$|t|<\epsilon$ with a suitable $0<\epsilon<\epsilon_{1}$, defined as follows.
Choose a neighborhood $\mathcal{W_{\gamma}}$ of the image of $\gamma$ in
$\mathcal{W}_{0}$ which does not contain other degeneracy points, that is,
$H_{t}^{-1}(\Sigma_{2})\cap\mathcal{W_{\gamma}}=\\{\gamma(t)\\}$ for all
$-\epsilon_{1}<t<\epsilon_{1}$. Consider the compact set
$\mbox{cl}(\mathcal{W}_{0})\setminus\mathcal{W}_{\gamma}$. Observe that there
is an $0<\epsilon<\epsilon_{1}$ such that $H_{t}(p)\notin\Sigma_{2}$ for
$p\in\mbox{cl}(\mathcal{W}_{0})\setminus\mathcal{W}_{\gamma}$ and
$0<|t|<\epsilon$. Indeed, otherwise there would be a series $(p_{i},t_{i})$
such that $t_{i}$ converges to $0$ and $H_{t_{i}}(p_{i})\in\Sigma_{2}$, and
the limit of a convergent subseries of $p_{i}$ would be a point
$q\in\mbox{cl}(\mathcal{W}_{0})\setminus\mathcal{W}_{\gamma}$ with
$H(q)\in\Sigma_{2}$. This proves point (a).

Part (b) follows from the transversality theorem [25, pg 68-69], see also [26,
Lemma 4.6.]. Consider a perturbation parametrized by the Hermitian matrices
$M\times\mbox{Herm}(n)\to\mbox{Herm}(n)$, defined as $(p,K)\mapsto
H_{K}(p):=H(p)+K$. This map is transverse to $\Sigma_{2}$, indeed, for a fixed
$p$ it is a translation of $\mbox{Herm}(n)$. By the transversality theorem,
those parameter values $K$ for which the map $p\mapsto H_{K}(p)$ is transverse
to $\Sigma_{2}$ form a dense subset in $\mbox{Herm}(n)$. This proves the
theorem.

∎

######  Remark 4.4.4.

In Corollary 3.4.8 we formulated the protected nature of the Weyl points in a
way which can be deduced from the properties of transversality. The goal is to
demonstrate the power of this approach in context of the degeneracy points.
However, there are several possible generalizations, whose rigorous proofs are
obstructed by technical difficulties.

  1. 1.

In part (b), one may expect a stronger result, namely, the existence of a one-
parameter perturbation $H_{t}$ of $H$ which has only Weyl points for
$0<t<\epsilon$. This does not follow from the transversality theorems
appearing in the literature. Indeed, the set of the ‘wrong’ perturbations (in
the particular form $H_{K}(p)=H(p)+K$, the set of those $K$ matrices for which
$H_{K}$ is non-transverse) might be very complicated in general, although its
complement is a dense set. If the map germ of $h$ at $p_{0}$ is ‘sufficiently
nice’, then the stronger statement holds. See the Appendix of [23] for related
results and examples.

  2. 2.

For a sufficiently small $\epsilon$, the Weyl points of $H_{K}$ in
$\mathcal{W}_{0}$ can be regarded as the Weyl points born from the non-generic
degeneracy point $p_{0}$ of $H$. If there is a one-parameter perturbation
$H_{t}$ with only Weyl points, then these Weyl points converge to $p_{0}$, as
$t$ tends to 0, and they can be separeted from other Weyl points similarly as
in the proof of part (a) of Corollary 3.4.8. In general it is harder to
formalize the concept of ‘Weyl points born from $p_{0}$’.

  3. 3.

Every Weyl point $p$ of $H_{t}$ has a sign ($\pm 1$), defined as the sign of
$\det((\mathrm{d}h_{t})_{p})$. The sum of the signs for the Weyl points born
from $p_{0}$ does not depend on the perturbation, it is an invariant of the
degeneracy point $p_{0}$ of $H$. Moreover, it is equal to the local degree
$\deg_{p_{0}}(h)$ of $h$ at $p_{0}$, and also with the Chern number of the
lowest eigenstate (up to sign). This number is the topological charge of the
non-generic degeneracy point $p_{0}$ of $H$.

  4. 4.

One can consider the global version, i.e., the transversality to the whole
degeneracy set $\Sigma$. However, $\Sigma$ is not a manifold. A map
$H:M^{3}\to\mbox{Herm}(n)$ defined on a 3-manifold $M^{3}$ is transverse to
$\Sigma$ if it is transverse to every stratum of $\Sigma$. In other words,
such map has only isolated transverse two-fold degeneracy points, but not
necessarily ground-state. A map which is non-transverse to $\Sigma$ can have
various degeneracy patterns, for example, multifold degeneracy points or non-
isolated degeneracies. Every map can be perturbed to a transverse one by an
arbitrary small perturbation, hence, any arbitrarily complicated degeneracy
splits into transverse two-fold degeneracy points (Weyl points).

##  5 Conclusions

Energy degeneracies in parameter-dependent quantum systems are often
accompanied by interesting phenomenology. In this work, we connected such
energy degeneracies, and the effects of perturbations breaking those
degeneracies, to the geometry of degeneracy submanifolds of the Euclidean
space of Hermitian matrices. One link we have found is that the Schrieffer-
Wolff transformation, a standard perturbative method to treat the eigenvalue
problem of quasidegenerate matrices, is, in fact, a local chart of the space
of Hamiltonians, which is aligned with the degeneracy submanifold. We also
established a distance theorem, relating the Frobenius distance of a matrix
from a degeneracy submanifold to the energy splitting of its energy
eigenvalues. As a consequence, we have found that the order of energy
splitting, caused by a one-parameter perturbation of a degenerate energy
level, is the same as the order of distancing of the perturbed Hamiltonian
from the degeneracy submanifold. Finally, as applications of our results, we
have proven the protected nature of Weyl points using the transversality
theorem, and shown that geometrical information on the degeneracy submanifold
can be obtained using results known in the domains of topological order and
quantum error correction. We anticipate that the connections established in
our work will further enhance the already-existing and fruitful cross-
fertilisation between physics — band-structure theory, quantum information,
quantum materials — and differential geometry.

## Author Contributions

G. P. and A. P. initiated the project. A. P. managed the project. G. P. wrote
the initial draft of the manuscript. G. P. and Gy. F. derived the rigorous
proofs with assistance from A. P. and D. V.. Gy. F. prepared the figures. All
authors took part in writing the final version of the manuscript.

