Optimal Topological Test for Degeneracies of Real Hamiltonians Niklas Johansson and Erik Sjo¨qvist ∗ Department of Quantum Chemistry, Uppsala University, Box 518, S-751 20 Sweden We consider adiabatic transport of eigenstates of real Hamiltonians around loops in parameter space. It is demonstrated that loops that map to nontrivial loops in the space of eigenbases must encircle degeneracies. Examples from Jahn-Teller theory are presented to illustrate the test. We show furthermore that the proposed test is optimal. PACSnumbers: 03.65.Vf,31.30.Gs,41.20.Jb,41.20.Cv Sign reversal of real electronic eigenfunctions when in the basis i of the n dimensional Hilbert space . {| i} H 4 continuously transported around a degeneracy was dis- We suppose that H(Q) is real, symmetric, and contin- 0 covered by Herzberg and Longuet-Higgins [1] and was uous for each Q = (Q1,...,Qd) in parameter space , 0 later used to construct a topological test for conical in- whichwe assumeto be a simply connectedsubset ofRQd. 2 tersections[2]. Suchintersectionsareabundantinmolec- TheeigenvectorsofH(Q)canalwaysbechosenreal. We n ular systems [3] and are important because they signal call the space of real vectors , which through the real a a breakdown of the Born-Oppenheimer approximation. expansion coefficients in the baHsis i can be identified J The topological test in [2] has been used to detect coni- with Rn. The set real of normali{z|edi}real vectors is the 9 calintersectionsinLiNaKandozone[4]. Thesignchange sphere Sn 1 in RnN. − 2 of the electronic eigenfunctions gives rise to the molec- Consider a simply connected surface S in , bounded Q v ular Aharonov-Bohm effect [5], which has recently been by the loop Γ. Longuet-Higgins’ theorem asserts that 8 experimentally tested [6] and theoretically investigated if a certain eigenvector ψ (Q) of H(Q) changes sign i 1 [7]. The structure of adiabatic wave functions and the when continuously transp|ortediaround Γ, then there is 0 sign change pattern in the vicinity of degeneracies have apointonS where ψ (Q) becomesdegeneratewithan- 0 | i i 1 also been analyzed in the context of quantum billiards. other state. We note that the theorem implies that if 3 Thebehavioroftherealwavefunctionsforsuchsystems, H(Q) is nondegenerate on S, then ψi(Q) represent 0 studied by analog experiments on microwave resonators two continuous functions from S to S±n| 1. i − h/ [8], has been interpreted in terms of both the standard As a first step towards a generalization of Longuet- p [9] and the off-diagonal [10] geometric phases. The mi- Higgins’ theorem we consider two-levelsystems. The set - crowave resonator experiments have motivated general of normalized vectors is then the circle S1. We nt theoretical treatments concerning both the concomitant provethatifaneigenvecNtoreraolfarealtwo-levelsystemrep- a geometric phases and structure of the wave functions resents a nontrivial loop in S1 when continuously trans- u q [9, 10, 11]. ported along a loop Γ, then Γ must encircle a degener- : InthisLetterwestudytheeigenvectorsofarealmatrix acy. Theproofisbyreductio ad absurdum. Supposethat v Hamiltonianonaloopinparameterspace. Weshowthat H(Q) is nondegenerate on the surface S bounded by Γ. i X the behavior of the eigenvectors may imply the presence A real eigenvector (Q) then represents a continuous r of a degeneracy even if none of the eigenvectors changes function F from S|t±o reial = S1. Since Γ is trivial (S a N sign around the loop. This result is a generalization of being simply connected), so is the loop traced out by Longuet-Higgins’ topologicaltest for intersections [2]. It (Q) . |± i