A Unified View on Geometric Phases and Exceptional Points

A Unified View on Geometric Phases and Exceptional
Points in Adiabatic Quantum Mechanics

Eric J. PAP ab, Daniël BOER b and Holger WAALKENS a E.J. Pap, D. Boer and H.
Waalkens

a) Bernoulli Institute, University of Groningen,
a) P.O. Box 407, 9700 AK Groningen, The Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS>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<EMAIL_ADDRESS>http://www.rug.nl/staff/d.boer/

Received July 23, 2021, in final form December 28, 2021; Published online
January 13, 2022

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 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 generalization 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.

adiabatic quantum mechanics; geometric phase; exceptional point; quantum
geometric tensor

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 eigenvalues 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 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 $\mathcal{PT}$ symmetric systems. Such systems are typically
described by non-Hermitian Hamiltonians, and the breaking of this symmetry is
often associated with an EP. Although $\mathcal{PT}$-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 Hamiltonian. In particular, the eigenstates
need not be cyclic and the Hamiltonian need not be Hermitian. 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 concerning 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 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 $A\colon
V\to V$, form the space $\operatorname{End}(V)$, which is a complex manifold
of complex dimension $n^{2}$. The set of all eigenvalues of an operator $A$ is
called the spectrum of $A$, and we denote it as $\operatorname{Spec}(A)$. If
$(v_{1},\dots,v_{n})$ is a frame, or basis, of $V$, we call it an eigenframe
of $A$ if each $v_{i}$ is an eigenvector of $A$. The set of all bases of $V$
we denote as $\operatorname{Fr}(V)$, and the set of all eigenframes of $A$ by
$\mathrm{EigFr}(A)$. Any basis $(v_{1},\dots,v_{n})$ of $V$ has a dual basis
$\big{(}\theta^{1},\dots,\theta^{n}\big{)}$ of the dual space $V^{\vee}$
defined by the condition

$\theta^{i}(v_{j})=\delta^{i}_{j}=\begin{cases}1&\text{if}\quad i=j,\\\
0&\text{if}\quad i\neq j.\end{cases}$

An eigencovector of $A$ is a non-zero covector $\theta$ such that $\theta
A=\lambda\theta$, where $\lambda\in\mathbb{C}$ is necessarily an eigenvalue of
$A$. An eigencoframe of $A$ is a frame of $V^{\vee}$ consisting of
eigencovectors of $A$. One can verify the following facts about eigenframes.

######  Lemma 2.1.

Let $A\in\operatorname{End}(V)$, then

  1. $(1)$

$A$ has an eigenframe if and only if $A$ is diagonalizable,

  2. $(2)$

$(v_{1},\dots,v_{n})$ is an eigenframe of $A$ if and only if the dual basis
$\big{(}\theta^{1},\dots,\theta^{n}\big{)}$ is an eigencoframe.

We will now focus on the operators $A$ which are non-degenerate, i.e., which
have $n$ distinct eigenvalues. The subspace in $\operatorname{End}(V)$ 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 $A\in\operatorname{End}(V)$ 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

$\displaystyle p\colon\ \operatorname{End}(V)$
$\displaystyle\to\mathbb{C}[z],$ $\displaystyle A$ $\displaystyle\mapsto
p(A,z):=\det(zI-A)$

and by evaluating the discriminant one obtains a composite function

$\displaystyle d\colon\ \operatorname{End}(V)$ $\displaystyle\to\mathbb{C},$
$\displaystyle A$ $\displaystyle\mapsto\operatorname{discrim}(p(A,z),z)$

known as the discriminant of operators. Its zero set

$\Delta(V)=\big{\\{}A\in\operatorname{End}(V)\mid d(A)=0\big{\\}}$

is called the discriminant set and consists of all matrices that are
degenerate. Clearly $N(V)$ is the complement of the discriminant set in
$\operatorname{End}(V)$, which yields our formal definition of $N(V)$.

######  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

$N(V)=\big{\\{}A\in\operatorname{End}(V)\mid d(A)\neq 0\big{\\}}.$

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
$\operatorname{End}(V)$. In particular, $N(V)$ is a submanifold of complex
dimension $n^{2}$ and real codimension $2$.

We next consider alternative characterizations of the non-degeneracy space.

###  2.2 Parametrizing the non-degeneracy space

Another way to describe $N(V)$ is by explicitly parametrizing its elements.
One such parametrization readily follows from the fact that any $A\in N(V)$ is
similar to a diagonal matrix where the diagonal entries are distinct. In other
words, there is a frame $\tilde{f}=(f_{1},\dots,f_{n})$ of $V$ so that the
matrix of $A$ w.r.t. $\tilde{f}$ takes the form
$\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$, where all the
$\lambda_{i}$ are distinct. We will denote the space of tuples
$\tilde{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ of $n$ distinct complex
numbers as $\mathbb{C}^{\underline{n}}$; more background on this space can be
found in Appendix A. We abbreviate
$\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$ as
$\operatorname{diag}\big{(}\tilde{\lambda}\big{)}$.

The above decomposition can be recast as the following parametrization. For
convenience, let us identify a frame $\tilde{f}\in\operatorname{Fr}(V)$ with
the map $S_{\tilde{f}}\colon\mathbb{C}^{n}\to V$ given by
$(z_{1},\dots,z_{n})\mapsto\sum_{i=1}^{n}z_{i}f_{i}$, which is a linear
isomorphism by definition of a frame. The parametrization of $N(V)$ is then
formally described using the map

$\displaystyle\Xi\colon\ \operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$
$\displaystyle\to N(V),$
$\displaystyle\big{(}\tilde{f},\tilde{\lambda}\big{)}$ $\displaystyle\mapsto
S_{\tilde{f}}\operatorname{diag}\big{(}\tilde{\lambda}\big{)}S_{\tilde{f}}^{-1},$

which is a smooth surjection. Clearly, this map is not injective for two
reasons; firstly, $\Xi$ is indifferent concerning a non-zero scaling of the
eigenvectors, and secondly, if we permute both the vectors of $\tilde{f}$ and
the values in $\tilde{\lambda}$ we obtain the same operator. This is a
symmetry of $\Xi$, which we phrase using group actions.

First, writing the group of non-zero complex numbers as $\mathbb{C}^{\times}$,
the scaling symmetry is given by the $(\mathbb{C}^{\times})^{n}$-action

$\tilde{z}\cdot\big{(}\tilde{f},\tilde{\lambda}\big{)}=\big{(}\tilde{z}\tilde{f},\tilde{\lambda}\big{)},$
(2.1)

where $\tilde{z}\tilde{f}$ is the entry-wise product
$(z_{1}f_{1},\dots,z_{n}f_{n})\in\operatorname{Fr}(V)$. To describe the
permutations, let us agree that a permutation acts on a tuple by permuting its
entries, e.g., for $\sigma\in S_{n}$ and a tuple
$\tilde{\lambda}\in\mathbb{C}^{\underline{n}}$,
$\sigma\cdot\tilde{\lambda}=\big{(}\lambda_{\sigma^{-1}(1)},\dots,\lambda_{\sigma^{-1}(n)}\big{)}$.
Then the permutation action on
$\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$ is simply

$\sigma\cdot\big{(}\tilde{f},\tilde{\lambda}\big{)}=\big{(}\sigma\tilde{f},\sigma\tilde{\lambda}\big{)}.$
(2.2)

The scaling and permutation actions merge into a single action of the wreath
product $\mathbb{C}^{\times}\wr I_{n}$, where $I_{n}:=\\{1,\dots,n\\}$ is an
index set with $n$ elements. This group is the semi-direct product
$(\mathbb{C}^{\times})^{n}\rtimes S_{n}$, whose defining action of $S_{n}$ on
$(\mathbb{C}^{\times})^{n}$ is exactly the tuple permutation as we used above.
The group multiplication of $\mathbb{C}^{\times}\wr I_{n}$ reads

$(\tilde{z}_{1},\sigma_{1})\cdot(\tilde{z}_{2},\sigma_{2})=\big{(}\tilde{z}_{1}(\sigma_{1}\tilde{z}_{2}),\sigma_{1}\sigma_{2}\big{)}.$

The action of the wreath product on
$\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$ is obtained by
performing first the permutation and then the scaling, i.e.,

$\displaystyle\mathbb{C}^{\times}\wr~{}I_{n}\times(\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}})$
$\displaystyle\to\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}},$
$\displaystyle(\tilde{z},\sigma)\cdot\big{(}\tilde{f},\tilde{\lambda}\big{)}$
$\displaystyle=\big{(}\tilde{z}\big{(}\sigma\tilde{f}\big{)},\sigma\tilde{\lambda}\big{)}.$
(2.3)

Let us make some remarks at this point. First,
$\mathbb{C}^{\times}\wr~{}I_{n}$ 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 $\Xi$, the appearance of
$\mathbb{C}^{\times}\wr~{}I_{n}$ 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 $\Xi$ defines a principal bundle for a wreath product group,
where the number field need not be $\mathbb{C}$. As we will argue now, the
parametrization $\Xi$ also defines a principal $\mathbb{C}^{\times}\wr
I_{n}$-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 $\Xi$ coincides with this
projection.

######  Lemma 2.4.

There is a principal bundle

${\mathbb{C}^{\times}\wr
I_{n}}$${\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}}$${N(V).}$$\scriptstyle{\Xi}$

###### Proof.

As stated, it suffices to show that $\Xi$ and the action quotient map are
isomorphic as bundle maps. As we already saw that the action preserves the
fibers of $\Xi$, it only remains to show that each group orbit exhausts the
fiber in which it lies. Therefore, assume
$\Xi\big{(}\tilde{f},\tilde{\lambda}\big{)}=\Xi\big{(}\tilde{f}^{\prime},\tilde{\lambda}^{\prime}\big{)}=:A$.
Then both $\tilde{\lambda}$ and $\tilde{\lambda}^{\prime}$ constitute
$\operatorname{Spec}(A)$, hence they are related by a permutation $\sigma\in
S_{n}$ as $\tilde{\lambda}=\sigma\tilde{\lambda}^{\prime}$. Then $f_{i}$ needs
to be parallel to $f_{\sigma^{-1}(i)}^{\prime}$ for each $i$, so $\tilde{f}$
and $\sigma\tilde{f}^{\prime}$ are equal up to an element of
$(\mathbb{C}^{\times})^{n}$. However, then
$\big{(}\tilde{f},\tilde{\lambda}\big{)}$ and
$\big{(}\tilde{f}^{\prime},\tilde{\lambda}^{\prime}\big{)}$ differ only up to
an element of $\mathbb{C}^{\times}\wr I_{n}$. ∎

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 $\Xi$. 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 $\operatorname{Fr}(V)$ with $\mathrm{GL}(n,\mathbb{C})$, the space
of non-degenerate operators is realized as a quotient space as

$N(V)\cong\frac{\mathrm{GL}(n,\mathbb{C})\times\mathbb{C}^{\underline{n}}}{\mathbb{C}^{\times}\wr
I_{n}}.$

######  Corollary 2.6.

Let $A_{0}\in N(V)$. There is a neighborhood $U$ of $A_{0}$ on which one has
smooth local eigenvalue functions $\lambda_{1},\dots,\lambda_{n}\colon
U\to\mathbb{C}$, exhausting the spectrum at each point, and smooth local
eigenvector functions $f_{1},\dots,f_{n}\colon U\to V$. That is, for all $A\in
U$ and $i\in I_{n}$ one has

$Af_{i}(A)=\lambda_{i}(A)f_{i}(A)$

such that $\operatorname{Spec}(A)=\\{\lambda_{1}(A),\dots,\lambda_{n}(A)\\}$
and $(f_{1}(A),\dots,f_{n}(A))$ is a basis of $V$. Moreover, the tuple
$(\lambda_{1}(A),\dots,\lambda_{n}(A))$ may be taken to be any given ordering
of $\operatorname{Spec}(A)$, and the tuple $(f_{1}(A),\dots,f_{n}(A))$ any
eigenframe of $A$ following the same ordering.

###### Proof.

The functions $f_{i}(A)$ and $\lambda_{i}(A)$ are the components of a local
section $s\colon U\to\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$,
which can be taken through an arbitrary point above $A$. ∎

###  2.3 Spectrum map on non-degeneracy space

Non-degenerate operators can also be characterized based on their spectrum.
Indeed, an operator $A\in\operatorname{End}(V)$ is non-degenerate if and only
if $\operatorname{Spec}(A)$ consists of $n$ distinct elements. In other words,
$\operatorname{Spec}(A)$ should not be any subset of $\mathbb{C}$, but belong
to the set $\binom{\mathbb{C}}{n}$ of all subsets of $\mathbb{C}$ consisting
of $n$ distinct elements. Taking the spectrum can thus be written as a map

$\displaystyle\operatorname{Spec}\colon N(V)$
$\displaystyle\to\binom{\mathbb{C}}{n},$ $\displaystyle A$
$\displaystyle\mapsto\operatorname{Spec}(A).$

We will now continue by showing that this map $\operatorname{Spec}$ 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 $\operatorname{Spec}$ 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 $\operatorname{Spec}$ is smooth. For the manifold
structure on $\binom{\mathbb{C}}{n}$, we follow the idea that the space
$\binom{\mathbb{C}}{n}$ can be obtained from $\mathbb{C}^{\underline{n}}$ by
reducing an (ordered) tuple to the (unordered) set of its elements. We write
$q\colon\mathbb{C}^{\underline{n}}\to\binom{\mathbb{C}}{n}$ for the quotient
map, and use the manifold structure on $\binom{\mathbb{C}}{n}$ for which $q$
defines a principal $S_{n}$-bundle with the standard permutation action of
$S_{n}$ on $\mathbb{C}^{\underline{n}}$. For more details, we refer to
Appendix A.

We can see that $\operatorname{Spec}$ is smooth using a geometric argument,
based on the projection $q$ and the map $\Xi$ from the previous part. The key
observation is the equality
$\operatorname{Spec}\big{(}\Xi\big{(}\tilde{f},\tilde{\lambda}\big{)}\big{)}=q\big{(}\tilde{\lambda}\big{)}$
for any
$\big{(}\tilde{f},\tilde{\lambda}\big{)}\in\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$;
by construction $\tilde{\lambda}$ lists the eigenvalues of the operator
$\Xi\big{(}\tilde{f},\tilde{\lambda}\big{)}$. In other words, we have the
following commutative diagram:

${\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}}$${\mathbb{C}^{\underline{n}}}$${N(V)}$${\binom{\mathbb{C}}{n}.}$$\scriptstyle{\mathrm{pr}_{\mathbb{C}^{\underline{n}}}}$$\scriptstyle{\Xi}$$\scriptstyle{q}$$\scriptstyle{\operatorname{Spec}}$
(2.4)

Smoothness of $\operatorname{Spec}$ is now clear; as the upper route in the
diagram is smooth and $\Xi$ is a surjective submersion the claim follows.