## Acknowledgements

We thank P. Vrana, A. Sándor for useful discussions. This research is
supported by the Ministry of Culture and Innovation and the National Research,
Development and Innovation Office within the Quantum Information National
Laboratory of Hungary (Grant No. 2022-2.1.1-NL-2022-00004), by the European
Union within the Horizon Europe research and innovation programme via the
ONCHIPS project under grant agreement No 101080022, and by the HUN-REN
Hungarian Research Network. D. V. was supported by the National Research,
Development and Innovation Office of Hungary under OTKA grant no. FK 146499.
and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germany’s Excellence Strategy through the Würzburg-Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter – ct.qmat (EXC 2147,
project-id 392019).

## Appendix: Further notes on SW decomposition

We compare our approach with the description of the SW transformation given in
[3]. Based on this we formulate the coordinate free version of Theorem 3.1.2,
we provide an important observation about the dependence on the base point
$H_{0}$ and we specify the validity of decomposition (3.1.4).

##  Appendix A Coordinate free description

In [3], $H_{0}$ is not assumed to be degenerate, neither diagonal. Dropping
the degeneracy of the base point does not cause an essential change in the SW
decomposition (3.1.4). Indeed, the effective Hamiltonian of $H$ with respect
to a non-degenerate Hermitian matrix $G$ is equal to
$G_{\Sigma}+H_{\textup{eff}}-G$, where $H_{\textup{eff}}$ is the effective
Hamiltonian of $H$ with respect to $H_{0}:=G_{\Sigma}$. However, from our
point of view the degeneracy of the base point is important, since we study
the geometry around $\Sigma_{k}$. Hence we everywhere assume that $H_{0}$ has
$k$-fold ground state degeneracy.

If $H_{0}\in\Sigma_{k}$ is not diagonal, then Theorem 3.1.2 can be formulated
by replacing the block diagonality conditions with their coordinate free
generalizations. Let $\mathcal{P}$ and $\mathcal{P}_{0}$ be the sum of the
eigenspaces corresponding to the lowest $k$ eigenvalues of $H$ and $H_{0}$
respectively. Their respective Hermitian orthogonal complements
$\mathcal{P}^{\perp}$ and $\mathcal{P}_{0}^{\perp}$ are the sum of the
eigenspaces corresponding to the $n-k$ highest eigenvalues. Let $P$ and
$P_{0}$ be the orthogonal projectors onto $\mathcal{P}$ and $\mathcal{P}_{0}$
respectively. Then $I-P$ and $I-P_{0}$ are the respective orthogonal
projectors onto $\mathcal{P}^{\perp}$ and $\mathcal{P}_{0}^{\perp}$, where $I$
is the $n\times n$ unitary matrix.

The general version of Theorem 3.1.2 induces exactly the same decomposition as
(3.1.4) except for replacing the conditions (1)–(4) with the following ones:

  1. (1)*

$P_{0}^{\perp}BP_{0}^{\perp}=B$.

  2. (2)*

$T=cP_{0}$ with a $c\in\mathbb{R}$.

  3. (3)*

$P_{0}H_{\textup{eff}}P_{0}=H_{\textup{eff}}$ and $\operatorname{{\rm
tr}}(H_{\textup{eff}})=0$.

  4. (4)*

$P_{0}SP_{0}=0,\;P_{0}^{\perp}SP_{0}^{\perp}=0$.

Although it can be deduced from [3, Section 3.1.], another way to see it is
the change to the eigenbasis of $H_{0}$ in $\mathbb{C}^{n}$. Let
$H_{0}^{\prime}=U_{0}^{-1}H_{0}U_{0}$ be a diagonalization of $H_{0}$ with
increasing order of the eigenvalues, and consider the SW decomposition (3.1.4)
of $H^{\prime}:=U_{0}^{-1}HU_{0}$ with respect to $H_{0}^{\prime}$, that is,

$H^{\prime}=e^{iS^{\prime}}\cdot(H^{\prime}_{0}+B^{\prime}+T^{\prime}+H^{\prime}_{\textup{eff}})\cdot
e^{-iS^{\prime}}.$ (A.0.1)

Then, $B:=U_{0}B^{\prime}U_{0}^{-1}$, $T:=U_{0}TU_{0}^{-1}$,
$H_{\textup{eff}}:=U_{0}H_{\textup{eff}}U_{0}^{-1}$ and
$S:=U_{0}S^{\prime}U_{0}^{-1}$ satisfy (1)*–(4)*, and

$H=e^{iS}\cdot(H_{0}+B+T+H_{\textup{eff}})\cdot e^{-iS},$ (A.0.2)

providing a SW decomposition of $H$ with respect to $H_{0}$ in the general
sense.

A SW chart around a non-diagonal $H_{0}\in\Sigma_{k}$ is formed by the
coordinates of the matrices $B^{\prime}$, $T^{\prime}$, $S^{\prime}$ (these
are the local ccordinates of $\Sigma_{k}$) and $H_{\textup{eff}}^{\prime}$
(transverse coordinates), generalizing Corollary 3.1.6. This chart obviously
depends on the choice of $U_{0}$.

##  Appendix B Dependence on the base point $H_{0}$

First of all, if two base points $H_{0},H_{0}^{\prime}\in\Sigma_{k}$ can be
diagonalized in the same basis with increasing order of the eigenvalues, then
the SW decomposition with respect to $H_{0}$ and $H_{0}^{\prime}$ are
essentially the same: the difference of the eigenvalues appears as a shift in
$B$, $T$, while $H_{\textup{eff}}$ and $S$ remain unchanged. Hence, in the
interesting case the base points have different eigenbases.

In the same way as we did it in case of diagonal $H_{0}$, one can define in
general (for non-diagonal $H_{0}$)

$H_{\textup{proj}}:=e^{iS}\cdot(H_{0}+B+T)\cdot e^{-iS}$ (B.0.1)

and prove that $H_{\textup{proj}}$ is equal to $H_{\Sigma}$, that is, the
closest point of $\Sigma_{k}$ to $H$ constructed independently of the SW
decomposition, see (3.1.10) and Section 4.2. Hence, $H_{\textup{proj}}$ does
not depend on the choice of the base point.

