# A brief review of mathematical foundation for analyzing topological
characteristics of quantum electronic states and matter phases

V. Nam Do<EMAIL_ADDRESS>Department of Basic Science, Phenikaa
Institute for Advanced Studies (PIAS), Floor 25th, A9 Building, Phenikaa
University, Hanoi 10000, Vietnam

###### Abstract

We briefly review the advanced mathematical language of fiber bundle
structures and how they can be used to classify two-level quantum systems
based on the analysis of the topological properties of their sets of state
vectors. The topological classes of quantum electronic states and matter
phases are characterized by topological invariants, which can be defined
geometrically as the integral of differential forms on the base manifold of
the fiber bundle structure. Specifically, we demonstrate that for one-
dimensional systems described by the Su-Schrieffer-Heeger (SSH) model, the set
of state vectors does not always have a fiber bundle structure directly on the
Brillouin zone. To classify the SSH systems, we use a technique based on the
concept of composite maps to decompose the set of electronic state vectors. As
a result, the SSH systems are classified based on the geometrical properties
of principal fiber bundles with different base manifolds.

##  I Introduction

For nearly two decades, there has been a growing use of topological invariant
concepts to characterize electronic and photonic states in various
media.Cayssol and Fuchs (2021); Price  _et al._ (2022); Lan  _et al._ (2022)
This has led to the understanding of two types of
semiconductors/superconductors: ordinary and topological
insulators/superconductors, as well as essential features in the electronic
structure of semimetals, among other examples.Bernevig and Hughes (2013)
Electronic states with nontrivial topological features, such as quantized
states localized at the edges of quantum Hall systems, are incredibly robust
and are not destroyed by perturbations, even by changing the atomic lattice of
material samples.von Klitzing  _et al._ (1980); Thouless  _et al._ (1982)
Therefore, nontrivial topological states are expected to have the potential to
create efficient qubits for quantum computers.A.Yu.Kitaev (2003)

Topology is a term that may be familiar to those learning mathematics but not
necessarily to those studying physics, particularly in the field of condensed
matter physics (CMP). Nevertheless, topology is now commonly used in CMP and
has implications for modern technologies.A.Yu.Kitaev (2003); Gilbert (2021)
Why is this? In fact, topology and related concepts are not unfamiliar to
physicists working in the domains of quantum field theories and general
relativity theories, as they are used to establish abstract structures of
space-times and the transformation of matter fields. The birth of topology is
attributed to Euler, who used graph theory to solve the famous problem of the
seven bridges of Königsberg in 1735.wol  It was then systematically developed
in Poincaré’s methodology in studying a series of basic problems in 1895.bri
(a) Many aspects of topology were also reflected in physics through Gauss’s
law (1835) and Ampère’s law (1825). bri (b, c) In the 20th century, since the
birth of quantum mechanics, topology has been used to solve many fundamental
issues in quantum mechanics, including the Dirac magnetic monopole
(1931),Dirac (1931); Wu and Yang (1975a, b) the Aharonov-Bohm effect
(1959),Aharonov and Bohm (1959); Chambers (1960) and the quantum Hall effect
(1980).von Klitzing  _et al._ (1980) The mystery of these phenomena was
revealed through Berry’s description of adiabatic evolution of physical
systems (1984). Berry ; Berry (1990) Through this description, it is clear
that topological and geometrical structures govern the quantum world.Resta
(2000); Shapere and (ed.)

Fundamentally, there are two problems in physics that we are concerned with.
The first is to characterize the existence of a system of matter, and the
second is to characterize its evolution over time. To tackle these issues, the
concept of degrees of freedom is introduced as a means of parameterizing the
system’s states. Using phase space transforms the physical problem into a
mathematical one, where each state is represented by a point in this space,
and the set of states of the system is established as a domain in this phase
space. However, it is essential to note that a geometric representation of
physical states is inadequate since different points in phase space may
correspond to the same physical state. The concept of “equivalence” is used in
mathematics to describe this situation, revealing the complex structure of the
set of physical states as a set of points in phase space. Consequently,
comprehending the transition of a system from one state to another
necessitates consideration of the structural properties of the set of points
in phase space. While calculus methods are typically used to perform specific
calculations, the frequent use of calculus may cause us to overlook the
natural meaning of the underlying operations. Calculus is built on the
fundamental concept of continuity, which is not a natural concept but depends
on the set of objects under consideration. Issues regarding the structure of
sets have been noticed and studied by mathematicians from early on, leading to
the formation of topology as a branch of modern mathematics. Although topology
theories are usually presented at advanced levels, topology is often described
as the study of the invariant properties of geometric objects under continuous
deformation. While a common illustration of this is the topological similarity
between a coffee cup and a doughnut or tire, this explanation is insufficient
for abstract objects such as sets of physical system states. Nevertheless, the
analogy of deforming one object into another may relate to the dynamics of the
physical system, as the transition of a system from one state to another
requires changing some parameters of the physical system. In CMP, the theory
of the energy band structure of electrons in periodic lattices is a direct and
significant achievement of quantum mechanics. The success of this theory
allows us to distinguish between two types of materials, metals and
insulators. However, this theory is built on the infinite extent of the
periodic lattice, so there are difficulties in describing certain physical
phenomena such as the charge polarization in the insulators. In fact, the
periodicity of the atomic lattice has been used to calculate the energy band
structure of electrons, which, mathematically, is a way of compactifying
infinite space. When rewriting the theory of charge polarization, a quantity
similar to the geometrical Berry phase appears, which is defined as the
integral of a vector field along a closed path in the Brillouin zone of the
crystalline lattice. This vector field is determined through the states of the
electronic system and has the meaning of guiding a “parallel motion” within
the set of electronic states.Asbóth  _et al._ (2016) Combining all of things,
one may wonder how all of these physical aspects have been resolved.
Specifically, how has mathematics been applied, and what reasoning method is
used here?

The purpose of this paper is to provide a brief review of the minimum basic
mathematical foundation and to demonstrate the application of topological
theories in condensed matter quantum physics. We employ two classical toy
models, one for general two-level systems and the Su-Schrieffer-Heeger (SSH)
model for one-dimensional electronic lattices, to demonstrate a procedure for
topologically characterizing and classifying quantum systems. These models are
commonly used in lectures and overview articles,Asbóth  _et al._ (2016);
Bernevig and Hughes (2013); Cayssol and Fuchs (2021); Lan  _et al._ (2022) but
they are usually presented in a mathematically loose manner. By employing
precise and rigorous mathematical language to correctly state physical
problems, we show that the set of state vectors does not always have the fiber
bundle structure defined on a compact manifold, such as the Brillouin zone, as
intuitively thought. Therefore, the analysis of the topological structure of
the set of state vectors is rather tricky. We aim to present the rigorous and
advanced mathematical language in a familiar way to resolve obstacles in the
“classical mindset”. We highlight the interpretation of expressions such as
“investigating behaviors of a physical system” as ”the investigation of
features of a map” or “a set of state vectors of a physical system” as a
vector field in a space of parameters where a set of equivalent states is
assigned at each point. Such a field of state vectors is determined by a
continuous map from a parameter space to a special space of the fiber bundle
structure. The movement in this space is guided by a quantity called the
connection, which is mathematically defined as a differential form in the
parameter space. The invariant characters of the set of quantum states
characterizing the movement are the geometrical properties of the fiber
bundle, which are usually determined by an integer number, the value of an
integral of the differential form over the entire parameter space.

This article, aside from the introductory and concluding sections, is divided
into two main parts. Part I, presented in Sec. II, provides an overview of the
preliminary mathematical concepts in topology that are necessary to understand
and apply the methodology. Part II, consisting of Secs. III and IV, presents
the methodology, along with detailed instructions, for analyzing two physical
models, each of which is presented in a separate section.

##  II Fundamental mathematical concepts

The aim of this section is to present a non-exhaustive list of fundamental
mathematical concepts that commonly appear in the analysis of the topological
properties of physical systems. Each concept is briefly introduced, but for
further details, readers are encouraged to refer to textbooks on topology such
as Nakahara, 2003; Hassani, 1999; Simon, 2021. These concepts serve as a basic
vocabulary for discussions in this field.

###  II.1 Topological spaces and continuous maps

Topology: It is the primary concept used to define other fundamental
mathematical concepts that describe physical objects as mathematical
structures from the viewpoint of the set of elementary elements. The topology
defines the neighborhood of an element in a set and the continuity in the
variation of a quantity with respect to another. The formal definition of
topology can be stated as follows: Given a set $X$ of objects, a topology
$T_{X}$ of $X$ is a collection of subsets $\\{U_{i}\\}$ of $X$ such that:

  1. 1.

Both $X$ and the empty set belong to $T_{X}$.

  2. 2.

Any union of elements in $T_{X}$ belongs to $T_{X}$.

  3. 3.

A finite intersection of elements in $T_{X}$ belongs to $T_{X}$.

Open sets: The subsets of $X$ that belong to the topology $T_{X}$ are called
open sets.

Topological spaces: Sets of objects endowed with an appropriate topology are
called topological spaces. A topological space is usually written as a pair
$(X,T_{X})$ or simply as the set $X$ if $T_{X}$ is a normal topology or if the
context makes it clear. Each object in a topological space is called a point.

Neighborhood: A subset $N$ of a set $X$ is called a neighborhood of a point
$p\in X$ if $N$ contains at least one open set that contains the point $p$,
i.e., $N\supset U_{i}$ such that $p\in U_{i}$ and $U_{i}\in T_{X}$. If $N$ is
an open set, it is called an open neighborhood.

Coverings: A collection of subsets ${U_{i}}$ of a set $X$ is called a covering
of $X$ if $\bigcup_{i}U_{i}=X$. If all $U_{i}$ belong to $T_{X}$, then the
covering is called an open covering.

Compactness: A set is called compact if it can be covered by a finite number
of open sets.

Connectedness: A set is called connected if it cannot be partitioned into two
non-empty subsets that are disjoint. In other words, the set cannot be
separated into two subsets without breaking the continuity of the set.

Continuous maps: A map $f$ between two topological spaces $X$ and $Y$ is
called continuous if the preimage of an open set in $Y$ is an open set in $X$.

Homeomorphisms: A map $f$ between two topological spaces $X$ and $Y$ is called
a homeomorphism if $f$ and its inverse are both continuous. This means that
there is a correspondence not only between elements in the two sets $X$ and
$Y$, but also between open sets in the two topologies $T_{X}$ and $T_{Y}$. If
there is a homeomorphism between two topological spaces, we say that the two
topological spaces are homeomorphic to each other. A homeomorphism defines an
equivalence relation on the set of all topological spaces. As a consequence,
given a homeomorphism, we can classify all topological spaces into equivalence
classes.

Topological invariants: Integer numbers are often used to characterize common
features of topological spaces in the same equivalence class under the
homeomorphism relation. These integer values are known as topological
invariants.

Homotopy and deformation: Homotopy is a relation between two continuous
functions that captures the idea of “continuous deformation”. More
specifically, given two continuous functions $f,g:X\to Y$ between two
topological spaces $X$ and $Y$, a homotopy between $f$ and $g$ is a continuous
function $H:X\times[0,1]\to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for
all $x$ in $X$.