Let us continue by discussing the model fiber of $\operatorname{Spec}$. This
is facilitated by viewing operators according to their spectral decomposition.
That is, any $A\in N(V)$ can be written as a sum
$\sum_{i=1}^{n}\lambda_{i}P_{i}$, where $\lambda_{i}\in\operatorname{Spec}(A)$
and each $P_{i}\colon V\to V$ is a projection. Geometrically, this expresses
that $A\in N(V)$ 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 $\operatorname{Spec}$ 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 $P_{i}$ projects to a one-
dimensional subspace in $V$, and together these projectors satisfy
$P_{i}P_{j}=\delta_{ij}P_{i}$ and $\sum_{i=1}^{n}P_{i}=\mathrm{id}_{V}$. That
is, the tuple $(P_{1},\dots,P_{n})$ must lie in the space

$\operatorname{PFr}(V):=\big{\\{}(P_{1},\dots,P_{n})\in\operatorname{End}(V)\mid
P_{i}P_{j}=\delta_{ij}P_{i},\,\dim_{\mathbb{C}}(\mathrm{im}(P_{i}))=1\big{\\}},$

which is thus also the model fiber for $\operatorname{Spec}$. One can think of
$\operatorname{PFr}(V)$ 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, $\operatorname{PFr}(V)$ relates to
$\operatorname{Fr}(V)$ by sending a basis $(f_{1},\dots,f_{n})$, with dual
basis $\big{(}\theta^{1},\dots,\theta^{n}\big{)}$, to the tuple
$\big{(}f_{1}\theta^{1},\dots,f_{n}\theta^{n}\big{)}\in\operatorname{PFr}(V)$.
This map is surjective, but not injective as individual scaling of the basis
vectors will yield the same projectors. Hence we find that
$\operatorname{Fr}(V)/(\mathbb{C}^{\times})^{n}$ is canonically isomorphic to
$\operatorname{PFr}(V)$, and it is the former form that we will see in the
upcoming proofs.

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 $\mathrm{GL}(V)$ on
$N(V)$ given by

$\displaystyle\mathrm{GL}(V)\times N(V)$ $\displaystyle\to N(V),$
$\displaystyle S\cdot A$ $\displaystyle=SAS^{-1}.$ (2.5)

The map $\operatorname{Spec}$ is invariant w.r.t. this action by the familiar
rule $\operatorname{Spec}\big{(}SAS^{-1}\big{)}=\operatorname{Spec}(A)$. Hence
each fiber of $\operatorname{Spec}$ is a $\mathrm{GL}(V)$-manifold as well,
and so we wish to view $\operatorname{PFr}(V)$ as a $\mathrm{GL}(V)$-manifold,
also equipped with the conjugation action. Observe that this action on
$\operatorname{PFr}(V)$ is naturally inherited from $\operatorname{Fr}(V)$, on
which it reads $S\cdot(f_{1},\dots,f_{n})=(Sf_{1},\dots,Sf_{n})$. We thus
arrive at the following result.

######  Lemma 2.7.

The $\mathrm{GL}(V)$-action on $N(V)$ is transitive on the fibers of
$\operatorname{Spec}$. Moreover, any fiber of $\operatorname{Spec}$ is
isomorphic to $\operatorname{PFr}(V)$ as $\mathrm{GL}(V)$-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
$\operatorname{Spec}$. Hence, every fiber of $\operatorname{Spec}$ is a
homogeneous $\mathrm{GL}(V)$-space. The stabilizer subgroup at $A\in N(V)$
consists of the maps that preserve all eigenrays of $A$ individually, hence is
isomorphic to $(\mathbb{C}^{\times})^{n}$. In order to parametrize this fiber,
let $\tilde{\lambda}$ be an ordering of $\operatorname{Spec}(A)$, and consider
$\Xi$ restricted to the subset
$\operatorname{Fr}(V)\times\\{\tilde{\lambda}\\}$. Clearly, this surjects on
the fiber of $\operatorname{Spec}$ containing $A$. Moreover, $\Xi$ is
equivariant w.r.t. the canonical $\mathrm{GL}(V)$-action on
$\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$. Hence, the fiber of
$\operatorname{Spec}$ containing $A$ is isomorphic, as
$\mathrm{GL}(V)$-manifold, to the quotient of $\operatorname{Fr}(V)$ by the
stabilizer. The stabilizer $(\mathbb{C}^{\times})^{n}$ is now straightforward;
it appears via the $(\mathbb{C}^{\times})^{n}$-action given in equation (2.1).
Hence we found a $\mathrm{GL}(V)$-equivariant isomorphism to
$\operatorname{Fr}(V)/(\mathbb{C}^{\times})^{n}$, and so to
$\operatorname{PFr}(V)$. ∎

The model fiber of $\operatorname{Spec}$ is thus $\operatorname{PFr}(V)$,
which we view as a $\mathrm{GL}(V)$-manifold, emphasizing its close relation
to similarity transformations. We thus wish to prove that
$\operatorname{Spec}$ defines a fiber bundle that respects the
$\mathrm{GL}(V)$-action. That is, $\operatorname{Spec}$ defines a
$\mathrm{GL}(V)$-manifold bundle in the language of [28]; both fiber and total
space are endowed with a $\mathrm{GL}(V)$-action, and local trivializations
can be taken $\mathrm{GL}(V)$-equivariant. We then arrive at the following
statement, which summarizes the results of this section.

######  Proposition 2.8.

The spectrum map induces the $\mathrm{GL}(V)$-manifold bundle

${\operatorname{PFr}(V)}$${N(V)}$${\binom{\mathbb{C}}{n}.}$$\scriptstyle{\operatorname{Spec}}$

###### Proof.

Pick a point $\\{\lambda_{1},\dots,\lambda_{n}\\}\in\binom{\mathbb{C}}{n}$,
and let $U\subset\binom{\mathbb{C}}{n}$ be a neighborhood of
$\\{\lambda_{1},\dots,\lambda_{n}\\}$ on which a local section $s\colon
U\to\mathbb{C}^{\underline{n}}$ of $q$ is defined. Consider the map

$\displaystyle U\times\operatorname{Fr}(V)$ $\displaystyle\to N(V),$
$\displaystyle\big{(}u,\tilde{f}\big{)}$
$\displaystyle\mapsto\Xi\big{(}\tilde{f},s(u)\big{)},$

which for each $u\in U$ surjects on the fiber of $\operatorname{Spec}$ above
$u$. By applying Lemma 2.7 fiber-wise we find that the reduced map
$U\times\operatorname{Fr}(V)/(\mathbb{C}^{\times})^{n}\to N(V)$ is well-
defined. In fact, if we look at the spectral decomposition, this map pairs the
tuple $(P_{1},\dots,P_{n})\in\operatorname{PFr}(V)$ determined by $\tilde{f}$
with the values in the tuple $s(u)$. We thus obtained a
$\mathrm{GL}(V)$-equivariant local trivialization of $\operatorname{Spec}$
around $\\{\lambda_{1},\dots,\lambda_{n}\\}$, hence the claim follows. ∎

###  2.4 Summarizing diagram

If one takes the bundles defined by $\Xi$ and $\operatorname{Spec}$ plus the
diagram in (2.4), then one readily obtains the following diagram of bundle
sequences:

${(\mathbb{C}^{\times})^{n}}$${\mathbb{C}^{\times}\wr
I_{n}}$${S_{n}}$${\operatorname{Fr}(V)}$${\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}}$${\mathbb{C}^{\underline{n}}}$${\operatorname{PFr}(V)}$${N(V)}$${\binom{\mathbb{C}}{n}.}$$\scriptstyle{\mathrm{pr}_{\mathbb{C}^{\underline{n}}}}$$\scriptstyle{\Xi}$$\scriptstyle{q}$$\scriptstyle{\operatorname{Spec}}$
(2.6)

The direct product $\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$ we
view as a bundle over $\mathbb{C}^{\underline{n}}$. The remaining bundles are
straightforward; on top is the defining decomposition of the wreath product
$\mathbb{C}^{\times}\wr I_{n}$, on the right the quotient map $q$, and on the
left the quotient realization of $\operatorname{PFr}(V)$. Observe that all
rows and columns are group-space bundles, i.e., each one is related to a group
action. With the exception of $\operatorname{Spec}$ all bundles are principal;
$\operatorname{Spec}$ itself defines a bundle of homogeneous
$\mathrm{GL}(V)$-spaces.

##  3 Eigenvalue bundle and exceptional points

Let us study the geometry that describes how eigenvalues and eigenvectors
depend on the operator. 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 particular
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 $\lambda=\lambda(A)$, with $\lambda(A)$ an
eigenvalue of the operator $A$, as a local section. Note that such functions
$\lambda(A)$ 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 $A\in N(V)$ 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 $\operatorname{End}(V)$ to $N(V)$ is
minimal in order to obtain a smooth structure.

######  Lemma 3.1.

Given $A\in\operatorname{End}(V)$, then $A\in N(V)$ if and only if

$\frac{\partial p}{\partial z}(A,\lambda_{i})\neq
0,\qquad\forall\lambda_{i}\in\operatorname{Spec}(A).$

We can now formalize the bundle which has the local eigenvalues as its local
sections. That is, its local sections are of the form $A\mapsto(A,\lambda(A))$
with $\lambda(A)$ 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 $(A,\lambda)$
such that $A\in N(V)$ and $\lambda\in\operatorname{Spec}(A)$. 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
definition111We remark the similarity with the space $\mathcal{M}$ in [21].
However, as we explicitly list the eigenvalue as a coordinate, it is not a
multi-valued function here. of the bundle, because of its algebraic
convenience. We use the name spectrum bundle: the fiber above $A\in N(V)$ is
simply $\operatorname{Spec}(A)$, 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

$\operatorname{Spec}(V)=\big{\\{}(A,\lambda)\in N(V)\times\mathbb{C}\mid
p(A,\lambda)=0\big{\\}}.$

Furthermore, we write $\pi_{\lambda}\colon\operatorname{Spec}(V)\to N(V)$ for
the projection $(A,\lambda)\mapsto A$.

Our first step in proving the bundle property of $\pi_{\lambda}$ is showing
that $\operatorname{Spec}(V)$ is a smooth manifold. This readily follows from
the derivative characterization of $N(V)$ in Lemma 3.1.

######  Proposition 3.3.

The space $\operatorname{Spec}(V)$ is a closed submanifold of
$N(V)\times\mathbb{C}$ of complex dimension $n^{2}$.

###### Proof.

Consider the restricted characteristic polynomial $p\colon
N(V)\times\mathbb{C}\to\mathbb{C}$. As $\operatorname{Spec}(V)$ 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 $\mathrm{d}p(A,z)$, which
contains the term $\frac{\partial p}{\partial z}(A,z)\mathrm{d}z$. This is a
surjection whenever $\frac{\partial p}{\partial z}(A,z)\neq 0$, which holds on
all of $\operatorname{Spec}(V)$ by Lemma 3.1. Hence $\operatorname{Spec}(V)$
is a closed submanifold, of the same dimension as $N(V)$. ∎

We are now in place to deduce the bundle structure of $\pi_{\lambda}$. A fiber
is of the form $\operatorname{Spec}(A)$ with $A\in N(V)$, hence by definition
a set of $n$ distinct points. We thus take the model fiber to be $I_{n}$.
Inspection of $\pi_{\lambda}$ yields the following argument.

######  Theorem 3.4.

The map $\pi_{\lambda}\colon\operatorname{Spec}(V)\to N(V)$ defines a fiber
bundle with model fiber $I_{n}$.

###### Proof.

First, $\pi_{\lambda}$ is a surjective submersion; the implicit function
theorem provides local eigenvalues, and so local sections, through any point
of $\operatorname{Spec}(V)$. By dimension count, $\pi_{\lambda}$ is a local
diffeomorphism. As $\pi_{\lambda}$ is also proper, it is a covering map [18].
Then, as each fiber has exactly $n$ elements, $\pi_{\lambda}$ is an
$I_{n}$-bundle. Indeed, a local trivialization is a map of the form

$\displaystyle\phi\colon\ U\times I_{n}$
$\displaystyle\to\operatorname{Spec}(V)|_{U},$ $\displaystyle(A,i)$
$\displaystyle\mapsto(A,\lambda_{i}(A))$ (3.1)

with $\lambda_{1},\dots,\lambda_{n}$ distinct local eigenvalues. ∎

This result formalizes the idea that $\operatorname{Spec}(V)$ is locally the
union of the graphs of $n$ local eigenvalues. We remark that the inverse of
the local trivialization $\phi$ in equation (3.1) above can be written as
$(A,\lambda)\mapsto(A,\\#(A,\lambda))$, where
$\\#\colon\operatorname{Spec}(V)|_{U}\to I_{n}$ 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. 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

$H\colon\ M\to\operatorname{End}(V),$

which sends a configuration of system parameters $x\in M$ to the Hamiltonian
$H(x)$ 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 $\gamma$ in $M$, whose initial point $x_{0}$ serves as a
reference. This results in the time-dependent Hamiltonian $H(\gamma(t))$. A
state $\psi(t)$ is called an instantaneous eigenstate at time $t$ if it
satisfies the eigenvalue problem $H(\gamma(t))\psi(t)=E(t)\psi(t)$, with
$E(t)$ an energy of $H(\gamma(t))$. 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 $\operatorname{Spec}(V)$. First, we remark that in order to unambiguously
follow a specific energy level of $H(\gamma(t))$, the energy level may not
become degenerate at any time. Hence we require $H(\gamma(t))$ to be non-
degenerate for all $t$, i.e., $t\mapsto H(\gamma(t))$ is a path in $N(V)$. Let
$E_{0}$ be the energy level of the initial Hamiltonian $H(x_{0})$ that we wish
to follow in time. Clearly, this defines the point $(H(x_{0}),E_{0})$ in the
fiber of $\operatorname{Spec}(V)$ above $H(x_{0})$. Hence, it specifies a
unique lift of $H\circ\gamma$ to $\operatorname{Spec}(V)$, which is of the
form $(H(\gamma(t)),E(t))$ 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 $\operatorname{Spec}(V)\to N(V)$ 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

$N(H):=H^{-1}(N(V))=\big{\\{}x\in M\mid H(x)\text{ has non-degenerate energy
levels}\big{\\}}.$

The energy bands of $H$, with the degenerate points omitted, then form the
space

$\operatorname{Spec}(H):=\big{\\{}(x,E)\in N(H)\times\mathbb{C}\mid
E\in\operatorname{Spec}(H(x))\big{\\}},$

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
$\operatorname{Spec}(H)$ the _spectrum bundle of $H$_; it has a natural
projection $\pi_{\lambda}^{H}\colon(x,E)\mapsto x$ to $N(H)$, so that the
fiber above $x$ is simply $\operatorname{Spec}(H(x))$. The bundle property of
$\pi_{\lambda}^{H}$ is guaranteed by the pull-back construction, and relates
to $\pi_{\lambda}$ as given by the pull-back diagram below:

${\operatorname{Spec}(H)}$${\operatorname{Spec}(V)}$${(x,E)}$${(H(x),E)}$${N(H)}$${N(V),}$${x}$${H(x).}$$\scriptstyle{\pi_{\lambda}^{H}}$$\scriptstyle{\pi_{\lambda}}$$\scriptstyle{H}$

We see that the path $\gamma$ can be lifted directly to
$\operatorname{Spec}(H)$, which is then both an intuitive and formally correct
way to obtain the function $E(t)$ directly. Indeed, by assumption $\gamma$
lies in $N(H)$, the initial point is now $(x_{0},E_{0})$, and the lift is of
the form $(\gamma(t),E(t))$.

