Structural phase transitions in low-dimensional ion crystals Shmuel Fishman,1 Gabriele De Chiara,2,3 Tommaso Calarco,4,5 and Giovanna Morigi2 1 Department of Physics, Technion, 32000 Haifa, Israel 2 Grup d’Optica, Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain 3 BEC-CNR-INFM & Physics Department, University of Trento, Via Sommarive 14, I-38050 Povo (TN) Italy 4 ITAMP, Harvard Smithsonian Center for Theoretical Atomic, Molecular, and Optical Physics, Cambridge, MA, USA 5 Abteilung Quanteninformationsverarbeitung, Universita¨t Ulm, 8 Albert-Einstein-Allee 11, D-89069 Ulm, Germany 0 (Dated: February 2, 2008) 0 Achainofsingly-chargedparticles,confinedbyaharmonicpotential,exhibitsasuddentransition 2 toazigzagconfigurationwhentheradialpotentialreachesacriticalvalue,dependingontheparticle n number. Thisstructuralchangeisaphasetransition of secondorder, whoseorderparameter isthe a crystaldisplacementfromthechainaxis. Westudyanalyticallythetransition usingLandautheory J andfindfull agreement with numerical predictionsbyJ. Schiffer[Phys. Rev. Lett. 70, 818 (1993)] 7 andPiacenteetal[Phys. Rev. B69,045324(2004)]. Ourtheoryallowsustodetermineanalytically thesystem’s behaviour at thetransition point. ] h PACSnumbers: 05.20.-y,52.27.Jt,61.50.-f c e m INTRODUCTION - t a t Wigner crystals of ions in Paulor Penning traps are a s remarkable example of selforganized matter at ultralow . t a temperatures[1]. Thesesystemsareusuallycomposedof m singly-charged particles, which are kept together by ex- - ternaltime-dependentradio-frequencyorstaticmagneto- d electric potentials, and which reach crystallization by n means of laser cooling. Among several important as- o pects, the transition from disorder to order for few ions c [ was studied in [2, 3, 4]; long-rangeorder in three dimen- sionalstructuresinPenningtrapswasfirstdemonstrated 2 in [5, 6]; and more complex crystalline structures have v 1 been realized, see for instance [7, 8, 9]. Most recently, FIG. 1: Structuralphase transition in a string of equidistant 3 these crystalline structureshave been attracting increas- trapped ions from a linear to a planar zigzag configuration. 8 ing attention for the realization of quantum information For ions in a harmonic trap,close to thetransition point the 1 zigzag configuration is evident about the center of the trap, processors [10, 11, 12] and simulators [13, 14, 15, 16]. . where thedensity of ion is larger [22]. 0 In this perspective, the clear understanding and char- 1 acterization of the structural properties would provide 7 the possibility to control at the microscopic level the 0 : dynamics of complex systems. Moreover, ion crys- conjectured that the structural change from a chain to v tals are systems characterized by truly long-range, un- a zigzag is a second order phase transition [22]. Further i X screened Coulomb interactions, and hence constitute in- numerical work showed that at this transition point the r terestingphysicalsystemswhereonecantestequilibrium groundstateenergyischaracterizedbyadiscontinuityin a and out-of-equilibrium statistical mechanics models for the second derivative with respect to the particles den- systems exhibiting non-extensive thermodynamic func- sity [23]. tions [17, 18]. Inthis article westudy the structuralphase transition Structural transitions in ion crystals are induced ei- of an ion crystal from a linear chain to a zigzag configu- ther by changing the external potential [19, 20] or by rationin a suitably defined thermodynamic limit, by de- introducing other forms of instabilities [21]. Structural velopingananalytictheorywhichallowsustodetermine transitionsinlowdimensionalioncrystalswerefirstchar- the behavior of the system at the critical point. From acterized experimentally in [19, 20]. Here, starting from symmetry considerations we conjecture the spontaneous a chain configuration, the sudden transition to a planar symmetry breaking. Applying Landau theory [24], we zigzag structure, as shown in Fig. 1, was observed when identify the order parameter and the soft mode driving the radial potential reached a critical value, dependent the instability, and demonstrate that the system under- on the ion number. In theoretical investigations it was goesa secondorderphasetransition. Ourtheoryisvalid 2 at T = 0, when the system exhibits long-range order. density of ions along the trap axis at equilibrium was It allows us to determine the system’s behavior at the determined in [27] for N 1 and using the local den- ≫ transitionpoint,andtheresultswefindareinagreement sity approximation. The linear fluctuations about the with the numerical results reported in [22] and in [23]. classical ground state of an ion chain in a harmonic This article is organized as follows. In Sec. we in- trap have been analytically studied in [28, 29]. This troduce the model and discuss first the transition for a study identified as well the value of the critical trans- chain of 3 ions from a linear to a zigzag configuration