isalsoproventhatthegeneralizedtestexhaustsalltopo- As an illustration, let us consider the coupling matrix logical information associated with the behavior of the Hamiltonian of the E ǫ Jahn-Teller system [12] ⊗ eigenvectors,concerning the presence of degeneracies. H(ρ,θ) We beginby the followingtopologicalfact. LetX and 1 2 1 2 Y be topological spaces. If Γ is a trivial loop in X, and kρcosθ+ gρ cos2θ kρsinθ gρ sin2θ = 12 2 − 21 2 , if F : X Y is continuous, then F(Γ) is a trivial loop (cid:18) kρsinθ gρ sin2θ kρcosθ gρ cos2θ (cid:19) 2 2 → − − − in Y. (1) To provethis, note that if G is a homotopy betweenΓ and a point x0 X, then F G is a homotopy between where ρ and θ are polar coordinates of parameter space F(Γ) and F(x0)∈ Y. ◦ = R2, and k and g are the linear and quadratic cou- ∈ Q pling strengths, respectively. In this system there are NowletH(Q)beann nmatrixHamiltonian,written × four degeneracies: one at the origin ρ = 0 and three at ρ = 2k/g and θ = π/3, π, and 5π/3. Continu- ous transport of an eigenvector around any single de- ∗Electronicaddress: [email protected] generacy produces a sign change [12]. Longuet-Higgins’ 2 test applied to such a loop would thus imply a de- twice. The fundamental group of SO(n 3) is the two- generacy. Consider instead a circular loop Γ given by elementgroupZ2. Inordertoapplythet≥estinthislatter ρ 2k/g and θ [0,2π], encircling all four degenera- situation, we need to know how to determine whether a ≫ ∈ cies. The eigenvectors (θ) sinθ 1 +cosθ 2 and loop in SO(n) with n 3 is trivial or not. In principle, |− i ≈ | i | i ≥ +(θ) cosθ 1 +sinθ 2 do not change sign as the this can be done by lifting the loop to the universal cov- | i ≈ − | i | i loop is traversed. However, viewed as elements of the eringspaceSpin(n)[13]ofSO(n). Inpractice,though,it circle S1, (θ) both make a complete clock-wise turn. seemsdifficult to findamethod, thatworksforallnand |± i This means that each eigenvectortraces out a nontrivial is easily implemented. Below we show explicitly how to loop in = S1. In this way the presence of degen- use the test in the cases n=3 and n=4. We use exam- real N eracies encircled by Γ can be detected. ples from Jahn-Teller theory as physical illustrations. Forthetwo-levelcasewehavethusobtainedageneral- First, let us consider the n = 3 case. SO(3) is home- ization of Longuet-Higgins’ test: also loops along which omorphic to the closed ball of radius π with antipodal the eigenvectorstrace out nontrivial loops in S1 encircle points on its surface identified [14]. A vector φvˆ in the degeneracies on every surface bounded by them. How- ballrepresentsarotationaroundtheunitvectorvˆbythe ever, for n 3, the space of real normalized vectors angle φ [0,π]. Antipodal points on the surface must = Sn ≥1 is simply connected, i.e., it contains only be identi∈fied since a rotation by the angle π is the same real − N trivial loops. Thus, following a single eigenvector along transformationregardlessof whether the rotationaxis is the loop is insufficient. To generalize the test in this plusorminusvˆ. AloopinSO(3)canthusbeviewedasa case, we consider instead a complete set ψ (Q) n of curvethat may exit the closedball atthe boundary,and {| i i}i=1 normalized eigenvectors of H(Q). enter again at the antipodal point. The loop is trivial If H(Q) is nondegenerate on S, then ψ (Q) are, if and only if the number of piercings of the boundary i ±| i for eachi, two continuous and globally defined functions is divisible by two. Such a piercing is characterized by from S to Sn 1. Without loss of generality we may as- φ=π and that vˆ abruptly changes sign [15]. − sourtmheontohramta{l|ψbia(sQis)io}fni=1 