###  II.2 Manifolds

Manifold: A manifold is a generalization of the concept of smooth curves and
surfaces to arbitrary dimensional objects. The smoothness of a topological
space allows the description of an arbitrary open neighborhood of a point by
an open set in an $n$-dimensional vector space $\mathbb{R}^{n}$. This allows
calculus on the topological space to be endowed thanks to the definition of
coordinates in $\mathbb{R}^{n}$. Formally, we have the following definition:
The topological space $\mathbb{M}$ is called a manifold if:

  1. 1.

There exists an open covering ${U_{i}}$ such that
$\mathbb{M}=\bigcup_{i}U_{i}$.

  2. 2.

For each open set $U_{i}$ in the $\mathbb{M}$-covering, there exists a
homeomorphism $f_{i}:U_{i}\to\mathbb{R}^{n}$. This map allows $U_{i}$ to be
described by an open set in $\mathbb{R}^{n}$.

  3. 3.

On the overlap of two open sets $U_{i}$ and $U_{j}$, the composite map
$t_{ij}=f_{i}^{-1}\circ f_{j}$ is differentiable. This is the compatibility
condition for the description using two different maps $f_{i}$ and $f_{j}$ in
$U_{i}\cap U_{j}$.

The pairs $(U_{i},f_{i})$ are called charts, and the set of all charts is
called an atlas of the manifold. $U_{i}$ is called the neighborhood, and the
map $f_{i}$ is called the coordinate function.

Dimension: The dimension of the vector space $\mathbb{R}^{n}$ that the
manifold locally resembles is called the dimension of the manifold.

Functions: Maps $f:\mathbb{M}\to\mathbb{R}$, where $p\in\mathbb{M}\mapsto
f(p)\in\mathbb{R}$, are called functions defined on the manifold. Each
bijective map $f_{i}:\mathbb{M}\to\mathbb{R}^{n}$ is represented by $n$
functions $p\in U_{i}\mapsto f_{i}(p)=(x^{1}(p),x^{2}(p),\dots,x^{n}(p))$.

Curves: A differentiable curve in the manifold $\mathbb{M}$ is a
$C^{\infty}$-map of an interval of $\mathbb{R}$ to $\mathbb{M}$, i.e.,
$c:[a,b]\subset\mathbb{R}\to\mathbb{M}$, where $t\in[a,b]\mapsto
c(t)=f_{i}^{-1}(x^{1}(c(t)),x^{2}(c(t)),\dots,x^{n}(c(t)))$.

Tangent vectors and tangent spaces: A tangent vector is an object $V(p)$
defined at each point of the manifold $\mathbb{M}$ that acts as the derivative
on functions at the point $p$, i.e., $V(p)=V^{\mu}(p)\partial/\partial
x^{\mu}$, where $V[f(p)]=V^{\mu}(p)\partial f(p)/\partial x^{\mu}$. Here $p\in
U_{i}$ and $f_{i}(p)=(x^{1}(p),x^{2}(p),\dots,x^{n}(p))$. The set of $n$
numbers $V^{\mu}\in\mathbb{R}$ is called the set of coordinates of the vector
$V(p)$.

Tangent bundle: The union of all tangent vectors over the manifold
$\mathbb{M}$ is called the tangent bundle, $T(\mathbb{M})=\bigcup_{p\in
M}T_{p}(\mathbb{M})$.

Vector fields: A vector field $V$ is a continuous map from $\mathbb{M}$ to the
tangent bundle $T(\mathbb{M})$, i.e., $V:\mathbb{M}\to T(\mathbb{M})$, where
$p\mapsto V(p)\in T_{p}(\mathbb{M})$.

One-forms and cotangent spaces: A one-form is a linear object $\omega$ defined
at each point $p\in\mathbb{M}$ that maps tangent vectors at $p$ to a number,
i.e., $\langle\omega(p),V(p)\rangle\in\mathbb{R}$. The set of all possible
one-forms defined at a point $p\in\mathbb{M}$ forms a vector space called the
cotangent space, denoted by $T_{p}^{\star}(\mathbb{M})$. If the partial
derivatives $\partial/\partial x^{\mu}$ are seen as the basis vectors of the
tangent space, the differential $dx^{\mu}$ plays the role of the basis vectors
of the cotangent space, i.e., $\omega(p)=\omega_{\mu}(p)dx^{\mu}$ where
$\omega_{\mu}(p)\in\mathbb{R}$, because $\langle dx^{\mu},\partial/\partial
x^{\nu}\rangle=\delta^{\mu}_{\nu}$. Here, $\delta^{\mu}_{\nu}$ denotes the
Kronecker delta symbol.

Tensor fields: A tensor of rank $(q,r)$ is a multilinear object that takes $q$
elements of $T^{\star}_{p}(\mathbb{M})$ and $r$ elements of
$T_{p}(\mathbb{M})$ and returns a number. The set of all tensors of rank
$(q,r)$ defined at the point $p$ on the manifold $\mathbb{M}$ is denoted by
$\mathcal{T}^{q}_{r,p}(\mathbb{M})$. The union
$\bigcup_{p\in\mathbb{M}}\mathcal{T}^{q}_{r,p}(\mathbb{M})$ is called the
tensor bundle. A tensor field of rank $(q,r)$ is a continuous map
$T:\mathbb{M}\to\mathcal{T}^{q}_{r}(\mathbb{M})$.

Connection: Intuitively, this concept is an instruction to a special motion in
a manifold, i.e., the so-called parallel transport. The connection is a
central quantity to describe the geometrical properties of manifolds. The
definition is quite technical, so readers should consult textbooks of
topology.Nakahara (2003); Hassani (1999); Simon (2021)

Curvature: Curvature is a concept used to characterize a geometrical feature
of manifolds. The general definition of this concept is quite technical, so we
ask readers to consult textbooks on geometry and topology.Nakahara (2003);
Hassani (1999); Simon (2021)

###  II.3 Fiber bundles

Fiber bundle: The tangent bundle and cotangent bundle are two natural
geometric objects associated with a manifold that allow for the definition of
vector fields on the manifold. A fiber bundle is a generalization and a
natural mathematical setting to describe physical fields. Technically, it is a
concept used to refer to a special kind of manifold that locally looks like
the direct product of two manifolds. Conversely, we can construct a fiber
bundle from some other manifolds. The formal definition is as follows: A
manifold $\mathbb{E}$ is said to have a fiber bundle structure over the base
manifold $\mathbb{B}$ with fiber manifold $\mathbb{F}$ if there exists a
surjective map $\hat{\pi}:\mathbb{E}\rightarrow\mathbb{B}$ satisfying the
following conditions:

  1. 1.

For all points $p\in\mathbb{B}$, the preimage $\hat{\pi}^{-1}(p)$ of $p$ by
the map $\hat{\pi}$ is homeomorphic to the manifold $\mathbb{F}$.

  2. 2.

For each open set $U_{i}$ of an open covering ${U_{i}}$ of $\mathbb{B}$, its
preimage $\hat{\pi}^{-1}(U_{i})$ is simply described as a direct product
$U_{i}\times\mathbb{F}$ by a diffeomorphism
$\phi_{i}:U_{i}\times\mathbb{F}\to\hat{\pi}^{-1}(U_{i})$. The maps $\phi_{i}$
are called the local trivializations.

  3. 3.

The description of $\hat{\pi}^{-1}(U_{i})$ as $U_{i}\times\mathbb{F}$ must be
consistent. This means that the composite map
$\phi_{i}^{-1}\circ\phi_{j}:(U_{i}\cap U_{j})\times\mathbb{F}\to(U_{i}\cap
U_{j})\times\mathbb{F}$ must satisfy the condition
$\phi_{i}^{-1}\circ\phi_{j}(p,f)=(p,g_{ij}(p)f)$, where $g_{ij}(p)$ are the
functions determined by the map $g_{ij}:U_{i}\cap U_{j}\to\mathbb{F}$ that
have the properties: $g_{ii}(p)=id_{U_{i}}$ and
$g_{ij}(p)g_{jk}(p)g_{ki}(p)=id_{U_{i}}$.

The set $\mathbb{E}$, called the total/entire space, is thus decomposed into
the fibers $\mathbb{F}(p)$, i.e.,
$\mathbb{E}=\bigcup_{p\in\mathbb{B}}\mathbb{F}(p)$, where
$\mathbb{F}(p)=\hat{\pi}^{-1}(p)$ is called the fiber attached to the base
manifold $\mathbb{B}$ at the point $p$.

Principal bundles: Principal bundles are fiber bundles in which the fiber
$\mathbb{F}$ is identical to the structure group $G$. In the next section, we
will work with this type of fiber bundle with the structure group $G=U(1)$
(the gauge $U(1)$ group).

Cross-sections: Let $\hat{\pi}:\mathbb{E}\to\mathbb{B}$ be a fiber bundle. A
cross-section of the fiber bundle is a smooth map $s:\mathbb{B}\to\mathbb{E}$.
Clearly, $p\mapsto s(p)$ is an element of $\mathbb{F}_{p}=\hat{\pi}^{-1}(p)$.

Connections: As mentioned in the previous subsection, it is rather technical
to define the connection. However, because of the special structure of the
fiber bundles, the idea of defining the connection is a way to separate the
tangent vector of the total space into the vertical (along the fiber) and
horizontal (along the base space) directions. Again, readers are asked to
consult textbooks.Nakahara (2003); Hassani (1999); Simon (2021)

##  III General model for two-level systems

This section presents an analysis of a toy model for generic two-level systems
to highlight the application of the fiber bundle structure in characterizing a
set of quantum states. Mathematically, this section illustrates the analysis
of a set of points defined by an appropriate map.

###  III.1 Physical model

The general model for the dynamics of an electron in a two-level system is
defined by a two-dimensional Hamiltonian matrix. Based on the Hermitian
property of physical observables, the Hamiltonian matrix reads

$\displaystyle H$ $\displaystyle=\begin{pmatrix}d_{0}+d_{z}&d_{x}-id_{y}\\\
d_{x}+id_{y}&d_{0}-d_{z}\end{pmatrix}$
$\displaystyle=d_{0}\sigma_{0}+d_{x}\sigma_{x}+d_{y}\sigma_{y}+d_{z}\sigma_{z},$
(1)

where $\sigma_{0}$ is the $2\times 2$ identity matrix, and
$\sigma_{x},\sigma_{y},\sigma_{z}$ are three conventional Pauli matrices. The
parameters $d_{x},d_{y},d_{z}$ and $d_{0}$ are real and determine the
properties of the system. Since the first term in Eq. (1) can be seen as the
energy reference, it can be ignored in the following analysis. Therefore, the
Hamiltonian matrix is determined by the remaining three terms. Defining the
3-dimensional vectors $\mathbf{d}=(d_{x},d_{y},d_{z})$ and
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$, the Hamiltonian
matrix can be expressed in a compact form as the scalar product of two
vectors:

$\displaystyle H=\mathbf{d}\cdot\boldsymbol{\sigma}.$ (2)

The vector $\mathbf{d}$ represents a point in the 3-dimensional linear space
$\mathbb{R}^{3}$. In this space, we do not consider the origin point $(0,0,0)$
as it defines a non-trivial physical system. Any other point in
$\mathbb{R}^{3}$ defines a particular 2-level system. In other words, the
existence condition of the 2-level systems is defined by the parameter vector
$\mathbf{d}$.