verse frequency ν(c) 3Nν/(4√logN), using an expan- t ≈ of charges. In Sec. we derive the dispersion relations sionatleadingorderin1/logN,andbyconsideringonly and eigenmodes at equilibrium of the linear chain and nearest-neighbourcontributions. Withinthisapproxima- of the zigzag configuration in the thermodynamic limit. tion this value is consistent with numerical results [22], In Sec. we focus onto the classical phase transition be- andis in goodagreementwith previousanalyticalevalu- tweenthe two configurations,identify the soft mode and ations in [30], which calculated the critical value taking study analytically the system around the critical point. into account the long-rangeinteraction between the ions InSec. weconcludeandintheappendiceswereportthe but assuming that the particles are equidistant. details of calculations presented in Sec. . When the transverse frequency is varied, so that ν < t (c) ν , the stable configuration is first a zigzag structure, t thenatsmallervaluesithasanabrupttransitiontoahe- ORDERED STRUCTURES OF IONS IN LOW licoidal one, and so on thereby acquiring more complex DIMENSIONS structures [19, 20, 30]. Eventually, for a large number of ions and for aspect ratios α sufficiently close to unity ThemodelweconsiderisconstitutedbyN particlesof the structure is expected to take the b.c.c. crystalline mass m and charge Q, which are confined by an exter- form [1]. In the following, we study the transition from nal harmonic potential along one axis. The particles are an ion chain to a zigzag structure for the most simple classical, and the Hamiltonian governing their dynamics model, namely three ions in a linear Paul trap. This reads system allows us to get some insight into the system, N p2 before considering the structural transition in the ther- H = j +V(r1,...,rN), (1) modynamic limit in Sec. . 2m j=1 X wherer =(x ,y ,z )andp arethepositionsandconju- j j j j j Structural stability of a three-ion chain gate momenta, with j =1,...,N. The term V accounts for the oscillator’s potential and the Coulomb repulsion, We considerN =3 ions inside atrapwith ν >ν, and t 1 N 1 N Q2 calculate their equilibrium positions as a function of the V = m ν2x2+ν2(y2+z2) + . aspect ratio α = ν /ν. We restrict for simplicity to two 2 j t j j 2 r r t j=1 j=1j6=i | i− j| dimensions, which we here identify with the x y plane, X (cid:2) (cid:3) XX − (2) and rewrite the potential (2) in dimensionless variables as Here,the potentialis characterizedbyharmonicconfine- 3 ment at frequency ν and νt in the axial and transverse V˜ = 1 (x′2+α2y′2)+ 1 , direction, respectively, whereby νt > ν for the case we 2 i i (x′ x′)2+(y′ y′)2 are going to study. We denote by α ν /ν the trap Xi=1 Xi<j i− j i− j t ≡ q (4) aspect ratio, such that α>1. where x′ = x /l, y′ = y /l , l3 = Q2/(mν2) and V˜ = At sufficiently low temperatures, the ions localize i i i i themselves at the equilibrium positions r(0) which solve V/(l2mν2). Throughout this section we drop the prime j superscript. the coupled equations describing the equilibrium of the The normal modes frequencies for the linear and the forces, zigzag structures are displayed in Fig. 2 as a function of ∂V α, when α is decreased across the value for which the =0. (3) ∂r linear chain becomes mechanically unstable. The transi- j(cid:12)(cid:12)rj=rj(0) tion to the zigzag configuration takes place at the value Whenthetransversefre(cid:12)(cid:12)quencyν exceedsacriticalvalue α = α∗, such that the smaller transverse frequency of t (c) the linear chain vanishes. We now study in detail the ν ,whichdepends ontheaxialtrapfrequencyν andon t classical equilibrium positions for α > 1. Assuming the the number of ions, the solutions of Eq. (3) are aligned convention x < x < x , the symmetry of the trap- along the x-axis, forming a string. Tables of the equi- 1 2 3 librium positions for string up to 10 ions have been re- ping potential imposes x(20) = 0, x(10) = −x(30) ≡ −x¯, ported in [25, 26]. An analytical form for the linear with x¯ > 0. From Eqs. (3) we also find y(0) = 2y¯ and 2 − 3 3 1.2 Zigzag Linear 2.5 1 2 0.8 1.5 0.6 Zigzag Linear 1 0.4 0.5 0.2 0 0 1.2 1.4 1.6 1.8 2 1.2 1.4 1.6 1.8 2 α* α α* α 3 Zigzag Linear 2.5 FIG. 3: Equilibrium position of the external ions of a string of 3 particles as a function of the trap anisotropy α. The 2 solidanddashedlinesdisplaythelongitudinalandtransverse 1.5 variables,x¯andy¯,inunitsofthecharacteristiclengthl. The ∗ vertical dotted line indicates the transition value α , where 1 theequilibriumconfigurationmakesanabruptchangefroma 0.5 linear chain to a zigzag structure. 