We see that the study of $\operatorname{Spec}(H)$ is a straightforward
generalization of Kato’s original setting [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 degeneracy was called an EP. Here, we see that
$\operatorname{Spec}(H)$ 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 $\mathcal{PT}$-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
$\operatorname{Spec}(H)$. In fact, the permutations of energies of $H(x_{0})$
are naturally described by the monodromy action at $x_{0}$, 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 $\gamma$ in $N(H)$ should return to our reference
$x_{0}\in N(H)$. In other words, $\gamma$ should be a loop based at $x_{0}$.
Let us write $\operatorname{Loop}(N(H),x_{0})$ for the set of such loops. We
are then interested in how the energies of $H(x_{0})$ return upon following
them along $\gamma$. For an energy $E\in\operatorname{Spec}(H(x_{0}))$, this
is given by the lift of $\gamma$ to $\operatorname{Spec}(H)$; the final energy
is the endpoint of the lift, which we write as $p_{\gamma}(E)$. Doing this for
every energy of $H(x_{0})$ yields the map
$p_{\gamma}\colon\operatorname{Spec}(H(x_{0}))\to\operatorname{Spec}(H(x_{0}))$.
This map is a bijection/permutation of $\operatorname{Spec}(H(x_{0}))$; its
inverse is given by traversing $\gamma$ in the opposite direction. Going over
all loops in $\operatorname{Loop}(N(H),x_{0})$ then yields the group of all
possible permutations of $\operatorname{Spec}(H(x_{0}))$, namely the group

$\operatorname{Hol}^{\operatorname{Spec}(H)}_{N(H)}(x_{0})=\big{\\{}p_{\gamma}\in\operatorname{Aut}(\operatorname{Spec}(H(x_{0})))\mid\gamma\in\operatorname{Loop}(N(H),x_{0})\big{\\}},$

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 $\Lambda(x_{0})$ 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
$H\colon\mathbb{C}\to\operatorname{End}\big{(}\mathbb{C}^{2}\big{)}$ given by

$H(x)=\begin{pmatrix}1&x\\\ x&-1\end{pmatrix}\\!.$

The eigenvalues are given by the multi-valued function $\sqrt{1+x^{2}}$, so
$N(H)=\mathbb{C}\setminus\\{\pm{\rm i}\\}$ and $\operatorname{Spec}(H)$ is the
graph of this multi-valued function, which in this case is a Riemann surface.
Let us take a branch cut, and write $E_{\pm}(x)$ for the two energy branches.
Clearly, $\operatorname{Spec}(H(x_{0}))=\\{E_{+}(x_{0}),E_{-}(x_{0})\\}$, and
if $\gamma$ is a loop based at $x_{0}$ encircling an EP once, it induces the
map

$p_{\gamma}\colon\ E_{\pm}(x_{0})\mapsto E_{\mp}(x_{0}).$

Of course, if $\gamma$ does not encircle any EP, we obtain
$p_{\gamma}=\mathrm{id}_{\operatorname{Spec}(H(x_{0}))}$.

The association $\gamma\mapsto p_{\gamma}$ brings us to a monodromy action as
follows. The map $p_{\gamma}$ is invariant under continuous deformation of
$\gamma$, i.e., it only depends on its homotopy class $[\gamma]$. The set of
such homotopy classes of loops based at $x_{0}$ is the (based) fundamental
group $\pi_{1}(N(H),x_{0})$. The exchange of energies can then be expressed by
the homomorphism
$\pi_{1}(N(H),x_{0})\to\operatorname{Hol}^{\operatorname{Spec}(H)}_{N(H)}(x_{0})$,
$[\gamma]\mapsto p_{\gamma}$, or equivalently as an action of
$\pi_{1}(N(H),x_{0})$ on $\operatorname{Spec}(H(x_{0}))$. This is known as the
monodromy action, which we write as

$\displaystyle\pi_{1}(N(H),x_{0})\times\operatorname{Spec}(H(x_{0}))$
$\displaystyle\to\operatorname{Spec}(H(x_{0})),$
$\displaystyle[\gamma]\cdot(x_{0},E)$ $\displaystyle=(x_{0},p_{\gamma}(E)).$

A picture to have in mind is Figure 1; following the energy level along
$\gamma$ defines a path in $\operatorname{Spec}(H)$, which may end at a
different energy. An explicit example we treat in Example 3.6 below. We note
that the group $\pi_{1}(N(H),x_{0})$ is typically non-trivial. Indeed, by
Lemma 2.3 the space $N(V)$ has real codimension 2 in $\operatorname{End}(V)$
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 $\operatorname{Spec}(H)$
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 $\mathbf{+}$ marks the EP of the
system related to the drawn exchange.

######  Example 3.6.

Continuing Example 3.5, it follows that $\pi_{1}(N(H),x_{0})$ is isomorphic to
the fundamental group of the “figure 8”, which is a free group on two
generators, for any reference $x_{0}$. Picking $x_{0}=0$, the monodromy action
maps both generators of $\pi_{1}(N(H),0)$ to the interchange of $+1$ and $-1$
in $\operatorname{Spec}(H(0))$.

The realization of $\operatorname{Spec}(H)$ as a pull-back of
$\operatorname{Spec}(V)$ allows us to distinguish between inherent geometric
properties coming from $\operatorname{Spec}(V)$ versus artifacts of the family
$H$. Such artifacts can result in different, i.e., non-isomorphic, spaces
$\operatorname{Spec}(H)$ for the different $H$. For example, we can explicitly
add an artifact to a particular $H\colon M\to\operatorname{End}(V)$ 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 $\operatorname{Spec}(H)$, 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 $H$.

####  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 $\gamma$ in $N(H)$. The spectrum then changes
according to the loop $\operatorname{Spec}(H(\gamma(t)))$ in
$\binom{\mathbb{C}}{n}$, with $n$ the dimension of the state space. To extract
the permutation of the energies, note that we cannot work in
$\binom{\mathbb{C}}{n}$ alone as the (unordered) spectrum will return to
itself regardless of the presence of EPs. We must thus lift the loop
$\operatorname{Spec}(H(\gamma(t)))$ to $\mathbb{C}^{\underline{n}}$ in order
to observe and extract a permutation. This argument translates directly into
the diagram

${\mathbb{C}^{\underline{n}}}$${N(H)}$${N(V)}$${\binom{\mathbb{C}}{n},}$$\scriptstyle{H}$$\scriptstyle{\operatorname{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 $\operatorname{Spec}(H)$ can have additional properties if the
family $H$ satisfies additional assumptions. One interesting assumption on $H$
is that $H(x)$ has real energies for all $x\in M$. 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 $\mathcal{PT}$ symmetry (see [24] on the relation between
these). In this case, $\operatorname{Spec}(H)$ 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
$\operatorname{Spec}(H)$. We remark that the argument also holds if the
eigenvalues are taken in another totally ordered subset of $\mathbb{C}$.

######  Proposition 3.7.

Write $R(V)$ for the subspace of $N(V)$ of matrices with real eigenvalues. The
bundle $\operatorname{Spec}(V)$ restricted to $R(V)$ is trivial.

###### Proof.

If $A\in R(V)$, then there is a unique ordering of $\operatorname{Spec}(A)$ by
indexing the eigenvalues from lowest to highest
$\lambda_{1}(A)<\dots<\lambda_{n}(A)$. This establishes a global labelling map
$\operatorname{Spec}(V)|_{R(V)}\to R(V)\times I_{n}$ defined as
$(A,\lambda_{i})\mapsto(A,i)$, which is a homeomorphism. ∎

######  Corollary 3.8.

If $H(x)$ has real eigenvalues for all $x\in N(H)$, then
$\operatorname{Spec}(H)$ is a trivial bundle. In particular, there are $n$
distinct energy bands $E_{1}(x),\dots,E_{n}(x)$ 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 $\operatorname{Spec}(V)$. We will now extend this formalism to
eigenstates, where again non-Hermitian Hamiltonians are allowed.

###  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
$A\in\operatorname{End}(V)$ corresponding to an eigenvalue
$\lambda\in\operatorname{Spec}(A)$ we denote as

$\operatorname{Eig}_{\lambda}(A)=\big{\\{}v\in V\setminus\\{0\\}\mid
Av=\lambda v\big{\\}}=\ker(A-\lambda I)\setminus\\{0\\}.$

We call this the space of eigenvectors, or eigenvector space, of $A$
corresponding to $\lambda$. This terminology emphasizes that the zero vector
is excluded. In this way, each $\operatorname{Eig}_{\lambda}(A)$ is a free
$\mathbb{C}^{\times}$-manifold; any non-zero multiple of an eigenvector is
again an eigenvector. Similarly, we define the eigenvector space
$\operatorname{Eig}(A)$ of $A$ to be the set of all eigenvectors of $A$.
Naturally,

$\operatorname{Eig}(A)=\bigsqcup_{\lambda\in\operatorname{Spec}(A)}\operatorname{Eig}_{\lambda}(A),$

which is immediately the partition of $\operatorname{Eig}(A)$ into its
connected components. We see that $\operatorname{Eig}(A)$ is also endowed with
$\mathbb{C}^{\times}$-scaling, but need not be a manifold as the eigenvector
spaces of the eigenvalues need not have the same dimension. However, in case
$A\in N(V)$ clearly $\operatorname{Eig}(A)$ is a
$\mathbb{C}^{\times}$-manifold of complex dimension 1, and the above
decomposition shows $\operatorname{Eig}(A)$ as a union of
$\mathbb{C}^{\times}$-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 $v(A)\in\operatorname{Eig}(A)$, 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 $\operatorname{Eig}(A)$ 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
$\operatorname{Eig}(A)$ 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

$\operatorname{Eig}(V)=\big{\\{}(A,\lambda,v)\in\operatorname{Spec}(V)\times
V\setminus\\{0\\}\mid v\in\operatorname{Eig}_{\lambda}(A)\big{\\}}.$

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 $\operatorname{Eig}(V)$ is a bundle with respect to
two different projections. The idea is that we can view
$\operatorname{Eig}(V)$ as a union of total eigenvector spaces or as a
disjoint union of individual eigenrays;

$\operatorname{Eig}(V)=\bigsqcup_{A\in
N(V)}\operatorname{Eig}(A)=\bigsqcup_{(A,\lambda)\in\operatorname{Spec}(V)}\operatorname{Eig}_{\lambda}(A).$

Both of these viewpoints come with their own projection. For the first, one
projects $(A,\lambda,v)\\!\mapsto\\!A$, which we write as the map
$\pi_{\lambda v}\colon\operatorname{Eig}(V)\to N(V)$. For the second, one only
omits the eigenvector and projects to $(A,\lambda)$, which defines a map
$\pi_{v}\colon\operatorname{Eig}(V)\to\operatorname{Spec}(V)$. These two
projections are related by the projection $\pi_{\lambda}$, as summarized in
the diagram below:

${\operatorname{Eig}(V)}$${\operatorname{Spec}(V)}$${(A,\lambda,v)}$${(A,\lambda)}$${N(V),}$${A.}$$\scriptstyle{\pi_{v}}$$\scriptstyle{\pi_{\lambda
v}}$$\scriptstyle{\pi_{\lambda}}$ (4.1)

Our first step in proving the bundle claims is to show that
$\operatorname{Eig}(V)$ is a smooth manifold. Clearly $\operatorname{Eig}(V)$
is the zero set of

$\displaystyle P\colon\ \operatorname{Spec}(V)\times V\setminus\\{0\\}$
$\displaystyle\to V,$ $\displaystyle(A,\lambda,v)$
$\displaystyle\mapsto(\lambda I-A)v.$

In this way we obtain the following structure on $\operatorname{Eig}(V)$.

######  Proposition 4.2.

The space $\operatorname{Eig}(V)$ is a closed algebraic
$\mathbb{C}^{\times}$-submanifold of $\operatorname{Spec}(V)\times
V\setminus\\{0\\}$ of complex dimension $n^{2}+1$. Moreover, the projection
maps $\pi_{\lambda v}$ and $\pi_{v}$ are smooth.

