Searching for degeneracies of real Hamiltonians using homotopy classification of loops in SO($n$)

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by  Niklas Johansson

Searching for degeneracies of real Hamiltonians using homotopy classification of loops
in SO(n)
Niklas Johansson and Erik Sjo¨qvist
∗ †
Department of Quantum Chemistry, Uppsala University, Box 518, Se-751 20 Sweden
(Dated: February 1, 2008)
Topological teststodetectdegeneraciesofHamiltonianshavebeenputforwardinthepast. Here,
we address the applicability of a recently proposed test [Phys. Rev. Lett. 92, 060406 (2004)] for
degeneracies of real Hamiltonian matrices. This test relies on the existence of nontrivial loops in
the space of eigenbases SO(n). We develop necessary means to determine the homotopy class of a
given loop in this space. Furthermore, in cases where the dimension of the relevant Hilbert space
is large the application of the original test may not be immediate. To remedy this deficiency, we
put forward a condition for when the test is applicable to a subspace of Hilbert space. Finally, we
5 demonstratethatapplyingthemethodologyof[Phys. Rev. Lett. 92,060406(2004)]tothecomplex
0 Hamiltonian case does not provideany new information.
0
2 PACSnumbers: 03.65.Vf,02.40.Re,31.50.Gh,41.20.Cv

n
a
J I. INTRODUCTION itation of the Longuet-Higgins theorem was resolved by
5 the present authors [23], who put forward a topological
2 test for degeneracies of real matrix Hamiltonians based
A particular instance of Berry’s discovery [1] of geo-
upon consideration of their eigenvectors on loops in pa-
metric phase factors accompanying adiabatic changes is
3 rameter space. This generalized test was further proved
the occurrence of sign reversal of eigenfunctions of real
v [23] to be optimal in the sense that it exhausts all topo-
5 Hamiltonians when transported around certain types of
logicalinformationcontainedinthe eigenvectors,related
0 degeneracies. Although already implied in a work by
to the presence of degeneracies.
1 Darboux [2] in the late 19th century, the significance
6 of such sign changes to physics was not realized until Continuouschangeoftheeigenvectorsofarealparam-
0 eterdependentn nmatrixHamiltonianaroundaclosed
Longuet-Higgins and coworkers pointed out their exis-
4 path in paramete×r space may, if all geometric phase fac-
tence in molecular theory [3, 4]. This latter insight
0 tors are unity, be viewed as a loop in the n dimensional
led Mead and Truhlar to the notion of the molecular
/
h Aharonov-Bohmeffect [5, 6], which has attracted exper- rotation group SO(n). Based upon this observation, it
p was proved [23] that if the eigenvectors correspond to a
imental [7] and theoretical [8, 9] interest recently.
- nontrivial loop in SO(n), then the loop must encircle at
t Signreversalhasbeennoted[10]incharacteristicfunc-
n leastonepointofdegeneracy. Thus,inordertoapplythe
tionsofvibratingmembraneswhoseboundaryischanged
a testweneedtofindaproceduretodeterminewhetherthe
u around closed paths. Such sign change patterns in the
changeoftheeigenvectorscorrespondstoatrivialloopin
q vicinityofdegeneracieshavebeenstudiedbyexperiments
SO(n)ornot. InRef. [23],methodsparticularlyadapted
: on microwave resonators [11, 12] and smectic films [13].
v to the special casesn=2,3,4were presented. The main
i The microwave resonator experiments have been inter-
X focus of the present paper is to provide a method that
preted in terms of both the standard [14] and the off-
makes the test applicable to any n.
r diagonal[15] geometricphases, andthey havemotivated
a Another important issue for the applicability of the
further theoretical studies concerning both the geomet-
test in Ref. [23] arises when noting that n may be very
ric phases and structure of the wave functions for real
large in many realistic scenarios. For example, the com-
Hamiltonians [14, 15, 16, 17].
