A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics
We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We star...
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2022 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2022
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211542 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics. Eric J. Pap, Daniël Boer and Holger Waalkens. SIGMA 18 (2022), 003, 42 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859609232317349888 |
|---|---|
| author | Pap, Eric J. Boer, Daniël Waalkens, Holger |
| author_facet | Pap, Eric J. Boer, Daniël Waalkens, Holger |
| citation_txt | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics. Eric J. Pap, Daniël Boer and Holger Waalkens. SIGMA 18 (2022), 003, 42 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we demonstrate that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection that yields the (generalized) geometric phase. This bundle also provides a natural generalization of the quantum geometric tensor and derived tensors, and we demonstrate how it can incorporate the non-geometric dynamical phase as well. We conclude by demonstrating how the bundle can be recast as a principal bundle, allowing both the geometric phases and the permutations of eigenstates to be expressed simultaneously using standard holonomy theory.
|
| first_indexed | 2026-03-14T04:12:43Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 003, 42 pages
A Unified View on Geometric Phases and Exceptional
Points in Adiabatic Quantum Mechanics
Eric J. PAP ab, Daniël BOER b and Holger WAALKENS a
a) Bernoulli Institute, University of Groningen,
P.O. Box 407, 9700 AK Groningen, The Netherlands
E-mail: e.j.pap@rug.nl, h.waalkens@rug.nl
URL: http://www.rug.nl/staff/e.j.pap/, http://www.rug.nl/staff/h.waalkens/
b) Van Swinderen Institute, University of Groningen, 9747 AG Groningen, The Netherlands
E-mail: d.boer@rug.nl
URL: http://www.rug.nl/staff/d.boer/
Received July 23, 2021, in final form December 28, 2021; Published online January 13, 2022
https://doi.org/10.3842/SIGMA.2022.003
Abstract. We present a formal geometric framework for the study of adiabatic quantum
mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework
generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states ap-
pearing for non-Hermitian Hamiltonians. We start with an investigation of the space of
non-degenerate operators on a finite-dimensional state space. We then show how the energy
bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays
form a bundle, a generalization of a principal bundle, which admits a natural connection
yielding the (generalized) geometric phase. This bundle provides in addition a natural gen-
eralization of the quantum geometric tensor and derived tensors, and we show how it can
incorporate the non-geometric dynamical phase as well. We finish by demonstrating how
the bundle can be recast as a principal bundle, so that both the geometric phases and the
permutations of eigenstates can be expressed simultaneously by means of standard holonomy
theory.
Key words: adiabatic quantum mechanics; geometric phase; exceptional point; quantum
geometric tensor
2020 Mathematics Subject Classification: 81Q70; 81Q12; 55R99
1 Introduction
The eigenvalues of a matrix or operator are of great interest in many fields of mathematics and
physics. In quantum mechanics, the eigenvalues of an observable are the possible measurement
outcomes for this observable. The dynamics is governed by the Hamilton operator, whose eigen-
values define the energy levels. This relation between quantum mechanics and linear algebra, or
more general functional analysis, is even tighter in the subfield of adiabatic quantum mechanics.
In general, if the Hamiltonian changes in time, then initial eigenstates need not evolve into
instantaneous eigenstates of the Hamiltonian at a later time. However, this could be realized
for sufficiently slow change of the Hamiltonian [7, 16], known as the adiabatic approximation.
The interest in adiabatic quantum mechanics gained momentum with the discovery of the
geometric phase by Berry [3]. He showed that the phase picked up by an eigenstate upon
varying parameters along a closed loop has in addition to the usual dynamical contribution
a purely geometric contribution. This geometric phase depends only on the traversed loop,
i.e., it is invariant under reparametrization of the path. Geometric phases have been found in
numerous physical systems, see the overviews in [8, 9, 31], where in fact the earliest observation
mailto:e.j.pap@rug.nl
mailto:h.waalkens@rug.nl
http://www.rug.nl/staff/e.j.pap/
http://www.rug.nl/staff/h.waalkens/
mailto:d.boer@rug.nl
http://www.rug.nl/staff/d.boer/
https://doi.org/10.3842/SIGMA.2022.003
2 E.J. Pap, D. Boer and H. Waalkens
of such a phase had been reported by Pancharatnam [26] in polarized light. On the theoretical
side, the geometric phase was readily identified to be the holonomy of a bundle of eigenstates
over system parameters [32]. In addition, generalizations have been studied, e.g., for degenerate
Hamiltonians [37] or for any so called cyclic state returning to its original ray in a Hilbert
space [1].
Geometric phase is also studied for non-Hermitian Hamiltonians, which allow for gain and
loss of energy. Using left-eigenstates instead of bra’s, a generalization of Berry’s phase for cyclic
eigenstates of a non-degenerate non-Hermitian Hamiltonian was presented in [12], and this was
related to a geometric model in [21]. Another prominent feature of a non-Hermitian Hamiltonian
family is that, when one follows a loop in the space of system parameters, the energies may swap
places, which renders the evolution non-cyclic. This effect indicates a non-trivial topology of
the energy bands; they need not be separated as in the Hermitian case. The interchange of
energies can be related to degeneracies in the space of system parameters, which are known as
exceptional points (EPs). These were first mentioned, with slightly different meaning, in [17],
and have been found in many experimental setups, see the overviews, e.g., in [14, 23]. Many
of these experiments revolve around parity-time or PT symmetric systems. Such systems are
typically described by non-Hermitian Hamiltonians, and the breaking of this symmetry is often
associated with an EP. Although PT -symmetry is not necessary for EP theory, it does provide
experimentally accessible realizations.
We remark that this phenomenology is not adiabatic in the standard way, as non-adiabatic
effects cannot be avoided. The adiabatic theorem in the non-Hermitian case only holds for the
state with highest relative gain [25]. This leads to significant limitations on how well the state
exchange around an EP can be measured [5, 36]. A solution is to consider quasi-static set-ups;
the path in parameter space is discretized into points, and one measures per configuration, see
also [15, 22].
In this paper we introduce a geometric formalism to properly describe the adiabatic evolution
of eigenstates of any non-degenerate finite-dimensional operator playing the role of a Hamilto-
nian. In particular, the eigenstates need not be cyclic and the Hamiltonian need not be Her-
mitian. To this end we introduce a bundle that directly follows from the eigenvalue problem
and consists of triples “matrix-eigenvalue-eigenvector”. This bundle will not be principal, rather
it has the structure of a semi-principal bundle as rigorously defined and studied in [28]. It is
naturally equipped with a connection, whose parallel transport corresponds to the generalized
geometric phase. As a result, both the geometric phase and the swaps of eigenstates arising from
EPs can be described in a single holonomy description. In fact, the same bundle also allows for
the incorporation of the non-geometric dynamical phase. We also treat its associated “frame”
bundle, which is principal and provides a rigorous geometric argument behind the matrices used
to describing state evolution.
The condition of non-degeneracy of the operator will be crucial, and hence we study the
space this defines in Section 2. The geometry of the eigenvalues of these operators we study
in Section 3. This will yield a natural way to treat the interchanges of energies around EPs.
In Section 4, this is extended to include eigenvectors, which yields a natural parallel transport
theory for geometric phases, also in the presence of EPs. Following up on this, we show in
Section 5 how these physical phenomena can be interpreted via a single holonomy description.
We finish with a discussion in Section 6.
2 The non-degeneracy space
We start with the mathematical objects that will provide the basis for all our arguments con-
cerning eigenvalues. At this moment, we do not consider any specific operator family, instead
only a complex vector space V of finite dimension n is given. An important remark is that we
A Unified View on Geometric Phases and Exceptional Points 3
do not endow V with an inner product; we only use the topological vector space and manifold
properties of V .
Let us establish some notation by reviewing the following definitions and facts. We recall
that endomorphisms or operators, i.e., linear maps A : V → V , form the space End(V ), which
is a complex manifold of complex dimension n2. The set of all eigenvalues of an operator A is
called the spectrum of A, and we denote it as Spec(A). If (v1, . . . , vn) is a frame, or basis, of V ,
we call it an eigenframe of A if each vi is an eigenvector of A. The set of all bases of V we
denote as Fr(V ), and the set of all eigenframes of A by EigFr(A). Any basis (v1, . . . , vn) of V
has a dual basis
(
θ1, . . . , θn
)
of the dual space V ∨ defined by the condition
θi(vj) = δij =
{
1 if i = j,
0 if i ̸= j.
An eigencovector of A is a non-zero covector θ such that θA = λθ, where λ ∈ C is necessarily an
eigenvalue of A. An eigencoframe of A is a frame of V ∨ consisting of eigencovectors of A. One
can verify the following facts about eigenframes.
Lemma 2.1. Let A ∈ End(V ), then
(1) A has an eigenframe if and only if A is diagonalizable,
(2) (v1, . . . , vn) is an eigenframe of A if and only if the dual basis
(
θ1, . . . , θn
)
is an eigen-
coframe.
We will now focus on the operators A which are non-degenerate, i.e., which have n distinct
eigenvalues. The subspace in End(V ) of non-degenerate operators we denote by N(V ). We will
go over different ways to formally define this subspace. First, we inspect an algebraic argument,
which allows for a straightforward result on the manifold properties of N(V ). Afterwards, we
consider other formulations of non-degeneracy, which will naturally guide us to symmetries and
bundle properties of N(V ).
2.1 Discriminant definition
The first characterisation of N(V ) we will inspect is based on the discriminant, and will form
our algebraic definition of N(V ). Naturally, the eigenvalues of A ∈ End(V ) are the zeros of the
characteristic polynomial p(A, z) of A, and whether or not the zeros of a polynomial are distinct
can be inferred from the discriminant. That is, one first has the map
p : End(V ) → C[z],
A 7→ p(A, z) := det(zI −A)
and by evaluating the discriminant one obtains a composite function
d : End(V ) → C,
A 7→ discrim(p(A, z), z)
known as the discriminant of operators. Its zero set
∆(V ) =
{
A ∈ End(V ) | d(A) = 0
}
is called the discriminant set and consists of all matrices that are degenerate. Clearly N(V ) is
the complement of the discriminant set in End(V ), which yields our formal definition of N(V ).
4 E.J. Pap, D. Boer and H. Waalkens
Definition 2.2 (non-degeneracy space). Given a finite-dimensional complex vector space V , its
space of non-degenerate operators, or non-degeneracy space, is
N(V ) =
{
A ∈ End(V ) | d(A) ̸= 0
}
.
It is well-known that d becomes a polynomial function in the matrix elements of the operator.
Hence one may readily conclude the following.
Lemma 2.3. The space N(V ) is an (algebraic) open and dense subset of End(V ). In particular,
N(V ) is a submanifold of complex dimension n2 and real codimension 2.
We next consider alternative characterizations of the non-degeneracy space.
2.2 Parametrizing the non-degeneracy space
Another way to describeN(V ) is by explicitly parametrizing its elements. One such parametriza-
tion readily follows from the fact that any A ∈ N(V ) is similar to a diagonal matrix where the
diagonal entries are distinct. In other words, there is a frame f̃ = (f1, . . . , fn) of V so that the
matrix of A w.r.t. f̃ takes the form diag(λ1, . . . , λn), where all the λi are distinct. We will denote
the space of tuples λ̃ = (λ1, . . . , λn) of n distinct complex numbers as Cn; more background on
this space can be found in Appendix A. We abbreviate diag(λ1, . . . , λn) as diag
(
λ̃
)
.
The above decomposition can be recast as the following parametrization. For convenience,
let us identify a frame f̃ ∈ Fr(V ) with the map Sf̃ : C
n → V given by (z1, . . . , zn) 7→
∑n
i=1 zifi,
which is a linear isomorphism by definition of a frame. The parametrization of N(V ) is then
formally described using the map
Ξ: Fr(V )× Cn → N(V ),(
f̃ , λ̃
)
7→ Sf̃ diag
(
λ̃
)
S−1
f̃
,
which is a smooth surjection. Clearly, this map is not injective for two reasons; firstly, Ξ is
indifferent concerning a non-zero scaling of the eigenvectors, and secondly, if we permute both
the vectors of f̃ and the values in λ̃ we obtain the same operator. This is a symmetry of Ξ,
which we phrase using group actions.
First, writing the group of non-zero complex numbers as C×, the scaling symmetry is given
by the (C×)n-action
z̃ ·
(
f̃ , λ̃
)
=
(
z̃f̃ , λ̃
)
, (2.1)
where z̃f̃ is the entry-wise product (z1f1, . . . , znfn) ∈ Fr(V ). To describe the permutations, let
us agree that a permutation acts on a tuple by permuting its entries, e.g., for σ ∈ Sn and a tuple
λ̃ ∈ Cn, σ · λ̃ =
(
λσ−1(1), . . . , λσ−1(n)
)
. Then the permutation action on Fr(V )× Cn is simply
σ ·
(
f̃ , λ̃
)
=
(
σf̃ , σλ̃
)
. (2.2)
The scaling and permutation actions merge into a single action of the wreath product C× ≀ In,
where In := {1, . . . , n} is an index set with n elements. This group is the semi-direct product
(C×)n ⋊ Sn, whose defining action of Sn on (C×)n is exactly the tuple permutation as we used
above. The group multiplication of C× ≀ In reads
(z̃1, σ1) · (z̃2, σ2) =
(
z̃1(σ1z̃2), σ1σ2
)
.
A Unified View on Geometric Phases and Exceptional Points 5
The action of the wreath product on Fr(V )×Cn is obtained by performing first the permutation
and then the scaling, i.e.,
C× ≀ In × (Fr(V )× Cn) → Fr(V )× Cn,
(z̃, σ) ·
(
f̃ , λ̃
)
=
(
z̃
(
σf̃
)
, σλ̃
)
. (2.3)
Let us make some remarks at this point. First, C× ≀ In has a faithful representation by
complex generalized permutation matrices. These are matrices with a single non-zero complex
number in every row and column, or, equivalently, products DP with D a diagonal matrix
without zeros on the diagonal and P a standard permutation matrix. Given the conjugating
nature of Ξ, the appearance of C× ≀ In is natural, as the generalized permutation matrices
form the stabilizer of the diagonal matrices. Let us also refer to some earlier reports on the
wreath product in combination with this parametrization. For example, in [38] there is firstly
a quotient by scaling and secondly the quotient by permutations. Although not reported as
such, we recognize the wreath product there. In [19, Lemma 1.1], it is shown that a restriction
of Ξ defines a principal bundle for a wreath product group, where the number field need not
be C. As we will argue now, the parametrization Ξ also defines a principal C× ≀ In-bundle. Let
us use that a projection is a principal bundle if it is given by the quotient of a free and proper
action. The latter is readily verified, hence it remains to be shown that Ξ coincides with this
projection.
Lemma 2.4. There is a principal bundle
C× ≀ In Fr(V )× Cn N(V ).Ξ
Proof. As stated, it suffices to show that Ξ and the action quotient map are isomorphic as
bundle maps. As we already saw that the action preserves the fibers of Ξ, it only remains to
show that each group orbit exhausts the fiber in which it lies. Therefore, assume Ξ
(
f̃ , λ̃
)
=
Ξ
(
f̃ ′, λ̃′
)
=: A. Then both λ̃ and λ̃′ constitute Spec(A), hence they are related by a permutation
σ ∈ Sn as λ̃ = σλ̃′. Then fi needs to be parallel to f ′σ−1(i) for each i, so f̃ and σf̃ ′ are equal
up to an element of (C×)n. However, then
(
f̃ , λ̃
)
and
(
f̃ ′, λ̃′
)
differ only up to an element
of C× ≀ In. ■
This result has various consequences. First, it shows that N(V ) can be realized as a quotient
space. Another important observation concerns the existence of local sections of Ξ. These
sections provide local moving eigenframes and corresponding smooth eigenvalues, as we will use
in Section 4. We state the details in the following corollaries.
Corollary 2.5. Identifying Fr(V ) with GL(n,C), the space of non-degenerate operators is rea-
lized as a quotient space as
N(V ) ∼=
GL(n,C)× Cn
C× ≀ In
.
Corollary 2.6. Let A0 ∈ N(V ). There is a neighborhood U of A0 on which one has smooth
local eigenvalue functions λ1, . . . , λn : U → C, exhausting the spectrum at each point, and smooth
local eigenvector functions f1, . . . , fn : U → V . That is, for all A ∈ U and i ∈ In one has
Afi(A) = λi(A)fi(A)
such that Spec(A) = {λ1(A), . . . , λn(A)} and (f1(A), . . . , fn(A)) is a basis of V . Moreover,
the tuple (λ1(A), . . . , λn(A)) may be taken to be any given ordering of Spec(A), and the tuple
(f1(A), . . . , fn(A)) any eigenframe of A following the same ordering.
Proof. The functions fi(A) and λi(A) are the components of a local section s : U → Fr(V )×Cn,
which can be taken through an arbitrary point above A. ■
6 E.J. Pap, D. Boer and H. Waalkens
2.3 Spectrum map on non-degeneracy space
Non-degenerate operators can also be characterized based on their spectrum. Indeed, an operator
A ∈ End(V ) is non-degenerate if and only if Spec(A) consists of n distinct elements. In other
words, Spec(A) should not be any subset of C, but belong to the set
(C
n
)
of all subsets of C
consisting of n distinct elements. Taking the spectrum can thus be written as a map
Spec: N(V ) →
(
C
n
)
,
A 7→ Spec(A).
We will now continue by showing that this map Spec defines a fiber bundle. In this way, we
find N(V ) realized as a total space instead of a base space. In addition, the map Spec will
reappear when we discuss EPs; formally it is this map that associates the change in spectrum
to a change in Hamiltonian.
To start, let us verify that Spec is smooth. For the manifold structure on
(C
n
)
, we follow
the idea that the space
(C
n
)
can be obtained from Cn by reducing an (ordered) tuple to the
(unordered) set of its elements. We write q : Cn →
(C
n
)
for the quotient map, and use the manifold
structure on
(C
n
)
for which q defines a principal Sn-bundle with the standard permutation action
of Sn on Cn. For more details, we refer to Appendix A.
We can see that Spec is smooth using a geometric argument, based on the projection q and
the map Ξ from the previous part. The key observation is the equality Spec
(
Ξ
(
f̃ , λ̃
))
= q
(
λ̃
)
for any
(
f̃ , λ̃
)
∈ Fr(V ) × Cn; by construction λ̃ lists the eigenvalues of the operator Ξ
(
f̃ , λ̃
)
.
In other words, we have the following commutative diagram:
Fr(V )× Cn Cn
N(V )
(C
n
)
.
prCn
Ξ
q
Spec
(2.4)
Smoothness of Spec is now clear; as the upper route in the diagram is smooth and Ξ is a surjective
submersion the claim follows.
