Discrete solitons of spin-orbit coupled Bose-Einstein condensates in optical lattices Mario Salerno1 and Fatkhulla Kh. Abdullaev2 1 Dipartimento di Fisica “E.R. Caianiello”, CNISM and INFN - Gruppo Collegato di Salerno, Universita´ di Salerno, Via Giovanni Paolo II, 84084 Fisciano (SA), Italy 2Deparment of Physics, Kulliyyah of Science, International Islamic University of Malaysia, 25200 Kuantan, Pahang, Malaysia (Dated: January 30, 2015) 5 1 We study localized nonlinear excitations of a dilute Bose-Einstein condensate (BEC) with spin- 0 orbit coupling in a deep optical lattice (OL). We use Wannier functions to derive a tight-binding 2 modelthat includesthespin-orbitcoupling (SOC)at thediscrete levelin theform of ageneralized discrete nonlinear Sch¨odinger equation. Spectral properties are investigated and the existence and n stability of discrete solitons and breathers with different symmetry properties with respect to the a OL is demonstrated. We show that the symmetry of the modes can be changed from on-site to J inter-site and to asymmetric modes simply by changing the interspecies interaction. Asymmetric 8 modes appear to benovel modes intrinsic of theSOC. 2 PACSnumbers: ] s a g Introduction. Presently there is a growing interest in in shallow optical lattices [13] and vortices in 2D opti- - the study of Bose-Einstein condensates (BEC) in the cal lattices [19] were also investigated. The extension to t n presenceofnonabeliangaugefieldsthatmimicmagnetic quasi-periodicOLandthe AndersonlocalizationofBEC a interactions. In particular, spin-orbit couplings (SOC) with effective SOC have been considered in [20]. These u q of Rashba and Dresselhaus types and their combina- studies refer mainly to the continuous case, and the ef- . tions,havebeenrecentlyrealizedinbinaryBECmixtures fectsofSOConlatticemodels(BECarrays)ispractically t a in the presence of trapping potentials V(x) of different uninvestigated. m types[1](seethereviewarticles[2,3]). Aswellknown,the - SOC represents a major source of intra-atomic magnetic The aim of the present letter is to consider BEC mix- d interaction. In solid state physics it plays an important tures with SOC in the presence of deep optical lattices. n o roleinthemagnetismofsolids,welldescribedintermsof WeuseWannierfunctionstoderiveatight-bindingmodel c individual ions, as it is for earth rare insulators. In gen- that includes the SOC at the discrete level in the form [ eral, however, in solids the SOC is a rather weak source of a generalized discrete nonlinear Scho¨dinger equation 1 of magnetic interaction (largely superseded by the elec- (SO-DNLS)withtypicaldoubleminimalineardispersion v trostaticeffects,whichisimpossibletomanage/enhance. relations, supporting nonlinear localized two component 6 excitations with chemicalpotentials inside the forbidden The situation is quite different in Bose-Einstein con- 9 zone of the band structure (gap-solitons), in the non- 2 densates where a variety of forms of synthetic spin-orbit linear case. The existence and stability of gap-solitons 7 couplings canbe easily generatedby externallaser fields (GS) has been investigated as a function of parameters 0 and the strength can be easily controlled. BEC with . such as the strength of the interatomic interactions and 1 SOC, indeed, now represent ideal systems to explore in- strengthoftheSOcoupling. Thebandstructureandthe 0 teresting phenomena that are difficult to achieve in solid location of the modes in the band structure is shown to 5 state, such as new quantum phases with unusual mag- 1 be symmetric with respect to the change of the signs of netic properties, existence