###  III.2 Determination of eigen-states

An eigen-state of a quantum system is determined by an energy value and a set
of objects known as state vectors. Mathematically, the eigen-energies and the
associated eigen-state vectors are determined by the eigen-value equation of
the Hamiltonian $H|\psi\rangle=E|\psi\rangle$. Specifically, the eigen-values
$E$ are determined by the secular equation:

$\displaystyle\det\begin{pmatrix}d_{z}-E&d_{x}-id_{y}\\\
d_{x}+id_{y}&-d_{z}-E\end{pmatrix}=0\quad$ $\displaystyle\rightarrow\quad$
$\displaystyle E^{2}-d_{z}^{2}-(d_{x}+id_{y})(d_{x}-id_{y})=0.$ (3)

This equation has two solutions for the unknown $E$ given by the formulae:

$\displaystyle E=\pm\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z}^{2}}=\pm|\mathbf{d}|=\pm
d=E_{\pm}(\mathbf{d}).$ (4)

The eigen-values of the Hamiltonian $H$ depend on the Euclidean length $d$ of
the vector $\mathbf{d}$. The parameter space $\mathbb{R}^{3}$ is therefore
identified as the Euclidean space $\mathbb{R}^{3}$. Since the point
$\mathbf{d}=(0,0,0)$ is not considered, we see that these two eigen-values are
always separated from each other by a finite amount:

$\displaystyle\Delta E(\mathbf{d})=E_{+}(\mathbf{d})-E_{-}(\mathbf{d})=2d>0.$
(5)

This confirms that the model (2) is appropriate for describing two-level
systems. To determine the eigenvectors associated with the two eigenvalues, we
define the generic state vector of the two-level systems as follows:

$\displaystyle|\Psi\rangle=\begin{pmatrix}\phi_{1}\\\ \phi_{2}\end{pmatrix},$
(6)

where $\phi_{1}$ and $\phi_{2}$ can take on complex values. The eigenvectors
of the Hamiltonian matrix are denoted by $|+,\mathbf{d}\rangle$ and
$|-,\mathbf{d}\rangle$, corresponding to the eigenvalues
$E_{\pm}(\mathbf{d})$, respectively. The equation
$[H-E_{-}(\mathbf{d})]|-,\mathbf{d}\rangle=0$ is specified as follows:

$\displaystyle\begin{pmatrix}d_{z}+d&d_{x}-id_{y}\\\
d_{x}+id_{y}&-d_{z}+d\end{pmatrix}\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=\begin{pmatrix}0\ 0\end{pmatrix}$
$\displaystyle\Rightarrow(d_{z}+d)\phi_{1}+(d_{x}-id_{y})\phi_{2}=0.$ (7)

To find $\phi_{1}$ and $\phi_{2}$, we need to consider the following cases:
(1) If $d_{z}=d$, then $d_{x}$ and $d_{y}$ are both zero, i.e.,
$d_{x}=d_{y}=0$. In this case, $\phi_{1}$ and $\phi_{2}$ can be arbitrarily
chosen, and the eigenvector is therefore ill-defined. (2) If $d_{z}\neq d$,
Eq. (III.2) allows us to find the relationship, but not a specific value,
between the two components $\phi_{1}$ and $\phi_{2}$ of the eigenvector.
Specifically, we can write:

$\displaystyle\left\\{\begin{array}[]{l}\phi_{1}=\gamma(-d_{x}+id_{y})\\\
\phi_{2}=\gamma(d+d_{z})\end{array}\right.,$ (10)

where $\gamma$ is a non-zero undetermined complex factor. Normalizing the
length of $|-,\mathbf{d}\rangle$, we calculate:

$\displaystyle|\phi_{1}|^{2}+|\phi_{2}|^{2}=2|\gamma|^{2}d(d+d_{z}).$ (11)

The expression of the eigenvector $|-,\mathbf{d}\rangle$ thus reads:

$\displaystyle|-,\mathbf{d}\rangle=\frac{e^{i\chi}}{\sqrt{2d(d+d_{z})}}\begin{pmatrix}-d_{x}+id_{y}\\\
d+d_{z}\end{pmatrix},$ (12)

Where $e^{i\chi}=\gamma/|\gamma|$ is the argument of the complex factor
$\gamma$. Here, $\chi$ is a real parameter. So, the eigen-vector
$|-,\mathbf{d}\rangle$ is not uniquely defined, but is up to a phase factor.
In other words, we can state that given a vector
$\mathbf{d}\in\mathbb{R}^{2}\backslash{(0,0,0)}$, Eq. (III.2) does not define
one state vector, but a set of state vectors that differ from each other by a
$U(1)$ gauge factor $e^{i\chi}$. Physically, all state vectors in this set
describe the same state of the system. Hence, they are classified into a
unique equivalence class with the $U(1)$ equivalence relation, i.e.,
$[\\![|-,\mathbf{d}\rangle_{0}]\\!]={g(\mathbf{d})|-,\mathbf{d}\rangle_{0},|,\forall,g(\mathbf{d})\in
U(1)}$, where the representative element $|-,\mathbf{d}\rangle_{0}$ is chosen
as:

$\displaystyle|-,\mathbf{d}\rangle_{0}=\frac{1}{\sqrt{2d(d+d_{z})}}\begin{pmatrix}d_{x}-id_{y}\\\
d-d_{z}\end{pmatrix}.$ (13)

The discussion is totally similar to the eigen-vector $|+,\mathbf{d}\rangle$,
which is given by the formula:

$\displaystyle|+,\mathbf{d}\rangle=\frac{e^{i\chi}}{\sqrt{2d(d-d_{z})}}\begin{pmatrix}d_{x}-id_{y}\\\
d-d_{z}\end{pmatrix}.$ (14)

Due to the dependence of the eigen-energies and state vectors on the length of
the $\mathbf{d}$ vector, we can change the parameterization of $\mathbf{d}$
from Cartesian coordinates $(d_{x},d_{y},d_{z})$ to spherical coordinates
$(d,\varphi,\theta)$, where $d>0$, $\varphi\in[0,2\pi]$, and
$\theta\in[0,\pi]$. The eigen-vectors $|\pm,\mathbf{d}\rangle$ can then be
rewritten in the form:

$\displaystyle|+,\varphi,\theta\rangle=e^{i\chi}\begin{pmatrix}e^{-i\varphi}\cos(\theta/2)\\\
\sin(\theta/2)\end{pmatrix},$ (15a)
$\displaystyle|-,\varphi,\theta\rangle=e^{i\chi}\begin{pmatrix}e^{-i\varphi}\sin(\theta/2)\\\
-\cos(\theta/2)\end{pmatrix}.$ (15b)

Using spherical coordinates, we find that the eigen-state vectors of the
2-level system do not depend on the length of the model parameter $\mathbf{d}$
vector, but only on the two angular coordinates. We can therefore classify the
set of all possible values of the vector $\mathbf{d}$ into a unique
equivalence class represented by the unit sphere $\mathbb{S}^{2}$ centered at
the origin of the Cartesian axes in the Euclidean $\mathbb{R}^{3}$ space (this
two-dimensional surface is embedded in the Euclidean space $\mathbb{R}^{3}$).
Each point on the unit sphere is parameterized by only two real variables
$\varphi\in[0,2\pi]$ and $\theta\in[0,\pi]$. A point on the sphere
$\mathbb{S}^{2}$ is determined by a point $(\varphi,\theta)$ in the
rectangular domain $[0,2\pi]\times[0,\pi]$. Topologically, we see that the
sphere $\mathbb{S}^{2}$ is homeomorphic to the rectangular domain
$[0,2\pi]\times[0,\pi]$.

###  III.3 Investigation of topological features of quantum states

Physically, we would like to know what happens when we drive the system.
According to the model given by Eq. (1), the condition for the existence of a
two-level quantum system is encoded in the parameter vector $\mathbf{d}$. Eq.
(4) shows the linear dependence of the energy values of the two eigen-states
of a system on the length of the $\mathbf{d}$ vector. Mathematically, the
energy spectrum of a system is determined by scalar fields on the manifold of
parameters. Some important features of such scalar fields can manifest through
the picture of the density of states such as the van Hove singularities.
However, these are local geometrical features of the energy isosurfaces, i.e.,
the existence of local extremal and/or saddle points of the surfaces. Driving
the $\mathbf{d}$ vector on the unit sphere $\mathbb{S}^{2}$, the energy
spectrum of the system does not change. Meanwhile, the state vectors
explicitly depend on the angle coordinates of the $\mathbf{d}$ vector, see
Eqs. (15a) and (15b), so they vary when driving $\mathbf{d}$ on
$\mathbb{S}^{2}$. Accordingly, it is natural to ask: what information can the
state vectors provide as the system is driven? To proceed with the discussion,
we consider the set of state vectors

$\displaystyle\mathbb{E}=\left\\{|-,\mathbf{d}\rangle=e^{i\chi}|-,\mathbf{d}\rangle_{0},|,\forall\chi\in\mathbb{R},\forall\mathbf{d}\in\mathbb{S}^{2}\right\\}.$
(16)

We consider this set due to the fact that physical systems tend to stay in
their lowest-energy state. The detailed discussion is presented in the
following subsections.

####  III.3.1 The fiber-bundle structure of the set of state vectors

Let us denote

$\displaystyle\mathbb{F}(\mathbf{d})=\left\\{|-,\mathbf{d}\rangle=\left.\frac{e^{i\chi}}{\sqrt{2d(d+d_{z})}}\begin{pmatrix}-d_{x}+id_{y}\\\
d+d_{z}\end{pmatrix},\right|,\forall\chi\in\mathbb{R}\right\\}.$ (17)

The set of state vectors $\mathbb{E}$ can thus be rewritten as:

$\displaystyle\mathbb{E}=\bigcup_{\mathbf{d}\in\mathbb{B}}\mathbb{F}(\mathbf{d}),$
(18)

where
$\mathbb{B}=\\{\mathbf{d}\in\mathbb{R}^{3},|,\text{norm}_{2}(\mathbf{d})=1\\}\equiv\mathbb{S}^{2}$.
We now show that the set $\mathbb{E}$ has a fiber bundle structure with the
fiber being the manifold $\mathbb{F}=\mathbb{S}^{1}$ endowed by the $U(1)$
structure group.

Indeed, first of all, let us show that there exists a surjective map from
$\hat{\pi}$ projecting $\mathbb{E}$ onto $\mathbb{B}$, i.e.,
$\hat{\pi}:\mathbb{E}\to\mathbb{B}$. To do so, we pick a generic element
$|-\rangle=(\phi_{1},\phi_{2})^{T}$ in $\mathbb{E}$ and identify it with an
element $|-,\mathbf{d}\rangle$ in a subset $\mathbb{F}(\mathbf{d})$. From Eq.
(12), we can construct the map $\hat{\pi}$, given by the following explicit
rule:

$\displaystyle\hat{\pi}:(\phi_{1},\phi_{2})\in\mathbb{E}\mapsto\left\\{\begin{array}[]{l}d_{x}=-2\text{Re}\left(|\phi_{2}|\frac{\phi_{1}}{\phi_{2}}\right)\\\
d_{y}=2\text{Im}\left(|\phi_{2}|\frac{\phi_{1}}{\phi_{2}}\right)\\\
d_{z}=2|\phi_{2}|-1\end{array}\right.\in\mathbb{B}.$ (22)

This means that the set $\mathbb{F}(\mathbf{d})$ is the preimage of
$\mathbf{d}$ under the map $\hat{\pi}$, i.e.,
$\mathbb{F}(\mathbf{d})=\hat{\pi}^{-1}(\mathbf{d})$.