0 1.2 1.4 1.6 1.8 2 α* α THE LINEAR AND THE ZIGZAG STRUCTURES FIG. 2: (color online) Excitation frequencies of the oscilla- tions modes of a three-ion string, in units of the axial trap In this section, we study the static properties of the frequencyν,asafunctionofthetrapanisotropyα. Theeigen- linear chain and of the zigzag configuration in the ther- ∗ modesfor α>α arethelongitunal(top)andthetransverse ones (bottom). For α< α∗ the eigenmodes are combination modynamic limit. For an ion chain inside a trap, a good thermodynamic limit is found by fixing the interparticle oflongitudinal andtransversemodeswith oppositeparityby reflection about x=0. spacing a at the chain center when N . This corre- →∞ sponds to the requirement that the axial trap frequency vanishesaccordingtotherelationν √logN/N [28,29]. y(0) = y(0) y¯, with y¯ 0. The linear configuration, ∼ 1 3 ≡ ≥ Inthis limit,the criticaltransversefrequencyν(c) iscon- namely the set of solutions with y¯ = 0, is found when t stant, and the behaviour at the mechanical instability is the aspect ratio α > α∗, where α∗ 12/5. When ≡ equivalent to that of a uniform chain with equal inter- α < α∗, then y¯ > 0 and the structure becomes planar. p particle distance a between neighbouring ions [31]. The We denote this case by ”zigzag configuration”, as it is uniform chain is the model we will use for determining indeedthemostelementaryinstanceofthestructureone the groundstate andthe motionofthe linear andzigzag observesformanyions. Here,forα<α∗ terms x¯, y¯take structure in the thermodynamic limit. the form α2 −31 1 3 23 x¯= 4 1 , y¯= x¯2, (5) − 3 3s α2 − The linear chain (cid:20) (cid:18) (cid:19)(cid:21) (cid:18) (cid:19) Theirfunctionaldependence onthe aspectratioαis dis- played in Fig. 3. One can observe the discontinuity of We assume a stable linear chain of ions, namely νt > the derivative at α∗, corresponding to the transition to ν(c). In this limit the equilibrium positions lie along the t a different equilibrium configuration. For α α∗ the x-axis,r(0) =(x(0),0,0),andweusethe conventionx > → j j i change is faster for the transverse displacement, as it is x fori>j. Forsmallvibrationsaroundthesepoints we j visible by the expansion of x¯,y¯at δα=α∗ α, approximate the potential in Eq. (2) by its second order − y¯ = y0 δα12 +O δα3/2 zTa.yIlonrtehxispalinmsiiotnthinetehqeuadtiisopnlascoefmmenottsioqnja=rexj−x(j0),yj, j x¯ = x¯ x δα(cid:16)+O δ(cid:17)α2 lin 0 − where x¯lin = (5/4)1/3 is the value ta(cid:0)ken(cid:1)by x¯ when the q¨i =−ν2qi− Kmi,j(qi−qj), (6) linear chain is stable, while y 0.74, x 1.85. We j6=i 0 0 X ≈ ≈ note that, aboutthe instability point ofthe linear chain, y¨ = ν2y + 1 Ki,j(y y ), (7) the transverse displacement y¯plays the role of the order i − t i 2 m i− j j6=i parameter, while the changes of the axial distance x¯ are X 1 induced by the changes of y¯, and therefore about the z¨ = ν2z + Ki,j(z z ), (8) value α∗ these are less dramatic. i − t i 2 m i− j j6=i X 4 3 with k = 2πn/Na and n = 0, 1, 2,...,N/2. The ± ± 2.5 spectrum corresponding to Eqs. (11) and (12) is shown ω ω 2 in Fig. 4. The axial eigenmodes at frequency ωk(k) are 0 1.5 Θ(±), such that k 1 2 0.5 qj = N Θ(k+)coskja+Θk(−)sinkja , (13) 0-1 -0.5 0 0.5 1 r kX>0(cid:16) (cid:17) (k a)/π where the superscript indicates parity by reflection ± k k. Analogously, we denote the transverse eigen- FIG. 4: (color online) Excitation spectrum of the uniform → − chain. The eigenfrequencies ω, in units of ω0 = pQ2/ma3, modes at frequency ω⊥(k) by Ψky(±) and Ψzk(±), where are plotted as a function of the quasimomentum k, in units of π/a. The axial spectrum (green solid line) and the trans- y = 2 Ψy(+)coskja+Ψy(−)sinkja ,(14) versespectrum (reddashedline)areobtained from Eqs.(11) j N k k and (12), respectively. Here, νt =1.1νt(c). r kX>0(cid:16) (cid:17) 2 z = Ψz(+)coskja+Ψz(−)sinkja .(15) j N k k and describe a system of coupled oscillators, with r kX>0(cid:16) (cid:17) long range interaction and position-dependent coupling We note thatthe modes atk =π/2areeven. Aclosein- strength. Here, the coefficients Ki,j ≡ −∂2V/∂xj∂xi|x0j spectiontoEq.(12)showsthattheremayexistvaluesof read the transverse trap frequency, at fixed interparticle dis- 2Q2 tance a, for which ω⊥2 < 0, that is, imaginary frequency Ki,j = |x(i0)−x(j0)|3 . (9) sTohluettiohnress.hoFlodrvsaulcuhe vνat(cl)u,essu,cthhutsh,atthfeorchνtai>n νist(cu)ntshteablilne-. We note that at secondorder in the harmonic expansion ear chain is stable, is found by solving mink(ω⊥) = 0 the axial and transverse vibrations are decoupled. It is (see Sec. ). The minimum is found at k = π/a and cor- easilyverifiedthatthecenter-of-massmotionisaneigen- respondingly mode ofthe secularequations(6)-(8) ateigenfrequencies 2Q2 N 1 j π ν andνtfortheaxialandtransversemotion,respectively. ν(c) 2 = 2 sin2 ThesolutiontoEqs.(6)-(8)havebeenstudiedin[28,29]. t ma3 j3 2 (cid:18) (cid:19)j=1 X Q2 7 For the purpose of studying the behaviour at the me- ζ(3), (16) → ma32 chanicalinstability,wenowconsiderthesimplifiedmodel of the uniform chain, where the interparticle distance at where result (16) is found for N using (2ℓ → ∞ ℓ>0 − equilibriumisfixed. Thiscaseisfoundbysettingν =0in 1)−p =(1 2−p)ζ(p), with ζ(p) the Riemann-zeta func- − P Eqs.