rep=resRennt.sEavperoysitsiuvcehlyboarsiiesnctaend sysWteema,pdpelsycrtihbeedmebtyhothdetcootuhpelinlignemaratTri⊗x [τ126]Jahn-Teller real H bethoughtofasanelementofthendimensionalrotation 0 Z Y group SO(n). We may thus define a continuous function H(R)= Z −0 −X , (3) F :S SO(n) as − − → Y X 0  − −  h1|ψ1(Q)i ... h1|ψn(Q)i where R=(X,Y,Z) parametrizes =R3. The Hamil- F(Q)= ... ... ... . (2) tonian is doubly degenerate on eighQt rays in R3 that go  nψ1(Q) ... nψn(Q)  out from the origin and into the middle of each octant. h | i h | i  At the origin there is a three-fold degeneracy. We con- Thus, if H(Q) is nondegenerate on S, then its eigenvec- sider a loop Γ lying completely in the X-Y plane. In tors represent a continuous function from S to SO(n). planar polar coordinates ρ and θ, the eigenvectors ar- Analogous to the two-level case it follows that the loop ranged according to increasing energy read ψ1(θ) = 1 | i in SO(n) traced out by F(Q) as Q varies along Γ, is a sinθ 1 +cosθ 2 + 3 , ψ2(θ) =cosθ 1 sinθ 2 , trivial loop. By reductio ad absurdum we consequently a√n2d(cid:0) ψ3(θ|)i= 1 |sini θ 1| i+(cid:1)c|osθ 2i 3 . N|oit−ethat|thie arrive at the following result. If the n eigenvectors of | i √2 | i | i−| i eigenvectorsare in(cid:0)dependent of ρ and th(cid:1)at none of them H(Q) represent a nontrivial loop in SO(n) when taken changessignaroundanyloopthatdoesnotpassthrough continuously around Γ, then there must be at least one the origin. The function F along the loop is degeneracyofH(Q)oneverysimplyconnectedsurfaceS 1 1 bounded by Γ. sinθ cosθ sinθ √2 √2 This result makes it possible to detect the presence of F(θ)= 1 cosθ sinθ 1 cosθ . (4) √2 − √2 aevdeengiefntehreaycydboyncootncshidanergiengsiegingeanrvoeucntdortshoenloaolpo.oIpticnonQ-  √12 0 −√12  stitutes the promised generalization of Longuet-Higgins’ We aim to determine which loops in the X-Y plane that test. map to nontrivial loops in SO(3) under F. The rotation WenotethatSO(2)ishomeomorphictoS1 andisthus angle φ and rotation vector vˆ of F(θ) are given by infinitely connected: it contains one class of nontrivial 1 1 loops for each nonzero integer. The fundamental group φ(θ) = arccos 1+ 1 (1 sinθ) , of SO(2) is the additive group of integers Z. Determin- (cid:18)− 2(cid:18) − √2(cid:19) − (cid:19) 1 ing whether a loop is trivial amounts to counting the vˆ(θ) = ( cosθ,sinθ 1,(1 √2)cosθ), (5) number of times the eigenvectorencirclesthe origin. For N − − − each n 3, however, SO(n) contains only one class of respectively, where N is a normalization constant. Eq. ≥ nontrivial loops, that all become trivial when traversed (5) shows that φ(θ) = π only for θ = π/2. It is also 3 3.5 3 3 2 φ φ 2.5 1 0 2 0 2 4 6 0 2 4 6 s θ ts θ nt 1 n 1 e e n n o o p p m m 0 0 o o c r c or o t t c−1 c−1 e 0 2 4 6 e 0 2 4 6 V θ V θ FIG. 2: The same plot as in Fig. 1 but φ and the vector FIG. 1: Rotation angle φ in radians and components of the componentsarecalculated fromthematrixA(θ)oftheG⊗g rotation vector vˆ, as functions of θ also in radians. φ and vˆ system along the loop Γ. Since φ equals π at one point, Γ are computed from the matrix F(θ) of Eq. (4) representing must encircle a degeneracy. eigenvectors in the T ⊗τ2 system. The first, second, and third component of vˆ are represented by the solid, dotted, and dashed curves, respectively. Note that φ equals π at out the matrix multiplication, yields θ=π/2, and that vˆ changes sign at that point. 