Next, we will show that the manifold $\mathbb{E}$ can be locally described as
the direct product of an open set of $\mathbb{B}$ and another manifold. We
first notice that the set $\mathbb{F}(\mathbf{d})$ is homeomorphic to the set
$\mathbb{F}=\mathbb{S}^{1}$ because all pairs of complex numbers
$(\phi_{1},\phi_{2})$ that are different from each other by a phase factor
$e^{i\varphi}$ are mapped onto the same $\mathbf{d}$ point in
$\mathbb{S}^{2}$. We also notice that $\mathbb{F}(\mathbf{d})$ is ill-defined
at $\mathbf{d}=(0,0,-d)$ because of the singularity of the factor
$1/\sqrt{2d(d+d_{z})}$ in the expression of the state vectors. However, this
singularity, as shown below, can be apparently avoided by choosing an
appropriate scheme of local parameterization. Indeed, using the spherical
coordinates as a specific parameterization of the $\mathbb{S}^{2}$ sphere
seems to eliminate the singularity because of the disappearance of the factor
$1/\sqrt{2d(d+d_{z})}$. Actually, the condition $d_{z}=-d$ exactly corresponds
to the value $\theta=\pi$, but $\varphi$ is still not uniquely determined.
Hence, the state vector $|-,\mathbf{d}\rangle_{0}$ is ill-defined at the south
pole of the sphere $\mathbb{S}^{2}$. For other state vectors,
$|-,\mathbf{d}\rangle=\exp(i\chi)|-,\mathbf{d}\rangle_{0}$, we see that if
$\chi$ is identical to $\varphi$ then:

$\displaystyle
e^{i\chi}|-,\mathbf{d}\rangle_{0}=e^{i\varphi}\begin{pmatrix}e^{-i\varphi}\sin(\theta/2)\\\
-\cos(\theta/2)\end{pmatrix}=\begin{pmatrix}\sin(\theta/2)\\\
-e^{i\varphi}\cos(\theta/2)\end{pmatrix}.$ (23)

These state vectors are clearly defined at the south pole, eliminating the
problem of ill-definition. However, we realize that the new state vectors
become ill-defined at the north pole with $\theta=0$ and arbitrary $\varphi$.
As we can see from Eq. (12), the singularity of the map $|-\rangle$ is
actually permanent and cannot be removed by choosing a particular
parameterization and a gauge transformation. The gauge transformation simply
moves the singularity point of the map
$|-\rangle_{0}:\mathbb{S}^{2}\to\mathbb{F}(\mathbf{d})$ from one point on the
domain manifold $\mathbb{S}^{2}$ to another point. In other words, the map
$|-\rangle_{0}$ is only locally defined in the entire manifold
$\mathbb{S}^{2}$.

To locally describe the manifold $\mathbb{E}$, we must use some open sets to
cover the manifold $\mathbb{B}=\mathbb{S}^{2}$. Concretely, we use two open
sets $U_{N}=[0,2\pi]\times[0,\pi/2+\epsilon)$ and
$U_{S}=[0,2\pi]\times(\pi/2-\epsilon,\pi]$ to cover the north and south half
spheres, respectively. The overlap of these two open sets is the ribbon
covering the equator, i.e., $U_{N}\cap
U_{S}=[0,2\pi]\times(\pi/2-\epsilon,\pi/2+\epsilon)$. We can then define the
maps $|-\rangle_{N/S}:U_{N/S}\times U(1)\to\hat{\pi}^{-1}(U_{N/S})$ in each
open set as follows:

$\displaystyle|-\rangle_{N}:(\varphi,\theta;e^{i\chi})\mapsto|-,\varphi,\theta\rangle_{N}=e^{i\chi}\begin{pmatrix}e^{-i\varphi}\sin(\theta/2)\\\
-\cos(\theta/2)\end{pmatrix},$ (24)
$\displaystyle|-\rangle_{S}:(\varphi,\theta;e^{i\chi})\mapsto|-,\varphi,\theta\rangle_{S}=e^{i\chi}\begin{pmatrix}\sin(\theta/2)\\\
-e^{i\varphi}\cos(\theta/2)\end{pmatrix}.$ (25)

Clearly, there is no problem of singularity of these maps on each chart
$U_{N}$ and $U_{S}$. These maps are diffeomorphic since their inverse maps,
e.g., $|-\rangle_{N}^{-1}$, given by the functions:

$\displaystyle\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}\mapsto\begin{pmatrix}-\frac{i}{2}\ln\left[\left(\frac{\phi_{1}}{\phi_{2}}\right)^{2}\cdot\frac{|\phi_{2}|^{2}}{1-|\phi_{2}|^{2}}\right],2\arccos{|\phi_{2}|};e^{i\chi}\end{pmatrix},$
(26)

is analytic. So, the pairs $(U_{N},|-\rangle_{N})$ and $(U_{S},|-\rangle_{S})$
are identified as the local trivializations.

The last point we would like to show is that the transition function $t_{NS}$
is an element of the gauge $U(1)$ group. Indeed, let $t_{NS}(\varphi,\theta)$
denote the function connecting $|-,\varphi,\theta\rangle_{S}$ to
$|-,\varphi,\theta\rangle_{N}$ for $(\varphi,\theta)\in U_{N}\cap U_{S}$. From
the requirement:

$\displaystyle e^{i\chi_{N}}\begin{pmatrix}e^{-i\varphi}\sin(\theta/2)\\\
-\cos(\theta/2)\end{pmatrix}=t_{NS}(\varphi,\theta)e^{i\chi_{S}}\begin{pmatrix}\sin(\theta/2)\\\
-e^{i\varphi}\cos(\theta/2)\end{pmatrix},$ (27)

it is easy to deduce the identification:

$t_{NS}(\varphi,\theta)=e^{-i\varphi+\Delta\chi_{NS}},$ (28)

where $\Delta\chi_{NS}=\chi_{N}-\chi_{S}$. Clearly, $t_{NS}(\varphi,\theta)$
belongs to $U(1)$ as expected.

To sum up, we have shown that the set $\mathbb{E}$ of state vectors can be
represented as a sphere $\mathbb{S}^{3}$ embedded in the space
$\mathbb{C}^{2}$. This set has a fiber bundle structure with the base manifold
$\mathbb{B}=\mathbb{S}^{2}$ and the fiber $\mathbb{F}=\mathbb{S}^{1}$ with the
structure group $U(1)$. The projection map
$\hat{\pi}:\mathbb{S}^{3}\to\mathbb{S}^{2}$ is given by Eq. (22). The local
trivializations of the bundle are given by the pairs $(U_{N},|-\rangle_{N})$
and $(U_{S},|-\rangle_{S})$, which are identified on the overlap $U_{N}\cap
U_{S}$ by the transition function $t_{NS}(\varphi,\theta)=\exp(i\varphi)$,
which belongs to the gauge group $U(1)$.

####  III.3.2 The state transition as the parallel motion in the principal
bundle

Physical processes always involve the transition of a system from one state to
another. The transition induced by varying the parameter vector $\mathbf{d}$
is mathematically translated into movement from one point to another in the
fiber bundle $(\hat{\pi}:\mathbb{E}\to\mathbb{B},\mathbb{F},U(1))$. While
moving in a flat space, a velocity field is enough to guide the motion, but in
a curved manifold, another quantity, named the connection, is needed to keep
the motion according to the curvature of the manifold, i.e., the parallel
motion. The parallel motion in the fiber bundle structure can be described as
the horizontal lift of the parallel motion in the base space. This description
highlights the holonomy, a typical geometrical phenomenon in nontrivial fiber
bundles.Nakahara (2003); El-Batanouny (2020); Simon (2021) To determine the
connection $\mathcal{A}_{-}$ in a fiber bundle, we need a vector field defined
as a local cross-section of the bundle and then track its variation along some
smooth curves. A concrete vector field $|-,\mathbf{d}\rangle_{N/S}$, locally
defined on the covering $U_{N/S}$, i.e., $|-\rangle_{N/S}:(\varphi,\theta)\in
U_{N/S}\mapsto e^{i\chi(\varphi,\theta)}|-,\varphi,\theta\rangle_{0,N/S}$, is
determined when a smooth function $\chi(\varphi,\theta)$ on $U_{N/S}$ is
given. Due to the scalar product of the state vectors defined in the Hilbert
space, the so-called Berry connection is determined as follows:El-Batanouny
(2020); Bernevig and Hughes (2013)

$\displaystyle\mathcal{A}_{-}(\mathbf{d})=i\langle-,\mathbf{d}|\hat{d}|-,\mathbf{d}\rangle,$
(29)

where $\hat{d}$ denotes the exterior derivative Nakahara (2003). Using the
parameterization of spherical coordinates $(\varphi,\theta)$, we obtain:

$\displaystyle\mathcal{A}_{-}(\varphi,\theta)$
$\displaystyle=i\langle-,\varphi,\theta|\partial_{\varphi}|-,\varphi,\theta\rangle
d\varphi$
$\displaystyle+i\langle-,\varphi,\theta|\partial_{\theta}|-,\varphi,\theta\rangle
d\theta.$ (30)

On each chart $U_{N,S}$ partially covering $\mathbb{S}^{2}$ we obtain:

$\displaystyle\partial_{\varphi}|-,\varphi,\theta\rangle_{N}=e^{i\chi}\begin{pmatrix}-ie^{-i\varphi}\sin(\theta/2)\\\
0\end{pmatrix},$ (31)
$\displaystyle\partial_{\theta}|-,\varphi,\theta\rangle_{N}=e^{i\chi}\frac{1}{2}\begin{pmatrix}-ie^{-i\varphi}\cos(\theta/2)\\\
\sin(\theta/2)\end{pmatrix}.$ (32)

Then, it yields the expression of the connection:

$\displaystyle\mathcal{A}_{-}(\varphi,\theta)_{N}=\sin^{2}(\theta/2)d\varphi=\frac{1}{2}(1-\cos\theta)d\varphi.$
(33)

Similarly, we have:

$\displaystyle\mathcal{A}_{-}(\varphi,\theta)_{S}=-\cos^{2}(\theta/2)d\varphi=-\frac{1}{2}(1+\cos\theta)d\varphi.$
(34)

We see that the connection is not uniquely defined, but it is associated with
each vector field under consideration. Since the vector fields are not
globally defined, the connection in each covering of the base manifold also
takes on its own form. However, in the overlap region $U_{N}\cap U_{S}$, the
local connections should be related to each other. It is easy to verify that
$\mathcal{A}_{-}(\varphi,\theta)N=\mathcal{A}_{-}(\varphi,\theta)_{S}+d\varphi$.
This relation does not depend on $d\chi$ but only on $d\varphi$, as it is
actually the consequence of the gauge $U(1)$ transformation.Nakahara (2003)
This relation therefore ensures the existence of a globally defined connection
one-form on the whole fiber bundle.

