# The geometry of the Hermitian matrix space and the Schrieffer–Wolff
transformation

Gergő Pintér  Department of Theoretical Physics, Institute of Physics,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111
Budapest, Hungary  György Frank  Department of Theoretical Physics, Institute
of Physics, Budapest University of Technology and Economics, Műegyetem rkp.
3., H-1111 Budapest, Hungary  Dániel Varjas  Department of Theoretical
Physics, Institute of Physics, Budapest University of Technology and
Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary  IFW Dresden and
Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtzstrasse 20, 01069
Dresden, Germany  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Strasse 38, 01187 Dresden, Germany  András Pályi  Department of
Theoretical Physics, Institute of Physics, Budapest University of Technology
and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary  HUN-REN–BME
Quantum Dynamics and Correlations Research Group, Műegyetem rkp. 3., H-1111
Budapest, Hungary

###### Abstract

In quantum mechanics, the Schrieffer–Wolff (SW) transformation (also called
quasi-degenerate perturbation theory) is known as an approximative method to
reduce the dimension of the Hamiltonian. We present a geometric interpretation
of the SW transformation: We prove that it induces a local coordinate chart in
the space of Hermitian matrices near a $k$-fold degeneracy submanifold.
Inspired by this result, we establish a ‘distance theorem’: we show that the
standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the
distance of this Hamiltonian from the corresponding $k$-fold degeneracy
submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter
perturbations of a degenerate Hamiltonian, and prove that the standard
deviation and the pairwise differences of the eigenvalues lead to the same
order of splitting of the energy eigenvalues, which in turn is the same as the
order of distancing from the degeneracy submanifold. As applications, we prove
the ‘protection’ of Weyl points using the transversality theorem, and infer
geometrical properties of certain degeneracy submanifolds based on results
from quantum error correction and topological order.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries, notations, and conventions
    1. 2.1 The space of the Hermitian matrices
    2. 2.2 Degeneracy of the eigenvalues
  3. 3 Summary of results
    1. 3.1 The Schrieffer–Wolff transformation induces a local chart
    2. 3.2 Energy splitting and the distance from $\Sigma_{k}$
    3. 3.3 Order of energy splitting of a degeneracy due to a perturbation
    4. 3.4 Parameter-dependent quantum systems and Weyl points
    5. 3.5 Examples
  4. 4 Proofs and details
    1. 4.1 The Schrieffer–Wolff transformation induces a local chart
    2. 4.2 Energy splitting and the distance from $\Sigma_{k}$
    3. 4.3 Order of energy splitting of a degeneracy due to a perturbation
    4. 4.4 Parameter-dependent quantum systems and Weyl points
  5. 5 Conclusions
  6. A Coordinate free description
  7. B Dependence on the base point $H_{0}$
  8. C Direct rotation
  9. D The range of validity of the SW decomposition

##  1 Introduction

Many physics problems are translated to matrix eigenvalue problems. For
example, the frequency spectrum and the spatial patterns (modes) of small
oscillations of a mechanical system are described by the eigenvalues and the
eigenvectors of the dynamical matrix, which is a real symmetric matrix.
Another example, which is the focus of this work, is quantum mechanics, where
the stationary states and the energies are given by the eigenvectors and
eigenvalues of the Hamiltonian, which is a Hermitian operator, often a finite-
dimensional matrix.

In certain cases, a physics problem translated to a matrix eigenvalue problem
is treated using perturbative methods. One of these is the Schrieffer-Wolff
(SW) transformation [1, 2, 3] — also known as quasidegenerate perturbation
theory [4, 5] Van Vleck perturbation theory [6], or Löwdin partitioning [7] —
which decouples the subspaces of the relevant and irrelevant energy
eigenstates through an appropriate unitary transformation.

In this work, motivated by many examples in quantum mechanics (see below), we
focus on the case, when the Hamiltonian ($n\times n$ Hermitian matrix) of
interest is in the vicinity of an _unperturbed Hamiltonian_ $H_{0}$ whose
energy spectrum hosts a $k$-fold degenerate eigenvalue, and we need to
determine only those eigenvalues and eigenstates of the perturbed Hamiltonian
$H$ that correspond to the degenerate subspace of $H_{0}$. In this case, the
Schrieffer-Wolff transformation takes the $n\times n$ Hermitian matrices
$H_{0}$ and $H$ as inputs, and outputs a $k\times k$ Hermitian matrix
$H_{\text{eff}}$, the _effective Hamiltonian_ , whose eigenvalues and
eigenvectors represent the relevant eigenvalues and eigenvectors of $H$. This
provides a practical, useful computational method: if $k$ is much smaller than
$n$, then computing the eigensystem of $H_{\text{eff}}$ may be much easier
than doing the same for $H$. Furthermore, the Schrieffer-Wolff transformation
is typically applied together with an approximate, truncated power expansion
of $H_{\text{eff}}$ in the perturbation parameter(s), i.e., $H_{\text{eff}}$
is approximated as a low-order polynomial.

In this work, we provide a geometrical interpretation of the Schrieffer-Wolff
transformation, relating the latter to the geometry of the submanifolds of
degenerate matrices. Our work is motivated by concrete examples in physics:
band-structure degeneracy points (including Weyl points) and degenerate ground
states of quantum spin models (including topologically ordered systems and
stabiliser code Hamiltonians). At the same time, we provide a purely
mathematical description of concepts and relations throughout the paper. The
geometrical view we develop here builds a new connection between differential
geometry and quantum mechanics, and hence enables the application of tools in
one domain to deepen the understanding of, or solve problems in, the other
domain.

Our first contribution is that we recognize that for any unperturbed
degenerate Hamiltonian $H_{0}$, the Schrieffer-Wolff transformation provides a
_canonical analytical local chart_ of the manifold of Hermitian matrices,
which is aligned with the corresponding degeneracy submanifold. Furthermore,
we identify the effective Hamiltonian as the collection of those coordinates
of this chart that lead out of the degeneracy submanifold.

Our second contribution is a ‘distance theorem’, which is a proportionality
relation between (i) the Frobenius (a.k.a. Hilbert-Schmidt) distance between a
generic Hamiltonian $H$ and a $k$-fold degeneracy submanifold, and (ii) the
standard deviation of the quasidegenerate eigenvalues of $H$ corresponding to
that degeneracy submanifold (sometimes simply called the ‘energy splitting of
the degeneracy’). More precisely, the standard deviation of the
quasidegenerate eigenvalues equals the distance divided by $\sqrt{k}$.

Third, we relate the first and second contributions: we show that the norm of
the effective Hamiltonian $H_{\text{eff}}$ obtained from the Schrieffer-Wolff
transformation is the same as the distance between $H$ and the degeneracy
submanifold.

Fourth, we consider the energy splitting of the degenerate energy eigenvalues
of $H_{0}$ due to a linear perturbation $H(t)=H_{0}+tH_{1}$. We define the
_order of energy splitting_ $r\in\\{1,2,\dots\\}\cup\\{\infty\\}$ for higher
degeneracies $k>2$ in multiple ways, and we prove that those definitions are
equivalent to each other. Moreover, as a consequence of our distance theorem,
the order of energy splitting coincides with the order at which the distance
of $H(t)$ from the degeneracy submanifold increases (_order of distancing_).
The order of energy splitting also generalises from linear to analytical
perturbations.

One application of our results is an alternative explanation of the
‘protection’of Weyl points, which are degeneracy points appearing, e.g., in
the electronic band structure of crystalline materials [8]. Making use of the
relations developed here, we show that the protection of Weyl points against
perturbations is analogous to the protection of the crossing point of two
lines drawn on a paper sheet, and we formalize this analogy by a common
underlying theorem, namely the transversality theorem.

We also demonstrate the applicability of our results at the intersection of
differential geometry, condensed matter physics, and quantum information
science. In the condensed-matter and quantum-information domains, quantum
systems with robust degeneracies are often desirable. In this particular
context, a robust degeneracy is such that physically relevant perturbations
break it with a high order $r$ of energy splitting; $r=1$ (_linear energy
splitting_) is not robust, $r=2$ (_quadratic energy splitting_) is already
robust, and the higher the $r$ the more robust the degeneracy.

Examples include the (1) the ground-state degeneracy of the quantum Ising
model at zero field, which is robust against a small tranverse field [9, 10,
11], (2) the zero-energy degeneracy of the edge modes of the fully dimerized
Su-Schrieffer-Heeger model [12, 13], which is robust against chiral-symmetric
local perturbations, (3) the zero-energy degeneracy of the Majorana edge modes
of the Kitaev chain [14], which is robust against local perturbations
respecting the particle-hole symmetry of the Bogoliubov-de Gennes formalism,
and (4) the ground-state degeneracy of the Toric Code [15], which is robust
against 1-local perturbations (Zeeman fields). In all four cases, the
considered perturbations cause an energy splitting whose order is proportional
to the linear size of the system. Further examples are (5) stabiliser code
Hamiltonians, which are Hamiltonians generalising (4) and corresponding to
quantum error correction codes [16]. Stabiliser code Hamiltonians have
degenerate ground states that are robust against 1-local perturbations,
exhibiting an order of energy splitting set by the code distance.

As we have described above (fourth contribution), an important finding of this
work is that the order of energy splitting of a degeneracy due to a linear
perturbation is the order of distancing of the perturbed Hamiltonian from the
degeneracy submanifold. As an application of this result, we show that the key
characteristics of a stabiliser code Hamiltonian, such as size, ground-state
degeneracy, and code distance, can be translated into geometric information
about the corresponding degeneracy submanifold in the space of Hamiltonians.

The rest of this paper is structured as follows. Sec. 2 summarizes relevant
preliminaries. Sec. 3 provides a summary of the key concepts and statements of
this work, including physics-related examples as well as applications. Sec. 4
contains the proofs and further details of the results, with subsections in
one-to-one correspondence with those of Sec. 3. The Appendix collects further
notes on the Schrieffer-Wolff transformation.

##  2 Preliminaries, notations, and conventions

Here we summarize some well known properties of the space of Hermitian
matrices. We aim to fix our notation and to provide the setting of our paper.

###  2.1 The space of the Hermitian matrices

The Hermitian matrices (of size $n\times n$) are the complex matrices $H$ with
$H=H^{\dagger}$. Their set $\mbox{Herm}(n)$ is a real vector space, it is a
subspace of $\mathbb{C}^{n\times n}$ of all $n\times n$ complex matrices.
Although $\mathbb{C}^{n\times n}$ is a complex vector space (of dimension
$n^{2}$), $\mbox{Herm}(n)$ inherits only real vector space structure, because
it is not closed under the multiplication by the imaginary unit $i$.

The dimension of $\mbox{Herm}(n)$ over $\mathbb{R}$ is $n^{2}$. We fix a basis
of $\mbox{Herm}(n)$, which we will refer to as ‘canonical basis’, which
enables to identify $\mbox{Herm}(n)$ with $\mathbb{R}^{n^{2}}$. This basis is
formed by three families of matrices, the real off-diagonal parts
($(n^{2}-n)/2$ matrices), imaginary parts ($(n^{2}-n)/2$ matrices) and the
diagonal parts ($n$ matrices):

$\sigma_{ab}^{(\textup{real})}=\frac{1}{\sqrt{2}}(e_{b}\cdot
e_{a}^{\dagger}+e_{a}\cdot
e_{b}^{\dagger})=\frac{1}{\sqrt{2}}\pNiceMatrix[last-row=6,last-
col=6]&\vdots\vdots\\\ \dots 0\dots 1\dots a\\\ \vdots\vdots\\\ \dots 1\dots
0\dots b\\\ \vdots\vdots\\\ ab\ \mbox{\hskip 28.45274pt(for $a<b$)},$ (2.1.1)
$\sigma_{ab}^{(\textup{im})}=\frac{i}{\sqrt{2}}(e_{b}\cdot
e_{a}^{\dagger}-e_{a}\cdot
e_{b}^{\dagger})=\frac{1}{\sqrt{2}}\pNiceMatrix[last-row=6,last-
col=6]&\vdots\vdots\\\ \dots 0\dots-i\dots a\\\ \vdots\vdots\\\ \dots i\dots
0\dots b\\\ \vdots\vdots\\\ ab\ \ \mbox{\hskip 28.45274pt(for $a<b$)},$
(2.1.2) $\sigma_{aa}^{(\textup{diag})}=e_{a}\cdot
e_{a}^{\dagger}=\pNiceMatrix[last-row=6,last-col=6]0&\\\ \ddots\\\ 1a\\\
\ddots\\\ 0\\\ a,$ (2.1.3)

where $e_{1},\dots,e_{n}$ is the standard basis of $\mathbb{C}^{n}$. For any
other unitary basis $u_{1},\dots,u_{n}$ of $\mathbb{C}^{n}$ (where
$u_{a}=Ue_{a}$ is the $a$-th column of a unitary matrix $U\in U(n)$), there is
an associated basis of $\mbox{Herm}(n)$ (over $\mathbb{R}$), obtained from
Equations (2.1.1), (2.1.2), (2.1.3) by replacing $e_{a}$ with $u_{a}$,
resulting $\sigma^{U}_{ab}=U\sigma_{ab}U^{\dagger}$.

For brevity, we denote the canonical basis of $\mathrm{Herm}(n)$ by
$C=(c_{1},\dots,c_{n^{2}})$, where $c_{a}$-s are the above defined $\sigma$
matrices in an appropriate order where the first $k^{2}$ elements generate the
upper left $k\times k$ block. For example,

$c_{1}=\sigma_{11}^{(\textup{diag})},\ c_{2}=\sigma_{12}^{(\textup{real})},\
c_{3}=\sigma_{12}^{(\textup{im})},\ c_{4}=\sigma_{22}^{(\textup{diag})},\
c_{5}=\sigma_{13}^{(\textup{real})},\ c_{6}=\sigma_{13}^{(\textup{im})},\
\dots$ (2.1.4)

Similarly, we denote the basis associated with a unitary matrix $U\in U(n)$ by
$C^{U}=(c^{U}_{1},\dots,c^{U}_{n^{2}})$, with $c^{U}_{a}=Uc_{a}U^{\dagger}$,
therefore, $c^{U}_{1},\dots,c^{U}_{k^{2}}$ span the subspace of
$\mbox{Herm}(n)$ consisting of the matrices acting on the subspace of
$\mathbb{C}^{n}$ spanned by $u_{1},\dots,u_{k}$.

For $k<n$, $\mbox{Herm}(k)$ is embedded in $\mbox{Herm}(n)$ as the subspace
that consists of the matrices having zero entries outside the upper left
$k\times k$ block, that is, the subspace spanned by $c_{1},\dots,c_{k^{2}}$.
We often identify $\mbox{Herm}(k)$ with this subspace of $\mbox{Herm}(n)$.

The complex vector space $\mathbb{C}^{n\times n}$ is endowed with a Hermitian
inner product (Frobenius inner product) defined as

$\langle M,N\rangle=\operatorname{{\rm tr}}(M^{\dagger}\cdot
N)=\sum_{a=1}^{n}\sum_{b=1}^{n}M^{\ast}_{ab}N_{ab},$ (2.1.5)

where $z^{\ast}$ denotes the complex conjugate of $z\in\mathbb{C}$. That is,
the inner product of two matrices agrees with the inner product of the vectors
formed by the entries.

Although, this inner product takes complex values, its restriction to
$\mbox{Herm}(n)$ is real valued because every summand on the right side of
Equation (2.1.5) has its complex conjugate too. Therefore, the Frobenius inner
product on $\mbox{Herm}(n)$ simplifies as $\langle
H,K\rangle=\operatorname{{\rm tr}}(H\cdot K)\in\mathbb{R}$, making
$\mbox{Herm}(n)$ a Euclidean space. The basis $C^{U}$ of $\mathrm{Herm}(n)$
associated to $U\in U(n)$ (in particular, the canonical basis $C$) is
orthonormal with respect to the Frobenius inner product. If the coordinates of
$H$ and $K$ in the basis $C^{U}$ are $h^{U}_{a}$ and $k^{U}_{b}$,
respectively, then

$\langle H,K\rangle=\sum_{a=1}^{n^{2}}h^{U}_{a}k^{U}_{a}.$ (2.1.6)

The induced Frobenius norm of a Hermitian matrix is
$\|H\|=\sqrt{\operatorname{{\rm tr}}(H^{2})}$. The distance of two matrices
$H,G\in\mbox{Herm}(n)$ induced by the Frobenius norm is denoted by
$d(H,G)=\|H-G\|$. This distance induces a topology on $\mbox{Herm}(n)$. An
open neighborhood, or simply a neighborhood of $H$ in $\mbox{Herm}(n)$ is an
open subset $A\subset\mbox{Herm}(n)$ which contains an open ball around $H$,
that is, there is a radius $0<r$ such that $d(H,G)<r$ implies that $G\in A$.

######  Lemma 2.1.7.

The scalar product is invariant under the conjugation by unitary matrices.
That is, $\langle UHU^{\dagger},UKU^{\dagger}\rangle=\langle H,K\rangle$ holds
for every $H,K\in\mbox{Herm}(n)$ and $U\in U(n)$.

###### Proof.

$\langle UHU^{\dagger},UKU^{\dagger}\rangle=\operatorname{{\rm
tr}}(UH^{\dagger}U^{\dagger}UKU^{\dagger})=\operatorname{{\rm
tr}}(U^{\dagger}UH^{\dagger}K)=\operatorname{{\rm tr}}(H^{\dagger}K)=\langle
H,K\rangle$. ∎

Let $\mbox{Herm}_{0}(n)\subset\mbox{Herm}(n)$ be the subspace consisting of
the matrices with zero trace. The dimension of $\mbox{Herm}_{0}(n)$ over
$\mathbb{R}$ is $n^{2}-1$. A basis $\breve{c}_{1},\dots,\breve{c}_{n^{2}-1}$
is formed by the $\sigma_{ij}^{\textup{(real)}}$ and
$\sigma_{ij}^{\textup{(im)}}$ matrices, extended with an orthonormal basis of
the diagonal matrices with zero trace. This latter basis of traceless diagonal
matrices replaces the matrices $\sigma_{ii}^{\textup{(diag)}}$, which are not
contained in $\mbox{Herm}_{0}(n)$. For example, for $n=2$ one can choose the
normalized Pauli matrices as a basis of $\mbox{Herm}_{0}(2)$:

$\displaystyle\frac{1}{\sqrt{2}}\sigma_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=\sigma_{12}^{\textup{(real)}},$ (2.1.8)
$\displaystyle\frac{1}{\sqrt{2}}\sigma_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}=\sigma_{12}^{\textup{(im)}},$ (2.1.9)
$\displaystyle\frac{1}{\sqrt{2}}\sigma_{z}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}=\frac{1}{\sqrt{2}}\left(\sigma_{11}^{\textup{(diag)}}-\sigma_{22}^{\textup{(diag)}}\right).$
(2.1.10)

Note that, in $\mbox{Herm}_{0}(n)$ with $n>2$, the differences
$\sigma_{a,a}^{\textup{(diag)}}-\sigma_{a+1,a+1}^{\textup{(diag)}}$ do form a
basis of the traceless diagonal matrices, however, they are not orthogonal to
each other. In order to obtain an orthonormal basis, one needs to perform the
Gram–Schmidt process. For example, for $n=3$ the normalized diagonal Gell–Mann
matrices read

$\displaystyle\frac{1}{\sqrt{2}}\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix}=\frac{1}{\sqrt{2}}\left(\sigma_{11}^{\textup{(diag)}}-\sigma_{22}^{\textup{(diag)}}\right),$
(2.1.11) $\displaystyle\frac{1}{\sqrt{2}}\lambda_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-2\end{pmatrix}=\frac{1}{\sqrt{6}}\left(\sigma_{11}^{\textup{(diag)}}+\sigma_{22}^{\textup{(diag)}}-2\sigma_{33}^{\textup{(diag)}}\right).$
(2.1.12)

(The other Gell-Mann matrices are the same as the off-diagonal matrices
defined in Equation (2.1.1) and (2.1.2), up to $\sqrt{2}$ factor.)