In contrast, the effective Hamiltonian deeply depends on the base point
$H_{0}$. If two base points $H_{0}$ and $H^{\prime}_{0}$ cannot be
simultaneously diagonalized, then the corresponding effective Hamiltonians
$H_{\textup{eff}}$ and $H^{\prime}_{\textup{eff}}$ of $H$ are obviously not
equal in general, since (3)* in Section A imposes a different condition on
them. Moreover, two matrices $H$ and $G$ having the same effective Hamiltonian
$H_{\textup{eff}}=G_{\textup{eff}}$ with respect to $H_{0}$ can have different
effective Hamiltonian $H^{\prime}_{\textup{eff}}\neq
G^{\prime}_{\textup{eff}}$ with respect to $H_{0}^{\prime}$, in general.
However, their spectra and norms are the same,
$\|H^{\prime}_{\textup{eff}}\|=\|G^{\prime}_{\textup{eff}}\|$. To see it,
observe that

$H-H_{\Sigma}=e^{iS(H)}\cdot H_{\textup{eff}}\cdot
e^{-iS(H)}=e^{iS^{\prime}(H)}\cdot H^{\prime}_{\textup{eff}}\cdot
e^{-iS^{\prime}(H)},$ (B.0.2) $G-G_{\Sigma}=e^{iS(G)}\cdot
G_{\textup{eff}}\cdot e^{-iS(G)}=e^{iS^{\prime}(G)}\cdot
G^{\prime}_{\textup{eff}}\cdot e^{-iS^{\prime}(G)},$ (B.0.3)

where $S(H)$ and $S(G)$ ($S^{\prime}(H)$ and $S^{\prime}(G)$, respectively)
denote the exponents coming from the SW decomposition of $H$ and $G$ with
respect to $H_{0}$ ($H^{\prime}_{0}$), see Section A. Then,
$H_{\textup{eff}}=G_{\textup{eff}}$ implies that

$G^{\prime}_{\textup{eff}}=e^{-iS^{\prime}(G)}\cdot e^{iS(G)}\cdot
e^{-iS(H)}\cdot e^{iS^{\prime}(H)}\cdot H^{\prime}_{\textup{eff}}\cdot
e^{-iS^{\prime}(H)}\cdot e^{iS(H)}\cdot e^{-iS(G)}\cdot e^{iS^{\prime}(G)},$
(B.0.4)

which does not imply $H^{\prime}_{\textup{eff}}=G^{\prime}_{\textup{eff}}$ in
general, however, it implies that the norms and spectra of
$H^{\prime}_{\textup{eff}}$ and $G^{\prime}_{\textup{eff}}$ are equal.
Furthermore, if $H_{0}$ and $H_{0}^{\prime}$ can be diagonalized in the same
basis (with increasing order of the eigenvalues), then
$H_{\textup{eff}}=G_{\textup{eff}}$ implies
$H^{\prime}_{\textup{eff}}=G^{\prime}_{\textup{eff}}$, since in this case
$e^{iS(H)}=e^{iS^{\prime}(H)}$ and $e^{iS(G)}=e^{iS^{\prime}(G)}$ in Equation
(B.0.4). Indeed, in this case
$H_{\textup{eff}}=G_{\textup{eff}}=H^{\prime}_{\textup{eff}}=G^{\prime}_{\textup{eff}}$
also holds.

##  Appendix C Direct rotation

$I-2P_{0}$, and $I-2P$ are the reflections with respect to
$\mathcal{P}_{0}^{\perp}$ and $\mathcal{P}^{\perp}$, respectively. Assuming
that $\sqrt{(I-2P)(I-2P_{0})}$ is defined (where the square root is taken so
that the eigenvalues are the closest to 1 from the two possibilities), it is a
unitary matrix called the direct rotation between the subspaces
$\mathcal{P}_{0}$ and $\mathcal{P}$, see [36, 3]. By [3, Section 2.2.],
$\sqrt{(I-2P)(I-2P_{0})}=e^{iS}$. It follows that $PSP=P^{\perp}SP^{\perp}=0$
also holds for $S$, that is, $S$ is off-block with respect to $H_{0}$, and
also with respect to $H$. Moreover, if $H_{0}$ is diagonal, then the direct
rotation $e^{iS}$ has a special form,

$e^{iS}=\begin{pmatrix}U_{1,1}&U_{1,2}\\\ -U_{1,2}^{\dagger}&U_{2,2}\\\
\end{pmatrix},$ (C.0.1)

where

$\displaystyle U_{1,1}$ $\displaystyle=$ $\displaystyle\sqrt{I_{k\times
k}-U_{1,2}U_{1,2}^{\dagger}},$ (C.0.2) $\displaystyle U_{2,2}$
$\displaystyle=$
$\displaystyle\sqrt{I_{(n-k)\times(n-k)}-U_{1,2}^{\dagger}U_{1,2}},$ (C.0.3)

are Hermitian positive definite matrices, see [3, Section 2.3.].

##  Appendix D The range of validity of the SW decomposition

The range of validity can be discussed in different senses, e.g. (1) Given an
$H_{0}$, which matrices $H$ have a SW decomposition with respect to $H_{0}$?
(2) How can we choose a neighborhood $\mathcal{V}_{0}$ of $H_{0}$ in
$\mbox{Herm}(n)$ and a neighborhood $\mathcal{U}_{0}$ of $(0,0,0,0)$ in the
$(S,B,T,H_{\textup{eff}})$ space such that every $H\in\mathcal{V}_{0}$ has a
unique SW decomposition with $(S,B,T,H_{\textup{eff}})\in\mathcal{U}_{0}$?
Which are the largest possible choices? (3) What is the radius of convergence
of the Taylor series expressing $H_{\textup{eff}}$ or $S$ in terms of $H$?
These problems lead to the analysis of the analytic map

$f:(S,B,T,H_{\textup{eff}})\mapsto
H=e^{iS}\cdot(H_{0}+B+T+S+H_{\textup{eff}})\cdot e^{-iS},$ (D.0.1)

determining its image and injective restrictions. The configurations
$(S,B,T,H_{\textup{eff}})$ where the Jacobian of the map has maximal rank have
a neighborhood on which the map is invertible, i.e., there is a unique
decomposition. An additional question is the ‘lowest $k$ state property’: for
given neighborhoods $\mathcal{U}_{0}$ and $\mathcal{V}_{0}$ such that
$f:\mathcal{U}_{0}\to\mathcal{V}_{0}$ is a bijection, is it true that the
first $k$ columns of $e^{iS}$ span the sum of the eigenspaces of $H$
corresponding to the lowest $k$ eigenvalues? Cf. the proof Theorem 3.1.2 in
Section 4.1.

This article is not intended to provide complete answers to these questions,
but we list some basic observations about them, and summarize the
corresponding results of [3]. Throughout the observations below, $f$ denotes
the map defined by Equation (D.0.1) with a fixed $H_{0}\in\Sigma_{k}$
specified in each case.

  * •

Question (1) is trivial: for every $H_{0}$ and $H$ there is a SW decomposition
of $H$ with respect to $H_{0}$. It can be constructed by taking a direct
rotation between the eigenspace of $H_{0}$ corresponding to the lowest $k$
eigenvalues to any $k$-dimensional eigenspace of $H$. This shows that $f$ is a
surjective map, and also shows that without any additional requirement
(formulated by the other questions) the SW decomposition itself is irrelevant.

