Composite objects, such as non- Abelian gauge fields localized on domain walls [11] and monopolesconfinedbyvortices [12]amongothers,ariseingauge theorieswithspontaneous symmetry breaking.I
Trang 1Contents lists available atScienceDirect
www.elsevier.com/locate/physletb
Department of Physics, University of Athens, GR-15784 Athens, Greece
Article history:
Received 3 April 2015
Accepted 26 June 2015
Available online 2 July 2015
Editor: A Ringwald
WepresenttwodifferentfamiliesofsolutionsoftheU(1)-Higgsmodelina(1+1)dimensionalsetting leadingtoalocalizationofthegaugefield.Firstweconsiderauniformbackground(theusualvacuum), whichcorrespondstothefullyhiggsed-superconductingphase.Thenwestudythecaseofanon-uniform background intheform ofadomainwall whichcould berelevantly closetothe criticalpoint ofthe associated spontaneoussymmetrybreaking.Forbothcasesweobtainapproximateanalyticalnodeless and nodalsolutionsforthegaugefieldresultingasboundstatesofaneffectivePöschl–Tellerpotential createdby the scalarfield.The twoscenaria differonlyin thescale of thecharacteristic localization length.Numericalsimulationsconfirmthevalidityoftheobtainedanalyticalsolutions.Additionallywe demonstratehowakinkmaybeused asamediatordriving thedynamicsfromthe criticalpointand beyond
©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense
(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3
1 Introduction
Solitons come in two “flavors” namely non-topological and
topologicalones Theirphysical meaning aswell as their
mathe-maticalpropertieshavebeenvastlystudiedintheliterature,both
in the context of field theories and cosmology but also in
con-densed matter physics Non-topological solitons are found as
lo-calized“lumps” [1], Q-balls [2]or oscillons [3]while topological
solitons may have the form of instantons [4], monopoles [5–7],
vortices [8,9] or domain walls [10] Composite objects, such as
(non-) Abelian gauge fields localized on domain walls [11] and
monopolesconfinedbyvortices [12]amongothers,ariseingauge
theorieswithspontaneous symmetry breaking.In manycases an
explicitanalytical solitonsolution of therespective theory is not
possible,andthepropertiesofsolitonsareobtainedbyperforming
numericalsimulations[13].Thelatterareusuallyaccompaniedby
someanalyticalapproximation[14]e.g.byconsideringthe
asymp-toticbehaviorofthefields
Thesimplesttopologicaldefectwithananalyticalexpressionis
adomainwall(aliaskink)in (1+1)dimensionsforasinglescalar
field,which isstudiedthoroughly inthesine-Gordon andthe φ4
model,andstillattractsinterest,seee.g.theveryrecentworksof
kink-kink interactions ofRefs [15,16] orof Ref [17] forkinksin
E-mail address:liakatsim@gmail.com (G.C Katsimiga).
a φ6 model.Domainwallsandtheir interactions are also consid-eredinsupersymmetrictheorieswheremorethanonescalarsare involved, andfamilies of such walls link various supersymmetric vacua[18,19].Kinksolutionsmayalsobeusedformodeling flux-ons [20] or describing phase-slipsin superconductors,where the phaseoftheorderparameterperiodicallydropsby2π inasingle point(seee.g.[21]orthemorerecentresultsof[22–24])
Animportantfeatureregardingtopologicaldefectsisthatthey may be used as a mechanism inducing localization Such exam-ples includethe localizationoffermions onakink [25,26],which constitutesa trapping mechanism forfermionic zero modes, and alsotheformationoflocalizedgaugebosonsonadomainwall[11]
withimplicationsintheprocessofdynamiccompactification.More recently the localizationof a spin-0 field [27] was induced by a kink-lumpsolutionoftwoscalarsleadingtoresonantbehavior rel-evanttogravityinwarpedspace–times[28]
Althoughlocalizedstructuresonanon-vanishingvacuumshare commonpropertieswith oscillonstrapped bytopological defects, thereisnolinkbetweenthesetwo differentsolutionsuptonow
In thepresentwork we will attempttoestablish such a connec-tion inthe framework of (1+1) dimensionalAbelian–Higgs [29, 30]model.Suchatheory,althoughsimple,candescribeboth non-topological (oscillon [31]) and topological (domain wall [10]) so-lutions As we will show below, both solutions (oscillons, kinks) generatean effectivePöschl–Teller[32,33]potential leadingtothe localizationofthe respectivegauge field.Additionally we provide
http://dx.doi.org/10.1016/j.physletb.2015.06.065
0370-2693/©2015 The Authors Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by 3
Trang 2awhole“family”ofgaugefield configurationsexhibitingnodesin