putedelectroniceigenvectorsinquantumchemicalappli-
It was provedby Longuet-Higgins [18] that signrever-
cations typically live in very large Hilbert spaces. This
sal of real electronic eigenfunctions when continuously
couldmakethetestdifficulttouseinthisimportantclass
transportedaroundaloopinnuclearconfigurationspace
ofproblemswheredegeneracypointsplayavitaldynami-
signalstheexistenceofdegeneracypointsinsidetheloop.
calrole. Toovercomethispotentialcomplication,weput
This topological result has been used to detect conical
forward a condition for when the test can be applied to
intersections in LiNaK [19] and ozone [20, 21]. On the
subspaces of the full Hilbert space.
other hand, there are cases where the Longuet-Higgins
Stone [24] demonstrated a topological test that ex-
test fails, such as, e.g., when the loop encircles an even
tendedLonguet-Higgins’originaltest[18]tothecomplex
number of conical intersections [22]. This apparent lim-
Hamiltonian case. In brief, this former test entails that
if the standardgeometric phase changescontinuously by
a nonzero integer multiple of 2π for a continuous set of
loops in parameter space, starting and ending with in-
∗Present address: Department of Theoretical Physics, Uppsala
University,Box803,Se-75108Uppsala,Sweden finitesimally small loops, then this set of loops must en-
†Electronicaddress: [email protected] close a degeneracy point. As for Longuet-Higgins’ test,
2
Stone’s test may fail to detect certainkinds of degenera- (see, e.g., Ref. [25], p. 172). Furthermore, the exponen-
cies. ThisapparentlyraisesthequestionwhetherStone’s tial map
test can be improved. The final concern of this work is
exactly to address the optimality of Stone’s test. exp: so(n) SO(n)
→
The outline of the paper is as follows. In the next ∞ Ak
section, we review the generalized topological test and A I+ (2)
7→ k!
develop a method to determine whether loops in SO(n) kX=1
for arbitraryn aretrivialor not. The mainresult ofthis
is continuous, onto, and exp(A) is well-defined for all
section is contained in Theorem 2. Sec. III contains the
A so(n). The function log : SO(n) so(n) is defined
conditionfortheapplicabilityofthetesttosubspaces,as ∈ →
asthe inverseofthe exponentialmap. Itismulti-valued,
summarized in Proposition 3. The optimality of Stone’s
i.e., there exist A=B such that exp(A)=exp(B).
test for complex Hamiltonians is discussed in Sec. IV. 6
By continuity of the exponential map, any curve F :
The paper ends with the conclusions.
[0,1] so(n) satisfying exp(F(0)) = exp(F(1)) corre-
→ e
sponds to a loop F in SO(n) via
e e
II. TOPOLOGICAL TEST FOR
DEGENERACIES F(t)=exp(F(t)). (3)
For brevity we say that continueous curves in so(n) for
Let H(Q) be an n n parameter dependent ma-
trix Hamiltonian, writte×n in the fixed basis i n of whichexp(F(0))=exp(F(1))arel-curves. Ourobjective
{| i}i=1 istofindacorrespondencebetweenclassesofl-curvesand
the n dimensional Hilbert space . We suppose that
e e
H thetwohomotopyclassesofloopsinSO(n 3). Thiswill
H(Q) is real, symmetric, and continuous for each Q =
≥
make it possible to deduce whether a loopF in SO(n) is
(Q ,...,Q ) in parameter space , which we assume to
1 d
be a simply connected subset ofQRd. Consider a loop trivial by studying its corresponding l-curve F. We first
Γ in . Let ψ (Q) n be a positively oriented set make a classification of the l-curves.