######  Remark 2.1.13.

We note that in quantum mechanics, the operator 2-norm $\|H\|_{2}$ is more
frequently used than the Frobenius norm, since it refers directly to the
2-norm of the state vectors $\Psi\in\mathbb{C}^{n}$. Indeed, it is defined as
$\|H\|_{2}=\mbox{sup}\\{\|H\cdot\Psi\|\ |\ \|\Psi\|=1\\}$, or equivalently
(for Hermitian matrices) $\|H\|_{2}=\max\\{|\lambda_{i}|\\}$, where
$\lambda_{i}$ are the eigenvalues of $H$. However, the Frobenius norm is more
suitable to study the geometry of $\mbox{Herm}(n)$: Importantly, the Frobenius
norm is induced by an inner product, which allows to study angles and
distances as well. In contrast, the operator 2-norm is not induced by an inner
product, as it does not satisfy the parallelogram law. By Lemma 2.1.7 the
Frobenius norm is $\|H\|=\sqrt{\sum_{i=1}^{n}\lambda_{i}^{2}}$, so we have the
relation $\|H\|_{2}\leq\|H\|\leq\sqrt{n}\cdot\|H\|_{2}$. This shows that the
operator 2-norm and the Frobenius norm on $\mbox{Herm}(n)$ are equivalent,
hence they induce the same topology. (Indeed, every norm on a finite
dimensional vector space is equivalent.)

In Ref. [3], the operator 2-norm is used, and we also use it to compare our
results to those of [3] in the Appendix.

###  2.2 Degeneracy of the eigenvalues

Hermitian matrices have real eigenvalues and they can be diagonalized by a
unitary basis transformation. Furthermore, for each $H\in\mbox{Herm}(n)$, one
can choose a unitary matrix $U\in U(n)$ such that

$\Lambda=U^{-1}\cdot H\cdot U$ (2.2.1)

is diagonal with the eigenvalues in increasing order
$\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{n}$. The ordered eigenvalues
are continuous functions $\lambda_{i}:\mbox{Herm}(n)\to\mathbb{R}$, as it can
be proven e.g. using Weyl’s inequality [17]. However, the diagonalizing matrix
$U$ is not unique; each column, in fact, can be independently multiplied by an
arbitrary phase factor. Moreover, in the case of degenerate eigenvalues
$\lambda_{i+1}=\dots=\lambda_{i+k}$, the corresponding columns of $U$ form a
unitary basis of the corresponding $k$-dimensional eigenspace. This basis can
be transformed by any unitary action of $U(k)$ to obtain a different matrix
$U^{\prime}$ that still diagonalizes $H$.

The _degeneracy set_ $\Sigma\subset\mbox{Herm}(n)$ is the set of matrices with
at least two coinciding eigenvalues. This degeneracy set $\Sigma$ is a
subvariety. Indeed, it can be defined by one polynomial equation, namely,
$\Sigma$ is the zero locus of the discriminant of the characteristic
polynomial. By the Neumann–Wigner theorem [18] the codimension of $\Sigma$ is
3. For convenience, we summarize the original proof in Table 1. Our work
provides an alternative proof, as a byproduct of the SW chart, see Corollary
3.1.6.

| Non-degenerate matrices |  $k$-fold ground state degenerate matrices
---|---|---
|  Diagonalization structure |  Number of parameters |  Diagonalization structure |  Number of parameters
$\Lambda$ | $\lambda_{1}<\dots<\lambda_{n}$ | $n$ | $\lambda_{1}=\dots=\lambda_{k}<\dots<\lambda_{n}$ | $n-k+1$
$U$ | $[U]\in\frac{\textstyle\text{U}(n)}{\underbrace{\text{U}(1)\times\dots\times\text{U}(1)}_{\textstyle n\text{ times}}}$ | $n^{2}-n$ | $[U]\in\frac{\textstyle\text{U}(n)}{\textstyle\text{U}(k)\times\underbrace{\text{U}(1)\times\dots\times\text{U}(1)}_{\textstyle n-k\text{ times}}}$ | $\begin{array}[]{c}n^{2}-k^{2}\\\ \;\;\;-(n-k)\end{array}$
$H$ |  | $n^{2}$ |  | $n^{2}-\left(k^{2}-1\right)$
Table 1: Summary of the proof of the Neumann–Wigner theorem for $k$-fold
(ground state) degeneracy. Consider a Hermitian matrix in form $H=U\Lambda
U^{-1}$, where $\Lambda$ is diagonal, containing the ordered eigenvalues, and
the columns of the unitary matrix $U\in U(n)$ are eigenvectors of $H$. The
dimension of $U(n)$ is $n^{2}$, since every unitary matrix sufficiently close
to the identity can be written as $e^{iK}$ with $K\in\mbox{Herm}(n)$ (in other
words, the Lie algebra of the Lie group $U(n)$ is the space
$\mathfrak{u}(n)=i\cdot\mbox{Herm}(n)$ of the anti-Hermitian matrices). If we
construct a non-degenerate $H$, every column of $U$ can be modified by a
$U(1)$ action providing the same matrix $H$. Hence the choice of $U$ has
$n^{2}-n$ free parameters, which together with the $n$ parameters of $\Lambda$
verifies that the dimension of the non-degenerate matrices (i.e., the generic
part of $\mbox{Herm}(n)$) is $n^{2}$. If we construct a $k$-fold degenerate
$H$ ($\lambda_{1}=\dots=\lambda_{k}<\lambda_{k+1}$), then the eigenvectors
$u_{1},\dots,u_{k}$ spanning the degenerate eigenspace can be modified by a
$U(k)$ action. Hence the choice of $U$ has $n^{2}-k^{2}-(n-k)$ free
parameters, which together with the $n-k+1$ parameters of $\Lambda$ verifies
that the dimension of the $k$-fold degeneracy is $n^{2}-k^{2}+1$, hence, its
codimension is $k^{2}-1$ independently of $n$. In particular, the codimension
of the two-fold degeneracy (i.e., the smooth part of $\Sigma$) is 3. The same
proof shows that the codimension of a general stratum $\Sigma_{\kappa}$ is
$\sum_{i=1}^{l}\left(k_{i}^{2}-1\right)$, where $\kappa=(k_{1},\dots,k_{l})$.

The set $\Sigma$ can be decomposed into the disjoint union of different
_strata_ based on the type of the degeneracy, i.e., which eigenvalues coincide
[19]. Each stratum can be labeled by an ordered partition of $n$ in the
following way. Let $\kappa=(k_{1},\dots,k_{l})$ be a sequence of integers
$1\leq k_{i}\leq n$ of length $1\leq l<n$ such that $n=\sum_{i=1}^{l}k_{i}$,
defining $\kappa$ as an ordered partition of $n$. The number of the ordered
partitions of $n$ is $2^{n-1}$, (including the case with $l=n$). The
associated stratum $\Sigma_{\kappa}$ of $\Sigma$ consists of the matrices with
coinciding eigenvalues
$\lambda_{1}=\dots=\lambda_{k_{1}}<\lambda_{k_{1}+1}=\dots=\lambda_{k_{1}+k_{2}}<\dots$.

The partition with $l=n$ and $k_{i}=1$ marks the complement of $\Sigma$, the
set of non-degenerate matrices. Each stratum $\Sigma_{\kappa}$ is a (not
closed) smooth submanifold of $\mbox{Herm}(n)$, whose dimension is
$\dim\Sigma_{\kappa}=n^{2}-\sum_{i=1}^{l}\left(k_{i}^{2}-1\right)$ by the
Neumann–Wigner theorem [18], cf. Table 1. The closure
$\mbox{cl}(\Sigma_{\kappa})$ contains the higher degeneracies. The smooth
points of $\Sigma$ are the strictly two-fold degenerate matrices, that is,
exactly 2 eigenvalues coincide and all the others are different. The set
$\Sigma$ is singular at all other points, see Figure 1. We refer to [19, 20]
for details.

Figure 1: The stratification of $\mbox{Herm}(4)$, the space of $4\times 4$
Hermitian matrices. (a) The strata are labeled with the ordered partitions of
4, whose number is $2^{3}=8$. The non-degenerate matrices form an open and
dense subset $\Sigma_{(1111)}$ in $\mbox{Herm}(4)$. Its complement is the
degeneracy set $\Sigma$ (grey), the set of matrices with at least two
coinciding eigenvalues. For example,
$\lambda_{1}=\lambda_{2}<\lambda_{3}<\lambda_{4}$ in $\Sigma_{(211)}$, and
$\lambda_{1}=\lambda_{2}<\lambda_{3}=\lambda_{4}$ in $\Sigma_{(22)}$. The two-
fold ground state degeneracy set $\Sigma_{2}$ (orange) consists of the
matrices with $\lambda_{1}=\lambda_{2}<\lambda_{3}$, that is,
$\Sigma_{2}=\Sigma_{(2,2)}\cup\Sigma_{(2,1,1)}$. (b) $\Sigma$ is a subvariety
of $\mbox{Herm}(4)$. Although, each stratum is a (non-closed) smooth
submanifold of $\mbox{Herm}(4)$, however, $\Sigma$ is not a smooth manifold.
The smooth points of $\Sigma$ are the exactly two-fold degenerate matrices in
$\Sigma_{(211)}\cup\Sigma_{(121)}\cup\Sigma_{(112)}$. The two-fold ground
state degeneracy set $\Sigma_{2}$ is a smooth submanifold in $\mbox{Herm}(4)$.
Its generic points belong to $\Sigma_{(211)}$. At the points of
$\Sigma_{(22)}$, $\Sigma_{2}$ is smooth, but $\Sigma$ is not smooth. Indeed,
$\Sigma_{(2,2)}$ is the (transverse) intersection of the smooth branches
$\Sigma_{2}$ and $\Sigma_{(1,1,2)}\cup\Sigma_{(2,2)}$.

In this article we restrict our study to the subset $\Sigma_{k}$ of $\Sigma$
consisting of $k$-fold ground-state degenerate matrices, that is, matrices
with $\lambda_{1}=\lambda_{2}=\dots=\lambda_{k}<\lambda_{k+1}$. Since higher
eigenvalues can coincide with each other, $\Sigma_{k}$ is the union of all
strata $\Sigma_{\kappa}$ corresponding to $\kappa=(k_{1},\dots,k_{l})$ with
$k_{1}=k$. The generic points of $\Sigma_{k}$ belong to the stratum
corresponding to $(k,1,\dots,1)$.

The set $\Sigma_{k}$ is a smooth, not closed submanifold in $\mbox{Herm}(n)$
of codimension $k^{2}-1$, that is, $\dim(\Sigma_{k})=n^{2}-(k^{2}-1)$, cf.
Figure 1. This follows from the Neumann-Wigner theorem, see Table 1 for a
sketch of the proof.

In addition to $\Sigma_{k}$, its closure $\mbox{cl}(\Sigma_{k})$ contains the
higher degeneracies as well, it precisely decomposes as the disjoint union
$\mbox{cl}(\Sigma_{k})=\Sigma_{k}\cup\Sigma_{k+1}\cup\dots\cup\Sigma_{n}$. The
proof is the following. If an infinite sequence $H_{1},H_{2},\dots$ of
elements $H_{i}\in\Sigma_{k}$ is convergent in $\mbox{Herm}(n)$, then its
limit $\lim_{i\to\infty}H_{i}$ is in $\mbox{cl}(\Sigma_{k})$ by the definition
of the closure. As a consequence of the continuity of the eigenvalues, the
lowest $k$ eigenvalues of $\lim_{i\to\infty}H_{i}$ are degenerate, and it
might happen that one or more other eigenvalues also converge to them.

Our results can be naturally adapted to arbitrary $k$-fold degeneracy, not
necessarily ground state, i.e., the set of matrices with
$\lambda_{i}<\lambda_{i+1}=\dots=\lambda_{i+k}<\lambda_{i+k+1}$.

We introduce one more notation. Let $\Sigma_{[k,k+1]}\subset\Sigma$ denote the
set of matrices $H\in\mbox{Herm}(n)$ with $\lambda_{k}=\lambda_{k+1}$, that
is, the closure of the two-fold degeneracy stratum corresponding to
$\lambda_{k}=\lambda_{k+1}$. For example, in $\mbox{Herm}(4)$ we have
$\Sigma_{[2,3]}=\Sigma_{(1,2,1)}\cup\Sigma_{(1,3)}\cup\Sigma_{(3,1)}\cup\Sigma_{(4)}$,
its complement is
$\Sigma_{(1,1,1,1)}\cup\Sigma_{(2,1,1)}\cup\Sigma_{(1,1,2)}\cup\Sigma_{(2,2)}$,
cf. Figure 1.

##  3 Summary of results

Here we summarize the results of this paper. The proofs and further details
can be found in Section 4, each one in the corresponding subsection with the
same title.

###  3.1 The Schrieffer–Wolff transformation induces a local chart

We consider matrices $H\in\mbox{Herm}(n)$ in a sufficiently small neighborhood
$\mathcal{V}_{0}$ of a fixed $k$-fold ground-state degenerate matrix
$H_{0}\in\Sigma_{k}$. For simplicity, we assume that $H_{0}$ is diagonal in
the canonical basis of $\mathbb{C}^{n}$ with increasing order of the
eigenvalues. For a more general setting (non-diagonal or non-degenerate
$H_{0}$) see the Appendix.

As we mentioned before, the choice of the unitary matrix $U\in U(n)$ in the
diagonalization (2.2.1) of $H$ is not unique. Furthermore, $U$ cannot be
chosen to depend continuously on $H$ in any (open) neighborhood of $H_{0}$ in
$\mbox{Herm}(n)$. This is because the eigenvectors corresponding to degenerate
eigenvalues cannot be chosen continuously111The eigenspaces form a complex
line bundle over the non-degenerate matrices, which is nontrivial. For
example, it is always possible to find a small 2-sphere $S^{2}$ around $H_{0}$
in $\mbox{Herm}(n)\setminus\Sigma$, such that the first Chern number of the
lowest eigenstate is non-zero, showing the impossibility of a continuous
choice of eigenvectors.. Instead, what happens if our objective is to attain
only a block diagonal structure

$\widetilde{B}=U^{-1}HU,$ (3.1.1)

where $\widetilde{B}$ consists of a $k\times k$ and a $(n-k)\times(n-k)$
block? It turns out that such a family of unitary matrices $U\in U(n)$ can be
chosen not only as a continuous, but an analytical function of $H$ as well.
Moreover, if $U$ is assumed to be of the form $U=e^{iS}$ with $S$ a block off-
diagonal Hermitian matrix, the decomposition (3.1.1) is essentially unique. It
is formulated by our first statement as follows.

######  Theorem 3.1.2 (Exact SW decomposition, cf. Figure 2).

Fix a diagonal matrix $H_{0}\in\Sigma_{k}$ with increasing order of its
eigenvalues along its diagonal. Then, there are neighborhoods
$\mathcal{V}_{0},\mathcal{W}_{0}\subset\mbox{Herm}(n)$ of $H_{0}$ and a
neighborhood $\mathcal{X}_{0}\subset\mbox{Herm}(n)$ of 0 such that for every
$H\in\mathcal{V}_{0}$ there is a unique decomposition

$H=e^{iS}\cdot\widetilde{B}\cdot e^{-iS},$ (3.1.3)

where $\widetilde{B}$ and $S$ are $n\times n$ Hermitian matrices with the
following special properties:

  1. 1.

$\widetilde{B}\in\mathcal{W}_{0}$ is a block diagonal matrix with $k\times k$
and $(n-k)\times(n-k)$ blocks.

  2. 2.

$S\in\mathcal{X}_{0}$ is an off-block matrix, i.e., its $k\times k$ and
$(n-k)\times(n-k)$ blocks along the diagonal are zero. Non-zero entries are in
the $k\times(n-k)$ and $(n-k)\times k$ off-diagonal blocks.

  3. 3.

The first $k$ columns of $e^{iS}$ span the sum of the eigenspaces of $H$
corresponding to the lowest $k$ eigenvalues.

Furthermore, the dependence of $S$ and $\widetilde{B}$ on
$H\in\mathcal{V}_{0}$ is (real) analytic.

Actually, the theorem gives an exact formulation of the well-known method
called Schrieffer–Wolff transformation [2, 1, 3, 4, 5], by breaking up
$\widetilde{B}$ into parts according to the blocks as follows, cf. Figure 2:

$H=e^{iS}\cdot(H_{0}+B+T+H_{\textup{eff}})\cdot e^{-iS},$ (3.1.4)

where $T$, $B$, $H_{\textup{eff}}$ and $S$ are $n\times n$ Hermitian matrices
with the following special properties:

  1. 1.

$B$ is block diagonal matrix and it has non-zero entries only in the
$(n-k)\times(n-k)$ bottom right block.

  2. 2.

$T$ is a scalar matrix in the $k\times k$ upper left block and all the other
entries are zero.

  3. 3.

The traceless _effective Hamiltonian_ $H_{\textup{eff}}$ has a traceless
$k\times k$ block and all the other entries are zero.

  4. 4.

$S$ is an off-block matrix, see above (point (2) in Theorem 3.1.2).

Assuming that $(S,T,B,H_{\textup{eff}})$ is in a sufficiently small
neighborhood of $(0,0,0,0)$, then the decomposition is unique, and the
dependence of $S$, $T$, $B$ and $H_{\textup{eff}}$ on $H\in\mathcal{V}_{0}$ is
(real) analytic, by Theorem 3.1.2. Note that in the literature the effective
Hamiltonian is usually defined together with its trace, i.e.,
$H_{\textup{eff}}+T$.

The proof of Theorem 3.1.2, and hence the unique decomposition (3.1.4) is
based on the analytic inverse function theorem, see Section 4.1. The unitary
transformation $e^{iS}$ is described in [3] as a ‘direct rotation’ between the
eigenspaces of $H$ and $H_{0}$ corresponding to the lowest $k$ eigenvalues. In
the we clarify the relation of [3] with Theorem 3.1.2, and we adapt the
results of [3] to specify the validity range of decomposition (3.1.4).

Figure 2: The exact SW decomposition (3.1.4). By Theorem 3.1.2 the block
diagonalization of a matrix $H$ close enough to $H_{0}$ is unique, if the
unitary matrix has the special form $e^{iS}$, where $S$ is off-block, and both
$S$ and the block diagonal matrix are sufficiently close to 0. Decomposing the
block diagonal matrix into parts gives the ‘non-interesting block’ $B$, the
scalar matrix $T$ in the ‘interesting block’ carrying the trace of this block,
and the traceless effective Hamiltonian $H_{\textup{eff}}$.

By the convention introduced in Section 2.1, $H_{\textup{eff}}$ can be
considered as a traceless $k\times k$ Hermitian matrix, that is,
$H_{\textup{eff}}\in\mbox{Herm}_{0}(k)\subset\mbox{Herm}(n)$. Let
$y_{1},\dots,y_{k^{2}-1}$ denote its coordinates in the orthonormal basis
$\breve{c}_{1},\dots,\breve{c}_{k^{2}-1}$ of $\mbox{Herm}_{0}(k)$ introduced
in Section 2. For example for two-fold degeneracy ($H_{0}\in\Sigma_{2}$), the
effective Hamiltonian is expressed as

$H_{\textup{eff}}=y_{1}\breve{c}_{1}+y_{2}\breve{c}_{2}+y_{3}\breve{c}_{3}=\frac{1}{\sqrt{2}}(y_{1}\sigma_{x}+y_{2}\sigma_{y}+y_{3}\sigma_{z}).$
(3.1.5)

The number of the (possibly) nonzero coordinates of the matrices $B$, $T$ and
$S$ in the canonical basis of $\mbox{Herm}(n)$ is

  1. 1.