(6)-(8)andassumingconstantinterparticledistance tion. The value in Eq. (16) depends on the interparti- a = x(0) x(0). Such condition can be realized for the cle spacing a and provides the range of validity of the j+1 − j results presented in this section. It coincides with the centralionsofalongionchaininsideofalinearPaultrap value reported in [30], where a similar model to the one [32] or for ions confined in a ring of large radius[19, 20]. discussed here was considered. It is close to the result Thissecondscenariocorrespondstotakeperiodicbound- ν(c),trap2 = 4Q2/ma (0)3 found at leading order in ary conditions, q1 = qN+1, etc. Crystallization is found t trap 1/logN in [29], where a(x) gives the interparticle dis- assuming, for instance, that one ion is pinned at the po- sition x(0) =0. Then, the classical equilibrium positions tance as a function of x in the local density approxi- 0 mation, and a(0) a is the value at the chain center. are x(0) = ja and the coupling strengths in Eqs. (6)-(8) ≡ j This result was obtained by considering the inhomoge- take the form neous distribution of ions along the chain, but keeping 2Q2 only the nearest-neighbours interaction. The small dis- uniform = . (10) Ki,j i j 3a3 ≡Ki−j crepancy between the two values is to be attributed to | − | the different approximations that have been applied in The dispersion relations are [33] each model. 2Q2 N 1 jka ω (k)2 =4 sin2 , (11) k ma3 j3 2 The zigzag structure (cid:18) (cid:19)j=1 X ω⊥(k)2 =νt2−2 m2Qa23 N j13 sin2 jk2a , (12) Forνt <νt(c),andsufficientlyclosetothecriticalvalue, (cid:18) (cid:19)j=1 the stable configuration is a zigzag structure. We now X 5 0.4 with χ = b/a. The coefficients Eq. (10), and the cor- responding equations of motion for the linear chain, 0.3 Eqs. (6), (7), are recoveredfor χ 0. → b/a0.2 Zigzag Linear In general, the structural change brings to a doubling of the unit cell d of the crystal, which from d = a in 0.1 the linear chain goes to d = 2a in the zigzag configura- 0 tion. Correspondingly, the Brillouin zone of the zigzag 0.9 0.95 1 1.05 1.1 is reduced by a factor 2, and the wave vectors now take νt/νt(c) the values k = 2πn/Na and n = 0, 1, 2,...,N/4. In ± ± Eqs. (18) and (19) one can easily verify that the bulk FIG. 5: Transverse equilibrium displacement b, in units of excitations are eigenmodes of the chain at frequencies ν the interparticle spacing a, as a function of the transverse and ν . The other eigenvalues and eigenfunctions can be t frequency νt in units of νt(c). On the right of the curve the found using the ansatz f(j,±), with n ion crystal is a linear chain. In the region on the left of the curveit exhibitsa zigzag structure. evaluate its dispersion relation and eigenmodes for ions f(j,±)(k)=( 1)ne−iωj,±t+ikna xˆ ie−inπǫ(j,±)yˆ ,(20) n ± ∓ k on a ring and for periodic boundary conditions. We h i assume the equilibrium positions to lie on the x y − plane with r(0) = (x(0),y(0),0). Then, x(0) = na and n n n n where j =1,2 and ka varies on the interval [ π/2,π/2]. y(0) = ( 1)nb/2, with b a real and positive constant, − wnhich is−determined from the equation In particular, we note the relation fn(j,−)(k)=fn(j,+)(k+ π/a). The corresponding eigenmodes are given by the mνt2 4 =0. (17) real and imaginary parts of these vectors. Using this Q2 − [(2ℓ 1)2a2+b2]3/2 ansatz, we obtain the coupled equations ℓ>0 − X Figure5displaysthetransverseequilibriumdisplacement b as a function of the transverse frequency ν , as it is t obtainedbysolvingnumericallyEq.(17). Assumingthat ω (k)2 =C(±)(k)+ǫ(j,±)B(k), (21) j,± 1 k txhne zxig(n0z)a,gwcno=nfiygnuraytin(o0)n,aisndstzanblteh,ewaxeiadleannodtetrbaynsqvners=e (νt2−ωj,±(k)2)=C2(±)(k)−ǫk(j,±)−1B(k), (22) − − displacements, and expand the potential of Eq. (2) up to second order. In this limit the motion along the z direction is decoupled from the vibrations on the plane, whereby andthe resultingequationsofmotionforq andw read n n mq¨ = x(q q ) (18) n − Kℓ n− n+ℓ 2 Xℓ6=0 B(k)= 2ℓ−1sin(2ℓ 1)ka, m Y − −(−1)n Yℓ(wn−wn+ℓ), Xℓ>0 Xℓ6=0 C(+)(k)= 4 xsin2 kℓa , mw¨ = mν2w + y(w w ) (19) 1 m Kℓ 2 n − t n Kℓ n− n+ℓ Xℓ>0 Xℓ6=0 C(+)(k)= 4 y sin2kℓa+ y cos2 (2ℓ−1)ka , ( 1)n (q q ). 2 m K2ℓ K2ℓ−1 2 − − Yℓ n− n+ℓ ℓ>0(cid:18) (cid:19) X ℓ6=0 X 4 (2ℓ 1)ka Thecoefficientsappearingintheseequationsdependonly C1(−)(k)= m K2xℓsin2kℓa+K2xℓ−1cos2 −2 , ontheinterparticledistance,ℓa=(n′ n)a,asthestruc- ℓ>0(cid:18) (cid:19) X − ture is periodic along x. In particular, for ℓ even they C(−)(k)= 4 ysin2 kℓa . read 2 m Kℓ 2 ℓ>0 2Q2 1 X x =2 y = , =0, Kℓ Kℓ a3 ℓ3 Yℓ | | while for ℓ odd they are given by Theeigenfrequenciesarefoundbyeliminatingtheparam- Kℓx = Qa32[ℓ22ℓ+2−χ2χ]52/2 , eetxehribǫik(tjs,±f)ofurrombrEanqcsh.e(2s1i)n-(2th2)e. nTehwe eBxrcililtoautiionnzsopneec,traunmd Q2 ℓ2 2χ2 their functional dependence on k is y = − , Kℓ a3 [ℓ2+χ2]5/2 Q2 3ℓχ = , Yℓ a3 [ℓ2+χ2]5/2 6 ν2+C(±)(k) C(±)(k) (ν2 C(±)(k) C(±)(k))2 ω (k)2 = t 1 − 2 +( 1)j t − 1 − 2 +B(k)2 , (23) j,± 2 − s 4 2.5 Indeed, one canobservethatthe transitionfroma lin- ω 2 ear to a zigzag configuration is characterized by a sym- metry breaking resulting in the increase of the