Q {| i i}i=1 e
of orthonormalized real eigenvectors of H(Q) for each Definition 1. Let F and F be l-curves in so(n). If
Q along Γ. If there are no points of degeneracy on the 0 1
there is a continuous function L : [0,1] [0,1]
loop and if all the concomitant geometric phase factors e e × →
so(n) such that L(t,0) = F (t), L(t,1) = F (t), and
of the eigenvectorsare unity, we may define the function 0 1
exp(L(0,s)) = exp(L(1,s)) holds for all s [0,1], then
F :Γ SO(n) as e ∈ e
→ F and F are called l-homotopic. The function L is
0 1
1ψ (Q) ... 1ψ (Q) called an l-homotopy between F and F .
h | 1 i h | n i e e 0 1
F(Q)= ... ... ... , (1) Two l-curves are thus l-homeotopic eif they can be de-
 nψ1(Q) ... nψn(Q)  formedinto eachother through a continuous family con-
h | i h | i 
sisting solely of l-curves. The following connection be-
suchthatF(Γ)isaloopinSO(n). Then,themainresult tweenl-homotopyinso(n)andhomotopyinSO(n)holds.
ofRef. [23]canbesummarizedinthe followingtheorem.
Proposition 1. If F and F are l-homotopic curves in
Theorem 1. If the n eigenvectors of H(Q) represent 0 1
so(n), thentheircorresponding loopsF andF inSO(n)
a nontrivial loop F(Γ) in SO(n) when taken continu- e e 0 1
are homotopic.
ously around Γ, then there must be at least one degen-
eracy point of H(Q) on every simply connectedsurface S Proof. A homotopy K : [0,1] [0,1] SO(n) between
× →
bounded by Γ. F and F is given by
0 1
The predictive power of Theorem 1 stems from the
K(t,s)=exp(L(t,s)), (4)
existence of nontrivial loops in SO(n 2). However, to
≥
determine to which homotopy class a given loop belongs where L is as in Definition 1. K defined in this way is
was only treated in the n = 2,3,4 cases in Ref. [23], continuous since it is the composition of two continuous
while the problem to find such a method for general n functions. Furthermore t K(t,s) is a loop for each s
was left open. Here, we resolve this deficiency and put since t L(t,s) is an l-cu7→rve for each s.
forwardanexplicitmethodthattreatsthegeneralncase. 7→
Let F : [0,1] SO(n) be a loop in SO(n). Without Note that we are working with free homotopies in
loss of general→ity we assume that F(0) = F(1) equals SO(n) rather than based ones. We remark that Propo-
the identity I on SO(n). This can always be achieved sition 1 implies that l-curves for which F(0)=F(1) cor-
by multiplying the whole loop by F(0) 1 = F(0)T, an respondto trivialloops in SO(n). This fact followssince
−
operationthatdoesnotchangethe homotopyclassofF. so(n) is simply connected. e e
ThespaceSO(n)ofrealorthogonalmatriceswithunit The restof this section is devotedto finding a method
determinant forms a Lie group whose corresponding Lie to use Proposition 1 to determine whether a given loop
algebraso(n) isthe spaceofrealantisymmetricmatrices F in SO(n) is trivial or not. Since we assume that
3
exp(F(0)) = exp(F(1)) = I, we may choose F(0) to be Proof. (a)Followssince so(n) is avectorspace,andthus
the zero matrix. F(1) is denoted K. We show that an simplyconnected. Toprove(b)wenotethatLiscontin-
e e e
uous and that
l-curveF betweenthe zeromatrixandK isl-homotopic,
e
either to a point, or to an l-curve connecting the zero
e exp(L(0,s)) = exp(A sX)=exp(A)exp( sX)
matrix to a matrix having only one nonzero 2 2 block − −
given by 2πiσ . In the first case the correspon×ding loop = exp(B)exp( sX)
y −
F = exp(F) in SO(n) is trivial, and in the second case = exp(B sX)=exp(L(1,s)). (8)
−
it is not. We also formulate a simple criterion that can
be used toedetermine to which “l-homotopy class” F be- For (c), note that since R acts nontrivially only on V, it
is possible to write R=exp(C), where C is an antisym-
longs.