####  III.3.3 Topological characters of the set of state vectors

The topological features of the set of state vectors $\mathbb{E}$ of two-level
systems are mathematically translated into the global geometrical features of
the fiber bundle $(\hat{\pi}:\mathbb{E}\to\mathbb{B},\mathbb{F},U(1))$. The
analysis presented in the previous subsection diagnoses such features: the
problem of singularity of the vector fields on the base manifold
$\mathbb{B}\equiv\mathbb{S}^{2}$ of a fiber-bundle structure. This feature is
quantitatively characterized by an index that is defined through an integral
over the base manifold. Since the base manifold is a two-dimensional surface,
we need a differential 2-form that is determined from the 1-form
$\mathcal{A}_{-}$ as follows:Nakahara (2003)

$\displaystyle\mathcal{F}_{-}(\varphi,\theta)$
$\displaystyle=d\mathcal{A}_{-}(\varphi,\theta)_{N}=d\mathcal{A}_{-}(\varphi,\theta)_{S}$
$\displaystyle=\partial_{\theta}\sin^{2}(\theta/2)d\theta\wedge
d\varphi=\frac{1}{2}\sin\theta d\theta\wedge d\varphi,$ (35)

where “$\wedge$” denotes the wedge product of two basis one-forms $d\theta$
and $d\varphi$ such that $d\theta\wedge d\varphi=-d\varphi\wedge d\theta$. The
2-form $\mathcal{F}_{-}(\varphi,\theta)$ is an antisymmetric 2-rank tensor and
does not depend on $\chi$ due to the identity $d^{2}\chi=0$. Geometrically,
this tensor determines the local curvature of the total manifold
$\mathbb{E}\equiv\mathbb{S}^{3}$. Therefore, the total curvature of the fiber-
bundle $\mathbb{E}$ is obtained by integrating the local curvature over the
entire base manifold:

$\displaystyle\mathcal{C}_{-}$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{S}^{2}}\mathcal{F}^{-}=\frac{1}{2}\int_{0}^{2\pi}\int_{0}^{\pi}\frac{1}{2}\sin\theta
d\theta\wedge d\varphi$ $\displaystyle=\frac{1}{4\pi}\int_{0}^{\pi}\sin\theta
d\theta\int_{0}^{2\pi}d\varphi=-1.$ (36)

This non-zero integer value of the integral characterizes the fact that the
vector field $|-,\mathbf{d}\rangle$ is not globally defined on the manifold
$\mathbb{S}^{2}$, but rather locally in each covering. The difference in the
connection $\mathcal{A}-$ on each covering directly results in the non-zero
value of the integral. Geometrically, we can understand this value as follows:
by moving around the whole manifold $\mathbb{S}^{2}$ of the parameter vector
$\mathbf{d}$ along the positive direction, the maps $|-\rangle_{N/S}$ allow us
to move around the whole sphere $\mathbb{E}\equiv\mathbb{S}^{3}$ in one round,
but via the negative direction. Thus, $\mathcal{C}_{-}$ (usually called the
Chern number) plays the role of the winding number characterizing the
topological features of the set of state vectors $\mathbb{E}$ of all two-level
systems described by the model (1).

##  IV Su-Shrieffer-Heeger model for one-dimensional lattices

In this section, we illustrate the analysis of deforming the band structure of
an atomic chain. We also demonstrate that the application of the fiber bundle
structure can be flexible and somewhat complex. Specifically, we will show
that the set of electronic state vectors of the atomic lattice does not always
possess the fiber bundle structure directly on the Brillouin zone.

###  IV.1 Physical model

Consider a one-dimensional lattice defined by two parameters $v$ and $w$,
which represent the hopping energies between the two nearest lattice nodes.
The lattice is periodic, with a unit cell containing two lattice nodes labeled
as $A$ and $B$. Let $a$ be the length of the unit cell. Assume that each
lattice node has only one electron orbital, denoted as $|\alpha,n\rangle$ for
node $\alpha$ ($\alpha=A,B$) in cell $n$, as shown in Fig. 1. The Hamiltonian
of an electron in the lattice is given by the following tight-binding model:

$\displaystyle H$ $\displaystyle=\sum_{n}|A,n\rangle\left(v\langle
B,n|+w\langle B,n+1|\right)$ $\displaystyle+\sum_{n}|B,n\rangle\left(v\langle
A,n|+w\langle A,n-1|\right).$ (37)

Using the Bloch theorem we can construct the so-called Bloch state vectors
$|\alpha,k\rangle$ for each value of $k$ in the Brillouin zone
$BZ=\\{k\in\mathbb{R}\,|\,-\pi/a\leq k\leq\pi/a\\}=[-\pi/a,\pi/a]$:

$\displaystyle|\alpha,k\rangle=\frac{1}{\sqrt{N}}\sum_{n}e^{ikan}|\alpha,n\rangle,$
(38)

Conversely,

$\displaystyle|\alpha,n\rangle=\frac{1}{\sqrt{N}}\sum_{k}^{BZ}e^{-ikan}|\alpha,k\rangle,$
(39)

Substitute Eq. (39) into the tight-binding Hamiltonian we get:

$\displaystyle H=\sum_{k}^{BZ}\sum_{\alpha,\beta}^{\\{A,B\\}}|\alpha,k\rangle
H_{\alpha\beta}(k)\langle\beta,k|,$ (40)

where $H_{\alpha\beta}(k)$ are the elements of the so-called Bloch-Hamiltonian
matrix that takes the following form:

$\displaystyle H(k)=\mathbf{d}(k)\cdot\boldsymbol{\sigma}.$ (41)

Here the dependence on $k$ is through a 2D vector $\mathbf{d}$:

$\displaystyle\mathbf{d}(k)=\left\\{\begin{array}[]{l}d_{x}(k)=v+w\cos(ka)\\\
d_{y}(k)=w\sin(ka)\\\ \end{array}\right.$ (44)

The Bloch-Hamiltonian is defined by three parameters $v,w$ and $ka$. While the
latter is real and given in the Brillouin zone
$BZ=[-\pi,\pi]\simeq\mathbb{S}^{1}$, the two former parameters can take
complex-values. For simplicity, we assume here that both $v,w$ are positive
real parameters. We need to distinguish the role of these parameters: $v$ and
$w$ define the physical system, and $k$ defines the state space of the
physical system; which is why we denote explicitly the dependence of the
Hamiltonian and the vector $\mathbf{d}$ on $k$.

Figure 1:  (a) The figure shows a one-dimensional lattice with two sub-
lattices labeled $A$ and $B$, with lattice constant $a$. Each lattice node has
one electronic orbital, and the kinetic energy of the electron is
characterized by two hopping parameters, $v$ and $w$, denoted in the figure.
The analysis presented in the text is consistent with the chosen unit cell.
(b) and (c) The images of the Brillouin zone $BZ=[-\pi,\pi]$ through the map
given by Eq. (44) in the cases where $w/v>1$ and $w/v<1$, respectively. In the
former case, the polar angle $\varphi$ can take on all values in the range
$[-\pi,\pi]$, while in the latter case it takes on values in the range
$[-\varphi_{0},\varphi_{0}]$, where $\varphi_{0}=\arcsin(w/v)$.

###  IV.2 Determination of eigen-states

The eigenvalues of the Hamiltonian can be obtained through the diagonalization
procedure. We can express the two eigenvalues $E_{\pm}(k)$ as follows:

$\displaystyle
E_{\pm}(k)=\pm\sqrt{d_{x}^{2}(k)+d_{y}^{2}(k)}=\pm\|\mathbf{d}(k)\|=\pm d(k),$
(45)

where $d(k)=\sqrt{v^{2}+w^{2}+2vw\cos(kb)}$. Knowing the eigen-values, the
corresponding eigen-vectors are determined by the equation
$(H-E)|\psi\rangle=0$, which is specified as:

$\displaystyle\begin{pmatrix}\pm d(k)&d_{x}(k)-id_{y}(k)\\\
d_{x}(k)+id_{y}(k)&\pm d(k)\end{pmatrix}\begin{pmatrix}\phi_{1}(k)\\\
\phi_{2}(k)\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$ (46)

This leads to the equation:

$\pm d(k)\phi_{1}(k)+[d_{x}(k)-id_{y}(k)]\phi_{2}(k)=0.$ (47)

To solve this equation, we need to consider two cases: (1) If $d(k)=0$, it
leads to $d_{x}(k)=d_{y}(k)=0$. So, $\phi_{1}(k)$ and $\phi_{2}(k)$ can be
arbitrarily chosen, and the state vectors are therefore ill-defined. Notice
that in Sec. II, we did not consider the case of $d=0$ because the
$\mathbf{d}$ vector determines the existence conditions of a particular
physical system directly. Here, the two parameters $v$ and $w$ play this role,
while the parameter $k$ determines the states of a particular 1D lattice.
Therefore, $d(k)=0$ can occur for a given physical system (i.e., with a given
set of $v$ and $w$) for a value of $k$ in the Brillouin zone. With $d_{x}(k)$
and $d_{y}(k)$ given by Eq. (44), the case $d(k)=0$ implies that:

$\displaystyle\left\\{\begin{array}[]{l}v+w\cos(ka)=0\\\
w\sin(ka)=0\end{array}\right.\to\left\\{\begin{array}[]{l}v=w\\\
ka=\pm\pi\end{array}\right.$ (52)

It shows that $d(k)=0$ occurs only for the configuration with $v=w$ at
$ka=\pm\pi$, i.e., at the edges of the Brillouin zone. (2) If $d(k)\neq 0$, we
obtain the following expressions for the state vectors:

$\displaystyle|\pm,k\rangle=\frac{e^{i\chi}}{\sqrt{2}}\begin{pmatrix}\pm\frac{d_{x}(k)-id_{y}(k)}{d(k)}\\\
1\end{pmatrix},$ (53)

where $\chi$ is an arbitrary real parameter defining the gauge $U(1)$ factor.
Since $\mathbf{d}(k)\in\mathbb{S}^{1}$ (a circle of radius $w$ embedded in the
plane $\mathbb{R}^{2}$ at the point $(v,0)$ as the center), it can be
parameterized by only one real parameter. There are many ways to parameterize
the circle $\mathbb{S}^{1}$, but, as we will see, the use of polar angle
coordinates $(d,\varphi)$ is more useful. Accordingly,
$d_{x}=d\cos\varphi,d_{y}=d\sin\varphi$. From Eq. (44) we deduce the equation
expressing the constraint of $d$ and $\varphi$ (see Fig. 1):

$\displaystyle(d\cos\varphi-v)^{2}+w^{2}\sin^{2}\varphi=w^{2}$
$\displaystyle\to$ $\displaystyle d^{2}-2dv\cos\varphi+(v^{2}-w^{2})=0.$ (54)

Using this parameterization, we can rewrite the state vector
$|\pm,\mathbf{d}(k)\rangle$ as:

$\displaystyle|\pm,k\rangle=\frac{e^{i\chi}}{\sqrt{2}}\begin{pmatrix}\pm
e^{-i\varphi(k)}\\\ 1\end{pmatrix}.$ (55)

This result shows that the state vectors do not depend on the radial
coordinate $d$, but only on the polar angle coordinate $\varphi$, which is
determined as a function of $k\in BZ$, see Eq. (61).

Normally, with the solution for the eigen-states, observable quantities that
determine physical properties of a system can be calculated. However, as
mentioned in the previous section, we are interested in what happens when we
drive the physical system from one state to another, i.e., vary the state
parameter $k$ in the Brillouin zone. The answer to this question is presented
in the next subsection.

###  IV.3 Investigation of topological features of quantum states

####  IV.3.1 The fiber-bundle structure of the set of state vectors

In Subsection II.2, we consider the set of state vectors for all possible two-
level systems. In contrast, in this subsection, we are interested in the
features of the set of state vectors for a generic 1D SSH system. We will then
compare the features of such sets to classify 1D systems described by the SSH
model into different classes.