theirprofilesemergingasboundstatesoftheaforementioned
po-tential Thesenodal solutions are long livedandrobust andmay
beinterpretedasoscillonexcitations.Furthermoreweshowhowa
“moving”kinkmaydynamicallylocalizea gaugefield inthebulk
dependingonitsinitialenergy.Wearguethatthisprocesscanbe
interpretedintermsofthedynamicsnearthecriticalpointandwe
demonstratehowthetravelingkinkcandrivethepathwayfroma
globallysymmetric vacuumstate tothephase ofanon-vanishing
vacuumwithgloballyspontaneouslybrokensymmetry
The paper is organized as follows: inSection 2 we write the
equationsofmotionandtheirexactvacuumsolutions,
correspond-ing totheuniform[the scalarfield profileattains a globally
con-stantvacuumexpectationvalue(vev)]andthenon-uniform
back-grounds (thescalarfield isadomainwall) InSection 3weshow
how the oscillons on top of the vev lead to localized solutions
forthe respectivegauge field, usingan approximateperturbation
method.AstandardperturbationschemeisemployedinSection4,
andanalytical solutions ofa smallamplitude gauge field, around
the domain wall of the non-uniform background, are presented
These families of solutions are shown to have different
charac-teristic length scale than that of the solutions in the uniform
background.Inbothsectionsnumericalsimulations verifyour
an-alyticalresultsandimplytherobustness ofthesolutions.Amore
detailedcomparison/connectionbetweenthetwodifferentregimes
andthe respective solutions is presented inSection 5where we
also show numericalresults demonstrating that a localized
solu-tionaroundthedomainwallmaybeobtainedbyamovingsoliton
inthebulk.OurconclusionsarepresentedinSection6
2 Lagrangian and equations of motion
WeconsiderclassicalelectrodynamicsinflatMinkowskispace–
timedescribedbythegaugeinvariantLagrangian L:
4F μν F
μν+ (D μ)∗(D μ) −V(∗), (1)
where isachargedscalarfieldinteractingwiththegaugefield
Aμ and V( ||) is the double well potential V( ||) = λ||4+
μ2||2.ThecovariantderivativeisdefinedasDμ= ∂μ +ie Aμ, e is
thecouplingconstant and Fμν istheelectromagnetictensor.The
Hamiltonian(energydensity)oftheabovesystemisgivenby:
∂ (∂0) ∂0 + ∂ L
∂ (∂0∗) ∂0∗+ ∂ L
∂ (∂0A ν) ∂0A ν− L
=1
2
|B|2+ |E|2
+ | π |2+ |D |2+V, (2)
whereinthelast expressionwe haveexplicitlyusedthephysical
fieldsE= −∂t A− ∇A0 (electric), B= ∇ ×A (magnetic)and π ≡
D0.Althoughouranalysiswillbegivenwithrespectto andA,
theconnectionwiththeelectromagneticfieldisnecessaryforthe
interpretationofourresults
For a (1+1) dimensional setting, we consider the following
ansatzforthegaugefield Aμ: A0=A1=A3=0, A2=A(x, t),i.e
a linearlypolarized (in the z axis) magnetic field propagating in
thex direction.For μ2<0 thesymmetryisspontaneouslybroken
andthescalarfield acquiresanon-vanishing vacuumexpectation
value (vev).Choosing the unitary gauge in which the field is
real,its vev is = ± υ, where υ2= − μ2/λ.Given the previous
assumptions, the electromagnetic tensor hasnon-vanishing
com-ponents F0ν≡F02= ∂t A2, F i ν≡F12= ∂x A2 andthe equationsof
motionstemmingfromtheaboveLagrangianare:
The potential in terms of the fields and A is V(, A) =
(λ/4)
2− υ22
+ (e2/2)2A2,wherewe haveaddeda constant
√
λ υ2/2 in order to complete the square in the first term The symmetricphase,withrespecttoreflectionsymmetry,corresponds
toasingleminimum:
andbreathersolutions arenot supportedby thesystem, whilein the brokenphase thesystemofEqs.(3)–(4) admitsthefollowing exactsolutions:
= ±υtanh√
2λ υx/
whereEq.(6)isthezeroenergysolutionE min=0 correspondingto
a homogeneousscalarfield(uniform vacuum).Ontheother hand
Eq (7) is an inhomogeneous solution (non-uniform background) withafiniteenergyper unitarea E kink=2√
2 υ3/3.Thelatteris the well known kink solution of the φ4 model, which has been studied in a variety of physicalcontexts (see e.g Refs [4,10] for
a field theory approach, Ref [34] for kinks in condensed mat-terphysics andRef.[9]forkinksandother topologicaldefectsin cosmology) Since theenergy differenceof theabove solutions is analogousto υ3,their energies are comparablefor υ →0+ near thecriticalpointi.e.justaftersymmetrybreaking
Belowwewillsearchforlocalizedlowenergysolutionsforboth the scalar and the gauge field in the following two cases: (i) in theuniformvacuumcaseand(ii)inthenon-uniformbackground aroundthekink’score
In what follows, we express Eqs (3)–(4) in a dimensionless form by rescaling space–time coordinates andfields asx→eυx,