unit cell ω0 1.5 by a factor of 2. It is combined with a transition from 1 a linear to a planar structure corresponding to the loss of rotational symmetry about the x-axis. Then, one can 0.5 identify the order parameter with the displacement of -00.5 0 0.5 the equilibrium position from the x-axis, while the con- (k a)/π trol parameter can be taken as the transverse frequency ν whenthe interparticle distance is fixed. Starting from FIG. 6: (color online) Branches of the excitation spectrum t of a zigzag structure for the modes on the x−y plane, as this educatedguessweapply Landautheoryto the tran- obtained from Eq. (23). The curves display the frequencies sition [24]. We focus on the situation in which the inter- ω2,+(k) (green solid), ω2,−(k) (blue dotted), ω1,+(k) (orange particle distance a is fixed, and study the crystal struc- dot-dashed), ω1,−(k) (red dashed), in units of ω0, as a func- turewhenthetransverseconfinementνt variesacrossthe tion of k, in units of π/a. The Brillouin zone is now half the criticalvalueν(c). Weexplicitlydeterminethecriticalex- t Brillouin zone of the linear chain due to the doubling of the ponent of the order parameter around the critical value, crystal periodicity. Here, νt =0.9νt(c). andfindthatitisinagreementwiththenumericalresults in [22]. with j = 1,2. The spectrum for the excitations on the x y plane is displayed in Fig. 6. − The soft mode We note that in the limit b 0 the branches of the → spectrum of the linear chain, Eqs. (11) and (12), are re- Let us now go back to the dispersion relation for the coveredfrom Eqs. (23). In fact, for b=0 we have B =0 transverse modes of the linear chain in Eq. (12). The and C1(±) = 2C2(∓), such that each solution has double structuraltransitiontakesplaceforthecriticalvalueν(c), degeneracy, with t Eq.(16),suchthatthefrequencyofthelowesttransverse mode of the linear chain vanishes, as shown in Fig. 7a. ω (k)2 =ω (k)2 =C(+)(k) 2,+ 2,− 1 The smallest transverse frequency ω is found at the b=0 b=0 b=0 ⊥ (cid:12) (cid:12) (cid:12) value of the wave vector k, at which the semipositive- (cid:12) (cid:12) (cid:12) and (cid:12) (cid:12) (cid:12) definite function ω1,+(k)2 =ω1,−(k)2 =νt2−C1(+)(k) b=0/2, F(ϕ)= N 1 sin2(jϕ) (cid:12) j3 (cid:12) j=1 which reproduce respectively Eqs. (11) an(cid:12)d (12) (note X that νt2−C1(±)(k)−C2(±)(k)<0). is maximum in the interval 0 ≤ ϕ ≤ π/2, as seen from Eq.(12)forka=[0,π]. Wefirstobservethat∂F/∂ϕ=0 atϕ=0,π/2. AsF(0)=0,atϕ=0thefunctionhasan LANDAU THEORY OF THE STRUCTURAL absolute minimum. The second-order derivative at ϕ = PHASE TRANSITION π/2 is negative, and one can simply prove analytically thatthispointisatleastarelativemaximum. Numerical If the ions are crystallized along a line, by lowering studies show that it is an absolute maximum, such that the transverse confinement νt the system will be led to thesmallesttransversefrequencyisfoundatwavevector a situation in which the linear chain gets unstable. In k π/a and takes the value 0 ≡ this regime, one observes experimentally a transition, in which the ions are crystallized on a plane, according to ω2 =ν2 ν(c) 2. (24) ⊥,min t − t a zigzag distribution ofparticles. In the literature it was conjectured that this is a second-order phase transition. This identifies the soft mode. The corresponding eigen- This conjecture is supported by the numerical results modesexhibitaperiodicdeformationofthechainatperi- in [22, 23]. odicity 2a,analogousto the zigzagstructure. We denote 7 3 the equilibrium positions of the chain, V = 4 V(l), l=1 2.5 where l labels the order. The zero order term at leading ω P 2 order in 1/N is [27] ω 0 1.5 1 Q2 1 V(0) = (N 1) γ ln2+ln(N)+O , 0.5 a − − N2 (cid:18) (cid:18) (cid:19)(cid:19) 0-1 -0.5 0 0.5 1 (26) a. (k a)/π where γ = 0.577216... is Euler’s constant. The first 3 order term vanishes as a result of the requirement that 2.5 ω we are looking for a minimum. Using the decomposition 2 ω into the eigenmodes of the linear chain, Eqs. (13)-(15), 0 1.5 thequadratictermoftheexpansionofpotential(2)(with 1 the use of (6-9) and (11-12)) takes the form 0.5 b. -00.5 (k a0)/π 0.5 V(2) = m2 ωk(k)2Θ(ks)2+β(k)(Ψky(s)2+Ψzk(s)2) , k>X0,s=±(cid:16) (cid:17) FIG. 7: (color online) Branches of the excitation spectrum (27) at (a.) νt = νtc+0+ (just above the critical value) and (b.) where ωk is given by Eq. (11), while νt = νtc +0− (just below the critical value). In (b.) the equilibrium structure is a zigzag, the periodicity is doubled with respect to the linear chain and the new Brillouin zone 2Q2 N 1 jka is halfed. The four branches of the spectrum are obtained at β(k)=ν2 2 sin2 (28) this point by “folding” the two branches of the linear chain t − ma3 j3 2 (cid:18) (cid:19)j=1 in (a.). The units and style codings in (a.) and (b.) are the X same as in Figs. 4 and 6, respectively. Here, ν(c) ≃2.05ω . t 0 and it coincides with ω2(k), Eq. (12), for ν >ν(c). The ⊥ t t third and fourth order terms, obtained by using this de- by b the amplitude of its oscillations, with