e metric matrix whose null space contains the orthogonal
On our way we need the following three lemmas, the
complement of V. This means that exp(rC) acts non-
first of which is a standard lemma from matrix theory
trivially only on V for any real number r. Lemma 2 is
[26].
thus applicable, and exp[exp( rC)Aexp(rC)] = exp(A)
−
Lemma 1. Let A be an antisymmetric matrix. Then for any r. We may thus define the l-homotopy
thereis an orthogonal matrix R and ablock diagonal ma-
trix DA such that L(t,s)=exp((1 t)sC)F(t)exp((t 1)sC). (9)
− −
A=RDART, (5) We see that L(t,0) = F(t),eL(0,1) = RART, and
L(1,1)=B as required.
where the blocks of DA are 2 2 matrices of the form e
×
Before we proceed it is convenient to introduce some
0 α
DA = i iα σ , (6) notationtodescribeblockdiagonalmatrices. First,A
i (cid:18)−αi 0 (cid:19)≡ i y ... Am willdenote ablockdiagonalmatrixwithblo1ck⊕s
⊕
A ,...,A . Secondly, for any numbers λ ,...,λ we
with α 0. If the dimension n is odd, there is also a 1 m 1 n/2
i
≥ define [λ ,...,λ ] (iλ σ ) ... (iλ σ ). Note
zero 1 1 block. 1 n/2 ≡ 1 y ⊕ ⊕ n/2 y
× thatanytwomatricesofthisformcommute. Thirdly,I
m
For notational convenience, we assume from now on and 0 denote the m-dimensional unit and zero matrix,
m
that n is even. The analysis for odd n is identical. respectively.
Now,wereturnto ourl-curveF connecting0 =F(0)
Lemma 2. Suppose that λ is a degenerate eigenvalue n
to K F(1). Let R be as in Lemma 1, so that K =
of exp(A), and that the corresponding eigenspace is V. ≡ e e
RDKRT, with DK = [α ,...,α ]. Note that since
Then exp(RART) = exp(A) for any orthogonal matrix e 1 n/2
exp(K) = I, we must have α = 2πk for some integers
R acting nontrivially only on V. i i
k . At the possible cost of having some k < 0 we may
i i
Proof. assumeR to haveunit determinant. This implies thatF
is l-homotopic to any l-curve connecting 0 and DK. To
n
exp(RART)=Rexp(A)RT =exp(A), (7) see this, let R=exp(C) and define e
where the last equality follows since exp(A) is λI on V.
L(t,s)=exp( sC)F(t)exp(sC). (10)
−
e
Lemma3. LetAandB beantisymmetricmatrices such Clearly L(t,0) = F(t) and L(t,1) goes from 0n to DK.
Furthermore, L is continuous, and
that exp(A) = exp(B), and let F be an l-curve between
e
them. Then the following statements hold.
e exp(L(0,s))=I =exp(L(1,s)). (11)
(a) F is l-homotopic to any other l-curve connecting A
to B. Specifically it is l-homotopic to the straight Thus, L is an l-homotopy, and by Lemma 3(a), F is l-
leine t (1 t)A+tB. homotopic to the l-curve
7→ − e
(b) If X is an antisymmetric matrix commuting with F (t)=tDK. (12)
block
both A and B, then L(t,s) = F(t) sX is an l-
homotopy. The l-curve L(t,1) conne−cts A X to Our goal is to makeeas many as possible of the k dis-
i
e −
B X. appear through l-homotopies. We begin by reducing
−
each of them to zero or one. Assume that k 2.
