# Reconstruction of event kinematics in semi-inclusive deep-inelastic
scattering using the hadronic final state and machine learning

Connor Pecar  Duke University  Anselm Vossen  Duke University  Presented at
DIS2022: XXIX International Workshop on Deep-Inelastic Scattering and Related
Subjects, Santiago de Compostela, Spain, May 2-6 2022

##  I Introduction

Deep-inelastic scattering (DIS) of electrons off ions is at the forefront of
experimental efforts to probe the internal structure of nucleons and nuclei
and will be a primary focus of study at the Electron-Ion Collider. In semi-
inclusive DIS, selected particles produced by the fragmentation of the struck
quark are observed in coincidence with the scattered electron,
$e(k)+N(P)\rightarrow e(k^{\prime})+h(p)+X$, resulting in observables which
provide access to a convolution of parton distribution functions (PDFs),
describing the momentum of partons within the nucleon, and fragmentation
functions (FFs), describing the probability of producing a final state
particle with some momentum from the struck quark in the factorization
approach. [1]

Figure 1: Definition of semi-inclusive kinematic variables in target rest
frame, reprinted from [1].

The kinematic variables describing the DIS process, with center of mass energy
squared $s=(P+k)^{2}$ can be defined in terms of the virtual photon four-
momentum $q$ as [2],

$Q^{2}=-q^{2},\;y=\frac{P\cdot q}{P\cdot k},\;x=\frac{Q^{2}}{sy}$ (1)

##  II SIDIS kinematic reconstruction

In semi-inclusive DIS, observables are extracted in the nucleon center of mass
frame, with the SIDIS cross-section a function of the inclusive DIS variables
as well as ($p_{h\perp},z,\phi_{h}$). The relevant transverse momentum is
defined with respect to the virtual photon axis, and the single-hadron
azimuthal angle $\phi_{h}$ is defined between the lepton scattering plane and
hadron production plane (figure 1). $z$ is defined as $z=\frac{p_{h}\cdot
P}{q\cdot P}$ The calculation of SIDIS kinematics therefore requires precise
reconstruction of the four-momenta of the selected hadron and the exchanged
virtual-photon.

###  II.1 Electron method

Extraction of SIDIS observables and multiplicities at the EIC presents a new
challenge, as fully multi-dimensional SIDIS studies have so far only been
carried out in lower energy fixed target experiments. In fixed target SIDIS
studies, $q$ has been determined using only the scattered electron,
$q=k-k^{\prime}$. However, studies done for the EIC yellow report and EIC
detector proposals have found that a significant contribution to uncertainty
in SIDIS kinematics is poor reconstruction of the virtual photon four-momentum
when using only the scattered electron. In particular, the electron method
fails in such regions of kinematic phase space at the EIC such as at low y
($y<0.05$), where the energy loss of the electron is small and not well
resolved. This is a significant issue for the study of TMD effects at e-p
colliders, as at low-$Q^{2}$ and large-x spin-orbit correlations are expected
to be most significant and higher twist effects are observable. Additionally,
the low-y region will be critical for overlapping the phase space covered by
the EIC and SIDIS studies carried out at other facilities such as Jefferson
Lab.

Figure 2: Momentum vs pseudorapidity in $x-Q^{2}$ bins for positive pions with
$z>0.2$ and requiring $y>0.05$, generated with Pythia-8. Different histogram
colors represent different $\sqrt{s}$ values. Red bands represent 3-$\sigma$
PID coverage based on ATHENA proposal projections.

###  II.2 Hadronic final state methods

Fast simulation studies for the EIC yellow report [3] and ATHENA (A Totally
Hermetic Electron Nucleon Apparatus) proposal [4] have demonstrated that DIS
reconstruction methods developed at past e-p colliders [2] can be used to
improve the reconstruction of inclusive DIS variables at the EIC. The DIS
reconstruction methods developed at HERA utilized combinations of measured
quantities from the scattered electron and the hadronic final state (HFS). The
use of the HFS allowed these additional methods, such as the double angle (DA)
and $\Sigma$-methods [2], to improve inclusive DIS kinematic reconstruction
for various regions of the HERA kinematic space, as well as to make the
reconstruction robust with respect to QED radiative effects [5, 6]. For the
studies planned at the EIC, methods utilizing the HFS must be extended to the
reconstruction of SIDIS kinematics.

The authors of this contribution conducted first studies of SIDIS kinematic
reconstruction for the EIC and demonstrated methods in which the hadronic
final state can be used to improve the reconstruction of the virtual photon
four momentum. This was carried out in the EIC yellow report and ATHENA
proposal [3, 4] by first obtaining the transverse component of $q$ from the
recoil of the HFS transverse to the beamline through a sum of the momenta of
HFS particles. Following the determination of this transverse recoil, the
remaining two components of $q$ can be constrained by the system of equations
including $q$ from the definitions of $Q^{2}$ and $y$,

$q_{x}=\sum_{i}^{N_{HFS}}p_{x,i},q_{y}=\sum_{i}^{N_{HFS}}p_{y,i}$ (2)
$q_{z},q_{t}\leftarrow\begin{cases}Q^{2}=-(q_{x}^{2}+q_{y}^{2}+q_{z}^{2}-q_{t}^{2})\\\
y=\frac{P_{x}q_{x}+P_{y}q_{y}+P_{z}q_{z}-P_{t}q_{t}}{P\cdot k}\end{cases}$ (3)

In the EIC yellow report and ATHENA proposal [3, 4], this procedure was
carried out using various inclusive DIS reconstruction methods developed at
HERA[2], in fast simulations showing improvements over the electron method in
some regions of the DIS kinematic space. As methods such as the Jacquet-
Blondel (JB) method [2] use only the hadronic final state information, this
also allows for the determination of $q$ from the HFS alone. Results using
this approach are shown in the next section compared to ML and electron
methods, with resolution using this method expected to improve with further
developed full simulations based on fast simulation results.