  * •

Without any restriction, the exponent $S$ is not unique in the SW
decomposition (3.1.4), since the exponential map $S\mapsto e^{iS}$ is not
injective.

  * •

If we fix $S$, the other terms $B$, $T$ and $H_{\textup{eff}}$ can be modified
as long as the resulting matrix $H$ does not reach $\Sigma_{[k,k+1]}$, that
is, until $\lambda_{k}\neq\lambda_{k+1}$. Furthermore, different
configurations of $B$, $T$ and $H_{\textup{eff}}$ provide different matrices
$H$, hence, the uniqueness of the SW decomposition cannot break down in this
way.

  * •

For a fixed $S$ and varying $B,T,H_{\textup{eff}}$, possibly $H$ crosses
$\Sigma_{[k,k+1]}$. Then, two eigenvalues of the two blocks of $\widetilde{B}$
are exchanged, hence, property (3) of Theorem 3.1.2 breaks down. A more
careful analysis shows that the unique decomposition property also fails when
$H$ reaches $\Sigma_{[k,k+1]}$, since a continuous family of exponents $S$
provides the same $H$. Indeed, the eigenspace $\mathcal{P}$ corresponding to
the lowest $k$ eigenvalues has a freedom if $\lambda_{k}=\lambda_{k+1}$.
Moreover, for the same reason, the Jacobian of the map $f$ has determinant 0
if $H\in\Sigma_{[k,k+1]}$. Cf. the proof of Theorem 3.1.2 in Section 4.1,
where the maximal rank of the Jacobian is crucial for the local uniqueness of
the SW decomposition.

  * •

Given $G\notin\Sigma_{[k,k+1]}$, choose $H_{0}:=G_{\Sigma}$. Then, $G$
obviously has a SW decomposition with respect to $H_{0}$ with $T=B=S=0$ and
$G_{\textup{eff}}=G-H_{0}$. Moreover, there is a neighborhood of $G$ in
$\mbox{Herm}(n)$ and $(0,0,0,G_{\textup{eff}})$ in the
$(T,B,S,H_{\textup{eff}})$ space, such that $f$ is an analytic diffeomorphism
between these neighborhoods. It can be seen as follows. Without losing
generality we can assume that $G$ and $H_{0}=G_{\Sigma}$ are diagonal. The
proof of Theorem 3.1.2 can be modified to show that the Jacobian of $f$ at
$(0,0,0,G_{\textup{eff}})$ has maximal rank, using that the eigenvalues of $G$
satisfies $\lambda_{k}<\lambda_{k+1}$.

  * •

Choose $G\notin\Sigma_{[k,k+1]}$ as above, and choose $H_{0}=G_{\Sigma}$. The
elements of the line segment joining $G$ and $H_{0}$ also have a SW
decomposition with $T=B=S=0$, only $H_{\textup{eff}}$ varies. We can show that
there is a neighborhood of the line segment joining $G$ and $H_{0}$ in
$\mbox{Herm}(n)$ whose elements admit a unique SW decomposition, with a
suitable restriction for its terms. This neighborhood can be imagined as a
‘capsule’ around the line segment, cf. Figure 9. The existence of such capsule
can be deduced as follows.

Consider the line segment $L$ joining $(0,0,0,H-H_{0})$ and $(0,0,0,0)$ in the
$(S,B,T,H_{\textup{eff}})$ space. $L\subset\mathbb{R}^{n^{2}}$ is a compact
submanifold and $f|_{L}$ is injective on $L$, furthermore, the Jacobian of $f$
has maximal rank at the points of $L$. By a generalization of the inverse
function theorem, see [25, Pg. 19., Exercise 10; Pg. 56., Exercise 14], $f$ is
a diffeomorphism between a neighborhood $\mathcal{L}_{0}$ of $L$ and its
image. The image of this neighborhood
$f(\mathcal{L}_{0})=\mathcal{V}_{0}\subset\mbox{Herm}(n)$ admits a unique SW
decomposition with $(S,B,T,H_{\textup{eff}})\in\mathcal{L}_{0}$.

Figure 9: ‘Capsule’: a neighborhood of the line segment joining
$G\notin\Sigma_{[k,k+1]}$ and $H_{0}=G_{\Sigma}$ in $\mbox{Herm}(n)$ whose
elements admit a unique SW decomposition.

Furthermore, we can define a ‘discus’ (of dimension $n^{2}$) extending the
capsule, with unique SW decomposition. We take a disc $\mathcal{B}$ of
dimension $k^{2}-1$ consisting of matrices $H$ with $H_{\Sigma}=H_{0}$ and
$d(H,H_{0})\leq d(G,H_{0})$. The disc $\mathcal{B}$ is orthogonal to
$\Sigma_{k}$ at $H_{0}$. Then, there is a neighborhood $\mathcal{V}_{0}$ of
$\mathcal{B}$ with a unique SW decompostition, resulting a discus shaped
volume in $\mathrm{Herm}(n)$.

We can give other conditions for the range of validity of the SW decomposition
(3.1.4) based on [3]. Let $r_{0}$ denote half of the spectral gap of $H_{0}$,
i.e.

$r_{0}=\frac{\lambda_{k+1}-\lambda_{k}}{2},$ (D.0.2)

where $\lambda_{i}$ denote the eigenvalues of $H_{0}$. Recall that
$\|\cdot\|_{2}$ denotes the operator 2-norm, cf. Remark 2.1.13. Then, based on
[3], we can deduce the following: The matrices $H$ satisfying
$\|H-H_{0}\|_{2}<r_{0}$ have a unique SW decomposition with $\|S\|_{2}<\pi/2$.
Since $\|H-H_{0}\|_{2}\leq\|H-H_{0}\|$, the more strict condition
$\|H-H_{0}\|<r_{0}$ also implies a unique SW decomposition with
$\|S\|_{2}<\pi/2$. Indeed, by [3, Lemma 3.1.], $\|H-H_{0}\|_{2}<r_{0}$ implies
that $\|P-P_{0}\|_{2}<1$, and in this case there is a unique $S$ with
$\|S\|_{2}<\pi/2$ such that $e^{iS}$ is the direct rotation between
$\mathcal{P}$ and $\mathcal{P}_{0}$, see [3, Corollary 2.2., Lemma 2.3.].