i
(c) Let λ be a degenerate eigenvalue of exp(A), and V The case k 1 can be treated similarly. D≥efine
i
be the corresponding eigenspace. If R is an orthog- Y = [0,...,0,≤2π,−0,...,0], where the 2π appears at the
i
onal transformation with unit determinant acting ith place. Y commutes with 0 and DK, i.e.,
i n
nontrivially only on V, then F is l-homotopic to
an l-curve connecting RART to B. L(t,s)=F (t) sY , (13)
e block − i
e
4
is an l-homotopy transforming F into F = (t Theorem 2. Suppose that F is a loop in SO(n) starting
block i
1)Yi +t(DK Yi). This l-curve starts at Yi and te−r- at I and that F is an l-curve that maps to F under the
− e −e
minates in the matrix DK Yi = 2π[k1,...,ki 1,ki exponential map Eq. (3). Then there is an orthogonal
1,ki+1,...,km]. Furthermor−e exp( Yi)=I. Thi−s make−s transformationeR so that
−
Lemma 3(c) applicable, V being the whole space. For
i 2 (the case i = 1 is similar), we choose the orthogo- F(1)=R2π(ik σ ) ... (ik σ )RT, (21)
1 y n/2 y
≥ ⊕ ⊕
nal transformation R as
for somee integers k ,...,k . Furthermore F is trivial
1 n/2
R=I2i 3 ( I1) σx In 2i, (14) if and only if h n/2k is even.
− ⊕ − ⊕ ⊕ − ≡ i=1 i
where σx is the x component of the standard Pauli ma- Note that in casPe of odd n, the matrix RTF(1)R has
trices. R thus defined has unit determinant, and
one zero 1 1 block that can be ignored.
× e
R( Yi)RT = 02i 2 ( i2πσxσyσx) 0n 2i Proof. The theorem follows from the above discussion,
− − ⊕ − ⊕ −
= 02i 2 (i2πσy) 0n 2i =Yi. (15) and from Proposition 1. If h is even, then F is l-
− ⊕ ⊕ −
homotopic to a point. Otherwise it is l-homotopic to
Consequently, by Lemma 3(c) F is l-homotopic to an e
i F (t)=tY forsomei,whichmakesexp(F (t))nontriv-
Y i Y
l-curve between Y and DK Y , and thus to the l-curve
i i ial.
− e
e e
Fblock,i (t)=t(DK 2Yi). (16) Theorem 2 reduces the task of determining the homo-
− −
topyclassofaloopinSO(n)thatstartsattheidentityto
IfwecomparetehistoEq. (12),andnotethatDK 2Y =
− i computing the logarithm F [27] and block diagonalizing
2π[k ,...,k ,k 2,k ,...,k ], we see that we have
redu1ced ki bi−y1twoi.− i+1 n itsWenediilnlugstproaitnet.the procedeure by determining the homo-
Proceeding in this way we may show that F is l-
block topy class of a loop in SO(3). Let
homotopic to F (t) = tP, where P = 2π[δ ,...,δ ],
red e1 n
and δi is defined by 1(cosθ+1) 1 sinθ 1(cosθ 1)
e 2 −√2 2 −
F(θ)= 1 sinθ cosθ 1 sinθ ,
δi =(cid:26)01 iiff kkii iiss eovdedn. (17)  12(√co2sθ−1) −√12sinθ 21(√co2sθ+1)
(22)
Our next task is to reduce the number of nonzero δi to where θ [0,2π] parametrizes the loop. This loop ap-
one or zero. We will show that any pair δi =δj =1 can pears in ∈the analysis [23] of the T τ Jahn-Teller sys-
2
be eliminated through l-homotopies. The procedure for tem [28]. In fact, Eq. (22) repres⊗ents the loop in Eq.