b = b (ν ), 0 0 0 t composition, are presented in App. . such that the transverse oscillations along y of the ion j are described by the function Thelinearchainbecomesmechanicallyunstablewhen, by varying ν , the frequency of the mode with wave vec- t ysoft =( 1)jb /2. (25) tor k = π/a, Eq. (25), vanishes. Starting from this ob- j − 0 0 servation, we study the behaviour of the corresponding In the following we assume zero temperature and study (c) mode close to the instability point, when ν ν . For theequilibriumpositionofthecrystalwiththetransverse convenience, we denote by Ψy and Ψz the ztig≃zagtmodes frequency varying in the interval [νt(c) − δν,νt(c) + δν], of the linear chain along the0y and z0 direction, respec- thus on both sides of the critical point, whereby δν is tively, at wave vector k . Around the instability point 0 a small positive quantity. Following Landau theory, we these modes will be coupled significantly to other quasi- demonstrate that the zigzag mode of the linear chain, degenerate modes by the third and fourth order terms given by Eq. (25), is indeed the soft mode, driving the V(3) and V(4). These quasi-degenerate modes are long instability across the critical point, and we evaluate the wavelengths axial modes Θ at wave vectors δk, such δk critical exponents for some quantities of interest. that δk a 1, and short wavelength transverse modes | | ≪ Ψσ at wave vector k =k +δk′, with δk′ a 1. k0+δk′ 0 | | ≪ Equilibrium positions around the critical point At first order in the small parameter δk a 1, the | | ≪ partV(3),thatcontainsthesummandsofthethirdorder k0 In order to determine the behaviour at the critical termV(3) givingthecouplingofthe modeatk withthe 0 point,wefirstexpandEq.(2)tillthefourthorderaround other quasi degenerate modes, has the form 21 Q2 V(3) = ζ(3) δk Ψσ Θ(+)Ψσ(−)+Θ(−)Ψσ(+) +O(δk2a2), (29) k0 2√2 a3√N 0 δk δk δk δk δXk>0 σX=y,z (cid:16) (cid:17) where we adopted for convenience the notation Ψσ :=Ψσ . We note that Eq. (29) is of first order in δk. The δk k=k0+δk 8 part V(4) of the fourth order term V(4), which is relevant to the dynamics of the soft mode at k , involves only the k0 0 transverse modes that are close in k to k , and has the form 0 V(4) = A(Ψy2+Ψz2)2+12A Ψσ2 Ψσ(+)2+Ψσ(−)2 +4A Ψσ2 Ψσ′(+)2+Ψσ′(−)2 k0 0 0 0 δk δk 0 δk δk σX=y,z δXk>0h i σ=yX,z;σ′6=σ δXk>0(cid:20) (cid:21) +16AΨyΨz Ψy(+)Ψz(+)+Ψy(−)Ψz(−) +A(Ψy+Ψz) Ψσ(+),Ψσ′(−),Ψσ′′(−) +O(δk2a2),(30) 0 0 δk δk δk δk 0 0 F δk1 δk2 δk1+δk2 δXk>0h i (cid:16) (cid:17) whereAiscalculatedfromthecoefficientsofV(4) atk = FromEq.(32)onefinds̺¯=0forβ >0,whileforβ <0 j 0 0 k (j = 1,2,3,4), see App. , and it takes the form A = 0/N with mβ0 1/2 A ̺¯= N . (33) − 4 331 Q2 (cid:18) A (cid:19) = ζ(5) . (31) This is indeed a minimum if we ignoreterms inV with A 232 a5 eff non-zero δk. It will be shown in what follows that this The function in Eq. (30) contains a sum of products minimumisstablewithrespecttoadditionofsuchterms. F σ(±) of three amplitudes Ψ for δk =0, and it is of no im- We now demonstrate that Eq. (33) is actually the δk 6 portanceforthefollowingconsiderations. Thenumerical transverse displacement, giving the equilibrium trans- factors appearing in Eq. (30), multiplying each term of verse positions of the zigzag structure, by verifying that thesum,accountforallpossiblepermutationsoftheam- Eq.(33),togetherwithΨ¯(±) =0forδk >0,yieldsasta- δk plitudes Ψσ in each summand (see App. and Eq. (47) ble solution. To check stability the matrix of the second δk therein). The coupling between the transverse modes derivatives of V with respect to the various variables eff at k0 and the axial modes does not appear explicitly in shouldbecalculated. ThesecondderivativeofVeff,given Eq. (30), as it scales with (δk a)2 1, and it is hence in Eq. (32). with respect to ̺¯is positive, ≪ of higher order with respect to the coupling among the ∂2V transverse modes. Since the third order term, Eq. (29), eff = 2mβ >0. (34) scales with δk a, at zeroth order in the expansion in ∂̺2 − 0 (cid:12){̺,Ψδk}={̺¯,0} δk a and close to the instability, the effective potential (cid:12) | | Inordertoinvest(cid:12)igatethecouplingofthesoftmodewith describing the dynamics of the mode at k is given by (cid:12) 0 the modes with δk = 0, one can calculate the second (see Eqs. (27) and (30)) 6 derivatives of V with respect to Ψσ. We find eff 0 m V = β Ψy2+Ψz2 (32) eff 2 0 0 0 ∂2Veff = 0. +m2 βδkh Ψσδk(+i)2+Ψδσk(−)2 +V0(4) , ∂Ψσ0∂Ψδσk′(±)(cid:12)(cid:12){̺,Ψδk}={̺¯,0} (cid:12) δXk>0 σX=y,zh i This result shows that t(cid:12)he derivatives with respect to (cid:12) whereβ β(k δk). Wenowallowthetransversefre- Ψσ form a sub-block of the stability matrix that can be δk ≡ 0− 0 quencyν totakevaluesintheinterval[ν(c) δν,ν(c)+δν], diagonalized separately. All its eigenvalues are found to suchthattβ maytakeonsmallbutnegtati−vevalutes. We be positive. The other secondderivativesat these points δk first determine the amplitude of the zigzag mode k and read 0 