Let us continue by discussing the model fiber of Spec. This is facilitated by viewing operators
according to their spectral decomposition. That is, any A ∈ N(V ) can be written as a sum∑n
i=1 λiPi, where λi ∈ Spec(A) and each Pi : V → V is a projection. Geometrically, this
expresses that A ∈ N(V ) is completely specified by the pairs of eigenvalue and the corresponding
eigenrays. Hence, if we fix the spectrum, then only the choice of eigenrays remains. This
means that any fiber of Spec is diffeomorphic to this space of possible choices of eigenrays, or,
equivalently, to the space of suitable tuples of projectors. The latter space we can describe in
more detail. Clearly, each Pi projects to a one-dimensional subspace in V , and together these
projectors satisfy PiPj = δijPi and
∑n
i=1 Pi = idV . That is, the tuple (P1, . . . , Pn) must lie in
the space
PFr(V ) :=
{
(P1, . . . , Pn) ∈ End(V ) | PiPj = δijPi, dimC(im(Pi)) = 1
}
,
which is thus also the model fiber for Spec. One can think of PFr(V ) as the space of all resolutions
of the identity which are compatible with non-degenerate operators, i.e., those corresponding
to n rays in V . Clearly, PFr(V ) relates to Fr(V ) by sending a basis (f1, . . . , fn), with dual basis(
θ1, . . . , θn
)
, to the tuple
(
f1θ
1, . . . , fnθ
n
)
∈ PFr(V ). This map is surjective, but not injective
as individual scaling of the basis vectors will yield the same projectors. Hence we find that
Fr(V )/(C×)n is canonically isomorphic to PFr(V ), and it is the former form that we will see in
the upcoming proofs.
A Unified View on Geometric Phases and Exceptional Points 7
At this point we have discussed the relevant spaces, but did not yet discuss their symmetry.
This symmetry can be found from the observation that operators with the same spectrum differ
by a similarity transformation. Let us implement these transformations in the language of the
action of GL(V ) on N(V ) given by
GL(V )×N(V ) → N(V ),
S ·A = SAS−1. (2.5)
The map Spec is invariant w.r.t. this action by the familiar rule Spec
(
SAS−1
)
= Spec(A). Hence
each fiber of Spec is a GL(V )-manifold as well, and so we wish to view PFr(V ) as a GL(V )-
manifold, also equipped with the conjugation action. Observe that this action on PFr(V ) is
naturally inherited from Fr(V ), on which it reads S · (f1, . . . , fn) = (Sf1, . . . , Sfn). We thus
arrive at the following result.
Lemma 2.7. The GL(V )-action on N(V ) is transitive on the fibers of Spec. Moreover, any
fiber of Spec is isomorphic to PFr(V ) as GL(V )-manifolds.
Proof. As any two non-degenerate operators with the same spectrum differ by a similarity
transformation, the action is transitive on the fibers of Spec. Hence, every fiber of Spec is a homo-
geneous GL(V )-space. The stabilizer subgroup at A ∈ N(V ) consists of the maps that preserve
all eigenrays of A individually, hence is isomorphic to (C×)n. In order to parametrize this fiber,
let λ̃ be an ordering of Spec(A), and consider Ξ restricted to the subset Fr(V ) × {λ̃}. Clearly,
this surjects on the fiber of Spec containing A. Moreover, Ξ is equivariant w.r.t. the canonical
GL(V )-action on Fr(V ) × Cn. Hence, the fiber of Spec containing A is isomorphic, as GL(V )-
manifold, to the quotient of Fr(V ) by the stabilizer. The stabilizer (C×)n is now straightforward;
it appears via the (C×)n-action given in equation (2.1). Hence we found a GL(V )-equivariant
isomorphism to Fr(V )/(C×)n, and so to PFr(V ). ■
The model fiber of Spec is thus PFr(V ), which we view as a GL(V )-manifold, emphasizing
its close relation to similarity transformations. We thus wish to prove that Spec defines a fiber
bundle that respects the GL(V )-action. That is, Spec defines a GL(V )-manifold bundle in
the language of [28]; both fiber and total space are endowed with a GL(V )-action, and local
trivializations can be taken GL(V )-equivariant. We then arrive at the following statement, which
summarizes the results of this section.
Proposition 2.8. The spectrum map induces the GL(V )-manifold bundle
PFr(V ) N(V )
(C
n
)
.
Spec
Proof. Pick a point {λ1, . . . , λn} ∈
(C
n
)
, and let U ⊂
(C
n
)
be a neighborhood of {λ1, . . . , λn} on
which a local section s : U → Cn of q is defined. Consider the map
U × Fr(V ) → N(V ),(
u, f̃
)
7→ Ξ
(
f̃ , s(u)
)
,
which for each u ∈ U surjects on the fiber of Spec above u. By applying Lemma 2.7 fiber-wise
we find that the reduced map U × Fr(V )/(C×)n → N(V ) is well-defined. In fact, if we look at
the spectral decomposition, this map pairs the tuple (P1, . . . , Pn) ∈ PFr(V ) determined by f̃
with the values in the tuple s(u). We thus obtained a GL(V )-equivariant local trivialization of
Spec around {λ1, . . . , λn}, hence the claim follows. ■
8 E.J. Pap, D. Boer and H. Waalkens
2.4 Summarizing diagram
If one takes the bundles defined by Ξ and Spec plus the diagram in (2.4), then one readily
obtains the following diagram of bundle sequences:
(C×)n C× ≀ In Sn
Fr(V ) Fr(V )× Cn Cn
PFr(V ) N(V )
(C
n
)
.
prCn
Ξ
q
Spec
(2.6)
The direct product Fr(V ) × Cn we view as a bundle over Cn. The remaining bundles are
straightforward; on top is the defining decomposition of the wreath product C× ≀In, on the right
the quotient map q, and on the left the quotient realization of PFr(V ). Observe that all rows and
columns are group-space bundles, i.e., each one is related to a group action. With the exception
of Spec all bundles are principal; Spec itself defines a bundle of homogeneous GL(V )-spaces.
3 Eigenvalue bundle and exceptional points
Let us study the geometry that describes how eigenvalues and eigenvectors depend on the op-
erator. We will find that eigenvalues and eigenvectors form bundles over the non-degenerate
operators. This space N(V ) of non-degenerate operators will reappear as the region free of
singularities. In this way, N(V ) shows us where we can use results from geometry, in partic-
ular concerning parallel transport, which in turn provides a framework for adiabatic quantum
mechanics. We will deal with the eigenvalues in this section, and provide an extended similar
argument in the next section concerning the eigenvectors.
3.1 The spectrum bundle
We start by describing a natural abstract model for the energy bands. Namely, when studying
EPs, we want to follow eigenvalues as a function of the operator. We wish to view this in
a geometric way. For example, we wish to view an eigenvalue function λ = λ(A), with λ(A)
an eigenvalue of the operator A, as a local section. Note that such functions λ(A) are necessarily
local; otherwise EPs could not exist. We will call such functions local eigenvalues.
We quickly come to the conclusion that we should restrict the operators. Namely, degenerate
operators pose a problem as they will form singularities. Indeed, around a degenerate energy,
the energy bands do not resemble a smooth manifold. On the other hand, for A ∈ N(V ) such
issues do not occur. As the following lemma shows, the implicit function theorem yields that
simple eigenvalues always admit an extension to a local eigenvalue. An immediate consequence
is that the restriction from End(V ) to N(V ) is minimal in order to obtain a smooth structure.
Lemma 3.1. Given A ∈ End(V ), then A ∈ N(V ) if and only if
∂p
∂z
(A, λi) ̸= 0, ∀λi ∈ Spec(A).
We can now formalize the bundle which has the local eigenvalues as its local sections. That
is, its local sections are of the form A 7→ (A, λ(A)) with λ(A) a local eigenvalue. This could be
used to define the bundle bottom-up, but we prefer to use the following more explicit top-down
method. Clearly, the total space of the bundle consists of all pairs (A, λ) such that A ∈ N(V )
A Unified View on Geometric Phases and Exceptional Points 9
and λ ∈ Spec(A). We observe that this set is the zero set of the characteristic polynomial map p,
restricted to non-degenerate operators. This viewpoint will form our primary definition1 of the
bundle, because of its algebraic convenience. We use the name spectrum bundle: the fiber above
A ∈ N(V ) is simply Spec(A), and the term bundle we will justify in Theorem 3.4.
Definition 3.2 (spectrum bundle). Given the vector space V , define its spectrum bundle to be
the space
Spec(V ) =
{
(A, λ) ∈ N(V )× C | p(A, λ) = 0
}
.
Furthermore, we write πλ : Spec(V ) → N(V ) for the projection (A, λ) 7→ A.
Our first step in proving the bundle property of πλ is showing that Spec(V ) is a smooth
manifold. This readily follows from the derivative characterization of N(V ) in Lemma 3.1.
Proposition 3.3. The space Spec(V ) is a closed submanifold of N(V )× C of complex dimen-
sion n2.
Proof. Consider the restricted characteristic polynomial p : N(V ) × C → C. As Spec(V ) is
the zero set of this map, by the Submersion Theorem it suffices to show that 0 is a regular
value. Hence we consider the differential dp(A, z), which contains the term ∂p
∂z (A, z)dz. This
is a surjection whenever ∂p
∂z (A, z) ̸= 0, which holds on all of Spec(V ) by Lemma 3.1. Hence
Spec(V ) is a closed submanifold, of the same dimension as N(V ). ■
We are now in place to deduce the bundle structure of πλ. A fiber is of the form Spec(A)
with A ∈ N(V ), hence by definition a set of n distinct points. We thus take the model fiber to
be In. Inspection of πλ yields the following argument.
Theorem 3.4. The map πλ : Spec(V ) → N(V ) defines a fiber bundle with model fiber In.
Proof. First, πλ is a surjective submersion; the implicit function theorem provides local eigen-
values, and so local sections, through any point of Spec(V ). By dimension count, πλ is a local
diffeomorphism. As πλ is also proper, it is a covering map [18]. Then, as each fiber has exactly n
elements, πλ is an In-bundle. Indeed, a local trivialization is a map of the form
ϕ : U × In → Spec(V )|U ,
(A, i) 7→ (A, λi(A)) (3.1)
with λ1, . . . , λn distinct local eigenvalues. ■
This result formalizes the idea that Spec(V ) is locally the union of the graphs of n local
eigenvalues. We remark that the inverse of the local trivialization ϕ in equation (3.1) above can
be written as (A, λ) 7→ (A,#(A, λ)), where #: Spec(V )|U → In yields the label of the graph in
which a point lies. This map # one can interpret as the “local labeling” induced by the chosen
local eigenvalues. We will see this map again when discussing the eigenvectors in Section 4.
3.2 Geometry behind swaps of energies
We will now treat how the abstract theory discussed above facilitates the study of instantaneous
energies in adiabatic quantum mechanics, including the swaps of energies related to exceptional
points (EPs). We assume that instantaneous eigenstates will remain instantaneous eigenstates.
1We remark the similarity with the space M in [21]. However, as we explicitly list the eigenvalue as a coor-
dinate, it is not a multi-valued function here.
10 E.J. Pap, D. Boer and H. Waalkens
Hence instantaneous energies are well-defined. The change of energy in time we will treat
formally using covering theory.
To start, we assume that an experimental set-up is captured by a Hamiltonian operator.
Typically, this Hamiltonian depends on the available system parameters, e.g., cavity size and
field strength in optics and photonics (see, e.g., the review in [23]). This induces a manifold
of system parameter values, which we denote by M . We thus obtain a family of experimental
set-ups, and so a family of Hamiltonians, described by a map
H : M → End(V ),
which sends a configuration of system parameters x ∈M to the HamiltonianH(x) corresponding
to that configuration. For simplicity, we will assume that M and H are smooth. We will refer
to H as the Hamiltonian family, where each H(x) is a Hamiltonian operator on V . We do
however not require the Hamiltonian operators to be Hermitian. We refer to the eigenvalues as
energies and to the eigenvectors as eigenstates.
The idea is now to vary the system parameters. In practice this means we follow a path γ
in M , whose initial point x0 serves as a reference. This results in the time-dependent Hamilto-
nian H(γ(t)). A state ψ(t) is called an instantaneous eigenstate at time t if it satisfies the
eigenvalue problem H(γ(t))ψ(t) = E(t)ψ(t), with E(t) an energy of H(γ(t)). The idea of
the adiabatic approximation is that this relation is preserved in time, at least approximately.
However, this means one first has to make sure that the function E(t) is well-defined.
This fundamental fact can now easily be deduced from the covering properties of Spec(V ).
First, we remark that in order to unambiguously follow a specific energy level of H(γ(t)), the
energy level may not become degenerate at any time. Hence we require H(γ(t)) to be non-
degenerate for all t, i.e., t 7→ H(γ(t)) is a path in N(V ). Let E0 be the energy level of the initial
Hamiltonian H(x0) that we wish to follow in time. Clearly, this defines the point (H(x0), E0) in
the fiber of Spec(V ) above H(x0). Hence, it specifies a unique lift of H ◦ γ to Spec(V ), which is
of the form (H(γ(t)), E(t)) for some function E = E(t). In this way, we obtain the instantaneous
energy for all relevant times in a formal way.
The intuition of this lifting argument is to follow the initial energy along the energy bands
of the family H. If H is fixed, it is convenient to consider these energy bands directly. This
can be done by taking the pull-back of the bundle Spec(V ) → N(V ) along H. To obtain this
pull-back, we must restrict the system parameters to those were H is non-degenerate, i.e., we
must restrict M to the subspace
N(H) := H−1(N(V )) =
{
x ∈M | H(x) has non-degenerate energy levels
}
.
The energy bands of H, with the degenerate points omitted, then form the space
Spec(H) :=
{
(x,E) ∈ N(H)× C | E ∈ Spec(H(x))
}
,
i.e., the space of all pairs of a specific configuration x of the system parameters and an energy E
of the system for this configuration. We call Spec(H) the spectrum bundle of H; it has a natural
projection πHλ : (x,E) 7→ x to N(H), so that the fiber above x is simply Spec(H(x)). The bundle
property of πHλ is guaranteed by the pull-back construction, and relates to πλ as given by the
pull-back diagram below:
Spec(H) Spec(V ) (x,E) (H(x), E)
N(H) N(V ), x H(x).
πH
λ
πλ
H
A Unified View on Geometric Phases and Exceptional Points 11
We see that the path γ can be lifted directly to Spec(H), which is then both an intuitive and
formally correct way to obtain the function E(t) directly. Indeed, by assumption γ lies in N(H),
the initial point is now (x0, E0), and the lift is of the form (γ(t), E(t)).
We see that the study of Spec(H) is a straightforward generalization of Kato’s original set-
ting [17]. There, H was assumed to depend analytically on a single complex variable, so that
the energies were locally given by analytic functions. By analytically continuing them along
a path around a degeneracy, the eigenvalues could return in a different order, and such a degen-
eracy was called an EP. Here, we see that Spec(H) allows us to drop the assumption of analytic
dependence; instead of analytic continuation we can use lifting along a covering map. Hence we
broadened the perspective from complex analytic to general continuous dependence. Note that
this is necessary to include Hermitian or PT -symmetric Hamiltonians, as these are defined by
a conjugate-linear operation and hence do not fit in an analytic setting.
The study of the permutations of energies in the family H is in this construction tantamount
to the study of the covering properties of Spec(H). In fact, the permutations of energies ofH(x0)
are naturally described by the monodromy action at x0, as we will argue now. First, as we wish
to compare energies, we should restore the original system configuration at the end (see also [27]
for a practical argument for this). That is, the path γ in N(H) should return to our reference
x0 ∈ N(H). In other words, γ should be a loop based at x0. Let us write Loop(N(H), x0) for
the set of such loops. We are then interested in how the energies of H(x0) return upon following
them along γ. For an energy E ∈ Spec(H(x0)), this is given by the lift of γ to Spec(H); the
final energy is the endpoint of the lift, which we write as pγ(E). Doing this for every energy
of H(x0) yields the map pγ : Spec(H(x0)) → Spec(H(x0)). This map is a bijection/permutation
of Spec(H(x0)); its inverse is given by traversing γ in the opposite direction. Going over all loops
in Loop(N(H), x0) then yields the group of all possible permutations of Spec(H(x0)), namely
the group
Hol
Spec(H)
N(H) (x0) =
{
pγ ∈ Aut(Spec(H(x0))) | γ ∈ Loop(N(H), x0)
}
,
where the automorphisms are simply bijections. This is indeed a holonomy group as lifting along
a covering can be regarded as parallel transport. We remark that this group was called Λ(x0)
in [27], which is a straightforward generalization of the permutation group studied by Kato [17]
from complex analytic to continuous Hamiltonian families.
Example 3.5. Let us consider a standard case, namely a family with EP2, where an EP2 is
an EP permuting 2 energies. Explicitly, consider H : C → End
(
C2
)
given by
H(x) =
(
1 x
x −1
)
.
The eigenvalues are given by the multi-valued function
√
1 + x2, so N(H) = C \ {±i} and
Spec(H) is the graph of this multi-valued function, which in this case is a Riemann surface. Let
us take a branch cut, and write E±(x) for the two energy branches. Clearly, Spec(H(x0)) =
{E+(x0), E−(x0)}, and if γ is a loop based at x0 encircling an EP once, it induces the map
pγ : E±(x0) 7→ E∓(x0).
Of course, if γ does not encircle any EP, we obtain pγ = idSpec(H(x0)).
The association γ 7→ pγ brings us to a monodromy action as follows. The map pγ is invariant
under continuous deformation of γ, i.e., it only depends on its homotopy class [γ]. The set of such
homotopy classes of loops based at x0 is the (based) fundamental group π1(N(H), x0). The ex-
change of energies can then be expressed by the homomorphism π1(N(H), x0) → Hol
Spec(H)
N(H) (x0),
12 E.J. Pap, D. Boer and H. Waalkens
[γ] 7→ pγ , or equivalently as an action of π1(N(H), x0) on Spec(H(x0)). This is known as the
monodromy action, which we write as
π1(N(H), x0)× Spec(H(x0)) → Spec(H(x0)),
[γ] · (x0, E) = (x0, pγ(E)).
A picture to have in mind is Figure 1; following the energy level along γ defines a path in
Spec(H), which may end at a different energy. An explicit example we treat in Example 3.6
below. We note that the group π1(N(H), x0) is typically non-trivial. Indeed, by Lemma 2.3 the
space N(V ) has real codimension 2 in End(V ) and so a non-trivial fundamental group, which
then holds for a generic N(H) as well. We remark that the relevance of the monodromy action
for EPs was reported in [33, 34]. The space used there is either Spec(H) or a related space we
obtain in Section 5.