###### Proof.

Pick a point $(A_{0},\lambda_{0},v_{0})\in\operatorname{Eig}(V)$ and pick
local coordinates $(A,\lambda)$ on $\operatorname{Spec}(V)$ around
$(A_{0},\lambda_{0})$; this implicitly defines $\lambda=\lambda(A)$ with
$\lambda(A_{0})=\lambda_{0}$. Consider now the differential

$\mathrm{d}P=(\lambda(A)I-A)\mathrm{d}v+\mathrm{d}(\lambda(A)I-A)v.$

As $P$ is constant on the $\mathbb{C}^{\times}$-orbits, $\mathrm{d}P$ has rank
at most $n-1$. On the other hand, the term $(\lambda(A)I-A)\mathrm{d}v$ has
the same rank as $\lambda(A)I-A$, which is $n-1$ by non-degeneracy. Hence
$\mathrm{d}P$ has constant rank $n-1$, and so by the constant rank theorem
[18] the fibers of $P$ are closed submanifolds of (complex) dimension
$\big{(}n^{2}+n\big{)}-(n-1)=n^{2}+1$. The maps $\pi_{\lambda v}$ and
$\pi_{v}$ are then restrictions of smooth projections to a submanifold, hence
smooth. ∎

Both $\pi_{\lambda v}$ and $\pi_{v}$ are invariant w.r.t. the
$\mathbb{C}^{\times}$-action on $\operatorname{Eig}(V)$, which hints at a
principal bundle structure. For $\pi_{v}$, this holds; its fibers are
eigenrays $\operatorname{Eig}_{\lambda}(A)$, which are
$\mathbb{C}^{\times}$-torsors, and $\pi_{v}$ coincides with the quotient of
the action on $\operatorname{Eig}(V)$. However, $\pi_{\lambda v}$ is clearly
not principal. The fiber of $\pi_{\lambda v}$ above $A\in N(V)$ is the total
eigenvector space $\operatorname{Eig}(A)$ of $A$, which is a union of
$\mathbb{C}^{\times}$-torsors, hence not a $\mathbb{C}^{\times}$-torsor itself
for $n>1$. Still, with the eye on adiabatic quantum mechanics, we are
motivated to describe $\pi_{\lambda v}$ 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 $G\times I$ 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 $G\times I_{n}$. More details on semi-
principal bundles can be found in [28].

Let us show that $\pi_{\lambda v}\colon\operatorname{Eig}(V)\to N(V)$ is a
semi-principal $\mathbb{C}^{\times}$-bundle with model fiber
$\mathbb{C}^{\times}\times I_{n}$. 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 $A\in N(V)$, there is an open neighborhood $U\subset N(V)$ of
$A$ and $n$ eigenvector functions $v_{1},\dots,v_{n}\colon U\to V$ 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 $\mathbb{C}^{\times}\wr~{}I_{n}$, as we did in
the proof of Lemma 2.4. ∎

######  Theorem 4.4.

The map $\pi_{\lambda v}\colon\operatorname{Eig}(V)\to N(V)$ defines a semi-
principal $\mathbb{C}^{\times}$-bundle with model fiber
$\mathbb{C}^{\times}\times I_{n}$.

###### Proof.

Pick $A_{0}\in N(V)$, let $U\subset N(V)$ be a neighborhood of $A_{0}$ and for
$i=1,\dots,n$ let $v_{i}\colon U\to V$ be a local eigenvector corresponding to
eigenvalue $\lambda_{i}(A_{0})$, according to Lemma 4.3. One has a map

$\displaystyle\Phi\colon\ U\times I_{n}\times\mathbb{C}^{\times}$
$\displaystyle\to\operatorname{Eig}(V)|_{U},$ $\displaystyle(A,i,z)$
$\displaystyle\mapsto(A,\lambda_{i}(A),zv_{i}(A)),$

which is smooth and clearly intertwines the $\mathbb{C}^{\times}$-actions. Its
inverse is

$(A,\lambda,v)\mapsto\big{(}A,\\#(A,\lambda),\big{[}v/v_{\\#(A,\lambda)}(A)\big{]}\big{)},$

which satisfies the same properties. Here $\\#$ is the local labelling, and
the last entry denotes the scalar $z$ such that $v=zv_{\\#(A,\lambda)}(A)$. It
follows that $\Phi$ is a $\mathbb{C}^{\times}$-equivariant local
trivialization, hence the bundle property follows. ∎

We observe that $\pi_{\lambda}$ and $\pi_{v}$ are closely tied to
$\pi_{\lambda v}$. 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 $\pi_{\lambda v}=\pi_{\lambda}\circ\pi_{v}$ is the decomposition of
$\pi_{\lambda v}$ in a principal bundle projection followed by a covering map.

###  4.2 Connection on the eigenvector bundle

Our next step is to show that $\operatorname{Eig}(V)$ supports a canonical
connection. This connection is compatible with both the semi-principal
projection $\pi_{\lambda v}$ and the principal projection $\pi_{v}$. 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 $\mathfrak{g}$; a 1-form
$\omega$ on a $G$-manifold $M$ is a $G$-connection if $\omega_{m}\circ
a_{m}=\mathrm{id}_{\mathfrak{g}}$ for all $m\in M$, with $a_{m}$ the
infinitesimal action at $m$, and
$L_{g}^{*}\omega=\operatorname{Ad}_{g}(\omega)$ for any $g\in G$. See also
[28] on this.

To describe the connection 1-form on $\operatorname{Eig}(V)$, it is helpful to
extend our notation. The set of eigencovectors of $A\in\operatorname{End}(V)$
we denote as $\operatorname{Eig}^{\vee}(A)$, and those corresponding to a
specific eigenvalue $\lambda$ by $\operatorname{Eig}^{\vee}_{\lambda}(A)$.
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
$\operatorname{Eig}(A)\to\operatorname{Eig}^{\vee}(A)$ sending
$v\in\operatorname{Eig}_{\lambda}(A)$ to the unique covector
$\theta\in\operatorname{Eig}^{\vee}_{\lambda}(A)$ defined by $\theta(v)=1$.

###### Proof.

Following the decomposition of $V$ by the eigenspaces of $A$, define the
covector $\theta$ to vanish on every eigenspace except the one of $\lambda$,
where it is fixed by setting $\theta(v)=1$. ∎

This correspondence shows that the scaling action on $\operatorname{Eig}(A)$
corresponds to inverse scaling on $\operatorname{Eig}^{\vee}(A)$, i.e., we
obtain the $\mathbb{C}^{\times}$-action on $\operatorname{Eig}^{\vee}(A)$
given by

$z\cdot\theta=\theta z^{-1}.$

The bijection $\operatorname{Eig}(A)\to\operatorname{Eig}^{\vee}(A)$ is then
an isomorphism of $\mathbb{C}^{\times}$-manifolds. In addition, as any triple
$(A,\lambda,v)\in\operatorname{Eig}(V)$ defines a unique $\theta$, we are free
to augment the triple to $(A,\lambda,v,\theta)$. This will be a convenient
notation when defining maps on $\operatorname{Eig}(V)$, such as the connection
1-form we seek at the moment. Similarly, one is free to omit $\lambda$ from
the triple. We will use these alternative notations frequently.

We are now set to describe the connection 1-form on $\operatorname{Eig}(V)$.
By definition, this 1-form is a right-inverse of the infinitesimal action,
which is specified by the vector field

$\frac{\partial}{\partial
v}\bigg{|}_{(A,v)}:=\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\exp(t)\cdot(A,v)=\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\big{(}A,{\rm
e}^{t}v\big{)},$

which can be thought of as a unit vector field pointing along the eigenray of
$A$ through $v$. Hence, given a path
$\Gamma(t)=(A(t),\lambda(t),v(t),\theta(t))$ in $\operatorname{Eig}(V)$, the
connection should measure the component of $\dot{v}$ 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 $\dot{v}$
along $v$ is simply $\theta(\dot{v})$.

Let us describe this argument formally. To obtain the quantity $\dot{v}$ from
$\dot{\Gamma}$, we can use the differential of the projection
$\mathrm{pr}_{V}\colon\operatorname{Eig}(V)\to V$, $(A,\lambda,v)\mapsto v$.
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 $\mathrm{d}v$ for
$(\mathrm{pr}_{V})_{*}$ with the image viewed in $V$. This map can be followed
up with $\theta$, which yields the connection 1-form.

######  Proposition 4.7.

The $1$-form $\omega\in\Omega^{1}(\operatorname{Eig}(V),\mathbb{C})$ given by

$\omega=\theta\mathrm{d}v$

is a $\mathbb{C}^{\times}$-connection.

###### Proof.

As $\mathbb{C}^{\times}$ is commutative, the equivariance of $\omega$ reduces
to invariance. This holds by the opposite scaling of $\theta$ and $v$: for
$z\in\mathbb{C}^{\times}$, $(L_{z})^{*}\omega=\big{(}\theta
z^{-1}\big{)}(\mathrm{d}(zv))=\theta\mathrm{d}v=\omega$. The left-inverse
property of $\omega$ follows as
$\omega(\partial_{v})=\theta\mathrm{d}v(\partial_{v})=\theta(v)=1$. ∎

For future reference, we also remark that this induces the global curvature
form $K$ on $\operatorname{Eig}(V)$ in the standard way, i.e.,
$K:=\mathrm{d}\omega+\frac{1}{2}[\omega,\omega]=\mathrm{d}\omega$. In
coordinates it reads $K=\mathrm{d}\theta\wedge\mathrm{d}v$, where
$\mathrm{d}\theta$ is defined similar to $\mathrm{d}v$. As
$\mathbb{C}^{\times}$ is commutative, the curvature $K$ is invariant under the
group action. Hence it admits push-forward along the quotient map $\pi_{v}$ to
$\operatorname{Spec}(V)$, resulting in the following.

######  Lemma 4.8.

The curvature form $K$ on $\operatorname{Eig}(V)$ reduces to a unique $2$-form
$k\in\Omega^{2}(\operatorname{Spec}(V),\mathbb{C})$, which satisfies
$K=\pi_{v}^{*}(k)$.

###  4.3 Geometry behind the geometric phase

We will now show that the connection $\omega$ yields the geometric phases in
adiabatic quantum mechanics in a natural way. That is, $\operatorname{Eig}(V)$
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 $\operatorname{Spec}(V)$ to
$\operatorname{Spec}(H)$ in Section 3.2 holds here. Namely, the space
$\operatorname{Eig}(V)$ 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 $\operatorname{Eig}(V)$ along the Hamiltonian family $H$ yields the
_eigenstate bundle of $H$_, given by

$\operatorname{Eig}(H)=\big{\\{}(x,E,\psi)\in\operatorname{Spec}(H)\times
V\setminus\\{0\\}\mid\psi\in\operatorname{Eig}_{E}(H(x))\big{\\}}.$

We observe that the pull-back construction of this space immediately
guarantees various properties of $\operatorname{Eig}(H)$. Clearly,
$\operatorname{Eig}(H)$ is a smooth manifold. One can also write an element as
$(x,E,\psi,\chi)$, where $\chi$ is the unique left-eigenstate of $H(x)$
corresponding to $E$ so that $\chi(\psi)=1$. Similarly, one may omit $E$ from
the tuple. The projection $\pi_{\lambda v}^{H}\colon\operatorname{Eig}(H)\to
N(H)$ is a semi-principal $\mathbb{C}^{\times}$-bundle, and the projection
$\pi_{v}^{H}\colon\operatorname{Eig}(H)\to\operatorname{Spec}(H)$ is a
principal $\mathbb{C}^{\times}$-bundle. The latter bundle confirms that also
$\operatorname{Eig}(H)$ can have non-trivial topology arising from the
permutations around the EPs of $H$.

The local sections of $\pi_{\lambda v}^{H}$ are in correspondence with local
eigenstates. Here, a local eigenstate is a smooth function $\psi\colon
U\to\operatorname{Eig}(H)$, with $U\subset N(H)$ open, such that $\psi(x)$ is
an eigenstate of $H(x)$. Clearly, this fixes a local eigenvalue $E(x)$. This
data is summarized in the corresponding local section $\hat{\psi}$ as

$\hat{\psi}\colon\ x\mapsto(x,E(x),\psi(x)).$

Observe that for each $x\in U$, knowing $E(x)$ and $\psi(x)$ fixes a unique
left-eigenstate $\chi(x)$ of $H(x)$, which depends smoothly on $x$. Hence we
can include $\chi(x)$ as the fourth component of the local section
$\hat{\psi}$, following our earlier remark on notation of elements of
$\operatorname{Eig}(H)$.

The space $\operatorname{Eig}(H)$ also inherits a connection, which we write
as $\omega_{H}$. Again, for any family $H$, the smoothness and other
properties of $\operatorname{Eig}(H)$ 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 $\hat{\psi}$, we can express $\omega_{H}$ locally as

$\hat{\psi}^{*}\omega_{H}=\chi\mathrm{d}\psi.$

This is indeed the expression reported by Garrison and Wright [12] as the
generalization of the Berry connection. That is, the parallel transport on
$\operatorname{Eig}(H)$ defined via the horizontal lifts w.r.t. $\omega_{H}$
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 $\operatorname{Eig}(H)$.

For example, the connection $\omega_{H}$ on $\operatorname{Eig}(H)$ 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 $\gamma$ is invariant
under continuous deformation of $\gamma$, which is equivalent to vanishing of
the curvature $K_{H}$ of $\operatorname{Eig}(H)$. 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 $\omega_{H}$ on $\operatorname{Eig}(H)$ 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 $x\in N(H)$.
Explicitly, if a local eigenstate $\psi$ is such that
$\psi^{\mathrm{T}}\psi=1$, then $\hat{\psi}^{*}\omega_{H}=0$.

###### Proof.

If $H=H(z)$ is analytic, we can find an analytic local eigenstate $\psi(z)$.
Consequently, $\hat{\psi}^{*}\omega_{H}=f(z)\mathrm{d}z$ for some analytic
function $f$. Using coordinates $z$ and $\bar{z}$ for the parameter space,
$\hat{\psi}^{*}K_{H}$ is proportional to $\frac{\partial
f(z)}{\partial\bar{z}}$. As this derivative vanishes, it follows that
$K_{H}=0$.

In the symmetric case, $\psi^{\mathrm{T}}$ is a non-zero multiple of the left-
eigenstate $\chi$ corresponding to $\psi$. Hence the function
$\psi^{\mathrm{T}}\psi$ is non-vanishing, and one may always (locally) scale
$\psi$ so that $\psi^{\mathrm{T}}\psi=1$, implying $\chi=\psi^{\mathrm{T}}$.
By partial integration,
$\hat{\psi}^{*}\omega_{H}=\tfrac{1}{2}(\chi\mathrm{d}\psi-\mathrm{d}\chi\psi)=\tfrac{1}{2}\big{(}\psi^{\mathrm{T}}\mathrm{d}\psi-\mathrm{d}\psi^{\mathrm{T}}\psi\big{)}=0.$
∎

####  4.3.1 Calculating the lift via an ansatz

An explicit calculation of the lift of a path to $\operatorname{Eig}(H)$ can
be done using an ansatz technique. For this technique, it does not matter if
the path lies in $N(H)$ or $\operatorname{Spec}(H)$. Indeed, if a path
$\gamma$ in $N(H)$ and an initial eigenstate $\psi_{0}$ of $H(x_{0})$ is
chosen, then $\psi_{0}$ has an energy $E_{0}$, and covering theory yields the
corresponding path $\bar{\gamma}(t)=(\gamma(t),E(t))$ in
$\operatorname{Spec}(H)$. Conversely, any path $\bar{\gamma}$ in
$\operatorname{Spec}(H)$ yields a path in $N(H)$, which obviously lifts to
$\bar{\gamma}$. If we restrict to loops the story changes, as we will see
after discussing the ansatz technique.