t→eυt, → υ φand A→ υA.Howeverfortheinterpretationof ourfindingswewillalwaysrefertothephysicalunits.After rescal-ingweobtainthefollowingsetofequations:
2φ +q2
2φ
3−q2
2φ + φA2=0, (8)
whereq≡ 2λ/e2 isthesingle parameterofthesystemandthe energydensitybecomes:
E=1
2(∂tφ)
2+1
2(∂xφ)
2+1
2(∂t A)
2+1
2(∂x A)
with
V=q2
8(φ
2−1)2+1
2φ
3 Solutions around the uniform vacuum
3.1 Analytical considerations – multiscale expansion
In this section we search for small amplitude localized solu-tionsinthebulkoftheclassicalvacuumEq.(6)andfarbeyondthe criticalpoint,i.e.when υ 1 andindimensionlessform φ = ±1 Althoughweshowresultsonlyfor φ =1 analogousresultsholdfor
φ = −1 duetothereflectionsymmetryofthepotential.This sce-nariocorresponds tothe fullyhiggsed–“superconducting” phase –[11]
Inorder tofindalocalizedsolution ofthenon-integrable sys-tem of Eqs (8)–(9), we will use a multiscale perturbation ex-pansion [35] While the details of thismethod are given in Ap-pendix A, herewe briefly commenton its basic ingredients The
Trang 3multiscale expansion introduces different space–time scales (fast
andslow)andacarrierwavesolutioninthefastscaleisobtained
inthelinearlimit.Thentheenvelopeofthiswave,whichis
con-sideredto evolve in the slowscales, is found to travel with the
groupvelocityoftheplanewaveandsatisfiesasolvablenonlinear
equationatsomehigherorder
Inourcasewewillusethefollowingasymptoticexpansion
φ =1+ φ(1)+ , A=0+ 2A (2)+ , (12)
where φ( i ) (i=1, 2, 3, )describetheperturbationsofthescalar
fieldontopofthevev, A ( i ) isthesmallamplitudegaugefieldand
1 isaformal smallparameter.Inthefirstorderofthe
expan-sion O( )for φandinthesecondorderforA, O( 2),thesolutions
correspondtothefollowingplanewaves:
φ(1)=u(x1,t1,x2,t2, )e i ( k1x−ω1t )+c.c, (13)
A (2)=v(x1,t1,x2,t2, )e i ( k2x−ω2t )+c.c, (14)
where“c.c” stands forthe complex conjugate.The wavenumbers
k1,2 and frequencies ω1,2 are connected through the dispersion
relations ω1= k2+q2, ω2= k2+1 The envelope functions
u and v are yet arbitraryin thisorder In the next orderof the
expansion(O( 2) forφand O( 3)for A),the compatibility
con-ditionsdictatethattheenvelopesmovewiththerespectivegroup
velocitiesv ( g1,2)=dω1,2/dk1,2.Inwhatfollowsandwithoutlossof
generalitywerestrictouranalysisinthecaseofzerogroup
veloc-ity(k1=k2=0) andthus theenvelopes arefunctionsof (x1, t2)
Howeverwenotethatforfinitegroupvelocitieswe obtainresults
thatarequantitativelythesamedescribingtravelingsolutions
Atthe orders O( 3) fortheenvelope u(x1, t2),and O( 4) for
theenvelopev(x1, t2)wefind:
iq∂t2u= −1
2∂
2
i∂t2v= −1
2∂
2
x1v+V(x1)v, V(x1) = − α |u|2, (16)
whereinEq.(16)α (q) =2(6−q2)/(4−q2) whileEq.(15)isthe
well knownfocusing (i.e withpositive relative signbetweenthe
dispersion andthe nonlinearity) NLS equation The latteradmits
brightsolitonsolutions[36]intheform:
whereu0 isa freeparameter characterizingtheamplitude ofthe
soliton, w= √3qu0 is its inversewidth This waywe have
con-structedalocalizedsolutionforthe φ(1)field[cf Eq.(13)]
For the above solutions of u(x1, t2), Eq (16) becomes a
lin-ear Schrödinger equation for the envelope v(x1, t2), in the
pres-enceoftheeffectivePöschl–Tellerpotential V(x1) ∼sech2(x1).The
strength α and in particular its sign, depend only on the
pa-rameterq. Inparticular for q∈ (0, 2) andq > √
6 the parameter
α is positive and V(x1) hasthe formof a sech-shaped well As
such,inthisparameterregime one canobtain localizedsolutions
forv(x1, t2) corresponding to a localized gauge field A. Bounded
solutions of Eq (16) can be found using the ansatz: v(x1, t2) =
ˆ
v(x1)exp[−i(E 2)t2] where E 2 is the energy eigenvalue
Substi-tutingtheaboveinEq.(16)weobtainaSturm–Liouvilleequation
ofthefollowingform:
∂x21vˆ (x1) + E+2αsech2(wx1)
ˆ
v(x1) =0. (18)
Equation (18) can be transformed into the associated Legendre
equation by making the substitution T =tanh(wx ) which can
Fig 1 (Coloronline.) Top panel: Total number of bound statesN ( q )for the uniform vacuum are shown with the solid line and for the non-uniform background with the dashed line Gray box indicates the region 2< q <√
6 where no bound states exist
in the uniform case Bottom panel: Profiles of a nodeless(n=0) state depicted with solid black line and the nodal ones, where the first excited state(n=1) is plotted with a dashed black line and the second excited(n=2) with the dotted black line For all casesq=1/2.