Figure 1. Exchange of energies around an EP is given by the monodromy of the chosen path. Plotted
is the real part of the energy levels of the Hamiltonian family in Example 3.5. Upon changing system
parameters (black circle) from the reference value (red dot below), the energy moves (blue line) along the
sheets and does not return to itself. Such behavior only happens around special degeneracies, namely
the EPs. Here the black + marks the EP of the system related to the drawn exchange.
Example 3.6. Continuing Example 3.5, it follows that π1(N(H), x0) is isomorphic to the fun-
damental group of the “figure 8”, which is a free group on two generators, for any reference x0.
Picking x0 = 0, the monodromy action maps both generators of π1(N(H), 0) to the interchange
of +1 and −1 in Spec(H(0)).
The realization of Spec(H) as a pull-back of Spec(V ) allows us to distinguish between inherent
geometric properties coming from Spec(V ) versus artifacts of the family H. Such artifacts can
result in different, i.e., non-isomorphic, spaces Spec(H) for the different H. For example, we
can explicitly add an artifact to a particular H : M → End(V ) by introducing a fictitious
system parameter, i.e., an artificial variable that does not correspond to any change in the
experimental set-up. This increases the dimensions of both M and Spec(H), but there is no
additional physical information. Another example is to include multiple degenerate levels in H
which do not depend on the system parameters. In this case N(H) becomes empty. Of course,
these examples are highly artificial, yet we see it is advisable to be careful not to include any
non-physical information in H.
A Unified View on Geometric Phases and Exceptional Points 13
3.2.1 Merging path method
A common way to demonstrate the existence of an EP in an experimental set-up is to follow
a loop in parameter space and trace the eigenvalues accordingly. If the eigenvalues do not return
to themselves, then there must be (at least one) EP structure inside the loop, provided that any
path in the parameter space M can be contracted to a point. Such an argument can already
be found in [13]. We observe that this method does not attribute a swap to a particular EP –
these are not even present in N(H) – but rather to a homotopy class of loops, which in turn
signals the presence of a degeneracy structure. In practice, the traversing of a loop is done in
a stroboscopic way by measuring the energy for sufficiently close discrete parameter values (e.g.,
in [39]). The formal mathematics behind this argument is straightforward.
Let us go through this merging path method step by step. The change of system parameters
is given by a loop γ in N(H). The spectrum then changes according to the loop Spec(H(γ(t)))
in
(C
n
)
, with n the dimension of the state space. To extract the permutation of the energies, note
that we cannot work in
(C
n
)
alone as the (unordered) spectrum will return to itself regardless of
the presence of EPs. We must thus lift the loop Spec(H(γ(t))) to Cn in order to observe and
extract a permutation. This argument translates directly into the diagram
Cn
N(H) N(V )
(C
n
)
,H Spec
(3.2)
which is the bottom right corner of diagram (2.6) plus the adaptation to the specific system
defined by H.
3.2.2 Real eigenvalue case
The space Spec(H) can have additional properties if the family H satisfies additional assump-
tions. One interesting assumption onH is thatH(x) has real energies for all x ∈M . This occurs,
e.g., when H(x) is always Hermitian w.r.t. some given inner product on V , or when all H(x)
have exact PT symmetry (see [24] on the relation between these). In this case, Spec(H) is
trivial over N(H), which means that all energy bands are globally disconnected. In particular,
no exchanges can occur and EPs cannot be present. We first prove this on the abstract level,
and then use the pull-back by H to go to Spec(H). We remark that the argument also holds if
the eigenvalues are taken in another totally ordered subset of C.
Proposition 3.7. Write R(V ) for the subspace of N(V ) of matrices with real eigenvalues. The
bundle Spec(V ) restricted to R(V ) is trivial.
Proof. If A ∈ R(V ), then there is a unique ordering of Spec(A) by indexing the eigenvalues from
lowest to highest λ1(A) < · · · < λn(A). This establishes a global labelling map Spec(V )|R(V ) →
R(V )× In defined as (A, λi) 7→ (A, i), which is a homeomorphism. ■
Corollary 3.8. If H(x) has real eigenvalues for all x ∈ N(H), then Spec(H) is a trivial
bundle. In particular, there are n distinct energy bands E1(x), . . . , En(x) defined continuously
on all of N(H), and the family H has no EPs.
4 Eigenvector bundle and geometric phases
In the previous section we showed how the energies of an adiabatic quantum system, in particular
their exchanges around EPs, can be treated using the covering space Spec(V ). We will now
extend this formalism to eigenstates, where again non-Hermitian Hamiltonians are allowed.
14 E.J. Pap, D. Boer and H. Waalkens
4.1 The eigenvector bundle
Let us start by studying how eigenvectors vary with the operator. We will use the following
notation. The set of all eigenvectors of an operator A ∈ End(V ) corresponding to an eigenvalue
λ ∈ Spec(A) we denote as
Eigλ(A) =
{
v ∈ V \ {0} | Av = λv
}
= ker(A− λI) \ {0}.
We call this the space of eigenvectors, or eigenvector space, of A corresponding to λ. This
terminology emphasizes that the zero vector is excluded. In this way, each Eigλ(A) is a free
C×-manifold; any non-zero multiple of an eigenvector is again an eigenvector. Similarly, we
define the eigenvector space Eig(A) of A to be the set of all eigenvectors of A. Naturally,
Eig(A) =
⊔
λ∈Spec(A)
Eigλ(A),
which is immediately the partition of Eig(A) into its connected components. We see that Eig(A)
is also endowed with C×-scaling, but need not be a manifold as the eigenvector spaces of the
eigenvalues need not have the same dimension. However, in case A ∈ N(V ) clearly Eig(A) is
a C×-manifold of complex dimension 1, and the above decomposition shows Eig(A) as a union
of C×-torsors.
Here we already found a hint that also concerning eigenvectors we need to restrict to non-
degenerate operators. Indeed, if we wish to view eigenvectors varying with the operator as local
sections of a bundle, i.e., locally defined functions v(A) so that v(A) ∈ Eig(A), then degenerate
operators will again give rise to singularities. We encounter the same situation as before, be it
with more facets. The problem with eigenvalues was that the number of distinct ones suddenly
drops at a degenerate operator. Here, this implies that Eig(A) suddenly consists of less than n
connected components, which renders a bundle structure impossible. In addition, we see for
degenerate yet diagonalizable operators that the eigenvector space can consist of parts with
different dimension, hence Eig(A) is not even a manifold for such operators. Hence we again
restrict to non-degenerate operators. The space that we obtain this way we state in a definition
because of its importance.
Definition 4.1 (eigenvector bundle). Given the vector space V , define its eigenvector bundle
to be the space
Eig(V ) =
{
(A, λ, v) ∈ Spec(V )× V \ {0} | v ∈ Eigλ(A)
}
.
Again, we have used the term bundle straightaway, and in the remainder of this section we
will justify this term.
In fact, we will find that Eig(V ) is a bundle with respect to two different projections. The
idea is that we can view Eig(V ) as a union of total eigenvector spaces or as a disjoint union of
individual eigenrays;
Eig(V ) =
⊔
A∈N(V )
Eig(A) =
⊔
(A,λ)∈Spec(V )
Eigλ(A).
Both of these viewpoints come with their own projection. For the first, one projects (A, λ, v) 7→A,
which we write as the map πλv : Eig(V ) → N(V ). For the second, one only omits the eigenvector
and projects to (A, λ), which defines a map πv : Eig(V ) → Spec(V ). These two projections are
related by the projection πλ, as summarized in the diagram below:
Eig(V ) Spec(V ) (A, λ, v) (A, λ)
N(V ), A.
πv
πλv πλ
(4.1)
A Unified View on Geometric Phases and Exceptional Points 15
Our first step in proving the bundle claims is to show that Eig(V ) is a smooth manifold.
Clearly Eig(V ) is the zero set of
P : Spec(V )× V \ {0} → V,
(A, λ, v) 7→ (λI −A)v.
In this way we obtain the following structure on Eig(V ).
Proposition 4.2. The space Eig(V ) is a closed algebraic C×-submanifold of Spec(V )×V \ {0}
of complex dimension n2 + 1. Moreover, the projection maps πλv and πv are smooth.
Proof. Pick a point (A0, λ0, v0) ∈ Eig(V ) and pick local coordinates (A, λ) on Spec(V ) around
(A0, λ0); this implicitly defines λ = λ(A) with λ(A0) = λ0. Consider now the differential
dP = (λ(A)I −A)dv + d(λ(A)I −A)v.
As P is constant on the C×-orbits, dP has rank at most n − 1. On the other hand, the term
(λ(A)I − A)dv has the same rank as λ(A)I − A, which is n − 1 by non-degeneracy. Hence dP
has constant rank n − 1, and so by the constant rank theorem [18] the fibers of P are closed
submanifolds of (complex) dimension
(
n2 + n
)
− (n − 1) = n2 + 1. The maps πλv and πv are
then restrictions of smooth projections to a submanifold, hence smooth. ■
Both πλv and πv are invariant w.r.t. the C×-action on Eig(V ), which hints at a principal
bundle structure. For πv, this holds; its fibers are eigenrays Eigλ(A), which are C×-torsors,
and πv coincides with the quotient of the action on Eig(V ). However, πλv is clearly not principal.
The fiber of πλv above A ∈ N(V ) is the total eigenvector space Eig(A) of A, which is a union
of C×-torsors, hence not a C×-torsor itself for n > 1. Still, with the eye on adiabatic quantum
mechanics, we are motivated to describe πλv in more detail.
We thus turn to a wider class of bundles, which extends the principal bundles. In short,
principal G-bundles are bundles endowed with a G-action so that the fibers are G-torsors (hence
diffeomorphic to G) and the projection admits G-equivariant local trivializations. Let us only
change the model fiber; we allow it to be a semi-torsor, i.e., a disjoint union of torsors (hence
diffeomorphic to G× I for some index set I). Bundles satisfying this more general condition we
call semi-principal G-bundles [28]. Clearly, as any torsor is a semi-torsor, this is an extension of
the principal bundles. In case the semi-torsor consists of n torsors, where n is finite, we write
the model fiber as G× In. More details on semi-principal bundles can be found in [28].
Let us show that πλv : Eig(V ) → N(V ) is a semi-principal C×-bundle with model fiber
C×× In. The main property left to show is the equivariant local triviality, for which we prepare
ourselves with the following extension property. We use the language of smooth maps, but they
may be taken to be algebraic.
Lemma 4.3. For every point A ∈ N(V ), there is an open neighborhood U ⊂ N(V ) of A and n
eigenvector functions v1, . . . , vn : U → V which constitute an eigenframe at each point of U .
Moreover, this moving eigenframe can be chosen to extend any eigenframe of A.
Proof. The local eigenvector functions exist by Corollary 2.6. The eigenframe they form at A
can be changed to any given eigenframe of A by acting with the appropriate group element
in C× ≀ In, as we did in the proof of Lemma 2.4. ■
Theorem 4.4. The map πλv : Eig(V ) → N(V ) defines a semi-principal C×-bundle with model
fiber C× × In.
16 E.J. Pap, D. Boer and H. Waalkens
Proof. Pick A0 ∈ N(V ), let U ⊂ N(V ) be a neighborhood of A0 and for i = 1, . . . , n let
vi : U → V be a local eigenvector corresponding to eigenvalue λi(A0), according to Lemma 4.3.
One has a map
Φ: U × In × C× → Eig(V )|U ,
(A, i, z) 7→ (A, λi(A), zvi(A)),
which is smooth and clearly intertwines the C×-actions. Its inverse is
(A, λ, v) 7→
(
A,#(A, λ),
[
v/v#(A,λ)(A)
])
,
which satisfies the same properties. Here # is the local labelling, and the last entry denotes the
scalar z such that v = zv#(A,λ)(A). It follows that Φ is a C×-equivariant local trivialization,
hence the bundle property follows. ■
We observe that πλ and πv are closely tied to πλv. Namely, it holds that every semi-principal
G-bundle decomposes naturally into a principal G-bundle and a covering space [28]. These maps
form exactly such a decomposition.
Proposition 4.5. The rule πλv = πλ ◦ πv is the decomposition of πλv in a principal bundle
projection followed by a covering map.
4.2 Connection on the eigenvector bundle
Our next step is to show that Eig(V ) supports a canonical connection. This connection is
compatible with both the semi-principal projection πλv and the principal projection πv. Hence,
we prefer to avoid the term “principal connection”, and make reference only to the action instead.
We will thus use the concept of a G-connection defined as follows. Let G be a Lie group with
algebra g; a 1-form ω on a G-manifold M is a G-connection if ωm ◦ am = idg for all m ∈ M ,
with am the infinitesimal action at m, and L∗
gω = Adg(ω) for any g ∈ G. See also [28] on this.
To describe the connection 1-form on Eig(V ), it is helpful to extend our notation. The set
of eigencovectors of A ∈ End(V ) we denote as Eig∨(A), and those corresponding to a specific
eigenvalue λ by Eig∨λ(A). Note that the zero covector is excluded. As we saw in Lemma 2.1,
for diagonalizable A, there is a bijection between eigenframes and eigencoframes. For non-
degenerate A a stronger statement holds, namely, the bijection already appears on the level of
individual eigenvectors and eigencovectors.
Lemma 4.6. For a non-degenerate matrix A, there is a canonical bijection Eig(A) → Eig∨(A)
sending v ∈ Eigλ(A) to the unique covector θ ∈ Eig∨λ(A) defined by θ(v) = 1.
Proof. Following the decomposition of V by the eigenspaces of A, define the covector θ to
vanish on every eigenspace except the one of λ, where it is fixed by setting θ(v) = 1. ■
This correspondence shows that the scaling action on Eig(A) corresponds to inverse scaling
on Eig∨(A), i.e., we obtain the C×-action on Eig∨(A) given by
z · θ = θz−1.
The bijection Eig(A) → Eig∨(A) is then an isomorphism of C×-manifolds. In addition, as any
triple (A, λ, v) ∈ Eig(V ) defines a unique θ, we are free to augment the triple to (A, λ, v, θ). This
will be a convenient notation when defining maps on Eig(V ), such as the connection 1-form we
seek at the moment. Similarly, one is free to omit λ from the triple. We will use these alternative
notations frequently.
A Unified View on Geometric Phases and Exceptional Points 17
We are now set to describe the connection 1-form on Eig(V ). By definition, this 1-form is
a right-inverse of the infinitesimal action, which is specified by the vector field
∂
∂v
∣∣∣∣
(A,v)
:=
d
dt
∣∣∣∣
t=0
exp(t) · (A, v) = d
dt
∣∣∣∣
t=0
(
A, etv
)
,
which can be thought of as a unit vector field pointing along the eigenray of A through v.
Hence, given a path Γ(t) = (A(t), λ(t), v(t), θ(t)) in Eig(V ), the connection should measure the
component of v̇ along v. This cannot be done with the vector space structure of V alone, but
given the operator this is possible. Indeed, the eigenrays of A(t) provide a natural decomposition
of V at time t. According to this choice, the coefficient of v̇ along v is simply θ(v̇).
Let us describe this argument formally. To obtain the quantity v̇ from Γ̇, we can use the
differential of the projection prV : Eig(V ) → V , (A, λ, v) 7→ v. Technically, this will take values
in the tangent space T(A,v)V , but this can be identified with V in a canonical way. Let us write
dv for (prV )∗ with the image viewed in V . This map can be followed up with θ, which yields
the connection 1-form.
Proposition 4.7. The 1-form ω ∈ Ω1(Eig(V ),C) given by
ω = θdv
is a C×-connection.
Proof. As C× is commutative, the equivariance of ω reduces to invariance. This holds by the
opposite scaling of θ and v: for z ∈ C×, (Lz)
∗ω =
(
θz−1
)
(d(zv)) = θdv = ω. The left-inverse
property of ω follows as ω(∂v) = θdv(∂v) = θ(v) = 1. ■
For future reference, we also remark that this induces the global curvature form K on Eig(V )
in the standard way, i.e., K := dω+ 1
2 [ω, ω] = dω. In coordinates it reads K = dθ∧dv, where dθ
is defined similar to dv. As C× is commutative, the curvature K is invariant under the group
action. Hence it admits push-forward along the quotient map πv to Spec(V ), resulting in the
following.
Lemma 4.8. The curvature form K on Eig(V ) reduces to a unique 2-form k ∈ Ω2(Spec(V ),C),
which satisfies K = π∗v(k).
4.3 Geometry behind the geometric phase
We will now show that the connection ω yields the geometric phases in adiabatic quantum
mechanics in a natural way. That is, Eig(V ) provides a geometric model for the geometric
phase. This is also applicable to non-cyclic states, which appear in the presence of EPs of
non-Hermitian Hamiltonian families.
An argument similar to the one that led us from Spec(V ) to Spec(H) in Section 3.2 holds
here. Namely, the space Eig(V ) can be seen as a general abstract model, which can be adjusted
for a specific Hamiltonian family using pull-back by H. This pull-back of Eig(V ) along the
Hamiltonian family H yields the eigenstate bundle of H, given by
Eig(H) =
{
(x,E, ψ) ∈ Spec(H)× V \ {0} | ψ ∈ EigE(H(x))
}
.
We observe that the pull-back construction of this space immediately guarantees various prop-
erties of Eig(H). Clearly, Eig(H) is a smooth manifold. One can also write an element as
(x,E, ψ, χ), where χ is the unique left-eigenstate of H(x) corresponding to E so that χ(ψ) = 1.
Similarly, one may omit E from the tuple. The projection πHλv : Eig(H) → N(H) is a semi-
principal C×-bundle, and the projection πHv : Eig(H) → Spec(H) is a principal C×-bundle.
18 E.J. Pap, D. Boer and H. Waalkens
The latter bundle confirms that also Eig(H) can have non-trivial topology arising from the
permutations around the EPs of H.
The local sections of πHλv are in correspondence with local eigenstates. Here, a local eigenstate
is a smooth function ψ : U → Eig(H), with U ⊂ N(H) open, such that ψ(x) is an eigenstate
ofH(x). Clearly, this fixes a local eigenvalue E(x). This data is summarized in the corresponding
local section ψ̂ as
ψ̂ : x 7→ (x,E(x), ψ(x)).
Observe that for each x ∈ U , knowing E(x) and ψ(x) fixes a unique left-eigenstate χ(x) of H(x),
which depends smoothly on x. Hence we can include χ(x) as the fourth component of the local
section ψ̂, following our earlier remark on notation of elements of Eig(H).
The space Eig(H) also inherits a connection, which we write as ωH . Again, for any family H,
the smoothness and other properties of Eig(H) are immediate from the pull-back construction.