of stripe modes[4], fractional : both inter- and intra- species interactions. We find that v topological insulators[5–7], Majorana fermions, etc. i forattractive(resp. repulsive)nonlinearinteractions,GS X Theinterplayofthespin-orbitcouplingwiththeinter- canexistinthesemi-infinitegapbelow(resp. above)the r atomic interactions (nonlinearity) and the periodicity of bottom (resp. top) band, as well as in the gap between a the OL also leads to the existence of SOC solitons[8, 9]. the two bands. The time evolution of attractive (resp. GapsolitonsinBECwith periodic Zeemanfieldwerere- repulsive) fundamental GSs, e.g. the one with the lower cently discussed in [10] (see also the review of BEC in (resp. higher) chemical potential, are very stable while OL with SOC in [11]). In particular, different settings the ones located in the intraband gap are typically un- for the spatial periodicity have been considered: peri- stable. Quite interestingly, by increasing the strength of odicity in each separate component[12, 13], periodicity the intra-species interaction γ, (for simplicity assumed in the Raman coupling[14], periodicity in the Zeeman equal for both species) away from the linear limit, for a field[10, 15–17]. Dispersion relations of one-dimensional fixed andequalsign ofinter-species nonlinearity,we find BECwithSOCinOLwereexperimentallyinvestigatedin three distinctive regions in which GS undergoes sponta- [18]andexistenceofflatbandsandsuperfluidityinBEC neouslysymmetrybreaking. Moreprecisely,intherange with SOC demonstrated in [12]. Array of vortex lattices 0<|γ|<|γ |theGSarefoundtobeasymmetricwithre- 1 2 specttothelatticepoints,intheinterval|γ |<|γ|<|γ | Inthefollowingwerestrictonlytooneband(theground 1 2 they display inter-site symmetry (eg symmetry with re- state band) and therefore we drop out the band index specttothemiddlepointbetweentwoconsecutivelattice m. By substituting (3) in Eq. (1) and projecting the sites) and above |γ | display the on-site symmetry, this resulting equations along the w(x−n) function, one ar- 2 behaviorbeingobservedbothforattractiveandrepulsive rives at the following coupled SO- DNLS system for the case. Asymmetric modes appearas novelmodes induced coefficients by the SOC. A possible physical interpretation of this phenomenon du σ issuggestedwhichcouldbevalidalsoforthecorrespond- i n = −Γ(u +u )+i (v −v )+Ωu + n+1 n−1 n+1 n−1 n dt 2 ing continuous case. (γ |u |2+γ|v |2)u , Tight-Binding Model. We consider BEC with equal 1 n n n contributions of Rashba and Dresselhaus SOC that can dvn σ i = −Γ(v +v )+i (u −u )−Ωv + be described in the mean field approximationby the fol- dt n+1 n−1 2 n+1 n−1 n lowing coupled Gross-Pitaevskiiequations[8]: (γ|u |2+γ |v |2)v , n 2 n n (5) ∂ψ ∂2 ∂ψ j 3−j i = (− +V(x))ψ −iα +Ω ψ + ∂t ∂x2 j ∂x j j with (g |ψ |2+g|ψ |2)ψ , j =1,2, (1) j j 3−j j ∂2 Γ≡Γ = w(x−n)∗ w(x−(n+1), with the linear coefficients α, Ω ≡ Ω = −Ω arising n,n+1 Z ∂x2 1 2 from the spin orbit interaction while the nonlinear ones, γ =g |w(x−n)|4dx, γ =g |w(x−n)|4dx, (6) g and gi,i = 1,2, are related to inter-species and intra- i iZ Z species scattering lengths, respectively. In the following ∂ weconsiderastrappingpotentialanopticallattice(OL), σ ≡σ(n,n+1)=2α w(x−n)∗ w(x−(n+1)). Z ∂x e.g. a periodic potential of the form V(x) = V cos(2x) 0 (equal for the two components), and concentrate on the InthederivationofEq. (5)thefollowingrelationsamong case of large amplitudes V >> 1 (deep optical lat- coefficients have been used: 0 tice) for which it is possible to develop the tight bind- ing approximation[21]. To this regard we expand the σ(n,n)=0, σ(n,n−1)=−σ(n−1,n)=−σ(n,n+1), two