.
In his case, he s a ic equa ions o mo ion educes o
[2ωh′]′= 2hω A2−A2
0+dV
dh ,(8)
A′
0=−2ωh2A , (9)
and he Gauss law
A′=−2ωh2A0,(10)
whe e he p ime deno es de i a ion wi h espec o x.
F om eqs. (9) and (10) we in e ha A0A′
0=AA′, so
ha ime and space componen s o he gauge iled a e
cons ained by
A2
0=A2−C , (11)
whe e Cis a eal cons an . Also, consis ency wi h eq. (8)
imposes a ela ion be ween he unc ion ω(h) and he po-
en ial V(h) exp essed as
d
dh"pV/ω
h#=−2ωh . (12)
p-2
De o ma ion me hod o Abelian HCS models
Now, he s abili y condi ion P1=P2= 0 leads o he
i s -o de equa ions [20]
h′=±hA (13)
A′=−2ωh2A0(14)
wi h
V=h2ωA2
0.(15)
Fo h≥0 and A≥0, he signal + (−) in eq. (13) co e-
sponds o he kink (an i-kink) like solu ion o he Higgs
ield, h(+) (h(−)). No e ha , he i s -o de equa ions (13)
and (14) sol e he equa ions o mo ions (2) and (3).
The s a ic solu ions a e physically cha ac e ized by hei
cha ge and ene gy. Now, e u ning o eq. (6), o non-
nega i e V(h) and ω(h), he ene gy o s a ic solu ions can
be ew i en in he o m
E=Z∞
−∞
dx T00
=Z∞
−∞
dx (V+ωh′+ 2ωh2A2
0+Cωh2)
=Z∞
−∞
dx h√V±√ωhA2
02+√ωh′±√ωhA2
+p−A0A′±√2ωhA02i
+Z∞
−∞
dx 2√ωV hA0±2ωhh′A
±2p−2ωA0A′hA0+A0A′−2ωh2A2
0,(16)
which is minimized i eqs. (13), (14), and (15) a e obeyed,
esul ing in
E=Z∞
−∞
dx (4V) = A2(−∞)−A2(+∞),(17)
o C= 0. In his case A2
0=A2, so he sys em o i s -
o de equa ions decouples and is sol ed simply by (13)
wi h
A(h) = −2Zωh dh +c , (18)
whe e cis an in eg a ion cons an sui able o he bounda y
condi ions equi ed o he gauge ield. And, om (5) and
(6), he elec ic cha ge, Q, and Noe he cha ge, P, a e
gi en by
Q=Z∞
−∞
dxρ(x) = A(−∞)−A(+∞),(19)
P=Z∞
−∞
dx T02 =1
2A2(−∞)−A2(+∞),(20)
which a e bo h conse ed due o he U(1) symme y and
he ansla ional in a iance along x2-di ec ion, espec-
i ely.
This shows ha , o hin a ange such ha ω(h)≥0
and V(h)≥0, he BPS solu ions o he i s -o de eqs. (13)
and (14), wi h (15), indeed co espond o solu ions o min-
imum ene gy and hei ene gy and cha ge can be calcu-
la ed knowing only he asymp o ic beha io o he gauge
ield. Co espondingly, he Higgs ield, o bo h kink and
an i-kink solu ions, connec s wo consecu i e acua o he
po en ial, while a lump-like solu ion s a s and e mina es
on he same acuum when x→ ±∞.
S anda d sel -dual domain walls. The simples
Abelian HCS model ha suppo s sel -dual domain wall
solu ions is he JLW model [10], which is de ined by he
Lag angian 1 wi h canonical kine ic e m (ω= 1) and he
(dimensionless) po en ial
V(h) = h2(1 −h2)2,(21)
plo ed in ig. 1. In his case, he use o eq. (18) (wi h
Fig. 1: The po en ial (21) as unc ion o he Higgs ield.