It is now easy to verify that parallel transport w.r.t. this connection is equivalent to studying
the geometric phase. Namely, using the local section ψ̂, we can express ωH locally as
ψ̂∗ωH = χdψ.
This is indeed the expression reported by Garrison and Wright [12] as the generalization of
the Berry connection. That is, the parallel transport on Eig(H) defined via the horizontal
lifts w.r.t. ωH is equivalent to the calculation of geometric phase, for both Hermitian and non-
Hermitian Hamiltonians. We may thus study geometric phases by studying the space Eig(H).
For example, the connection ωH on Eig(H) can be flat, in which case the geometric phase
is actually of a topological nature. By this we mean that the geometric phase due to a loop γ
is invariant under continuous deformation of γ, which is equivalent to vanishing of the curva-
ture KH of Eig(H). Hence, only non-contractible loops can yield a non-trivial geometric phase.
Flatness is also of practical relevance, as the following result shows.
Proposition 4.9. The connection ωH on Eig(H) is flat in case
� the Hamiltonian family H is an analytic function in a single complex variable z.
� H(x) is a symmetric matrix w.r.t. a fixed basis of V for all x ∈ N(H). Explicitly, if a local
eigenstate ψ is such that ψTψ = 1, then ψ̂∗ωH = 0.
Proof. If H = H(z) is analytic, we can find an analytic local eigenstate ψ(z). Consequently,
ψ̂∗ωH = f(z)dz for some analytic function f . Using coordinates z and z̄ for the parameter
space, ψ̂∗KH is proportional to ∂f(z)
∂z̄ . As this derivative vanishes, it follows that KH = 0.
In the symmetric case, ψT is a non-zero multiple of the left-eigenstate χ corresponding to ψ.
Hence the function ψTψ is non-vanishing, and one may always (locally) scale ψ so that ψTψ = 1,
implying χ = ψT. By partial integration, ψ̂∗ωH = 1
2(χdψ − dχψ) = 1
2
(
ψTdψ − dψTψ
)
= 0. ■
4.3.1 Calculating the lift via an ansatz
An explicit calculation of the lift of a path to Eig(H) can be done using an ansatz technique.
For this technique, it does not matter if the path lies in N(H) or Spec(H). Indeed, if a path γ
in N(H) and an initial eigenstate ψ0 of H(x0) is chosen, then ψ0 has an energy E0, and covering
theory yields the corresponding path γ̄(t) = (γ(t), E(t)) in Spec(H). Conversely, any path γ̄
in Spec(H) yields a path in N(H), which obviously lifts to γ̄. If we restrict to loops the story
changes, as we will see after discussing the ansatz technique.
A Unified View on Geometric Phases and Exceptional Points 19
The ansatz technique here is simply the application of the usual one to the parallel transport
equation. Let us continue with the path γ in N(V ) and the initial state ψ0. This data fixes
a unique lift to Eig(H), which we denote by Γ and which is given by
Γ(t) = (γ(t), E(t),Ψ(t)),
where Ψ(t) is the adiabatically evolved eigenstate at time t, up to dynamical phase. Naturally,
this lift is also given as (γ̄(t),Ψ(t)) in the perspective of lifting a path γ̄ from Spec(V ) to Eig(V ).
In practice, one would not calculate Ψ(t) directly, but instead approach this problem using
reference instantaneous eigenstates ψ(t). That is, for each t, one finds an eigenstate ψ(t) of H(t)
with energy E(t), or ψ(t) ∈ EigE(t)(H(t)) for short. For example, ψ(t) can be obtained by
explicitly calculating an eigenstate of H(γ(t)), and then letting t vary. We may assume that
t 7→ ψ(t) is differentiable and satisfies the initial condition ψ(0) = ψ0. These instantaneous
eigenstates define a path Γ0(t) = (γ(t), E(t), ψ(t), χ(t)) in Eig(H), where χ(t) is the path of
accompanying left-eigenstates.
Of course, Γ0 does not need to be the lift Γ, as equivalently ψ(t) does not need to be equal
the actual state Ψ(t). However, Γ0 can function as an ansatz to calculate Γ explicitly. Indeed,
both ψ(t) and Ψ(t) must lie in EigE(t)(H(t)), and so differ only by a scale factor ef(t). That is,
one has
Γ(t) = ef(t) · Γ0(t) =
(
γ(t), E(t), ef(t)ψ(t), χ(t)e−f(t)
)
with f(t) some differentiable complex-valued function satisfying f(0) = 0. This f depends solely
on the ansatz ψ by imposing the lift condition
0 = (ωH)Γ(t)
(
Γ̇(t)
)
= χ(t)e−f(t)
d
dt
(
ef(t)ψ(t)
)
= ḟ(t) + χ(t)ψ̇(t).
Solving for f(t), we thus find the actual state from the ansatz by application of the scale
factor ef(t);
Ψ(t) = exp
(
−
∫ t
0
χ(t′)ψ̇(t′)dt′
)
ψ(t). (4.2)
This is the general expression for a state ψ0 that undergoes parallel transport along a
path γ or γ̄, also in the non-Hermitian case, expressed using an ansatz. Although the integral∫ t
0 χ(t
′)ψ̇(t′)dt′ is famous for its relation to the geometric phase, without additional assumptions
it does not have physical significance. Indeed, the integral is not ansatz independent. That
is, given another ansatz ψ′(t), then ψ′(t) = exp(a(t))ψ(t) for some complex-valued function
a = a(t), and
∫ t
0 χ(t
′)ψ̇(t′)dt′ becomes
∫ t
0 χ
′(t′)ψ̇′(t′)dt′ + [a(t)− a(0)], which can in principle be
any complex-valued continuous function. The actual state Ψ(t) is of course ansatz independent;
assuming the same initial condition, i.e., ψ′(0) = ψ0, then a(0) = 0 and as desired we obtain
Ψ′(t) = exp
(
−
∫ t
0
χ′(t′)ψ̇′(t′)dt′
)
ψ′(t) = exp
(
−
∫ t
0
χ(t′)ψ̇(t′)dt′ − a(t)
)
ea(t)ψ(t) = Ψ(t).
This confirms that the integral is, in general, only a correction factor to the specific ansatz ψ(t);
it merely quantifies how much our ansatz ψ(t) was away from being the lift Ψ(t).
In fact, just from physical arguments one should not expect the integral to yield an observable;
we did not assume the state to be cyclic, so that there is no particular phase the integral can
be equal to. Of course, if the state is cyclic, say the state returns at time T , then a geometric
phase is well-defined. With an additional assumption on the ansatz ψ, namely ψ(T ) = ψ0,
this geometric phase is indeed given by the integral at t = T . However, even in this cyclic
20 E.J. Pap, D. Boer and H. Waalkens
case, if we consider intermediate times t, i.e., 0 < t < T , the integral does not yield a physical
quantity; we fixed a(T ) = 0, but for intermediate times a(t) can still be anything. In other
words, concerning a state acquiring a geometric phase, there is no well-defined rate at which
this happens. This reflects that a geometric phase depends only on the locus of a (closed) path,
not its parametrization. Still, the integral
∫ t
0 χ
′(t′)ψ̇′(t′)dt′ can be related to path “lengths”, as
we consider in Section 4.6.3.
We will consider the state evolution in more detail in the following. We will distinguish
between the cyclic and the non-cyclic case. The approach is summarized in diagram (4.3)
below, which is the pullback of diagram (4.1) along H. Namely, we see that the projection
πHλv : Eig(H) → N(H) can be considered as the main bundle to model the evolution of states.
Indeed, given any loop in N(H), it will certainly induce a holonomy operation on Eig(H),
regardless of states being cyclic or non-cyclic. In contrast, one can also consider the bundle
πHv : Eig(H) → Spec(H). Clearly, loops in Spec(H) correspond to cyclic evolution only.2 How-
ever, this bundle is principal and so has the advantage that its holonomies are easier to describe.
We thus remark that for cyclic states one uses πHv , and one uses πHλv primarily for non-cyclic
states. We also emphasize that the calculation of Ψ(t) above can be used in both cases:
Eig(H) Spec(H) (x,E, ψ) (x,E)
N(H), x.
cyclic states
non-cyclic states energy swaps
πH
v
πH
λv πH
λ
(4.3)
4.3.2 The cyclic case
Let us start with the cyclic case. As said, this concerns the bundle πHv : Eig(H) → Spec(H).
We remark that the only difference between a path in N(H) and a path in Spec(H) is that
the latter not only specifies the change in system parameters, but also which energy level is
of interest. Moreover, as we have seen, fixing an initial energy E0 uniquely specifies a path in
Spec(H) given a path in N(H). It is thus natural to consider a loop γ̄ in Spec(H) as the input
data for cyclic evolution.
Let us now consider the evolution of the eigenstates. Denote the basepoint of γ̄ again by
(x0, E0), and assume γ̄ returns to this point at time t = T . The state evolution is then described
by the parallel transport map Pγ̄ , e.g., Pγ̄(ψ0) = Ψ(T ) following the previous calculation. As πHv
defines a principal bundle and ωH is a principal connection, Pγ̄ follows from standard holonomy
theory. That is, Pγ̄ is an automorphism of the C×-torsor EigE0
(H(x0)), and thus amounts to
scaling by a unique element in C×. This element is clearly the geometric phase factor that any
state in EigE0
(H(x0)) acquires due to following γ̄. As we allow for non-Hermitian Hamiltonians,
this phase factor need not be unitary. We hence obtain a definition of (generalized) geometric
phase from holonomy as follows.
Definition 4.10 (geometric phase). Let γ̄ be a loop in Spec(H) based at (x0, E0). The geometric
phase γgeo due to γ̄ is defined, modulo 2π, via
Pγ̄ = eiγgeo · idEigE0
(H(x0)).
It now remains to calculate γgeo explicitly. For this, we can use the earlier result from
the ansatz technique. As usual, one only has to compare the phase difference between the
final state Ψ(T ) and the initial state ψ0, i.e., Ψ(T ) = Pγ̄(ψ0) = exp(iγgeo)ψ0. We thus con-
sider the unique non-zero complex scalar [Ψ(T )/ψ0] such that Ψ(T ) = [Ψ(T )/ψ0]ψ0. Equating
2In an adiabatic setting, with cyclic we also assume that the system parameters are restored.
A Unified View on Geometric Phases and Exceptional Points 21
exp(iγgeo) = [Ψ(T )/Ψ(0)] and substituting our expressing for Ψ(t) in terms of our ansatz ψ(t),
we find the expression
γgeo = i
∫ T
0
χ(t)ψ̇(t)dt− i ln([ψ(T )/ψ0]), (4.4)
where the logarithm term yields the usual modulo 2π of a phase. By construction this is invariant
under both replacing ψ0 with another state in the same ray and picking another ansatz ψ.
We see two terms of different nature in equation (4.4). The integral term is clearly the usual
integral for the geometric phase, and corrects for changes of ψ along its own direction. The
logarithm term is a correction for ψ not closing on itself. Observe that only the two terms
together are invariant under changing the choice of the ansatz ψ. The logarithm term vanishes
whenever ψ(1) = ψ(0), which happens, e.g., if ψ is built using a local eigenstate. On the other
hand, the integral term vanishes, e.g., when ψ is chosen to be the lift Ψ, in which case the
integrand is identically zero. In this case γgeo can be computed using the end points only, see
Example 4.11 below for an explicit example.
Example 4.11. Let us consider a typical system with a diabolic point (DP), named after the
diabolo shape of the energy bands [6]. Let us pick V = C2 with standard basis (e1, e2), in which
the Hamiltonian family reads
H(a, b) =
(
a b
b −a
)
,
where a, b are real numbers, i.e., M = R2. The energy bands are given as λ±(a, b) = ±
√
a2 + b2,
which have a single degeneracy at the origin. This is located at the apex of the diabolo, and is
the DP of this system.
Let us show how the geometric phase integral can be seen as a correction factor. Therefore,
let us traverse the unit circle using the path γ(t) = (cos(t), sin(t)) with time interval [0, 2π],
and consider the λ+ level (λ− is similar). By looking at H(γ(t))− λ+(γ(t))I, one readily finds
an nsatz ψ(t) for the evolution of the eigenstate, together with a left-eigenstate path χ(t), as
ψ(t) =
1
2
(
1 + cos(t)
sin(t)
)
, χ(t) =
1
1 + cos(t)
(
1 + cos(t)
sin(t)
)T
.
The factor 1/2 is introduced to have ψ(0) = e1. Note that ψ and χ are defined only for t ∈ [0, π);
being a (left-)eigenstate, they are not allowed to vanish. Still, for these times we can calculate
the lift of γ(t) starting at e1.
We can correct the ansatz ψ(t) following equation (4.2). We thus evaluate the geometric
phase integral, whose integrand is
χ(t)ψ̇(t) =
− sin(t)
2(1 + cos(t))
=
1
2
d
dt
ln(1 + cos(t)).
Hence we find the lift of γ starting at e1 to be
Ψ(t) =
√
1
2(1 + cos(t))
(
1 + cos(t)
sin(t)
)
=
(
cos(t/2)
sin(t/2)
)
.
As the scale factor is real, we can interpret it as a length correction. This length interpretation
clearly shows that our original ψ(t) was varying along itself.
22 E.J. Pap, D. Boer and H. Waalkens
Note that our final expression of Ψ(t) can be extended to arbitrary times, hence yields the
full lift of γ. Consequently, it is now convenient to obtain the geometric phase via Ψ(t). For
lifts, only the logarithm term in equation (4.4) contributes, and we find the phase (modulo 2π)
γgeo = −i ln([Ψ(2π)/ψ0]) = −i ln([−e1/e1]) = −i ln(−1) = π.
In the perspective of loops in Spec(H), the above concerns the loop γ̄+(t) := (γ(t), λ+(γ(t))) =
(γ(t), 1), revealing that Pγ̄+ = −id. A similar argument shows that the loop γ̄−(t) = (γ(t),−1),
i.e., considering the other energy band, yields Pγ̄− = −id as well.
In some situations, one can express the geometric phase alternatively as a curvature integral.
A standard argument is as follows. Let U ⊂ N(H) be a neighborhood on which a local eigenstate
ψ = ψ(x) is defined, which has as partner the local left-eigenstate χ. If γ is a loop in U that
forms the boundary of a surface S also contained in U , then the geometric phase is given by
γgeo = i
∫
γ
χdψ = i
∫
∂S
ψ̂∗ωH = i
∫
S
ψ̂∗(dωH) = i
∫
ψ̂(S)
KH .
We see that the assumption of γ being a boundary is essential. If γ is a loop but not a boundary,
then the geometric phase is not given by a curvature integral. This motivates us to consider
homology theory.
In order to do this, it is convenient to work on Spec(H) instead. Indeed, the integral above can
be rewritten as an integral over kH , which is the reduced curvature on Spec(H). Clearly, the local
eigenstate ψ(x) defines an energy function E(x). The map Ê : U → Spec(H), x 7→ (x,E(x))
is then a local section of Spec(H). Moreover, as Ê = πHv ◦ ψ̂, where πHv is the projection
Eig(H) → Spec(H), we have
γgeo = i
∫
S
ψ̂∗(KH) = i
∫
S
ψ̂∗((πHv )∗
(kH)
)
= i
∫
S
Ê∗(kH) = i
∫
Ê(S)
kH .
Let us interpret this result. First, we remark that the integral over kH is manifestly inde-
pendent of the chosen local eigenstate ψ. More precisely, only the energy bands matter, not the
exact eigenstates. In addition, we now consider surfaces in Spec(H), hence the above argument
can be used whenever our path γ̄ is a boundary. We thus find that the homology of Spec(H),
i.e., the homology of the energy bands, plays a key role in the relation of geometric phase to cur-
vature. Moreover, in case Eig(H) is flat, the homology theory allows one to see the topological
nature of the geometric phase in an explicit way, as shown in the following.
Proposition 4.12. Let γ̄ be a loop in Spec(H). If γ̄ is the boundary of a surface Σ in Spec(H),
then the geometric phase acquired by traversing γ̄ equals i
∫
Σ kH .
Corollary 4.13. If Eig(H) is flat, then the geometric phase due to a loop in Spec(H) only
depends on the class of the loop in the first homology group H1(Spec(H)).
Example 4.14. Let us continue Example 4.11. As H is symmetric, by Proposition 4.9 the con-
nection is flat. Hence the geometric phases only depend on the homology of Spec(H), and thus
are of topological nature. Clearly, Spec(H) is the diabolo minus the DP, which is homeomorphic
to two punctured planes. Hence H1(Spec(H)) ∼= Z2, generated by (the classes of) the images
of the unit circle under λ+ and λ−. It thus suffices to know the phase due to each generator,
which is π following our earlier calculation in Example 4.11.
It is now straightforward to recognize the bundles introduced by Simon [32] in the bundle
Eig(H) → Spec(H). According to Corollary 3.8, for Hermitian H the bundle Spec(H) → N(H)
is trivial, i.e., Spec(H) separates into n distinct energy bands. Let us label these energy bands
A Unified View on Geometric Phases and Exceptional Points 23
by 1, . . . , n and write Speck(H) for energy band k, and similarly Eigk(H) for the subbundle
over Speck(H). Then Eigk(H) → Speck(H) is a principal C×-bundle, which is the bundle
used by Simon for energy level k adapted to our language. It is thus geometrically clear why
this approach does not apply to non-Hermitian Hamiltonians; in that case the energy bands in
Spec(H) need not be separated but connected via “spiral staircase” like structures. If this is
the case, i.e., if non-cyclic states appear, then the bundle Eig(H) → Spec(H) can still model
these states by parallel transport, but as the path γ̄ is then not a loop this does not fit in the
framework of holonomy.
4.3.3 The non-cyclic case
Let us now demonstrate how non-cyclic states do have a holonomy description when using the
other projection, i.e., the bundle πHλv : Eig(H) → N(H). Here, holonomy is to be understood
in the context of a semi-principal bundle. Similarly to the case above, parallel transport along
a loop γ in N(H) based at x0 induces the parallel transport map Pγ , which is an automorphism
of the C×-semi-torsor Eig(H(x0)). The set of all Pγ for a fixed base point x0 then defines the
holonomy group
Hol
Eig(H)
N(H) (x0) =
{
Pγ ∈ Aut(Eig(H(x0))) | γ ∈ Loop(N(H), x0)
}
.
However, these automorphisms are harder to describe. Whereas we could previously identify
a map Pγ̄ with an element in C×, this need not be for Pγ . In particular, by definition a non-cyclic
state does not return to the same group orbit, i.e., the initial eigenray, hence no element in C×
can relate the initial and final states. Instead, one must find how the eigenrays are transported
individually, as illustrated in the following example.