then be solved analytically In fact, for each value of the pa-rameter q thereexist a total numberof N boundsolutions with
E n (n=0, 1 , N−1) discreteenergy eigenvalues,both givenin termsofthefunctions:
f N(z) = (z+1 4)1/2−1 2
f E(z) = −1
z
(z+1 4)1/2− (n−1 2)
2
wherein Eq.(19),“[ ]” denotes theinteger part.Fromthe afore-mentionedsubstitutionsitfollowsthatinthecaseoftheuniform backgroundboth N and E n aregivenbytheexpressions:
w2
, E n= −2αf E 2α
w2
Furthermore the localized solutions of the envelope v(x1, t2) are givenbytheso-called associated Legendre functions[37]asfollows:
ˆ
vn(x1) =P σ(tanh(wx1)) , (22)
where σ and ρ arerelatedtotheenergyandpotentialcoefficients throughtherelations: σ2= −E n,and ρ2+ ρ =2α /w2.Inthetop panel of Fig 1, we show the total numberof bound states N as
a function ofq. In the region q <2, as q decreases the number
of bound statesincreases, for q > √
6 onlyone such state exists, whilefor2 <q < √
6 onlyscatteringstatesarefound(indicatedby thegraybox)[30]
Wecannowwritetheapproximatesolutionsforfields φandA
as:
φ (x,t) ≈1+ u0sech(w x)
e−i ( ω1− 2 w2/ 2q ) t+c.c
A(x,t) ≈ 2vˆn( x)
e−i ( ω2+ 2E n /2) t+c.c
and vˆn is given by Eq (22) We have thus shown that, in this regime ofa smallgauge field A,thelocalized perturbations of φ
(duetoselfinteractions)uponthevev,actasaneffectivepotential within which the gauge field can be localized More importantly
we notethefollowing: ourresultofEq.(24)inthe caseofn=0 corresponds toan “oscillon”solutionforthe gaugefield withthe usual sech-shaped(nodeless) envelope Such oscillons have been shown to exist in various settings [38,39] and their properties (stabilityandrobustness)havebeenextensivelystudied.However
Trang 4Fig 2 (Coloronline.) Top row: 3d plots showingt=2T oscillationperiods for A ( x , t )forn=0,q=1 (left),n=1,q=3/4 (middle) andn=2,q=1/2 (right) Bottom row: 3d plots depicting the fieldφ( x , t )for each of the aboveA’s.For the uniform vacuum.
for larger values of n=1, 2, 3, the solutions in Eq (24)
cor-respond to localized oscillating structures with a finite number
of nodes-nodal oscillons-which asfar aswe know have not been
yet recognized as such in the literature The analytical resultof
Eq.(24)fort=0 is plottedinthe bottompanel ofFig 1 forthe
nodeless case n=0, and for the first and second excited states
(n=1 and n=2 respectively) Belowwe willemploy direct
nu-mericalsimulationsinordertostudytherobustnessandlongevity
of these structures It is worthwhile at this point to stress out
thattheoscillonsolutionspresentedabove,owetheirexistenceto
thespontaneousbreakingoftheglobalreflectionsymmetry
lead-ing to =0, incontrast tothe symmetric phase ofEq.(5) for
which =0 andnobreathersolutionsexist.Furthermore,since
thescalarfieldattaineditsnon-vanishingvev,thisscenario
corre-spondstothefully-higgsedsuperconductingphasefarbeyondthe
associatedcriticalpoint.Thelocalizedexcitationsofthescalarfield
inducelocalizationtotherespectivegaugefieldleadinginturnto
avanishingmagneticfieldinthisregion
3.2 Numerical results: uniform vacuum
Inthissectionnumericalresultsarepresented,concerningthe
evolutionoftheapproximatesolutionsobtainedinEqs.(23)–(24)
InparticularweperformdirectintegrationofEqs.(8)–(9)usingas
initial conditions Eqs.(23)–(24) att=0 In all numerical results
presented belowwe use lattice spacingdx=0.2, time step dt=
0.01 and the total time of integration is oforder t∼104 which
correspondsto∼104oscillationsforthefields
InFig 2weshowtheevolutionduringtwofullperiodsintime
t=2T i, where T1,2=2π / ω1,2 forthe scalarand thegauge field
respectively.Theleftcolumncorrespondstothecaseofthe
node-lessoscillon (n=0) for q=1 Both fields have similar structure
butdifferentfrequencies;alsonotethatinorderforthemultiscale
expansiontobevalid,thewidthofbothfieldsisrestrictedtobeof
theorder O(101).Themiddlecolumnofthisfigureshowsa new
mode for the gauge field A with one node (n=1) for q=3/4,
whiletherightcolumnshowsthesecond excitedstate (n=2) of
Eq.(18)inbothcasesthescalarfieldisstillsech-shaped
Wehaveconfirmedtherobustevolutionofsuchstatesfortimes
uptot∼104.InFig 3contourplotsoftheenergydensityEq.(10)
fordifferentvaluesofq andforboththenodelessandnodal
oscil-lonsaregiven.Althoughthemoreusualcaseofanodelesssoliton
issomehowexpectedtoberobustintheone-dimensionalsetting,
the robustness ofthe higher excited statesis not necessarily
ex-pected
Fig 3 (Coloronline.) 3d plots showing the normalized, with respect to its maximum value, energy density E ( x , t )for different values of the parameterq andfor total time of integration in each case oft=10 4 in the uniform vacuum.