##  III Machine learning kinematic reconstruction

###  III.1 Network architecture

Multiple studies have been conducted demonstrating an improved resolution of
inclusive DIS variables $Q^{2},y,x$ through deep learning approaches [7, 8],
but these have not yet been extended to reconstruction of semi-inclusive DIS
kinematics. In this study, we demonstrate that machine learning models which
learn from the full HFS and scattered electron can be used to improve on
current reconstruction methods to provide reliable reconstruction of the
virtual photon axis across all of the DIS kinematic coverage at the EIC.

This approach to semi-inclusive DIS reconstruction is centered on the use of
deep neural networks to better leverage the full hadronic final state at the
level of reconstructed tracks. While previous applications of deep learning to
inclusive DIS reconstruction directly regressed the kinematic variables of
interest [7, 8], this study aims to improve kinematics by directly regressing
the virtual photon four-momentum in the lab frame.

Improvements to the HFS reconstruction are carried out through the use of
Particle Flow Networks [9]. Particle Flow Networks are an application of the
deep sets neural network architecture, which learns a function of an unordered
set of objects rather than from a fixed size input. The network consists of
fully connected linear neural network layers which take as input the features
of each particle individually, the outputs of which are summed over all
particles to create a latent space representation of the event. The latent
space variables and supplied global features of the event are then passed to
another set of dense layers which produce the final output of the network [9].
Particle flow networks have seen particular success in tasks such as jet
classification at the LHC. Particle flow networks implemented in Keras [10]
are included in the EnergyFlow python package. [9]

Figure 3: Network diagram of Particle Flow Networks [9] with global event
features. Features of each HFS particle supplied individually to layers
$\Phi$, then summed over to form a latent space representation. Latent space
features and global features of event (green), including reconstructed
inclusive DIS variables and DIS electron momentum, supplied to layer $F$ which
produces final output.

###  III.2 Variables and dataset

Figure 4: SIDIS $p_{h\perp}$ resolution mean (left) and RMS (right) as a
function of $y_{true}$ for positive pions with $z>0.2$, $p_{h\perp}>0.1GeV$.
HFS methods surpass electron method for very low $y$, while PFN equals or
outperforms electron method for all $y$.

Figure 5: SIDIS $\phi_{h}$ resolution mean (left) and RMS (right) as a
function of $y_{true}$ for positive pions with $z>0.2$, $p_{h\perp}>0.1GeV$.

The features of the hadronic final state reconstructed particles provided to
the particle flow network include the four-momentum of each particle, as well
as the lab frame azimuthal angle and pseudorapidity to provide direct
information on angular acceptance in addition to momentum in each direction.

The global features used for training include the four-momentum of the
scattered electron and the DIS variables $x$ and $Q^{2}$ from the electron,
DA, and JB methods. By supplying the full electron four-momentum following the
single-particle layers $\Phi$, the model is intended to learn corrections to
the electron method based on the hadronic final state latent space variables.
When a greater amount of fully simulated EIC simulated data is available, the
DIS methods could also be replaced by the output of the deep learning models
for inclusive DIS variables described previously.

The particle flow network was trained to predict the full four-momentum $q$ in
the lab frame. The particle flow network, implemented in Keras [10] and
available in the EnergyFlow python package, is used with per-particle dense
layer units $\phi=(500,500,500)$, $l=512$, and final dense layer units
$F=(200,200,200)$. Both the layers making up $\phi$ and $F$ employ a relu
activation function, with the final output layer having linear activation.

Figure 6: SIDIS $p_{h\perp}$ resolution mean (left) and RMS (right) as a
function of $p_{h\perp,true}$ for positive pions with $z>0.2$.

Figure 7: SIDIS $\phi_{h}$ resolution mean (left) and RMS (right) as a
function of $p_{h\perp,true}$ for positive pions with $z>0.2$.

The dataset used for the training and testing of this model was the ATHENA
full simulation developed for the ATHENA detector proposal for the first
interaction region at the EIC. ATHENA was developed with the objective of
meeting the resolution and physics goals laid out in the EIC yellow report.
The ATHENA full simulation was implemented in DD4hep, Geant4, and Juggler [11,
12, 13]. At the time of the detector proposal, PID algorithms were not fully
implemented, meaning PID information was not included in this model.
Additionally, the scattered electron was taken as always correctly identified
by matching the scattered electron with the MC truth information.

The simulated event sample used for model training and testing was a neutral
current DIS sample generated using Pythia-8 [14], with additional beam
smearing and crossing angle effects implemented. 3 million events with
$Q^{2}>1\>\mathrm{GeV}^{2}$ and 2 million events with
$Q^{2}>10\>\mathrm{GeV}^{2}$ were used for training with 1 million
$Q^{2}>1\>\mathrm{GeV}^{2}$ set aside for model validation.

##  IV Results

As a function of $y$ (figures 4 and 5), using the virtual photon four-momentum
as predicted by the neural network model results in significantly improved
reconstruction of $p_{h\perp}$, $\phi_{h}$ and $z$ for low-y, when compared to
both the electron method and methods utilizing information from the hadronic
final state. The neural network reconstruction of $q$ results in a
distribution of the SIDIS variables which is both better centered around the
true value, and with a significantly smaller RMS where the electron method
begins to fail at low-y. At large-y, the neural network achieves performance
only slightly surpassing that of the electron method, which is expected based
on the projected energy and tracking resolution for the scattered electron
with ATHENA.

As a function of $p_{h\perp,true}$, we also observe a significant improvement
in kinematic reconstruction for both transverse momentum and for the semi-
inclusive azimuthal angle. As the electron method begins to degrade for lower
values of $p_{h\perp}$, the neural network reconstruction of $q$ results in
stable performance to the lowest values of $p_{h\perp}$ in the dataset.

