BERRY PHASE AND THE SYMMETRY OF THE VIBRONIC GROUND STATE IN DYNAMICAL JAHN-TELLER SYSTEMS NICOLA MANINIAND PAOLODE LOS RIOS European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble C´edex, 9 France Institut de Physique Th´eorique, Universit´e de Fribourg, 1700-CH Fribourg, 9 Switzerland 9 1 Due to the frequent presence of aBerryphase, inmost cases of dynamical Jahn- Tellersystemsthesymmetryofthegroundstateisthesameasthatoftheelectronic n state. However, the H⊗h icosahedral case, relevant for the physics of fullerene a ions, provides a first example of linear coupling leading, at strong coupling, to J a change in symmetry of the ground state to a totally symmetric nondegenerate 2 state. Wegeneralizethisobservationandshowthroughdetailedexamplesthatthe 1 absence of a Berry phase can, but does not necessarily, lead to a nondegenerate groundstate. 1 v 7 The traditional field of degenerate electron-lattice interactions (Jahn- 9 Teller effect) in molecules and impurity centers in solids1,2 has attracted new 0 interest in recent years, excited by the realization of new systems which call 1 for a revision of a number of commonly accepted beliefs. A whole range of 0 icosahedralmolecular systems including C ions and some higher fullerenes, 9 60 9 thanks to the rich structure of the symmetry group, are characterized by up / to fivefold-degenerate representations of the electronic and vibrational states t a of the isolated molecule/ion. Novel Jahn-Teller (JT) systems have therefore m been considered theoretically,2,3,4 disclosing intriguing features,4,5,6,7,8 often - related to the rˆole of a Berry phase9 in the coupled dynamics. d As it is well known, the molecular symmetry, reduced by the JT distor- n o tion with the splitting of the electronic-state degeneracy, is restored when c the coherent tunneling between equivalent distortions is considered, in the : dynamical Jahn-Teller (DJT) effect. In this context it was commonly ac- v i cepted an empirical “symmetry conservation rule”, sometimes referred to as X “Ham’s theorem”, stating that the symmetry of the vibronic DJT ground r state, at all coupling strengths, remains the same as that of the electronic a multiplet prior to coupling:2 all linear JT systems knowntill a few yearsago, for single-electron occupancy, systematically satisfy this empiric rule. It was understood recently that this phenomenon, not automatically implied by the DJT physics, is in reality a fingerprint of a Berry phase9 in the entangled electronic-phononic dynamics.4,7,10 Consequently, this geometrical phase ap- peared as a universal feature of the DJT systems. In this context, it came unexpected the discovery of the first linear JT erice98: submitted to World Scientific on February 1, 2008 1 system showing a nondegenerate ground state in the strong-coupling limit.7,8 This result was demonstrated for the model that in spherical symmetry is indicated as D(2) ⊗d(2), where electrons of angular momentum L = 2 inter- act with vibrations also belonging to an l = 2 representation. This system is relevant to the physics of fullerene ions C+, where the 5-fold degenerate 60 electronicstatehasH icosahedrallabelandthe quadrupolardistortionscor- u respond to some of the h modes.7 It has been shown by different methods g and independent groups that, for increasing coupling, a nondegenerate state in the vibronic spectrum moves down, to cross the 5-fold ground state at some finite value of the coupling parameter, thus becoming the ground state at strong coupling .7,8 This phenomenon is related to the absence of a Berry phase entanglement in the coupled dynamics.7 TherˆolegenerallyattributedtotheBerryphaseisthereforetoguarantee a “symmetry conservation rule” for the ground state from weak to strong coupling of DJT systems. The absence of this geometrical phase allows the strong-coupling ground state to become the “natural” nondegenerate totally symmetrical representation that a naive picture, ignoring this geometrical phase, would predict in all cases. In this work we reconsider in detail the connectionbetweenthe symmetry/degeneracyofthe vibronicgroundstate of a large class of DJT systems, and the presence/absence of a Berry phase in thecoupleddynamics,findingthattherelationsketchedabovedoesnotapply automatically to all cases. In the general formalism of the JT effect, a