Let’s denote

$\displaystyle\mathbb{F}(k)=\left\\{|-,k\rangle=\frac{e^{i\chi}}{\sqrt{2}}\begin{pmatrix}-\frac{d_{x}(k)-id_{y}(k)}{d(k)}\\\
1\end{pmatrix}\,|\,\forall\,\chi\in\mathbb{R}\right\\},$ (56)

The set of state vectors describes the same eigen-state with the energy
$E(k)=-d(k)$, as shown in Eq. (45). It is important to note that the set
$\mathbb{F}(k)$ is not defined for $k$ such that $d(k)=0$, as stated in Eq.
(53). This condition is satisfied only for configurations with $v=w$ at
$k=\pm\pi/a$, as shown in Eq. (52).

For configurations where $v\neq w$, the sets $\mathbb{F}(k)$ are well-defined
for all $k\in BZ$. Therefore, we can consider the set

$\displaystyle\mathbb{E}=\bigcup_{k\in\mathbb{B}}\mathbb{F}(k),$ (57)

where $\mathbb{B}=BZ\simeq\mathbb{S}^{1}\subset\mathbb{R}^{2}$. Since
$\mathbb{F}(k)$ is the set of all physically equivalent state vectors,
$\mathbb{E}$ mathematically represents the set of equivalent classes. To
decompose $\mathbb{E}$ according to $\mathbb{B}$, we need to determine a
surjective map $\hat{\pi}:\mathbb{E}\to\mathbb{B}$. To do so, we take an
element $|-\rangle=(\phi_{1},\phi_{2})^{T}$ in $\mathbb{E}$ and identify it
with one element in a subset $\mathbb{F}(k)$:

$\displaystyle\left\\{\begin{array}[]{l}\phi_{1}=-\frac{e^{i\chi}}{\sqrt{2}}\frac{d_{x}(k)-id_{y}(k)}{d(k)}\\\
\phi_{2}=\frac{e^{i\chi}}{\sqrt{2}}\end{array}\right.$ (60)

It results in the equation for $kb$:

$\displaystyle\sin(ka+\varphi)=-\frac{v}{w}\sin(\varphi),$ (61)

where $\varphi=\arg(\phi_{1}/\phi_{2})\in[-\pi,\pi]$. It is clear that this
equation does not always have a solution for $kb$ because of the factor $v/w$.
Specifically, there are two cases to consider:

  1. 1.

$v/w<1$: Eq. (61) always has a solution for $ka$ for all
$\varphi\in[-\pi,\pi]$. It means that when taking an arbitrary element of
$\mathbb{E}$, we can always identify it as belonging to a subset
$\mathbb{F}(k)$. This identification procedure defines the map
$\hat{\pi}:\mathbb{E}\to\mathbb{B}$, which is given by the rule:
$ka=-\varphi-\arcsin(v/w\sin(\varphi))$.

  2. 2.

$v/w>1$: Eq. (61) only has a solution for elements $(\phi_{1},\phi_{2})$ such
that $|\sin(\varphi)|\leq w/v<1$. In other words, the map
$\hat{\pi}:\mathbb{E}\to\mathbb{B}$ is not entirely, or globally, defined in
$\mathbb{B}$.

Therefore, based on this rough analysis, 1D SSH systems can be classified into
three categories according to the values of the ratio $v/w$, i.e., (1)
$v/w=1$, (2) $v/w<1$, and (3) $v/w>1$. The first case is special because the
two energy bands $E_{-}(k)$ and $E_{+}(k)$ touch each other at the two edge
points of the Brillouin zone, and the energy value $E=0$ is degenerate.
Physically, the SSH configuration with $v=w$ behaves as a semi-metallic
system.

To quantify the classification of the second and third cases, we move away
from considering the set $\mathbb{E}$ given by Eq. (57) and instead consider:

$\displaystyle\mathbb{E}=\bigcup_{\varphi\in\mathbb{B}}\mathbb{F}(\varphi),$
(62)

where $\mathbb{B}$ is any open set in the interval $[-\pi,\pi]$ and

$\displaystyle\mathbb{F}(\varphi)=\left\\{|-,\varphi\rangle=\left.\frac{e^{i\chi}}{\sqrt{2}}\begin{pmatrix}-e^{-i\varphi}\\\
1\end{pmatrix}\,\right|\,\forall\chi\in\mathbb{R}\right\\}.$ (63)

To decompose the set $\mathbb{E}$ according to $\mathbb{B}$ we determine a map
$\hat{\pi}:\mathbb{E}\to\mathbb{B}$. It is easy to find:

$\displaystyle\hat{\pi}:(\phi_{1},\phi_{2})\mapsto\varphi=i\ln\left(-\frac{\phi_{1}}{\phi_{2}}\right).$
(64)

This map is surjective since it maps all pairs of complex variables
$(\phi_{1},\phi_{2})$ and
$(\phi_{1}^{\prime},\phi_{2}^{\prime})=e^{i\chi}(\phi_{1},\phi_{2})$ for all
$\chi\in\mathbb{R}$ to the same value of $\varphi$. It means that, given a
value of $\varphi$, its preimage
$\hat{\pi}^{-1}(\varphi)=\mathbb{F}(\varphi)=\mathbb{S}^{1}$. The fiber bundle
structure of $\mathbb{E}$ is confirmed by a trivialization map
$\phi:\mathbb{B}\times\mathbb{S}^{1}\to\hat{\pi}^{-1}(\mathbb{B})$. This map
is easily defined by the assignment:

$\displaystyle\phi:(\varphi,e^{i\chi})\mapsto|-,\varphi\rangle=\frac{e^{i\chi}}{\sqrt{2}}\begin{pmatrix}-e^{-i\varphi}\\\
1\end{pmatrix}.$ (65)

It is a diffeomorphism because its inverse
$\phi^{-1}:\hat{\pi}^{-1}(\mathbb{B})\to\mathbb{B}\times\mathbb{S}^{1}$
determined by the decomposition
$\phi^{-1}(|-,\varphi\rangle)=(\varphi,e^{i\chi})$ also differentiable. Note
that, in the problem under consideration, we do not need a local description
of the set $\mathbb{E}$ but a global one. We thus conclude that the set
$\mathbb{E}$ is a fiber bundle over the base manifold $\mathbb{B}$ with the
fiber $\mathbb{F}=\mathbb{S}^{1}\simeq U(1)$ according to the decomposition
map $\hat{\pi}$.

The issue now is to determine the set $\mathbb{B}$ as the image of the
Brillouin zone $BZ$ through some map $\varphi:BZ\to\mathbb{B}=\varphi(BZ)$.
From Eq. (44) we have:

$\displaystyle\varphi=\arg\left[v+w\cos(ka)+iw\sin(ka)\right].$ (66)

We see that depending on the value of $v/w$ the range of $\varphi$ is
determined as follows (see Fig. 2):

  1. 1.

$v/w>1$:
$\varphi\in[-\varphi_{0},\varphi_{0}]\equiv\varphi(BZ)\subset[-\pi,\pi]$,
where $\varphi_{0}=\arcsin(w/v)$.

  2. 2.

$v/w<1$: $\varphi\in[-\pi,\pi]\simeq\mathbb{S}^{1}$.

To summarize, we distinguish two different fiber bundle structures for the set
of state vectors $\mathbb{E}$. One has the base manifold
$\mathbb{B}=[-\pi,\pi]$ (for $v/w<1$) homeomorphic to the circle
$\mathbb{S}^{1}$, and the other has the base manifold
$\mathbb{B}=[-\varphi_{0},\varphi_{0}]$ (for $v/w>1$) simply a part of the
interval $[-\pi,\pi]$.

Figure 2:  The variation of $\varphi$ with respect to $ka$ is shown. The blue
and red curves correspond to the cases $v/w=0.8<1$ and $v/w=1.25>1$,
respectively.

####  IV.3.2 The connection of the fiber bundle

After the determination of the fiber bundle structure of the set of state
vectors $\mathbb{E}$ we can proceed to analyze its geometrical features by
performing the “exploration travelling” in $\mathbb{E}$. Because of the
principal fiber bundle structure, the parallel motion in $\mathbb{E}$ is
realized by a horizontal lift of a parallel motion in the base manifold
$\mathbb{B}$ up to $\mathbb{E}$. So, we need to perform a motion in
$\mathbb{B}$ first. We consider separately two cases:

  1. 1.

$v/w>1$: When varying $k$ from $-\pi/a$ to $\pi/a$, $\varphi$ continuously
moves from $0$ to $-\varphi_{0}$ then back to $0$. After that it moves from
$0$ to $\varphi_{0}$ and then to $0$. The entire path that $\varphi$ draws is
$\varphi:0\to-\varphi_{0}\to 0\to\varphi_{0}\to 0$. This path can be seen as
consisting of two loops of opposite direction, one is
$\varphi:0\to\varphi_{0}\to 0$ with the positive direction, and the other is
$\varphi:0\to-\varphi_{0}\to 0$ with the negative direction.

  2. 2.

$v/w<1$: When varying $k$ from $-\pi/b$ to $\pi/a$, $\varphi$ continuously
moves from $-\pi$ to $\pi$, drawing a loop homotopic to the circle
$\mathbb{S}^{1}$.

In Fig. 2 we graphically illustrate the range of $\varphi$ and movement of
$\varphi$ with respect to $ka\in[-\pi,\pi]$ according to these two cases.

After choosing a vector field as a smooth map
$|-\rangle:\mathbb{B}\to\mathbb{E}$, given by
$|-\rangle:\varphi\mapsto|-,\varphi\rangle$, we can determine a connection in
order to horizontally lift the motion in $\mathbb{B}$ up to $\mathbb{E}$. The
so-called Berry connection is then given by
$\mathcal{A}-=i\langle-,\varphi|\hat{d}|-,\varphi\rangle$, where $\hat{d}$
denotes the exterior derivative. Using Eq. (44), we can write:

$\displaystyle\hat{d}|-,\varphi\rangle=\frac{e^{i\chi}}{\sqrt{2}}\begin{pmatrix}ie^{-i\varphi}\\\
0\end{pmatrix}d\varphi,$ (67)

which allows us to obtain the expression for the Berry connection:

$\mathcal{A}_{-}(\varphi)=\frac{1}{2}d\varphi.$ (68)

####  IV.3.3 Topological characters

Some topological features of the set of state vectors $\mathbb{E}$ are
characterized by the global geometrical features of the fiber-bundle
$(\hat{\pi}:\mathbb{E}\to\mathbb{B},\mathbb{F},U(1))$. Since the base manifold
$\mathbb{B}$ is one-dimensional, all the higher-rank differential forms
deduced from the one-form $\mathcal{A}_{-}$ vanish, e.g.,
$d\mathcal{A}_{-}=0$. A topological quantity can be defined as an integral of
$\mathcal{A}_{-}$ over the whole base manifold $\mathbb{B}$ to characterize
the geometrical properties of the manifold $\mathbb{E}$, as follows:

$\displaystyle\gamma=\int_{\mathbb{B}}\mathcal{A}_{-}=\left\\{\begin{array}[]{ll}0&\frac{v}{w}>1,\\\
\pi&\frac{v}{w}<1.\end{array}\right.$ (71)

This integral is physically named the Zak phase, which determines the phase
accumulated by the state vector when it completes its motion in the fiber
bundle $\mathbb{E}$ according to the motion of $k$ in the entire Brillouin
zone. The values of the Zak phase are the topological characteristic of the
three categories of 1D lattices. If we consider the energy band structure, we
realize only two phases: the semiconducting phase with a finite band gap and
the semimetallic phase with a zero-band gap. However, by looking deeply into
the set of state vectors, we realize that the semiconducting phase should be
classified further into two different categories, characterized by two
different values of the Zak phase.