1 0 0 0  0 A11(t) A12(t) A13(t) T(t)F(t)= . (7) clear that vˆ is continuous everywhere except on the ray  0 A21(t) A22(t) A23(t)  0 A31(t) A32(t) A33(t) θ =π/2,whereitabruptlychangessign. Aplotofφand   vˆ as functions of θ appears in Fig. 1. Thus, exactly the Thus, the loop T(t)F(t) is trivial if and only if the loop loops in the X-Y plane that cross the ray θ = π/2 an A(t)=(A (t)) in SO(3) is trivial. ij odd number of times are mapped to nontrivial loops in We have applied the above procedure numerically to SO(3). This is a concrete manifestation of the validity the G g Jahn-Teller system. The coupling matrix for of the test, since any loop that passes θ = π/2 an odd this sy⊗stem reads [17] number of times must encircle the degenerate subset of . H(g)= qkG√2 Q − g Consider now the case n = 4. Let F(t) = (Fij(t)) g3 g4 g1 g3 g2+g4 be a loop in SO(4) parametrized by t R. The first  g4 g3 g2−+g4 g1+g3  ∈ − − , (8) columnf(t) (F11(t),F21(t),F31(t),F41(t))ofF(t)then × g1 g3 g2+g4 g1 g2 represents an≡element of S3 tracing out a loop. Define g2+−g4 −g1+g3 −g2 g1    the matrix whereg =(qg1,...,qg4)arenormalmodesandkgG isthe linearcouplingconstant. We consideraloopΓdescribed T(f(t)) by qg1 = qg4 = cosθ and qg2 = qg3 = sinθ, θ [0,2π]. ∈ F11(t) F21(t) F31(t) F41(t) This loop encircles a four-fold degeneracy at the origin  F21(t) F11(t) F41(t) F31(t)  of parameter space and it can be checked that none of = −F31(t) F41(t) −F11(t) F21(t) , (6) the eigenvectors changes sign along Γ. Fig. 2 shows the  −F41(t) F31(t) F21(t) −F11(t)  angle and vector of rotation of the matrix A(θ) along  − −  Γ. φ equals π once, meaning that the loop in SO(4) traced out by the eigenvectors is nontrivial. The four- beingorthogonalandcontinuousasafunctionoff S3. fold degeneracy at the origin is thus detected. ∈ It follows that T(t) T(f(t)) SO(4) since it can Let us now consider whether there can be a better ≡ ∈ be continuously connected to the identity matrix, cor- topological test than the present one. It is easy to find responding to f = (1,0,0,0) = ˆe1 S3. Furthermore, loopsencircling degeneraciesthat mapto trivialloops in since f(t) is a loop in S3, T(t) is a t∈rivial loop. We con- SO(n). This means that the test does not find every de- sider the loop in SO(4) given by T(t)F(t). This loop is generacy, so there may be room for improvement. The nontrivialexactlywhenF(t)is,sinceT(t)istrivial. How- restofthisLetterwedevotetoanargumentshowingthat ever, the loop T(t)F(t) is simpler than F(t). Carrying ifweareonlyallowedtoconsidereigenvectorsonaloopin 4 parameter space , and if the only information we have ofrealHamiltoniansbasedonthebehavioroftheireigen- Q abouttheHamiltonianisthatitiscontinuous,thenthere vectors on a loop in parameter space. If one considers a isnotestthatcandobetterinimplyingdegeneracies. To complete set of eigenstates, one may detect a degener- prove this, assume that the eigenvectors ψ (Q(t)) n acy even if none of the corresponding vectors changes {| i i}i=1 and corresponding eigenvalues λ (Q(t)) n are known sign. The test works for all real quantum systems with along the loop Γ parametrized{ biy t }i=R1. Suppose finitedimensionalHilbertspaces,includingquantumbil- ∈ that the eigenvectors are nondegenerate along Γ so that liards and Jahn-Teller systems, and could find explicit λ1(Q(t))<λ2(Q(t))<...