The ansatz technique here is simply the application of the usual one to the
parallel transport equation. Let us continue with the path $\gamma$ in $N(V)$
and the initial state $\psi_{0}$. This data fixes a unique lift to
$\operatorname{Eig}(H)$, which we denote by $\Gamma$ and which is given by

$\Gamma(t)=(\gamma(t),E(t),\Psi(t)),$

where $\Psi(t)$ is the adiabatically evolved eigenstate at time $t$, up to
dynamical phase. Naturally, this lift is also given as
$(\bar{\gamma}(t),\Psi(t))$ in the perspective of lifting a path
$\bar{\gamma}$ from $\operatorname{Spec}(V)$ to $\operatorname{Eig}(V)$.

In practice, one would not calculate $\Psi(t)$ directly, but instead approach
this problem using reference instantaneous eigenstates $\psi(t)$. That is, for
each $t$, one finds an eigenstate $\psi(t)$ of $H(t)$ with energy $E(t)$, or
$\psi(t)\in\operatorname{Eig}_{E(t)}(H(t))$ for short. For example, $\psi(t)$
can be obtained by explicitly calculating an eigenstate of $H(\gamma(t))$, and
then letting $t$ vary. We may assume that $t\mapsto\psi(t)$ is differentiable
and satisfies the initial condition $\psi(0)=\psi_{0}$. These instantaneous
eigenstates define a path $\Gamma_{0}(t)=(\gamma(t),E(t),\psi(t),\chi(t))$ in
$\operatorname{Eig}(H)$, where $\chi(t)$ is the path of accompanying left-
eigenstates.

Of course, $\Gamma_{0}$ does not need to be the lift $\Gamma$, as equivalently
$\psi(t)$ does not need to be equal the actual state $\Psi(t)$. However,
$\Gamma_{0}$ can function as an ansatz to calculate $\Gamma$ explicitly.
Indeed, both $\psi(t)$ and $\Psi(t)$ must lie in
$\operatorname{Eig}_{E(t)}(H(t))$, and so differ only by a scale factor ${\rm
e}^{f(t)}$. That is, one has

$\Gamma(t)={\rm e}^{f(t)}\cdot\Gamma_{0}(t)=\big{(}\gamma(t),E(t),{\rm
e}^{f(t)}\psi(t),\chi(t){\rm e}^{-f(t)}\big{)}$

with $f(t)$ some differentiable complex-valued function satisfying $f(0)=0$.
This $f$ depends solely on the ansatz $\psi$ by imposing the lift condition

$0=(\omega_{H})_{\Gamma(t)}\big{(}\dot{\Gamma}(t)\big{)}=\chi(t){\rm
e}^{-f(t)}\frac{\mathrm{d}}{\mathrm{d}t}\big{(}{\rm
e}^{f(t)}\psi(t)\big{)}=\dot{f}(t)+\chi(t)\dot{\psi}(t).$

Solving for $f(t)$, we thus find the actual state from the ansatz by
application of the scale factor ${\rm e}^{f(t)}$;

$\Psi(t)=\exp\bigg({-}\int_{0}^{t}\chi(t^{\prime})\dot{\psi}(t^{\prime})\mathrm{d}t^{\prime}\bigg{missing})\psi(t).$
(4.2)

This is the general expression for a state $\psi_{0}$ that undergoes parallel
transport along a path $\gamma$ or $\bar{\gamma}$, also in the non-Hermitian
case, expressed using an ansatz. Although the integral
$\int_{0}^{t}\chi(t^{\prime})\dot{\psi}(t^{\prime})\mathrm{d}t^{\prime}$ 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 $\psi^{\prime}(t)$, then
$\psi^{\prime}(t)=\exp(a(t))\psi(t)$ for some complex-valued function
$a=a(t)$, and
$\int_{0}^{t}\chi(t^{\prime})\dot{\psi}(t^{\prime})\mathrm{d}t^{\prime}$
becomes
$\int_{0}^{t}\chi^{\prime}(t^{\prime})\dot{\psi}^{\prime}(t^{\prime})\mathrm{d}t^{\prime}+[a(t)-a(0)]$,
which can in principle be any complex-valued continuous function. The actual
state $\Psi(t)$ is of course ansatz independent; assuming the same initial
condition, i.e., $\psi^{\prime}(0)=\psi_{0}$, then $a(0)=0$ and as desired we
obtain

$\displaystyle\Psi^{\prime}(t)=\exp\bigg({-}\\!\int_{0}^{t}\chi^{\prime}(t^{\prime})\dot{\psi}^{\prime}(t^{\prime})\mathrm{d}t^{\prime}\bigg{missing})\psi^{\prime}(t)=\exp\bigg({-}\\!\int_{0}^{t}\chi(t^{\prime})\dot{\psi}(t^{\prime})\mathrm{d}t^{\prime}-a(t)\bigg{missing}){\rm
e}^{a(t)}\psi(t)=\Psi(t).$

This confirms that the integral is, in general, only a correction factor to
the specific ansatz $\psi(t)$; it merely quantifies how much our ansatz
$\psi(t)$ was away from being the lift $\Psi(t)$.

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 $\psi$, namely
$\psi(T)=\psi_{0}$, this geometric phase is indeed given by the integral at
$t=T$. However, even in this cyclic 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
$\int_{0}^{t}\chi^{\prime}(t^{\prime})\dot{\psi}^{\prime}(t^{\prime})\mathrm{d}t^{\prime}$
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 $\pi_{\lambda
v}^{H}\colon\operatorname{Eig}(H)\to N(H)$ 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 $\operatorname{Eig}(H)$,
regardless of states being cyclic or non-cyclic. In contrast, one can also
consider the bundle
$\pi_{v}^{H}\colon\operatorname{Eig}(H)\to\operatorname{Spec}(H)$. Clearly,
loops in $\operatorname{Spec}(H)$ correspond to cyclic evolution only.222In an
adiabatic setting, with cyclic we also assume that the system parameters are
restored. However, 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 $\pi_{v}^{H}$, and one uses $\pi_{\lambda v}^{H}$ primarily for non-
cyclic states. We also emphasize that the calculation of $\Psi(t)$ above can
be used in both cases:

${\operatorname{Eig}(H)}$${\operatorname{Spec}(H)}$${(x,E,\psi)}$${(x,E)}$${N(H),}$${x.}$cyclic
statesnon-cyclic statesenergy
swaps$\scriptstyle{\pi_{v}^{H}}$$\scriptstyle{\pi_{\lambda
v}^{H}}$$\scriptstyle{\pi_{\lambda}^{H}}$ (4.3)

####  4.3.2 The cyclic case

Let us start with the cyclic case. As said, this concerns the bundle
$\pi_{v}^{H}\colon\operatorname{Eig}(H)\to\operatorname{Spec}(H)$. We remark
that the only difference between a path in $N(H)$ and a path in
$\operatorname{Spec}(H)$ 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 $E_{0}$ uniquely specifies a path in
$\operatorname{Spec}(H)$ given a path in $N(H)$. It is thus natural to
consider a loop $\bar{\gamma}$ in $\operatorname{Spec}(H)$ as the input data
for cyclic evolution.

Let us now consider the evolution of the eigenstates. Denote the basepoint of
$\bar{\gamma}$ again by $(x_{0},E_{0})$, and assume $\bar{\gamma}$ returns to
this point at time $t=T$. The state evolution is then described by the
parallel transport map $P_{\bar{\gamma}}$, e.g.,
$P_{\bar{\gamma}}(\psi_{0})=\Psi(T)$ following the previous calculation. As
$\pi_{v}^{H}$ defines a principal bundle and $\omega_{H}$ is a principal
connection, $P_{\bar{\gamma}}$ follows from standard holonomy theory. That is,
$P_{\bar{\gamma}}$ is an automorphism of the $\mathbb{C}^{\times}$-torsor
$\operatorname{Eig}_{E_{0}}(H(x_{0}))$, and thus amounts to scaling by a
unique element in $\mathbb{C}^{\times}$. This element is clearly the geometric
phase factor that any state in $\operatorname{Eig}_{E_{0}}(H(x_{0}))$ acquires
due to following $\bar{\gamma}$. 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 $\bar{\gamma}$ be a loop in $\operatorname{Spec}(H)$ based at
$(x_{0},E_{0})$. The geometric phase $\gamma_{\mathrm{geo}}$ due to
$\bar{\gamma}$ is defined, modulo $2\pi$, via

$P_{\bar{\gamma}}={\rm e}^{{\rm
i}\gamma_{\mathrm{geo}}}\cdot\mathrm{id}_{\operatorname{Eig}_{E_{0}}(H(x_{0}))}.$

It now remains to calculate $\gamma_{\mathrm{geo}}$ 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 $\Psi(T)$ and the
initial state $\psi_{0}$, i.e., $\Psi(T)=P_{\bar{\gamma}}(\psi_{0})=\exp({\rm
i}\gamma_{\mathrm{geo}})\psi_{0}$. We thus consider the unique non-zero
complex scalar $[\Psi(T)/\psi_{0}]$ such that
$\Psi(T)=[\Psi(T)/\psi_{0}]\psi_{0}$. Equating $\exp({\rm
i}\gamma_{\mathrm{geo}})=[\Psi(T)/\Psi(0)]$ and substituting our expressing
for $\Psi(t)$ in terms of our ansatz $\psi(t)$, we find the expression

$\gamma_{\mathrm{geo}}={\rm i}\int_{0}^{T}\chi(t)\dot{\psi}(t)\mathrm{d}t-{\rm
i}\ln([\psi(T)/\psi_{0}]),$ (4.4)

where the logarithm term yields the usual modulo $2\pi$ of a phase. By
construction this is invariant under both replacing $\psi_{0}$ with another
state in the same ray and picking another ansatz $\psi$.

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 $\psi$ along its own direction. The logarithm term is a correction for
$\psi$ not closing on itself. Observe that only the two terms together are
invariant under changing the choice of the ansatz $\psi$. The logarithm term
vanishes whenever $\psi(1)=\psi(0)$, which happens, e.g., if $\psi$ is built
using a local eigenstate. On the other hand, the integral term vanishes, e.g.,
when $\psi$ is chosen to be the lift $\Psi$, in which case the integrand is
identically zero. In this case $\gamma_{\mathrm{geo}}$ 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 $V=\mathbb{C}^{2}$ with
standard basis $(e_{1},e_{2})$, in which the Hamiltonian family reads

$H(a,b)=\begin{pmatrix}a&b\\\ b&-a\end{pmatrix}\\!,$

where $a$, $b$ are real numbers, i.e., $M=\mathbb{R}^{2}$. The energy bands
are given as $\lambda_{\pm}(a,b)=\pm\sqrt{a^{2}+b^{2}}$, 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
$\gamma(t)=(\cos(t),\sin(t))$ with time interval $[0,2\pi]$, and consider the
$\lambda_{+}$ level ($\lambda_{-}$ is similar). By looking at
$H(\gamma(t))-\lambda_{+}(\gamma(t))I$, one readily finds an nsatz $\psi(t)$
for the evolution of the eigenstate, together with a left-eigenstate path
$\chi(t)$, as

$\psi(t)=\frac{1}{2}\begin{pmatrix}1+\cos(t)\\\
\sin(t)\end{pmatrix}\\!,\qquad\chi(t)=\frac{1}{1+\cos(t)}\begin{pmatrix}1+\cos(t)\\\
\sin(t)\end{pmatrix}^{\mathrm{T}}\\!.$

The factor $1/2$ is introduced to have $\psi(0)=e_{1}$. Note that $\psi$ and
$\chi$ are defined only for $t\in[0,\pi)$; being a (left-)eigenstate, they are
not allowed to vanish. Still, for these times we can calculate the lift of
$\gamma(t)$ starting at $e_{1}$.

We can correct the ansatz $\psi(t)$ following equation (4.2). We thus evaluate
the geometric phase integral, whose integrand is

$\chi(t)\dot{\psi}(t)=\frac{-\sin(t)}{2(1+\cos(t))}=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\ln(1+\cos(t)).$

Hence we find the lift of $\gamma$ starting at $e_{1}$ to be

$\Psi(t)=\sqrt{\frac{1}{2(1+\cos(t))}}\begin{pmatrix}1+\cos(t)\\\
\sin(t)\end{pmatrix}=\begin{pmatrix}\cos(t/2)\\\ \sin(t/2)\end{pmatrix}\\!.$

As the scale factor is real, we can interpret it as a length correction. This
length interpretation clearly shows that our original $\psi(t)$ was varying
along itself.

Note that our final expression of $\Psi(t)$ can be extended to arbitrary
times, hence yields the full lift of $\gamma$. Consequently, it is now
convenient to obtain the geometric phase via $\Psi(t)$. For lifts, only the
logarithm term in equation (4.4) contributes, and we find the phase (modulo
$2\pi$)

$\gamma_{\mathrm{geo}}=-{\rm i}\ln([\Psi(2\pi)/\psi_{0}])=-{\rm
i}\ln([-e_{1}/e_{1}])=-{\rm i}\ln(-1)=\pi.$

In the perspective of loops in $\operatorname{Spec}(H)$, the above concerns
the loop
$\bar{\gamma}_{+}(t):=(\gamma(t),\lambda_{+}(\gamma(t)))=(\gamma(t),1)$,
revealing that $P_{\bar{\gamma}_{+}}=-\mathrm{id}$. A similar argument shows
that the loop $\bar{\gamma}_{-}(t)=(\gamma(t),-1)$, i.e., considering the
other energy band, yields $P_{\bar{\gamma}_{-}}=-\mathrm{id}$ as well.