degenerate electronic state correspondingto arepresentationΓ ofthe symmetrygroupG ofthe molecule caninteractwiththevibrationalmodescorrespondingtorepresentations{Λ} contained in the symmetric part of the direct product Γ⊗Γ (excluding the identical representationwhich is trivial). In the case where exactly one mode of each symmetry label Λ, of frequency ω and coordinates q , interacts Λ Λi linearly with strength g with the |Γ|-fold degenerate electronic level (with a Λ fermion operator c ), the Hamiltonian may be written: Γk |Λ| 1 H = 2X¯hωΛX(p2Λi+qΛ2i)+He−v , (1) Λ i=1 with |Λ| |Γ| 1 He−v = 2XgΛ¯hωΛX X qΛic†ΓjcΓkhΛi|ΓjΓki, (2) Λ i=1j,k=1 where hΛi|ΓjΓki are the Clebsch-Gordan coefficients for the group G.11 In erice98: submitted to World Scientific on February 1, 2008 2 Eq. (1) we choose the real representation for the vibrational degrees of free- dom, and a second-quantized notation for the electrons. Inthegeneralcaseofarbitraryfrequenciesω andcouplingsg ,thepoint Λ Λ group symmetry G is reflected in the JTM, constituted of isolated minima, separated by saddle points. However, the continuous JTM of the special equal-coupling equal-frequencies case is invariant for transformations in the group SO(|Γ|) of the electronic manifold. Indeed, the whole problem reduces to a single-mode JT coupling between two representations of that group of |Γ|-dimensional rotations.12,13 In such a case, it is well known14 that the set of minima of the Born-Oppenheimer (BO) potential, corresponding to the most energetically-favorable classical distortions, constitute a continuous manifold,referredtoasJahn-Tellermanifold(JTM).TheJTcouplinginduces an adiabatic mapping of the vibrationalspace into the electronic space. Here we only sketchthis mapping, whichis describedin greaterdetailelsewhere.15 In the traditional BO scheme there are assumed much larger separations betweenconsecutiveelectroniclevelsthanthetypicalvibrationalenergiesh¯ω. In a JT problem, each electronic eigenvector |ψ i of the coupling matrix (2), ξ of eigenvalue λ , generates a BO potential sheet V (~q). At strong coupling ξ ξ g, the separation of the potential sheets becomes so large that the adiabatic motioncanbe safely assumedto alwaysfollowthe lowestBO potentialsheet, while virtual electronic excitations may be treated as a small correction. On the other side, due to time-reversal invariance of H, the space of all possible (normalized) electronic eigenstates can be representedby an (hyper- )sphere in the |Γ|-dimensional real space (see Fig. 1). The BO dynamics realizes an adiabatic mapping of the vibrational space into this electronic sphere:14 everypoint~q onthe JTM(in the vibrationalspace)is associatedto theelectronicwavefunction|ψ (~q)i, correspondingtothe lowesteigenvalue min λ of the interaction matrix. min Thisadiabaticmappingistwo-valued,sinceoppositepoints±|ψ (~q)ion min theelectronicspheregivethesameJTstabilizationenergy,thuscorresponding tothesameoptimaldistortionontheJTM.Thisidentificationoftheantipodal points through the mapping is the mechanism allowing the JTM to have a different(topology)withrespecttotheelectronicsphere. Thelatterisalways simply connected, i.e. any closed path on it can be smoothly contracted to a single point. The JTM, instead, may well be multiply connected, i.e. it can have intrinsic “holes” in its topology, related to the nontrivial class of those loops mapped on a path going from a point to its antipode on the electronic sphere, such as π in Fig. 1. This electronic sign change is a case of Berry 2 phase.9 This geometric phase acts as a boundary condition for the quantization erice98: submitted to World Scientific on February 1, 2008 3 π 1 A’ π 2 A ∆ 0.6 Figure 1. A sketch of the electronic sphere. The picture individuates the two classes of paths mapping onto closed loops in the JTM: paths of the type π1 may be contracted continuouslytoasinglepoint,whilethoseoftypeπ2 involveasignchange(fromAtoA’) oftheelectronicstate(aBerryphase). of the vibrational motion. As a consequence, the motion on the JTM is constrained by special selection rules. For example the JTM of the simple E ⊗ e system is a circle: the low-energy vibronic spectrum is indeed a j2 spectrum as for a circular rotor, but the Berry phase implies j =±1,±3,..., 2 2 instead of j = 0,±1,±2,... as for an ordinary quantum rotor.2,16 Similarly, the