$B$: $(n-k)^{2}$,

  2. 2.

$T$: $1$,

  3. 3.

$S$: $2\cdot k\cdot(n-k)$.

Together these are $n^{2}-k^{2}+1$ coordinates, let
$x_{1},\dots,x_{n^{2}-k^{2}+1}$ denote them.

######  Corollary 3.1.6 (SW decomposition induces a local chart, cf. Figure
3).

  1. (a)

The map $\varphi:\mathcal{V}_{0}\to\mathbb{R}^{n^{2}}$ with coordinate
functions $x_{i}$ and $y_{j}$ is a local chart on $\mbox{Herm}(n)$ around
$H_{0}$, and $\varphi$ is analytic in the canonical coordinates on
$\mbox{Herm}(n)$.

  2. (b)

In this local chart the set $\Sigma_{k}$ of $k$-fold ground state degenerate
matrices is the common zero locus of the coordinates $y_{j}$, that is,

$\Sigma_{k}\cap\mathcal{V}_{0}=\\{H\in\mathcal{V}_{0}\ |\
y_{1}(H)=y_{2}(H)=\dots=y_{k^{2}-1}(H)=0\\}.$ (3.1.7)

Figure 3: A schematic picture showing the local chart induced by the SW
decomposition around the fixed $H_{0}$. A matrix $H\in\mathcal{V}_{0}$ is
endowed with two sets of coordinates: (1) the projection
$H_{\text{proj}}=H_{\Sigma}$ of $H$ to $\Sigma_{k}$ is described by the
coordinates of $S$, $B$ and $T$ (these are $n^{2}-k^{2}+1$ coordinates denoted
by $x_{i}$), (2) the traceless effective Hamiltonian $H_{\textup{eff}}$ is
described by $k^{2}-1$ coordinates denoted by $y_{i}$. The constant level set
of the $y$ and $x$ coordinates through $H$ is denoted by $X_{H}$ and $Y_{H}$,
respectively. The $Y_{H}$ sets are affine subspaces in $\mbox{Herm}(n)$ of
dimension $k^{2}-1$. As an essential step of the proof of Theorem 3.2.2 (see
Proposition 4.2.6) we will show that the line joining $H$ and $H_{\Sigma}$ is
orthogonal to $\Sigma_{k}$ at $H_{\Sigma}$, or equivalently, the $Y_{H}$
subspaces are orthogonal to $\Sigma_{k}$, moreover, $H_{\Sigma}$ is the
closest point of $\Sigma_{k}$ to $H$.

This chart shows that $\Sigma_{k}$ is a codimension $k^{2}-1$ submanifold in
$\mbox{Herm}(n)$, providing an alternative proof for the Neumann–Wigner
theorem (whose original proof is summarised in Table 1).

Consider the projection $H_{\textup{proj}}$ of $H$ to $\Sigma_{k}$ by omitting
$H_{\text{eff}}$ from the SW decomposition (3.1.4), that is,

$H_{\textup{proj}}=e^{iS}\cdot(H_{0}+B+T)\cdot e^{-iS}.$ (3.1.8)

This can be rephrased in the SW chart picture as making the $y_{j}$
coordinates 0, see Figure 3. We show that $H_{\textup{proj}}$ can be
constructed without using the SW decomposition. The following construction
works for the large set $H\in\mbox{Herm}(n)\setminus\Sigma_{[k,k+1]}$ of
Hermitian matrices, not only on the domain $\mathcal{V}_{0}$ of the SW
decomposition around an $H_{0}$. Let $\Lambda=U^{-1}HU$ be a diagonalization
of $H$, containing the eigenvalues
$\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{k}<\lambda_{k+1}\leq\dots\leq\lambda_{n}$
of $H$. Let

$\overline{\lambda}=\frac{\sum_{j=1}^{k}\lambda_{j}}{k}.$ (3.1.9)

be the mean of the lowest $k$ eigenvalues. Let $\Lambda_{\Sigma}$ be the
diagonal matrix obtained from $\Lambda$ by replacing the first $k$ entries
$\lambda_{j}$ ($j=1,\dots,k$) with $\overline{\lambda}$. Define

$H_{\Sigma}=U\cdot\Lambda_{\Sigma}\cdot U^{-1},$ (3.1.10)

the matrix obtained from $H$ by ‘collapsing the lowest $k$ eigenvalues’.
Observe that $H_{\Sigma}\in\Sigma_{k}$. In Section 4.1 we show that
$H_{\Sigma}$ does not depend on the choice of the unitary matrix $U$, and
$H_{\Sigma}$ depends on $H$ in analytic way, although $U$ cannot be chosen
continuously. The following statement will be also proved in Section 4.1.

######  Theorem 3.1.11 (Projection to $\Sigma_{k}$, cf. Figure 3).

If the SW decomposition (3.1.4) of $H$ with respect to $H_{0}\in\Sigma_{k}$ is
defined, then it holds that $H_{\Sigma}=H_{\textup{proj}}$.

###  3.2 Energy splitting and the distance from $\Sigma_{k}$

Recall the distance of two matrices $H,G\in\mbox{Herm}(n)$ induced by the
Frobenius metric is denoted by $d(H,G)=\|H-G\|$. The distance $d(H,A)$ of an
element $H\in\mbox{Herm}(n)$ and a subset $A\subset\mbox{Herm}(n)$ is the
infimum of the distances $d(H,G)$, $G\in A$.

Consider the standard deviation of the lowest $k$ eigenvalues of $H$, that is,

$D_{k}(H)=\sigma(\lambda_{1},\dots,\lambda_{k})=\sqrt{\frac{\sum_{j=1}^{k}(\lambda_{j}-\overline{\lambda})^{2}}{k}},$
(3.2.1)

where $\overline{\lambda}=\sum_{j=1}^{k}\lambda_{j}/k$ is the mean of the
eigenvalues.

######  Theorem 3.2.2 (Distance from $\Sigma_{k}$).

For every $H\in\mbox{Herm}(n)\setminus\Sigma_{[k,k+1]}$

$d(H,\Sigma_{k})=d(H,H_{\Sigma})=\sqrt{k}\cdot D_{k}(H)=\|H_{\textup{eff}}\|$
(3.2.3)

holds, where $H_{\textup{eff}}$ is the effective Hamiltonian of $H$ with
respect to any $H_{0}\in\Sigma_{k}$ for which the exact SW decomposition
(3.1.4) of $H$ is defined.

We prove the theorem in Section 4.2. The first equation expresses that
$H_{\Sigma}$ is the closest point of $\Sigma_{k}$ to $H$. We note that the
first and second equations hold also for $H\in\Sigma_{[k,k+1]}$, although the
projection $H_{\Sigma}$ is not unique, as it depends on the choice of $U$.

In particular, for the special case $k=2$, Eq. (3.2.3) takes the following
form:

$d(H,\Sigma_{2})=d(H,H_{\Sigma})=\frac{1}{\sqrt{2}}\cdot|\lambda_{2}-\lambda_{1}|=\|H_{\textup{eff}}\|,$
(3.2.4)

where $\lambda_{1}$ and $\lambda_{2}$ are the lowest two eigenvalues of $H$.

###  3.3 Order of energy splitting of a degeneracy due to a perturbation

For $k>2$ one can also consider the pairwise differences
$\lambda_{i}-\lambda_{j}$. We describe these functions only for one-parameter
families. Consider a one-parameter perturbation of $H_{0}\in\Sigma_{k}$, also
called a one-parameter family, or a curve of Hermitian matrices. This is an
analytic function222For simplicity, we formulate the results for analytic
families, but smooth ($\mathcal{C}^{\infty}$) is enough, with the obvious
modification of the statements. $H:\mathbb{R}\to\mbox{Herm}(n)$ defined in a
neighborhood of $0\in\mathbb{R}$ with $H(0)=H_{0}\in\Sigma_{k}$. An example is
a linear perturbation in the form $H(t)=H_{0}+tH_{1}$ with a choice of $0\neq
H_{1}\in\mbox{Herm}(n)$. (Note the slight abuse of notation: so far $H$
denoted a single matrix, here it denotes a map into the matrix space.)

The frequently used quantity ‘order of energy splitting’ can be defined in
several different ways. On the one hand, the energy splitting along $H(t)$ is
measured by the standard deviation function

$\mathfrak{s}_{k}(t):=D_{k}(H(t)).$ (3.3.1)

By Theorem 3.2.2 this agrees with the distance function up to a scalar factor,
that is,

$\sqrt{k}\cdot\mathfrak{s}_{k}(t)=d(H(t),\Sigma_{k}).$ (3.3.2)

On the other hand, one can consider the pairwise differences of the
eigenvalues, and possibly define the order of the energy splitting either as
(1) the minimum of the orders of the pairwise differences or (2) the minimum
of the orders of the differences of the neighbours or (3) the order of the
difference of the two extrema $\lambda_{1}$ and $\lambda_{k}$ or (4) the
minimum of the orders of the differences from the mean value. Theorem 3.3.6
below clarifies that these orders are equal. Moreover, any of them agrees with
the order of the standard deviation of the lowest $k$ eigenvalues, and hence,
the order of the distancing of $H(t)$ from $\Sigma_{k}$.

To formalize the statements of the previous paragraph, we define the pairwise
energy splitting functions $\mathfrak{s}_{i,j}$ and the splitting from the
mean of the eigenvalues $\overline{\mathfrak{s}}_{i}$, respectively, as

$\displaystyle\mathfrak{s}_{i,j}(t)$ $\displaystyle=$
$\displaystyle\lambda_{i}(t)-\lambda_{j}(t),$ (3.3.3)
$\displaystyle\overline{\mathfrak{s}}_{i}(t)$ $\displaystyle=$
$\displaystyle\lambda_{i}(t)-\overline{\lambda}(t),$ (3.3.4)

where $1\leq i<j\leq k$. Although the functions $\mathfrak{s}_{i,j}$ and
$\overline{\mathfrak{s}}_{i}$ defined in this way are not differentiable at
$t=0$ in general (cf. Figure 8), for positive $t$ they are given by a power
series of $t$ centered at 0, hence their order is well defined in this sense.
Indeed, they can be made analytic around 0 by a slightly modified definition,
presented in Section 4.3. For this, we use the following well-known fact, and
we provide a new proof for it in Section 4.3, based on the SW decomposition:

######  Theorem 3.3.5 (The eigenvalues are analytic).

The eigenvalues of one-parameter analytic Hermitian matrix families form
analytic functions, after a suitable re-ordering of the indexing for negative
$t$.

Based on these preliminaries, the equality of the order of the standard
deviation function defined in Eq. (3.3.1), and the further orders listed as
(1)-(4) below that, is formalized in the following theorem.

######  Theorem 3.3.6 (Order of energy splitting).

$\displaystyle\mathrm{ord}_{0}(\mathfrak{s}_{k})$ $\displaystyle=$
$\displaystyle\min_{1\leq i<j\leq
k}\\{\mathrm{ord}_{0}(\mathfrak{s}_{i,j})\\}$ $\displaystyle=$
$\displaystyle\min_{1\leq i\leq
k-1}\\{\mathrm{ord}_{0}(\mathfrak{s}_{i,i+1})\\}$ $\displaystyle=$
$\displaystyle\mathrm{ord}_{0}(\mathfrak{s}_{1,k})$ $\displaystyle=$
$\displaystyle\min_{1\leq i\leq
k}\\{\mathrm{ord}_{0}(\overline{\mathfrak{s}}_{i})\\},$

where $\mathrm{ord}_{0}(f)$ denotes the order of the function $f$ at $t=0$.

Theorem 3.3.6 together with Equation (3.3.2) provides a geometric description
for the order of the splitting. Namely, it is independent of the way we
measure it, and it agrees with the order of distancing from $\Sigma_{k}$.
Moreover, it also agrees with the leading order of the effective Hamiltonian,
which is the minimum of the orders of the coordinates
$y_{1},\dots,y_{k^{2}-1}$.

######  Theorem 3.3.8 (Energy splitting and distancing have the same order).

Let $r$ denote the order of energy splitting, that is, any of the five equal
quantities in Theorem 3.3.6. Then

$r=\mathrm{ord}_{0}(t\mapsto d(H(t),\Sigma_{k}))=\mathrm{ord}_{0}(t\mapsto
H_{\textup{eff}}(t))=\min_{1\leq j\leq k^{2}-1}\\{\mathrm{ord}_{0}(t\mapsto
y_{j}(t))\\},$ (3.3.9)

where $H_{\textup{eff}}(t)$ is the effective Hamiltonian of $H(t)$ with
respect to any diagonal $H^{\prime}_{0}\in\Sigma_{k}$ (not necessarily equal
to $H_{0}$) for which the exact SW decomposition (3.1.4) of $H(t)$ is defined.

######  Corollary 3.3.10.

For a linear family $H(t)=H_{0}+tH_{1}$ the order of energy splitting is

  * •

$r=1$ if $H_{1}$ is not tangent to $\Sigma_{k}$ at $H_{0}$, i.e. $H_{1}\notin
T_{H_{0}}\Sigma_{k}$,

  * •

$r\geq 2$ if $H_{1}$ is tangent to $\Sigma_{k}$ at $H_{0}$, i.e. $H_{1}\in
T_{H_{0}}\Sigma_{k}$.

According to the above proposition, $r>2$ means stronger ‘stickiness’ of the
tangent vector $H_{1}$ to the degeneracy submanifold $\Sigma_{k}$ at $H_{0}$.

###  3.4 Parameter-dependent quantum systems and Weyl points

Parameter-dependent quantum systems are described by a smooth
($\mathcal{C}^{\infty}$) map from a manifold $M$ of dimension $m$ to the space
of Hamiltonians, i.e., Hermitian matrices $\mbox{Herm}(n)$. Slightly abusing
the notation again, we denote this map by $H$. So from now on
$H:M\to\mbox{Herm}(n)$ is a smooth map with $H(p_{0})=H_{0}\in\Sigma_{k}$ for
a point $p_{0}\in M$, which is called degeneracy point. For simplicity, assume
that $H_{0}$ is diagonal — it can be reached by a unitary change of basis in
$\mathbb{C}^{n}$, see App. A.

As above, let $\mathcal{V}_{0}$ denote a neighborhood of $H_{0}$ where the SW
decomposition is unique. Then, consider the corresponding neighborhood
$\mathcal{W}_{0}\subset H^{-1}(\mathcal{V}_{0})$ of $p_{0}$ in $M$. On this
$\mathcal{W}_{0}$, the (traceless) effective Hamiltonian map is defined by the
SW decomposition, that is,

$H_{\textup{eff}}:\mathcal{W}_{0}\to\mbox{Herm}_{0}(k).$ (3.4.1)

By introducing a local chart in $\mathcal{W}_{0}$ centered at $p_{0}$ (i.e.,
$p_{0}=0$) and expressing $H_{\textup{eff}}$ in the basis $\breve{c}_{i}$ of
$\mbox{Herm}_{0}(k)$ (cf. Sec. 2), we obtain a map

$h:\mathbb{R}^{m}\to\mathbb{R}^{k^{2}-1}$ (3.4.2)

defined in a neighborhood of the origin, satisfying $h(0)=0$. The $i$-th
component of $h$ is $h_{i}=y_{i}\circ H$, where $y_{i}$ are the effective
coordinates of the SW chart in Corollary 3.1.6, $i=1,\dots,k^{2}-1$.

If we want to describe the ‘type’ of the degeneracy point $p_{0}$, this
problem leads to the description of the intersection of $H$ and $\Sigma_{k}$
at $H_{0}$, which can be reduced to the description of the ‘type’ of the root
of $h$ at 0. For this it is necessary to know $h$ in an arbitrarily small
neighborhood of the origin. This description of the degeneracy point leads to
the study of map germs $h:(\mathbb{R}^{m},0)\to(\mathbb{R}^{k^{2}-1},0)$, and
their classifications in singularity theory. Although this relation between
degeneracy points and singularities implicitly appears in several works, see
e.g. [21, 22, 23, 24], this relation is essentially based on the geometric
picture described here. We illustrate this relation on the characterization of
Weyl points. Weyl points represent the simplest type of degeneracy, and we
show that they correspond to the simplest type of intersection called
transverse intersection of $H$ and $\Sigma$, and also to the simplest
singularity type of $h$. These relations explain the protected nature of Weyl
points, as a straightforward consequence of transversality.

Recall that in the physics terminology, Weyl points are isolated twofold
degeneracy points in a 3-dimensional parameter space, with linear energy
splitting in every direction. We formalize this informal description below.
According to this description, we restrict our study to twofold (ground state)
degeneracy — that is, $H(p_{0})=H_{0}\in\Sigma_{2}$ – and $m=3$. Hence, the
resulting map (germ) is $h:(\mathbb{R}^{3},0)\to(\mathbb{R}^{3},0)$.

######  Remark 3.4.3.

Observe that the equality of the dimensions is coming from the fact that the
codimension of $\Sigma_{2}$ in $\mbox{Herm}(n)$ is 3, and we also choose 3 for
the dimension of the parameter space $M$. This is the case in many physical
examples (3D crystals, magnetic fields, see Example 3.5.13), hence, this
coincidence of the (co)dimensions explains the appearance of Weyl points.

Recall the notion of transversality, see e.g. [25, 26]. Consider smooth
manifolds $M$ and $N$, a smooth submanifold $Z\subset N$ of $N$, and a smooth
map $f:M\to N$ with $f(p)\in Z$ for a point $p\in M$. The tangent map
$\mathrm{d}f_{p}$ is a linear map from the tangent space $T_{p}M$ of $M$ at
$p$ to the tangent space $T_{f(p)}N$ of $N$ at $f(p)$. In local coordinates
$(\mathrm{d}f)_{p}$ is the Jacobian matrix. Then, $f$ is _transverse_ to $Z$
at $p$ if

$T_{f(p)}Z+(\mathrm{d}f)_{p}(T_{p}M)=T_{f(p)}N$ (3.4.4)

holds. That is, the tangent space of the submanifold $Z$ and the image of the
tangent map $(\mathrm{d}f)_{p}$ span the tangent space of $N$ at $f(p)$, see
panel (a) and (d) in Figure 4.

Figure 4: Transversality, as a characterization of Weyl points, cf. Equation
(3.4.4) and Theorem 3.4.5. On each panel the submanifold is orange, and the
image of the map is blue. (a) Transverse intersection of two curves at a
single point on the plane. For a small perturbation the intersection point
slightly moves away, but it remains transverse. (b) Non-transverse
intersection of two curves on the plane at a single point. Non-transversality
follows from the fact that their tangent lines coincide, hence they do not
span the tangent plane of the plane, thus Equation (3.4.4) does not hold for
them. For a small perturbation the intersection splits into transverse
intersection points, whose number is either 0 or 2 in this case. (c) Non-
transverse intersection of two curves in 3-space at a single point. Non-
transversality follows from the inequality $1+1<3$, that is, the sum of the
dimensions is less than the dimension of the ambient space. In this case
Equation (3.4.4) cannot hold at an intersection point. According to this, the
intersection point disappears for a small perturbation. (d) Transverse
intersection of a surface and a curve at a single point in 3-space. Any small
perturbation preserves transversality, the intersection point only slightly
moves away. In the context of parameter-dependent quantum systems
$H:M\to\mbox{Herm}(n)$, the submanifold $\Sigma_{2}$ has codimension 3, the
same as the dimension of the parameter space $M$. The red point is the image
of a twofold ground state degeneracy point $H(p_{0})=H_{0}\in\Sigma_{2}$. It
is a Weyl point if $H$ is transverse to $\Sigma_{2}$ in $\mbox{Herm}(n)$ at
$p_{0}$ (like in panel (a) and (d)), otherwise it is a non-generic degeneracy
point, which splits into Weyl points (yellow points) or disappears for a small
perturbation of $H$ (dashed blue line).