##  V Summary

The EIC will provide the first opportunity for semi-inclusive DIS measurements
in an e-A collider context, giving access to new kinematic regions in which to
precisely explore the 3-dimensional spin structure of nucleons. The
development of reliable kinematic reconstruction methods will be critical to
enabling precision extraction of SIDIS observables, especially at low-y. This
can be achieved through the use of information from the hadronic final state
alongside the scattered electron. As demonstrated in this contribution,
machine learning, here using particle flow networks, can combine the
information from the scattered electron and full HFS to provide reliable SIDIS
kinematic reconstruction across the DIS variable space. Further steps in this
work will include the consideration of QED radiative effects on SIDIS
reconstruction, as well as possible extension to other neural network
architectures exploiting correlations between particles. Additionally, this
approach will continue to be studied and validated as more detailed full
detector simulations are developed for the EIC.

##  VI Acknowledgements

We thank Markus Diefenthaler for helpful discussions throughout the
development of the described machine learning approach and for comments on
this submission.

## References

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# On the Minkowski inequality near the sphere

Otis Chodosh  Otis Chodosh
Stanford University
Department of Mathematics
Building 380
Stanford, CA 94305, USA
<EMAIL_ADDRESS>,  Michael Eichmair  Michael Eichmair
University of Vienna
Faculty of Mathematics
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
https://orcid.org/0000-0001-7993-9536
<EMAIL_ADDRESS>and  Thomas Koerber  Thomas Koerber
University of Vienna
Faculty of Mathematics
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
https://orcid.org/0000-0003-1676-0824
<EMAIL_ADDRESS>
(Date: August 30, 2024)

###### Abstract.

We construct a sequence $\\{\Sigma_{\ell}\\}_{\ell=1}^{\infty}$ of closed,
axially symmetric surfaces $\Sigma_{\ell}\subset\mathbb{R}^{3}$ that converges
to the unit sphere in $W^{2,p}\cap C^{1}$ for every $p\in[1,\infty)$ and such
that, for every $\ell$,

$\int_{\Sigma_{\ell}}H_{\Sigma_{\ell}}-\sqrt{16\,\pi\,|\Sigma_{\ell}|}<0$

where $H_{\Sigma_{\ell}}$ is the mean curvature of $\Sigma_{\ell}$. This shows
that the Minkowski inequality with optimal constant fails even for
perturbations of a round sphere that are small in $W^{2,p}\cap C^{1}$ unless
additional convexity assumptions are imposed.

##  1\. Introduction

Let $\Sigma\subset\mathbb{R}^{3}$ be a closed surface and

$\mathcal{M}(\Sigma)=\int_{\Sigma}H-\sqrt{16\,\pi\,|\Sigma|}$

where $|\Sigma|$ is the area of $\Sigma$ and $H\in C^{\infty}(\Sigma)$ is the
mean curvature computed with respect to the outward unit normal.
Let $S=\\{x\in\mathbb{R}^{3}:|x|=1\\}$ be the unit sphere centered at the
origin. Given $u\in C^{\infty}(S)$, we denote by

$\Sigma(u)=\\{(1+u(x))\,x:x\in S\\}$

the normal graph of $u$ above $S$.
Note that $\mathcal{M}(S)=0$. A classical result of H. Minkowski [Minkowski,
§7] asserts that, provided that $\Sigma$ is convex,

(1) $\displaystyle\mathcal{M}(\Sigma)\geq 0\text{ with equality if and only if
}\Sigma\text{ is a round sphere.}$

P. Guan and J. Li [GuanLi, Theorem 2] have proven that (1) holds provided that
$\Sigma$ is star-shaped and mean-convex. G. Huisken has shown that (1) holds
for outward-minimizing surfaces; see [GuanLi, Theorem 6]. J. Dalphin, A.
Henrot, S. Masnou, and T. Takahashi [DalphinHenrotMasnouTakahashi, Theorem
1.1] have established (1) in the case where $\Sigma$ is axially symmetric and
such that $\Sigma\cap\Pi$ is connected for every affine plane $\Pi$ orthogonal
to the axis of symmetry.
By contrast, there are closed surfaces $\Sigma\subset\mathbb{R}^{3}$ with
$\mathcal{M}(\Sigma)<0$. The following examples of such surfaces have been
constructed in [DalphinHenrotMasnouTakahashi, Theorem 1.2].

######  Theorem 1.

There is a sequence $\\{\Sigma_{\ell}\\}_{\ell=1}^{\infty}$ of closed surfaces
$\Sigma_{\ell}\subset\mathbb{R}^{3}$ with $|\Sigma_{\ell}|=4\,\pi$ and
$\mathcal{M}(\Sigma_{\ell})\to-\infty$. There is a sequence
$\\{\tilde{\Sigma}_{\ell}\\}_{\ell=1}^{\infty}$ of closed axially symmetric
surfaces $\tilde{\Sigma}_{\ell}\subset\mathbb{R}^{3}$ with
$|\tilde{\Sigma}_{\ell}|=4\,\pi$ and
$\mathcal{M}(\tilde{\Sigma}_{\ell})\to-8\,\pi$.

It is natural to ask whether the Minkowski inequality (1) holds for surfaces
in a neighborhood of $S$. Since every $C^{2}$-perturbation of the round sphere
is strictly convex, we consider lower regularity perturbations. In this
direction, F. Glaudo [Glaudo, Appendix A] has recently proved the following
result.

######  Theorem 2.

There is a sequence $\\{u_{\ell}\\}_{\ell=1}^{\infty}$ of functions
$u_{\ell}\in C^{\infty}(S)$ with the following properties.