JTMofthe T⊗h(i.e.D(1)⊗d(2), inthe sphericallanguage)is equivalent to a sphere,4,17 but out of all the states, labeled by J,M, of a particle on a sphere, the Berry phase retains only the odd-J ones.2,4,17 Note in particular that in these examples the presence of a Berry phase rules out the “natural” nondegenerategroundstate,andenforces,tothestrong-couplingDJTground erice98: submitted to World Scientific on February 1, 2008 4 state, the same original symmetry Γ of the degenerate electronic state. The Berry phase, though not automatically implied by linear JT Hamil- tonians (1), is indeed a very common feature. The double-valuedness of the adiabatic mapping described above is unavoidable. For the Berry-phase–free cases, the mechanism leading to equivalence of the paths in the class 1 and 2 needs to coexist with it. As demonstrated in earlier work,7,15,18 the solution of the riddle is provided by a point ~q on the JTM where the mapping is de- d generate,i.e.itlinks~q notjusttoapairofoppositepoints±|ψ (~q )ionthe d min d electronic sphere, but to the whole circle (such as, for example, ∆ in Fig. 1) oflinearcombinationscosθ |ψ (~q )i+sinθ |ψ (~q )ioftwodegenerateorthog- 1 d 2 d onalelectroniceigenstates. Where suchapointispresent,itallowstodeform smoothly any loop of class 2 on the JTM, until its image on the electronic sphere becomes half this circle, thus shrinks to the single point ~q . All loops d are therefore contractable, thus equivalent to one another and, therefore, the JTM is simply connected. No Berry phase is possible in such a case.15,18 Suchatangency pointisthe originofthe inversionofthe low-lyinglevels in the H ⊗ h JT problem,7,15 leading to a nondegenerate ground state at strong coupling. Similar tangential points were demonstrated15,18 in other spherically symmetric linear models, the D(L) ⊗ d(L), with L = 2,4,6,.... AllthesesystemsarethereforeBerry-phasefree,with,inparticular,astrong- couplingnon-degeneratevibronicgroundstate. Anumericaltestconfirmsthis result in the D(4) ⊗d(4) case. On the contrary, these tangencies are absent in most DJT cases (E ⊗e, T ⊗h, D(2) ⊗d(4), ...), whence the Berry phase, whence the degenerate ground state at strong coupling. The systemsD(L)⊗d(l), withL>l,areremarkableinhavingatangency point,thusnoBerryphaseastheprecedingexample,butnosymmetrychange of the ground state, which remains degenerate to all couplings.15 This case shouldbe keptasa warningagainstthe simplistic equation: absenceofBerry phase = nondegenerate strong-coupling ground state. Wemoveonnowtotheinvestigationoftherelationsbetweengroundstate symmetry,Berryphasesandtangenciesofpotentialsheetsinthemoregeneral case of H⊗(2h⊕g) Jahn-Teller coupling in icosahedralsymmetry. The two h and the g modes can be classified according to their spherical parentage d(2) →h [2] d(4) →h ⊕g . (3) [4] The existence of two different couplings g and g to modes h reflects h[2] h[4] the fact that the icosahedral group is not simply reducible:19 two indepen- dent sets of Clebsch-Gordan coefficients for the coupling of h and h to h are necessary.8,11,20 In the special case when ω =ω =ω and g =g =g h[4] g 4 h[4] g 4 erice98: submitted to World Scientific on February 1, 2008 5 the SO(3) symmetry of the linear problem is restored, and it can be labeled accordingly: D(2) ⊗(d(2) ⊕d(4))). In the limit g = 0 we recover the Berry- 4 phase–free D(2)⊗d(2) model discussed above. In the completely equal-coupling equal-frequencies limiting case ω =ω 4 2 andg =g ,asanticipatedaboveforthegeneralcase,thesymmetryrisesfur- 4 2 ther to SO(5):13 the modelmaybe describedas[1,0] ⊗ [2,0]inthe notation ofSO(5)representations. Fortheequal-couplingcase,thepresenceofaBerry phase has been explicitly demonstrated,7 together with its consequences for the selection rules on the levels: it favors in the low-energy end of the spec- trum [k,0] levels with odd k. In particular, it was verified that the ground state remains 5-folddegenerate([1,0] in SO(5) notation, i.e. D(2) as a SO(3) representation), and the first excited is a 30-fold degenerate [3,0] level. Inthegeneralcaseg 6=g ,thesymmetryreducestoSO(3),thusthelarge 2 4 SO(5) representations split into their spherical components. In sweeping the value of g from g down to 0, the system passes smoothly from a regular 4 2 Berry-phase–relateddegenerate ground state to the