Example 4.15. Let us consider a standard EP2 example, using the Hamiltonian family of
Example 3.5. We take x0 = 0 again as our reference point, and from there we encircle the EP
at x = +i in positive direction by following a path γ. One can take γ to be a circular path, but
as the connection is flat by Proposition 4.9, the exact shape of γ does not play a role.
The evolution of the states due to encircling γ is captured by the map Pγ : Eig(H(0)) →
Eig(H(0)). Clearly, Eig(H(0)) is the disjoint union of the two eigenrays, i.e.,
Eig(H(0)) = C×e1 ⊔ C×e2,
where e1 = (1, 0)T and e2 = (0, 1)T are the standard basis vectors. Hence, an element of
Eig(H(0)) is of the form ze1 or ze2 with z ∈ C×. We emphasize that Eig(H(0)) is a semi-
torsor and not a vector space; linear combinations of e1 and e2 are not present, as these are not
eigenstates of H(0). Accordingly, Pγ is equivariant and not linear.
We can now calculate Pγ as follows. First, it is sufficient to know Pγ(e1) and Pγ(e2) due
to equivariance; Pγ(zek) = zPγ(ek) for k = 1, 2. That is, it is sufficient to follow e1 and e2
around the EP. This can be done using an explicit parametrization of Spec(H). One can easily
find eigenstates depending on x, and following Proposition 4.9 we “normalize” them so that no
geometric phase will appear.3 Hence we arrive at the expressions
ψ±(x) = ∓ 1√
2(1 + x2 ∓
√
1 + x2)
(
−x
1∓
√
1 + x2
)
,
where we remark that ψ+ has a removable singularity at x = 0; the limit reads limx→0 ψ+(x)
= e1. Hence at x0 = 0, ψ+(0) = e1 and ψ−(0) = e2 as desired. The lifts of these states to
3Note that this uses flatness of the connection. In the non-flat case the geometric phase depends on the exact
shape of γ and so cannot be expressed using local eigenstates. One should then use, e.g., the ansatz method
instead.
24 E.J. Pap, D. Boer and H. Waalkens
Eig(H) can thus be found by inspection of ψ±(x). Encircling the EP in positive direction swaps
the ± signs, and in addition the overall root in the denominator obtains a factor of i. Thus,
after following ψ±(x) around the EP, we return with i−1ψ∓(x). At our reference x0 = 0, this
rule becomes e1 7→ −ie2, e2 7→ −ie1. This information is enough to specify Pγ , which we thus
find to be given by
Pγ : ze1 7→ −ize2, ze2 7→ −ize1. (4.5)
In order to study Pγ , let us consider the following statement showing how a general auto-
morphism of a semi-torsor can be studied by its invariant subspaces, similar to linear algebra.
Proposition 4.16. Let G be a Lie group, F a G-semi-torsor and ϕ : F → F an automorphism.
Then F = ⊔i∈I′Fi, for some index set I ′, decomposes F into minimal ϕ-invariant subspaces,
i.e., ϕ(Fi) = Fi for all i. Moreover, if G is Abelian and a particular Fi consists of k <∞ orbits,
then (ϕ|Fi)
k equals translation by an element of G.
Proof. Clearly, the index set I ′ is the original orbit space modulo the relation that orbits
mapped into one another by ϕ are identified. If Fi consists of k orbits, then by minimality
(ϕ|Fi)
k preserves the orbits in Fi. Picking f ∈ Fi, we find ϕk(f) = g(f)f for some g(f) ∈ G.
If G is Abelian, then g(f) is constant on the orbit through f . In addition, as equivariance yields
ϕk(ϕ(f)) = g(f)ϕ(f), we see ϕ(f) is scaled by the same element. It follows that g(f) is constant
on the subspace Fi. ■
For the map Pγ , it is clear that a minimal subspace is any minimal union of eigenrays whose
energies are permuted upon traversing γ. The element of C× associated to such a minimal union
is a phase factor. Indeed, if there are k rays in the union, then a ray first returns to itself by
following γ exactly k times, after which is has obtained precisely this phase factor. Again, if
k > 1 the state is non-cyclic and there is no definite phase between an initial state ψ0 in the
union and its transport Pγ(ψ0). We come back to this in Section 5, where we treat how Pγ can
be expressed via a holonomy matrix. Of course, if k = 1, i.e., ψ0 is cyclic, then we do recover
the geometric phase. We summarize these findings in the following.
Corollary 4.17. Given a loop γ, the invariant subspaces of Pγ are the minimal unions of
eigenrays. If a union consists of k eigenrays, then the characteristic phase of the union equals
the geometric phase obtained by traversing the loop γ̄k, where γ̄ is the lift of γ to Spec(H) with
an energy corresponding to any of the eigenrays in the union.
It is also clear that Pγ contains the information of the underlying permutation pγ of the
energies. This is formally expressed by πHv being a bundle map that is equivariant w.r.t. the
collapsing quotient homomorphism C× → 1. As πHv maps Eig(H) to Spec(H), this quotient
reduces Pγ to pγ .
Proposition 4.18. For any γ ∈ Loop(N(H), x0), the map Pγ induces the following commutative
diagram:
Eig(H(x0)) Eig(H(x0))
Spec(H(x0)) Spec(H(x0)).
Pγ
πH
v πH
v
pγ
A Unified View on Geometric Phases and Exceptional Points 25
4.4 Including the dynamical phase
The geometric properties of adiabatic dynamics for non-degenerate operators, Hermitian or non-
Hermitian, can be described on Eig(V ) using the connection ω. This leaves out an important
non-geometrical property, namely the dynamical phase. Nevertheless, Eig(V ) does support
a calculation of the dynamical phase. This builds on a particular complex-valued function,
which simply extracts the eigenvalue from the elements in Eig(V ), i.e.,
E : Eig(V ) → C,
(A, λ, v) 7→ λ.
Given a lift Γ in Eig(V ), the corresponding dynamical phase is then the integral
γdyn = − i
ℏ
∫
E(Γ(t))dt.
It is even possible to put the dynamical phase explicitly in the lift. Clearly, the lift Γ without
dynamical phase vanishes under the covariant derivative
D = d + ω.
If one modifies this to
Dtot = d + ω +
i
ℏ
E
and calculates the lift Γtot of γ given by DtotΓtot = 0, then Γtot is the path of the eigenstate,
including the dynamical phase, assuming γ is parametrized by physical time.
4.5 Relation with the work of Aharonov and Anandan
We found that Eig(V ) provides a framework suitable for any non-degenerate finite-dimensional
Hamiltonian, including non-Hermitian Hamiltonians in particular. However, this brings us to the
question how the theory of Hermitian systems relates to Eig(V ). The geometric framework for
such Hermitian cases was pioneered by Aharonov and Anandan in [1] (see also the elaboration
in [2]). The geometric spaces found there differ significantly from the spaces obtained here.
Nevertheless, we will show that they can be obtained from Eig(V ).
Let us summarize the theory of [1], rephrasing it in line with our approach to Eig(V ). First,
the state space V is now assumed to be a Hilbert space, i.e., V should be equipped with a Hermi-
tian inner product ⟨ | ⟩. This allows one to define the unit sphere S1(V ) inside V by restricting
to norm 1 states. As norm 1 fixes a state up to a U(1)-phase, S1(V ) is naturally a U(1)-manifold,
and the quotient is the projective space P (V ) of V . This quotient defines the principal bundle
U(1) S1(V ) P (V ), (4.6)
which is the central object in the formalism. The base P (V ) can be viewed as the space of rays
in V , but also as the set of projectors projecting to a line. That is, the ray through |ψ⟩ ∈ S1(V )
can be identified with the projection operator |ψ⟩ ⟨ψ|. The metric on V obtained from the inner
product ⟨ | ⟩ induces a metric tensor on S1(V ), and hence a connection 1-form. On a path |ψ(t)⟩
in S1(V ), this connection yields ⟨ψ(t)|ψ̇(t)⟩. If one lifts a loop from P (V ) to S1(V ), it follows
that the final state must lie in the same ray as the initial state. In this case, one says that the
state is cyclic. If the path |ψ(t)⟩ satisfies |ψ(1)⟩ = |ψ(0)⟩, then the obtained Aharonov–Anan-
dan (AA) phase is given by the integral i
∫ 1
0 ⟨ψ(t)|ψ̇(t)⟩dt. This coincides with the adiabatic
Berry phase if |ψ(t)⟩ evolves adiabatically.
26 E.J. Pap, D. Boer and H. Waalkens
The AA phase is thus a generalization of the Berry phase from adiabatic state evolution to
any path of states. Hence the AA phase is viewed as a non-adiabatic generalization. Still, we
argue that it can be obtained from Eig(V ), equipped with the “adiabatic” connection ω. That
is, we will show that the bundle in equation (4.6), including connection, can be obtained by
collapsing the principal bundle πv : Eig(V ) → Spec(V ). This can be done in two steps. First,
we use the inner product to restrict the operators to the Hermitian ones and the vectors to
normalized ones. After this, listing the operator will be redundant, and the second step is to
discard it.
Let us describe the first step. The only additional ingredient we use is the chosen inner
product ⟨ | ⟩ on V . It allows us to talk about Hermitian operators, and we restrict N(V )
accordingly to the closed subset
N⟨ | ⟩(V ) :=
{
A ∈ N(V ) | A† = A with respect to ⟨ | ⟩
}
of all non-degenerate Hermitian operators on V . This is a non-canonical subset of N(V ); a dif-
ferent inner product may yield a different subset. Consequently, there are the subbundles over
N⟨ | ⟩(V ) given by
Spec⟨ | ⟩(V ) :=
{
(A, λ) ∈ Spec(V ) | A† = A with respect to ⟨ | ⟩
}
,
Eig⟨ | ⟩(V ) :=
{
(A, λ, v) ∈ Eig(V ) | A† = A, ∥v∥ = 1 with respect to ⟨ | ⟩
}
.
The C×-action on Eig(V ) reduces to U(1)-phase rotation on Eig⟨ | ⟩(V ), which has quotient space
Spec⟨ | ⟩(V ).
The key observation now is that, after this restriction, we no longer need to know A and λ
to compute the eigencovector and eigenprojectors. Indeed, given the normalized vector |v⟩, the
covector θ is ⟨v| and the eigenprojector is |v⟩ ⟨v|, regardless of the exact A and λ. Hence, we
are motivated to discard A and λ. To do so, it is convenient to describe Spec(V ) using pro-
jectors. Clearly, any pair (A, λ) ∈ Spec(V ) defines an eigenprojector P(A,λ) projecting on the
eigenspace of A corresponding to λ. Conversely, given an eigenprojector P , the eigenvalue can
be retrieved from the identity AP = PA = λP . Hence we may write (A, λ) ∈ Spec(V ) equiva-
lently as (A,P(A,λ)). For Spec⟨ | ⟩(V ), we may even write an element as (A, |v⟩ ⟨v|), where |v⟩ is
a normalized eigenvector of A.
We can now perform the second step, i.e., the reduction. Reducing Spec⟨ | ⟩(V ) to the
space P (V ) is straightforward using the projector description where the element (A, |v⟩ ⟨v|)
goes to |v⟩ ⟨v|. Reducing Eig⟨ | ⟩(V ) to S1(V ) is similar; we only keep the vector. Together, these
maps define a morphism of bundles as follows:
Eig⟨ | ⟩(V ) S1(V ) (A, λ, |v⟩) |v⟩
Spec⟨ | ⟩(V ) P (V ), (A, |v⟩ ⟨v|) |v⟩ ⟨v| .
πv
Our final claim is that the connection ω on Eig(V ) also carries over to S1(V ). This again
follows the same two steps. First, clearly ω restricts to a U(1)-connection on Eig⟨ | ⟩(V ). Second,
this restricted form admits push-forward to S1(V ). Concerning explicit formulas, this push-
forward is given by substituting θ = ⟨v|, which yields exactly the connection used to define
the AA phase. The following then summarizes these findings.
Proposition 4.19. The restricted projection Eig⟨ | ⟩(V ) → Spec⟨ | ⟩(V ) has a canonical reduction
to the bundle S1(V ) → P (V ). Moreover, the restricted connection on Eig⟨ | ⟩(V ) admits push-
forward to S1(V ), yielding the standard U(1)-connection as used for the AA phase.
A Unified View on Geometric Phases and Exceptional Points 27
We conclude that the above projection allows us to translate the general theory of Eig(V )
to more specific results, enabled by a Hermitian inner product. We will use the phrase “in the
Hermitian case” to indicate such a passage has happened. We already saw that notationally
this amounts to replacing θ by ⟨v|, but the broad picture contains more. For example, Eig(V )
is locally partitioned according to eigenvalues, while this is completely absent in S1(V ).
4.6 Quantum geometric tensor
We will show that the covariant derivatives naturally define a tensor on Eig(V ), which is
a straightforward generalization of the quantum geometric tensor. We also consider its reduced
version on Spec(V ). Before we discuss the tensor itself, we first introduce convenient bases of
tangent spaces of Eig(V ) and Spec(V ). Afterwards we comment on a relation between geometric
phase and distance.
4.6.1 Bases for tangent space of spectrum and eigenvector bundles
Let us formulate (complex) bases for the tangent spaces T(A,λ,v) Eig(V ) and T(A,λ) Spec(V ). The
idea is that we pass the v, λ and A components one-by-one. In this way, both tangent spaces
can be described in a similar way.
Let us start with the eigenvector part. Clearly, the main difference between the two tan-
gent spaces is that T(A,λ,v) Eig(V ) has a tangent along the eigenray Eigλ(A). This direction is
naturally spanned by the fundamental tangent vector ∂v originating from the scaling action.
An advantage of picking ∂v is that we may pick the remaining tangent vectors in T(A,λ,v) Eig(V )
to be horizontal. These tangent are then the horizontal lifts of unique tangents in T(A,λ) Spec(V ),
which means we cover both tangent spaces simultaneously.
We continue with the eigenvalue part. Obviously, a change of eigenvalue must be accompanied
by a change of operator. Hence, let us change the operator only by what is absolutely necessary.
Writing P = vθ for the eigenprojector of A corresponding to λ, a shift in λ is then given by the
tangent
∂λ =
d
dt
∣∣∣∣
t=0
(A+ tP, λ+ t, v, θ).
We must then find n − 1 other tangent vectors for the other eigenvalues of A. These can
be obtained via paths of the form (A + tP ′, λ, v, θ), with P ′ the corresponding eigenprojector.
Note that λ is constant as we vary another eigenvalue, but still consider tangents at (A, λ, v).
The combination of these n tangent vectors defines the tuple ∂λ̃, where we pick an ordering of
Spec(A). Writing the coefficients of these vectors as ∆λ̃, again cf the ordering, we obtain the
linear combination ∆λ̃∂λ̃. The corresponding tangents in T(A,λ) Spec(V ) are similar.
It thus remains to describe all changes in operator and eigenvector, where the eigenvector
should not change along itself. As we should now avoid to change the spectrum of A, let us use
similarity transformations. That is, we conjugate by the operator eitX , where iX ∈ End(V ) can
be viewed as a generator; the extra i is for later convenience. We then obtain the infinitesimal
conjugation action (extending equation (2.5)), which we view as the linear map End(V ) →
T(A,λ,v) Eig(V ) given by
iX 7→ d
dt
∣∣∣∣
t=0
(
eitXAe−itX , λ, eitXv, θe−itX
)
.
As is, this parametrization of T(A,λ,v) Eig(V ) is not compatible with our earlier choices. For
instance, if iX = P we retrieve ∂v, and for other eigenprojectors the obtained tangent vanishes.
Hence we require X to be free of eigenprojectors of A, which means that the matrix of X w.r.t.
28 E.J. Pap, D. Boer and H. Waalkens
any eigenframe of A has zero diagonal. In this case, we say X is A-free. Note that the obtained
tangent is horizontal if and only if θXv = 0, which is automatically satisfied for A-free X.
In summary, we may express a general element v ∈ T(A,λ,v,θ) Eig(V ) as
v = iX +∆λ̃∂λ̃ + z∂v,
where for T(A,λ) Spec(V ) the first two terms suffice. This is a basis if we impose X to be A-free,
in which case the terms span a subspace of dimension n2 − n, n and 1, respectively. The values
of dv and dθ then read
dv(v) =
d
dt
∣∣∣∣
t=0
eitXetzv = (iX + z)v, dθ(v) = −θ(iX + z).
We finally remark that it is possible to not impose X to be A-free; the map to T(A,λ,v) Eig(V ) is
then still surjective, but no longer injective. This can be convenient in practice; X can play the
role of a Schrödinger Hamiltonian, which need not be A-free. We will keep this in mind when
describing the tensors in the following.
4.6.2 Generalized quantum geometric tensor
On the space Eig(V ) there is a canonical tensor which in the Hermitian case reduces to the
quantum geometric tensor (QGT). This QGT was first reported in [30], where it was found by
looking at infinitesimal distance between states. Its anti-symmetric part was later recognized
to essentially be the Berry curvature, demonstrating its relevance to the geometric phase, while
its symmetric part yields a metric on parameter space [4]. We now show that Eig(V ) supports
a more general tensor, which relates directly to covariant derivatives. Moreover, this generalized
tensor makes no reference to an inner product. It is thus also incorporates a generalization of
the QGT based on PT -symmetry as reported in [40].
Let us start from the standard expression. Fixing local coordinates on N(H) and a local
normalized eigenstate |ψ⟩ = |ψ(x)⟩, the QGT is given by
Tij = ⟨∂iψ| (1− |ψ⟩ ⟨ψ|) |∂jψ⟩.
We regard this to be the pull-back of a more abstract tensor G on Eig(V ). Clearly, this G is
simply given by
G(A,v,θ)(v1, v2) = dθ(v1)(1− vθ)dv(v2), v1, v2 ∈ T(A,v,θ) Eig(V ). (4.7)
Because of this straightforward generalization from the Hermitian case to Eig(V ), we refer to G
as the (generalized) quantum geometric tensor. In addition, G is a natural tensor on Eig(V )
in the following sense. Observe that the projector 1 − vθ is naturally obtained by taking the
covariant derivative, as
∇v(v) = dv(v)− ω(v)v = dv(v)− θ(dv(v))v = (1− vθ)dv(v), v ∈ T(A,v,θ) Eig(V ),
and similarly ∇θ = dθ(1− vθ). We thus observe that G is the natural combination
G(A,v,θ)(v1, v2) = ∇θ(v1)(∇v(v2)), v1, v2 ∈ T(A,v,θ) Eig(V ),
which directly displays its scale invariance.
We can write G more explicitly using our description of the tangent space T(A,λ,v,θ) Eig(V ).
Introducing labels 1 and 2 according to the two tangent vectors, this yields
G(A,v,θ)(v1, v2) = θ(X1X2)v − (θX1v)(θX2v).