4 Solutions around the non-uniform background
4.1 Analytical considerations-perturbation around kink’s core
Main subjectofthissection is toobtain localizedsolutions to the system of Eqs (8)–(9) around the core of the domain wall Since thewidthofthedomainwall isoforderO(1),andweare interested infindinglocalizedsolutionsforthegauge fielddueto thepresenceofthedomainwall,insteadofusingaslow-scale ap-proximationweconsiderthefollowingperturbationexpansion:
In the above expression A (1) is the unknown, small amplitude gaugefield, φkistheexactkinksolutionofEq.(7),and ˜φdescribes higher orderperturbations upon thekink dueto thepresence of
A (1)[cf Eqs.(8)–(9)].NotethatinEq.(25),correctionstothescalar fieldoforder arenotincluded,sincesuchtermsdescribe pertur-bations around the kink, decoupled from the gauge field, which were studiedinRef.[40].Ontheother handouranalysis, aswell
asourexpansion,concernstheeffectsofasmallamplitudegauge field A (1)
Substituting the expansion of Eq (25) into the system of Eqs (8)–(9),at leading order(i.e to the order ) we obtain the followingequationforthesmallamplitudegaugefield:
2A (1)+ φ2
Notice thatthestationarykinksolution φk actsasaneffective potential forthe field A (1) Furthermore,welookforsolutions of
Trang 5theform: A (1)(x, t) =exp[−iωt] ˆA (1)(x),where ωisthefrequency
while Aˆ(1)(x) isafunctiondependingon thespatialcoordinatex.
SubstitutingtheaforementionedansatzintoEq.(26)weobtainthe
eigenvalueproblem:
∂x2Aˆ(1)+ E+sech2(qx/ )
ˆ
where E= ω2−1 is the corresponding eigenvalue It is readily
seenthat Eq.(27)is identicaltoequation (18) andthuslocalized
solutions of Aˆ(1) can be found as the bound statesof the above
equationgivenby:
ˆ
Thetotalnumberofbound states(cf dashedblacklineinFig 1)
andtheenergyspectrumarenowgivenby
q2
, E n=f E 4
q2
whilethecorrespondingapproximatesolutionsforA canbe
writ-tenas:
A(x,t) ≈ Aˆ(1)
n (x)
e−i E n t+c.c
Theabove solutions correspondto a familyoflocalized gauge
fieldscenteredatthedomainwallhavingtheformofnodelessand
nodal oscillon-like structures supported by an effective potential
dueto thepresenceofthekink Theseoscillonsareofsmall
am-plitudeandhaveaspatialwidthoftheorderofthecorresponding
domain wall (which dependents on q). Although these solutions
bare many similarities with the solutions obtained in Section 3,
theyarecharacterized byinadifferentlengthscale,andtheir
lo-calizationmechanismisfundamentally different Inparticularthe
effectivepotential inEq.(27),isduetothepresenceofthe
time-independent, exact solution of the original system of equations,
whiletheeffectivepotential inEq.(18)is dueto anapproximate
oscillon.Inparticularthecaseofthenon-uniformbackground
dis-cussed in this section corresponds to a scenario just after the
symmetrybreaking,andthusclosetothecriticalpoint φ =0.The
kink interpolatesbetween the justformed wells of the potential
connectingthem,andsinceq isrelativelysmall,thesolutions
ob-tained above are not energeticallydisfavored Furthermore, since
thevacuaaredegenerateandclosetothevacuumoftheunbroken
phase,the orderparameter,having theform ofthekink, induces
localization of the respective gauge field leading to a vanishing
magnetic field around kink’s core Below we will attemptto
es-tablishaconnectionbetweenthesetwoscenaria
4.2 Numerical results: non-uniform background
Intheprecedingsection weobtainedfamilies ofnodelessand
nodaloscillon-likestructures.Inwhatfollows,wewillelaborateon
howthe aforementionedsolutions evolve intime, so asto verify
thevalidityaswellastherobustnessofouranalyticalfindings.In
particularwewillperformnumericalintegrationofthesystemof
equations(8)–(9) using asinitial conditions,(at t=0), the exact
domainwallsolutionofEq.(7)andEq.(30)
IntheleftcolumnofFig 4a3dplotshowsthefirsttwo
oscil-lationsfora sech-shaped gaugefield (top),i.e.anodeless (n=0)
oscillon, for q=1 and its profile at t =0 is indicated with a
solid black line The scalar field corresponding to the above
os-cillon is also depicted inthe bottom panel of the same column