  * $\circ$

$\mathcal{M}(\Sigma(u_{\ell}))\to-\infty$

  * $\circ$

$|u_{\ell}|_{C^{1}(S)}\to 0$

The goal of this paper is to establish the following related result.

######  Theorem 3.

There is a sequence $\\{u_{\ell}\\}_{\ell=1}^{\infty}$ of axially symmetric
functions $u_{\ell}\in C^{\infty}(S)$ with the following properties.

$\displaystyle\circ\qquad|u_{\ell}|_{C^{1}(S)}\to
0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\circ\qquad|u_{\ell}|_{W^{2,p}(S)}\to 0\text{ for every
}p\in[1,\infty)$
$\displaystyle\circ\qquad\int_{\Sigma(u_{\ell})}|\mathring{h}|^{2}\to 0$
$\displaystyle\circ\qquad\mathcal{M}(\Sigma(u_{\ell}))<0\text{ for every
}\ell$

Our interest in pursuing Theorem 3 stems from the following result due to the
first-named and the second-named author; see [angstnomore, Proposition E.4].

######  Theorem 4.

Let $\\{u_{\ell}\\}_{\ell=1}^{\infty}$ be a sequence of functions $u_{\ell}\in
C^{\infty}(S)$ with $|u_{\ell}|_{C^{1}(S)}\to 0$. Then, as $\ell\to\infty$,

$\mathcal{M}(\Sigma(u_{\ell}))\geq-o(1)\int_{\Sigma(u_{\ell})}|\mathring{h}|^{2}.$

C. De Lellis and S. Müller [DeLellisMuller, Theorem 1.1] have proven the
following quantitative Schur estimate.

######  Theorem 5.

There exists a constant $c\geq 2$ such that, for every closed surface
$\Sigma\subset\mathbb{R}^{3}$,

(2)
$\displaystyle\int_{\Sigma}\left(H-|\Sigma|^{-1}\,\int_{\Sigma}H\right)^{2}\leq
c\,\int_{\Sigma}|\mathring{h}|^{2}.$

Note that, if $\Sigma$ has genus zero, then

$\int_{\Sigma}\left(H-|\Sigma|^{-1}\,\int_{\Sigma}H\right)^{2}=2\,\int_{\Sigma}|\mathring{h}|^{2}-\sqrt{64\,\pi}\,|\Sigma|^{-1/2}\,\mathcal{M}(\Sigma)-|\Sigma|^{-1}\mathcal{M}(\Sigma)^{2}$

by the Gauss-Bonnet theorem. Consequently, given $\varepsilon>0$, Theorem 4
shows that (2) holds with $c=2+\varepsilon$ provided that $\Sigma$ is
sufficiently close to $S$ in $W^{2,2}\cap C^{1}$. By contrast, in view of
Theorem 3, (2) is false with $c=2$, even for closed surfaces
$\Sigma\subset\mathbb{R}^{3}$ that are $W^{2,p}\cap C^{1}$-close to $S$ where
$p\in[1,\infty)$. See also [Perez:thesis, Chapter 3] and [Glaudo] for other,
related considerations.

######  Remark 6.

Theorem 4 implies that the surfaces $\Sigma(u_{\ell})$ constructed by F.
Glaudo as described in Theorem 2 satisfy

$\int_{\Sigma(u_{\ell})}|\mathring{h}|^{2}\to\infty.$

Theorem 4 has played a pivotal role in the classification of large stable
constant mean curvature surfaces in initial data sets; see [angstnomore, Lemma
4.8] and also [EichmairKoerberMetzgerSchulze, Lemma 22]. This has led the
authors to hope that there might exist $\varepsilon_{0}>0$ such that (1) holds
provided that

$\int_{\Sigma}|\mathring{h}|^{2}<\varepsilon_{0}.$

Theorem 3 dashes this hope.
By contrast, S. Brendle [Brendle] has recently proved the Michael-Simon
Sobolev inequality [MichaelSimon, Theorem 2.1] with optimal constant; see also
the alternative proof of S. Brendle and the second-named author
[BrendleEichmair]. This result implies the following Minkowski inequality with
non-optimal constant for mean-convex surfaces.

######  Theorem 7 ([Brendle, Corollary 2]).

Let $\Sigma\subset\mathbb{R}^{3}$ be a closed surface with non-negative mean
curvature. There holds

$\int_{\Sigma}H\geq\sqrt{4\,\pi\,|\Sigma|}.$

Let $\Sigma\subset\mathbb{R}^{3}$ be a closed surface with non-negative mean
curvature such that $|\Sigma|=4\,\pi$. Theorem 7 contrasts Theorem 1 and
Theorem 2 in the sense that $\mathcal{M}(\Sigma)\geq-4\,\pi$. Moreover,
[GuanLi, Theorem 2] contrasts Theorem 3 in the sense that
$\mathcal{M}(\Sigma)\geq 0$ provided that $\Sigma$ is $C^{1}$-close to
$S_{1}(0)$. These reasons support the following conjecture; see also
[DalphinHenrotMasnouTakahashi, p. 2732].

######  Conjecture 8.

Let $\Sigma\subset\mathbb{R}^{3}$ be a closed surface with non-negative mean
curvature. There holds $\mathcal{M}(\Sigma)\geq 0$ with equality if and only
if $\Sigma$ is a round sphere.

######  Remark 9.

By [DalphinHenrotMasnouTakahashi, Theorem 1.1], Conjecture 8 holds if
$\Sigma\subset\mathbb{R}^{3}$ is assumed to be closed, axially symmetric, and
with non-negative mean curvature. Indeed, by the maximum principle,
$\Sigma\cap\Pi$ is connected for every affine plane $\Pi$ orthogonal to the
axis of symmetry.