A Unified View on Geometric Phases and Exceptional Points 29
One can also write this using scale invariant quantities only. Writing points of Eig(V ) as
(A,P, v, θ), we find
G(A,P,v,θ)(v1, v2) = tr(PX1X2)− tr(PX1) tr(PX2).
This also provides an explicit form of the reduced QGT defined on Spec(V ). We observe that
in these expressions we need not impose the Xi to be A-free. This is similar to correcting for
a non-zero mean in probability theory; the second term provides a correction that vanishes for
A-free Xi. Hence we choose not to impose the A-free condition, and instead keep the second
term.
In the Hermitian case, the QGT is the sum of a symmetric tensor, known as the quantum
metric, plus an anti-symmetric part proportional to the Berry curvature [4]. We find a similar
decomposition here. The anti-symmetric part of G is readily seen to be K/2, as
Galt
(A,v,θ)(v1, v2) = θ
[X1, X2]
2
v =
1
2
dθ ∧ dv(v1, v2) =
1
2
K(A,v,θ)(v1, v2).
We may also write this as tr
(
P [X1,X2]
2
)
, which in addition yields an explicit form of the reduced
curvature k. The symmetric part M := Gsym of G is then a tensor on Eig(V ) generalizing the
standard quantum metric tensor (QMT). Hence we will refer to M using the same name. A scale
invariant explicit form is
M(A,P,v,θ)(v1, v2) = covP (X1, X2) := tr
(
P
{X1, X2}
2
)
− tr(PX1) tr(PX2),
where covP is the covariance of non-commutative operators w.r.t. the density P . We hence obtain
a decomposition similar to [4, equation (30)], but now for the generalized tensors on Eig(V ).
Proposition 4.20. The QGT on Eig(V ) decomposes as a linear combination of the QMT and
the curvature as
G = M+
1
2
K,
which is the decomposition of G into its symmetric and anti-symmetric part, respectively.
Notable properties that do not generalize from the Hermitian case are the following.
Clearly M and K are complex rather than real-valued tensors, and no longer the real resp.
imaginary part of G. In addition, M is a degenerate form, hence does not follow a standard
metric interpretation. We also wish to comment on reducing the QMT to a “metric” on pa-
rameter space. In the Hermitian case this can be done for each energy band separately. That
is, for a fixed energy band, one can define a global eigenstate and so obtain a metric on N(H)
by pull-back of M. In the non-Hermitian case, or better, whenever Spec(H) is non-trivial, this
need not be possible as a global eigenstate could be unavailable.
4.6.3 Relation between geometric phase and distance
Let us comment on a relation between geometric phase and distance. This was pointed out
in [29] for the Hermitian case, but was also considered for the non-Hermitian case in [11]. The
idea is to compare two distance functions defined on normalized states. Using a given inner
product on V and the corresponding bundle S1(V ) → P (V ) from Section 4.5, these distances
are given by
L(ψ,ψ′) =
∥∥ψ − ψ′∥∥, D(ψ,ψ′) =
√
2− 2|⟨ψ|ψ′⟩|, ψ, ψ′ ∈ S1(V ).
30 E.J. Pap, D. Boer and H. Waalkens
Here, L is the inner product metric on V restricted to S1(V ). The ray space P (V ) also inherits
a metric, whose pull-back to S1(V ) is D. In this way, L measures total length, whereas D only
measures the underlying change of rays. Their difference is then related to movement along the
ray, which is where we find geometric phase. To make this concrete, one works infinitesimally.
Setting ψ′ = ψ+ ψ̇dt+ 1
2 ψ̈dt
2 + · · · and keeping terms up to dt2 only, one finds dL2 = ⟨ψ̇|ψ̇⟩dt2
and dD2 =
[
⟨ψ̇|ψ̇⟩+ ⟨ψ|ψ̇⟩2
]
dt2. One thus finds dL2−dD2 = (i⟨ψ|ψ̇⟩dt)2, so that the geometric
phase integrand equals
√
dL2 − dD2, which is one of the main results in [29]. It is tempting
to write “dγgeo” for i⟨ψ|ψ̇⟩dt, but we refrain from doing that. Following the discussion after
equation (4.2), there is no definite rate at which a geometric phase is acquired, although the
contrary is suggested by such notation. There is also no need to introduce more notation; the
tensor field corresponding to this quantity is simply iω.
We now generalize to the non-Hermitian setting of Eig(V ). First, the distances L and D do
not carry over. Concerning L, we see L2 = 2 − ⟨ψ|ψ′⟩ − ⟨ψ′|ψ⟩ generalizes to the function 2 −
θ1(v2)− θ2(v1) on Eig(V ), but its square root would be multi-valued on Eig(V ), meaning that L
itself does not generalize. Similar issues appear for D2 = 2 − 2
√
⟨ψ|ψ′⟩⟨ψ′|ψ⟩. Nevertheless,
the tensor fields dL2 and dD2 do have a generalization to Eig(V ). One can check that dD2
is actually M for the Hermitian case, see also [10, 11]. Finally, dL2 generalizes to the tensor
field L := (dθdv)sym, which is the non-covariant counterpart of M. The relation between L, M
and ω is readily found from equation (4.7); first rewriting this to
G = dθdv + (−dθv)⊗ (θdv) = dθdv + ω ⊗ ω
and then taking the symmetric part yields
M = L+ ω ⊗ ω.
This is thus an equality of tensor fields on Eig(V ), generalizing the relation “dD2 = dL2−dγ2geo”.
5 Explicit description of the holonomy
In Sections 3 and 4 we saw how the bundles Spec(V ) and Eig(V ) provide a holonomy interpreta-
tion for the physics of geometric phases and exceptional points. However, in the non-cyclic case,
we did not provide an easier description of the maps pγ and Pγ in terms of explicit permutations
like (12) resp. holonomy matrices. We will now treat how these can be obtained, and afterwards
show that this naturally follows holonomy theory as well.
5.1 Explicit permutations and holonomy matrices
Let us start with the permutations of energies. Our goal is to find a formalism so that a permu-
tation pγ of the eigenvalues of some A ∈ N(V ) can be expressed using an standard permutation
σ ∈ Sn. A first attempt would be to label the eigenvalues, i.e., call them λ1, . . . , λn, and define σ
by the relation pγ(λi) = λσ(i). However, this approach has a serious disadvantage. Although
this works fine if we consider the eigenvalues of only the operator A, as the authors did in [27],
this method does not extend to all of N(V ). The reason is that knowing the action of Sn on
a spectrum is equivalent to having a labelling of the eigenvalues [28]. For example, if there are
3 eigenvalues, then from the action of (12) one finds which eigenvalue is labeled 3. Hence, if the
above extends to a global Sn-action, then Spec(H) admits a global labelling and is thus trivial.
We thus wish to obtain a method that also applies to non-trivial Spec(H). Instead of con-
sidering eigenvalues separately, let us consider specific tuples of eigenvalues. Namely, for an
operator A, we consider the n-tuples that list all eigenvalues exactly once, which brings us to
the set
Spec(A)! =
{
λ̃ = (λ1, . . . , λn) ∈ Spec(A)n | all eigenvalues of A appear once in λ̃
}
.
A Unified View on Geometric Phases and Exceptional Points 31
Observe that this set has n! elements, hence the factorial notation. In addition, this space is
naturally endowed with the Sn-action on n-tuples, cf. our earlier notation in equation (2.2). It is
this Sn-action that will help us with describing the map pγ . Of course, one can act with pγ on
a tuple λ̃ entry-wise, which leads to the map p̃γ : (λ1, . . . , λn) 7→ (pγ(λ1), . . . , pγ(λn)). As p̃γ
(
λ̃
)
is a reordering of λ̃, we obtain a unique permutation σ, depending on λ̃, by stating that
p̃γ
(
λ̃
)
= σ · λ̃.
Let us say a few more words on how the tuple method is subtly yet significantly different from
the labelling method we started with. First, although we write the eigenvalues with indices (just
to distinguish them), in principle they are never labelled. Indeed, like in Example 5.1 below, we
are interested in the values of the energies, not the names we use for them. Of course, any tuple
induces a labelling by assigning label k to the energy appearing in position k, like we also used
above. However, σ is not based on these labels, but instead on the tuple positions. That is,
σ will send the energy at place k to place σ(k), regardless of what energy is located at place k
or which value we designated as λk.
Example 5.1. Let us continue with the EP2 example of Example 3.5, where E+(x0) and
E−(x0) are exchanged when one encircles an EP. Picking the ordering of Spec(H(x0)) to be
(E+(x0), E−(x0)), the above reads
p̃γ(E+(x0), E−(x0)) = (E−(x0), E+(x0)) = (12) · (E+(x0), E−(x0))
so that σ = (12). In this case, as n = 2, one also obtains (12) for the alternative ordering
(E−(x0), E+(x0)).
Consider now the case where we add a separate constant energy level E0(x) = 0, i.e., con-
sider the Hamiltonian H ′(x) = H(x) ⊕ 0 which brings us to n = 3. Picking the ordering
(E+(x0), E0(x0), E−(x0)) of Spec(H
′(x0)) one finds
p̃γ(E+(x0), E0(x0), E−(x0))= (E−(x0), E0(x0), E+(x0))= (13) · (E+(x0), E0(x0), E−(x0)),
which thus yields (13). Of course, in the ordering (E+(x0), E−(x0), E0(x0)) we would again
obtain (12).
Finally, let us demonstrate how the tuple method differs from labelling the energies. Consider
we label E+(x0) = E1, E0(x0) = E2 and E−(x0) = E3. In the ordering (E1, E2, E3), clearly σ
will replace the labels as expected (up to a convention; here the label Ek will become Eσ−1(k)
instead of Eσ(k)). However, in another ordering this need not be so, for example
(12) · (E3, E1, E2) = (E1, E3, E2)
instead of the tuple (E3, E2, E1) one would obtain by checking the labels. The method of using
a reference tuple thus circumvents the need for explicit labels.
We can extend this approach to describe the evolution of states, as given by a map Pγ ,
by a holonomy matrix. Similarly to the previous approach with eigenvalues, we will consider
a tuple formulation. In this case, for an operator A ∈ N(V ), we consider tuples f̃ = (f1, . . . , fn)
of eigenvectors of A such that each eigenray is represented exactly once. That is, f̃ should be
an eigenframe of A, and we write EigFr(A) for the space of all eigenframes of A. This space has
a natural action by the wreath product C× ≀ In, isomorphic to the group of complex generalized
permutation matrices, which we already encountered in Section 2.2. In this case, the action reads
C× ≀ In × EigFr(A) → EigFr(A),
((z1, . . . , zn), σ) · (f1, . . . , fn) = (z1fσ−1(1), . . . , znfσ−1(n)),
32 E.J. Pap, D. Boer and H. Waalkens
i.e., the action is similar to the earlier C× ≀ In-action in equation (2.3). Using this action, we can
express a map Pγ by an element (z̃, σ) ∈ C× ≀ In once we have chosen a reference eigenframe f̃ .
Writing P̃γ for the map on EigFr(A) obtained by applying Pγ entry-wise, we obtain the group
element by stating
P̃γ(f̃) = (z̃, σ) · f̃ .
If one represents (z̃, σ) by a permutation matrix, which we will denote by Mf̃ , then Mf̃ is the
matrix of the linear extension of Pγ to the entire state space. The dependence ofMf̃ on f̃ (again
keeping the loop γ fixed) follows the familiar conjugation rule.
Example 5.2. In case of the EP2, we found Pγ as given by equation (4.5). With respect to the
eigenframe (e1, e2) of H(0), we find
P̃γ(e1, e2) = (Pγ(e1), Pγ(e2)) = (−ie2,−ie1) = ((−i,−i), (12)) · (e1, e2),
i.e., we find ((−i,−i), (12)) ∈ C× ≀ I2. The holonomy matrix M(e1,e2) is thus
M(e1,e2) =
(
−i 0
0 −i
)(
0 1
1 0
)
=
(
0 −i
−i 0
)
.
Or, alternatively,(
Pγ(e1)
Pγ(e2)
)
=
(
−ie2
−ie1
)
=
(
0 −i
−i 0
)(
e1
e2
)
.
The more familiar pattern ψ1 7→ ψ2 7→ −ψ1 is obtained by picking, e.g., (ie1, e2) as the
reference frame. Indeed, as
P̃γ(ie1, e2) = (e2,−ie1) = ((1,−1), (12)) · (ie1, e2)
one finds the group element ((1,−1), (12)) and the holonomy matrix reads
M(ie1,e2) =
(
1 0
0 −1
)(
0 1
1 0
)
=
(
0 1
−1 0
)
.
The matrices are different only because we picked a different reference frame; both matrices still
describe the map Pγ .
5.2 Holonomy interpretation on the frame bundle
We now discuss how the approach of Section 5.1 complies with a more general theory on semi-
torsors. As shown in [28], one can define the notion of a frame of a semi-torsor. Let us briefly
review this here for completeness. If G is a Lie group and F a G-semi-torsor consisting of n
orbits, then G×In ∼= F as (left) G-spaces. Such an isomorphism is always of the form (g, i) 7→ gfi
for a unique tuple f̃ = (f1, . . . , fn) of elements in F . This tuple f̃ we call a basis of F , similar
to linear algebra. Observe that n elements of F form a basis if and only if each orbit in F
is represented exactly once. The set of all frames of F we denote by Fr(F ). Clearly, one can
translate or permute the elements of a basis. This is captured by a natural action of the wreath
product G ≀ In, i.e.,
G ≀ In × Fr(F ) → Fr(F ),
((g1, . . . , gn), σ) · (f1, . . . , fn) =
(
g1fσ−1(1), . . . , gnfσ−1(n)
)
.
In fact, this action is free and transitive, so that, in other words, Fr(F ) is a G ≀ In-torsor.
A Unified View on Geometric Phases and Exceptional Points 33
This is exactly what we did in the previous section. For the eigenvalues, we see that Spec(A) is
a set containing n points, hence trivially a semi-torsor with G the trivial group. Each eigenvalue
is thus an orbit by itself, and a frame is thus any tuple λ̃ = (λ1, . . . , λn) listing all elements of
Spec(A) once. These are exactly the tuples we considered before, i.e.,
Fr(Spec(A)) = Spec(A)!.
This space is naturally endowed with the Sn-action on n-tuples, making it an Sn-torsor. For the
eigenstates, we consider the space Eig(A), which we already found to be a C×-semi-torsor. The
orbits in this case are the eigenrays, and hence a frame of Eig(A) is a tuple f̃ of eigenvectors
such that each eigenray is represented exactly once. Indeed, we again find that f̃ should be an
eigenframe of A, i.e.,
Fr(Eig(A)) = EigFr(A).
And this space is clearly a C× ≀ In-torsor.
The next step is to lift this procedure of taking frames to the level of semi-principal bundles.
We remark that this theory closely resembles that of the frame bundle of a vector bundle.
As shown in [28], the operation of taking the frame space induces a functor from semi-principal
bundles to principal bundles. Given a semi-principal G-bundle π : B →M , we obtain its frame
bundle π! : Fr(B) →M by applying the above fiber-wise;
Fr(B) =
⊔
m∈M
Fr(Bm).
It holds that π! : Fr(B) → M is a principal G ≀ In-bundle. Moreover, any G-connection on B
becomes a G ≀ In-connection on Fr(B) by stating that a path in Fr(B) is horizontal if and only if
all frame elements traverse horizontal paths. The map B 7→ Fr(B) induces a retracting functor
from the semi-principal bundles to the principal bundles. The holonomy interpretation of the
explicit permutations and holonomy matrices we found above can now be obtained by applying
this frame bundle functor to Spec(V ) resp. Eig(V ).
5.2.1 Frame bundle of the spectrum bundle
Let us start with Spec(V ). We found in Theorem 3.4 that πv : Spec(V ) → N(V ) is an In-bundle,
hence it is a semi-principal bundle with trivial structure group. The frame bundle functor then
yields the following.
Proposition 5.3. The frame bundle of Spec(V ) is the principal Sn-bundle
E(V ) := Fr(Spec(V )) =
{(
A, λ̃
)
∈ N(V )× Cn | λ̃ ∈ Fr(Spec(A))
}
,
with action
σ · (λ1, . . . , λn) =
(
λσ−1(1), . . . , λσ−1(n)
)
and projection
πλ! : E(V ) → N(V ),(
A, λ̃
)
7→ A.
Remarkably, we observe that πλ! is also the pull-back of q : Cn →
(C
n
)
along the map
Spec: N(V ) →
(C
n
)
; the fiber Fr(Spec(A)) of E(V ) equals q−1(Spec(A)). In addition, when
34 E.J. Pap, D. Boer and H. Waalkens
a Hamiltonian operator family H : M → End(V ) is given, it is again natural to consider the
pull-back along H. This yields the bundle πHλ ! : E(H) → N(H), (x, Ẽ) 7→ x, where Ẽ is then
a tuple listing the energies of H(x), and is the frame bundle of πHλ . Observe that both pull-
backs together result in the following commutative diagram, where all vertical maps are principal
Sn-bundles:
E(H) E(V ) Cn (x, Ẽ) (H(x), Ẽ) Ẽ
N(H) N(V )
(C
n
)
, x H(x) Spec(H(x)).
πH
λ ! πλ!
q
H Spec
We observe that E(H) and E(V ) are also the natural spaces behind the merging path method for
EP detection, which we reviewed in Section 3.2.1. Namely, the above diagram can alternatively
be obtained by completing diagram (3.2) using pull-backs.
The holonomy interpretation behind the permutations can be explained as follows. Fix
a loop γ in N(H) based at x0, and let Ẽ be the ordering of Spec(H(x0)) w.r.t. which we find
the permutation σ expressing pγ . Observe that the picking of the ordering Ẽ = (E1, . . . , En)
of Spec(H(x0)) is equivalent to picking the point
(
x0, Ẽ
)
in the fiber of E(H) above x0. This
point
(
x0, Ẽ
)
fixes a unique lift of γ to E(H). Physically, this lift traces all of the energies
simultaneously and in a particular order. The permutation σ now naturally appears via the
holonomy theory of principal bundles; the end point of the lift is σ
(
x0, Ẽ
)
, which fixes σ uniquely.
One can alternatively argue by means of holonomy groups. We already found that pγ lies in the
holonomy group Hol
Spec(H)
N(H) (x0). Clearly, pγ extends to a map p̃γ on frames by applying pγ on
each frame element separately, and so p̃γ is an element of Hol
E(H)
N(H)(x0). Both of these groups are
based on a point in the base space, and so consist of “abstract” fiber automorphisms. An explicit
group element is obtained by considering the holonomy group at a point in the total space. In this
case, we move to the holonomy group at
(
x0, Ẽ
)
defined as
HolE(H)
(
x0, Ẽ
)
=
{
σ ∈ Sn |
(
x0, Ẽ
)
∼ σ
(
x0, Ẽ
)}
,
where the equivalence relation ∼ holds if and only if the two points can be connected by a path.