####  IV.3.4 A view from symmetry as global constraints

Assuming that the two parameters $v$ and $w$ are real, the Hamiltonian of the
SSH atomic chain does not change under time-reversal operations. This is known
as time-reversal symmetry, and it implies that the eigenvalues of the Bloch-
Hamiltonian matrix $H(k)$ are even functions of $k$, as shown in Eq. (45). The
time-reversal symmetry plays this role specifically for the SSH model. If we
assume further that the hopping parameters $v$ and $w$ are equal, then the
Hamiltonian also possesses an inversion symmetry. Together with the chiral
symmetry of the Bloch-Hamiltonian matrix due to the nearest-neighbor
approximation for the hopping, we find a zero-energy state with a doubly
degenerate band structure. This means that the SSH atomic chain is a metal
with the conduction and valence bands touch each other at $k=0$. The inversion
symmetry is rigorously enforced for all possible configurations of the chain
as long as $v=w$. The touching point of two energy bands is a consequence of
this symmetry. It is therefore said to be protected by inversion symmetry.
Breaking this inversion symmetry simply requires deforming the atomic chain
such that $v\neq w$. In this case, the touching point of two energy bands is
broken as expected, an energy gap appears, and the system becomes an
insulator. With only two parameters, $v$ and $w$, there are two distinct
insulating phases by setting $v/w<1$ and $v/w>1$. The question is whether
these two insulating phases are equivalent or have any distinguishing
features. The answer, as shown in the previous sections, is that there is a
difference, and it is expressed in the structure of the set of state vectors
of the system. This difference is characterized by a topological invariant
quantity named the Zak phase.

##  V Conclusion

Topology is a fundamental concept that not only establishes the foundation for
mathematical theories but also enables physicists to analyze the structure of
the physical world, including space-time. However, the abstract nature and
high degree of generalization of topology often pose challenges when applying
topological concepts and related methods to analyze physical problems,
particularly in the domain of condensed matter physics where the practical
application is of high interest. Therefore, it is crucial to use mathematical
concepts and rigorous mathematical language accurately to state physical
problems when solving problems. We have carefully selected and presented a
concise list of essential mathematical concepts as the vocabulary to discuss
this topic. We have demonstrated that topological methods can be used to study
problems induced by driving a quantum system through a systematic procedure.
To illustrate this idea, we have analyzed two classical toy models.

The first model is general for two-level quantum systems, and the second model
is for one-dimensional crystalline lattices consisting of two sub-lattices.
The analysis procedure starts with determining the system’s states as the
objects under study. While the energies associated with quantum states are
realized as scalar fields defined on a manifold of the model parameters, the
state vectors are treated as points in a special manifold with a fiber bundle
structure. The set of state vectors is thus realized as vector fields defined
as the cross-sections of the fiber bundle. The transition of the system from
one state to another is thus translated as the connectivity of points in a
topological space through a mechanism of parallel transport. This movement in
a curved manifold is guided by a quantity called a connection. Mathematically,
the connection is defined as a one-form, which is a differential form of rank
one, that is a vector field of the dual space of the tangent bundle of the
manifold.

A topological index can be defined by integrating a differential form of
appropriate rank over the entire manifold of parameters to characterize the
geometric features of the fiber bundle. The two models considered exhibit
different levels of sophistication in characterizing the nature governing the
set of physical states. In the general two-level model, the topological
properties are evident in the singularity of the state vector fields defined
over the entire manifold of the model parameters. In contrast, for the SSH
model, greater sophistication is required to discern the presence of a
topological structure in a set of state vectors at an intermediate level of
analysis. Additionally, we show that the set of state vectors does not always
exhibit the fiber bundle structure directly over the Brillouin zone, contrary
to intuitive thinking. Therefore, semiconducting atomic chains can be
classified by the topological properties of two fiber bundles with the same
total and fiber spaces but different base spaces.

This brief article cannot encompass all aspects of topology and its
applications in physics, nor can it address the diverse range of physical
problems to which topology can be applied. However, we hope that it has
provided insight into the use of topological theories to characterize the
fundamental structure of material phases in the quantum realm.

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#  : Decoupling and Dynamic Queries for Oriented Object Detection with
Transformers

###### Abstract

Despite the promising results, existing oriented object detection methods
usually involve heuristically designed rules, e.g., RRoI generation, rotated
NMS. In this paper, we propose an end-to-end framework for oriented object
detection, which simplifies the model pipeline and obtains superior
performance. Our framework is based on DETR, with the box regression head
replaced with a points prediction head. The learning of points is more
flexible, and the distribution of points can reflect the angle and size of the
target rotated box. We further propose to decouple the query features into
classification and regression features, which significantly improves the model
precision. Aerial images usually contain thousands of instances. To better
balance model precision and efficiency, we propose a novel dynamic query
design, which reduces the number of object queries in stacked decoder layers
without sacrificing model performance. Finally, we rethink the label
assignment strategy of existing DETR-like detectors and propose an effective
label re-assignment strategy for improved performance. We name our method
D2Q-DETR. Experiments on the largest and challenging DOTA-v1.0 and DOTA-v1.5
datasets show that D2Q-DETR outperforms existing NMS-based and NMS-free
oriented object detection methods and achieves the new state-of-the-art.

Index Terms—  Rotated Object Detection, Transformer

Fig. 1: Overall architecture of the proposed D2Q-DETR for oriented object
detection which consists of four main parts: the points prediction head, query
feature decoupling, dynamic queries, and label re-assignment.

##  1 Introduction

Oriented object detection has recently attracted increasing attention for its
important application across different scenarios, including aerial images,
scene text, and faces, etc. Especially in aerial images, many well-designed
oriented object detectors [1, 2, 3, 4, 5, 6, 7, 8] have been proposed and
reported promising results on large datasets [9, 10]. Compared with natural
images, objects in aerial images usually exhibit dense distribution, large
aspect ratios, and arbitrary orientations. These characteristics make the
model pipeline of the existing oriented object detectors complicated. Our work
here focuses on simplifying oriented object detection, eliminating the need
for complex hand-crafted components, including but not limited to rule-based
training target assignment, rotated RoI generation [1, 8], rotated non-maximum
suppression (NMS) and rotated RoI feature extractor [1, 7].

Our method is based on DETR [11], which is a recently proposed end-to-end
object detection framework using transformers. The most relevant work to us is
O2DETR [12], which adds additional angle regression to DETR for oriented
object detection. As pointed out in [3], the direct regression of angle will
cause two problems: one is boundary discontinuity caused by the periodicity of
angle, and the other is the unit mismatch between box angle (in radians) and
size (in pixels). Unlike O2DETR, we propose predicting representative points
for each rotated box. The learning of points is more flexible, and the
distribution of points can reflect the angle and size of the target rotated
box. To our knowledge, we are the first to combine representative points
prediction with transformers and achieve superior performance for oriented
object detection.

It is common for aerial images to contain thousands of instances, which is
different from natural images. For example, in the DOTA-v1.0 dataset [9], the
number of object instances per image can reach 2000. In order to improve model
precision, we need to increase the number of queries, which will inevitably
increase the computational complexity and memory usage of the model. To better
balance model precision and efficiency, in this work, we propose a novel
dynamic query design, which gradually reduces the number of object queries in
the stacked decoder layers without sacrificing model performance.

Many object detection works prior to DETR have shown that the decoupling of
classification and regression features is beneficial in improving performance.
For example, Double Head [13] proposes a double-head structure. However,
current DETR-like detectors do not consider the feature decoupling of
classification and regression tasks, especially in the design of the decoder
layer. In this work, we propose to decouple the query features into
classification and regression features at the decoder layer. To our knowledge,
we are the first to consider query feature decoupling at the decoder layer for
DETR-like detectors.

Furthermore, we rethink the label assignment of existing DETR-like detectors.
DETR-like detectors use bipartite matching between object queries and target
boxes for label assignment and do not further impose any restrictions on the
queries that match the target boxes. In contrast, we propose a label re-
assignment strategy, which is used after bipartite matching to filter out low
quality queries and effectively improves the model performance.

All the above modules (i.e., points prediction head, query feature decoupling,
dynamic queries, and label re-assignment) together constitute our end-to-end
framework for oriented object detection, named D2Q-DETR. We achieve the new
state-of-the-art on challenging DOTA-v1.0 and DOTA-v1.5 datasets compared with
existing NMS-based and NMS-free oriented object detection methods.

##  2 Proposed Framework

In this section, we illustrate our proposed D2Q-DETR for oriented object
detection.

###  2.1 Points Prediction Head

As we stated in the introduction, to avoid problems raised by angle
regression, our D2Q-DETR adopts an angle-free approach. Specifically, we
predict a set of representative points
$\hat{\mathcal{C}}_{i}=\\{(\hat{x}_{i}^{k},\hat{y}_{i}^{k}),~{}k=1,\cdots,K\\}$
for each query, as shown in Fig. 1. $K$ denotes the number of predicted points
for each query and is set to 9 by default. The learning of points is more
flexible, and the distribution of points can reflect the angle and size of the
target box. During inference, for each query, we convert the predicted point
set $\hat{\mathcal{C}}_{i}$ to a rotated box with minAreaRect function
provided in OpenCV [14].

###  2.2 Query Feature Decoupling

This section illustrates the proposed query feature decoupling, where the
query features in the decoder layer are decoupled into two sets of class
queries and box queries after the self-attention module, as shown in Fig. 1.
Class queries and box queries are responsible for object class prediction and
points regression, respectively. When stacking multiple decoder layers, the
input query feature of each decoder layer is obtained by fusing the class-
query feature and box-query feature of the previous decoder layer with
element-wise addition. In this way, the self-attention module also plays a
role in exchanging information between class-query and box-query features.

###  2.3 Dynamic Queries

DETR-like detectors stack multiple decoder layers, and each decoder layer uses
the same number of object queries $N$. The increase of $N$ will inevitably
increase computational complexity and memory usage. To better balance model
performance and efficiency, we propose to dynamically reduce the number of
queries in decoder layers. The core idea is that we set the number of object
queries of the first decoder layer to an initial value of $N_{\text{first}}$,
while dynamically reducing the number of queries in subsequent decoder layers:

$N_{i}=(N_{\text{first}}-N_{\text{last}})*\rho^{i}+N_{\text{last}},$ (1)

where $N_{\text{first}}$ denotes the number of object queries in the first
decoder layer, $N_{\text{last}}$ ($<N_{\text{first}}$) denotes the number of
object queries in the last decoder layer, $N_{i}$ denotes the number of object
queries in the $i$-th decoder layer. $\rho$ ($<1$) is a hyper-parameter. The
prediction of class probability directly implies whether the object query
corresponds to an existing object or background. Hence, we select the top
$N_{i}$ object queries with larger class probability predictions in the
$(i-1)$-th decoder layer.