Next we show that geometrically $r_{0}$ is the radius of the largest open ball
around $H_{0}$ in operator 2-norm which does not intersect $\Sigma_{[k,k+1]}$.
First, take a closest element $G_{0}$ of $\Sigma_{[k,k+1]}$ to $H_{0}$ in the
Frobenius norm. Then $r_{0}$ is equal to the distance of $H_{0}$ and $G_{0}$
in the operator 2-norm. Indeed, in a similar way we constructed $H_{\Sigma}$
and we proved in Section 4.3 that it is the closest element of $\Sigma_{k}$ to
$H$, one can construct the closest element of $\Sigma_{[k,k+1]}$ to $H_{0}$ by
collapsing $\lambda_{k}$ and $\lambda_{k+1}$ to their mean value
$(\lambda_{k}+\lambda_{k+1})/2$ (and leave the other eigenvalues unchanged).
Because the $k$-th eigenvalue of $H_{0}$ is degenerate, the closest element
$G_{0}$ of $\Sigma_{[k,k+1]}$ is not unique: the construction depends on the
choice of the eigenspace corresponding to $\lambda_{k}$, which can be any one
dimensional subspace of $\mathcal{P}_{0}$. The distance of $H_{0}$ and $G_{0}$
in the operator 2-norm is obviously $r_{0}$.

Moreover, $G_{0}$ is a closest point of $\Sigma_{[k,k+1]}$ to $H_{0}$ in the
operator 2-norm as well. To show this, assume that
$H_{0}+K\in\Sigma_{[k,k+1]}$ for a Hermitian matrix $K$. Let $\mu$ denote the
$k$-th eigenvalue of $H_{0}+K$, which is equal to the $k+1$-th eigenvalue of
$H_{0}+K$. One can deduce from Weyl’s inequality [17] that
$|\mu-\lambda_{k}|\leq\|K\|_{2}$ and also $|\mu-\lambda_{k+1}|\leq\|K\|_{2}$.
But at least one of the left sides is at least $r_{0}$, i.e.,
$r_{0}\leq|\mu-\lambda_{k}|$ or $r_{0}\leq|\mu-\lambda_{k+1}|$ holds, implying
that $r_{0}\leq\|K\|_{2}$.

Summarizing the above arguments,

$\mathcal{V}_{0}=\\{H\in\mbox{Herm}(n)\ |\ \|H-H_{0}\|_{2}<r_{0}\\}$ (D.0.3)

is the largest open ball around $H_{0}$ in operator 2-norm which does not
intersect $\Sigma_{[k,k+1]}$. There is a unique SW decomposition on
$\mathcal{V}_{0}$ if we restrict $(S,B,T,H_{\textup{eff}})$ such that
$\|S\|_{2}<\pi/2$. The decomposition automatically satisfies the lowest $k$
state property, since Ref. [3] uses the direct rotation between the
eigenspaces of $H_{0}$ and $H$ corresponding to the lowest $k$ eigenvalues.

We do not know, whether the Jacobian of $f$ has maximal rank in the domains
indicated by [3]. There is a weaker result in [3] about analytic one parameter
families $H(t)=H_{0}+tH_{1}$: The radius of convergence of the power series
expressing $H_{\textup{eff}}(t)$ and $S(t)$ in terms of $H(t)$ around $H_{0}$
is – surpsrisingly, only – at least $r_{0}/4$.

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# White-light Continuum Observation of the Off-limb Loops of the SOL2017-09-10
X8.2 Flare: Temporal and Spatial Variations

Junwei Zhao11affiliation: W. W. Hansen Experimental Physics Laboratory,
Stanford University, Stanford, CA 94305-4085, USA , Wei Liu22affiliation:
Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto, CA94304, USA
33affiliation: Bay Area Environmental Research Institute, NASA Research Park,
Moffett Field, CA 94035, USA , and Jean-Claude Vial44affiliation: Institut
d’Astrophysique Spatiale, Université Paris-Saclay, C.N.R.S., Batiment 121,
ORSAY 91405, France

###### Abstract

Observations of the Sun’s off-limb white-light (WL) flares offer rare
opportunities to study the energy release and transport mechanisms in flare
loops. One of the best such events was SOL2017-09-10, an X8.2 flare that
occurred near the Sun’s west limb on 2017 September 10 and produced a WL loop
system lasting more than 60 minutes and reaching an altitude higher than 30
Mm. The event was well observed by a suite of ground- and space-based
instruments, including the Solar Dynamics Observatory/Helioseismic and
Magnetic Imager (SDO/HMI) that captured its off-limb loops in WL continuum
near Fe I 6173 Å, and the Atmospheric Imager Assembly (SDO/AIA) that observed
its ultraviolet (UV) and extreme-ultraviolet (EUV) counterparts. We found
quasi-periodic pulsations in the WL and UV emissions at the flare loop-top
with a period around 8.0 min. Each pulsation appears to have an EUV
counterpart that occurs earlier in time and higher in altitude. Despite many
similarities in the WL and UV images and light curves, the WL flux at the
loop-top continues to grow for about 16 minutes while the UV fluxes gradually
decay. We discuss the implication of these unprecedented observations on the
understanding of the enigmatic off-limb WL flare emission mechanisms.

Sun: activity — Sun: corona — Sun: flares — oscillations — UV radiation

##  1 Introduction

White-light flares (WLFs; Hudson et al., 2006; Hudson, 2011) are rare flaring
events that cause continuum emission in excess of the photospheric background.
Off-limb WLFs, with flare-loop brightening observed in white light (WL) beyond
the solar limb, are even rarer with only a few cases reported (e.g., Hiei et
al., 1992; Martínez Oliveros et al., 2014). Three off-limb WLFs were so far
observed in the continuum near Fe I 6173 Å by the Helioseismic and Magnetic
Imager (HMI; Scherrer et al., 2012; Schou et al., 2012) onboard the spacecraft
Solar Dynamics Observatory (SDO; Pesnell et al., 2012), i.e.,
SOL2013-05-13T02:16 (X1.7), SOL2013-05-13T16:01 (X2.8), and
SOL2017-09-10T15:35 (X8.2). These observations offered unprecedented
opportunities for solar physicists to study properties of WLFs that are
otherwise impossible: e.g., Martínez Oliveros et al. (2014) compared the WL
brightening sources with those observed in extreme ultraviolet (EUV) and soft
X-ray; Saint-Hilaire et al. (2014) analyzed the linear polarization observed
in the WL flare loops; and Heinzel et al. (2017) studied emission mechanisms
of the WL off the solar limb.