Since φ (x, t) is non-oscillating, the main contribution forsuch a
statecomes fromtheleading orderkinksolution φk.Accordingly,
inthe middle and rightcolumns of Fig 4 top panels depict the
field A(x, t) corresponding to the first excited state, (n=1 solu-tion), for q=3/4 and second excited state (n=2) for q=1/2 respectively.Thecorrespondingscalarfieldsarealsoplottedinthe bottom panel of each column In both cases the kink solutionis slightlyaffected bythe smallamplitude perturbations considered here.Both nodelessandnodaloscillonsremainrobustforatleast
t∼104 total time of integration and for different values of the parameter q. In order to highlight the longevity as well as the robustness of the oscillons obtained in this limit, in Fig 5 a 3d plotconsistingfromthreeenergydensitycontoursisdepictedfor
t=104
Wehavethusverifiedthatinthenon-uniformcaseasmall am-plitude gauge field alters theexact kink solutionat order 2, as per ouranalyticalfindings ofEq.(30).Assuch, thekinksupports not onlythe standardnodeless oscillonsbutalsothe nodalones, whichinturnremainlocalizedthroughoutalloursimulations Ad-ditionally, thesenovel structures seem to expelsmaller amounts
ofradiationwhencomparedtothenodaloscillonsoftheuniform vacuumcase
5 Dynamical localization of the gauge field
Ourpreviousanalysiswasguidedbytwo configurationsofthe scalarfield: theuniformnon-zero vacuumandthekink state.As already mentioned, these two states describe different physical scenaria andseemtobe dynamicallydisconnected.However, just afteraspontaneoussymmetrybreaking,when υisverysmall,the uniformnon-zerovacuumisenergeticallyalmostdegeneratewith thekinkconfigurationandthedynamicsmaysupportmixed con-figurationscombiningcharacteristicsofbothscenaria
Inordertodevelopaphysicalpictureforthisparticularcase,let
usfocusonthescalarfield anditsgroundstate.Whenthe reflec-tionsymmetry φ → −φisrestoredthegroundstateistheuniform configuration φ =0.Tomakeaclearerconnectionwiththephase transitioninduced by thespontaneous symmetry breakingofthe reflectionsymmetry inthisself-interactingscalar fieldtheory,let
usdefine asanorder parameterofthe transitionthespace aver-aged valueof thescalar field: V = 1
V
V dxφ (x),where φ (x) is
astaticfieldconfigurationminimizingtheenergyfunctionalofthe field φ.Forthesymmetricphase( =0)anylocalizedexcitation
ofthisconfigurationintheformofanoscillonisunstableanddies outastimeevolves
Consider nowthecasewhenthereflectionsymmetryis spon-taneouslyjustbroken.Thenthegroundstateofthefieldbecomes doublydegenerate(± υ)with υ closeto zero.Theuniformstates
φ (x) = υ or φ (x) = − υ are energeticallyalmost degenerate with thekink configuration φ (x) = υtanh√
2 υx 2 Due tothis ap-proximate degeneracywe extend the definition ofthe order pa-rameter allowing in the averaging also the use ofthe kink con-figuration.Ofcoursealsotheorderparametervaluesforall these configurations are almost degenerate However, an important is-sueconcerning symmetry breaking isthat the kink configuration
is characterized by vanishing order parameter =0 while the two degenerate vacua have =0 Thus the kink isa topologi-calstructureallowingthecommunicationbetweenthetwovacua, breakingthe symmetrylocallybutnotglobally i.e.atthe levelof the order parameter In thissense one can interpret the kink as
a fluctuation ofthe reflectionsymmetric vacuum φ (x) =0 lead-ing toa localsymmetry breaking beforea globalbreakingof the reflectionsymmetryestablishes
states (± υ) and the kink is very small, then the latter may be entropically favored constituting the representative of the field fluctuations driving the transitionfrom the globally unbroken to
Trang 6Fig 4 (Coloronline.) Top row: 3d plots showingt=2T oscillationperiods for A ( x , t )forn=0,q=1 (left),n=1,q=3/4 (middle) andn=2,q=1/2 (right) Bottom row: 3d plots depicting the fieldφ( x , t )for each of the aboveA’s.For the non-uniform background.
Fig 5 (Color online.) Same asFig 3 but for the non-uniform background.