D(2)⊗d(2) Berry-phase– free nondegenerate groundstate (for large enough g ). In this final situation, 2 the degenerate D(2) state takes the rˆole of the lowestexcited state, separated by a finite energy gap from the nondegenerate D(0) ground state.7,8 Thus, a level crossing takes place between the low-lying levels, at some intermediate valueofg : wecandefineacrossovervaluegc (dependentong )forwhichthe 4 4 2 ground-state symmetry changes. At strong coupling, the energy gap E[L = 2]−E[L=0]=c /g2+O(g−4)forg =0andE[L=2]−E[L=0]=−c /g2+ 2 2 2 4 4 4 O(g−4) for g = 0, where c and c are positive constants depending on ω 4 2 2 4 2 andω . Thus,thecrossovercurve(g ,gc(g ))shouldgetasymptoticallyclose 4 2 4 2 tothe straightlineg =g (c /c )1/2 intheplaneofthe couplingparameters. 4 2 4 2 These considerations, as well as some exact diagonalizations on a trun- catedbasis,permittodrawthequalitativezero-temperature“phasediagram” representedinFig.2. Itstrikesforcontainingawholeregion(0<g <gc(g )) 4 4 2 where a nondegenerate L = 0 ground state coexists with the presence of a Berry phase. This example should stand as a warning against the simplistic equation: Berryphase=degeneratestrong-couplinggroundstateofthesame symmetry as the non-interacting electronic state. We conclude, accordingly, that the presence of a Berry phase in many-mode DJT systems is not a suf- ficient condition for the degeneracy of the ground state. Indeed, even if the overallsystemhas aphase entanglement,the absenceofa Berryphase inone ofthesingle-modecouplingsallowsforanon-degenerategroundstateinsome regions of the coupling-parameters space. At this point it is necessary to reconcile the gradual, smooth lowering of the nondegenerate state as g /g is reduced from equal coupling towards 4 2 erice98: submitted to World Scientific on February 1, 2008 6 1155 degenerate 1100 4 g not 55 degenerate 00 00 55 1100 1155 g 2 0.6 Figure 2. The zero-temperature “phase diagram” of the D(2)⊗(d(2)⊕d(4)) JT system inthespaceofthecouplingparameters g2 andg4,forfixedfrequenciesω2 andω4. Atthe solidlineg4=g4c theL=0andL=2groundstates become(accidentally) degenerate. zero,withtheabruptdisappearanceoftheBerryphase(whichisatopological effect, intrinsically non-perturbative)for g =0. The originof the nondegen- 4 erate state is to be traced back to the 30-fold degenerate first-excited state ([3,0] according to SO(5)) of the equal-coupling “hypersymmetrical” spec- trum which splits into its L = 0,3,4,6 components (SO(3) representations) as soon as g 6= g . In particular, this L = 0 fragment is the lowest when 4 2 g /g < 1. For small enough g /g , this nondegenerate state has the oppor- 4 2 4 2 tunity to localize as much as possible in the potential well in the d(2) vibron space (corresponding to the JTM in the space of d(2) vibrations), eventu- ally crossingdownbelow the L=2 groundstate, to become itself the ground erice98: submitted to World Scientific on February 1, 2008 7 state. Eveninthisregion,however,theBerry-phaseprescriptionintheSO(5) language is respected, since the L = 0 state is indeed a fragment of an odd ([3,0]) – Berry-phase allowed – level: the ground state still fulfills the par- ity constraint imposed to the low-energy SO(5) representations by the Berry phase in the global space. Note, incidentally, that, although the D(2)⊗d(2) problem is only SO(3)- symmetric,its JTMhasSO(5)symmetry. Therefore,inthe limitofinfinitely large g and vanishing g , where the motion is essentially restricted to the 2 4 JTM in the d(2) vibration space, that same nondegenerate ground state may also be classifiedas an [0,0]state for the symmetry groupof the JTM, where it complies therefore with the absence of Berry phase. As a first remark, we note that our treatment calls for a revision of the customary association of Berry’s phase to a breakdown of the BO approxi- mation. Indeed, the geometrical phase originates at the conical intersections of the lowest two BO sheets. At strong coupling, such points lie at high en- ergy and the system explores them with extremely small probability. On the contrary, here we relate the absence of the geometrical phase to tangential contacts of the adiabatic sheets, on the JTM, thus affecting low-potential re- gions which the system occupies currently. Thus, in these systems, it is not theBerryphasewhichisconnectedtoabreakdownoftheBOapproximation, but