In some situations, one can express the geometric phase alternatively as a
curvature integral. A standard argument is as follows. Let $U\subset N(H)$ be
a neighborhood on which a local eigenstate $\psi=\psi(x)$ is defined, which
has as partner the local left-eigenstate $\chi$. If $\gamma$ is a loop in $U$
that forms the boundary of a surface $S$ also contained in $U$, then the
geometric phase is given by

$\gamma_{\mathrm{geo}}={\rm i}\int_{\gamma}\chi\mathrm{d}\psi={\rm
i}\int_{\partial S}\hat{\psi}^{*}\omega_{H}={\rm
i}\int_{S}\hat{\psi}^{*}(\mathrm{d}\omega_{H})={\rm
i}\int_{\hat{\psi}(S)}K_{H}.$

We see that the assumption of $\gamma$ being a boundary is essential. If
$\gamma$ 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 $\operatorname{Spec}(H)$
instead. Indeed, the integral above can be rewritten as an integral over
$k_{H}$, which is the reduced curvature on $\operatorname{Spec}(H)$. Clearly,
the local eigenstate $\psi(x)$ defines an energy function $E(x)$. The map
$\hat{E}\colon U\to\operatorname{Spec}(H)$, $x\mapsto(x,E(x))$ is then a local
section of $\operatorname{Spec}(H)$. Moreover, as
$\hat{E}=\pi_{v}^{H}\circ\hat{\psi}$, where $\pi_{v}^{H}$ is the projection
$\operatorname{Eig}(H)\to\operatorname{Spec}(H)$, we have

$\gamma_{\mathrm{geo}}={\rm i}\int_{S}\hat{\psi}^{*}(K_{H})={\rm
i}\int_{S}\hat{\psi}^{*}\big{(}\big{(}\pi_{v}^{H}\big{)}^{*}(k_{H})\big{)}={\rm
i}\int_{S}\hat{E}^{*}(k_{H})={\rm i}\int_{\hat{E}(S)}k_{H}.$

Let us interpret this result. First, we remark that the integral over $k_{H}$
is manifestly independent of the chosen local eigenstate $\psi$. More
precisely, only the energy bands matter, not the exact eigenstates. In
addition, we now consider surfaces in $\operatorname{Spec}(H)$, hence the
above argument can be used whenever our path $\bar{\gamma}$ is a boundary. We
thus find that the homology of $\operatorname{Spec}(H)$, i.e., the homology of
the energy bands, plays a key role in the relation of geometric phase to
curvature. Moreover, in case $\operatorname{Eig}(H)$ 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 $\bar{\gamma}$ be a loop in $\operatorname{Spec}(H)$. If $\bar{\gamma}$ is
the boundary of a surface $\Sigma$ in $\operatorname{Spec}(H)$, then the
geometric phase acquired by traversing $\bar{\gamma}$ equals ${\rm
i}\int_{\Sigma}k_{H}$.

######  Corollary 4.13.

If $\operatorname{Eig}(H)$ is flat, then the geometric phase due to a loop in
$\operatorname{Spec}(H)$ only depends on the class of the loop in the first
homology group $H_{1}(\operatorname{Spec}(H))$.

######  Example 4.14.

Let us continue Example 4.11. As $H$ is symmetric, by Proposition 4.9 the
connection is flat. Hence the geometric phases only depend on the homology of
$\operatorname{Spec}(H)$, and thus are of topological nature. Clearly,
$\operatorname{Spec}(H)$ is the diabolo minus the DP, which is homeomorphic to
two punctured planes. Hence
$H_{1}(\operatorname{Spec}(H))\cong\mathbb{Z}^{2}$, generated by (the classes
of) the images of the unit circle under $\lambda_{+}$ and $\lambda_{-}$. It
thus suffices to know the phase due to each generator, which is $\pi$
following our earlier calculation in Example 4.11.

It is now straightforward to recognize the bundles introduced by Simon [32] in
the bundle $\operatorname{Eig}(H)\to\operatorname{Spec}(H)$. According to
Corollary 3.8, for Hermitian $H$ the bundle $\operatorname{Spec}(H)\to N(H)$
is trivial, i.e., $\operatorname{Spec}(H)$ separates into $n$ distinct energy
bands. Let us label these energy bands by $1,\dots,n$ and write
$\operatorname{Spec}_{k}(H)$ for energy band $k$, and similarly
$\operatorname{Eig}_{k}(H)$ for the subbundle over
$\operatorname{Spec}_{k}(H)$. Then
$\operatorname{Eig}_{k}(H)\to\operatorname{Spec}_{k}(H)$ is a principal
$\mathbb{C}^{\times}$-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 $\operatorname{Spec}(H)$ 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
$\operatorname{Eig}(H)\to\operatorname{Spec}(H)$ can still model these states
by parallel transport, but as the path $\bar{\gamma}$ 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 $\pi_{\lambda
v}^{H}\colon\operatorname{Eig}(H)\to N(H)$. Here, holonomy is to be understood
in the context of a semi-principal bundle. Similarly to the case above,
parallel transport along a loop $\gamma$ in $N(H)$ based at $x_{0}$ induces
the parallel transport map $P_{\gamma}$, which is an automorphism of the
$\mathbb{C}^{\times}$-semi-torsor $\operatorname{Eig}(H(x_{0}))$. The set of
all $P_{\gamma}$ for a fixed base point $x_{0}$ then defines the holonomy
group

$\operatorname{Hol}^{\operatorname{Eig}(H)}_{N(H)}(x_{0})=\big{\\{}P_{\gamma}\in\operatorname{Aut}(\operatorname{Eig}(H(x_{0})))\mid\gamma\in\mathrm{Loop}(N(H),x_{0})\big{\\}}.$

However, these automorphisms are harder to describe. Whereas we could
previously identify a map $P_{\bar{\gamma}}$ with an element in
$\mathbb{C}^{\times}$, this need not be for $P_{\gamma}$. 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 $\mathbb{C}^{\times}$ 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 $x_{0}=0$ again as our reference point, and from there we
encircle the EP at $x=+{\rm i}$ in positive direction by following a path
$\gamma$. One can take $\gamma$ to be a circular path, but as the connection
is flat by Proposition 4.9, the exact shape of $\gamma$ does not play a role.

The evolution of the states due to encircling $\gamma$ is captured by the map
$P_{\gamma}\colon\operatorname{Eig}(H(0))\to\operatorname{Eig}(H(0))$.
Clearly, $\operatorname{Eig}(H(0))$ is the disjoint union of the two
eigenrays, i.e.,

$\operatorname{Eig}(H(0))=\mathbb{C}^{\times}e_{1}\sqcup\mathbb{C}^{\times}e_{2},$

where $e_{1}=(1,0)^{T}$ and $e_{2}=(0,1)^{T}$ are the standard basis vectors.
Hence, an element of $\operatorname{Eig}(H(0))$ is of the form $ze_{1}$ or
$ze_{2}$ with $z\in\mathbb{C}^{\times}$. We emphasize that
$\operatorname{Eig}(H(0))$ is a semi-torsor and not a vector space; linear
combinations of $e_{1}$ and $e_{2}$ are not present, as these are not
eigenstates of $H(0)$. Accordingly, $P_{\gamma}$ is equivariant and not
linear.

We can now calculate $P_{\gamma}$ as follows. First, it is sufficient to know
$P_{\gamma}(e_{1})$ and $P_{\gamma}(e_{2})$ due to equivariance;
$P_{\gamma}(ze_{k})=zP_{\gamma}(e_{k})$ for $k=1,2$. That is, it is sufficient
to follow $e_{1}$ and $e_{2}$ around the EP. This can be done using an
explicit parametrization of $\operatorname{Spec}(H)$. One can easily find
eigenstates depending on $x$, and following Proposition 4.9 we “normalize”
them so that no geometric phase will appear.333Note that this uses flatness of
the connection. In the non-flat case the geometric phase depends on the exact
shape of $\gamma$ and so cannot be expressed using local eigenstates. One
should then use, e.g., the ansatz method instead. Hence we arrive at the
expressions

$\psi_{\pm}(x)=\mp\frac{1}{\sqrt{2(1+x^{2}\mp\sqrt{1+x^{2}}})}\begin{pmatrix}-x\\\
1\mp\sqrt{1+x^{2}}\end{pmatrix}\\!,$

where we remark that $\psi_{+}$ has a removable singularity at $x=0$; the
limit reads $\lim_{x\to 0}\psi_{+}(x)\allowbreak=e_{1}$. Hence at $x_{0}=0$,
$\psi_{+}(0)=e_{1}$ and $\psi_{-}(0)=e_{2}$ as desired. The lifts of these
states to $\operatorname{Eig}(H)$ can thus be found by inspection of
$\psi_{\pm}(x)$. Encircling the EP in positive direction swaps the $\pm$
signs, and in addition the overall root in the denominator obtains a factor of
${\rm i}$. Thus, after following $\psi_{\pm}(x)$ around the EP, we return with
${\rm i}^{-1}\psi_{\mp}(x)$. At our reference $x_{0}=0$, this rule becomes
$e_{1}\mapsto-{\rm i}e_{2}$, $e_{2}\mapsto-{\rm i}e_{1}$. This information is
enough to specify $P_{\gamma}$, which we thus find to be given by

$P_{\gamma}\colon\quad ze_{1}\mapsto-{\rm i}ze_{2},\qquad ze_{2}\mapsto-{\rm
i}ze_{1}.$ (4.5)

In order to study $P_{\gamma}$, let us consider the following statement
showing how a general automorphism 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 $\phi\colon F\to F$ an
automorphism. Then $F=\sqcup_{i\in I^{\prime}}F_{i}$, for some index set
$I^{\prime}$, decomposes $F$ into minimal $\phi$-invariant subspaces, i.e.,
$\phi(F_{i})=F_{i}$ for all $i$. Moreover, if $G$ is Abelian and a particular
$F_{i}$ consists of $k<\infty$ orbits, then $(\phi|_{F_{i}})^{k}$ equals
translation by an element of $G$.

###### Proof.

Clearly, the index set $I^{\prime}$ is the original orbit space modulo the
relation that orbits mapped into one another by $\phi$ are identified. If
$F_{i}$ consists of $k$ orbits, then by minimality $(\phi|_{F_{i}})^{k}$
preserves the orbits in $F_{i}$. Picking $f\in F_{i}$, we find
$\phi^{k}(f)=g(f)f$ for some $g(f)\in G$. If $G$ is Abelian, then $g(f)$ is
constant on the orbit through $f$. In addition, as equivariance yields
$\phi^{k}(\phi(f))=g(f)\phi(f)$, we see $\phi(f)$ is scaled by the same
element. It follows that $g(f)$ is constant on the subspace $F_{i}$. ∎

For the map $P_{\gamma}$, it is clear that a minimal subspace is any minimal
union of eigenrays whose energies are permuted upon traversing $\gamma$. The
element of $\mathbb{C}^{\times}$ 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 $\gamma$ 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 $\psi_{0}$ in the union and its
transport $P_{\gamma}(\psi_{0})$. We come back to this in Section 5, where we
treat how $P_{\gamma}$ can be expressed via a holonomy matrix. Of course, if
$k=1$, i.e., $\psi_{0}$ is cyclic, then we do recover the geometric phase. We
summarize these findings in the following.

######  Corollary 4.17.

Given a loop $\gamma$, the invariant subspaces of $P_{\gamma}$ 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 $\bar{\gamma}^{k}$, where $\bar{\gamma}$ is the lift of
$\gamma$ to $\operatorname{Spec}(H)$ with an energy corresponding to any of
the eigenrays in the union.

It is also clear that $P_{\gamma}$ contains the information of the underlying
permutation $p_{\gamma}$ of the energies. This is formally expressed by
$\pi_{v}^{H}$ being a bundle map that is equivariant w.r.t. the collapsing
quotient homomorphism $\mathbb{C}^{\times}\to 1$. As $\pi_{v}^{H}$ maps
$\operatorname{Eig}(H)$ to $\operatorname{Spec}(H)$, this quotient reduces
$P_{\gamma}$ to $p_{\gamma}$.

######  Proposition 4.18.

For any $\gamma\in\operatorname{Loop}(N(H),x_{0})$, the map $P_{\gamma}$
induces the following commutative diagram:

${\operatorname{Eig}(H(x_{0}))}$${\operatorname{Eig}(H(x_{0}))}$${\operatorname{Spec}(H(x_{0}))}$${\operatorname{Spec}(H(x_{0})).}$$\scriptstyle{P_{\gamma}}$$\scriptstyle{\pi_{v}^{H}}$$\scriptstyle{\pi_{v}^{H}}$$\scriptstyle{p_{\gamma}}$

###  4.4 Including the dynamical phase

The geometric properties of adiabatic dynamics for non-degenerate operators,
Hermitian or non-Hermitian, can be described on $\operatorname{Eig}(V)$ using
the connection $\omega$. This leaves out an important non-geometrical
property, namely the dynamical phase. Nevertheless, $\operatorname{Eig}(V)$
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 $\operatorname{Eig}(V)$, i.e.,

$\displaystyle\mathcal{E}\colon\ \operatorname{Eig}(V)$
$\displaystyle\to\mathbb{C},$ $\displaystyle(A,\lambda,v)$
$\displaystyle\mapsto\lambda.$

Given a lift $\Gamma$ in $\operatorname{Eig}(V)$, the corresponding dynamical
phase is then the integral

$\gamma_{\mathrm{dyn}}=-\frac{\rm
i}{\hbar}\int\mathcal{E}(\Gamma(t))\mathrm{d}t.$

It is even possible to put the dynamical phase explicitly in the lift.
Clearly, the lift $\Gamma$ without dynamical phase vanishes under the
covariant derivative

$D={\rm d}+\omega.$

If one modifies this to

$D_{\mathrm{tot}}={\rm d}+\omega+\frac{\rm i}{\hbar}\mathcal{E}$

and calculates the lift $\Gamma_{\mathrm{tot}}$ of $\gamma$ given by
$D_{\mathrm{tot}}\Gamma_{\mathrm{tot}}=0$, then $\Gamma_{\mathrm{tot}}$ is the
path of the eigenstate, including the dynamical phase, assuming $\gamma$ is
parametrized by physical time.

###  4.5 Relation with the work of Aharonov and Anandan

We found that $\operatorname{Eig}(V)$ 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 $\operatorname{Eig}(V)$. 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 $\operatorname{Eig}(V)$.