Transversality implies the expected dimension of the pre-image, namely, if $f$
is transverse to $Z$ at every point of $f^{-1}(Z)$, then $f^{-1}(Z)$ is a
submanifold of $M$ of dimension $\dim(f^{-1}(Z))=\dim(M)+\dim(Z)-\dim(N)$, see
[25, pg. 28]. By the statements closely related to the transversality theorem
[25, pg. 35, 68-69], transversality with respect to a fixed submanifold is a
stable and generic property of smooth maps, cf. Figure 4.

######  Theorem 3.4.5 (Characterization of Weyl points).

Given a parameter-dependent quantum system by a $\mathcal{C}^{\infty}$ map
$H:M^{3}\to\mbox{Herm}(n)$ and a two-fold ground state degeneracy point
$p_{0}\in M^{3}$, $H(p_{0})=H_{0}\in\Sigma_{2}$ and the induced effective map
germ $h:(\mathbb{R}^{3},0)\to(\mathbb{R}^{3},0)$, the following properties are
equivalent:

  1. 1.

The map $H$ is transverse to $\Sigma_{2}$ at $p_{0}$,

  2. 2.

The rank of the Jacobian of $h$ at 0 has maximal rank, i.e.
$\mbox{rk}((\mathrm{d}h)_{0})=3$.

  3. 3.

Taking any curve $\gamma:(\mathbb{R},0)\to(M^{3},p_{0})$ with
$\gamma(0)=p_{0}$ and $\gamma^{\prime}(0)\neq 0$, for the composition
$(h\circ\gamma)^{\prime}(0)\neq 0$ holds.

  4. 4.

Taking any curve $\gamma:(\mathbb{R},0)\to(M^{3},p_{0})$ with
$\gamma(0)=p_{0}$ and $\gamma^{\prime}(0)\neq 0$, for the composition
$H\circ\gamma$ the order of energy splitting is 1.

A two-fold degeneracy point $p_{0}$ satisfying any, hence all of the
properties (1)–(4) is called Weyl point. Note that (4) formulates the
physicist definition.

######  Remark 3.4.6.

By (2), the characterization of Weyl points is already determined by the
first-order part of the exact effective map $h$. It implies that in the
approximate computation of the SW transformation (see e.g. [4, 5]) the first-
order term is sufficient to decide whether a degeneracy point $p_{0}\in M^{3}$
is a Weyl point or not. Namely, it can be decided by the following steps, cf.
Example 3.5.13:

  1. 1.

Take the first-order SW transformation, i.e. the upper-left traceless $2\times
2$ block of $H(p)$.

  2. 2.

Via the Pauli decomposition it can be considered as a map
${h}^{(1)}:\mathbb{R}^{3}\to\mathbb{R}^{3}$, defined in a neighborhood of the
origin $p_{0}=0$.

  3. 3.

$p_{0}$ is a Weyl point if and only if $\mbox{rk}((\mathrm{d}h^{(1)})_{0}=3$.

The properties of the transversality mentioned above imply the following
corollaries.

######  Corollary 3.4.7 (Weyl points are isolated).

Every Weyl point $p_{0}$ is an isolated degeneracy point, in the sense that
there is a neighborhood $\mathcal{W}_{0}\subset M^{3}$ of $p_{0}$ such that
$H(p)\notin\Sigma_{2}$ for $p_{0}\neq p\in\mathcal{W}_{0}$.

Intuitively, the protected nature of Weyl points includes the following
phenomena:

  1. (a)

_Weyl points are stable:_ For any small perturbation, a Weyl point does not
disappear, it only gets displaced by a small amount in $M$ (if at all).

  2. (b)

_Weyl points are generic:_ For a generic perturbation, a non-generic
degeneracy point splits into Weyl points (or possibly disappears).

To translate (a) and (b) into rigorous claims, we consider one-parameter
perturbations $H_{t}$ of $H$. Formally, these are $\mathcal{C}^{\infty}$ maps
(germs) from $M^{3}\times\mathbb{R}$ to $\mbox{Herm}(n)$ defined on a
neighborhood of $(p_{0},0)$, such that $H_{t=0}=H$. For simplicity we
formulate the statement in a local version for isolated twofold degeneracy
points, although, it can be generalized for non-isolated or multifold
degeneracy points, see Remark 4.4.4.

######  Corollary 3.4.8 (Weyl points are stable and generic).

Let $H:M^{3}\to\mbox{Herm}(n)$ be a parameter-dependent quantum system with an
isolated two-fold ground state degeneracy point $p_{0}\in M^{3}$,
$H(p_{0})=H_{0}\in\Sigma_{2}$. Let $\mathcal{W}_{0}\subset M^{3}$ be a
neighborhood of $p_{0}$ whose closure $\mbox{cl}(\mathcal{W}_{0})$ is compact
and it does not contain other ground state degeneracy points, that is,
$H^{-1}(\mbox{cl}(\Sigma_{2}))\cap\mbox{cl}(\mathcal{W}_{0})=\\{p_{0}\\}$.

  1. (a)

If $p_{0}$ is a Weyl point, then for every one-parameter perturbation $H_{t}$
of $H_{t=0}=H$, there is an $0<\epsilon$, such that for $|t|<\epsilon$ the
perturbed Hamiltonian $H_{t}$ has exactly one degeneracy point in
$\mathcal{W}_{0}$, and it is a Weyl point. Moreover, there is a
$\mathcal{C}^{\infty}$ curve $\gamma:(-\epsilon,\epsilon)\to M^{3}$ such that
$\gamma(t)$ is the unique Weyl point of $H_{t}$ in $\mathcal{W}_{0}$.

  2. (b)

If $p_{0}$ is not a Weyl point, then for every $0<\epsilon$ there is a
$K\in\mbox{Herm}(n)$ with $\|K\|<\epsilon$, such that every degeneracy point
of the perturbed map $H_{K}:p\mapsto H_{K}(p)=H(p)+K$ in $\mathcal{W}_{0}$ is
a Weyl point.

Finally, Property (2) of Theorem 3.4.5 implies that the topological charge of
a Weyl point is $\pm 1$. Indeed, the topological charge is equal to the local
degree $\deg_{0}h$ of $h$ at $0$, which is $\pm 1$, if
$\mbox{rk}(\mathrm{d}h_{0})=3$, see [23]. (Note that in the physics literature
the topological charge is defined as the first Chern number of the eigenvector
bundle corresponding to the lowest eigenvalue, evaluated on a small sphere in
$M^{3}$ around $p_{0}$, but it is equal to the local degree $\deg_{0}h$ of $h$
at $0$.)

###  3.5 Examples

######  Example 3.5.1 ($\mbox{Herm}(3)$).

In the case of $3\times 3$ matrices, the exact SW decomposition 3.1.2 can be
given in a closed form using Cardano’s formula to determine the eigenvalues,
then, to get the block-diagonalized form, one needs to perform the exact
direct rotation between the near-degenerate subspaces, cf. Appendix C.
However, the resulting expressions are extremely complicated. As an
alternative to exact decomposition, one might use a series expansion [4] to
approximate the terms of the decomposition.

As an example, we take a general $H\in\mbox{Herm}(3)$ around
$H_{0}=\mathrm{diag}(0,0,1)\in\Sigma_{2}$ with the elements as coordinates
such that

$H=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&1\end{pmatrix}+\begin{pmatrix}v+z&x-iy&p-iq\\\ x+iy&v-z&r-is\\\
p+iq&r+is&w\end{pmatrix}.$ (3.5.2)

The SW decomposition up to second order reads

$\displaystyle iS$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&0&(1+v-w+z)(p-iq)+(x-iy)(r-is)\\\
0&0&(1+v-w-z)(r-is)+(x+iy)(p-iq)\\\
-\text{h.c.}&-\text{h.c.}&0\end{pmatrix}+\dots,$ (3.5.3) $\displaystyle B$
$\displaystyle=$
$\displaystyle\left(1+w+p^{2}+q^{2}+r^{2}+s^{2}+\dots\right)\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&1\end{pmatrix},$ (3.5.4) $\displaystyle T$ $\displaystyle=$
$\displaystyle\left(v-\frac{p^{2}+q^{2}+r^{2}+s^{2}}{2}+\dots\right)\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&0\end{pmatrix},\mbox{ and}$ (3.5.5) $\displaystyle
H_{\textup{eff}}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}z&x-iy&0\\\
x+iy&-z&0\\\
0&0&0\end{pmatrix}-\frac{1}{2}\begin{pmatrix}p^{2}+q^{2}-r^{2}-s^{2}&2(p-iq)(r+is)&0\\\
2(p+iq)(r-is)&-p^{2}-q^{2}+r^{2}+s^{2}&0\\\ 0&0&0\end{pmatrix}+\dots$
$\displaystyle=$ $\displaystyle\left(x-pr-qs+\dots\right)\sigma_{x}$
$\displaystyle+$ $\displaystyle\left(y+ps-qr+\dots\right)\sigma_{y}$
$\displaystyle+$
$\displaystyle\left(z-\frac{p^{2}+q^{2}-r^{2}-s^{2}}{2}+\dots\right)\sigma_{z}.$

Recall that in the last equation $H_{\textup{eff}}$ is considered as a
$2\times 2$ matrix of trace zero. Note that the effective Hamiltonian in the
first-order $H_{\text{eff}}^{(1)}$ is the truncation of $H$ to its near-
degenerate upper-left $2\times 2$ block.

######  Example 3.5.7 (Exact SW decomposition).

In the special case $p=q=r=s=0$ in Eq. (3.5.2) it is trivial to perform the SW
decomposition, as $H$ is already block diagonal, and $S=0$.

For a simple non-trivial example we take the 2 dimensional section of
$\mbox{Herm}(n)$

$\displaystyle H(p,r)=\begin{pmatrix}0&0&p\\\ 0&0&r\\\ p&r&1\\\
\end{pmatrix}.$ (3.5.8)

The components $S$, $B$, $T$, $H_{\textup{eff}}$ of the SW decomposition
(3.1.4) of $H$ with respect to $H_{0}$ can be expressed explicitly as
functions of $p$ and $r$ as:

$\displaystyle iS(p,r)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{p^{2}+r^{2}}}\tan^{-1}\left(\frac{\sqrt{1+4p^{2}+4r^{2}}-1}{2\sqrt{p^{2}+r^{2}}}\right)\begin{pmatrix}0&0&p\\\
0&0&r\\\ -p&-r&0\end{pmatrix},$ (3.5.9) $\displaystyle B(p,r)$
$\displaystyle=$
$\displaystyle\frac{1+\sqrt{1+4p^{2}+4r^{2}}}{2}\begin{pmatrix}0&0&0\\\
0&0&0\\\ 0&0&1\end{pmatrix},$ (3.5.10) $\displaystyle T(p,r)$ $\displaystyle=$
$\displaystyle\frac{1-\sqrt{1+4p^{2}+4r^{2}}}{4}\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&0\end{pmatrix},\mbox{ and}$ (3.5.11) $\displaystyle
H_{\text{eff}}(p,r)$ $\displaystyle=$
$\displaystyle\frac{1-\sqrt{1+4p^{2}+4r^{2}}}{4(p^{2}+r^{2})}\begin{pmatrix}p^{2}-r^{2}&2pr&0\\\
2pr&-p^{2}+r^{2}&0\\\ 0&0&0\end{pmatrix}$ (3.5.12) $\displaystyle=$
$\displaystyle\frac{1-\sqrt{1+4p^{2}+4r^{2}}}{4(p^{2}+r^{2})}\left(2pr\sigma_{x}+(p^{2}-r^{2})\sigma_{z}\right).$

Observe that every term $S$, $B$, $T$ and $H_{\textup{eff}}$ is an analytic
function of $(p,r)$ in a neighborhood of $(0,0)$, although this is not obvious
at first sight. The non-analytic behaviour of these maps far from $(0,0)$
shows that the SW decomposition can only be defined locally.

######  Example 3.5.13 (Parameter-dependent quantum systems exhibiting Weyl
points.).

In Sec. 3.4 we have discussed the generic nature of Weyl points in a
mathematical context. This discussion is relevant to many physical setups.
Weyl points arise as spectral features in the electronic, phononic, photonic,
magnonic band structures of crystalline materials, or metamaterials. More
generally, Weyl points also arise in quantum systems described by a
Hamiltonian depending on three parameters. One example is an interacting spin
system in a homogeneous magnetic field, where the manifold of parameters is
$M=\mathbb{R}^{3}$, corresponding to the external magnetic field vector [27,
28]. Another example is a multiterminal Josephson junction, where the manifold
of parameters is the three-dimensional torus, hosting the values of three
magnetic flux biases piercing the loops of the superconducting circuit [29,
30, 31].

An explicit $3\times 3$ example for a Hamiltonian with a Weyl point is the
following:

$H(x,y,z)=\begin{pmatrix}z&x-iy&y-ixz\\\ x+iy&-z&x-iyz\\\
y+ixz&x+iyz&1+xyz\end{pmatrix}.$ (3.5.14)

This matrix has a two-fold ground state degeneracy at $x=y=z=0$. This is a
Weyl point according to Remark 3.4.6 as the first-order effective Hamiltonian
is

$H_{\mathrm{eff}}^{(1)}(x,y,z)=x\sigma_{x}+y\sigma_{y}+z\sigma_{z},$ (3.5.15)

which, expanded in the orthonormal Pauli basis, corresponds to the effective
map

$h^{(1)}(x,y,z)=\sqrt{2}\left(x,y,z\right).$ (3.5.16)

The Jacobian of $h^{(1)}$ is $\sqrt{2}$ times the identity, hence it has
maximal rank 3. Note that the off-block matrix elements can be arbitrary
functions of $x$,$y$ and $z$ with constant term 0, so that the resulting
Hamiltonian still describes a Weyl point at the origin. Moreover, one can also
perturb the $2\times 2$ block with higher-order terms without destroying the
Weyl point, as they do not change the Jacobian (and hence its rank) of the
first-order effective map $h^{(1)}$ at the origin. Off-block and higher-order
terms have significant effect, with the possibility of creating new degeneracy
points, only far from the origin.

A $3\times 3$ counterexample which is a point-like two-fold ground state
degeneracy that is not a Weyl point is described in Eq. (3.5.8). It can be
considered as a $\mathcal{C}^{\infty}$ map
$H:\mathbb{R}^{2}\to\mbox{Herm}(3)$. The effective map
$h:(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ defined in a neighborhood of the
origin reads

$h(p,r)=\frac{1-\sqrt{1+4p^{2}+4r^{2}}}{2\sqrt{2}(p^{2}+r^{2})}\left(2pr,0,(p^{2}-r^{2})\right).$
(3.5.17)

This shows that the order of the energy splitting is 2 along every curve
$\gamma:\mathbb{R}\to\mathbb{R}^{2}$ with $\gamma(0)=(0,0)$ and
$\gamma^{\prime}(0)$. Therefore, the order of distancing
$\mathrm{ord}_{0}(t\mapsto d(H(\gamma(t)),\Sigma_{2}))$ is 2 along every such
curve. Moreover, this map is not even equidimensional, meaning that a generic
small perturbation lifts the degeneracy, cf. panel (c) in Figure 4.

######  Example 3.5.18 (Parameter-dependent non-interacting quantum systems
exhibiting higher-order energy splitting of a degeneracy).

In Sec. 1, we have mentioned physical systems having energy degeneracies that
exhibit higher-order energy splitting for physically relevant perturbations.
Here, we illustrate how those examples relate to our formalism and results.
Our first example is the Su-Schrieffer-Heeger (SSH) model [12, 13]. As an
application of our results, we translate a well-known property of the SSH
model into information about the geometry of the corresponding degeneracy
submanifold of the Hermitian matrix space.

We describe the SSH model (or _SSH chain_) as a tight-binding model of a
single electron on a one-dimensional bipartite crystal lattice with a unit
cell of two atoms (orbitals), translational invariance, and open boundary
conditions, i.e., the lattice terminates at both ends. The SSH Hamiltonian is
in $\text{Herm}(2N)$, where $N$ denotes the number of unit cells. For example,
an SSH chain of $N=4$ unit cells is described by the following $8\times 8$
Hamiltonian matrix:

$H_{\text{SSH}}=\left(\begin{array}[]{cccccccc}0&v&0&0&0&0&0&0\\\
v&0&w&0&0&0&0&0\\\ 0&w&0&v&0&0&0&0\\\ 0&0&v&0&w&0&0&0\\\ 0&0&0&w&0&v&0&0\\\
0&0&0&0&v&0&w&0\\\ 0&0&0&0&0&w&0&v\\\ 0&0&0&0&0&0&v&0\end{array}\right).$
(3.5.19)

We define the unperturbed Hamiltonian as $H_{0}=H_{\text{SSH}}(v=0,w=1)$,
which is referred to as the _topological fully dimerized limit_ of the SSH
chain. It is well known, and straightforward to show from the block diagonal
structure of $H_{0}$, that $H_{0}$ has a twofold degenerate eigenenergy at
zero, separated from the other two eigenenergies $1$ and $-1$ which both have
$(N-1)$-fold degeneracy. In this section, we will use $\Sigma_{\text{SSH}}$ to
denote the degeneracy submanifold of $\text{Herm}(2N)$ where the $N$th and
$(N+1)$th ordered eigenvalues are degenerate, but different from the others,
i.e.,
$\\{\dots\leq\lambda_{N-1}<\lambda_{N}=\lambda_{N+1}<\lambda_{N+2}\leq\dots\\}$.
Clearly, $H_{0}$ is on this degeneracy submanifold $\Sigma_{\text{SSH}}$.

The twofold zero-energy degeneracy is robust against special perturbations.
Consider a perturbation $H_{1}$ that is referred to as _disordered nearest-
neighbor hopping_ in physics terminology. This means that $H_{1}$ is a
tridiagonal Hermitian matrix with zeros on the diagonal, i.e., it has nonzero
elements only on the first diagonal above the main diagonal, and the complex
conjugates of those on the first diagonal below the main diagonal. Such
perturbation defines a $(4N-2)$-dimensional subspace in the Hermitian matrix
space. This perturbation causes an energy splitting of the zero-energy
degeneracy, with an order of energy splitting of $N$ (at least), a fact which
can be shown, e.g., by SW perturbation theory after diagonalizing $H_{0}$.
This result, well known in the subfield of physics studying _topological
insulators_ , combined with Corollary 3.3.10 and the first equality of Theorem
3.3.8, implies the following geometrical property of the submanifold
$\Sigma_{\text{SSH}}\subset\text{Herm}(2N)$: in the point
$H_{0}\in\Sigma_{\text{SSH}}$, the $(4N-2)$-dimensional subspace corresponding
to the perturbation $H_{1}$ is in the $(4N^{2}-3)$-dimensional tangent space
of $\Sigma_{\text{SSH}}$ at $H_{0}$ (according to Corollary 3.3.10, since
$r=N\geq 2$), and this subspace of the tangent space is sticking to the to the
degeneracy submanifold especially strongly, as the distancing function in this
subspace has order $N$ (according to Theorem 3.3.8).

######  Example 3.5.20 (Parameter-dependent interacting quantum systems
exhibiting high-order energy splitting of a degeneracy.).

A further example of a robust degeneracy is the twofold ground-state
degeneracy of the quantum mechanical Ising model, a model of interacting
qubits (or spins), in the presence of a transverse-field perturbation.