The groups are related by the isomorphism Hol
E(H)
N(H)(x0) → HolE(H)
(
x0, Ẽ
)
, p̃γ 7→
[
p̃γ
(
Ẽ
)
/Ẽ
]
,
where [−/−] denotes the unique group element translating the right entry to the left one.
We summarize these findings in the following.
Lemma 5.4. Given a loop γ in N(H) based at x0, the permutation σ expressing pγ w.r.t. the or-
dering Ẽ of Spec(H(x0)) is the holonomy element at the point
(
x0, Ẽ
)
in the frame bundle E(H).
It follows that we should consider σ to lie in the group HolE(H)
(
x0, Ẽ
)
. This automatically
accounts for the change in σ if we choose another ordering Ẽ′. Geometrically, we see this amounts
to picking a different reference point in E(H), namely (x0, Ẽ
′) instead of
(
x0, Ẽ
)
. Hence, the
permutation σ′ obtained w.r.t. Ẽ′ lies in HolE(H)
(
x0, Ẽ
′) rather than HolE(H)
(
x0, Ẽ
)
. We
observe that the relation between σ and σ′ is given by the usual conjugation relation between
holonomy groups. Namely, there is a unique τ ∈ Sn such that Ẽ′ = τẼ, and consequently
σ′ = τστ−1. Note that the frame dependence of σ can also be regarded as a gauge dependence.
The gauge freedom is then in the choice of frame of Spec(A), and different gauges are related by
an Sn-symmetry. Of course, the gauge invariant behind the permutations σ and σ′ is the map
of eigenvalues pγ , or equivalently its frame version p̃γ .
Using the framework of holonomy groups, we can be more explicit on how permutations
arising from different loops γ are related, as the authors also studied in [27]. Like Spec(H), also
A Unified View on Geometric Phases and Exceptional Points 35
E(H) is a covering space of N(H), and so the permutations are prescribed by the monodromy
action. With respect to the ordering Ẽ of Spec(H(x0)), this is captured by the homomorphism
π1(N(H), x0) → HolE(H)
(
x0, Ẽ
)
,
[γ] 7→
[
p̃γ
(
Ẽ
)
/Ẽ
]
.
Clearly this homomorphism is surjective, and its kernel consists of all classes whose paths induce
a cyclic change of eigenvectors. In addition, it shows how the holonomy matrix of a complicated
loop can be decomposed in holonomy matrices of more elementary loops. This present argument
hence improves on a previous proof of this fact given in [27].
Let us briefly compare the holonomy groups of E(V ) and E(H). For E(V ), every holonomy
group exhausts all of Sn as all possible paths are present. Indeed, the bundle πλ! : E(V ) → N(V )
does not admit a reduction of the structure group. However, E(H) can have smaller holonomy
groups, and admit a reduction of the structure group. In an extreme case, e.g., if the Hamiltonian
family is always Hermitian w.r.t. a given inner product on V , then E(H) is trivial as Spec(H)
is,4 and the structure group can be reduced to the trivial group 1. Intermediate cases are also
possible, and express in what way the energy bands are connected. For example, if k energy
bands are disconnected from the remaining n−k, clearly the structure group Sn can be reduced
to Sk×Sn−k. Practically, this can be achieved by restricting to tuples Ẽ in which these k energy
bands appear only in the first k entries.
Example 5.5. Let us continue the study of the standard EP2 system as we described in Ex-
ample 3.5. We found N(H) = C \ {±i}, which means π1(N(H), 0) has two generators. Define
generators a± to wind once around ±i in positive direction. We found in Example 3.6 that both
generators exchange the eigenvalues. Let us label the spectrum Spec(H(0)) = {±1} as (+1,−1).
Then the permutations arising from the EPs are expressed by the homomorphism
π1(N(H), 0) → HolE(H)(0, (+1,−1)),
a± 7→ (12).
Clearly, this confirms that a permutation occurs if and only if a loop decomposes into an odd
number of a±. As seen in Example 5.1, if an extra level is present, the exact form of the
permutation does indeed depend on the labelling. In addition, we observe that for the extended
system H ′ presented there, the principal S3-bundle E(H ′) reduces to a S2-bundle, which is
isomorphic to E(H).
We note that the space E(V ) has appeared in previous papers. We already remarked that [34]
mentioned the relevance of the monodromy action in the theory of EPs. The covering used there
is of the form E(H), using the one-to-one correspondence between eigenvalues and eigenpro-
jectors for non-degenerate operators. We also recognize E(V ) as the space M̃ defined in [21].
However, where [21] views E(V ) primarily as an extended parameter space, we consider it rather
as the bundle over N(V ) which yields the permutations due to EPs by its holonomy.
5.2.2 Frame bundle of the eigenvector bundle
We now proceed by considering the frame bundle of πλv : Eig(V ) → N(V ). This is a semi-
principal C×-bundle by Theorem 4.4 and endowed with the C×-connection ω as in Proposi-
tion 4.7. Applying the frame bundle functor then yields the following.
4An alternative argument: if the energies of H are real, then E(H) is a pull-back of Rn →
(R
n
)
, hence trivial
by Proposition A.3.
36 E.J. Pap, D. Boer and H. Waalkens
Proposition 5.6. The frame bundle of Eig(V ) is the principal C× ≀ In-bundle
EigFr(V ) =
{(
A, f̃
)
∈ N(V )× Fr(V ) | f̃ is an eigenframe of A
}
,
where the element
(
A, f̃
)
can also be written as
(
A, λ̃, f̃
)
with λ̃ listing the eigenvalues of A in
the order of f̃ . The C× ≀ In-action on EigFr(V ) reads
(z̃, σ) ·
(
A, λ̃, f̃
)
=
(
A, σλ̃, z̃
(
σf̃
))
and the projection is
πλv! : EigFr(V ) → N(V ),(
A, λ̃, f̃
)
7→ A.
Furthermore, EigFr(V ) is naturally endowed with the C× ≀ In-connection
ω!(A,λ̃,f̃) = ω(A,λ1,f1) ⊕ · · · ⊕ ω(A,λn,fn).
Recall that we already found the principal C× ≀ In-bundle Ξ: Fr(V ) × Cn → N(V ) in
Section 2.2. We observe that Ξ and πλv! are reformulations of each other, i.e., they are canonically
isomorphic bundles.
Lemma 5.7. The map
EigFr(V ) → Fr(V )× Cn,(
A, λ̃, f̃
)
7→
(
f̃ , λ̃
)
is a canonical isomorphism of bundles over N(V ). Hence, in particular, EigFr(V ) is canonically
isomorphic to Fr(V ) × Cn as a C× ≀ In-manifold, and consequently E(V ) is canonically diffeo-
morphic to PFr(V )× Cn.
Again, given a Hamiltonian family H, it is more natural to study the pull-back bundle. This
yields the space EigFr(H), whose elements we write as
(
x, Ẽ, ψ̃
)
. The connection form ω! also
carries over to the connection form ωH !. Note that EigFr(H) may admit a reduction of the
structure group, e.g., to U(1)n when H is Hermitian. We can summarize the pull-back and
relation with Ξ in diagram (5.1).
EigFr(H) EigFr(V ) Fr(V )× Cn
(
x, Ẽ, ψ̃
) (
H(x), Ẽ, ψ̃
) (
ψ̃, Ẽ
)
N(H) N(V ) N(V ), x H(x) H(x).
πH
λv !
∼
πλv ! Ξ
H id
(5.1)
We can now view the holonomy interpretation of the map on eigenstates as follows. Again, we
fix a family H and a loop γ in N(H) based at some reference x0. We then pick an eigenframe ψ̃
of H(x0), which fixes the tuple Ẽ of corresponding energies. That is, we pick a point
(
x0, Ẽ, ψ̃
)
in the fiber of EigFr(V ) above x0. This point fixes a unique lift of γ to EigFr(V ), which traces
all states in the tuple ψ simultaneously. The end point of this lift then equals (z̃, σ)
(
x0, Ẽ, ψ̃
)
,
where (z̃, σ) ∈ C× ≀ In we obtained as before. Hence it lies in the holonomy group
HolEigFr(H)
(
x0, Ẽ, ψ̃
)
=
{
(z̃, σ) ∈ C× ≀ In |
(
x0, Ẽ, ψ̃
)
∼ (z̃, σ)
(
x0, Ẽ, ψ̃
)}
,
where the equivalence relation ∼ means that the points can be connected by a horizontal path
in EigFr(H), or equivalently, that there is a path in N(H) whose lift to EigFr(H) through one
A Unified View on Geometric Phases and Exceptional Points 37
point ends at the other. Again, we consider the holonomy group at a point in the total space
rather than in the base space.
Clearly, σ is the same permutation as we would find when looking only at the energies.
This we already found in Proposition 4.18, which is based on the map πv : Eig(V ) → Spec(V ).
However, we can also phrase this on the frame bundles as follows. Namely, the frame bun-
dle functor turns πv into the bundle map πv! : EigFr(V ) → E(V ),
(
A, λ̃, f̃
)
7→
(
A, λ̃
)
. This
map is equivariant w.r.t. the canonical quotient C× ≀ In → Sn, and is compatible with the
connections. Fixing a point
(
A, λ̃, f̃
)
, this quotient reappears as a map on holonomy groups
HolEigFr(V )
(
A, λ̃, f̃
)
→ HolE(V )
(
A, λ̃
)
. This map simply extracts σ, which confirms that σ is
the holonomy from E(V ). We summarize the above findings as follows.
Lemma 5.8. Given a loop γ in N(H) based at x0, the element (z̃, σ) expressing Pγ w.r.t.
the eigenframe ψ̃ of H(x0) is the holonomy element at the point (x0, ψ̃) in the frame bundle
EigFr(H). Writing Ẽ for the corresponding ordering of Spec(H(x0)), σ is the permutation
representing pγ w.r.t. Ẽ.
The element (z̃, σ) thus lies in the group HolEigFr(H)(x0, ψ̃). We can view the picking of ψ̃
as a gauge freedom, just like the picking of Ẽ earlier. A change of gauge thus corresponds to
switching holonomy group, with the usual similarity transformation of the holonomy matrix.
The gauge invariant in this case is the map Pγ . Indeed, Pγ is defined without using any gauge,
hence is trivially independent of gauge.
Let us then take a closer look at the phase factors z̃ appearing in the holonomy matrix.
Given the parallel transport formalism, each of these may be called a “geometric” phase factor.
However, we also see that they are gauge dependent. Namely, the order gets permuted by
changing the order of the eigenframe ψ̃, and scaling of the individual states in ψ̃ affects the
phase factors in z̃ corresponding to non-cyclic evolution. On the other hand, z̃ is subject to
constraints coming from the characteristic phases of Pγ . Parametrizing zj = exp(iαj), if we
follow a state until it returns to its original ray, its acquired phase is
∑
cycle αi, where we sum
over the indices belonging to the corresponding cycle. Hence
∑
cycle αi must be equal to the
characteristic phase of the cycle, independent of the chosen gauge. Note however that this does
not yield a canonical phase for a non-cyclic state after following γ once; in one gauge all αi in
the cycle could be equal, whereas in another all except one could vanish.
In terms of the holonomy matrix, we can express this as follows. By permuting ψ̃ the phase
factors can be moved within the matrix, which is subject to the underlying cycle structure of the
permutation. The matrix is block-diagonal if and only if states from the same minimal union
are listed adjacently. The size of a block is k × k, with k the length of the corresponding cycle.
The k individual phase factors within this block can be gauged to arbitrary values, with the
only condition that their product is fixed.
Let us treat in more detail how a seemingly Abelian connection ωH ! can still induce non-
Abelian holonomy. That is, given the technique of path-ordered exponential and the diagonal
form of ωH !, one may wonder how the end result can be non-diagonal. In short, the reason
is that if the loop γ lifts to non-cyclic states, then any instantaneous eigenframe returns with
a twist, and this twist yields an additional permutation matrix making the theory non-Abelian.
That is, in order to write ωH ! as a diagonal gauge potential, in practice one would need to
extend the initial eigenframe ψ̃ to a path of instantaneous eigenframes over γ. Upon returning
to x0, one cannot use the initial eigenframe ψ̃ again due to the presence of non-cyclic states.
Consequently, one needs to correct for the difference between final and initial eigenframe. This
basis transformation can be expressed by an element in C× ≀ In, and thus permutation matrices
enter the end result. Naturally, this issue can only occur when E(H) is non-trivial and γ is
a non-contractible loop, and so is completely absent in case π1(N(H), x0) is trivial.
38 E.J. Pap, D. Boer and H. Waalkens
Let us also compare EigFr(V ) to earlier work in [21], which is likewise aimed to provide
a holonomy description for geometric phases in the presence of EPs. There, one considers
cyclic states under the adiabatic evolution due to a change of a Hamiltonian, which may be
non-Hermitian. The general geometric model consists of n complex line bundles L1, . . . , Ln
over E(V ), there called M̃, where each line bundle corresponds to a certain self-chosen part of
the energy bands. A main disadvantage of this approach is that the connection information is
spread over the Lk. Hence, calculation of a lift can in general not be done in a single Lk, i.e.,
multiple line bundles need to be used. That is, formally the Lk do not support a holonomy
interpretation for geometric phases in the presence of EPs. In contrast, we found that EigFr(V )
is a single bundle allowing to lift a path directly from N(V ), which can moreover be used to
treat non-cyclic states.
It is also the case that the theory of EigFr(V ) provides a geometric model for the off-diagonal
phases reported by Manini and Pistolesi [20], and extends the generalized permutation matrices
found by Tanaka, Cheon and Kim [35]. The generalization of their work to non-Hermitian
Hamiltonians can be obtained from EigFr(H) in the following way.
First, let us clarify that there are 3 main classes of paths that are considered in this theory.
In the above, we considered loops γ only, i.e., the final point equals the initial point x0. In con-
trast, [20] starts by considering general paths, so that the final point x1 need not equal x0.
They then consider a more special case, which is also treated in [35] and there called an
“adiabatic loop”. This term refers to a path γ for which x1 need not equal x0, but still
Eig(H(x0)) = Eig(H(x1)) (as subsets of V , strictly speaking not as fibers of Eig(H)). We thus
observe that an “adiabatic loop” need not be a “loop” in the sense we use, and forms an inter-
mediate subset:{
loops;
x0 = x1
}
⊂
{
“adiabatic loops”;
Eig(H(x0)) = Eig(H(x1))
}
⊂
{
general paths;
no relation x0, x1
}
.
Let us now demonstrate that EigFr(V ) also describes the theory of the general paths, so that
we obtain the “adiabatic loops” as a special case. Let γ be any path in N(H), so belonging to
the most right set in the above comparison. Lifting to EigFr(V ), it induces the parallel transport
map
P̃γ : EigFr(H(x0)) → EigFr(H(x1)).
More explicitly, given an initial eigenframe ψ̃ of H(x0), one obtains a lift Γ̃ of γ starting at ψ̃,
and the final frame is ψ̃′ = P̃γ(ψ̃). However, as ψ̃′ and ψ̃ are (generally) not in the same fiber,
the two frames cannot be compared using the bundle structure of EigFr(H). The solution is
to consider the frames by themselves; one neglects the fact that they are associated to different
values of the system parameters. Clearly, any two frames of V are related by a unique matrix;
in this case there is a unique U ∈ GL(n,C) such that
ψ̃′ = Uψ̃.
Mathematically, this argument is captured by employing the canonical projection EigFr(V ) →
Fr(V ), and using that Fr(V ) is a GL(n,C)-torsor. We also remark that if both ψ̃ and ψ̃′ are
orthonormal w.r.t. some inner product, then U is indeed unitary, and can be calculated by taking
inner products between the states, as used in [20].
Of course, the matrix U is not unique; if we change ψ̃ to Gψ̃, with G ∈ C× ≀ In a generalized
permutation matrix, then ψ̃′ = P̃γ
(
ψ̃
)
becomes P̃γ
(
Gψ̃
)
= GP̃γ
(
ψ̃
)
= Gψ̃′, and so U becomes
GUG−1. That is, U is unique up to conjugation by C× ≀ In. When looking for invariants, if we
write G = (g̃, τ), we may set τ = id; non-trivial τ simply rearrange the elements of U . It follows
that Uij becomes giUijg
−1
j . Hence, one obtains an invariant quantity for each cycle in Sn
A Unified View on Geometric Phases and Exceptional Points 39
by taking the corresponding product, e.g., (12) yields the invariant U12U21 and (123) yields
the invariant U12U23U31, in addition to the diagonal elements Ukk which are also invariant.
Whenever such a product is non-zero, its argument is a gauge-invariant phase, as reported
in [20].
The “adiabatic loops” are then the special case where U is also a generalized permutation
matrix, which was studied further in [35]. The similarity of this case with our study of the holo-
nomy on EigFr(V ) is straightforward to explain; in both cases EigFr(H(x0)) = EigFr(H(x1)),
so that we are studying an automorphism of a semi-torsor. The method above, based on the
matrix U , then reduces to the argument of Section 5.1, and invariant phases follow from Propo-
sition 4.16.
6 Discussion
In this paper we introduced a framework that facilitates the mathematical description of the
adiabatic quantum mechanics of finite-dimensional, non-degenerate but otherwise arbitrary and
hence not necessarily Hermitian Hamiltonians. We started with a preparatory study of the
space of non-degenerate operators, which was summarized in diagram (2.6). The following
step was to find the geometry behind the adiabatic change of energies of a Hamiltonian fam-
ily H. This is captured by the covering space formed by the energy bands, where we re-
strict to system parameter values where H is non-degenerate. Moreover, the specific covering
needed for H we found as the pull-back of an abstract yet explicit model covering, which makes
the covering property and hence bundle structure immediate for any smooth Hamiltonian fa-
mily H.
We continued by introducing a bundle for the eigenstates. A remarkable property of this
bundle is that it is not a principal bundle, but has the structure of a semi-principal bundle,
which is described in [28]. That is, each of its fibers consists of a collection of eigenrays rather
than a single eigenray. We found a natural connection on this bundle, which provides a par-
allel transport description of the adiabatic geometric phase as found by Garrison and Wright
[12]. Because of the semi-principal structure, this incorporates non-cyclic states as well. This
formalism hence extends previous holonomy formulations, which are based on principal bun-
dles. Among these are the first holonomy interpretation reported by Simon [32], which treats
the Hermitian case with separate energy bands, and the formalism in [21], which considered
the non-Hermitian case restricted to cyclic states. We furthermore found that the Aharonov–
Anandan formalism [1] for general Hermitian systems can be obtained by a reduction from
the present formalism using a given inner product. Moreover, we found that the dynami-
cal phase can be included as a non-geometric addition, which means that the full adiabatic
evolution of a state can be calculated naturally on this bundle. We then obtained a gener-
alized quantum geometric tensor, which we found to be directly related to covariant deriva-
tives.