its absence. Our analysis considers for simplicity spherical DJT models: however, it can be extended to molecular point groups. For example, the Berry phase and ground-state symmetry switch of the H ⊗ h +h are completely (cid:0) [2] [4](cid:1) analogous to those of D(2)⊗(d(2)⊕d(4)) described above.8 Also, we assume a linear JT coupling scheme (Hamiltonian (1)), which is thelessrealistic,thestrongertheJTdistortion. Theintroductionofquadratic andhigher-ordercouplingshasusuallyeffectssimilartothoseproducedbyun- equal linear couplings and/or frequencies in T ⊗(e+h) in cubic symmetry,21 i.e. of “warping” the JTM, reducing its symmetry. Yet, the connectedness properties are topological properties, thus robust against warping, as long as itcanbetreatedasaperturbations. Toquotethesimplestexample,theintro- duction of quadraticterms in the e⊗E Hamiltonian22 does notsubstantially changethepictureasfarastheBerryphaseandthesymmetry/degeneracyof the groundstate are concerned. In fact, even at strong JT coupling, the tun- nelingamongratherdeepisolatedminimaisaffectedbytheelectronicphase,2 and, as a result, the lowest tunnel-split state retains the same symmetry and degeneracy as in the purely linear-coupling case. Of course, if the quadratic and higher-order couplings dominate over the linear term, new conical inter- sections may appear, thus affecting the Berry phase and, consequently the erice98: submitted to World Scientific on February 1, 2008 8 ground-state symmetry.23 Insummary,the standardrˆoleofthe Berryphaseistoguaranteea“sym- metry conservation rule” for the ground state from weak to strong coupling of linear DJT systems. Here, we propose two counterexamples to this simple pattern: (i) a wholefamily, ofBerry-phase–freedynamicalJT systems witha degenerate ground state at all couplings, the D(L)⊗d(l) models, with l < L; and (ii) the case of many modes coupled at the same time to an electronic state, some with a Berry phase entanglement, and some without it, in the region where the coupling to the seconds prevail, the strong-coupling ground state can switch to nondegenerate, as we illustrate for D(2) ⊗(d(2) ⊕d(4)). This second point, in particular, for those cases, such as positive fullerene ions, where Berry-phase–free modes are present, underlines the relevance of the actual values of the coupling strengths between degenerate electrons and vibrations,whichonlypermittomakeaprevisionabouttheactualsymmetry ofthevibronicgroundstate. Inthisperspective,theexperimentalorab-initio determinationofthe detailedvalues ofsuchcouplingsis ofthe utmostimpor- tance for this class ofsystems. Finally,the rˆoleof the Berryphase being that oforderingthestrong-couplingspectrum,itisconceivableasystemwherethe geometric phase enforces a non-totally symmetrical vibronic state of symme- try other than Γ, thatofthe originalelectronicstate: further investigationod the icosahedral JT zoology may find a realization of this possibility. Acknowledgement We thank Arnout Ceulemans, Brian R. Judd, Erio Tosatti, and Lu Yu for useful discussions. References 1. R. Englman, The Jahn Teller Effect in Molecules and Crystals (Wiley, London, 1972). 2. I.B.BersukerandV.Z.Polinger,Vibronic Interactions in Molecules and Crystals (Springer Verlag, Berlin, 1989). 3. J. Ihm, Phys. Rev. B 49, 10726 (1994). 4. A. Auerbach,N. Manini, andE.Tosatti, Phys.Rev. B49, 12998(1994). 5. C. A. Mead, Rev. Mod. Phys. 64, 51 (1992). 6. Geometric Phases In Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989). 7. 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Manini, in Recent Advances in the Chemistry and Physics of Fullerenes and Related Materials: Volume 5, edited by K. M. Kadish and R. S. Ruoff (The Electrochemical Society, Pennington, NJ, 1997), p. 468. 19. M. Hamermesh, Group Theory and its applications to physical problems (Addison-Wesley, London, 1962). 20. In the literature, different choices of these Clebsch-Gordan coefficients are available, according to the combination of h and h being taken [2] [4] as basis set. 21. M. C. M. O’Brien, Phys. Rev. 187, 407 (1969). 22. F. S. Ham, Phys. Rev. Lett. 58, 725 (1987). 23. I. B. Bersuker, Proceedings of the XIV International Symposium on Electron-Phonon Dynamics and Jahn-Teller Effect (Erice, Italy, July 7- 13, 1998). erice98: submitted to World Scientific on February 1, 2008 10