owthtehenedrestmheoromwdientsheaattrheienscttaohbrerleevc.ticioFinnoisrtyΨt¯hoδyikf(s±tp)h,ueΨr¯pfrzδoek(±sqe)ueftonorcytβhδνekt(ce)<quni0o- ∂∂Ψ2zδVk(e±ff)2(cid:12)(cid:12)(cid:12){̺,Ψδk}={̺¯,0} = mβδk+8A̺¯2+16AΨ¯(z0325,) librium positions of the linear chain using Eq. (32), as- ∂2V (cid:12) suming that these give rise to a small displacement b ∂Ψy(e±ff)2(cid:12)(cid:12) = mβδk+8A̺¯2+16AΨ¯(y0326,) with respect to the equilibrium interparticle distance a, δk (cid:12){̺,Ψδk}={̺¯,0} (cid:12) b a. In particular, following our hypothesis that close ∂2V (cid:12) to≪the transition point the soft mode is unique, and it is eff (cid:12) = 0, (37) ∂Ψy(±)∂Ψz(∓)(cid:12) the zigzag mode, we consider the set of solutions where δk δk (cid:12){̺,Ψδk}={̺¯,0} (cid:12) Ψ¯(±) =0forδk >0,andintroducetheFourieramplitude ∂2V (cid:12) δk eff (cid:12) = 16AΨ¯yΨ¯z , (38) of the displacement in the transverse plane ̺¯= √Nb/2, ∂Ψy(±)∂Ψz(±)(cid:12) 0 0 as indicated from Eq. (13), such that δk δk (cid:12){̺,Ψδk}={̺¯,0} (cid:12) wherewehaveu(cid:12)sedthat0>β >β . Thisresultshows (cid:12) δk 0 ̺¯= Ψ¯y 2+ Ψ¯z 2. thatthemodesΨy(±) andΨz(±) arecoupledinpairs. All 0 0 δk δk q (cid:0) (cid:1) (cid:0) (cid:1) 9 contributions resulting of differentiation of the function where = 112 ζ(3)/[93 ζ(5)], and whose second deriva- C in Eq. (30) vanish. The stability matrix splits into tive with respect to ν is clearly discontinuous at the t F 2 2 blocks that can be diagonalized separately. Using critical point. This result is consistent with the result × Eq. (33) and β < β one finds that the eigenvalues of presented in [23], where a discontinuity in the second 0 δk eachblockarem(β 2β )andm(β 6β ),henceboth derivative ofthe groundstate energywith respectto the δk 0 δk 0 − − positive. Therefore, a gap opens between the soft mode particles density was found. frequency and the frequency of the modes at δk = 0 6 in the vicinity of the transition point. Therefore, the instability is driven by the soft modes with wave vector k , determining the order of the zigzag phase [34, 35]. 0 Discussion Behaviour at the critical point Usingsymmetryargumentswehavedemonstratedthat thetransitionfromalinearchaintoazigzagstructure,in a systemofanisotropicallyconfinedcharges,is a second- From Eq. (32), using the results of the previous sec- orderphasetransition,whoseorderparameteris the dis- tion we can now write the effective potential for the soft placement from the trap axis. This theory has been de- modes, which reads veloped in the thermodynamic limit, fixing the interpar- 2 ticle distance a as the number of ions was let to infin- Vsoft = (Ψy)2+(Ψz)2 +A (Ψy)2+(Ψz)2 ,(39) V 0 0 0 0 ity. In this limit, we found that the soft modes are the (cid:16) (cid:17) (cid:16) (cid:17) zigzag modes of the linear chain, whose periodicity is where A is given by Eq. (31) and equal to twice the interparticle distance a. The insta- bility is thus driven by these modes as the transverse = mβ = 1m ν2 ν(c)2 . (40) potential is changed across the critical value ν(c). V 2 0 2 t − t t (cid:16) (cid:17) TheseconsiderationsarestrictlyvalidforN ,but →∞ Here, we have used that β = ω2 , which in turn is can still be useful for finite systems, and in particular 0 ⊥,min given by Eq. (24). Hence, for > 0 the potential Vsoft whenthe ionsareconfinedina trap,whichprovidesalso has a single minimum with ΨσV=0, and the linear chain axialharmonicconfinement. While detailedquantitative 0 is the ground state structure, while for < 0 the po- predictions can be only made by accurately evaluating tential landscape has the characteristic foVrm of a Mexi- thefinite-sizecorrections,wecanstillmakesomereason- can hat with degenerate zigzag ground states at differ- able conjectures, based on previous results in the litera- ent angles around the symmetry axis. Indeed, while the ture and on our theory. Inside a harmonic trapping po- order parameter ̺¯ is fixed by condition (33), the ratio tential,the interparticledistance betweenthe ionsvaries Ψ¯y/Ψ¯z is arbitrary. The system hence possesses ”Gold- along the chain and it is minimal at the center. Numer- 0 0 stonemodes”atzerofrequency,whichareaconsequence ical results, based on molecular dynamics simulations, of the symmetry by rotations around the trap axis. showed that in this case the zigzag structure appears at The transverse displacement from the trap is given the center of the chain where the density is highest [22]. from Eq. (33) by using Eq. (25) and the relation b/2 = Analyticalstudiesfoundthattheshortwavelengthmodes ̺¯/√N, which links the displacement in real space with are characterized by largest displacements at the chain itsFourierdecomposition. Hence,forν <ν(c) thetrans- center,whiletheionsattheedgealmostdonotmove[29]. t t In this case, hence, we can still identify the zigzag mode verse displacement from the trap depends on ν as t of the ion chain with the soft mode. In the presence of axial confinement, however, both the transverse as well b=¯b ν(c) ν , (41) t − t