We highlight that the presented formalism, as it allows one to also describe non-cyclic states
by means of parallel transport, is a natural formalism for the study of exceptional points. The
description including geometric phases is then given on the full eigenstate bundle, whereas
the permutations of energies can already be accurately described with the monodromy theory
of the energy band covering. Both of these spaces have a semi-principal bundle structure,
which is the key to incorporate non-cyclic behavior. However, by switching to a multi-state
approach, one can describe the permutations of energies and the accompanying geometric phase
factors simultaneously in a more explicit way, namely via holonomy matrices. This is given by
a straightforward application of the frame bundle technique as described in [28]. This provides
a more explicit holonomy description of state evolution, also in the non-cyclic case enabled by
exceptional points.
40 E.J. Pap, D. Boer and H. Waalkens
A Bundle structure of tuples of distinct complex numbers
If one considers n distinct complex numbers, one can do so with or without an ordering of these
numbers. Let us start with the ordered one. This brings us to the space
Cn =
{
(z1, . . . , zn) ∈ Cn | i ̸= j =⇒ zi ̸= zj
}
,
where we borrow the notation of falling factorials; if we would replace C by a finite set F , then
|Fn| = |F |(|F | − 1) · · · (|F | − n), which in combinatorics is known as the falling factorial |F |n.
The space Cn is a submanifold of Cn, and more is true.
Lemma A.1. The space Cn is an (algebraic) open and dense submanifold of Cn.
Proof. For i ̸= j, define the function mij : Cn → C, (z1, . . . , zn) 7→ zi− zj , and write D(mij) ={
(z1, . . . , zn) ∈ Cn | mij(z1, . . . , zn) ̸= 0
}
. Then Cn = ∩i ̸=jD(mij) expresses the space as
a finite intersection of (algebraic) open subsets, hence is open in Cn and so a submanifold. It is
also dense; given a tuple with repeated numbers, these become unequal by an arbitrarily small
perturbation. ■
We can now proceed to sets of n distinct complex numbers, which is the unordered variant of
the ordered tuple case as just discussed. The idea is to view the unordered sets as a quotient of
the tuple space. This quotient is specified by a permutation action; upon identifying rearranged
tuples, clearly the ordering vanishes. The action is the usual permutation of slots, i.e., for σ ∈ Sn
and (z1, . . . , zn) ∈ Cn we have
σ · (z1, . . . , zn) = (zσ−1(1), . . . , zσ−1(n)).
This action is smooth as it is the restriction of the smooth permutation action on Cn to an
invariant submanifold. Taking the quotient results in the space of subsets of C with n distinct
complex numbers, which we write as(
C
n
)
:= Cn/Sn =
{
S ⊂ C | |S| = n
}
.
Again, we borrow our notation from combinatorics. We write q for the quotient map Cn →
(C
n
)
.
Clearly, the fiber of q above a certain set in
(C
n
)
consists of all the possible orderings of its
elements. Via q one obtains a unique manifold structure on
(C
n
)
, and in this way q is principal
bundle.
Proposition A.2. There is a unique manifold structure on
(C
n
)
such that the quotient map
q : Cn →
(C
n
)
is a surjective submersion. The permutation action on Cn thus induces the principal
bundle
Sn Cn
(C
n
)
.
q
Moreover, this bundle is non-trivial for n > 1.
Proof. Clearly the Sn-action on Cn is free and proper, hence the quotient map defines a prin-
cipal bundle. Assume it is trivial for n > 1 for contradiction. This would imply that Cn
and
(C
n
)
×Sn are homeomorphic. However, as Cn and
(C
n
)
are path-connected but Sn is not, this
cannot be. ■
A Unified View on Geometric Phases and Exceptional Points 41
The non-triviality of the bundle is related to the topology of C not being compatible with
a total ordering. To expand on this, observe that for any manifold M one can define the space
of distinct tuples Mn. This similarly has a Sn-action defined on it, and one obtains a principal
bundle over the space
(
M
n
)
of unordered subsets of n elements. IfM has an order topology given
by a total ordering, such as R, this bundle is trivial. Indeed, given any element S ∈
(
M
n
)
, then
there is a standard way to list the elements of S, e.g., from lowest to highest. This induces
triviality as follows.
Proposition A.3. If a manifold M is orderable, then the principal Sn-bundle M
n →
(
M
n
)
is
trivial for all n.
Proof. Let ≤ be a total ordering on M compatible with the topology. Define an ordering map
o :
(
M
n
)
→Mn by the condition that
o({m1, . . . ,mn}) = (m1, . . . ,mn), where m1 < · · · < mn.
As all the mi are distinct this is well-defined, and by assumption on the topology of M also
continuous. It follows that o is a global section of the principal bundle, which is thus trivial. ■
Acknowledgements
The authors thank the anonymous referees whose careful remarks contributed to the quality of
the paper.
References
[1] Aharonov Y., Anandan J., Phase change during a cyclic quantum evolution, Phys. Rev. Lett. 58 (1987),
1593–1596.
[2] Anandan J., Aharonov Y., Geometry of quantum evolution, Phys. Rev. Lett. 65 (1990), 1697–1700.
[3] Berry M.V., Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392
(1984), 45–57.
[4] Berry M.V., The quantum phase, five years after, in Geometric Phases in Physics, Editors A. Shapere,
F. Wilczeck, Adv. Ser. Math. Phys., Vol. 5, World Sci. Publ., Teaneck, NJ, 1989, 7–28.
[5] Berry M.V., Uzdin R., Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon, J. Phys. A:
Math. Theor. 44 (2011), 435303, 26 pages.
[6] Berry M.V., Wilkinson M., Diabolical points in the spectra of triangles, Proc. Roy. Soc. London Ser. A 392
(1984), 15–43.
[7] Born M., Fock V., Beweis des Adiabatensatzes, Z. Phys. 51 (1928), 165–180.
[8] Chruściński D., Jamio lkowski A., Geometric phases in classical and quantum mechanics, Progress in Mathe-
matical Physics, Vol. 36, Birkhäuser Boston, Inc., Boston, MA, 2004.
[9] Cohen E., Larocque H., Bouchard F., Nejadsattari F., Gefen Y., Karimi E., Geometric phase from Aharonov–
Bohm to Pancharatnam–Berry and beyond, Nature Rev. Phys. 1 (2019), 437–449, arXiv:1912.12596.
[10] Cui X.-D., Zheng Y., Geometric phases in non-Hermitian quantum mechanics, Phys. Rev. A 86 (2012),
064104, 4 pages.
[11] Cui X.-D., Zheng Y., Unification of the family of Garrison–Wright’s phases, Sci. Rep. 4 (2014), 5813, 8 pages.
[12] Garrison J.C., Wright E.M., Complex geometrical phases for dissipative systems, Phys. Lett. A 128 (1988),
177–181.
[13] Heiss W.D., Phases of wave functions and level repulsion, Eur. Phys. J. D 7 (1999), 1–4, arXiv:quant-
ph/9901023.
[14] Heiss W.D., The physics of exceptional points, J. Phys. A: Math. Theor. 45 (2012), 444016, 11 pages,
arXiv:1210.7536.
https://doi.org/10.1103/PhysRevLett.58.1593
https://doi.org/10.1103/PhysRevLett.65.1697
https://doi.org/10.1098/rspa.1984.0023
https://doi.org/10.1088/1751-8113/44/43/435303
https://doi.org/10.1088/1751-8113/44/43/435303
https://doi.org/10.1098/rspa.1984.0022
https://doi.org/10.1007/BF01343193
https://doi.org/10.1007/978-0-8176-8176-0
https://doi.org/10.1007/978-0-8176-8176-0
https://doi.org/10.1038/s42254-019-0071-1
https://arxiv.org/abs/1912.12596
https://doi.org/10.1103/PhysRevA.86.064104
https://doi.org/10.1038/srep05813
https://doi.org/10.1016/0375-9601(88)90905-X
https://doi.org/10.1007/s100530050339
https://arxiv.org/abs/quant-ph/9901023
https://arxiv.org/abs/quant-ph/9901023
https://doi.org/10.1088/1751-8113/45/44/444016
https://arxiv.org/abs/1210.7536
42 E.J. Pap, D. Boer and H. Waalkens
[15] Höller J., Read N., Harris J.G.E., Non-Hermitian adiabatic transport in spaces of exceptional points, Phys.
Rev. A 102 (2020), 032216, 9 pages, arXiv:1809.07175.
[16] Kato T., On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan 5 (1950), 435–439.
[17] Kato T., Perturbation theory for linear operators, Grundleheren der Mathematischen Wissenschaften,
Vol. 132, Springer, Berlin – Heidelberg, 1966.
[18] Lee J.M., Introduction to smooth manifolds, Graduate Texts in Mathematics, Vol. 218, Springer, New York,
2012.
[19] Mandel A., Gonçalves J.Z., Construction of open sets of free k-tuples of matrices, Proc. Edinburgh Math.
Soc. 30 (1987), 121–131.
[20] Manini N., Pistolesi F., Off-diagonal geometric phases, Phys. Rev. Lett. 85 (2000), 3067–3071, arXiv:quant-
ph/9911083.
[21] Mehri-Dehnavi H., Mostafazadeh A., Geometric phase for non-Hermitian Hamiltonians and its holonomy
interpretation, J. Math. Phys. 49 (2008), 082105, 17 pages, arXiv:0807.3405.
[22] Milburn T.J., Doppler J., Holmes C.A., Portolan S., Rotter S., Rabl P., General description of quasiadiabatic
dynamical phenomena near exceptional points, Phys. Rev. A 92 (2015), 052124, 12 pages, arXiv:1410.1882.
[23] Miri M.-A., Alù A., Exceptional points in optics and photonics, Science 363 (2019), 42 pages.
[24] Mostafazadeh A., Exact PT -symmetry is equivalent to Hermiticity, J. Phys. A: Math. Gen. 36 (2003),
7081–7091, arXiv:quant-ph/0304080.
[25] Nenciu G., Rasche G., On the adiabatic theorem for nonselfadjoint Hamiltonians, J. Phys. A: Math. Gen.
25 (1992), 5741–5751.
[26] Pancharatnam S., Generalized theory of interference, and its applications. I. Coherent pencils, Proc. Indian
Acad. Sci. Sect. A 44 (1956), 247–262.
[27] Pap E.J., Boer D., Waalkens H., Non-Abelian nature of systems with multiple exceptional points, Phys.
Rev. A 98 (2018), 023818, 9 pages, arXiv:1805.04291.
[28] Pap E.J., Waalkens H., Frames of group-sets and their application in bundle theory, arXiv:2010.03913.
[29] Pati A.K., Relation between “phases” and “distance” in quantum evolution, Phys. Lett. A 159 (1991),
105–112.
[30] Provost J.P., Vallee G., Riemannian structure on manifolds of quantum states, Comm. Math. Phys. 76
(1980), 289–301.
[31] Shapere A., Wilczek F. (Editors), Geometric phases in physics, Advanced Series in Mathematical Physics,
Vol. 5, World Sci. Publ. Co., Inc., Teaneck, NJ, 1989.
[32] Simon B., Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51 (1983), 2167–
2170.
[33] Tanaka A., Cheon T., Bloch vector, disclination and exotic quantum holonomy, Phys. Lett. A 379 (2015),
1693–1698, arXiv:1409.5211.
[34] Tanaka A., Cheon T., Path topology dependence of adiabatic time evolution, in Functional Analysis and
Operator Theory for Quantum Physics, Editors J. Dittrich, H. Kovař́ık, A. Laptev, EMS Ser. Congr. Rep.,
Eur. Math. Soc., Zürich, 2017, 531–542, arXiv:1512.06983.
[35] Tanaka A., Cheon T., Kim S.W., Gauge invariants of eigenspace and eigenvalue anholonomies: examples in
hierarchical quantum circuits, J. Phys. A: Math. Theor. 45 (2012), 335305, 20 pages, arXiv:1203.5412.
[36] Uzdin R., Mailybaev A., Moiseyev N., On the observability and asymmetry of adiabatic state flips generated
by exceptional points, J. Phys. A: Math. Theor. 44 (2011), 435302, 8 pages.
[37] Wilczek F., Zee A., Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52 (1984),
2111–2114.
[38] Wojcik C.C., Sun X.-Q., Bzdušek T., Fan S., Homotopy characterization of non-Hermitian Hamiltonians,
Phys. Rev. B 101 (2020), 205417, 17 pages, arXiv:1911.12748.
[39] Xu H., Mason D., Jiang L., Harris J.G.E., Topological energy transfer in an optomechanical system with
exceptional points, Nature 537 (2016), 80–83, arXiv:1602.06881.
[40] Zhang D.-J., Wang Q.-H., Gong J., Quantum geometric tensor in PT-symmetric quantum mechanics, Phys.
Rev. A 99 (2018), 042104, 13 pages, arXiv:1811.04638.
https://doi.org/10.1103/physreva.102.032216
https://doi.org/10.1103/physreva.102.032216
https://arxiv.org/abs/1809.07175
https://doi.org/10.1143/JPSJ.5.435
https://doi.org/10.1007/978-3-662-12678-3
https://doi.org/10.1007/978-1-4419-9982-5
https://doi.org/10.1017/S0013091500018046
https://doi.org/10.1017/S0013091500018046
https://doi.org/10.1103/PhysRevLett.85.3067
https://arxiv.org/abs/quant-ph/9911083
https://arxiv.org/abs/quant-ph/9911083
https://doi.org/10.1063/1.2968344
https://arxiv.org/abs/0807.3405
https://doi.org/10.1103/physreva.92.052124
https://arxiv.org/abs/1410.1882
https://doi.org/10.1126/science.aar7709
https://doi.org/10.1088/0305-4470/36/25/312
https://arxiv.org/abs/quant-ph/0304080
https://doi.org/10.1088/0305-4470/25/21/027
https://doi.org/10.1007/BF03046050
https://doi.org/10.1007/BF03046050
https://doi.org/10.1103/physreva.98.023818
https://doi.org/10.1103/physreva.98.023818
https://arxiv.org/abs/1805.04291
https://arxiv.org/abs/2010.03913
https://doi.org/10.1016/0375-9601(91)90255-7
https://doi.org/10.1007/BF02193559
https://doi.org/10.1142/0613
https://doi.org/10.1103/PhysRevLett.51.2167
https://doi.org/10.1016/j.physleta.2015.05.009
https://arxiv.org/abs/1409.5211
https://doi.org/10.4171/175-1/26
https://arxiv.org/abs/1512.06983
https://doi.org/10.1088/1751-8113/45/33/335305
https://arxiv.org/abs/1203.5412
https://doi.org/10.1088/1751-8113/44/43/435302
https://doi.org/10.1103/PhysRevLett.52.2111
https://doi.org/10.1103/physrevb.101.205417
https://arxiv.org/abs/1911.12748
https://doi.org/10.1038/nature18604
https://arxiv.org/abs/1602.06881
https://doi.org/10.1103/PhysRevA.99.042104
https://doi.org/10.1103/PhysRevA.99.042104
https://arxiv.org/abs/1811.04638
1 Introduction
2 The non-degeneracy space
2.1 Discriminant definition
2.2 Parametrizing the non-degeneracy space
2.3 Spectrum map on non-degeneracy space
2.4 Summarizing diagram
3 Eigenvalue bundle and exceptional points
3.1 The spectrum bundle
3.2 Geometry behind swaps of energies
3.2.1 Merging path method
3.2.2 Real eigenvalue case
4 Eigenvector bundle and geometric phases
4.1 The eigenvector bundle
4.2 Connection on the eigenvector bundle
4.3 Geometry behind the geometric phase
4.3.1 Calculating the lift via an ansatz
4.3.2 The cyclic case
4.3.3 The non-cyclic case
4.4 Including the dynamical phase
4.5 Relation with the work of Aharonov and Anandan
4.6 Quantum geometric tensor
4.6.1 Bases for tangent space of spectrum and eigenvector bundles
4.6.2 Generalized quantum geometric tensor
4.6.3 Relation between geometric phase and distance
5 Explicit description of the holonomy
5.1 Explicit permutations and holonomy matrices
5.2 Holonomy interpretation on the frame bundle
5.2.1 Frame bundle of the spectrum bundle
5.2.2 Frame bundle of the eigenvector bundle
6 Discussion
A Bundle structure of tuples of distinct complex numbers
References
|
| id | nasplib_isofts_kiev_ua-123456789-211542 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T04:12:43Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Pap, Eric J. Boer, Daniël Waalkens, Holger 2026-01-05T12:30:38Z 2022 A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics. Eric J. Pap, Daniël Boer and Holger Waalkens. SIGMA 18 (2022), 003, 42 pages 1815-0659 2020 Mathematics Subject Classification: 81Q70; 81Q12; 55R99 arXiv:2107.02497 https://nasplib.isofts.kiev.ua/handle/123456789/211542 https://doi.org/10.3842/SIGMA.2022.003 We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we demonstrate that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection that yields the (generalized) geometric phase. This bundle also provides a natural generalization of the quantum geometric tensor and derived tensors, and we demonstrate how it can incorporate the non-geometric dynamical phase as well. We conclude by demonstrating how the bundle can be recast as a principal bundle, allowing both the geometric phases and the permutations of eigenstates to be expressed simultaneously using standard holonomy theory. The authors thank the anonymous referees whose careful remarks contributed to the quality of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics Article published earlier |
| spellingShingle | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics Pap, Eric J. Boer, Daniël Waalkens, Holger |
| title | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_full | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_fullStr | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_full_unstemmed | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_short | A Unified View on Geometric Phases and Exceptional Points in Adiabatic Quantum Mechanics |
| title_sort | unified view on geometric phases and exceptional points in adiabatic quantum mechanics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211542 |
| work_keys_str_mv | AT papericj aunifiedviewongeometricphasesandexceptionalpointsinadiabaticquantummechanics AT boerdaniel aunifiedviewongeometricphasesandexceptionalpointsinadiabaticquantummechanics AT waalkensholger aunifiedviewongeometricphasesandexceptionalpointsinadiabaticquantummechanics AT papericj unifiedviewongeometricphasesandexceptionalpointsinadiabaticquantummechanics AT boerdaniel unifiedviewongeometricphasesandexceptionalpointsinadiabaticquantummechanics AT waalkensholger unifiedviewongeometricphasesandexceptionalpointsinadiabaticquantummechanics |