###  2.4 Query Label Re-Assignment

Fig. 2: Cumulative distribution function plot of convex-hull IoU in different
decoder layers.

To investigate the quality of queries after bipartite matching, we use a
D2Q-DETR model after 50 epochs of training to statistic the convex-hull IoU
between queries and matched target boxes on DOTA-v1.0 validation set. As shown
in Fig. 2, in the last decoder layer, about 10% queries and their matched
target boxes have convex-hull IoU below 0.5. And the ratio increases to about
25% in the first decoder layer. Obviously, the model performance and
convergence speed will be affected by low-quality queries. To alleviate the
negative effect of low-quality queries, we propose query label re-assignment
after bipartite matching. As shown in Eq. 2, after bipartite matching, we
check the convex-hull IoU between query predictions $\hat{\mathcal{C}}_{i}$
and matched target boxes $\mathcal{B}_{\hat{\sigma}(i)}$, and set the assigned
label $y_{\hat{\sigma}(i)}$ of queries whose convex-hull IoU are lower than
threshold $\tau$ to $\emptyset$ (no object or background).

$y_{\hat{\sigma}(i)}^{*}=\begin{cases}y_{\hat{\sigma}(i)},&\text{ if
}~{}{\frac{|\Gamma(\hat{\mathcal{C}}_{i})\cap\Gamma(\mathcal{B}_{\hat{\sigma}(i)})|}{|\Gamma(\hat{\mathcal{C}}_{i})\cup\Gamma(\mathcal{B}_{\hat{\sigma}(i)})|}}>\tau,\\\
\emptyset,&\text{ otherwise. }\end{cases}$ (2)

Where the hyper-parameter $\tau$ is set to 0.5 if not specified. As stated in
Eq. 3, our label re-assignment only affects the classification loss
$\mathcal{L}_{\text{cls}}$ with re-assigned class labels
$y_{\hat{\sigma}(i)}^{*}$ .

###  2.5 Training Loss

In this section, we describe the training loss used in D2Q-DETR. As shown in
Eq. 3, the total loss $\mathcal{L}$ includes one classification loss
$\mathcal{L}_{\text{cls}}$ and two regression losses $\mathcal{L}_{L1}$ and
$\mathcal{L}_{\text{iou}}$:

$\begin{split}\mathcal{L}=\frac{1}{N_{\text{pos}}}&\sum_{i=1}^{N}\Big{[}~{}\lambda_{\text{cls}}~{}\mathcal{L}_{\text{cls}}\big{(}\hat{\mathbf{p}}_{i},y_{\hat{\sigma}(i)}^{*}\big{)}\\\
&+\mathbbm{1}_{y_{\hat{\sigma}(i)}\neq\emptyset}~{}\lambda_{L1}~{}\mathcal{L}_{L1}\big{(}\Lambda(\hat{\mathcal{C}}_{i}),\Lambda(\mathcal{B}_{\hat{\sigma}(i)})\big{)}\\\
&+\mathbbm{1}_{y_{\hat{\sigma}(i)}\neq\emptyset}~{}\lambda_{\text{iou}}~{}\mathcal{L}_{\text{iou}}\big{(}\Gamma(\hat{\mathcal{C}}_{i}),\Gamma(\mathcal{B}_{\hat{\sigma}(i)})\big{)}~{}\Big{]}.\end{split}$
(3)

where $\hat{\sigma}$ is the optimal one-to-one matching between the queries
and the ground-truths without duplicates. $N$ denotes the total number of
queries, $N_{\text{pos}}$ denotes the number of queries matched with ground-
truths.

Fig. 3: Schematic diagram of regression losses (including $\mathcal{L}_{L1}$
and $\mathcal{L}_{\text{iou}}$) for predicted points.

When calculating the classification loss $\mathcal{L}_{\text{cls}}$, we use
the Focal Loss [15]. When calculating the regression losses (including
$\mathcal{L}_{L1}$ and $\mathcal{L}_{\text{iou}}$), we make some changes since
now we predict a point set for each query. Concretely, we first use
$\mathcal{L}_{L1}$ to narrow the distance between the two point sets of
$\hat{\mathcal{C}}_{i}$ and $\mathcal{B}_{\hat{\sigma}(i)}$. As shown in Fig.
3, $\mathcal{L}_{L1}$ represents the $L$1 loss between the center points of
the predicted point set $\hat{\mathcal{C}}_{i}$ and the target point set
$\mathcal{B}_{\hat{\sigma}(i)}$. The defined $\mathcal{L}_{L1}$ loss helps the
convergence of the learned point set, however, it can not drive the predicted
point set $\hat{\mathcal{C}}_{i}$ to align well with the target bounding
polygon $\mathcal{B}_{\hat{\sigma}(i)}$. For better alignment of point sets,
we further use the Convex-hull GIoU loss [2] to compute
$\mathcal{L}_{\text{iou}}$, which takes as input two convex hulls and can be
formulated as:

$\begin{split}\mathcal{L}_{\text{iou}}\big{(}\Gamma(\hat{\mathcal{C}}_{i}),\Gamma(\mathcal{B}_{\hat{\sigma}(i)})\big{)}&=1-\frac{\big{|}\Gamma(\hat{\mathcal{C}}_{i})\cap\Gamma(\mathcal{B}_{\hat{\sigma}(i)})\big{|}}{\big{|}\Gamma(\hat{\mathcal{C}}_{i})\cup\Gamma(\mathcal{B}_{\hat{\sigma}(i)})\big{|}}\\\
&+\frac{\big{|}\mathcal{R}_{i}\setminus\big{(}\Gamma(\hat{\mathcal{C}}_{i})\cup\Gamma(\mathcal{B}_{\hat{\sigma}(i)})\big{)}\big{|}}{\big{|}\mathcal{R}_{i}\big{|}},\end{split}$
(4)

where $\Gamma(\cdot)$ represents the Jarvis March algorithm [16] to convert a
point set to a minimal convex-hull. $\mathcal{R}_{i}$ is the minimum bounding
polygon of convex-hulls $\Gamma(\hat{\mathcal{C}}_{i})$ and
$\Gamma(\mathcal{B}_{\hat{\sigma}(i)})$, as shown in Fig. 3.

Fig. 4: Examples of detection results on the DOTA-v1.5 test set.

##  3 Experiments

###  3.1 Dataset

DOTA [9] is the largest dataset for oriented object detection in aerial images
with two commonly used versions: DOTA-v1.0 and DOTA-v1.5. DOTA-v1.0 contains
2806 large aerial images with the size ranges from $800\times 800$ to
$4000\times 4000$ and 188, 282 instances among 15 common categories. DOTA-v1.5
is released for DOAI Challenge 2019 with a new category and more extremely
small instances (less than 10 pixels). DOTA-v1.5 contains 402, 089 instances.
Compared with DOTA-v1.0, DOTA-v1.5 is more challenging but stable during
training.

###  3.2 Implementation Details

We use 8 V100 GPUs with a total batch size of 8 for training. AdamW [17]
optimizer is adopted with the base learning rate of $1.5\times 10^{-4}$,
$\beta_{1}=0.9$, $\beta_{2}=0.999$, and weight decay of 0.05. Like DETR, we
use six encoder layers, six decoder layers, and multi-scale features with
downsampling rates of 8, 16, 32, and 64. In the inference phase, we use
OpenCV’s minAreaRect function to convert the predicted points of each object
query into a rotated box. No other post-processing operations, e.g., rotated
NMS, are required.

###  3.3 Comparison with the State-of-the-art

Method | Backbone | mAP (%)
---|---|---
NMS-based method |  |
RoI Trans.∗ [1] | R101-FPN | 69.56
O2-DNet∗ [18] | H104 | 72.80
DRN∗ [19] | H104 | 73.23
Gliding Vertex∗ [20] | R101-FPN | 75.02
R3DET∗ [6] | R-152 | 76.47
SCRDet++∗ [5] | R152-FPN | 76.56
CFA∗ [2] | R-152 | 76.67
R3DET-DCL∗ [3] | R-152 | 77.37
HSP∗ [21] | R-101 | 78.01
ReDet∗ [7] | ReR50-ReFPN | 80.10
NMS-free method |  |
O2DETR∗♯ [12] | R50-FPN | 72.15
D2Q-DETR∗ (Ours) | R50-FPN | 76.58
D2Q-DETR∗ (Ours) | Swin-L-FPN | 79.60
D2Q-DETR†∗ (Ours) | R50-FPN | 79.40
D2Q-DETR†∗ (Ours) | Swin-L-FPN | 81.24

Table 1: Performance comparisons on DOTA-v1.0 test set. ∗ denotes multi-scale
training and testing. † indicates that we finetune an additional RCNN head to
compare with two-stage methods like ReDet [7].

Method | Backbone | mAP (%)
---|---|---
NMS-based method |  |
RetinaNet-O [15] | R50-FPN | 59.16
FR-O [9] | R50-FPN | 62.00
Mask R-CNN [22] | R50-FPN | 62.67
HTC [23] | R50-FPN | 63.40
OWSR∗ [24] | R101-FPN | 74.90
ReDet∗ [7] | ReR50-ReFPN | 76.80
NMS-free method |  |
D2Q-DETR∗ (Ours) | Swin-L-FPN | 75.23
D2Q-DETR†∗ (Ours) | Swin-L-FPN | 78.85

Table 2: Performance comparisons on DOTA-v1.5 test set.

Results on DOTA-v1.0. As shown in Table 1, we compare our D2Q-DETR with
existing methods on DOTA-v1.0 OBB Task and achieve the new state-of-the-art
performance of 81.24 mAP on the test set.

Results on DOTA-v1.5. Compared with DOTA-v1.0, DOTA-v1.5 contains many
extremely small instances, which increases the difficulty of object detection.
We report OBB results on DOTA-v1.5 test set in Table 2. Compared to the
previous best result of ReDet [7], our model achieves state-of-the-art
performance of 78.85 mAP. Fig. 4 shows some detection results on the DOTA-v1.5
dataset.

###  3.4 Ablations

Points Head | Decoupling-Q | LR | Dynamic-Q | mAP (%) | GFLOPs
---|---|---|---|---|---
|  |  |  | 70.72 | 194.52
$\checkmark$ |  |  |  | 71.44 ($\uparrow$) | 194.60
$\checkmark$ | $\checkmark$ |  |  | 73.25 ($\uparrow$) | 211.44
$\checkmark$ | $\checkmark$ | $\checkmark$ |  | 74.66 ($\uparrow$) | 211.44
$\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 74.57 | 197.63 ($\downarrow$)

Table 3:  Effectiveness of proposed components including points prediction for
oriented box (Points Head), decoupling query features (Decoupling-Q), dynamic
query design (Dynamic-Q) and query label re-assignment (LR).

We conduct ablation experiments on the DOTA-v1.0 dataset, with the training
set for training and the validation set for testing. All models are trained
for 50 epochs and the learning rate is decayed at the 40-th epoch by a factor
of 0.1. As shown in Table 3, each component we proposed positively impacts the
model performance, and combining all these components achieves the best
performance.

##  4 Conclusion

We have proposed an end-to-end detector, named D2Q-DETR, for arbitrary-
oriented object detection. Experimental results on DOTA-v1.0 and DOTA-v1.5
datasets demonstrate the superiority of our D2Q-DETR framework against current
advanced NMS-based and NMS-free detectors.

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