Here, we focus on the 1D Ising model, i.e., the Ising chain, with open
boundary conditions. The Ising chain consists of $N$ qubits that are nearest-
neighbor coupled with Ising interaction:

$H_{0}=-\left(\sigma_{z}\otimes\sigma_{z}\otimes I\otimes
I\otimes\dots+I\otimes\sigma_{z}\otimes\sigma_{z}\otimes
I\otimes\dots+\dots\right)\equiv-\sum_{i=1}^{N-1}Z_{i}Z_{i+1}.$ (3.5.21)

Here $Z=\ket{1}\bra{1}-\ket{0}\bra{0}$ is the single-qubit Pauli $z$ operator
defined using the orthonormal basis states $\ket{0}$ and $\ket{1}$ of the
qubit. Furthermore, we use the usual shorthand notation for the tensor
products of operators, e.g.,
$Z_{1}Z_{2}\equiv\sigma_{z}\otimes\sigma_{z}\otimes I\otimes I\otimes\dots$,
etc. The unperturbed Hamiltonian $H_{0}$ is in $\text{Herm}(2^{N})$, and
$H_{0}$ has a twofold degenerate ground-state energy at $-N+1$, with two
orthogonal ground states given as $\ket{00\dots}$ and $\ket{11\dots}$.

Then, a special type of perturbation is the disordered transverse field

$H_{1}=\sum_{i=1}^{N}\left(x_{i}X_{i}+y_{i}Y_{i}\right),$ (3.5.22)

which is described by $2N$ real parameters, namely, the _local transverse
fields_ $x_{i}$ and $y_{i}$ multiplying the Pauli operators $X$ and $Y$ of
each qubit. Hence, the parameter space of this perturbation $H_{1}$ is
$\mathbb{R}^{2N}$. In this case, the order of energy splitting due to this
perturbation is $N$ (at least), which can be proven, e.g., by mapping this
model to the SSH model via the Jordan-Wigner transformation (see, e.g., Eq.
(6) of [11]). The fact that the order of energy splitting is $N$ is
translated, using Corollary 3.3.10 and the first equality of Theorem 3.3.8, to
a geometry result, as follows: The parameter space $\mathbb{R}^{2N}$ of the
perturbation correspond to a $2N$-dimensional subspace of the
$(2^{2N}-3)$-dimensional tangent space of the twofold ground-state degenerate
submanifold at its point $H_{0}$ (according to Corollary 3.3.10, since
$r=N\geq 2$), and this $2N$-dimensional subspace is sticking to the degeneracy
submanifold especially strongly, as the distance function in this subspace has
order $N$ (according to Theorem 3.3.8).

######  Example 3.5.23 (Hamiltonians defined from stabiliser quantum error
correction codes).

Stabiliser-code Hamiltonians describe interacting spins, and are directly
related to $[[n,k,d]]$ stabiliser quantum error correction codes [16]. Here,
$n$ is the number of physical qubits, $k$ is the number of logical qubits, and
$d$ is the code distance. Examples are the Toric Code [15] or the five-qubit
code [16]. For a stabiliser-code Hamiltonian derived from an $[[n,k,d]]$
stabiliser code, (i) the ground-state degeneracy is $2^{k}$-fold, and (ii) the
order of ground-state energy splitting caused by 1-local perturbations are of
order $d$. (The latter claim is indicated in [15] for the Toric Code, and is
proven for $[[n,1,3]]$ stabiliser-code Hamiltonians in [32]; we are not aware
of a formal proof for general stabiliser-code Hamiltonians though.) Combining
this information with our results, we deduce geometric information for the
$2^{k}$-fold ground-state degeneracy submanifolds. We illustrate this on the
five-qubit code, but a similar analysis can also be done for other stabiliser
code Hamiltonians, such as the Toric Code.

In the $[[5,1,3]]$ five-qubit code, 5 physical qubits are used to encode 1
logical qubit. The sum of the so-called generators defines the stabiliser-code
Hamiltonian $H_{0}$, i.e., a Hermitian matrix in $\mathrm{Herm}(32)$,

$H_{0}=X_{1}Z_{2}Z_{3}X_{4}+X_{2}Z_{3}Z_{4}X_{5}+X_{1}X_{3}Z_{4}Z_{5}+Z_{1}X_{2}X_{4}Z_{5}.$
(3.5.24)

$H_{0}$ has twofold ground-state degeneracy, that is,
$\lambda_{1}=\lambda_{2}<\lambda_{3}\leq\dots$, and its two-dimensional
ground-state subspace encodes the logical qubit. The logical qubit is robust
against errors of the physical qubits, in the following sense. Consider one-
parameter perturbations $H(t)=H_{0}+tH_{1}$ where $H_{1}$ is 1-local, that is,

$H_{1}=A_{1}+A_{2}+A_{3}+A_{4}+A_{5},$ (3.5.25)

where $A_{i}$-s are linear combinations of the three Paulis acting on site
$i$. Then the robustness means that the order of energy splitting
$\mathrm{ord}_{0}(\mathfrak{s}_{2}(t))$ of the lowest two eigenvalues along
$H(t)$ is at least 3, i.e., the so-called ‘code distance’ of the five-qubit
code.

Using our results connecting the order of energy splitting and the order of
distancing, we can infer geometrical information as follows. In the space
$\mbox{Herm}(32)$ of dimension $32^{2}=1024$, the twofold ground state
degeneracy submanifold $\Sigma_{2}$ has dimension $1021$. The 15-dimensional
space of perturbations described by $H_{1}$ correspond to a 15-dimensional
subspace of the 1021-dimensional tangent space of the twofold ground-state
degenerate submanifold at its point $H_{0}$ (according to Corollary 3.3.10,
since $r=3\geq 2$), and this 15-dimensional subspace is sticking to the
degeneracy submanifold strongly, as the distance function in this subspace has
order $3$ (according to Theorem 3.3.8).

Examples 3.5.18, 3.5.20, and 3.5.23 demonstrate that known properties of
energy splittings in tight-binding models and interacting spin models can be
used to illustrate the geometrical description of the degeneracy submanifolds
in the space of Hermitian matrices. An exciting application of our results
would be to make use of this connection in the reverse direction: to describe
the degeneracy submanifold using geometrical tools, translate that information
to the language of quantum Hamiltonians, and thereby enable the construction
of interacting quantum systems with robust energy degeneracies, or even novel
quantum error correction codes.

##  4 Proofs and details

This section contains the proofs and further details of the statements in
Section 3. Every subsection here has the same title as the corresponding
subsection in Section 3.

###  4.1 The Schrieffer–Wolff transformation induces a local chart

###### Proof of Theorem 3.1.2 on the exact SW decomposition.

Recall that the diagonal matrix $H_{0}\in\Sigma_{k}$ is fixed, and define
$\breve{B}:=B+T+H_{\textup{eff}}=\widetilde{B}-H_{0}$, see decomposition
(3.1.4) and Figure 2 for notations. $\breve{B}$ is a block diagonal matrix of
$k\times k$ and $(n-k)\times(n-k)$ blocks. The correspondence

$(S,\breve{B})\mapsto H=f(S,\breve{B}):=e^{iS}\cdot(H_{0}+\breve{B})\cdot
e^{-iS}$ (4.1.1)

defines a real analytic map $f:\mathbb{R}^{n^{2}}\to\mathbb{R}^{n^{2}}$, in
fact, the dimension of the space of $(S,\breve{B})$ is $(n-k)^{2}+k^{2}+2\cdot
k\cdot(n-k)=n^{2}$. We apply the analytic inverse function theorem [33, pg.
47, Thm. 2.5.1.].

We show that the Jacobian of $H$ at $0$ has maximal rank, that is,
$\mbox{rk}((\mathrm{d}f)_{0})=n^{2}$. This is equivalent to the fact that for
every fix $(S,\breve{B})\neq(0,0)$, the first-order part of
$f(tS,t\breve{B})=e^{itS}\cdot(H_{0}+t\breve{B})\cdot e^{-itS}$ at $t=0$ is
nonzero. That is,

$\left.\frac{\mathrm{d}}{\mathrm{d}t}\left(e^{itS}\cdot(H_{0}+t\breve{B})\cdot
e^{-itS}\right)\right|_{t=0}=i[S,H_{0}]+\breve{B}\neq 0.$ (4.1.2)

To show that $i[S,H_{0}]+\breve{B}\neq 0$, consider the entry $a,b$ of the
commutator:

$[S,H_{0}]_{a,b}=\sum_{l=1}^{n}(S_{a,l}H_{0;l,b}-H_{0;a,l}S_{l,b})=S_{a,b}(\lambda_{b}-\lambda_{a}).$
(4.1.3)

It shows that $[S,H_{0}]$ is an off-block matrix, and it is nonzero if $S\neq
0$. Indeed, $S_{a,b}\neq 0$ can happen only if the index $a,b$ satisfies
$a\leq k<b$ or $b\leq k<a$, hence in this case $\lambda_{b}-\lambda_{a}\neq
0$. Since $\breve{B}$ is block diagonal, for this $a,b$ index
$\breve{B}_{a,b}=0$, therefore it cannot cancel $[S,H_{0}]_{a,b}\neq 0$. That
is, $(i[S,H_{0}]+\breve{B})_{a,b}\neq 0$, if $S_{a,b}\neq 0$, implying that
$i[S,H_{0}]+\breve{B}=0$ only if $S=0$ and $\breve{B}=0$.

By the analytic inverse function theorem, there is a neighborhood
$\widetilde{\mathcal{W}}_{0}$ of $(0,0)\in\mathbb{R}^{n^{2}}$ (in the
$(S,\breve{B})$ space) and $\mathcal{V}_{0}$ of $H_{0}\in\mbox{Herm}(n)$ such
that
$f|_{\widetilde{\mathcal{W}}_{0}}:\widetilde{\mathcal{W}}_{0}\to\mathcal{V}_{0}$
is a bijection, whose inverse is also analytic. This gives the unique
decomposition if $H\in\mathcal{V}_{0}$ and
$(\breve{B},S)\in\widetilde{W}_{0}$, and the analytic dependence of
$\breve{B}$ and $S$ on $H$.

Moreover, we have to show that the ‘lowest $k$ state property’ is satisfied,
that is, the first $k$ columns of $e^{iS}$ span the sum of the eigenspaces of
$H$ corresponding to the lowest $k$ eigenvalues. This is an additional
property, which possibly requires the choice of smaller neighborhoods
$\mathcal{V}_{0}$ and $\widetilde{W}_{0}$. Observe the following:

  1. (a)

The eigenvalues of $\widetilde{B}$ and $H$ are equal.

  2. (b)

The subspace spanned by the first $k$ columns of $e^{iS}$ is exactly the sum
of the eigenspaces of the $k\times k$ block of $\widetilde{B}$.

Therefore, the lowest $k$ state property is equivalent to the statement that
$\lambda<\mu$ holds for each eigenvalue $\lambda$ of the $k\times k$ block of
$\widetilde{B}$ and $\mu$ of the $(n-k)\times(n-k)$ block of $\widetilde{B}$.
The inequality $\lambda<\mu$ holds for $H=H_{0}$, and the eigenvalues are
continuous functions of $H$. Therefore, the inequality, and hence, the lowest
$k$ state property holds for every $H$ in a sufficiently small neighborhood of
$H_{0}$.

∎

######  Remark 4.1.4.

Surprisingly, in our proof the _continuous_ behaviour of the eigenvalues
implies the _analytic_ behaviour of the sum of the eigenspaces corresponding
to the lowest $k$ eigenvalues around $H_{0}\in\Sigma_{k}$ (more precisely, the
projector onto this eigenspace). Indeed, the dependence of $S$ on $H$ is
analytic, and the sum of the eigenspaces is equal to the subspace spanned by
the first $k$ columns of $e^{iS}$, which is deduced from the continuous
behaviour of the eigenvalues, and it implies the analytic dependence of the
projector.

In the Appendix we provide exact conditions for the neighborhoods
$\mathcal{V}_{0}$, $\mathcal{W}_{0}$ and $\mathcal{S}_{0}$ based on the
results in [3].

###### Proof of Corollary 3.1.6 on the local chart induced by the SW
decomposition.

(a) The proof of Theorem 3.1.2 shows that the correspondence
$H\mapsto(x_{i},y_{j})$ is an analytic bijection with analytic inverse between
the neighborhood $\mathcal{V}_{0}\subset\mbox{Herm}(n)$ of $H_{0}$ and a
neighborhood of $0\in\mathbb{R}^{n^{2}}$. (b) Equation (3.1.7) is a
straightforward consequence of the construction. Indeed, $y_{j}=0$ means that
$H_{\textup{eff}}=0$, and in this case $H\in\Sigma_{k}$ by equation (3.1.4).

∎

######  Remark 4.1.5.

The existence of the SW chart fitting to $\Sigma_{k}$ in the sense of
Corollary 3.1.6 gives an alternative proof for the Neumann–Wigner theorem,
stating that $\Sigma_{k}$ is a submanifold of codimension $k^{2}-1$ (see Table
1 for the sketch of the original proof).

###### Proof of Theorem 3.1.11 on the projection to $\Sigma_{k}$.

Let $\mathcal{P}$ denote the subspace spanned by the first $k$ columns of
$e^{iS}$, which is equal to the sum of the eigenspaces of the lowest $k$
eigenvalues of $H$. Let $\mathcal{P}^{\perp}$ denote its Hermitian complement,
this is the subspace spanned by the last $n-k$ columns of $e^{iS}$, and it
also agrees with the sum of the eigenspaces of the highest $n-k$ eigenvalues
of $H$. In particular, $\mathcal{P}$ and $\mathcal{P}^{\perp}$ are invariant
subspaces of $H_{\Sigma}$ and $H_{\textup{proj}}$. We show that the
restrictions of $H_{\textup{proj}}$ and $H_{\Sigma}$ to both $\mathcal{P}$ and
$\mathcal{P}^{\perp}$ are equal.

Both $H_{\Sigma}$ and $H_{\textup{proj}}$ has $k$-fold degeneracy, and
$\mathcal{P}$ is the eigenspace corresponding to the lowest $k$ degenerate
eigenvalues of $H_{\textup{proj}}$, and also $H_{\Sigma}$. Moreover the
degenerate eigenvalues of these matrices are also equal. Indeed, the trace of
the $k\times k$ block of $H_{\textup{proj}}$ is equal to
$\sum_{i=1}^{k}\lambda_{i}$ (where $\lambda_{i}$ denotes the eigenvalues of
$H$), since $H_{\textup{eff}}$ has trace 0. Hence the restrictions of
$H_{\textup{proj}}$ and $H_{\Sigma}$ to their common degenerate eigensubspace
$\mathcal{P}$ are equal.

On the other hand, their restrictions to $\mathcal{P}^{\perp}$ are equal to
the restriction of $H$. Then $H_{\textup{proj}}$ and $H_{\Sigma}$ agree on
both $\mathcal{P}$ and $\mathcal{P}^{\perp}$, hence
$H_{\textup{proj}}=H_{\Sigma}$.

∎

###  4.2 Energy splitting and the distance from $\Sigma_{k}$

In this subsection we prove Theorem 3.2.2 on the distance from $\Sigma_{k}$.
We start with its easy parts.

######  Proposition 4.2.1.

$d(H,H_{\Sigma})=\|H_{\textup{eff}}\|=\sqrt{k}\cdot D_{k}(H)$.

###### Proof.

$d(H,H_{\Sigma})=\|H-H_{\Sigma}\|=\|e^{iS}\cdot H_{\textup{eff}}\cdot
e^{-iS}\|=\|H_{\textup{eff}}\|$, the last equation follows from Lemma 2.1.7.
Applying it to Equation (3.1.10) implies that
$d(H,H_{\Sigma})=d(\Lambda,\Lambda_{\Sigma})$, which is obviously equal to

$\|\Lambda-\Lambda_{\Sigma}\|=\sqrt{\sum_{i=1}^{k}(\lambda_{i}-\overline{\lambda})^{2}}=\sqrt{k}\cdot\sigma(\lambda_{1},\dots,\lambda_{k})=\sqrt{k}\cdot
D_{k}(H),$ (4.2.2)

where the last equality defines $D_{k}(H)$ as the standard deviation of the
lowest $k$ eigenvalues of $H$. ∎

Next we show the harder part, namely, for every
$H\in\mbox{Herm}(n)\setminus\Sigma_{[k,k+1]}$ the unique closest point of
$\Sigma_{k}$ to $H$ is $H_{\Sigma}$. First of all, notice that $\Sigma_{k}$ is
not a compact set, hence, theoretically it might happen that $\Sigma_{k}$ does
not have closest point to $H$, more precisely, the distance function
$d_{H}:\mbox{Herm}(n)\to\mathbb{R}$ defined as $d_{H}(G)=d(H,G)$ does not have
a minimum on $\Sigma_{k}$. Avoiding this possibility causes several
complications, which are managed at the end of this subsection. But before
this, we first observe that a minimum point of the restricted function
$d_{H}|_{\Sigma_{k}}$ is also critical point of it, therefore we have the
following.

######  Proposition 4.2.3.

If $\mu$ is a minimum value of $d_{H}$ on $\Sigma_{k}$, that is,
$\mu=\min\\{d(H,G)\ |\ G\in\Sigma_{k}\\}$, and $K\in\Sigma_{k}$ satisfies
$d(H,K)=\mu$, then the line $\\{tK+(1-t)H\ |\ t\in\mathbb{R}\\}$ is orthogonal
to $\Sigma_{k}$ at $K$.

###### Proof.

$d_{H}$ has a global minimum at $K$, in particular, it is a local minimum.
Since $\Sigma_{k}$ is a smooth manifold, every local minimum of a smooth
function is a critical point, meaning that the differential
$(\mathrm{d}(d_{H}|_{\Sigma_{k}}))_{K}$ of $d_{H}|_{\Sigma_{k}}$ at $K$ is 0.
It means that the gradient of $d_{H}$ at $K$ is orthogonal to $\Sigma_{k}$.
Indeed, for any tangent vector $v\in T_{K}\Sigma_{k}$ of $\Sigma_{k}$ at $K$
the evaluation of the differential is

$(\mathrm{d}(d_{H}|_{\Sigma_{k}}))_{K}(v)=(\mathrm{d}(d_{H}))_{K}(v)=\langle\mbox{grad}_{K}(d_{H}),v\rangle,$
(4.2.4)

where the first equality comes from the definition of the restriction, and
second equality is definition of the gradient. Hence
$(\mathrm{d}(d_{H}|_{\Sigma_{k}}))_{K}$ is zero for every $v\in
T_{K}\Sigma_{k}$ if and only if $\mbox{grad}_{K}(d_{H})$ is orthogonal to
$T_{K}\Sigma_{k}$.

On the other hand, since $d_{H}$ is the distance from $H$, the gradient
$\mbox{grad}_{K}(d_{H})$ is parallel to $K-H$, hence it is parallel to the
line joining $H$ and $K$. Therefore, this line is orthogonal to the tangent
space $T_{K}\Sigma_{k}$, hence to $\Sigma_{k}$ at $K$.

∎

From now on we look for all the lines through $H$ orthogonal to $\Sigma_{k}$.
According to Proposition 4.2.3, the intersection points of these lines with
$\Sigma_{k}$ are the candidates for the closest point of $\Sigma_{k}$ to $H$.
Let

$L_{H}=\\{tH_{\Sigma}+(1-t)H\ |\ t\in\mathbb{R}\\}$ (4.2.5)

be the line joining $H$ and $H_{\Sigma}$.

######  Proposition 4.2.6.

$L_{H}$ is orthogonal to $\Sigma_{k}$ at the intersection point $H_{\Sigma}$.

Before the proof we highlight its essential step, the descripiton of the
tangent space.

######  Lemma 4.2.7.

Let $H_{0}\in\Sigma_{k}$ be a diagonal matrix. Then the tangent space
$T_{H_{0}}\Sigma_{k}$ of $\Sigma_{k}$ at $H_{0}$ consists of the Hermitian
matrices whose upper-left $k\times k$ block is a scalar matrix. See Figure 5.