One popular phenomenon that is associated with solar flaring events and has
been widely studied by various authors is quasi-periodic pulsations (QPPs) in
the time-dependent intensity curves observed in different wavelengths during
flares (Nakariakov & Melnikov, 2009; Van Doorsselaere et al., 2016). This
phenomenon, with a repeating period of a few seconds to a few minutes, has
been observed almost across all the electromagnetic spectrum, such as in
$\gamma$-rays (Nakariakov et al., 2010), EUV and soft X-rays (e.g., Dennis et
al., 2017; Dominique et al., 2018), chromospheric lines (e.g., Brosius et al.,
2016; Tian et al., 2016), and also in radio bursts (e.g., Li et al., 2015).
Due to the rare observation of WLFs, no known QPP has yet been reported in
white light. The physical mechanism for the generation of QPP is not very
clear, but it is believed that this phenomenon is either due to the
characteristics of the flaring loops that exhibit magnetohydrodynamic (MHD)
oscillations at the flaring sites or MHD waves passing through the flaring
loops (e.g., Nakariakov & Melnikov, 2009), or due to the quasi-periodic
magnetic reconnection and energy release that power the intensity enhancements
of the spectrum lines (e.g., Murray et al., 2009).

The X8.2 limb flare of 2017 September 10 (SOL2017-09-10T15:35), whose loops
rose right above the Sun’s west limb and which was well observed by a suite of
ground- and space-based instruments, has been studied from many different
perspectives to tackle various physics problems related to flare dynamics and
energy releases. For example, using EUV observations Warren et al. (2018)
studied the formation and evolution of the current sheet following the
eruption; Coupling hard X-ray and ground-based radio observations, Gary et al.
(2018) studied the evolving spatial and energy distribution of high-energy
electrons in the flaring region; Coupling with GOES X-ray and EUV
observations, Hayes et al. (2019) found QPPs with periods of about 65 s and
150 s; Using the SDO/HMI’s WL observations, Jejčič et al. (2018) modeled the
physical conditions inside the flare loops and explored the WL emission
mechanisms; Heinzel et al. (2020) identified signatures of He continuum in the
cool flare loops; Using Ca II Stokes components observed by the Swedish 1m
Solar Telescope, Kuridze et al. (2019) reported a strong magnetic field of 350
G at the flare’s loop-top; Using microwave observations, Fleishman et al.
(2020) estimated that the coronal magnetic field in the flaring loops decayed
with a rate of $\sim 5$ G s-1 during the flare, and Yu et al. (2020) analyzed
the bidirectional outflows from the magnetic reconnection sites.

In this Letter, combining simultaneous observations of the SOL2017-09-10 flare
obtained in the SDO/HMI’s continuum intensity and SDO/AIA’s (Atmospheric
Imaging Assembly; Lemen et al., 2012) ultraviolet (UV) and EUV observations,
along with data from the Geostationary Operational Environmental Satellite
(GOES) and Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI; Lin
et al., 2002), we analyze the temporal and spatial variations of the
brightening observed in different wavelength channels, as well as the QPPs in
all these channels. It is believed that these unprecedented simultaneous
observations in different wavelength channels offer us a rare opportunity to
understand the emission mechanism of white light, heat dissipation process,
and energy releases during the flare. This Letter is organized as follows: we
introduce observations in Section 2, present our analysis and results in
Section 3, and discuss our results in Section 4.

##  2 Observations

Figure 1: Selected white-light images from the SDO/HMI observations show
temporal evolution of the flare’s off-limb loops, after a long-time-averaged
pre-flare background is removed. White area on the left side of each panel
indicates the Sun’s photospheric disk and limb. Yellow dashed lines in panel
(e) delimit the area inside which intensities are integrated for further
analysis, and the magenta contour showing the WL loop-top is used in Figure 2c
for a location comparison. (An animation showing the entire sequence of the
event is available online, with a total running time of 7 seconds.)

The SDO/HMI uses Fe I 6173 Å line to observe the Sun with 1″ spatial
resolution and 45 sec temporal cadence, and provides full-disk continuum
intensity data, along with Doppler velocity and magnetic field. The field of
view of SDO/HMI is nominally $45\arcsec$ wider than the photospheric solar
disk, allowing detection of features bright in visible light immediately
beyond the solar limb. The SOL2017-09-10 X8.2 flare started at approximately
15:35 UT in NOAA Active Region 12673, when most of the active region rotated
around the west limb into far side of the Sun. The sudden brightening at the
footpoints of the flare in the chromosphere and the rise of the bright flare
loops, up to about $40\arcsec$ above the limb, were observed by the SDO/HMI in
its continuum intensity, i.e., white light. As shown in Figure 1 and its
associated online animation, the flare loops become visible above the limb at
around 16:03:00 UT. Its loop-top brightness gradually enhances with the
increase of height, reaching an intensity maximum at about 16:12:00 UT when
the SDO/AIA’s UV channels also reach their respective maxima (Cf. Figure 4b).
While continuing to rise in height, the WL intensity drops for about 5 min,
and then starts to enhance again, reaching a even brighter maximum at about
16:27:45 UT. The WL brightening of the flare loops then gradually fade away,
lasting a total of about 70 min.

Figure 2: Selected images of the flare loops at the wavelength channels of (a)
304 Å near its peak intensity at 16:07:05 UT; (b) 1700 Å near its peak
intensity at 16:12:29UT; and (c) 335 Å at 16:27:36UT, overplotted with a
magenta WL contour near the WL peak time (see Figure 1e). (d)-(e) Selected
composite images of HMI WL (blue-red background image) and AIA 335 Å (orange-
white semi-transparent foreground image), with RHESSI 10-16 keV intensity
over-plotted as white contours.

The flare was also observed in all the SDO/AIA wavelength channels. However,
due to the powerful intensity of this flare, a number of AIA channels either
saturated near and during the peak of the flare, mostly at the location of the
loop-top, or had irregular exposure times in order not to saturate the
observations. The data from those channels with irregular exposure times may
still be useful for some types of analyses, but cannot be used in this
analysis, which focuses on temporal variations of the loop-top intensity and
is sensitive to the alteration of exposure times. This leaves only
observations from the wavelengths of two UV channels (1700 Å and 1600 Å) and
two EUV channels (335 Å and 304 Å) useful for the analysis in this study.
According to Lemen et al. (2012), the UV channel 1700 Å (1600 Å) is sensitive
to the temperature of $10^{3.7}$ K ($10^{5.0}$ K) corresponding to the upper
photosphere (the upper photosphere and transition region), the EUV channel 304
Å (335 Å) is sensitive to the temperature of $10^{4.7}$ K ($10^{6.4}$ K)
corresponding to the chromosphere and transition region (active region
corona). However, it must be stressed that these properties of different
channels are assessed mostly for on-disk observations, and may not strictly
apply to the off-limb observations. Figure 2 shows key snapshots of the flare
loops for some wavelengths, along with selected composite images showing the
relative spatial and temporal relations of WL and EUV channels. Particularly,
as seen in Figure 2c, the WL loop-top, at its peak brightness, is located
below the 335Å loop-top in its absorption area.