Let us summarize the theory of [1], rephrasing it in line with our approach to
$\operatorname{Eig}(V)$. First, the state space $V$ is now assumed to be a
Hilbert space, i.e., $V$ should be equipped with a Hermitian inner product
$\langle\;|\;\rangle$. This allows one to define the unit sphere $S_{1}(V)$
inside $V$ by restricting to norm 1 states. As norm 1 fixes a state up to a
$U(1)$-phase, $S_{1}(V)$ 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)}$${S_{1}(V)}$${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 $\ket{\psi}\in S_{1}(V)$ can be identified with
the projection operator $\ket{\psi}\bra{\psi}$. The metric on $V$ obtained
from the inner product $\langle\;|\;\rangle$ induces a metric tensor on
$S_{1}(V)$, and hence a connection 1-form. On a path $\ket{\psi(t)}$ in
$S_{1}(V)$, this connection yields $\langle\psi(t)|\dot{\psi}(t)\rangle$. If
one lifts a loop from $P(V)$ to $S_{1}(V)$, 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 $\ket{\psi(t)}$ satisfies
$\ket{\psi(1)}=\ket{\psi(0)}$, then the obtained Aharonov–Anandan (AA) phase
is given by the integral ${\rm
i}\int_{0}^{1}\langle\psi(t)|\dot{\psi}(t)\rangle\mathrm{d}t$. This coincides
with the adiabatic Berry phase if $\ket{\psi(t)}$ evolves adiabatically.

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
$\operatorname{Eig}(V)$, equipped with the “adiabatic” connection $\omega$.
That is, we will show that the bundle in equation (4.6), including connection,
can be obtained by collapsing the principal bundle
$\pi_{v}\colon\operatorname{Eig}(V)\to\operatorname{Spec}(V)$. 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 $\langle\;|\;\rangle$ on $V$. It allows us to talk about
Hermitian operators, and we restrict $N(V)$ accordingly to the closed subset

$N_{\langle\;|\;\rangle}(V):=\big{\\{}A\in N(V)\mid A^{\dagger}=A\text{ with
respect to }\langle\;|\;\rangle\big{\\}}$

of all non-degenerate Hermitian operators on $V$. This is a non-canonical
subset of $N(V)$; a different inner product may yield a different subset.
Consequently, there are the subbundles over $N_{\langle\;|\;\rangle}(V)$ given
by

$\displaystyle\operatorname{Spec}_{\langle\;|\;\rangle}(V):=\big{\\{}(A,\lambda)\in\operatorname{Spec}(V)\mid
A^{\dagger}=A\text{ with respect to }\langle\;|\;\rangle\big{\\}},$
$\displaystyle\operatorname{Eig}_{\langle\;|\;\rangle}(V):=\big{\\{}(A,\lambda,v)\in\operatorname{Eig}(V)\mid
A^{\dagger}=A,\norm{v}=1\text{ with respect to }\langle\;|\;\rangle\big{\\}}.$

The $\mathbb{C}^{\times}$-action on $\operatorname{Eig}(V)$ reduces to
$U(1)$-phase rotation on $\operatorname{Eig}_{\langle\;|\;\rangle}(V)$, which
has quotient space $\operatorname{Spec}_{\langle\;|\;\rangle}(V)$.

The key observation now is that, after this restriction, we no longer need to
know $A$ and $\lambda$ to compute the eigencovector and eigenprojectors.
Indeed, given the normalized vector $\ket{v}$, the covector $\theta$ is
$\bra{v}$ and the eigenprojector is $\ket{v}\bra{v}$, regardless of the exact
$A$ and $\lambda$. Hence, we are motivated to discard $A$ and $\lambda$. To do
so, it is convenient to describe $\operatorname{Spec}(V)$ using projectors.
Clearly, any pair $(A,\lambda)\in\operatorname{Spec}(V)$ defines an
eigenprojector $P_{(A,\lambda)}$ projecting on the eigenspace of $A$
corresponding to $\lambda$. Conversely, given an eigenprojector $P$, the
eigenvalue can be retrieved from the identity $AP=PA=\lambda P$. Hence we may
write $(A,\lambda)\in\operatorname{Spec}(V)$ equivalently as
$(A,P_{(A,\lambda)})$. For $\operatorname{Spec}_{\langle\;|\;\rangle}(V)$, we
may even write an element as $(A,\ket{v}\bra{v})$, where $\ket{v}$ is a
normalized eigenvector of $A$.

We can now perform the second step, i.e., the reduction. Reducing
$\operatorname{Spec}_{\langle\;|\;\rangle}(V)$ to the space $P(V)$ is
straightforward using the projector description where the element
$(A,\ket{v}\bra{v})$ goes to $\ket{v}\bra{v}$. Reducing
$\operatorname{Eig}_{\langle\;|\;\rangle}(V)$ to $S_{1}(V)$ is similar; we
only keep the vector. Together, these maps define a morphism of bundles as
follows:

${\operatorname{Eig}_{\langle\;|\;\rangle}(V)}$${S_{1}(V)}$${(A,\lambda,\ket{v})}$${\ket{v}}$${\operatorname{Spec}_{\langle\;|\;\rangle}(V)}$${P(V),}$${(A,\ket{v}\bra{v})}$${\ket{v}\bra{v}.}$$\scriptstyle{\pi_{v}}$

Our final claim is that the connection $\omega$ on $\operatorname{Eig}(V)$
also carries over to $S_{1}(V)$. This again follows the same two steps. First,
clearly $\omega$ restricts to a $U(1)$-connection on
$\operatorname{Eig}_{\langle\;|\;\rangle}(V)$. Second, this restricted form
admits push-forward to $S_{1}(V)$. Concerning explicit formulas, this push-
forward is given by substituting $\theta=\bra{v}$, which yields exactly the
connection used to define the AA phase. The following then summarizes these
findings.

######  Proposition 4.19.

The restricted projection
$\operatorname{Eig}_{\langle\;|\;\rangle}(V)\to\operatorname{Spec}_{\langle\;|\;\rangle}(V)$
has a canonical reduction to the bundle $S_{1}(V)\to P(V)$. Moreover, the
restricted connection on $\operatorname{Eig}_{\langle\;|\;\rangle}(V)$ admits
push-forward to $S_{1}(V)$, yielding the standard $U(1)$-connection as used
for the AA phase.

We conclude that the above projection allows us to translate the general
theory of $\operatorname{Eig}(V)$ 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 $\theta$ by $\bra{v}$, but the broad picture contains
more. For example, $\operatorname{Eig}(V)$ is locally partitioned according to
eigenvalues, while this is completely absent in $S_{1}(V)$.

###  4.6 Quantum geometric tensor

We will show that the covariant derivatives naturally define a tensor on
$\operatorname{Eig}(V)$, which is a straightforward generalization of the
quantum geometric tensor. We also consider its reduced version on
$\operatorname{Spec}(V)$. Before we discuss the tensor itself, we first
introduce convenient bases of tangent spaces of $\operatorname{Eig}(V)$ and
$\operatorname{Spec}(V)$. 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
$T_{(A,\lambda,v)}\operatorname{Eig}(V)$ and
$T_{(A,\lambda)}\operatorname{Spec}(V)$. The idea is that we pass the $v$,
$\lambda$ 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 tangent spaces is that $T_{(A,\lambda,v)}\operatorname{Eig}(V)$ has a
tangent along the eigenray $\operatorname{Eig}_{\lambda}(A)$. This direction
is naturally spanned by the fundamental tangent vector $\partial_{v}$
originating from the scaling action. An advantage of picking $\partial_{v}$ is
that we may pick the remaining tangent vectors in
$T_{(A,\lambda,v)}\operatorname{Eig}(V)$ to be horizontal. These tangent are
then the horizontal lifts of unique tangents in
$T_{(A,\lambda)}\operatorname{Spec}(V)$, 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 $P=v\theta$ for the eigenprojector of
$A$ corresponding to $\lambda$, a shift in $\lambda$ is then given by the
tangent

$\partial_{\lambda}=\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}(A+tP,\lambda+t,v,\theta).$

We must then find $n-1$ other tangent vectors for the other eigenvalues of
$A$. These can be obtained via paths of the form
$(A+tP^{\prime},\lambda,v,\theta)$, with $P^{\prime}$ the corresponding
eigenprojector. Note that $\lambda$ is constant as we vary another eigenvalue,
but still consider tangents at $(A,\lambda,v)$. The combination of these $n$
tangent vectors defines the tuple $\partial_{\tilde{\lambda}}$, where we pick
an ordering of $\operatorname{Spec}(A)$. Writing the coefficients of these
vectors as $\Delta\tilde{\lambda}$, again cf the ordering, we obtain the
linear combination $\Delta\tilde{\lambda}\partial_{\tilde{\lambda}}$. The
corresponding tangents in $T_{(A,\lambda)}\operatorname{Spec}(V)$ 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 ${\rm e}^{{\rm i}tX}$, where ${\rm
i}X\in\operatorname{End}(V)$ can be viewed as a generator; the extra $\rm i$
is for later convenience. We then obtain the infinitesimal conjugation action
(extending equation (2.5)), which we view as the linear map
$\operatorname{End}(V)\to T_{(A,\lambda,v)}\operatorname{Eig}(V)$ given by

${\rm i}X\mapsto\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}\big{(}{\rm
e}^{{\rm i}tX}A{\rm e}^{-{\rm i}tX},\lambda,{\rm e}^{{\rm i}tX}v,\theta{\rm
e}^{-{\rm i}tX}\big{)}.$

As is, this parametrization of $T_{(A,\lambda,v)}\operatorname{Eig}(V)$ is not
compatible with our earlier choices. For instance, if ${\rm i}X=P$ we retrieve
$\partial_{v}$, 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. 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 $\theta Xv=0$, which is automatically satisfied for $A$-free $X$.

In summary, we may express a general element $\mathfrak{v}\in
T_{(A,\lambda,v,\theta)}\operatorname{Eig}(V)$ as

$\mathfrak{v}={\rm
i}X+\Delta\tilde{\lambda}\partial_{\tilde{\lambda}}+z\partial_{v},$

where for $T_{(A,\lambda)}\operatorname{Spec}(V)$ 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 $n^{2}-n$, $n$ and 1, respectively. The values of
$\mathrm{d}v$ and $\mathrm{d}\theta$ then read

$\mathrm{d}v(\mathfrak{v})=\frac{\mathrm{d}}{\mathrm{d}t}\bigg{|}_{t=0}{\rm
e}^{{\rm i}tX}{\rm e}^{tz}v=({\rm
i}X+z)v,\qquad\mathrm{d}\theta(\mathfrak{v})=-\theta({\rm i}X+z).$

We finally remark that it is possible to not impose $X$ to be $A$-free; the
map to $T_{(A,\lambda,v)}\operatorname{Eig}(V)$ 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 $\operatorname{Eig}(V)$ 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 $\operatorname{Eig}(V)$ 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 $\mathcal{PT}$-symmetry as reported in
[40].

Let us start from the standard expression. Fixing local coordinates on $N(H)$
and a local normalized eigenstate $\ket{\psi}=\ket{\psi(x)}$, the QGT is given
by

$T_{ij}=\bra{\partial_{i}\psi}(1-\ket{\psi}\bra{\psi})\ket{\partial_{j}\psi}\\!.$

We regard this to be the pull-back of a more abstract tensor $\mathcal{G}$ on
$\operatorname{Eig}(V)$. Clearly, this $\mathcal{G}$ is simply given by

$\mathcal{G}_{(A,v,\theta)}(\mathfrak{v}_{1},\mathfrak{v}_{2})=\mathrm{d}\theta(\mathfrak{v}_{1})(1-v\theta)\mathrm{d}v(\mathfrak{v}_{2}),\qquad\mathfrak{v}_{1},\mathfrak{v}_{2}\in
T_{(A,v,\theta)}\operatorname{Eig}(V).$ (4.7)

Because of this straightforward generalization from the Hermitian case to
$\operatorname{Eig}(V)$, we refer to $\mathcal{G}$ as the _(generalized)
quantum geometric tensor_. In addition, $\mathcal{G}$ is a natural tensor on
$\operatorname{Eig}(V)$ in the following sense. Observe that the projector
$1-v\theta$ is naturally obtained by taking the covariant derivative, as

$\nabla
v(\mathfrak{v})=\mathrm{d}v(\mathfrak{v})-\omega(\mathfrak{v})v=\mathrm{d}v(\mathfrak{v})-\theta(\mathrm{d}v(\mathfrak{v}))v=(1-v\theta)\mathrm{d}v(\mathfrak{v}),\qquad\mathfrak{v}\in
T_{(A,v,\theta)}\operatorname{Eig}(V),$

and similarly $\nabla\theta=\mathrm{d}\theta(1-v\theta)$. We thus observe that
$\mathcal{G}$ is the natural combination

$\mathcal{G}_{(A,v,\theta)}(\mathfrak{v}_{1},\mathfrak{v}_{2})=\nabla\theta(\mathfrak{v}_{1})(\nabla
v(\mathfrak{v}_{2})),\qquad\mathfrak{v}_{1},\mathfrak{v}_{2}\in
T_{(A,v,\theta)}\operatorname{Eig}(V),$

which directly displays its scale invariance.

We can write $\mathcal{G}$ more explicitly using our description of the
tangent space $T_{(A,\lambda,v,\theta)}\operatorname{Eig}(V)$. Introducing
labels 1 and 2 according to the two tangent vectors, this yields

$\mathcal{G}_{(A,v,\theta)}(\mathfrak{v}_{1},\mathfrak{v}_{2})=\theta(X_{1}X_{2})v-(\theta
X_{1}v)(\theta X_{2}v).$

One can also write this using scale invariant quantities only. Writing points
of $\operatorname{Eig}(V)$ as $(A,P,v,\theta)$, we find

$\mathcal{G}_{(A,P,v,\theta)}(\mathfrak{v}_{1},\mathfrak{v}_{2})=\tr(PX_{1}X_{2})-\tr(PX_{1})\tr(PX_{2}).$

This also provides an explicit form of the reduced QGT defined on
$\operatorname{Spec}(V)$. We observe that in these expressions we need not
impose the $X_{i}$ 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 $X_{i}$. 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 $\mathcal{G}$ is readily seen to be $K/2$, as

$\mathcal{G}^{\mathrm{alt}}_{(A,v,\theta)}(\mathfrak{v}_{1},\mathfrak{v}_{2})=\theta\frac{[X_{1},X_{2}]}{2}v=\frac{1}{2}\mathrm{d}\theta\wedge\mathrm{d}v(\mathfrak{v}_{1},\mathfrak{v}_{2})=\frac{1}{2}K_{(A,v,\theta)}(\mathfrak{v}_{1},\mathfrak{v}_{2}).$

We may also write this as $\tr\big(P\frac{[X_{1},X_{2}]}{2}\big{missing})$,
which in addition yields an explicit form of the reduced curvature $k$. The
symmetric part $\mathcal{M}:=\mathcal{G}^{\mathrm{sym}}$ of $\mathcal{G}$ is
then a tensor on $\operatorname{Eig}(V)$ generalizing the standard quantum
metric tensor (QMT). Hence we will refer to $\mathcal{M}$ using the same name.
A scale invariant explicit form is

$\mathcal{M}_{(A,P,v,\theta)}(\mathfrak{v}_{1},\mathfrak{v}_{2})=\operatorname{cov}_{P}(X_{1},X_{2}):=\tr\bigg(P\frac{\\{X_{1},X_{2}\\}}{2}\bigg{missing})-\tr(PX_{1})\tr(PX_{2}),$

where $\operatorname{cov}_{P}$ 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
$\operatorname{Eig}(V)$.