component fields in terms of the Wannier functions w(x−n)localizedaroundlatticesitesnoftheunderlying Γ(n,n+1)=Γ(n,n−1). uncoupled linear periodic eigenvalue problem, e.g. with α=Ω=0,g =g =0, in Eq. (1), i Moreover, due to the strong localization of the Wannier functions around the lattice sites, the sums on n have ∂2 − +V(x) ϕ =ε (k)ϕ . (2) been restricted to onsite and to next neighbor sites only, ∂x2 m,k m m,k h i for non diagonal and diagonal terms, respectively. Here ϕ and ε (k) denote, as usual, Bloch (Floquet) Linear Case. Let us first consider the case with all m,k m nonlinear coefficients detuned to zero, γ =γ =0, functionsandenergyband,withmthebandindexandk i tFhoeraBldoecehpwopavtiecnaulmlabtteircetaitkeisncionntvheenifienrstttoBruislleotuhineWzoanne-. iddutn =−Γ(un+1+un−1)+iσ2(vn+1−vn−1)+Ωun,(7) nier basis and expand the fields as idvn =−Γ(vn+1+vn−1)+iσ(un+1−un−1)−Ωvn.(8) dt 2 ψ = u (t)w (x−n), ψ = v (t)w (x−n), (3) 1 n m 2 n m corresponding to the case of a linear chain with spin- Xn,m Xn,m orbit coupling. From physical point of view this corre- wherew (x−n)denoteWannierfunctionsassociatedto spondtoaone-dimensionalarrayofBECindeepoptical m theperiodicproblem(2),andu ,v aretimedependent latticeinthe presenceofSOinteraction. Tofindthe dis- n n expansion coefficients to be fixed in such a manner that persionrelationofthis linearchainweconsidersolutions Eq. (1) are satisfied. Notice that in the expansions (3) of the form appear the same Wannier functions since the underlying u =Aei(kna−ωt), v =Bei(kna−ωt) (9) periodicproblemisthesameforbothcomponents. Also, n n we remark that Wannier functions are orthonormalwith with wavenumber k = 2π/Ln varying in the first Bril- respect to both the band index and the lattice site n n louin zone [−π/a,π/a], with −N/2≤n≤N/2, L=Na around which they are centered: thelengthofthechain,athelatticeconstantfixedbelow without loss of generality to a = 1. Substituting (9) in Z wj(x−n)∗wj′(x−n′)dx=δj,j′δn,n′ (4) Eq. (8) one obtains a homogeneous system of equations 3 0.5 a 3.0 ç b -00..50 --22..2250 22..05çôçôçôçôçôçôçôçôçôçôçôçôçôçôçôôìçôçôçôçôçôçôçôçôçôçôçôçôçôçôçô Μ-1.0 --22..3350 1.5ìòìòìòìòìòìòìòìòìòìòìòìòìòòòìòìòìòìòìòìòìòìòìòìòìòìòìòìòìò -1.5 -2.40 1.0 ììò -0.7-0.6-0.5-0.4-0.3 æ -2.0 0.5 -2.5 0.0æàæàæàæàæàæàæàæàæàæàæàæàæàæàæààæàæàæàæàæàæàæàæàæàæàæàæàæàæàæà -1.0 -0.8 -0.6 -0.4 -0.2 0.0 -15 -10 -5 0 5 10 15 Γ n 3çôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôcçôçôç 3çôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôçôdçôçôçô FIG.1: Chemicalpotentialvskinthereciprocalspaceforthe 2 ìì 2 ìì DγN=L0SacnhdaipnawraimthetSeOr vcaoluupeslinΓg=fo0r.3t,hΩe l=in1ea.3r5c2a,sσe=γ11.=5.γT2h=e 1ìòìòìòìòìòìòìòìòìòìòìòìòìòìòìòìòòòìòìòìòìòìòìòìòìòìòìòìòìòìòì 1ìòìòìòìòìòìòìòìòìòìòìòìòìòìòìòìòòòìòìòìòìòìòìòìòìòìòìòìòìòìò ææ æ rmeoddaensdfobrlutheednootnslirneeparresceanstesc:heγm1i=calγ2po=te−nt0i.a6l5s,oγf12lo=ca−liz1e.d8 0æàæàæàæàæàæàæàæàæàæàæàæàæàæàæàààæàæàæàæàæàæàæàæàæàæàæàæàæàæà 0æàæàæàæàæàæàæàæàæàæàæàæàæàæàæààæàæàæàæàæàæàæàæàæàæàæàæàæàæàæà (attractive case), and γ1 = γ2 = 0.65,γ12 = 1.8 (repulsive -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 n n case), respectively. Modes are plotted on the k = 0 line for graphical convenience. FIG. 2: Panel (a). Existence curves of the localized modes of the DNLS with SO coupling as a function of the equally for the coefficients A,B, whose compatibility conditions attractiveintraspeciesinteractionγ ≡γ1=γ2 forparameters Γ=0.3,Ω=1.352,σ =1.5 and fixed intraspecies interaction directly leads to the dispersion relation γ12 = −1.8. The inset display details of the bottom curves, enlarged along the