GOES captured the total soft X-ray flux of the Sun in its both channels: 1–8 Å
and 0.5–4 Å (Figure 4a). RHESSI’s X-ray images did not cover the entire flare
event, but its coverage during the early impulsive phase of the flare provides
useful information that can be compared with the WL/UV/EUV locations (Figure
3f). Selected RHESSI X-ray images are also plotted as contours with the WL and
EUV images for comparison of loop-top locations (Figure 2d-f).

##  3 Results

###  3.1 Time–Height Relation

To analyze the time–height relations of the loop-top observed in all the five
visible, UV, and EUV channels through the period of interest from 15:56 UT to
17:12 UT, for each time step and for each altitude we integrate the intensity
for the total flux in a $10\arcsec$-wide band between the dashed lines marked
in Figure 1d, which best includes the brightest part of the loop-top. The
integrated brightness for all the five wavelength channels, as well as their
comparison, are shown in Figure 3 as a function of time and height above the
photospheric limb. Note that the HMI, AIA’s UV and EUV channels have different
temporal cadences of 45 s, 24 s, and 12 s, and HMI and AIA have different
pixel sizes of about $0.5\arcsec$ and $0.6\arcsec$; therefore, the time–height
plots in Figure 3 are shown after all the AIA images are interpolated to match
the WL image resolution and are aligned with the WL images in space and time.

Figure 3: Time–height relations obtained for the flare’s loop-top after an
integration over a $10\arcsec$-wide band for each time step, using
observations from (a) HMI WL, (b) 1700 Å, (c) 1600 Å, (d) 335 Å, and (e) 304
Å. Contours in each of these images mark the 75% of intensity level at the
peak of their corresponding brightness enhancement patches. The WL contours
are also plotted in panel (d) to better compare the WL emission relative to
EUV emissions. (f) Contours in panels (a) – (e) are plotted together and
numbers “1” – “4” are marked corresponding to the four brightening patches.
The centroid locations obtained from RHESSI’s 6-10 keV and 25-60 keV soft
X-ray data are also overplotted as ‘+’ signs for a comparison.

It can be seen from Figure 3 that through all the five wavelength channels,
the loop-top shows four distinct brightness-enhancements patches, each of
which is marked by one contour. Within each wavelength channel, every
brightness-enhancement patch differs each other in both enhancement amount and
duration, therefore we plot the contours using 75% of the peak intensities of
their respective patches. As shown in Figure 3f, based on the time and
location of the brightness-enhancement patches, the five wavelength channels
form two groups, with one group being the WL and UV channels (1700 Å and 1600
Å), and another group the EUV channels (335 Å and 304 Å). Within each group,
the height and the duration of the four brightness enhancements observed in
different channels are quite similar but do not exactly overlap. The
brightness enhancements in the EUV channels occur earlier in time and higher
in altitude than those in the WL and UV channels, and the time lags and height
differences between the two groups of enhancement patches increase as the
flare loop-top evolves to higher altitude. The X-ray centroid locations of the
loop-top, obtained from the RHESSI’s 6-10 keV and 25-60 keV channels during
15:53UT and 16:17UT, can be found at least 10″ higher and at least 10 minutes
earlier than the AIA’s EUV channels (Figure 3f).

Figure 4: (a) Soft X-ray flux from GOES during the flare’s impulsive and
gradual phases. Vertical dotted lines indicate the time window for the curves
plotted in panel (b). (b) Temporal evolution of the flux for the HMI and AIA
wavelength channels, integrated in the stripe shown in Figure 1e. All fluxes
are normalized relative to their respective peak fluxes during 16:05 –
16:15UT.

To better compare the temporal variations of the intensity for all the
WL/UV/EUV channels, we further integrate the time–height diagrams over
heights, and study the total flux change over time. In order to focus only on
the temporal changes in the loop-top, we integrate the intensity between the
heights of $6\arcsec$ and $45\arcsec$ above the limb, excluding the upper
photosphere that keeps elevated brightness throughout the analyzed period in
the WL and UV channels. Figure 4b shows all the flux curves, which are
normalized to 1.0 at their respective major brightness peak between 16:05 –
16:15 UT. It can be seen that all the UV and EUV channels show similar trend
of growth and decay, despite that the UV channels are grouped together with
the WL in the brightness-enhancement time and altitude. Showing a growth trend
similar to the UV channels before the first major peak around 16:12 UT, the WL
light flux continues to grow to a brighter peak around 16:28 UT after a 5-min
dip in brightness following the first peak. This can also well be seen in
Figure 1 and its accompanied animation. That is, the WL and UV channels show
sharply different behaviors after about 16:17 UT despite their many
similarities in other aspects, highlighting the different emission mechanisms
in the WL and UV wavelengths at the flare’s loop-top. For comparison, the GOES
soft X-ray flux is plotted in Figure 4a. We need to keep in mind that the GOES
flux is an integration of the X-ray flux of the entire Sun, including the
flare’s footpoints, loops, and elsewhere, while the fluxes shown in Figure 4b
are obtained only from the flare’s off-limb loop-top.

###  3.2 Quasi-Periodic Pulsations

To study the shorter-period pulsations in the brightness observed in all the
wavelengths, we detrend the brightness curves shown in Figure 4b by
subtracting their respective running averages with a window width of 300 sec,
a practice commonly used in studying quasi-periodic pulsations (e.g., Hayes et
al., 2019; Yuan et al., 2019). We have experimented with different window
widths, and found that the 300-sec window width only suppresses long-period
pulsations with little impact on the short-period pulsations that are of
interest in this study. The left panels of Figure 5 show the detrended flux
curve of the WL and its comparisons with the detrended curves from the UV and
EUV channels, and the right panels of Figure 5 show the wavelet power of the
detrended curves of the WL, one UV channel (1600 Å), and one EUV channel (335
Å).

Figure 5: (a) Detrended HMI WL flux curve, and (b) wavelet power of the
detrended WL curve. (c) Comparison of detrended flux curves between the WL and
the UV channels, and (d) wavelet power of the detrended 1600 Å curve. (e)
Comparison of detrended flux curves between the WL and the EUV channels, and
(f) wavelet power of the detrended 335 Å curve. Dark arrows in the left panels
point to the four peak times in the detrended WL curve, and the magenta arrows
indicate the peak times of the detrended 335 Å curve. The shown wavelet power
maps are part of a map that has a longer time sequence of 2 hr with a longer
period range, thus the influence cones locate outside of the plotted range.