Figure 5: The matrix form of the elements of the tangent space
$T_{H_{0}}\Sigma_{k}$ and normal space $N$ of $\Sigma_{k}$ at a diagonal
matrix $H_{0}\in\Sigma_{k}$. By Lemma 4.2.7, the tangent space consists of
matrices whose upper-left $k\times k$ block is a scalar matrix. The normal
space, i.e. the orthogonal complement of the tangent space agrees with the
space of effective Hamiltonians $\mbox{Herm}_{0}(k)\subset\mbox{Herm}(n)$: its
elements have only a traceless upper left $k\times k$ block and zero entries
everywhere else.

###### Proof of Lemma 4.2.7.

By the SW decomposition (3.1.4) and Corollary 3.1.6, $\Sigma_{k}$ is locally
given by the equation $H_{\textup{eff}}=0$, that is,
$y_{1}=\dots=y_{k^{2}-1}=0$. Hence the tangent space $T_{H_{0}}\Sigma_{k}$
consists of the directions orthogonal to the $H_{\textup{eff}}$ directions,
which includes everything in the $(n-k)\times(n-k)$ block, in the off-block,
and the trace of the $k\times k$ block, proving the lemma.

∎

######  Remark 4.2.8.

A more precise analysis also highlights the role of the off-block form of the
exponent $S$ in SW decomposition (3.1.4): its variation changes the off-block
elements of $H_{0}$ up to first order. Indeed, by Equation (4.1.2), every
tangent vector can be written in form $i[S,H_{0}]+\breve{B}$ with an off-block
$S$ and a block diagonal $\breve{B}$, where now $\breve{B}=B+T$, that is,
$H_{\textup{eff}}=0$, since we are in $\Sigma_{k}$. By Equation (4.1.3), the
entries of this tangent vector are
$iS_{a,b}(\lambda_{b}-\lambda_{a})+\breve{B}_{a,b}$, showing that $S$
generates the off-block elements, and $\breve{B}$ generates the
$(n-k)\times(n-k)$ block and the trace of the $k\times k$ block.

Moreover, one can prove Lemma 4.2.7 without referring to the SW decomposition,
but starting from an over-parametrization of $\Sigma_{k}$ around $H_{0}$ of
the form $e^{iG}(H_{0}+\breve{B})e^{-iG}$, where $\breve{B}=B+T$ as above, but
now $G$ can be any element of $\mbox{Herm}(n)$. This leads to tangent vectors
of the form $i[G,H_{0}]+\breve{B}$ with entries
$iG_{a,b}(\lambda_{b}-\lambda_{a})+\breve{B}_{a,b}$, which also shows that the
off-diagonal elements of a tangent vector depend on the off-block entries of
$G$ and it can be arbitrary, since $\lambda_{b}-\lambda_{a}\neq 0$ between
different blocks; the $k\times k$ block of $G$ is irrelevent, since
$\lambda_{b}-\lambda_{a}=0$ if $a,b\leq k$; and the $(n-k)\times(n-k)$ block
of $G$ contributes only to the $(n-k)\times(n-k)$ block of the tangent vector,
which can be arbitrary by a choice of $\breve{B}$, as well as the trace of the
$k\times k$ block.

###### Proof of Proposition 4.2.6.

It is enough to show that $H-H_{\Sigma}$ is orthogonal to the tangent space
$T_{H_{\Sigma}}\Sigma_{k}$ of $\Sigma_{k}$ at $H_{\Sigma}$. Because of Lemma
2.1.7 it is enough to show the orthogonality in the diagonal case, namely,
$\Lambda-\Lambda_{\Sigma}$ is orthogonal to $T_{\Lambda_{\Sigma}}\Sigma_{k}$.
But $\Lambda-\Lambda_{\Sigma}$ is a diagonal matrix with nonzero elements only
in the $k\times k$ block, and its trace is 0. More precisely, the diagonal
elements are $\lambda_{i}-\overline{\lambda}$ ($i=1,\dots,k$) where
$\lambda_{i}$ are lowest $k$ eigenvalues of $H$ and $\overline{\lambda}$ is
their mean value. By Lemma 4.2.7, the $k\times k$ block of each tangent vector
$K\in T_{\Lambda_{\Sigma}}\Sigma_{k}$ is a scalar matrix, we denote its
diagonal entries by $c$. Then

$\langle\Lambda-\Lambda_{\Sigma},K\rangle=\sum_{i=1}^{k}c(\lambda_{i}-\overline{\lambda})=0,$
(4.2.9)

proving the proposition. ∎

######  Remark 4.2.10.

If $H$ is sufficiently close to $\Sigma_{k}$, then the orthogonality of
$L_{H}$ to $\Sigma_{k}$ implies that $H_{\Sigma}$ is the unique closest point
of $\Sigma_{k}$ to $H$. This follows from the tubular neighborhood theorem
[34] (see also [25, pg. 74, exercise 3.]): if a point $p\in\mathbb{R}^{n}$ is
sufficiently close to a submanifold $N\subset\mathbb{R}^{n}$, then it has a
unique closest point $p^{\prime}\in N$, characterised by the orthogonality of
the line joining $p$ and $p^{\prime}$ to $N$. However, it is not enough for
the global version of Theorem 3.2.2, that is, for every
$H\in\mbox{Herm}(n)\setminus\Sigma_{[k,k+1]}$.

In the following we characterise the lines orthogonal to $\Sigma_{k}$ at a
point.

######  Proposition 4.2.11.

Let $L$ be a line in $\mbox{Herm}(n)$ through a point $H_{0}\in\Sigma_{k}$.
Then the following are equivalent:

  1. 1.

$L=L_{H}$ for an element $H\in L\setminus\\{H_{0}\\}$ (in particular,
$H_{\Sigma}=H_{0}$).

  2. 2.

$L$ is orthogonal to $\Sigma_{k}$ at $H_{0}$.

  3. 3.

$L$ can be parametrized as follows: Starting with any diagonalization
$H_{0}=U\Lambda_{0}U^{-1}$ with increasing order of the eigenvalues, we choose
a $k\times k$ diagonal matrix $D$ of trace 0 with increasing order of the
diagonal elements in the upper-left block, and take the parametrization

$t\mapsto U(\Lambda_{0}+tD)U^{-1}.$ (4.2.12)

  4. 4.

$L=L_{H}$ for every $H\in L\setminus\\{H_{0}\\}$ sufficiently close to $H_{0}$
(in particular, $H_{\Sigma}=H_{0}$).

###### Proof.

(1) $\Rightarrow$ (2): Follows from Proposition 4.2.6.

(1) $\Rightarrow$ (3): Follows directly from the construction of $H_{\Sigma}$,
by choosing $D=\Lambda-\Lambda_{0}$, with any choice of $U$ diagonalizing $H$
(hence, also $H_{0}$), where $\Lambda=U^{-1}HU$ is diagonal.

(3) $\Rightarrow$ (4): Consider an element $G=U(\Lambda_{0}+tD)U^{-1}\in L$,
then $G_{\Sigma}=H_{0}$ if the diagonal elements of $\Lambda_{\Sigma}+tD$ are
still in increasing order, that is, until the line reaches the
$\Sigma_{[k,k+1]}$ degeneracy stratum. This holds for sufficiently small $t$.

(4) $\Rightarrow$ (1) is obvious. Until now we proved the equivalence of (1),
(3) and (4), and any of them implies (2).

(2) $\Rightarrow$ (3): It follows by counting the dimensions, cf. Table 1 for
a similar method. First note that (3) implies (2), that is, the
parametrization (4.2.12) provides orthogonal lines to $\Sigma_{k}$ at $H_{0}$.
Then we count how many dimensions can be covered by such a parametrization.
The choice of $D$ up to a real scalar factor gives $k-1$ dimensions. The
choice of the lowest $k$ eigenspaces, that is, the first $k$ columns of $U$ up
to $U(1)$ rotations gives $k^{2}-k$ dimensions. Hence, the dimension of the
subspace of matrices which can be reached by a parametrization in form
(4.2.12) has $k-1+k^{2}-k=k^{2}-1$ dimensions, therefore it covers the whole
normal space of $\Sigma_{k}$ at $H_{0}$.

The above implications imply the equivalence, proving the theorem.

∎

######  Remark 4.2.13.

By point (3) of the above proposition, the orthogonal lines to $\Sigma_{k}$ at
$H_{0}$ can be described as ‘spreading the eigenvalues linearly’. Although
this construction seems to be insightful, its difficulty is that the direction
of the line depends on both $D$ and the unitary matrix $U$ – more precisely,
on the subspaces spanned by its first $k$ columns. Another way to characterize
the orthogonal lines via SW decomposition is ‘turning an effective Hamiltonian
on’: Starting from the unique SW decomposition (3.1.8) of $H_{0}$ (with
respect to a possibly different base point $H_{0}^{\prime}$), we choose a
$k\times k$ Hermitian matrix $H_{\textup{eff}}$ of trace 0 in the upper left
block. The corresponding line is parametrized as

$t\mapsto H_{0}+t\cdot e^{iS}\cdot H_{\textup{eff}}\cdot e^{-iS},$ (4.2.14)

which is orthogonal to $\Sigma_{k}$ at $H_{0}$, cf. Lemma 4.2.7.

Figure 6: The projections of $H$ onto $\Sigma_{k}$ for $n=3$, $k=2$,
illustrating the proof of Proposition 4.2.20.
Panel $(a)$ ($(b)$ respectively): $H_{\Sigma}=H_{\Sigma}^{\\{1,2\\}}$
($H_{\Sigma}^{I}=H_{\Sigma}^{\\{1,3\\}}$) is constructed by contracting the
eigenvalues $\lambda_{1}$ and $\lambda_{2}$ ($\lambda_{1}$ and $\lambda_{3}$)
of $H$ to their mean value. The linear motion of the eigenvalues (red, green
and blues line segments) realizes the line segments of $L_{H}$ ($L_{H}^{I}$),
joining $H$ with $H_{\Sigma}$ ($H_{\Sigma}^{I}$).
Panel $(c)$: The crossing of the eigenvalues is resolved (see colors),
providing another matrix $\widetilde{H}_{\Sigma}^{I}\in\Sigma_{k}$. The non-
linear motion of the eigenvalues traces a broken-line (consisting of two line
segments) joining $H$ and $\widetilde{H}_{\Sigma}^{I}$. The vertex of the
broken line is denoted by $K$.
Panel $(d)$: Illustration of the whole configuration in $\mbox{Herm}(n)$. By
construction, the distances $d(K,H_{\Sigma}^{I})$ and
$d(K,\widetilde{H}_{\Sigma}^{I})$ are equal, it is denoted by $d_{2}$. With
the notations $d_{1}=d(H,K)$ and $d_{3}=d(H,\widetilde{H}_{\Sigma}^{I})$
(dashed line), $d_{3}<d_{1}+d_{2}$ holds by the triangle inequality, showing
that $\widetilde{H}_{\Sigma}^{I}$ is a closer point of $\Sigma_{k}$ to $H$
than $H_{\Sigma}^{I}$

Next we characterise the lines through a point
$H\in\mbox{Herm}(n)\setminus\Sigma_{[k,k+1]}$ which intersect $\Sigma_{k}$
orthogonally. $L_{H}$ is one of these lines. To find the others, first
consider an example, a matrix $H\in\mbox{Herm}(3)$ with eigenvalues
$\lambda_{1}<\lambda_{2}<\lambda_{3}$. To obtain its projection
$H_{\Sigma}^{\\{1,2\\}}=H_{\Sigma}\in\Sigma_{2}$, we contract $\lambda_{1}$
and $\lambda_{2}$ to their mean value
$\overline{\lambda}^{\\{1,2\\}}=(\lambda_{1}+\lambda_{2})/2$, then the line
through $H_{\Sigma}$ and $H$ is $L_{H}^{\\{1,2\\}}=L_{H}$.

Another possibility is the contraction of $\lambda_{1}$ and $\lambda_{3}$ to
$\overline{\lambda}^{\\{1,3\\}}=(\lambda_{1}+\lambda_{3})/2$. If
$\overline{\lambda}^{\\{1,3\\}}<\lambda_{2}$, then we obtain a point
$H_{\Sigma}^{\\{1,3\\}}\in\Sigma_{2}$. Consider the line

$L_{H}^{\\{1,3\\}}=\\{tH_{\Sigma}^{\\{1,3\\}}+(1-t)H\ |\ t\in\mathbb{R}\\}$
(4.2.15)

joining $H_{\Sigma}^{\\{1,3\\}}$ and $H$. It consists of the matrices

$\displaystyle tH_{\Sigma}^{\\{1,3\\}}+(1-t)H$ $\displaystyle=$ $\displaystyle
U\begin{pmatrix}\lambda_{1}+\frac{t}{2}(\lambda_{3}-\lambda_{1})&0&0\\\
0&\lambda_{2}&0\\\
0&0&\lambda_{3}+\frac{t}{2}(\lambda_{1}-\lambda_{3})\end{pmatrix}U^{-1}$
(4.2.16) $\displaystyle=$ $\displaystyle
U^{\prime}\begin{pmatrix}\lambda_{1}+\frac{t}{2}(\lambda_{3}-\lambda_{1})&0&0\\\
0&\lambda_{3}+\frac{t}{2}(\lambda_{1}-\lambda_{3})&0\\\
0&0&\lambda_{2}\end{pmatrix}(U^{\prime})^{-1},$

where $U^{\prime}$ is the product of $U$ with the transposition of the 2nd and
3th basis elements. The role of $U^{\prime}$ is to satisfy the increasing
order of the eigenvalues, if we want to have a conventional form. Then, by
Proposition 4.2.11, $L_{H}^{\\{1,3\\}}$ is orthogonal to $\Sigma_{2}$ at
$H_{\Sigma}^{\\{1,3\\}}$. However, it crosses the stratum $\Sigma_{(12)}$ of
$\Sigma$ (corresponding to the degeneracy $\lambda_{2}=\lambda_{3}$) between
$H_{\Sigma}^{\\{1,3\\}}$ and $H$, namely, for
$t=2(\lambda_{3}-\lambda_{2})/(\lambda_{3}-\lambda_{1})$. See Figure 6.

The third possibility, the contraction of $\lambda_{2}$ and $\lambda_{3}$ to
$\overline{\lambda}^{\\{2,3\\}}$ results a matrix $H_{\Sigma}^{\\{2,3\\}}$,
which is not in $\Sigma_{2}$, since
$\overline{\lambda}^{\\{2,3\\}}>\lambda_{1}$.

The same can be done in general, for an
$H\in\mbox{Herm}(n)\setminus\Sigma_{[k,k+1]}$. Choose $k$ indices $1\leq
i_{1}<i_{2}<\dots<i_{k}\leq n$, and let $I=\\{i_{1},\dots,i_{k}\\}$ denote
their set. Let $H_{\Sigma}^{I}$ denote the matrix obtained from $H$ by
replacing the eigenvalues $\lambda_{i_{1}},\dots,\lambda_{i_{k}}$ with their
mean $\overline{\lambda}^{I}=\sum_{j=1}^{k}\lambda_{i_{j}}/k$. More precisely,
let $\Lambda_{\Sigma}^{I}$ be the diagonal matrix obtained from
$\Lambda=U^{-1}HU=\mbox{diag}(\lambda_{1},\dots,\lambda_{n})$ by replacing
$\lambda_{i}$ with $\overline{\lambda}^{I}$ for $i\in I$, and define

$H_{\Sigma}^{I}=U\cdot\Lambda_{\Sigma}^{I}\cdot U^{-1}.$ (4.2.17)

In general, $H_{\Sigma}^{I}$ can depend on the choice of $U$. In fact, if
there is a degeneracy $\lambda_{i}=\lambda_{i^{\prime}}$ of eigenvalues of $H$
with $i\in I$ and $i^{\prime}\notin I$, then their separation depends on the
choice of the diagonalization inside the degenerate eigenspace. For simplicity
we omit the $U$ dependence from the notation, and $H_{\Sigma}^{I}$ denotes any
of the possible choices.

If $\overline{\lambda}^{I}$ is smaller then the lowest omitted eigenvalue
$\lambda_{m}$, where $m=\min(\\{1,\dots,n\\}\setminus I)$, then
$H_{\Sigma}^{I}\in\Sigma_{k}$. Let

$L_{H}^{I}=\\{tH_{\Sigma}^{I}+(1-t)H\ |\ t\in\mathbb{R}\\}$ (4.2.18)

be the line joining $H_{\Sigma}^{I}$ and $H$ Obviously, $L_{H}^{I}$ crosses
other strata of $\Sigma$ between $H$ and $H_{\Sigma}^{I}$. By Proposition
4.2.11, the lines $L_{H}^{I}$ are orthogonal to $\Sigma_{k}$ at
$H_{\Sigma}^{I}$, moreover:

######  Proposition 4.2.19.

The only lines through $H\in\mbox{Herm}(n)\setminus\Sigma_{[k,k+1]}$ which are
orthogonal to $\Sigma_{k}$ are the $L_{H}^{I}$ lines with
$I\subset\\{1,\dots,n\\}$ satisfying $H_{\Sigma}^{I}\in\Sigma_{k}$.

###### Proof.

Assume that a line $L$ through $H$ intersects $\Sigma_{k}$ orthogonally at
$H_{0}\in\Sigma_{k}$. By point (3) of Proposition 4.2.11, $L$ has a special
parametrization in form (4.2.12). The linear motion of the degenerate
eigenvalues of $H_{0}$ arrives at $k$ eigenvalues of $H$ corresponding to the
indices $\\{i_{1},\dots,i_{k}\\}=I$, and then, $H_{0}=H_{\Sigma}^{I}$ by the
construction. ∎

Recall that our goal is to prove that $H_{\Sigma}$ is the closest point of
$\Sigma_{k}$ to $H$. Towards this goal the next step is to show that
$H^{I}_{\Sigma}$ cannot be a closest point of $\Sigma_{k}$ to $H$ if
$I\neq\\{1,2,\dots,k\\}$. Recall that it does not imply directly that
$H_{\Sigma}$ is the closest point, until we prove that there is a closest
point, which will be the last step.

######  Proposition 4.2.20.

Consider an index set $I\neq\\{1,2,\dots,k\\}$ and the corresponding point
$H_{\Sigma}^{I}\in\Sigma_{k}$ (with a fixed unitary matrix $U$ diagonalizing
$H$, cf. Equation (4.2.17)). Then, there is a point
$\widetilde{H}_{\Sigma}^{I}\in\Sigma_{k}$ such that
$d(H,\widetilde{H}_{\Sigma}^{I})<d(H,H_{\Sigma}^{I})$.

###### Proof.

We fix $U$ during the construction, hence we actually work in the space
$\mathbb{R}^{n}$ of the diagonal entries, endowed with the usual inner
product. The lines correspond to the linear motion of the diagonal entries.