c= 1) p o ides he esul
A= 1 −h2,(22)
which, subs i u ing in (13), gi es he solu ions
h(+)(x) = 1/p1 + e−2x, A(−)(x) = 1/(1 + e2x),(23)
and
h(−)(x) = 1/p1 + e2x, A(+)(x) = 1/(1 + e−2x),(24)
which a e displayed in ig. 2. We see ha he scala
ield, in bo h cases, in e pola es be ween he symme ic
and he asymme ic acua. ig. 3shows he ene gy and
Fig. 2: The Higgs ield (solid line) and he gauge ield
(dashed line), (h(+)(x), A(−)(x)) om eq. (23) on he le , and
(h(−)(x), A(+)(x)) om eq. (24) on he igh .
elec ic-cha ge densi ies o bo h wall solu ions. We ind
ha he spa ial dis ibu ion o he elec ic cha ge is sym-
me ic a ound he o igin, while o he ene gy he axis
o symme y a e displaced om he o igin. And, om
eqs. (17), (19) and (20), o he solu ions (h(+), A(−)) and
(h(−), A(+)), we ha e he cha ges Q= 1, P= 1/2, and
Q=−1, P=−1/2, espec i ely, and he same ene gy,
E= 1.
p-3
L. Losano e al.
Fig. 3: Ene gy densi y o he solu ions (h(+)(x), A(−)(x))
(dashed line) and (h(−)(x), A(+)(x)) (dashed-do ed line), and
module o elec ic-cha ge densi y o bo h solu ions (solid line).
The de o ma ion me hod. – Le us now de elop
he de o ma ion me hod o gene alized Abelian HCS
models ollowing he spi i o he p ocedu e in oduced
o scala ields [15]. As we shall show, by de o ming si-
mul aneously he Higgs and he CS ields, we a e able o
cons uc many new gene alized HCS models and hei
s a ic domain-wall solu ions. The o iginal and he de-
o med models a e mapped in o each o he h ough he
de o ma ion unc ion.
Deno e by ˜
φ(x) and e
A(x) new ields whose dynamics is
go e ned by he (dimensionless) Lag angian densi y
e
L=eω(|˜
φ|)|Dµ˜
φ|2−e
V(|˜
φ|) + 1
4ǫαβγ e
Aαe
Fβγ ,(25)
whe e e
V(|˜
φ|) and eω(|˜
φ|) a e new unc ions speci ying his
model. As in sec. II, we assume ha he sel -dual BPS
domain-wall solu ions o his model ake he o m
˜
φ=˜
h(x),e
Aµ=e
A0(x),e
A1= 0,e
A2=e
A(x),(26)
and sa is y he i s -o de equa ions o mo ion
˜
h′=±˜
he
A , (27)
e
A′=−2eω˜
h2e
A0,(28)
whe e ˜
h′≡d˜
h/dx and e
A′≡de
A/dx, wi h he cons ain s
e
V=h2eωe
A2
0and e
A2
0=e
A2.
Now, in oduce he de o ma ion unc ion such ha
he Higgs ields o he wo models a e mapped in o each
o he , h= (˜
h), which is assumed o be in e ible (in a
p esc ibed domain o de ini ion) and di e en iable. Also,
conside ha he de o med CS-gauge ield is ob ained om
Aby he p esc ip ion
e
A(˜
h) = (˜
h)A[h→ (˜
h)]
˜
h e
h
,(29)
whe e e
h=d /deh. Then, i ollows om eqs. (27) and
(28), using eq. (29), ha he model de ined by Lag angian
densi y (25), wi h he de o med unc ion eωand he de-
o med po en ial e
Vgi en by
eω(˜
h) = 1
2e
A˜
h
˜
h,e
V(˜
h) = ˜
h2e
A2eω(˜
h),(30)
whe e e
A˜
h=de
A/d˜
h, possesses s a ic BPS solu ions
˜
h(x) = −1[h(x)] ,e
A(x) = e
A −1[h(x)],(31)
whe e h(x) is a s a ic solu ion o he o iginal model (1).
I should be no ed ha all he conside a ions and e-
la ions p esen ed in sec. II, ela i e o ene gy and con-
se ed cha ges, a e held unchanged o he de o med sys-
em. In he ollowing, aking as he s a ing poin he JLW
domain-wall solu ions desc ibed in sec. II.A, we conside
some illus a i e examples o he me hod.