doing this is similar to the reduction of the ki. (4) of Ref. [23] multiplied by its inverse at θ = 0, so
Suppose that i<j. First deform Fred by that F(0) = I. For each θ, F(θ) is a three-dimensional
s rotation whose angle φ and axis vˆ of rotation are given
e
L(t,s)=Fred(t) (Yi+Yj), (18) by
− 2
e
yieldingF (t)startingat Y 1(Y +Y ). Notethat φ(θ) = θ,
exp( Y)iis,jfour-folddegene−rate≡w−ith2 eiigenvjalue 1. The 1
eigen−spacee is supp(Y). We apply Lemma 3(c)−with R vˆ(θ) = (vˆ1(θ),vˆ2(θ),vˆ3(θ))= √2(−1,0,1). (23)
defined by
Note that vˆ is undefined at θ = 0 and 2π, since
R=I σ I σ I . (19) F(0) = F(2π) = I. It is straightforward to check that
2(i 1) x 2(j i 1) x n 2j
− ⊕ ⊕ −− ⊕ ⊕ − exp(F(θ))=F(θ) if we define
This matrix is orthogonal and has unit determinant.
Also, we may verify that e 0 vˆ (θ) vˆ (θ)
3 2
−
R( Y)RT =Y. (20) F(θ) = φ(θ) vˆ3(θ) 0 −vˆ1(θ) 
− vˆ2(θ) vˆ1(θ) 0
e  − 
Consequently, as we went from F (t) = tDK to 0 1 0
block θ −
F (t)=t(DK 2Y ), we can go from F (t)=tP =  1 0 1. (24)
block,i− − i e red √2 0 1 0
tteorixFrPed,−iY,−j(t)Y=hta(Ps t−he2Ysa)m=ets(tPru−ctYurie−aYsej)P.,Tbhuetmhaas-  − 
e − i − j This curve is continuous everywhere, starts at the zero
δ = δ = 0. Continuing in this fashion we can reduce
i j matrix and ends, by continuity, at
thenumberofnonzeroδ toone(zero)ifthisnumberwas
i
odd (even) to begin with. Note that this number is odd 0 1 0
(even) exactly when ni=/12ki is odd (even). At long last K =π√21 −0 1. (25)
we arrive at the followPing main result. 0 1 0
 − 
5
The orthogonaltransformation Proposition 2. Suppose that
1 0 1 1
R= 1  −0 √2 0  (26) hφi|φii>1− p (30)
√2
1 0 1
  holds for all i. Then φ is a linearly independent set.
i
{| i}
block diagonalizes K. Explicitly, we have
Proof. Assume that Eq. (30) holds for each i, and that
the vectors φ are linearly dependent. We show that
0 0 0 | ii
DK =RTKR=0 0 2π . (27) this leads to a contradiction. With |φii=|φNi i hφi|φii,
0 2π 0 there are by Lemma 4 j =k such that p
6
 − 
Thus, h=k =1 and the loop is nontrivial. φ φ = φ φ φ φ φN φN
1 |h j| ki| qh j| jiph k| ki|h j | k i|
1 1 1
> 1 = . (31)
(cid:18) − p(cid:19)p 1 p
III. APPLICATION TO SUBSPACES
−
Note however that
Here, we demonstrate how to apply the test to a sub-
spp<acdeimof Heiilgbeenrtvescptoarcse.ofaSppaecriafimcaetlleyr,dwepeenshdoewntHthaamtili-f 0=hψj|ψki=hφj|φki+hφ⊥j |φ⊥ki (32)
H
tonian can be well approximated by their projections in and thus that
somefixedpdimensionalsubspaceofHilbertspace,then
1/2
the p dimensional version of the test in Ref. [23] is ap- |hφj|φki| = |hφ⊥j |φ⊥ki|≤ hφ⊥j |φ⊥j ihφ⊥k|φ⊥ki
plicable. This result may be of use in quantum chemical (cid:8) 1/(cid:9)2
= (1 φ φ )(1 φ φ )
applications, where the detection of degeneracy points −h j| ji −h k| ki
(cid:8)1 (cid:9)
for electronic Hamiltonian matrices of large dimension
< , (33)
becomes pertinent. p
Suppose that i p is an orthonormal set of fixed
real vectors. We{c|oin}sii=d1er p eigenvectors ψ (Q) p of contradicting Eq. (31).