Consider two indices $i\in I$ and $i^{\prime}\notin I$ with $i>i^{\prime}$.
Then $\lambda_{i}>\lambda_{i^{\prime}}$ holds for the eigenvalues of $H$, and
$\overline{\lambda}^{I}<\lambda_{i^{\prime}}$, since
$H^{I}_{\Sigma}\in\Sigma_{k}$. Define $\widetilde{H}_{\Sigma}^{I}$ as

$\widetilde{H}_{\Sigma}^{I}=U\cdot\widetilde{\Lambda}_{\Sigma}^{I}\cdot
U^{-1},$ (4.2.21)

where $\widetilde{\Lambda}_{\Sigma}^{I}$ is the diagonal matrix containing the
same entries as ${\Lambda}_{\Sigma}^{I}$, but the $i$-th and $i^{\prime}$-th
eigenvalues swapped. Namely, the $i$-th diagonal element of
$\widetilde{\Lambda}_{\Sigma}^{I}$ is $\lambda_{i^{\prime}}$, and the
$i^{\prime}$-th diagonal element of $\widetilde{\Lambda}_{\Sigma}^{I}$ is
$\overline{\lambda}^{I}$. See Figure 6 and Table 2.

|  $H$ |  $H_{\Sigma}^{I}$ |  $\widetilde{H}_{\Sigma}^{I}$ |  $K$
---|---|---|---|---
$j\in I$, $j\neq i$ |  $\lambda_{j}$ |  $\overline{\lambda}^{I}$ |  $\overline{\lambda}^{I}$ |  $t\overline{\lambda}^{I}+(1-t)\lambda_{j}$
$j^{\prime}\notin I$, $j^{\prime}\neq i^{\prime}$ |  $\lambda_{j^{\prime}}$ |  $\lambda_{j^{\prime}}$ |  $\lambda_{j^{\prime}}$ |  $\lambda_{j^{\prime}}$
$i\ (\in I)$ |  $\lambda_{i}$ |  $\overline{\lambda}^{I}$ |  $\lambda_{i^{\prime}}$ |  $\lambda_{i^{\prime}}$
$i^{\prime}\ (\notin I)$ |  $\lambda_{i^{\prime}}$ |  $\lambda_{i^{\prime}}$ |  $\overline{\lambda}^{I}$ |  $\lambda_{i^{\prime}}$
Table 2: The diagonal entries of the diagonalizations of $H$,
$H_{\Sigma}^{I}$, $\widetilde{H}_{\Sigma}^{I}$ and $K$ in the proof of
Proposition 4.2.20. The four matrices are diagonalized by the same unitary
matrix $U$, and the diagonal entries of $H$ are in increasing order.
$H_{\Sigma}^{I}$ is constructed by replacing the eigenvalues of $H$
corresponding to the indices in $I$ with their mean value
$\overline{\lambda}^{I}$. For two fixed indices $i\in I$ and $i^{\prime}\notin
I$ with $i>i^{\prime}$, we swap the $i$-th and $i^{\prime}$-th diagonal
entries of ${H}_{\Sigma}^{I}$, obtaining $\widetilde{H}_{\Sigma}^{I}$. $K$ is
defined as $K=tH_{\Sigma}^{I}+(1-t)H$ with
$t\overline{\lambda}^{I}+(1-t)\lambda_{i}=\lambda_{i^{\prime}}$.

We show that $d(H,\widetilde{H}_{\Sigma}^{I})<d(H,H_{\Sigma}^{I})$. Along the
line segment of $L_{H}^{I}$ joining $H$ and $H_{\Sigma}^{I}$, the eigenvalue
$\lambda_{i}$ moves to $\overline{\lambda}^{I}$ linearly, and it crosses
$\lambda_{i^{\prime}}$ at a point $K\in L_{H}^{I}$. More precisely,
$K=tH_{\Sigma}^{I}+(1-t)H$ with
$t\overline{\lambda}^{I}+(1-t)\lambda_{i}=\lambda_{i^{\prime}}$. Both the
$i$-th and $i^{\prime}$-th diagonal entry of $K$ is $\lambda_{i^{\prime}}$,
see Figure 6 and Table 2. Then we have

$d(K,\widetilde{H}_{\Sigma}^{I})=d(K,{H}_{\Sigma}^{I}).$ (4.2.22)

Indeed, the only difference in the distances might come from the $i$-th and
$i^{\prime}$-th diagonal entries (see Table 2), which gives

$(d(K,\widetilde{H}_{\Sigma}^{I}))^{2}-(d(K,{H}_{\Sigma}^{I}))^{2}=[(\lambda_{i^{\prime}}-\lambda_{i^{\prime}})^{2}+(\lambda_{i^{\prime}}-\overline{\lambda}^{I})^{2}]-[(\lambda_{i^{\prime}}-\overline{\lambda}^{I})^{2}+(\lambda_{i^{\prime}}-\lambda_{i^{\prime}})^{2}]=0.$
(4.2.23)

Then,

$d(H,H_{\Sigma}^{I})=d(H,K)+d(K,H_{\Sigma}^{I})=d(H,K)+d(K,\widetilde{H}_{\Sigma}^{I})>d(H,\widetilde{H}_{\Sigma}^{I}),$
(4.2.24)

where the last inequality is the triangle inequality applied to the non-
collinear points $H$, $K$ and $\widetilde{H}_{\Sigma}^{I}$, see Figure 6. This
completes the proof.

∎

Comparing Proposition 4.2.3, 4.2.6, 4.2.19 and 4.2.20, we conclude the
following.

######  Corollary 4.2.25.

If the distance function $d_{H}$ has a global minimum $\mu$ on $\Sigma_{k}$,
then $d(H,H_{\Sigma})=\mu$ and $H_{\Sigma}$ is the unique closest point of
$\Sigma_{k}$ to $H$. In particular, $d(H,\Sigma_{k})=d(H,H_{\Sigma})$ holds.

In the following we show that $d_{H}$ has a global minimum on $\Sigma_{k}$.
First we take a weaker observation.

######  Proposition 4.2.26.

The distance function $d_{H}$ has a global minimum on the closure
$\mbox{cl}(\Sigma_{k})$ of $\Sigma_{k}$ in $\mbox{Herm}(n)$.

###### Proof.

Actually it is a classical fact for any closed subset $A$ of $\mathbb{R}^{n}$
and a point $P\in\mathbb{R}^{n}$ that $A$ has a point with minimal distance
from $P$. For completeness we write a proof for our particular situation,
which works in general.

Take a closed ball $\mathcal{B}\subset\mbox{Herm}(n)$ of radius $R$ centered
at $H$, that is, $\mathcal{B}=\\{G\in\mbox{Herm}(n)\ |\ d(H,G)\leq R\\}$. The
radius $R$ has to be chosen such that $\mbox{cl}(\Sigma_{k})\cap\mathcal{B}$
is non-empty, e.g., $R>d(H,H_{\Sigma})$ is good. If $d_{H}$ has a global
minimum on $\mbox{cl}(\Sigma_{k})\cap\mathcal{B}$, then it is the global
minimum on $\mbox{cl}(\Sigma_{k})$ as well.

$\mbox{cl}(\Sigma_{k})\cap\mathcal{B}$ is a closed and bounded subset of
$\mathbb{R}^{n}$, hence it is compact, therefore, any continuous function on
it has a global minimum, proving the proposition. ∎

######  Proposition 4.2.27.

$H_{\Sigma}$ is the unique closest point of $\Sigma_{k}$ to $H$. In
particular, $d(H,\Sigma_{k})=d(H,H_{\Sigma})$ holds.

###### Proof.

According to Proposition 4.2.26, take a point $K\in\mbox{cl}(\Sigma_{k})$ with
minimal distance from $H$. Recall from Section 2.2 the disjoint decomposition
$\mbox{cl}(\Sigma_{k})=\Sigma_{k}\cup\Sigma_{k+1}\cup\dots\cup\Sigma_{n}$,
hence, $K\in\Sigma_{k^{\prime}}$ holds with a $k^{\prime}\geq k$. Then, by
applying Corollary 4.2.25 to $\Sigma_{k^{\prime}}$, it follows that $K$ is the
projection of $H$ to $\Sigma_{k^{\prime}}$ by contracting the lowest
$k^{\prime}$ eigenvalues of $H$ to their mean value.

Assume indirectly that $k^{\prime}>k$. The triangle of vertices $H$,
$H_{\Sigma}$, $K$ has a right angle at $H_{\Sigma}$. Indeed, by Proposition
4.2.6, the line $L_{H}$ joining $H$ and $H_{\Sigma}$ is orthogonal to
$\Sigma_{k}$. Since the line joining $H_{\Sigma}$ and $K$ lies in
$\Sigma_{k}$, it is orthogonal to $L_{H}$. See Figure 7. It follows that
$d(H,H_{\Sigma})<d(H,K)$, which contradicts with the premise that $K$ is the
closest point of $\mbox{cl}(\Sigma_{k})$ to $H$.

Figure 7: The proof of Proposition 4.2.27. Since $k^{\prime}>k$, the closure
of $\Sigma_{k}$ contains $\Sigma_{k^{\prime}}$. The projections of $H$ to
$\Sigma_{k}$ and $\Sigma_{k^{\prime}}$, (obtained by contracting the lowest
$k$ and the lowest $k^{\prime}$ eigenvalues of $H$ to their mean value) is
denoted by $H_{\Sigma}$ and $K$, respectively. $K$ is also the projection of
$H_{\Sigma}$ to $\Sigma_{k^{\prime}}$ in the same sense. Moreover, the line
segment joining $H_{\Sigma}$ and $K$ lies in $\Sigma_{k}$, implying that it is
orthogonal to the line segment joining $H$ and $H_{\Sigma}$, by Proposition
4.2.6. Therefore, the right triangle with vertices $H$, $H_{\Sigma}$ and $K$
shows that $d(H,H_{\Sigma})<d(H,K)$. (The orthogonality of the line segment
joining $H$ and $K$ to $\Sigma_{k^{\prime}}$ and $H_{\Sigma}$, respectively,
and $K$ to $\Sigma_{k^{\prime}}$ is not used in the proof.)

Therefore, $k^{\prime}=k$ and $K=H_{\Sigma}$, which proves the proposition. ∎

###### Proof of Theorem 3.2.2 on the distance from $\Sigma_{k}$.

Proposition 4.2.27 and Corollary 4.2.1 prove Theorem 3.2.2.

∎

######  Remark 4.2.28.

The same argument holds for arbitrary (not necessarily ground state) $k$-fold
degeneracy with the obvious modifications. The degeneracy set
$\Sigma^{(a)}_{k}$ of matrices with
$\lambda_{a}<\lambda_{a+1}=\lambda_{a+2}=\dots=\lambda_{a+k}<\lambda_{a+k+1}$
is a submanifold of $\mbox{Herm}(n)$, and if $\lambda_{a}<\lambda_{a+1}$ and
$\lambda_{a+k}<\lambda_{a+k+1}$ holds for a matrix $H\in\mbox{Herm}(n)$, then
$\Sigma^{(a)}_{k}$ has a unique closest element $H_{\Sigma}^{(a)}$ to $H$. The
matrix $H_{\Sigma}^{(a)}$ is constructed from $H$ by replacing its eigenvalues
$\lambda_{a+1},\dots,\lambda_{a+k}$ with their mean value. The proof presented
here can be generalized with the straightforward modifications.

######  Remark 4.2.29.

If $H\in\Sigma_{[k,k+1]}$, then $H_{\Sigma}$ is not unique, it depends on the
choice of the unitary matrix diagonalizing $H$, more precisely, on the
subspace generated by the $k$-th column of $U$. However, $d(H,H_{\Sigma})$ is
the same for every $H_{\Sigma}$, and it is the global minimum of the distance
function $d_{H}$ on $\Sigma_{k}$. Therefore, $d(H,\Sigma_{k})=d(H,H_{\Sigma})$
holds in this case with any choice of $H_{\Sigma}$.

###  4.3 Order of energy splitting of a degeneracy due to a perturbation

In this subsection we prove Theorem 3.3.5 on the analyticity of the
eigenvalues, Theorem 3.3.6, Theorem 3.3.8 and Corollary 3.3.10 on the order of
energy splitting and distancing.

Consider a one-parameter analytic perturbation
$H:\mathbb{R}\to\mbox{Herm}(n)$, $H(0)=H_{0}\in\Sigma_{k}$. The standard
deviation $D_{k}(H(t))$ is not differentiable at 0 in general. We want to
modify it slightly to obtain an analytic function $\mathfrak{s}_{k}(t)$ which
agrees with $D_{k}(H(t))$ up to a sign for every $t$, and for $t\geq 0$ they
are equal. For this we use the following observation:

######  Lemma 4.3.1.

Let $f_{1},\dots,f_{l}:\mathbb{R}\to\mathbb{R}$ be analytic functions defined
in a neighborhood of 0, that is, locally convergent power series centered at
0. Let $r=\min_{1\leq i\leq l}\\{\mathrm{ord}_{0}(f_{i}(t))\\}$ be the minimum
of the orders. Then

$F(t):=(\mbox{sgn}(t))^{r}\cdot\sqrt{\sum_{i=1}^{l}(f_{i}(t))^{2}}$ (4.3.2)

is also analytic at 0, and its order $\mathrm{ord}_{0}(F(t))$ is equal to $r$.

###### Proof.

The proof can be done by factoring out the lowest degree term under the root
sign. We can write

$f_{i}(t)=t^{r_{i}}\cdot\widetilde{f}_{i}(t),$ (4.3.3)

where $r_{i}=\mathrm{ord}_{0}(f_{i})$ hence $\widetilde{f}_{i}$ is a power
series with nonzero constant term, that is, $\widetilde{f}_{i}(0)\neq 0$.
Taking $r:=\min\\{r_{i}\\}$, we can write

$F(t)=(\mbox{sgn}(t))^{r}\cdot|t^{r}|\cdot\sqrt{\sum_{i=1}^{l}t^{2r_{i}-2r}\cdot(\widetilde{f}_{i}(t))^{2}}=:t^{r}\sqrt{\widetilde{F}(t)}.$
(4.3.4)

The expression under the square root (denoted by $\widetilde{F}(t)$) is a
power series with nonzero constant term, that is,
$\mathrm{ord}_{0}(\widetilde{F}(t))=0$. Indeed, (1) all the exponents
$2r_{i}-2r$ are bigger or equal to $0$, and (2) at least one of them is zero,
moreover (3) the coefficients corresponding to the zero exponents — that is,
the constant terms of the corresponding $(\widetilde{f}_{i}(t))^{2}$ — are
positive, hence they cannot cancel each other.

Hence $\widetilde{F}(0)\neq 0$, which implies that $\sqrt{\widetilde{F}(t)}$
is also an analytic function (a locally convergent power series), and its rank
is 0. Therefore $F(t)$ is analytic too. The order of $F(t)$ is equal to $r$,
the minimum of the orders of $f_{i}$, proving Lemma 4.3.1. ∎

By SW decomposition theorem 3.1.2 the effective Hamiltonian $H_{\textup{eff}}$
depends on $H$ analytically, the real matrix elements $y_{j}(H(t))$ of the
effective Hamiltonian $H_{\textup{eff}}(t)$ of $H(t)$ are analytic functions
of $t$. Let $r$ be the minimum of the orders of $y_{j}(H(t))$. Define

$\mathfrak{s}_{k}(t):=(\mbox{sgn}(t))^{r}\cdot D_{k}(H(t)).$ (4.3.5)

Note that for $t>0$ this agrees with the function $\mathfrak{s}_{k}(t)$
defined in Equation 3.3.1, indeed, the present version is its analytic
extension for every $t$.

######  Corollary 4.3.6.

$\mathfrak{s}_{k}(t)$ is an analytic function at $0$ of order $r$.

###### Proof.

By Theorem 3.2.2,

$\sqrt{k}\cdot\mathfrak{s}_{k}(t)=(\mbox{sgn}(t))^{r}\cdot\|H_{\textup{eff}}(t)\|=(\mbox{sgn}(t))^{r}\cdot\sqrt{\sum_{j=1}^{k^{2}-1}(y_{j}(H(t)))^{2}},$
(4.3.7)

and by Lemma 4.3.1, the right hand side is analytic and its order is $r$. ∎

In the following we want to compare the order of $\mathfrak{s}_{k}(t)$ with
the order of the pairwise differences of the eigenvalues, but in general,
these differences in the form $\lambda_{i}(t)-\lambda_{j}(t)$ are not
differentiable at the origin, cf. Figure 8. Similarly to
$\mathfrak{s}_{k}(t)$, we can modify the pairwise differences by a suitable
reordering of the eigenvalues using Theorem 3.3.5, which is proved here in the
following reformulated form.

######  Theorem 4.3.8 (The eigenvalues are analytic).

Let $H(t)$ be an analytic one-parameter family of Hermitian matrices. There
are analytic functions
$\widetilde{\lambda}_{1}(t),\dots,\widetilde{\lambda}_{n}(t)$ such that for
all $t$ the eigenvalues of $H(t)$ are $\widetilde{\lambda}_{i}(t)$
($i=1,\dots,n$).

Equivalently, there is a permutation $i\mapsto i^{\prime}$ of the indices
$i\in\\{1,\dots,n\\}$ such that the functions

$\widetilde{\lambda}_{i}(t)=\left\\{\begin{array}[]{lcc}\lambda_{i}(t)&\mbox{if}&t\geq
0,\\\ \lambda_{i^{\prime}}(t)&\mbox{if}&t<0\par\end{array}\right.$ (4.3.9)

are analytic.

Figure 8: Separating the degenerate eigenvalues in the proof of Theorem 4.3.8.
(a) The eigenvalues $\lambda_{i}$ of $H(t)$. The degenerate eigenvalues in
increasing order are not analytic functions, however, after a suitable
permutation on the negative part the eigenvalues form analytic functions. (b)
To find the right permutation, we consider the eigenvalues $\lambda_{i}^{(1)}$
of $H_{\textup{eff}}(t)/t$. These are ‘less degenerate’ than the eigenvalues
of $H$ in the sense that the order of the pairwise energy splittings decreases
by one, in particular, the first-order intersections become non-degenerate.

Although it follows from a more general argument [35, Theorem 1.10.], we
present here a new direct proof based on the exact SW decomposition Theorem
3.1.2. In the interesting case $H(0)$ is degenerate, e.g.,
$H(0)\in\Sigma_{k}$. By Remark 4.3.10 below, the permutation is the identity
for non-degenerate eigenvalues, i.e., $i^{\prime}=i$ for a non-degenerate
eigenvalue $\lambda_{i}$.

######  Remark 4.3.10.

The non-degenerate eigenvalues are always analytic functions of the parameters
for arbitrary matrix families with arbitrary (finite) number of parameters.
More precisely, consider a family $H(t)$ of complex matrices depending
analitically on the parameter $t\in\mathbb{R}^{m}$ defined in a neighborhood
of the origin. If $\lambda(0)$ is a non-degenerate eigenvalue of $H(0)$, then
there is an analytic family $\lambda(t)$ such that $\lambda(t)$ is an
eigenvalue of $H(t)$. Classically it is proved via the following steps:

  * •

The coefficients of the characteristic polynomial are analytic functions of
the entries of the matrix $H(t)$, hence, they are analytic functions of the
parameter $t$.

  * •

A root of a polynomial with multiplicity 1 depends analytically on the
coefficients of the polynomial. It follows from the analytic implicit function
theorem [33, pg. 47, Thm. 2.5.1.], since a single root is not a root of the
derivative of the polynomial.

However, for a Hermitian matrix family $H(t)$ the analytic dependence of a
non-degenerate eigenvalue $\lambda$ can be deduced alternatively from the SW
decomposition theorem 3.1.2, more precisely, applying the same argument for
$k=1$ as follows. Consider an $H_{0}$ with a non-degenerate eigenvalue
$\lambda$, for simplicity assume that $H(0)$ is diagonal and $\lambda$ is in
its $1\times 1$ upper-left corner (it can be reached by a unitary change of
the basis of $\mathbb{C}^{n}$). The correspondence $f:(S,\breve{B})\mapsto
e^{iS}(H_{0}+\breve{B})e^{-iS}$ defines an analytic map from
$\mathbb{R}^{n^{2}}$ to $\mathbb{R}^{n^{2}}$, and its Jacobian has maximal
rank at $(0,0)$, as one can show in the same way as we proved Theorem 3.1.2 in
Section 4.1. Hence, locally $f$ has an analytic inverse, and it provides a SW
type decomposition for the nearby matrices $H$, where the analytically
dependent $1\times 1$ upper-left block $T$ is an eigenvalue of $H$, and it is
equal to $\lambda$ for $H=H_{0}$.