<λn(Q(t))holdsforallt,and useinquantumchemistryappliedtocomputedeigenvec- that the corresponding loop F(Q(t)) in SO(n) is trivial. tors. We show also that no other topologicaltest can do Undertheseconditionsweshowthattherealwaysexistsa better in detecting degeneracies. HamiltonianH(Q),continuousandnondegenerateonall , having exactly the eigenvectors ψ (Q(t)) n and Q {| i i}i=1 eigenvalues λ (Q(t)) n on Γ. Thus, if F(Q(t)) is triv- We wish to thank Tobias Ekholm for discussions. The { i }i=1 ial, the Hamiltonian can be nondegenerate everywhere, work by E.S. was financed by the Swedish Research meaning that no test can imply a degeneracy. Council. For the proof we need to assume that the loop Γ is homeomorphictoS1 andthatthereisahomeomorphism D from Rd onto itself, such that D(Γ) is the unit circle in the X1-X2 plane. (X1,...,Xd) = X = D(Q) denotes the coordinatesofthe imageofapointQ Rd underD. [1] G.HerzbergandH.C.Longuet-Higgins,Disc.Farad.Soc. ∈ 35, 77 (1963). Such a D exists for all physically interesting Γ. [2] H.C. Longuet-Higgins, Proc. R. Soc. Lond. Ser. A 344, We consider first the unit disc in the X1-X2 plane, 147 (1975). parametrized by the usual polar coordinates ρ [0,1] [3] D.G. Truhlar and C.A. Mead, Phys. Rev. A 68, 032501 ∈ and θ [0,2π]. By the homeomorphism D we (2003). ∈ may view F(Q(t)) and λ (Q(t)) n as functions of θ, [4] A.J.C. Varandas, J. Tennyson, and J.N. Murrell, Chem. i.e., F(θ) F D 1(co{sθ,isinθ,0},i.=.1.,0) and λ (θ) Phys. Lett. 61, 431 (1979); S. Xantheas, S.T. Elbert, λ D 1(cos≡θ,sin(cid:0)θ,−0,...,0) . The eigen(cid:1)values cian b≡e andK.Ruedenberg,J.Chem.Phys.93,7519(1990);M. i − Ceotto and F.A. Gianturco, J. Chem. Phys. 112, 5820 co(cid:0)ntinuously extended to t(cid:1)he unit disc preserving the (2000). nondegeneracy by defining λ˜i(ρ,θ)=ρλi(θ)+(1 ρ)Ai, [5] C.A. Mead and D.G. Truhlar, J. Chem. Phys. 70, 2284 − where Ai are any real numbers satisfying A1 < A2 < (1979); C.A. Mead, Chem. Phys. 49, 23 (1980); 49, 33 ...<An. Let Λ˜(ρ,θ)=diag[λ˜1(ρ,θ),...,λ˜n(ρ,θ)]. Also (1980). theeigenvectorscanbecontinuouslyextendedtothedisc. [6] H. von Busch, V. Dev, H.-A. Eckel, S. Kasahara, J. Wang, and W. Demtr¨oder, Phys. Rev. Lett. 81, 4584 This is exactly because they represent a trivial loop in (1998). SO(n). Let the function G : [0,1] [0,1] SO(n) be a × → [7] B. Kendrick, Phys. Rev. Lett. 79, 2431 (1997); E. homotopybetweentheloopF(θ)andaconstantelement Sj¨oqvist, Phys. Rev.Lett. 89, 210401 (2002). R0 of SO(n) such that G(0,s)=R0 for all s [0,1] and [8] H.-M.Lauber,P.Weidenhammer,andD.Dubbers,Phys. ∈ G(1,θ/2π) = F(θ). A continuous extension of F to the Rev. Lett.72, 1004 (1994). whole disc is F˜(ρ,θ)=G(ρ,θ/2π). [9] D.E.ManolopoulosandM.S.Child,Phys.Rev.Lett.82, WearenowinapositiontoconstructasuitableHamil- 2223 (1999). [10] F. Pistolesi and N. Manini, Phys. Rev. Lett. 85, 1585 tonianas a function of the coordinateX. Let ρ take any (2000). value in [0,+ ). Define ∞ [11] J. Samuel and A. Dhar, Phys. Rev. Lett. 87, 260401 (2001); Phys.Rev.A 66, 044102 (2002). F˜(ρ,θ)Λ˜(ρ,θ)F˜(ρ,θ)T, if ρ 1 [12] J.W.ZwanzigerandE.R.Grant,J.Chem.Phys.87,2954 H˜(X)= ≤ (9) (1987). TheHamiltonian in the present Letterdiffers by  F˜(1,θ)Λ˜(1,θ)F˜(1,θ)T, if ρ>1, a unitary transformation from the one in this reference. [13] J. Jost, Riemannian Geometry and Geometric Analysis  where T denotes matrix transposition. Note that H˜(X) (Springer-Verlag, Berlin, 1998), 2nd ed., pp.55-60. defined in this way is independent of the coordinates [14] W.-K. Tung, Group theory in physics (World Scientific, Singapore, 1985), pp.96-97. X3,...,Xd, that it is continuous and nondegenerate for [15] Note that if vˆ does not change sign for φ = π, this cor- all X, and that it has ψ (θ) n as eigenvectors and {| i i}i=1 responds to a gentle touch of the boundary of the ball, {λi(θ)}ni=1 aseigenvaluesonthe unit circleinthe X1-X2 and does therefore not count as a piercing. plane. To obtain the desired Hamiltonian as a function [16] M.C.M. O’Brien, J. Phys. A 22, 1779 (1989). of Q, simply define H(Q)=H˜(D(Q)). [17] J.P. Cullerne and M.C.M. O’Brien, J. Phys: Condens. Inconclusion,wehavedescribedatestfordegeneracies Matter 6, 9017 (1994).