######  Remark 10.

Let $\Sigma\subset\mathbb{R}^{3}$ be a closed surface with non-negative mean
curvature and $\Omega\subset\mathbb{R}^{3}$ the compact domain bounded by
$\Sigma$. V. Agostiniani, M. Fogagnolo, and L. Mazzieri
[AgostinianiFogagnoloMazzieri, Theorem 1.4] have shown that

(3) $\displaystyle\int_{\Sigma}H\geq(384\,\pi^{2}\,|\Omega|)^{1/3}$

with equality if and only if $\Sigma$ is a round sphere. Note that (3) would
follow from Conjecture 8 and the isoperimetric inequality.

F. Glaudo’s proof of Theorem 2 is based on adding dents with very negative
mean curvature to the sphere. In view of Theorem 4, the traceless second
fundamental form of such dents is large in $L^{2}$. The construction of the
examples in Theorem 3 is necessarily more subtle.
We now outline the proof of Theorem 3. Let $\\{u_{\ell}\\}_{\ell=1}^{\infty}$
be a sequence of functions $u_{\ell}\in C^{\infty}(\Sigma)$ with zero mean
such that $|u_{\ell}|_{C^{1}(S)}\to 0$. As in [Glaudo, Appendix A], our
starting point is a Taylor expansion of the functional $\mathcal{M}$ off the
unit sphere $S$. Indeed, we have

(4) $\displaystyle\mathcal{M}(\Sigma(u_{\ell}))=-\int_{S}|\nabla
u_{\ell}|^{2}\,\Delta
u_{\ell}+O(1)\,|u_{\ell}|^{2}_{W^{1,2}(S)}+O(1)\,|u_{\ell}|^{2}_{C^{1}(S)}\,|u_{\ell}|_{W^{1,2}(S)}\,|u_{\ell}|_{W^{2,2}(S)};$

see Lemma 12. For ease of exposition, we focus on the first two terms on the
right-hand side of (4). The key idea is to exploit the cubic nature of the
first of those terms. Given $\alpha\in(0,1)$, we consider

$u_{\ell}=-\ell^{-2-\alpha}\,\sum_{i=0}^{2^{-5}\,\ell}v_{\ell_{i}}$

where $v_{\ell_{i}}\in C^{\infty}(S)$ are spherical harmonics whose frequency
is comparable to $\ell$. A straightforward computation shows that
$|u_{\ell}|_{W^{2,2}(S)}\to 0$. What is more, given $p\in[1,\infty)$, we show
that $|u_{\ell}|_{W^{2,p}(S)}\to 0$ provided that $\alpha>1-2/p$; see Lemma
15. Next, we observe that

$\int_{S}|\nabla u_{\ell}|^{2}\,\Delta
u_{\ell}\sim\ell^{-2-3\,\alpha}\sum_{i_{1},\,i_{2},\,i_{3}=1}^{2^{-5}\,\ell}\int_{S}v_{\ell_{i_{1}}}\,v_{\ell_{i_{2}}}\,v_{\ell_{i_{3}}}\qquad\text{while}\qquad\int_{S}|\nabla
u_{\ell}|^{2}\sim\ell^{-2-2\,\alpha}.$

Choosing the spherical harmonics $v_{\ell_{i}}$ carefully, we obtain that

$\inf_{\ell,\,i_{1},\,i_{2},\,i_{3}}\ell^{2}\,\int_{S}v_{\ell_{i_{1}}}\,v_{\ell_{i_{2}}}\,v_{\ell_{i_{3}}}>0;$

see Lemma 16. We note that these estimates build on precise quantitative
estimates for E. Wigner’s $3j$-symbol; see Appendix B.

### Acknowledgments

Otis Chodosh was supported by a Terman Fellowship. Michael Eichmair
acknowledges the support of the START Programme Y963 of the Austrian Science
Fund. Thomas Koerber acknowledges the support of the Lise Meitner Programme
M3184 of the Austrian Science Fund.

##  2\. Graphs over Euclidean spheres

Let $u\in C^{\infty}(S)$ and recall that $\Sigma(u)$ is the normal graph of
$u$ over $S$. Let $g(u)$ be the first fundamental form, $\nu(u)$ the outward
unit normal, $h(u)$ the second fundamental form, and $H(u)$ the mean curvature
of $\Sigma(u)$. Let $g$ be the first fundamental form of $S$ and
$f(u)=\sqrt{(1+u)^{2}+|\nabla u|^{2}}$.

######  Lemma 11.

The following identities hold.

$\displaystyle\circ\qquad g(u)=(1+u)^{2}\,g+du\otimes
du\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\circ\qquad g(u)^{-1}=(1+u)^{-2}\,\left(g^{-1}-f(u)^{-2}\,\nabla
u\otimes\nabla u\right)$
$\displaystyle\circ\qquad\nu(u)=f(u)^{-1}\,((1+u)\,x-\nabla u)$
$\displaystyle\circ\qquad h(u)=f(u)^{-1}\,\big{[}(1+u)^{2}\,g+2\,du\otimes
du-(1+u)\,\nabla du\big{]}$

######  Lemma 12.

Let $\\{u_{\ell}\\}_{\ell=1}^{\infty}$ be a sequence of functions $u_{\ell}\in
C^{\infty}(S)$ with $|u_{\ell}|_{C^{1}(S)}\to 0$ and

$\displaystyle\int_{S}u_{\ell}=0$

for all $\ell$. Then

$|\Sigma(u_{\ell})|=4\,\pi+O(1)\,|u_{\ell}|^{2}_{W^{1,2}(S)}$

and

$\int_{\Sigma(u_{\ell})}H=8\,\pi+\int_{S}\nabla du_{\ell}(\nabla
u_{\ell},\nabla
u_{\ell})+O(1)\,|u_{\ell}|^{2}_{W^{1,2}(S)}+O(1)\,|u_{\ell}|^{2}_{C^{1}(S)}\,|u_{\ell}|_{W^{1,2}(S)}\,|u_{\ell}|_{W^{2,2}(S)}.$