vertical axis. Panels (b), c and (d) show ω(k)± =−2Γcos(k)± Ω2+σ2sin2(k). (10) un (blue lines) and vn (red lines) profiles of gap solitons at q γ = −0.65 (panel (b)), γ = −0.52 (panel (c)) and γ = −0.3 (panel (d)), corresponding to red points (bottom profiles), blue points (middle profiles) and black points (top profiles) It is worth to note that the upper and lower branches shown in panel (a) (in all cases vn is purely real while un of the dispersion curve are related as follows is purely imaginary). Middle and top profiles in all panels areshiftedupwardby1.5and2.5,respectively,toavoidover- ω (k)=−ω (k+π). (11) ± ∓ lappings. Vertical dotted lines separate regions of different symmetry type. All plotted quantities are in dimensionless Also notice the presence of two degenerated minima at units. ±k on the ground state branch and two degenerated − maxima at ±k on the upper branch, with k given by + ± and intra-species interatomic interactions, gap-solitons 2Γ σ2+Ω2 to form with chemical potentials just below local min- k±=±arcos ± . (12) σ rσ2+4Γ2 ima (maxima) of the dispersion relation. This is indeed h i whatonefindsfromanumericalself-consistentdiagonal- ThedispersioncurveinthefirstBrillouinzoneisdepicted ization of the stationary eigenvalue problem associated for typical parameter values in Fig. 1. One can readily to Eq. (1) (see Fig. (1) for typical examples). show that the amplitudes of the two components must In the panel a) of Fig. 2 are shown existence curves be related by of gap-solitons of the DNLS with SO coupling for the case γ < 0 and for equally attractive intra-species in- B csc(k) 12 (A)± = σ (Ω∓qΩ2+σ2sin2(k)). (13) teractions γ1 = γ2 ≡ γ < 0 . The lower two branches correspond to the ground localized modes in the lower By combining the k and k modes one readily obtain semi-infinite gapwhile the topcurvereferstoa modein- + − stationary stripe solutions for ω (top panels of Fig. 2) side the interbandgap. In remainingpanels ofthe figure + and ω (bottom panels of Fig.2). areshowntheimaginaryandrealpartsofthecomponent − Nonlinear Case. In the presence of the nonlinearity profiles un (blue lines) and vn (red lines) of the gap soli- theexistenceoflocalizedmodeswithchemicalpotentials tons for the different values of γ corresponding to black, intheforbiddenzonesofthe bandgapstructure(discrete blue and red points depicted in panel (a). gap-solitons) become possible. It is worth to note that there is a phase difference of In particular, in analogy with gap solitons or intrin- eiπ/2 between the u and v components (v being real n n n sic localized modes of continuous and discrete Gross- and u purely imaginary) as well as different symmetry n Pitaevskii equations in absence of spin orbit coupling, properties with respect to the lattice sites for the two one expects that for attractive (resp. repulsive) inter components. Also notice that the symmetry properties 4 2 2 2 2 |un| |vn| |un| |vn| 250 250 250 250 200 200 200 200 150 150 150 150 t t t t 100 100 100 100 50 50 50 50 0 0 0 0 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 n n n n 2 2 2 2 |un| |vn| |un| |vn| 200 200 200 200 t t t t 100 100 100 100 0 0 0 0 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 n n n n FIG.3: Timeevolutionofthelocalizeddiscretemodesshown FIG.5: Timeevolutionofthelocalizeddiscretemodesshown in panels b of Fig. 2 to the parameter value γ = −0.65.Top inthepaneldofFig. 2forthecaseγ =−0.3. Topandbottom andbottompanelsrefertothegroundstateandtothefirstex- panels refer totheground state and tothefirst excited state cited state in the lower semi-infinite gap, respectively. Other inthelowersemi-infinitegap,respectively. Otherparameters parameters are fixedas in Fig. 2. are fixed as in Fig. 2. of the modes change as the intra-species interactions are varied. In general, the following situation is observed. 