Example I. Fi s ly, we conside he couple o de o -
ma ion unc ion
(˜
h)(±)= (±)1−˜
h2
1 + ˜
h2,(32)
which, using eqs. (22) and (29), gi es e
A(±)(˜
h) = (˜
h)(±);
and, om eq. (30), i ollows ha
˜ω=2
(1 + ˜
h2)2,e
V=2˜
h2(1 −˜
h2)2
(1 + ˜
h2)4.(33)
These unc ions, which a e plo ed in ig. 4, de ine he gen-
e alized Abelian HCS model employed in Re . [12]. No e
ha , he h ee acua a ˜
h= 0,1,+∞es ablish wo walls,
one be ween ˜
h= 0 and ˜
h= 1, and o he be ween ˜
h= 1
and ˜
h= +∞. F om he in e se o he de o ma ion unc-
Fig. 4: The po en ial (33) ( op panel) and he co esponding
unc ion w(bo om panel), as unc ion o he Higgs ield.
Fig. 5: The Higgs ield (solid line) and he gauge ield
(dashed line), (˜
h(+)(x),
e
A(−)(x)) om eq. (34) on he le , and
(˜
h(−)(x),
e
A(+)(x)) om eq. (35) on he igh , o 0 ≤˜
h≤1
ion (32) and eqs. (23) and (24), o he ange 0 ≤˜
h≤1,
we ob ain he solu ions
˜
h(+)(x) = p1 + e−2x−e−x,e
A(−)(x) = 1/p1 + e2x,
(34)
˜
h(−)(x) = 1/p1 + 2e2x,e
A(+)(x) = 1/(1 + e−2x),(35)
p-4
De o ma ion me hod o Abelian HCS models
while o ˜
h≥1 we ha e
˜
h(+)(x) = p1 + 2e2x,e
A(+)(x) = 1/(1 + e−2x),(36)
˜
h(−)(x) = p1 + e−2x+e−x,e
A(−)(x) = 1/(p1 + e2x).
(37)
In igs. 5 and 6, we display hese domain wall solu ions.
The walls o 0 ≤˜
h≤1 and ˜
h≥1 ha e he same gauge
ields, bu wi h he asymp o ic alue o x=±∞ changed.
Then, o bo h anges he walls ha e he same ene gy,
E= 1, and cha ges Q= 1 and P= 1/2, o e
A(−), and
Q=−1 and P=−1/2, o e
A(+). This makes possible o
ha e a ac i e o epulsi e o ce be ween he wo walls.
In ig. 7, we display he ene gy and cha ge densi ies
Fig. 6: The ields (˜
h(+)(x),
e
A(+)(x)), eq. (36) (solid line), and
(˜
h(−)(x),
e
A(−)(x)), eq. (37) (dashed line), o ˜
h≥1.
Fig. 7: Module o elec ic cha ge o solu ions A(−)(x) (dashed
line) and A(+)(x) (dashed-do ed line), and ene gy densi y
(solid line) o bo h walls.
o he wo walls. The compa ison wi h he walls o he
JLW model shows ha , no wi hs anding he walls ha e
he same cha ges and ene gy, he JLW walls ha e sym-
me ic spa ial dis ibu ions o ene gy and cha ge, while
he e only he dis ibu ion o ene gy is symme ic and all
he co esponding dis ibu ions a e mo e sp ead ou . The
model de ined by eqs. (33), which was ob ained by de o m-
ing he JLW model, was s udied in Re . [12] bu only he
solu ion sa is ying 0 ≤˜
h≤1 was conside ed.
The de o ma ion unc ion (32) is a pa icula case o
he de o ma ion unc ion (˜
h) = cos[αa c an(˜
h)], co e-
sponding o α= 2; om ha new amily o models can
be gene a ed o αin ege .