{| i i}i=1
H(Q) and their projections in Span i . Define
{| i} We are now in a position to give a condition for when
the p dimensional test is applicable to a subspace of
p
P = i i, Hilbertspace. LetS be asimplyconnectedsurfacein ,
Q
| ih | bounded by the loop Γ, and let the p eigenvectors along
Xi=1
Γ be denoted ψ (Q) p . Assume that for φ (Q) de-
φi(Q) = P ψi(Q) , {| i i}i=1 | i i
| i | i fined by Eq. (28), the inequality
φ (Q) = ψ (Q) φ (Q) (28)
| ⊥i i | i i−| i i
1
The eigenvectors |ψi(Q)i are mutually orthogonal, but hφi(Q)|φi(Q)i>1− p, (34)
φ (Q) neednotbe. Infacttheymayevenbelinearlyde-
i
| i
ipsensudffiencti.enTthlyis,smhoawllevfoerr,scoamneocic.urTohnilsyiisf hfφoir(mQa)l|iφzei(dQi)ni hisoltdhsefnorleinaecharil,yaninddfeopreenadcehnQt ∈byS.PTrohpeosseitti{o|nφi2(Q. )iT}pih=i1s
Proposition 2. For the proof we need the following. means that the Gram-Schmidt orthonormalization pro-
cedure can be applied to φ (Q) p , for any Q in S.
Lemma 4. Let v1,...,vp be linearly dependent unit vec- Thisprocedureis continuo{u|s,iandip}ri=od1ucesanorthonor-
torsinarealvectorspace. Thenforsomepair vj,vk with mal set φGS(Q) p , which can be interpreted as an
j =k we have {| i i}i=1
6 element F(Q) SO(p). Thus, given that Eq. (34) holds
∈
andthat H(Q)is nondegenerateonS,we have a contin-
1
|vj ·vk|≥ p 1. (29) uous function F :S →SO(p). By the same reasoning as
− in Ref. [23] we arrive at the following result.
Proof. Let Span v ,...,v have dimension m. It is
enough to prove{th1e statepm}ent for m = p 1, since if Proposition 3. Suppose that the projections |φi(Q)i of
− the p eigenvectors of H(Q) satisfy Eq. (34) for each
it is false for some m, then it is false for all higher m.
Q S. Suppose furthermore that F(Q) defined as above
Intuitively,tomakeallscalarproductsassmallaspos- ∈
traces out a nontrivial loop in SO(p) as Q varies along
sible, we need to “spread” the vectors as much as pos-
Γ. Then H(Q) becomes degenerate somewhere in S.
sible, i.e., the vectors should point to the vertices of a
regular(p 1)-simplex [29]. The scalarproductbetween NotethattheresultinProposition3concernstheexact
any two di−stinct vectors is then 1 [30]. This proves HamiltonianH(Q),butisbaseduponthebehaviorofthe
−p 1
the statement. − approximate eigenvectors φ (Q) .
i
| i
6
IV. STONE’S TEST quently,Stone’stestexhaustsalltopologicalinformation
contained in the eigenvectors around S.
The original test by Longuet-Higgins [18], as well as We conclude this sectionby noting that, while Stone’s
its generalization [23], suffers from the limitation of be- testisoptimalforgeneralHamiltonians,topologicaltests
ing applicable only to real Hamiltonians. When the test similartothatofRef. [23]maybeconstructedforHamil-
of Longuet-Higgins is applicable, the loop in parameter tonians obeying additional symmetries.