###### Proof.

Note that

$\sqrt{\det g(u_{\ell})}=(1+u_{\ell})\,f(u_{\ell})$

and

$\displaystyle H(u_{\ell})$
$\displaystyle=2\,f(u_{\ell})^{-1}+f(u_{\ell})^{-3}\,|\nabla u_{\ell}|^{2}$
$\displaystyle\qquad-(1+u_{\ell})^{-1}\,f(u_{\ell})^{-1}\,\Delta
u_{\ell}+(1+u_{\ell})^{-1}\,f(u_{\ell})^{-2}\,\nabla du_{\ell}(\nabla
u_{\ell},\nabla u_{\ell}).$

Using Taylor’s theorem, we obtain

$\sqrt{\det g(u_{\ell})}=1+2\,u_{\ell}+O(1)\,(|u_{\ell}|^{2}+|\nabla
u_{\ell}|^{2})$

and

$\displaystyle H(u_{\ell})\,\sqrt{\det g(u_{\ell})}$
$\displaystyle=2\,(1+u_{\ell})+(1+u_{\ell})\,f(u_{\ell})^{-2}\,|\nabla
u_{\ell}|^{2}-\Delta u_{\ell}+f(u_{\ell})^{-1}\,\nabla du_{\ell}(\nabla
u_{\ell},\nabla u_{\ell})$ $\displaystyle=2\,(1+u_{\ell})-\Delta
u_{\ell}+\nabla du_{\ell}(\nabla u_{\ell},\nabla u_{\ell})$
$\displaystyle\qquad+O(1)\,|\nabla u_{\ell}|^{2}+O(1)\,(|u_{\ell}|+|\nabla
u_{\ell}|^{2})\,|\nabla u_{\ell}|^{2}\,|\nabla^{2}u_{\ell}|.$

The assertion follows from these estimates, using that

$\int_{S}\Delta u_{\ell}=0$

and

$\displaystyle\int_{S}(|u_{\ell}|+|\nabla u_{\ell}|^{2})\,|\nabla
u_{\ell}|^{2}\,|\nabla^{2}u_{\ell}|$
$\displaystyle=O(1)\,|u_{\ell}|^{2}_{C^{1}(S)}\,\int_{S}|\nabla
u_{\ell}|\,|\nabla^{2}u_{\ell}|$
$\displaystyle=O(1)\,|u_{\ell}|^{2}_{C^{1}(S)}\,|u_{\ell}|_{W^{1,2}(S)}\,|u_{\ell}|_{W^{2,2}(S)}.$

∎

##  3\. Proof of Theorem 3

Let $k\geq 7$ and $\ell=2^{k}$.
Given an integer $0\leq i\leq 2^{-5}\,\ell,$ we let $\ell_{i}=\ell+4\,i$. Note
that

$\displaystyle\ell_{i}=0\,\,\text{mod}\,\,4\qquad\text{and}\qquad\ell\leq\ell_{i}\leq\tfrac{9}{8}\,\ell.$

Recall the definitions of $\Lambda_{\ell_{i}}\subset C^{\infty}(S)$ and
$v_{\ell_{i}}\in\Lambda_{\ell_{i}}$ from Appendix A and the definition of E.
Wigner’s $3j$-symbol from Appendix B.
Let $\alpha\in(0,1)$. We define $u_{\ell}\in C^{\infty}(S)$ by

$u_{\ell}=-\ell^{-2-\alpha}\sum_{i=0}^{2^{-5}\,\ell}v_{\ell_{i}}.$

######  Lemma 13.

There holds $|u_{\ell}|_{W^{1,2}(S)}=O(1)\,\ell^{-1-\alpha}.$

###### Proof.

By Lemma 17, as $\ell\to\infty$,

$\int_{S}|\nabla u_{\ell}|^{2}=O(1)\,\ell^{-2-2\,\alpha}$

and

$\int_{S}u_{\ell}^{2}=O(1)\,\ell^{-4-2\,\alpha}.$

∎

######  Lemma 14.

There holds $|u_{\ell}|_{C^{1}(S)}=O(1)\,\ell^{-\alpha}$.

###### Proof.

Using Lemma 19 and Lemma 20, we obtain

$|u_{\ell}|_{C^{1}(S)}=O(1)\,\ell^{-2-\alpha}\sum_{i=0}^{2^{-5}\,\ell}|v_{\ell_{i}}|_{C^{1}(S)}=O(1)\,\ell^{-2-\alpha}\sum_{i=0}^{2^{-5}\,\ell}\ell_{i}=O(1)\,\ell^{-\alpha}.$

∎

######  Lemma 15.

Let $m\geq 2$ be an even integer. There holds
$|u_{\ell}|_{W^{2,m}(S)}=O(\ell^{1-\alpha-2/m}).$

###### Proof.

Since $u_{\ell}$ has zero mean,

$|u_{\ell}|^{m}_{W^{2,m}(S)}=O(1)\,\int_{S}(\Delta u_{\ell})^{m}$

by elliptic regularity; see, e.g., [GilbargTrudinger, Theorem 9.11].
Assume that $m=2$. By Lemma 17,

$\int_{S}(\Delta
u_{\ell})^{2}=\ell^{-4-2\,\alpha}\,\sum_{i=0}^{2^{-5}\,\ell}\int_{S}(\Delta
v_{\ell_{i}})^{2}=O(1)\,\ell^{-4-2\,\alpha}\,\sum_{i=0}^{2^{-5}\,\ell}\ell_{i}^{3}=O(1)\,\ell^{-2\,\alpha}.$

Assume that $m\geq 4$. Using that $v_{\ell_{i}}\in\Lambda_{\ell_{i}}$, we
obtain that