the globally broken phase of the reflection symmetry Adopting
thispoint ofview,onecan nownaturallyaskhowlocalized,time
dependent fluctuations(breathers) of the kink,which may cause
thedynamical establishment ofthenon-vanishing order
parame-tervalue,evolveintime
Thisistheissuewewillconsiderinthissectiontakinginto
ac-count also the presence ofthe gauge field Furthermore we will
consider also the case when the kink is traveling with a
veloc-ityv k.Thisisadynamicalprocessconnectingsnapshotsconsisting
of different static kink configurations with equal energy These
differentconfigurations could be interpreted asthe origin ofthe
entropicdominance ofthe kinksolitonclosetothecriticalpoint
Stability oflocalized fluctuationsis such a time dependent
back-groundwouldsignalthevalidityoftheattemptedcriticaldynamics
description.Note that the solitonsolutions of the preceding
sec-tionsmayhavetheformoftravelingwavesbyapplyingaLorentz
boost Inour numericalsimulations a moving kink is realized as
follows: φk=tanh [γq(x−v k t)/2],where v kisthevelocityofthe
kink and γ =1/
1−v2 is the Lorentz factor [10].The relevant nodelesssolitons inthebulk are givenby Eqs.(23)–(24) andare
nottraveling
We haveperformedvarious realizations ofthe above
configu-rationsinanumericalexperimentfordifferentvaluesofq andfor
differentvelocities.Theresultsaresummarizedasfollows.Adirect
relationbetweenthepossibleoutcomeofthecollisionandthe
ve-locityofthedomainwall (thusitskinetic energy)isobserved.In
factwefoundthatforanyq,thereisalowercriticalvelocity,above
which thegauge field is localized onthe domain wall
Addition-ally,dependingonthevalueofq andthepossiblenodal(excited)
states[cf toppanelofFig 1],asthevelocityofthekinkincreases the higherexcited stateis realized.Theabove resultcan,atleast qualitatively,beexplainedfromenergeticconsiderations.Forsmall velocities,thekinetic energyofthekinkisnot sufficientinorder
togeneratethelowerpossibleboundstateoftheeigenvalue prob-lem (27),andthus a lowercriticalvelocity exists.Alsothe larger thevelocityofthemovingdomainwall,themoreenergyis trans-ferredtothegaugefieldandthehigherexcitedstatescanthenbe formed.Importantlyafterthecollisiontheoriginallocalized oscil-lon inthebulk remains intact, andundergoes a phase-shift [41] Thisshiftisfoundtobevelocitydependentinasimilarmannerto theoutcomeofsolitoncollisions(seeRefs.[15,42])
The above resultsare illustrated inthe snapshots ofthe field profilesshowninFig 6.Inparticularweshowthreedoublets,each depictingtheprofileofthegaugefield(upperpanel)andthescalar field (lowerpanel).Top,middleandbottomdoubletsshowresults forkinkvelocities v k=0.2,v k=0.3 and v k=0.4 respectively,for
q=0.4.Theinitialcondition (grayline)att=t0 correspondstoa kink located at x= −220 and a bulk oscillonat x= −150.At t1
a localized gaugefield is shownto travelalong withthe domain wall,whilethebulkoscillonisphase-shifted(thinblackline).We also show an additional profile at t2 (thick black line) in order
to illustratetheoscillations ofbothnodeless andnodal solutions Fromtoptobottomweobservethegenerationofalocalizedgauge fieldwithnonodes,onenodeandtwonodesrespectively
6 Concluding remarks
Inthe presentwork weanalyticallyobtainedfamilies ofnodal and nodelesslocalized structuresof the classical electromagnetic
Trang 7Fig 6 (Coloronline.) Profiles of the fieldsA and φforq=0.4, for a collision of
a moving kink with the bulk oscillons are depicted Gray (solid) lines indicate the
initial condition att=t0 , before the collision takes place, while black (solid) lines
att=t1 show the profiles after the dynamical localization of the gauge field at
kink’s core To illustrate the oscillations that these structures undergo, profiles of
both fields are also plotted att=t2 From top to bottom each field doublet refers
to zero, one, and two nodes respectively while the relevant velocities of the kink
are also depicted in the yellow box (bottom right of each).
sectorina (1+1)dimensionalsetting.Theinteractionbetweenthe
gaugeandthescalarfieldwasshowntobereducedtoaneffective
Pöschl–Tellerpotential, responsiblefor thelocalizationof asmall
amplitudegaugefield
In particular two different cases were studied: (i) a uniform
vacuumφ = υ and (ii)a non-uniform background(domain wall)
φ = υtanh√
2 υx 2
In the uniform, fully higgsed-“supercon-ducting”phase,families ofsmallamplitudelocalizedsolutionsfor
bothfieldswere found, withtheenvelopeof thescalarfield
sat-isfyinga focusing NLS equation leading to sech-shapedoscillons
Accordingly,theenvelopeofthegaugefieldonasufficientlylarge
length scale, was found to satisfy a linear Schrödinger equation