Example II. As a second example, conside he se o
de o ma ion unc ions [16]
α(˜
h)] = cos[αa ccos(˜
h)] = Tα(˜
h),(38)
whe e he in ege α > 2 and Tαis he Chebyshew poly-
nomials o i s kind. Using his de o ma ion in eq. (29),
wi h eq. (22), we ha e he gauge ield
e
Aα(˜
h) = (1 −˜
h2)1/2sin[2αa ccos(˜
h)] /2α˜
h ,
= (1 −˜
h2)U2α−1(˜
h)/2α˜
h , (39)
whe e Uis he Chebyshew polynomials o second kind;
which explici esul s, o α= 2,3, a e
e
A2(˜
h) = (1 −˜
h2)(2˜
h2−1) ,(40)
e
A3(˜
h) = 1
3(1 −˜
h2)(1 −2˜
h2)(3 −4˜
h2).(41)
In his case, om eqs. (30) and (38), we ha e a amily o
models de ined by he unc ion eωα(˜
h) and he po en ial
e
Vα(˜
h) w i en in polynomial o m as
eωα(˜
h) = |[2α˜
h T2α(˜
h) + U2α−1(˜
h)]/4α˜
h3|,(42)
e
Vα(˜
h) = (1 −˜
h2)2U2
2α−1(˜
h)eωα(˜
h)/4α2.(43)
Then, each alue o he pa ame e αspeci ies a model o
his amily. The explici esul s o α= 2,3 a e
eω2(˜
h) = 3 −4˜
h2,(44)
e
V2(˜
h) = ˜
h2(1 −˜
h2)2(1 −2˜
h2)2eω2(˜
h),(45)
eω3(˜
h) = 1
3(19 −64˜
h2+ 48˜
h4),(46)
e
V3(˜
h) = 1
9˜
h2(1 −˜
h2)2(1 −2˜
h2)2(3 −4˜
h2)2eω3(˜
h).
(47)
Fo hese models, om he in e se o de o ma ion unc ion
Fig. 8: The po en ial (45) ( op panel) and he unc ion w(44)
(bo om panel), as unc ion o ˜
h.
(38), we ob ain he s a ic Higgs ield solu ions in he o m
˜
h(±)(x) = cos h(a ccos(h(±)(x)) + (m−1)π)/αi,(48)
whe e h(±)(x) is gi en by eqs. (23) and (24), and mis an
in ege , which gene a es dis inc solu ions only o m=
0, ..., α −1 .
Fi s ly, we examine he model o α= 2, de ined by
eqs. (44) and (45) displayed in ig. 8. We see ha , he
p-5

L. Losano e al.
po en ial is posi i e only o ˜
h≤p3/2. Then, he e
a e wo kind o s a ic solu ions o he Higgs ield, one
pai kink/an i-kink like solu ion be ween 0 ≤˜
h≤1/√2,
and a lump-like solu ion be ween 1/√2≤˜
h≤p3/2. In
Re . [21] is conside ed a model ha p esen s a cha ged
lump-like solu ion. He e, he lump-like solu ion p esen s
anishing cha ges and ene gy, hence we examine only he
wall o 0 ≤˜
h≤1/√2. In ig. 9, we display he Higgs ield
(48) and he gauge ield (39) solu ions. These walls ha e
he same o al ene gy and cha ges o he walls o he s an-
da d JLW model, bu wi h di e en spacial dis ibu ion o
he ene gy and cha ge densi ies, as shown in ig. 10.
Fig. 9: The Higgs ield (48) o m= 0 (solid line) and
he ma ching gauge ield (40) (dashed line), o ˜
h(+)(x) and
e
A(−)(x), on he le , and o ˜
h(−)(x) and
e
A(+)(x), on he igh .
Fig. 10: The ene gy densi y (on he le ) and he module o
cha ge densi y (on he igh ), o solu ion A(−)(x) (solid line),
and o A(+)(x) (dashed-line).
Ending Commen s. – The examples p esen ed
abo e illus a e how he de o ma ion me hod may be used
o gene a e many new gene alized Abelian HCS models
and hei de ec solu ions. This is achie ed wi hou e-
qui ing o di ec ly sol e he nonlinea equa ions o mo ion
o he new models. The me hod also allows he cons uc-
ion o new de ec solu ions con olling impo an ea u es
such as hei heigh , wid h o he opological cha ac e .
Such esul s a e o di ec in e es o applica ions o do-
main walls in se e al con ex s, such as high-ene gy o
condensed-ma e physics.
∗∗∗
Two o us (LL and JMCM) hank CAPES and CNPq
(B azilian agencies) o inancial suppo .
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