$\int_{S}(\Delta
u_{\ell})^{m}=\ell^{-2\,m-\alpha\,m}\,\sum_{i_{1},\ldots,i_{m}=0}^{2^{-5}\,\ell}\,\int_{S}\,\prod_{j=1}^{m}\Delta
v_{\ell_{i_{j}}}=\ell^{-\alpha\,m}\,\sum_{i_{1},\ldots,i_{m}=0}^{2^{-5}\,\ell}\,O(1)\,\bigg{|}\int_{S}\,\prod_{j=1}^{m}v_{\ell_{i_{j}}}\bigg{|}.$

Recall the definition of $\Gamma(\ell_{i_{1}},\ldots,\ell_{i_{m-1}})$ from
(8). By Lemma 22,

$\displaystyle\int_{S}\,\prod_{j=1}^{m}v_{\ell_{i_{j}}}$
$\displaystyle=\sum_{z\in\Gamma(\ell_{i_{1}},\ldots,\ell_{i_{m-1}})}\delta_{z_{{m-1}}\ell_{i_{m}}}\,(2\,\ell_{i_{m}}+1)^{-1}\,\prod_{j=1}^{m-2}(2\,z_{j+1}+1)\,\begin{pmatrix}z_{{j}}&\ell_{i_{j+1}}&z_{j+1}\\\
0&0&0\end{pmatrix}^{2}$
$\displaystyle=O(1)\,\ell^{-1}\,\sum_{z\in\Gamma(\ell_{i_{1}},\ldots,\ell_{i_{m-1}})}\,\delta_{z_{{m-1}}\,\ell_{i_{m}}}\,\prod_{j=1}^{m-2}(2\,z_{j+1}+1)\,\begin{pmatrix}z_{{j}}&\ell_{i_{j+1}}&z_{j+1}\\\
0&0&0\end{pmatrix}^{2}.$

Clearly,

$\displaystyle\,\sum_{i_{m}=0}^{2^{-5}\,\ell}\,\sum_{z\in\Gamma(\ell_{i_{1}},\ldots,\ell_{i_{m-1}})}\,\delta_{z_{{m-1}}\,\ell_{i_{m}}}\,\prod_{j=1}^{m-2}(2\,z_{j+1}+1)\,\begin{pmatrix}z_{{j}}&\ell_{i_{j+1}}&z_{j+1}\\\
0&0&0\end{pmatrix}^{2}$
$\displaystyle\qquad\leq\sum_{z\in\Gamma(\ell_{i_{1}},\ldots,\ell_{i_{m-1}})}\,\prod_{j=1}^{m-2}(2\,z_{j+1}+1)\,\begin{pmatrix}z_{{j}}&\ell_{i_{j+1}}&z_{j+1}\\\
0&0&0\end{pmatrix}^{2}.$

Applying Lemma 25, we see that

$\sum_{z\in\Gamma(\ell_{i_{1}},\ldots,\ell_{i_{m-1}})}\,\prod_{j=1}^{m-2}(2\,z_{j+1}+1)\,\begin{pmatrix}z_{{j}}&\ell_{i_{j+1}}&z_{j+1}\\\
0&0&0\end{pmatrix}^{2}=O(1).$

The assertion follows, using that

$\sum_{i_{1},\ldots,i_{m-1}=0}^{2^{-5}\,\ell}1=O(1)\,\ell^{m-1}.$

∎

######  Lemma 16.

There holds

$\liminf_{\ell\to\infty}\ell^{1+3\,\alpha}\,\int_{S}|\nabla
u_{\ell}|^{2}\,\Delta u_{\ell}>0.$

###### Proof.

Using Lemma 18 and Lemma 22, we have

$\displaystyle\frac{1}{4\,\pi}\,\int_{S}|\nabla u_{\ell}|^{2}\,\Delta
u_{\ell}=\frac{1}{2}\,\ell^{-6-3\,\alpha}\,\sum_{i_{1},\,i_{2},\,i_{3}=0}^{2^{-5}\,\ell}\,\ell_{i_{1}}\,(\ell_{i_{1}}+1)$
$\displaystyle\,\big{[}\ell_{i_{2}}\,(\ell_{i_{2}}+1)+\ell_{i_{3}}\,(\ell_{i_{3}}+1)-\ell_{i_{1}}\,(\ell_{i_{1}}+1)\big{]}$
$\displaystyle\,\times\begin{pmatrix}\ell_{i_{1}}&\ell_{i_{2}}&\ell_{i_{3}}\\\
0&0&0\end{pmatrix}^{2}.$

Note that

$\ell_{i_{2}}\,(\ell_{i_{2}}+1)+\ell_{i_{3}}\,(\ell_{i_{3}}+1)-\ell_{i_{1}}\,(\ell_{i_{1}}+1)\geq\ell^{2}/2$

so that

$\ell_{i_{1}}\,(\ell_{i_{1}}+1)\,\big{[}\ell_{i_{2}}\,(\ell_{i_{2}}+1)+\ell_{i_{3}}\,(\ell_{i_{3}}+1)-\ell_{i_{1}}\,(\ell_{i_{1}}+1)\big{]}\geq\ell^{4}/2.$

In conjunction with Lemma 23, as $\ell\to\infty$,

$\int_{S}|\nabla u_{\ell}|^{2}\,\Delta
u_{\ell}\geq\left(2^{-18}+o(1)\right)\,\ell^{-1-3\,\alpha}.$

∎

###### Proof of Theorem 3.

Using Lemma 12, Lemma 13, Lemma 14, and Lemma 15, we see that

$\mathcal{M}(\Sigma(u_{\ell}))=-\int_{S}|\nabla u_{\ell}|^{2}\,\Delta
u_{\ell}+O(1)\,\ell^{-2-2\,\alpha}+O(1)\,\ell^{-1-4\,\alpha}.$