withaneffectivepotentialofthePöschl–Tellerform.Boundstates
ofthelatter,correspondtoalocalizedgaugefield witheitherthe
usual sech-shaped (nodeless) form, or the form of excited nodal
localizedstructures In a non-uniformbackground we also found
localizedgaugefieldsolutions,stemmingfromaneffectivePöschl–
Tellerpotential aroundthekink’score.Inthiscasehoweversince
thekinkisresponsibleforthelocalization, thelength-scale ofthe
respectiveboundstatesisthesameasthedomain wall,whichis
atleastanorderofmagnitudesmallerthanintheuniformcase
Numericalsimulations were presentedshowing the long time
evolutionoftheabove obtainedfamiliesofsolutions, whereboth
theground state andexcited statesof the effectivePöschl–Teller
potential, were found to be robust Furthermore, by inducing a
collision between the core of the domain wall and an oscillon
structureinthebulk,weestablishedadirectconnectionbetween
thetwodifferentstates.Inthisrespect,the mainoutcomeofour
analysiswastwofold:(i)Itwasdemonstratedhowlocalized
fluctu-ationsofthekinkbecomestable.Thisisanimportantresult
sup-portingthepreviouslydescribedpicture,concerningthedynamics
closetothecriticalpoint ofthespontaneousreflectionsymmetry
breaking: the kinkconfiguration acts asmediator of theglobally
brokenphasesincefluctuationsleadingtoanon-vanishingvalueof
theorderparameterbecomestableusingthekinkasabackground
field.(ii)Adynamicalmechanismforthelocalizationofthegauge field was alsodemonstrated Thisprocess mayhave phenomeno-logicalimpactonthedynamicsrelatedtotheMeissnereffectclose
to the criticalpoint Thus theuse oftopological defects as back-ground fields mayserve as a way of driving the dynamics from thecriticalpointandbeyondandviceversa
Appendix A Multiscale expansion
Inthissectionwe presentthe MSPTexpansioninmoredetail
We expand space–time coordinates and their derivatives as fol-lows: x0=x, x1= x, x2= 2x, , t0=t, t1= t, t2= 2t .,
∂x= ∂x0+ ∂x1+ , ∂t= ∂t0+ ∂t1+ , while the asymptotic expansion for both fields is given by Eq (12) Substituting the above expansionsfor the coordinates and Eq (12) for the fields intoEqs.(8)–(9)weobtainthefollowingequationsforboth φand
A uptoorder O( 4):
O ( ) : ˆL φφ(1)=0, (A.1)
O ( 2) : ˆL φφ(2)= −2∂μ0∂μ1φ(1)−3q2
2 φ
(1)2, (A.2)
ˆ
L A A (2)=0, (A.3)
O ( 3) : ˆL φφ(3)= −2∂μ0∂μ1φ(2)− 21+2∂μ0∂μ2
φ(1)
−3q2φ(1)φ(2)−q2
2φ
(1)3, (A.4)
ˆ
L A A (3)= −2∂μ0∂μ1A (2)−2φ(1) A (2), (A.5)
O ( 4) : ˆL A A (4)= −2∂μ0∂μ1A (3)− 21+2∂μ0∂μ2
A (2)
−2
φ(1) A (3)+ φ(2) A (2)
− φ(1)2A (2), (A.6)
where we introduced the operators Lˆφ≡ (20+q2) and LˆA≡
( 20+1).The lowestorderequationsforthefields φ(1) and A (2), i.e equations (A.1) and (A.3), admit plane wave solutions given
by Eqs.(13)–(14),whileeach satisfies therelevant dispersion re-lation: ω1= k2+q2, ω2= k2+1 respectively.Inordertosolve
Eq (A.2) we first eliminate the term ∼ ∂μ0∂μ1 Such a secular
term, resonates with the operator ˆL φ leading to solutions that growlinearlywithtimehaving theform:∼te iCt,withC the fre-quencyof the driver.Thus, we constrain the function u(x i, t i)by demanding that it depends on the variables x1, t1 only through
X1=x1− υ( g1) t1, where υ( g1) is the group velocity Taking into accountthissolvabilitycondition,theformofthesolution φ(2) be-comes:
φ(2)=u2
2 e
−2i ω1t+u∗2
2 e
2i ω1t−3|u|2. (A.7)
Continuing our analysis, to order O( 3) the solvabilitycondition leadstothefollowingequationforthescalarfield:
21+2∂μ0∂μ2
φ(1)−3q2φ(1)φ(2)−q2
2φ
(1)3=0. (A.8)
Substitutingintheabove φ(1), φ(2) fromequations(13)and(A.7)
respectively,weobtaintheNLSEq.(15)fortheenvelopeu(x1, t2)
Inthesameorder,byrepeatingtheaforementionedarguments, andgoingtoaframeofreferencemovingwithgroupvelocity υ( g2), thefield A (3)becomes:
2+−1uve
−i +t+ 2
2−−1uv
∗e−i −t+c.c, (A.9)
withfrequency ±= ω2± ω1 Finally to order O( 4) the solvability condition leads to the SchrödingerEq.(16)fortheunknownfunctionv(x , t )
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... phenomeno-logicalimpactonthedynamicsrelatedtotheMeissnereffectcloseto the criticalpoint Thus theuse oftopological defects as back-ground fields mayserve as a way of driving the dynamics from thecriticalpointandbeyondandviceversa...
sup-portingthepreviouslydescribedpicture,concerningthedynamics
closetothecriticalpoint ofthespontaneousreflectionsymmetry
breaking: the kinkconfiguration acts asmediator of theglobally... velocities,thekinetic energyofthekinkisnot sufficientinorder
togeneratethelowerpossibleboundstateoftheeigenvalue prob-lem (27),andthus a lowercriticalvelocity exists.Alsothe larger thevelocityofthemovingdomainwall,themoreenergyis