It can be seen that both the WL and UV channels show four near-simultaneous
brightness peaks, with a seemingly one-to-one correspondence to the four
brightness-enhancement patches shown in Figure 3. The WL brightness peaks,
marked by the dark arrows, show quasi-periodic pulsations with a period of
approximately 8.0 min. The pulsations in the two UV channels occurred slightly
earlier than those in the WL. The wavelet power calculated from the detrended
WL flux curves peaks around the period of 8.0 min at the beginning, and
gradually shifts to the period of 5 min, lasting a total of about 40 min. The
wavelet power calculated from the 1600 Å shows a wider period range between
$\sim$7 and $\sim$11 min and lasts for about 30 min.

The pulsations in the two EUV channels, indicated by the four magenta arrows
in Figure 5e, lead the WL pulsations by $\sim$2.4 – 6.5 min. These EUV
pulsations are less regular in periodicity with an averaged period of
$\sim$6.6 min, significantly shorter than that of the WL pulsations. Between
the two EUV channels, the pulsations in 335 Å channel lead those in 304 Å
channel slightly by about 28 sec, estimated through cross-correlation. It is
rather interesting that there are four episodes of emission enhancements
across EUV, UV, and WL channels. However, it is not clear whether there exists
an one-to-one physical correspondence among the pulsations of these channels,
because of their different emission mechanisms and the complex flare structure
involving an arcade of loops distributed along the line-of-sight that can
contribute to the EUV emission due to its optically-thin nature.

##  4 Discussion

We have analyzed the rare white-light observation of the evolution of an off-
limb loop system in the SOL2017-09-10 X8.2 flare, and found that (1) there are
quasi-periodic pulsations in the WL and UV channels with a period of around
8.0 min, and less regular pulsations with an average period of 6.6 min in the
EUV channels; (2) intensity-enhancements appear sequentially in WL and UV/EUV
channels with time delays and altitude decreases corresponding to
progressively lower temperatures of the emitting plasma; and (3) the WL flux
of the flare’s loop-top continues to grow for about 16 more minutes while the
UV/EUV fluxes decay.

The quasi-periodic pulsations observed across the WL/UV/EUV channels, with
different periods and irregular time delays between different channels,
implies that these pulsations are unlikely of the MHD wave nature or
oscillatory properties of the flaring plasma. Through analyzing the evolution
of a strong on-disk flare, Asai et al. (2004) suggested that repeated
enhancements of X-ray/EUV emissions were indicative of energy release and
particle acceleration episodes resulting from magnetic reconnection. Our
observation of the limb-flare loop presents a different view angle of a
similar phenomenon, whose quasi-periodic brightenings are also possibly caused
by episodic magnetic reconnection above. For this same flaring event, Reeves
et al. (2021) found damped oscillations in Doppler velocity with a period
$\sim$400 s in the region south of the the main flare arcade, and suggested
that the oscillations are caused by magnetic reconnection outflows. Despite
different areas of the same event analyzed, the loop-top of the main arcade in
our study and the loop-top south of the main arcade in theirs, the
similarities in the pulsation periods may point to a common physical cause –
pulsed magnetic reconnection.

The brightness-enhancement patches across different wavelengths can serve as
tracers of the flare’s energy release and subsequent thermodynamic evolution
across different temperature and density ranges. As Figure 3f shows, the
emission generally progresses with time toward lower altitudes from X-ray to
EUV and then UV/WL, corresponding to the emitting plasma at progressively
lower temperatures. This is a well-known trend that can result from the
interplay of (a) upward development of magnetic reconnection toward higher
altitude with time in the standard flare model, (b) downward contraction of
newly reconnected hot flaring loops that undergo cooling at the same time.
Such contractions have various observational manifestations, such as shrinkage
of flare loops (Švestka et al/, 1987) at speeds of the order of 10 km s-1 seen
in soft X-rays (Forbes & Acton, 1996; Reeves et al., 2008), and the so-called
supra-arcade downflows (e.g., McKenzie & Hudson, 1999; Savage & McKenzie,
2011; Liu et al., 2013). In our analysis (see Figure 3f), the altitudes for
the first two groups of brightenings, around 16:02UT and 16:10UT across all
wavelengths, are similar at about 10″ and 18″ above the limb, while the time
delays between the EUV and UV emissions are about 2 and 4 min, respectively.
For the other two groups of later brightenings, the EUV emission originates
about 7″ higher than the UV/WL emission with time delays of 3 – 5 min, which
translates to a speed of $\sim 17-28$ km s-1 if we assume this results from
loop shrinkage. This trend agrees with the increase with time of the
separation between the 6-10 and 25-60 keV X-ray emission centroids (especially
after about 16:05UT; cf., Liu et al., 2004).

Despite many similarities in WL and UV observations, their integrated light
curves show different trends: the WL flux continues to grow while the UV flux
starts to decay after the peak around 16:10UT. This implies, unsurprisingly,
that the UV and WL have different emission mechanisms. Off-limb flare loops
are visible in WL primarily because of a combination of the following
processes: hydrogen Paschen and Brackett recombination, hydrogen free-free
continuum emission, and Thomson scattering of the solar radiation (Heinzel et
al., 2017; Jejčič et al., 2018). It is likely that before 16:10UT plasma
temperature is relatively high at the flare’s loop-top, and all the WL
emission mechanisms play a role. After 16:10UT and above the height of
approximately $22\arcsec$ (see Figure 3a), the temperature may have dropped
substantially and resulted in reduced UV/EUV emissions; however, the
increasing electron density may still keep the WL intensity growing for a
longer time, primarily because the Paschen and Brackett recombination and
free-free emission depend quadratically on the density. However, one should
also realize that high densities also correspond to non-negligible opacity
leading to a decrease of emission. In fact, a detailed comparison of WL and
EUV 335Å emissions in Figures 2c and 3d, along with the sequence of 335Å
images, shows that during the EUV decay period, the WL emissions are located
in an area of reduced 335 Å emission that appears to be a dark absorption.
This is also evident on other AIA EUV channels, such as 193Å and 211Å (cf.
Figure 1 of Heinzel et al. 2020 and Figure 3 of Jejčič et al. 2018), providing
another piece of evidence that plasma has cooled substantially in the areas
where WL intensity enhances.

We thank an anonymous referee for constructive comments that helped shape this
paper to the current version. Both SDO and RHESSI are NASA space missions, and
GOES are NOAA satellites. J. Z. is partly sponsored by NASA grant
80NSSC18K0668. W. L. was supported by NASA grants NNX16AF78G, 80NSSC21K1327,
and 80NSSC21K1687.

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