2 2 |un| |vn| For a fixed attractive inter-species nonlinearity, we find 250 250 three distinctive regions in which GS undergoes sponta- 200 200 neouslysymmetrybreakingasthestrengthoftheattrac- 150 150 tiveintra-speciesinteractions(assumedequale.g. γi ≡γ, t t forsimplicity)awayfromtheγ =0limit. Moreprecisely, 100 100 intherange−0.35<γ <0theGSarefoundtobeasym- 50 50 metric with respect to the lattice points, in the interval −0.6 < |γ| < −0.35 they display a symmetry with re- 0 0 -20 -10 0 10 20 -20 -10 0 10 20 n n specttothemiddlepointbetweentwoconsecutivelattice sites (inter-site symmetry) and for γ < 0.6 they display the on-site symmetry. Notice that the borders of these 2 2 |un| |vn| regions correspond to the appearance of small kinks in 500 500 the existence curves µ versus γ and have been evidenced 400 400 by dashed verticallines (see inset in panel a) ofand Fig. 300 300 2). Also notice that the kinks at the left border of the t t middle regionseparatingthe modes with on-site symme- 200 200 try (see panel (b)) from the ones with intra-site symme- 100 100 try (see panel (c)), both of symmetric or anti-symmetric 0 0 type,aresharperthantheonesontherightborderwhere -20 -10 0 10 20 -20 -10 0 10 20 n n modes becoming asymmetric (see panel (c)). This also FIG.4: Timeevolutionofthelocalizeddiscretemodesshown correlateswith the factthat ingeneralone wouldexpect in the panel c of Fig. 2 for the case γ = −0.52. Top and that a lost of symmetry being a smoother process with bottom panels refer to the ground state and to the first ex- respect to a spontaneously change of a symmetry. cited state in the lower semi-infinite gap, respectively. Other The dynamical properties of the nonlinear modes de- parameters are fixedas in Fig. 2. picted in Fig. 2(c)-(d)) have been investigated by direct 5 2 2 2.5 |un| |vn| 2.0 2.40 3.0 ô b 200 200 Μ 11..05 222...233505 122...505ìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçôìçòôìçòìçôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòô t t a0.5 2.20 0.3 0.4 0.5 0.6 0.7 1.0 ò àò 0.0 0.5 100 100 -0.5 -00..05æàæàæàæàæàæàæàæàæàæàæàæàæàæàæàææàæàæàæàæàæàæàæàæàæàæàæàæàæàæà 0.0 0.2 0.4 0.6 0.8 -15 -10 -5 0 5 10 15 Γ n FipnrIsoGifid.lee6st:)hTefoiimrntetehre-e2-v0bocaal-un1s0tedion0γgna=po1f0s−thh02o00e.w6t5nh-.20einOo-1ttn0hh-esne0irtpepa1and0reisalcm20br0eeotteferFgsaigap.res2ofil(ixttooenpd çôçôçô123ìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôcìçòôìçòôììç ì123ìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôìçòôdìçòôìçòôìçòô à as in Fig. 2. 0æàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæææàæàæàæàæàæàæàæàæàæàæàæàæàæà 0æàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàææàæàæàæàæàæàæàæàæàæàæàæàæàæà à à -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 n n numerical integrations of the DNLS-SO, by taking them as initial conditions with a small noise component im- FIG. 7: Panel (a). Existence curves of the localized modes posedto check their stability under time evolution. This of the DNLS with SO coupling as a function of the equally isshowninFigs. 