######  Remark 4.3.11.

For the eigenvalues degenerate at $t=0$ the situation is more complicated.
Their analytic behaviour in a one-parameter family is a unique property of
normal (in particular, Hermitian) matrices. This is based on the fact that
they are unitarily diagonalizable. However, in general the degenerate
eigenvalues of $H(0)$ are not analytic functions of $t$ if $H(t)$ is a family
of matrices with more than one parameter. Cf. [35, Chapter Two].

###### Proof of Theorem 4.3.8.

We prove the analiticity of eigenvalues with an iterated process. First, take
the eigenvalue $\lambda_{i}(0)$ of $H(0)$. There are two possibilities.

_Case 1:_ If the eigenvalue $\lambda_{i}(0)$ of $H(0)$ is non-degenerate,
Remark 4.3.10 implies that

$\widetilde{\lambda}_{i}(t):=\lambda_{i}(t)$ (4.3.12)

is an analytic function of $t$ around $t=0$.

_Case 2:_ If the eigenvalues
$\lambda_{j+1}(0)=\lambda_{j+2}(0)=\dots=\lambda_{j+k}(0)$ of $H(0)$ are
$k$-fold degenerate we can apply the SW decomposition to the $k$-fold
degeneracy. For simplicity assume that $j=0$, i.e., $H(0)\in\Sigma_{k}$. By
Theorem 3.1.2, the traceless effective Hamiltonian $H_{\textup{eff}}(t)$ of
$H(t)$ is an analytic function of $t$. Each $H_{\textup{eff}}(t)$ is a
Hermitian matrix with zero trace, and $H_{\textup{eff}}(0)=0$, since
$H(0)\in\Sigma_{k}$. Hence $H_{\textup{eff}}(t)$ can be expressed as

$H_{\textup{eff}}(t)=tH_{1}+t^{2}H_{2}+t^{3}H_{3}+\dots$ (4.3.13)

with traceless Hermitian matrices $H_{j}\in\mbox{Herm}_{0}(k)$.

Define

$H^{(1)}(t)=\frac{H_{\textup{eff}}(t)}{t}=H_{1}+tH_{2}+t^{2}H_{3}+\dots,$
(4.3.14)

it is an analytic function of $t$. We need to show that the eigenvalues of
$H^{(1)}(t)$ can be expressed as analytic functions
$\widetilde{\lambda}_{i}^{(1)}(t)$. Indeed, the eigenvalues of
$H_{\textup{eff}}(t)$ are $t\cdot\widetilde{\lambda}_{i}^{(1)}(t)$ and we can
define

$\widetilde{\lambda}_{i}(t)=\left\\{\begin{array}[]{lcc}\overline{\lambda}(t)+t\cdot\widetilde{\lambda}_{i}^{(1)}(t),&\mbox{if}&t\geq
0,\\\
\overline{\lambda}(t)+t\cdot\widetilde{\lambda}_{k+1-i}^{(1)}(t),&\mbox{if}&t<0,\end{array}\right.$
(4.3.15)

where $\overline{\lambda}(t)$ is the diagonal entry of the scalar matrix
$T(t)$ in Equation (3.1.4), hence it is analytic. One can verify that
$\widetilde{\lambda}_{i}(t)$ is an eigenvalue of $H(t)$. Note that the indices
$i$ of the functions $\widetilde{\lambda}_{i}(t)$ do not correspond to the
increasing order of the eigenvalues, in fact,, the multiplication by a
negative $t$ reverses their order.

If $H(0)$ has an arbitrary (not ground state) $k$-fold degeneracy, then the
straight-forward generalization of the SW decomposition can be used.

If $H_{1}(0)$ is still degenerate, that is, some of the eigenvalues
$\lambda_{i}^{(1)}(0)$ coincide, we iterate the previous steps with
$H^{(1)}(t)$ in place of $H(t)$. We create the traceless effective Hamiltonian
$H_{\textup{eff}}^{(1)}(t)$ for each degenerate subspace, and since it is
divisible by $t$, we can define the analytic family
$H^{(2)}(t):=H^{(1)}_{\textup{eff}}(t)/t$. If an eigenvalue
$\lambda_{i}^{(2)}(0)$ of $H^{(2)}(0)$ is non-degenerate (case 1), then the
corresponding eigenvalue of $H^{(1)}(t)$ and $H(t)$ is an analytic function of
$t$.

For a degenerate eigenvalue $\lambda_{i}^{(2)}(0)$ we iterate the above steps
obtaining the analytic matrix families $H^{(j)}(t)$ and the eigenvalues
$\lambda_{i}^{(j)}(t)$ (where $j=1,2,3,\dots$). If an eigenvalue becomes non-
degenerate in finite steps, that is, there is a $j$ such that
$\lambda_{i}^{(j)}(0)$ is a non-degenerate eigenvalue of $H^{(j)}(0)$, then by
induction the corresponding eigenvalue of $H(t)$ is an analytic function of
$t$.

_Case 3:_ Some eigenvalues may not split in finite steps. More precisely, for
some index $i$ we have
$\lambda_{i}^{(j)}(0)=\lambda_{i+1}^{(j)}(0)=\dots=\lambda_{i+l-1}^{(j)}(0)$
holds for every step $j$. In this case consider the first $j=j_{0}$ such that
the other eigenvalues of $H^{(j_{0})}(0)$ are different from them. Then for
every $j>j_{0}$ we have $H_{\textup{eff}}^{(j)}(t)=H^{(j)}(t)$, hence
$H^{(j+1)}(t)=H^{(j)}(t)/t$. Thus (each entry of) $H^{(j)}(t)$ is divisible by
an arbitrary power of $t$, in other words its order is infinite, implying that
$H^{(j)}(t)=0$ identically. Hence the eigenvalues of $H^{(j)}(t)=0$ are
analytic (they are constant zero functions), therefore the corresponding
eigenvalues of $H(t)$ are analytic. ∎

For $0<i<j\leq k$ define the pairwise splitting of the eigenvalues in analytic
way as

$\mathfrak{s}_{i,j}(t)=\widetilde{\lambda}_{i}(t)-\widetilde{\lambda}_{j}(t).$
(4.3.16)

The proof of Theorem 3.3.6 on the order of energy splitting is based on the
following lemma.

######  Lemma 4.3.17.

Let $f_{1},\dots,f_{l}:\mathbb{R}\to\mathbb{R}$ be analytic functions defined
in a neighborhood of 0. Let $\overline{f}=\sum_{i=1}^{l}f_{i}/l$ be their
mean. Then

$\min_{1\leq i,j\leq l}\\{\mathrm{ord}_{0}(f_{i}-f_{j})\\}=\min_{1\leq i\leq
l-1}\\{\mathrm{ord}_{0}(f_{i}-f_{i+1})\\}=\min_{1\leq i\leq
l}\\{\mathrm{ord}_{0}(f_{i}-\overline{f})\\}.$ (4.3.18)

###### Proof of Lemma 4.3.17.

Let

$f_{i}(t)=\sum_{r=0}^{\infty}a_{i,r}t^{r}\mbox{, and }\
\overline{f}(t)=\sum_{r=0}^{\infty}\overline{a}_{r}t^{r}$

be the Taylor series of $f_{i}$ and $\overline{f}$, respectively.

For a fixed integer $r$ the following are equivalent:

  1. 1.

Not all the coefficients $a_{i,r}$ are equal ($1\leq i\leq l$).

  2. 2.

There are indices $1\leq i,j\leq l$ such that $a_{i,r}\neq a_{j,r}$.

  3. 3.

There is an index $1\leq i\leq l-1$ such that $a_{i,r}\neq a_{i+1,r}$.

  4. 4.

There is an index $1\leq i\leq l$ such that $a_{i,r}\neq\overline{a}_{r}$.

By definition, the left hand side of Equation (4.3.18) is the smallest integer
$r$ for which (2) holds, the middle part is the smallest integer $r$ for which
(3) holds, and the right hand side is the smallest integer such that (4)
holds. This proves the lemma.

∎

###### Proof of Theorem 3.3.6 and Theorem 3.3.8 on the order of energy
splitting and distancing.

By Lemma 4.3.17 we get

$\min_{1\leq i,j\leq
k}\\{\mathrm{ord}_{0}(\widetilde{\lambda}_{i}(t)-\widetilde{\lambda}_{j}(t))\\}=\min_{1\leq
i\leq
k-1}\\{\mathrm{ord}_{0}(\widetilde{\lambda}_{i}(t)-\widetilde{\lambda}_{i+1}(t))=\min_{1\leq
i\leq
k}\\{\mathrm{ord}_{0}(\widetilde{\lambda}_{i}(t)-\overline{\lambda}(t))\\},$
(4.3.19)

that is,

$\min_{1\leq i<j\leq k}\\{\mathrm{ord}_{0}(\mathfrak{s}_{i,j})\\}=\min_{1\leq
i\leq k-1}\\{\mathrm{ord}_{0}(\mathfrak{s}_{i,i+1})\\}=\min_{1\leq i\leq
k}\\{\mathrm{ord}_{0}(\overline{\mathfrak{s}}_{i})\\}.$ (4.3.20)

Let $r$ denote this minimum. In particular, considering only the right side,
$r$ is the minimum of the orders of the functions
$\overline{\mathfrak{s}}_{i}=\widetilde{\lambda}_{i}(t)-\overline{\lambda}(t)$.
Applying Lemma 4.3.1 to these functions (i.e.,
$f_{i}=\overline{\mathfrak{s}}_{i}$) gives that

$t\mapsto\frac{(\mbox{sgn}(t))^{r}}{\sqrt{k}}\cdot\sqrt{\sum_{i=1}^{k}(\widetilde{\lambda}_{i}(t)-\overline{\lambda}(t))^{2}}$
(4.3.21)

is an analytic function of order $r$. But this function is equal to
$\mathfrak{s}_{k}(t)$ (defined by Equation (4.3.5)). This shows that
$\mathrm{ord}_{0}(\mathfrak{s}_{k})=\min_{1\leq i\leq
k}\\{\mathrm{ord}_{0}(\overline{\mathfrak{s}}_{i})\\}=r$. Together with
Corollary 4.3.6, it also shows that $r$ agrees with the order of the effective
Hamiltonian, i.e., the last two expressions of Equation (3.3.9).

Next we show that $\mathrm{ord}_{0}(\mathfrak{s}_{1,k})=r$. Clearly,

$\mathrm{ord}_{0}(\mathfrak{s}_{1,k})\geq\min_{1\leq i<j\leq
k}\\{\mathrm{ord}_{0}(\mathfrak{s}_{i,j})\\}$ (4.3.22)

holds by definition. Assume indirectly that the inequality is strict. Take
$i,j$ such that $\mathrm{ord}_{0}(\mathfrak{s}_{i,j})=r$. Then there is an
$\epsilon>0$ such that $|\mathfrak{s}_{i,j}(t)|>|\mathfrak{s}_{1,k}(t)|$ holds
for $0<|t|<\epsilon$. This is a contradiction, since
$|\mathfrak{s}_{i,j}(t)|<|\mathfrak{s}_{1,k}(t)|$ holds for sufficiently small
positive $t$, since the eigenvalues are in increasing order. This proves the
theorem. ∎

###### Proof of Corollary 3.3.10 on the order of energy splitting in linear
families.

Consider the SW chart (Corollary 3.1.6) on the neighborhood $\mathcal{V}_{0}$
of $H_{0}$ in $\mbox{Herm}(n)$, with coordinates
$\phi(H)=(x,y)\in\mathbb{R}^{n^{2}-k^{2}+1}\times\mathbb{R}^{k^{2}-1}$.
Consider the map
$\phi_{y}=\mathrm{pr}_{2}\circ\phi:\mathcal{V}_{0}\to\mathbb{R}^{k^{2}-1}$,
that is, $\phi_{y}(H)=y$. Since
$\Sigma_{k}\cap\mathcal{V}_{0}=\phi_{y}^{-1}(0)$, its tangent space at $H_{0}$
is $T_{H_{0}}\Sigma_{k}=\ker((\mathrm{d}\phi_{y})_{H_{0}})$. Then, $H_{1}\in
T_{H_{0}}\Sigma_{k}$ if and only if

$0=(\mathrm{d}\phi_{y})_{H_{0}}(H_{1})=\left.\frac{\mathrm{d}}{\mathrm{d}t}(\phi_{y}(H_{0}+tH_{1}))\right|_{t=0},$
(4.3.23)

which is equivalent to the fact that the order of every component
$(\phi_{y})_{j}(H_{0}+tH_{1})=y_{j}(H_{0}+tH_{1})$ is bigger than 1, that is,

$\min_{1\leq j\leq k^{2}-1}\\{\mbox{ord}_{0}(t\mapsto
y_{j}(H_{0}+tH_{1}))\\}>1$ (4.3.24)

and the left side is equal to the order of energy splitting $r$ by Theorem
3.3.8. This proves the corollary.

∎

###  4.4 Parameter-dependent quantum systems and Weyl points

###### Proof of Theorem 3.4.5 on the characterization of Weyl points.

The equivalence of (1) and (2) is a well-known property of transversality, see
e.g. [25, pg. 28] or [26, Lemma 4.3.]. To see it in our particular situation,
consider a chart on $M^{3}$ around $p_{0}=0$, and consider the SW chart
(Corollary 3.1.6) on the neighborhood $\mathcal{V}_{0}$ of $H_{0}$ in
$\mbox{Herm}(n)$, with coordinates
$\phi(H)=(x,y)\in\mathbb{R}^{n^{2}-3}\times\mathbb{R}^{3}$. Let
$\phi_{y}:\mathcal{V}_{0}\to\mathbb{R}^{3}$ denote the map
$\phi_{y}(H)=(y_{1},y_{2},y_{3})$. Obviously, $h=\phi_{y}\circ H$, and the
tangent space of $\Sigma_{2}$ at $H_{0}$ is
$T_{H_{0}}\Sigma_{2}=\ker((\mathrm{d}\phi_{y})_{H_{0}})$. The transversality
of $H$ to $\Sigma_{2}$ at $H(p_{0})=H_{0}$ means that

$T_{H_{0}}\Sigma_{2}+(\mathrm{d}H)_{p_{0}}(T_{p_{0}}M^{3})=T_{H_{0}}\mbox{Herm}(n),$
(4.4.1)

or equivalently,

$\ker((\mathrm{d}\phi_{y})_{H_{0}})+(\mathrm{d}H)_{p_{0}}(T_{p_{0}}M^{3})=T_{H_{0}}\mbox{Herm}(n).$
(4.4.2)

(Recall the definitions from Section 3.4.) By applying
$(\mathrm{d}\phi_{y})_{H_{0}}$ on both sides, we get

$(\mathrm{d}\phi_{y})_{H_{0}}\circ(\mathrm{d}H)_{p_{0}}(T_{p_{0}}M^{3})=(\mathrm{d}\phi_{y})_{H_{0}}(T_{H_{0}}\mbox{Herm}(n)).$
(4.4.3)

On the left side,
$(\mathrm{d}\phi_{y})_{H_{0}}\circ(\mathrm{d}H)_{p_{0}}=(\mathrm{d}h)_{p_{0}}$
by the chain rule, and the right side is equal to $T_{0}\mathbb{R}^{3}$, since
$(\mathrm{d}\phi_{y})_{H_{0}}$ is surjective. Therefore, the transversality is
equivalent to the fact that $(\mathrm{d}\phi_{y})_{H_{0}}$ is surjective on
the the image of $(\mathrm{d}H)_{p_{0}}$, which is equivalent to the fact that
$(\mathrm{d}h)_{p_{0}}$ has maximal rank 3. This proves (1) $\Leftrightarrow$
(2).

Point (3) is clearly equivalent to (2), since it is equivalent to the
following: for every $\gamma$ with $\gamma(0)=p_{0}$, $\gamma^{\prime}(0)\neq
0$, $\gamma^{\prime}(0)\notin\ker((\mathrm{d}h)_{p_{0}})$. Point (3) is also
equivalent to (4), since, by Theorem 3.3.8, the order of energy splitting is
equal to the order of $h(\gamma(t))$, which is 1 by (3). ∎

###### Proof of Corollary 3.4.7 on Weyl points being isolated.

First we show that $p_{0}$ has a neighborhood $\widetilde{\mathcal{W}}_{0}$ in
$M$ such that the restriction $H|_{\widetilde{\mathcal{W}}_{0}}$ is transverse
to $\Sigma_{2}$. The differential of $h$ has maximal rank at $p_{0}$, that is,
its determinant is non-zero. But the determinant
$p\mapsto\det((\mathrm{d}h)_{p})$ is a continuous map, hence $p_{0}$ has a
neighborhood $\widetilde{\mathcal{W}}_{0}$ such that
$\det((\mathrm{d}h)_{p})\neq 0$ if $p\in\widetilde{\mathcal{W}}_{0}$, that is,
the rank of the differential of $h$ has maximal rank at every point
$p\in\widetilde{\mathcal{W}}_{0}$. Then, by point (2) of Theorem 3.4.5, the
restriction
$H|_{\widetilde{\mathcal{W}}_{0}}:\widetilde{\mathcal{W}}_{0}\to\mbox{Herm}(n)$
is transverse to $\Sigma_{2}$, i.e., at every
$p\in\widetilde{\mathcal{W}}_{0}$, either $H$ is transverse to $\Sigma_{2}$ at
$H(p)\in\Sigma_{2}$, or $H(p)\notin\Sigma_{2}$. Then, by the theorem in [25,
pg. 30] (see also [26, Thm. 4.4.]),
$H^{-1}(\Sigma_{2})\cap\widetilde{\mathcal{W}}_{0}$ is a submanifold of
dimension $\dim(M^{3})-\mbox{codim}(\Sigma_{2})=0$ that is, it consists only
of isolated points, proving the corollary.

∎

###### Proof of Corollary 3.4.8 on Weyl points being stable and generic.

Part (a) is essentially the stability theorem [25, 35], see also [26, pg. 59,
exercise (1) (a)]. We formulate the proof in our particular situation.

The perturbation $H_{t}$ induces a perturbation $h_{t}$ of $h_{t=0}=h$, that
is, a smooth map $(p,t)\mapsto h_{t}(p)$ defined in a neighborhood of
$(p_{0},0)$. The determinant map $(p,t)\mapsto\det((\mathrm{d}h_{t})_{p})$ is
continuous, hence, it is non-zero in a neighborhood of $(p_{0},0)$. It implies
that there is an $\epsilon_{1}>0$ such that for a fixed $|t|<\epsilon_{1}$ the
map $p\mapsto H_{t}(p)$ is transverse to $\Sigma_{2}$ in a sufficiently small
neighborhood of $p_{0}$ in $M$. Therefore, the map $(p,t)\mapsto H_{t}(p)$ is
also transverse to $\Sigma_{2}$, hence the preimage of $\Sigma_{2}$ is a
smooth manifold $\Gamma$ in a neighborhood of $(p,0)$ in $M\times\mathbb{R}$
of dimension
$\dim(\Gamma)=\dim(M\times\mathbb{R})-\mbox{codim}(\Sigma_{2})=1$. Let
$\Gamma_{0}$ be the component of $\Gamma$ containing $(p_{0},0)$. Applying the
projection $(p,t)\mapsto p$ to $\Gamma_{0}$ gives the $\mathcal{C}^{\infty}$
curve $\gamma:(-\epsilon_{1},\epsilon_{1})\to M$ through $\gamma(0)=p_{0}$,
satisfying that $\gamma(t)\in H_{t}^{-1}(\Sigma_{2})$ is a Weyl point of
$H_{t}$.

Other degeneracy points can be avoided by a further restriction of $t$, i.e.,