space maps to an open curve in Rn representing the V. CONCLUSIONS
Hilbert space. In this case the correspondingloop in the
space of states RPn−1 is nontrivial. Similarly, the gen- The need to demonstrate whether or not there exist
eralizationmakes use of the existence of nontrivialloops degeneracy points in the spectra of Hamiltonians is of
in SO(n), the space of eigenbases. A pertinent question relevance in many fields of physics. For example, such
is whether there exist analogous results for the general points are abundant in molecular systems [32] and are
complex case. importantbecause they signala breakdownof the Born-
For a generic Hamiltonian, the degenerate subsets of Oppenheimer approximation. Another instance where
parameter space have co-dimension 3 [24, 31], meaning the presence of degeneracy points become pertinent is
thatanysuchtestmustconsidereigenvectorsonaclosed in the recently proposed paradigm of adiabatic quan-
surface, rather than on a closed loop. An eigenstate tumcomputation[33,34],whoseefficiencyreliescrucially
taken around the surface represents a 2-loop in projec- uponthepresenceofnonvanishingenergygapsalongcer-
tive Hilbert space, being the n 1 dimensional complex tain paths in parameter space.
projective space CPn−1. − Topologicaltests to detectdegeneracieshavebeen put
Stone [24] put forward a topological test relating the forward in the past by Longuet-Higgins [18] and Stone
behavior of a single eigenvector on a closed surface S in [24]. More recently, the present authors [23] extended
parameter space, to the presence of degeneracies inside Longuet-Higgins’ test by consideration of complete sets
the surface. Potentially, this test might be possible to ofeigenvectorsofrealparameterdependentHamiltonian
generalize in the same manner as the one by Longuet- matrices as paths in SO(n). This extended test can de-
Higgins. However, as shown below, this is impossible. tect degeneracies even in cases where Longuet-Higgins’
Despite that it preceded Berry’s work [1] by eight original test fails.
years, Stone’s test is conveniently formulated in the lan-
In this paper, we have put forward a method that
guage of geometric phases. The surface S is swept out
makes the topological test in Ref. [23] applicable to any
by a continuous set L N of loops, where L and L
{ i}i=1 1 N dimension n of the Hamiltonian matrix. This method
are infinitesimally small. The cyclic geometric phases
is based upon the multi-valuedness of the function log :
{γi}Ni=1alongtheseloopsaremodulo2πquantities. How- SO(n) so(n),thatconnectsSO(n)withitscorrespond-
ever, if we require continuity in the index i and choose ingLie7→algebraso(n)ofrealantisymmetricmatrices. We
γ = 0, γ becomes uniquely determined, and equal to
1 N have further demonstrated under what conditions the
2πk for some integer k. Stone proved that if k =0, then
topological test in Ref. [23] is applicable to subspaces
6
there must be a degeneracy point somewhere inside S.
of the full Hilbert space. These two major findings of
The integer k can be topologically interpreted as label-
the present paper open up the possibility to use the test
ing the homotopy class to which the 2-loop represented
in realistic scenarios, such as, e.g., for computed elec-
by the states around S belongs. Equivalently, k charac-
tronic eigenvectors in various molecular systems or for
terizes the topological structure of the monopole bundle
the eigenfunctions of quantum billiards. Our final result
withfiberU(1)representingstatevectorsandbasespace
concernsStone’s test for degeneraciesof complex Hamil-
S2representingthesurfaceS(see,e.g.,Ref. [25],p. 320).
tonians. We have shown that Stone’s test is optimal in
It turns out that it is possible to define a global and
the sense that no other topological test can do better in
continuous state vector around S if and only if k = 0,
detectingdegeneraciesinsystemsthatneedadescription
i.e., if and only if Stone’s test does not signal a degen-
in terms of general complex Hamiltonians.
eracy. Let us try to construct a test that works even
for some cases when k = 0 by considering a complete
set of eigenvectors around S. We then have a continu-
ous function from the surface S to the space of eigen- Acknowledgments
bases U(n). This can contain topological information
only if U(n) contains nontrivial 2-loops. This, however, We wish to thank David Kult and Johan ˚Aberg for
is not the case (see, e.g., Ref. [25], pp. 120-121). Conse- discussions and useful comments.
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