astheaxialequilibriumpointswillchange. Inparticular, q when going to the zigzag structure the axial density of with ¯b = 2mν(c)/ . This behaviour is in agreement ions in the center will increase. Close to the transition t A point, one finds that the axial corrections to the linear with the nqumerical results in [22]. chainpositionsaremuchsmallerthanthetransversedis- From Eq. (41) we evaluate the difference between the placements from the trap axis as this is a quantity that groundstateenergyofthe linearandofthezigzagstruc- followstheorderparameter. Thisisalsoconfirmedbythe ture. Considering the energy per particle, from Eq. (39) analysis made for the simple case of three ions in Sec. , we find where close to the critical value of the aspect ratio the Vsoft(ν ν(c)−) Vsoft(ν ν(c)+) transverse displacement varies faster than the axial one. ∆E = t → t − t → t Therefore, we expect that our theory will still provide N 1 reasonable predictions close to the critical point, also in =−2mCa2(νt−νt(c))2 (42) presence of axial confinement. 10 Optical Physics at the Smithsonian Center for Astro- 104 physics and Harvard Department of Physics. S.F. ac- ν(c) (MHz) t knowledges the US-Israel Binational Science Foundation 102 Linear (BSF),TheIsraelScienceFoundation(ISF),theMinerva Zigzag CenterofNonlinearPhysicsofComplexSystems,andthe 100 fund for Promotion of Research at the Technion. G.M. thanks Herbert Walther, who motivated this work. 101-20-2 10-1 100 101 102 a (µm) EXPANSION ABOUT THE EQUILIBRIUM FIG. 8: Phase diagram close to the linear-zigzag transition POSISIONS OF THE LINEAR CHAIN in the thermodynamic limit, for 40Ca+ ions. The horizontal axisistheinterparticle spacingain µm,thevertical axisthe In this Appendix we evaluate the higher order terms corresponding critical frequency ν(c) in MHz. This graphic t of the expansion of the potential in Eq. (2) about the does not report further curves in the left region, giving the equilibrium position of the linear chain. For this pur- transition to more complex structures. A detailed study of pose we rewrite the interparticle distance as r r = the transitions to these structures can be found in [19, 20] | i − j| and [23]. Aij +τij +ǫij with p A =(i j)2a2, ij − CONCLUSIONS AND OUTLOOK τ =2a(i j)(q q ), ij i j − − ǫ =(q q )2+δ2 , ij i− j ij The structural phase transition from a linear chain to a zigzag configuration,in a system composed of trapped and δi2j = (yi−yj)2+(zi−zj)2 and we have used xj = singly-charged particles, is a second order phase tran- ja+qj. We now expand in the parameters ǫij and τij, sition. Using a mean field approach we have derived assuming that they are small with respect to Aij, i.e., a classical model, describing the system at the critical tothe axialequilibriumdistancesbetweenthe ionswhen point and its vicinity. Our theory is analytical and its the chainis stable. We willchecklater forconsistencyof predictions agree with the numerical simulations of [22] this assumption. We can write and [23]. Q2 The corresponding phase diagram is shown in Fig. 8, V(l) = W(l) , (43) 2 ij it shows the regions of stability of the linear chain as a i,j6=i X functionofthe interparticlespacinga andthe transverse frequencyν . Thephasediagramisevaluatedinthether- with t modynamic limit, corresponding in keeping a fixed as N 1 and the chain length go to infinity. The analysis is valid W(0) = , ij i j a for T = 0, where long-range order in one-dimensional | − | σ structuresexists. The quantumstatistics ofthe particles W(1) = ij (q q ), ij −(i j)2a2 i− j at these densities seem irrelevant even at these ultralow − temperatures since the interaction energy at all stages W(2) = 1 (2(q q )2 δ2), muchlargerthanthekineticenergy,andtheparticlescan ij 2i j 3a3 i− j − ij | − | be considered distinguishable at all effects [36]. On the W(3) = σij (q q ) 3δ2 2(q q )2 , otherhand,atthecriticalpoint,wherelargefluctuations ij 2i j 4a4 i− j ij − i− j | − | of the transverse motion classically occur, quantum ef- 1 3 (cid:2) (cid:3) W(4) = δ4 +(q q )4 3δ2(q q )2 , fects mayberelevantandcouldbe inprincipleobserved. ij i j 5a5 8 ij i− j − ij i− j The authors thank Efrat Shimshoni, Grigory As- | − | (cid:16) (cid:17) trakharchik, Eugene Demler, Bert Halperin, and Tom- wherewehaveintroducedσij =(i j)/i j . Wenotice − | − | masoRoscildeforstimulatingdiscussionsandusefulcom- that ments. ThisworkwaspartlysupportedbytheEuropean Q2 q q Commission(EMALI,MRTN-CT-2006-035369;SCALA, V(1) = σ i− j =0 (44) ContractNo.015714;QOQIP,MOIF-CT-2005-8688),by −2a2 ij(i j)2 i j6=i − XX the Spanish Ministerio de Educaci´on y Ciencia (Con- solider Ingenio 2010 ”QOIT”, CSD2006-00019; QLIQS, as one can easily verify by using the definition of σ . ij FIS2005-08257; Ramon-y-Cajal individual fellowship), This is satisfied also in the ion chain in presence of an andbythe NationalScience Foundationthrougha grant axial trapping potential, since V(1) = 0 determines the to the Institute for Theoretical Atomic, Molecular, and equilibrium positions.