######  Proposition 4.20.

The QGT on $\operatorname{Eig}(V)$ decomposes as a linear combination of the
QMT and the curvature as

$\mathcal{G}=\mathcal{M}+\frac{1}{2}K,$

which is the decomposition of $\mathcal{G}$ into its symmetric and anti-
symmetric part, respectively.

Notable properties that do not generalize from the Hermitian case are the
following. Clearly $\mathcal{M}$ and $K$ are complex rather than real-valued
tensors, and no longer the real resp. imaginary part of $\mathcal{G}$. In
addition, $\mathcal{M}$ 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 parameter 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
$\mathcal{M}$. In the non-Hermitian case, or better, whenever
$\operatorname{Spec}(H)$ 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 $S_{1}(V)\to P(V)$ from Section 4.5, these distances are
given by

$L(\psi,\psi^{\prime})=\norm{\psi-\psi^{\prime}},\qquad
D(\psi,\psi^{\prime})=\sqrt{2-2\absolutevalue{\langle\psi|\psi^{\prime}\rangle}},\qquad\psi,\psi^{\prime}\in
S_{1}(V).$

Here, $L$ is the inner product metric on $V$ restricted to $S_{1}(V)$. The ray
space $P(V)$ also inherits a metric, whose pull-back to $S_{1}(V)$ 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
$\psi^{\prime}=\psi+\dot{\psi}\mathrm{d}t+\tfrac{1}{2}\ddot{\psi}\mathrm{d}t^{2}+\cdots$
and keeping terms up to $\mathrm{d}t^{2}$ only, one finds
$\mathrm{d}L^{2}=\langle\dot{\psi}|\dot{\psi}\rangle\mathrm{d}t^{2}$ and
$\mathrm{d}D^{2}=\big{[}\langle\dot{\psi}|\dot{\psi}\rangle+\langle\psi|\dot{\psi}\rangle^{2}\big{]}\mathrm{d}t^{2}$.
One thus finds $\mathrm{d}L^{2}-\mathrm{d}D^{2}=({\rm
i}\langle\psi|\dot{\psi}\rangle\mathrm{d}t)^{2}$, so that the geometric phase
integrand equals $\sqrt{\mathrm{d}L^{2}-\mathrm{d}D^{2}}$, which is one of the
main results in [29]. It is tempting to write
“$\mathrm{d}\gamma_{\mathrm{geo}}$” for ${\rm
i}\langle\psi|\dot{\psi}\rangle\mathrm{d}t$, 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 ${\rm i}\omega$.

We now generalize to the non-Hermitian setting of $\operatorname{Eig}(V)$.
First, the distances $L$ and $D$ do not carry over. Concerning $L$, we see
$L^{2}=2-\langle\psi|\psi^{\prime}\rangle-\langle\psi^{\prime}|\psi\rangle$
generalizes to the function $2-\theta_{1}(v_{2})-\theta_{2}(v_{1})$ on
$\operatorname{Eig}(V)$, but its square root would be multi-valued on
$\operatorname{Eig}(V)$, meaning that $L$ itself does not generalize. Similar
issues appear for
$D^{2}=2-2\sqrt{\langle\psi|\psi^{\prime}\rangle\langle\psi^{\prime}|\psi\rangle}$.
Nevertheless, the tensor fields $\mathrm{d}L^{2}$ and $\mathrm{d}D^{2}$ do
have a generalization to $\operatorname{Eig}(V)$. One can check that
$\mathrm{d}D^{2}$ is actually $\mathcal{M}$ for the Hermitian case, see also
[10, 11]. Finally, $\mathrm{d}L^{2}$ generalizes to the tensor field
$\mathcal{L}:=(\mathrm{d}\theta\mathrm{d}v)^{\mathrm{sym}}$, which is the non-
covariant counterpart of $\mathcal{M}$. The relation between $\mathcal{L}$,
$\mathcal{M}$ and $\omega$ is readily found from equation (4.7); first
rewriting this to

$\mathcal{G}=\mathrm{d}\theta\mathrm{d}v+(-\mathrm{d}\theta
v)\otimes(\theta\mathrm{d}v)=\mathrm{d}\theta\mathrm{d}v+\omega\otimes\omega$

and then taking the symmetric part yields

$\mathcal{M}=\mathcal{L}+\omega\otimes\omega.$

This is thus an equality of tensor fields on $\operatorname{Eig}(V)$,
generalizing the relation
“$\mathrm{d}D^{2}=\mathrm{d}L^{2}-\mathrm{d}\gamma_{\mathrm{geo}}^{2}$”.

##  5 Explicit description of the holonomy

In Sections 3 and 4 we saw how the bundles $\operatorname{Spec}(V)$ and
$\operatorname{Eig}(V)$ provide a holonomy interpretation 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 $p_{\gamma}$ and
$P_{\gamma}$ 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 permutation $p_{\gamma}$ of the eigenvalues of some $A\in
N(V)$ can be expressed using an standard permutation $\sigma\in S_{n}$. A
first attempt would be to label the eigenvalues, i.e., call them
$\lambda_{1},\dots,\lambda_{n}$, and define $\sigma$ by the relation
$p_{\gamma}(\lambda_{i})=\lambda_{\sigma(i)}$. 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 $S_{n}$ 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
$S_{n}$-action, then $\operatorname{Spec}(H)$ admits a global labelling and is
thus trivial.

We thus wish to obtain a method that also applies to non-trivial
$\operatorname{Spec}(H)$. Instead of considering 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

$\operatorname{Spec}(A)!=\big{\\{}\tilde{\lambda}=(\lambda_{1},\dots,\lambda_{n})\in\operatorname{Spec}(A)^{n}\mid\text{all
eigenvalues of $A$ appear once in $\tilde{\lambda}$}\big{\\}}.$

Observe that this set has $n!$ elements, hence the factorial notation. In
addition, this space is naturally endowed with the $S_{n}$-action on
$n$-tuples, cf. our earlier notation in equation (2.2). It is this
$S_{n}$-action that will help us with describing the map $p_{\gamma}$. Of
course, one can act with $p_{\gamma}$ on a tuple $\tilde{\lambda}$ entry-wise,
which leads to the map
$\tilde{p}_{\gamma}\colon(\lambda_{1},\dots,\lambda_{n})\mapsto(p_{\gamma}(\lambda_{1}),\dots,p_{\gamma}(\lambda_{n}))$.
As $\tilde{p}_{\gamma}\big{(}\tilde{\lambda}\big{)}$ is a reordering of
$\tilde{\lambda}$, we obtain a unique permutation $\sigma$, depending on
$\tilde{\lambda}$, by stating that

$\tilde{p}_{\gamma}\big{(}\tilde{\lambda}\big{)}=\sigma\cdot\tilde{\lambda}.$

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, $\sigma$ is not
based on these labels, but instead on the tuple positions. That is, $\sigma$
will send the energy at place $k$ to place $\sigma(k)$, regardless of what
energy is located at place $k$ or which value we designated as $\lambda_{k}$.

######  Example 5.1.

Let us continue with the EP2 example of Example 3.5, where $E_{+}(x_{0})$ and
$E_{-}(x_{0})$ are exchanged when one encircles an EP. Picking the ordering of
$\operatorname{Spec}(H(x_{0}))$ to be $(E_{+}(x_{0}),E_{-}(x_{0}))$, the above
reads

$\tilde{p}_{\gamma}(E_{+}(x_{0}),E_{-}(x_{0}))=(E_{-}(x_{0}),E_{+}(x_{0}))=(12)\cdot(E_{+}(x_{0}),E_{-}(x_{0}))$

so that $\sigma=(12)$. In this case, as $n=2$, one also obtains $(12)$ for the
alternative ordering $(E_{-}(x_{0}),E_{+}(x_{0}))$.

Consider now the case where we add a separate constant energy level
$E_{0}(x)=0$, i.e., consider the Hamiltonian $H^{\prime}(x)=H(x)\oplus 0$
which brings us to $n=3$. Picking the ordering
$(E_{+}(x_{0}),E_{0}(x_{0}),E_{-}(x_{0}))$ of
$\operatorname{Spec}(H^{\prime}(x_{0}))$ one finds

$\displaystyle\tilde{p}_{\gamma}(E_{+}(x_{0}),E_{0}(x_{0}),E_{-}(x_{0}))\\!=(E_{-}(x_{0}),E_{0}(x_{0}),E_{+}(x_{0}))\\!=(13)\cdot(E_{+}(x_{0}),E_{0}(x_{0}),E_{-}(x_{0})),$

which thus yields $(13)$. Of course, in the ordering
$(E_{+}(x_{0}),E_{-}(x_{0}),E_{0}(x_{0}))$ we would again obtain $(12)$.

Finally, let us demonstrate how the tuple method differs from labelling the
energies. Consider we label $E_{+}(x_{0})=E_{1}$, $E_{0}(x_{0})=E_{2}$ and
$E_{-}(x_{0})=E_{3}$. In the ordering $(E_{1},E_{2},E_{3})$, clearly $\sigma$
will replace the labels as expected (up to a convention; here the label
$E_{k}$ will become $E_{\sigma^{-1}(k)}$ instead of $E_{\sigma(k)}$). However,
in another ordering this need not be so, for example

$(12)\cdot(E_{3},E_{1},E_{2})=(E_{1},E_{3},E_{2})$

instead of the tuple $(E_{3},E_{2},E_{1})$ 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 $P_{\gamma}$, by a holonomy matrix. Similarly to the previous approach
with eigenvalues, we will consider a tuple formulation. In this case, for an
operator $A\in N(V)$, we consider tuples $\tilde{f}=(f_{1},\dots,f_{n})$ of
eigenvectors of $A$ such that each eigenray is represented exactly once. That
is, $\tilde{f}$ should be an eigenframe of $A$, and we write
$\mathrm{EigFr}(A)$ for the space of all eigenframes of $A$. This space has a
natural action by the wreath product $\mathbb{C}^{\times}\wr~{}I_{n}$,
isomorphic to the group of complex generalized permutation matrices, which we
already encountered in Section 2.2. In this case, the action reads

$\displaystyle\mathbb{C}^{\times}\wr~{}I_{n}\times\mathrm{EigFr}(A)$
$\displaystyle\to\mathrm{EigFr}(A),$
$\displaystyle((z_{1},\dots,z_{n}),\sigma)\cdot(f_{1},\dots,f_{n})$
$\displaystyle=(z_{1}f_{\sigma^{-1}(1)},\dots,z_{n}f_{\sigma^{-1}(n)}),$

i.e., the action is similar to the earlier
$\mathbb{C}^{\times}\wr~{}I_{n}$-action in equation (2.3). Using this action,
we can express a map $P_{\gamma}$ by an element
$(\tilde{z},\sigma)\in\mathbb{C}^{\times}\wr~{}I_{n}$ once we have chosen a
reference eigenframe $\tilde{f}$. Writing $\tilde{P}_{\gamma}$ for the map on
$\mathrm{EigFr}(A)$ obtained by applying $P_{\gamma}$ entry-wise, we obtain
the group element by stating

$\tilde{P}_{\gamma}(\tilde{f})=(\tilde{z},\sigma)\cdot\tilde{f}.$

If one represents $(\tilde{z},\sigma)$ by a permutation matrix, which we will
denote by $M_{\tilde{f}}$, then $M_{\tilde{f}}$ is the matrix of the linear
extension of $P_{\gamma}$ to the entire state space. The dependence of
$M_{\tilde{f}}$ on $\tilde{f}$ (again keeping the loop $\gamma$ fixed) follows
the familiar conjugation rule.

######  Example 5.2.

In case of the EP2, we found $P_{\gamma}$ as given by equation (4.5). With
respect to the eigenframe $(e_{1},e_{2})$ of $H(0)$, we find

$\tilde{P}_{\gamma}(e_{1},e_{2})=(P_{\gamma}(e_{1}),P_{\gamma}(e_{2}))=(-{\rm
i}e_{2},-{\rm i}e_{1})=((-{\rm i},-{\rm i}),(12))\cdot(e_{1},e_{2}),$

i.e., we find $((-{\rm i},-{\rm i}),(12))\in\mathbb{C}^{\times}\wr I_{2}$. The
holonomy matrix $M_{(e_{1},e_{2})}$ is thus

$M_{(e_{1},e_{2})}=\begin{pmatrix}-{\rm i}&0\\\ 0&-{\rm i}\\\
\end{pmatrix}\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix}=\begin{pmatrix}0&-{\rm
i}\\\ -{\rm i}&0\end{pmatrix}\\!.$

Or, alternatively,

$\begin{pmatrix}P_{\gamma}(e_{1})\\\ P_{\gamma}(e_{2})\\\
\end{pmatrix}=\begin{pmatrix}-{\rm i}e_{2}\\\ -{\rm i}e_{1}\\\
\end{pmatrix}=\begin{pmatrix}0&-{\rm i}\\\ -{\rm
i}&0\end{pmatrix}\begin{pmatrix}e_{1}\\\ e_{2}\\\ \end{pmatrix}\\!.$

The more familiar pattern $\psi_{1}\mapsto\psi_{2}\mapsto-\psi_{1}$ is
obtained by picking, e.g., $({\rm i}e_{1},e_{2})$ as the reference frame.
Indeed, as

$\tilde{P}_{\gamma}({\rm i}e_{1},e_{2})=(e_{2},-{\rm
i}e_{1})=((1,-1),(12))\cdot({\rm i}e_{1},e_{2})$

one finds the group element $((1,-1),(12))$ and the holonomy matrix reads

$M_{({\rm i}e_{1},e_{2})}=\begin{pmatrix}1&0\\\ 0&-1\\\
\end{pmatrix}\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix}=\begin{pmatrix}0&1\\\
-1&0\\\ \end{pmatrix}\\!.$

The matrices are different only because we picked a different reference frame;
both matrices still describe the map $P_{\gamma}$.

###  5.2 Holonomy interpretation on the frame bundle