3-5forthetwolowestnodesinthesemi- repulsive intraspecies interaction γ ≡γ1 =γ2 for parameters infinite gap. As one can see, for the chosen parameters γ =0.3,Ω=1.352,σ =1.5 and fixed intraspecies interaction the on-site symmetric modes are both very stable (see γ12 =1.8. Theinsetdisplaydetailsofthetopcurves,enlarged Fig. 3) while for inter-site symmetric modes stability is alongtheverticalaxis. Panels(b),(c)and(d)showun (blue lines) and vn (red lines) profiles at γ =0.65 (panel (b)), γ = achievedonly forthe groundstate, the firstexcitedstate 0.52 (panel (c)) and γ = 0.3 (panel (d)), corresponding to beingmetastable. Notice,inthelastcase,thatthemode black points (bottom profiles), blue points (middle profiles) decaysintoanon-site-symmetricmodeplusbackground and red points (top profiles) in panel (a), respectively (in all radiation,similarlytotheinter-site-on-sitetransitionob- cases vn is purely real while un is purely imaginary). Middle servedfordiscretebreathersofcoupledDNLSinabsence andtopprofilesintheb,c,d,panelsareshiftedupwardby1.5 of SO coupling. Quite surprisingly, asymmetric modes and 2.5, respectively, to avoid overlapping. Vertical dotted linesseparateregionsofdifferentsymmetrytype. Allplotted of the region γ < −0.3 also appear to be stable (or long quantitiesare in dimensionless units. lived) under time evolution, as one can see form Fig. 5. The existence ofsuchmodes appearsto be typicalofthe SO coupling, since they would not be possible just in ordinary coupled DNLS. The stability properties of lo- results. For this one can adopt an experimental setting calized modes in the semi-infinite gap, however, appear similar to the one in [1]. In particular, the SOC can be to be critical. In particular, for the parameterranges we realized in 87Rb using two counter-propagating Raman have explored and independently from their symmetry laserswithλR =804nm,spin-orbitcouplingα=~kR/m, types,none ofthemwasfoundto be stable (see Fig.6for and~Ω=2ER(hereER =~2kR2/2mistherecoilenergy). a typical example). The optical lattice can be generated by means of two Similar results are found for the case of all repul- additional beams with λL =1540nm[18]. Typical values sive interactions. By reversing signs of all the interac- of the OL potential strength V0 suitable for the tight- tions chemicalpotentialsofthe discrete gapsolitonsalso bindingcaseweconsideredcanbeV0/ER >10[21,22]. In change their signs (see Fig. 1). Shape and symmetries thecaseofallattractiveinterspeciesinteractionnonlinear of the modes, however, are different in the two cases as modes below the bands should appear for wide range one can see by comparing Figs. 7 with 2. Also notice of number of atoms, being typically stable. Change of the symmetry of corresponding existence curves in the the interspecies scattering length to observe the change µ−γ plane, with appearance of kinks at the change of of symmetry of the discussed modes can be achieve by symmetry points. The stability properties under time means of the Feshbach resonance technique. evolution, look also similar to the all attractive case. In In conclusion we have derived a tight-binding model particular,modesinsidetheinter-bandgaparealsofound for BEC with SOC in deep optical lattices and demon- to be unstable, while the ones with highest chemical po- stratedforthismodeltheexistenceandstabilityofdiffer- tentials inside the upper semi-infinite gap are typically ent types of discrete solitons. We showed that nonlinear stable (not shown for brevity). modes can change symmetry from on-site symmetric to In closing this Letter we provide parameter estimates inter-sitesymmetricandtofullyasymmetricastheinter- for possible experimental implementations of the above species interaction is varied. Asymmetric modes appear 6 to be intrinsic novel excitations of the BEC with SOC. [8] V. Achilleos, D. J. Frantzeskakis, P. G. Kevrekidis, and The critical values at which the symmetry changes are D. E. 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