By Lemma 16, the right-hand side is negative for all sufficiently large. Using
Lemma 13 and Lemma 11, we see that

$\int_{\Sigma(u_{\ell})}|\mathring{h}|^{2}\to 0.$

Let $m\geq 2$ be an even integer. By Lemma 15, $|u_{\ell}|_{W^{2,m}(S)}\to 0$
provided that $\alpha>1-2/m$. Using a diagonal argument, the assertion
follows. ∎

##  Appendix A Spherical harmonics and Legendre polynomials

Recall that the eigenvalues of the operator

$-\Delta:H^{2}(S)\to L^{2}(S)$

are given by $\ell\,(\ell+1)$ where $\ell\geq 0$ is an integer. We let
$\Lambda_{\ell}$ denote the eigenspace corresponding to the eigenvalue
$\ell\,(\ell+1)$ and recall that $\dim\,\Lambda_{\ell}=2\,\ell+1$.
The Legendre polynomials $P_{\ell}$ may be defined via a generating function
as follows. Given $s\in[0,1]$ and $t\in[0,1)$, there holds

(5)
$\displaystyle(1-2\,s\,t+t^{2})^{-\frac{1}{2}}=\sum_{\ell=0}^{\infty}P_{\ell}(s)\,t^{\ell}.$

We define $v_{\ell}\in C^{\infty}(S)$ by

$v_{\ell}(x^{1},x^{2},x^{3})=P_{\ell}(-x^{3}).$

Recall from, e.g., [CourantHilbert, §8] that $v_{\ell}\in\Lambda_{\ell}$,
namely, $-\Delta v_{\ell}=\ell\,(\ell+1)\,v_{\ell}$.

######  Lemma 17 ([CourantHilbert, §8]).

Let $\ell_{1},\ell_{2}\geq 0$ be integers. The following identities holds.

$\displaystyle\circ\qquad\frac{1}{4\,\pi}\,\int_{S}v_{\ell_{1}}\,v_{\ell_{2}}=\frac{1}{2\,\ell_{1}+1}\,\delta_{\ell_{1}\ell_{2}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\circ\qquad\frac{1}{4\,\pi}\,\int_{S}\nabla
v_{\ell_{1}}\cdot\nabla
v_{\ell_{2}}=\frac{\ell_{1}\,(\ell_{1}+1)}{2\,\ell_{1}+1}\,\delta_{\ell_{1}\ell_{2}}$
$\displaystyle\circ\qquad\frac{1}{4\,\pi}\,\int_{S}\Delta v_{\ell_{1}}\,\Delta
v_{\ell_{2}}=\frac{\ell_{1}^{2}\,(\ell_{1}+1)^{2}}{2\,\ell_{1}+1}\,\delta_{\ell_{1}\ell_{2}}$

######  Lemma 18.

Let $\ell_{1},\,\ell_{2},\,\ell_{3}\geq 0$ be integers. There holds

$\int_{S}\Delta v_{\ell_{1}}\,\nabla v_{\ell_{2}}\cdot\nabla
v_{\ell_{3}}=-\frac{1}{2}\,\ell_{1}\,(\ell_{1}+1)\,\big{[}\ell_{2}\,(\ell_{2}+1)+\ell_{3}\,(\ell_{3}+1)-\ell_{1}\,(\ell_{1}+1)\big{]}\int_{S}v_{\ell_{1}}\,v_{\ell_{2}}\,v_{\ell_{3}}.$

###### Proof.

We have

$\displaystyle\operatorname{div}\,(v_{\ell_{1}}\,v_{\ell_{2}}\,\nabla
v_{\ell_{3}}+v_{\ell_{1}}\,v_{\ell_{3}}\,\nabla
v_{\ell_{2}}-v_{\ell_{2}}\,v_{\ell_{3}}\,\nabla v_{\ell_{1}})$
$\displaystyle\qquad=2\,v_{\ell_{1}}\,\nabla v_{\ell_{2}}\cdot\nabla
v_{\ell_{3}}+v_{\ell_{1}}\,v_{\ell_{2}}\,\Delta
v_{\ell_{3}}+v_{\ell_{1}}\,v_{\ell_{3}}\,\Delta
v_{\ell_{2}}-v_{\ell_{2}}\,v_{\ell_{3}}\,\Delta v_{\ell_{1}}.$

Integrating and using that $v_{\ell_{i}}\in\Lambda_{i},\,i=1,\,2,\,3$, the
assertion follows. ∎

The proof of the following lemmas is based on [SteinWeiss, Chapter IV, Lemma
2.8 and Corollary 2.9].

######  Lemma 19.

There holds, as $\ell\to\infty$,

$|v_{\ell}|_{C^{0}(S)}=O(1).$

###### Proof.

Given $x\in S$, consider the linear map $\Lambda_{\ell}\to\mathbb{R}$ given by
$f\mapsto f(x)$. By the Riesz representation theorem, for every $x\in S$,
there is a unique $F_{x}\in\Lambda_{\ell}$ so that

$f(x)=\int_{S}f\,F_{x}.$

In particular,

$|f(x)|\leq|f|_{L^{2}(S)}\,|F_{x}|_{L^{2}(S)}.$

Since $SO(3)$ acts transitively on $S$, it follows that
$|F_{x}|_{L^{2}(S)}^{2}$ is independent of $x$.

Let $\\{f_{i}\\}_{i=1}^{2\,\ell+1}$ be an orthonormal basis of
$\Lambda_{\ell}$, we write

$F_{x}=\sum_{i=1}^{2\,\ell+1}\left(\int_{S}f_{i}\,F_{x}\right)\,f_{i}=\sum_{i=1}^{2\,\ell+1}f_{i}(x)\,f_{i}$

so that

$|F_{x}|_{L^{2}(S)}^{2}=\sum_{i=1}^{2\,\ell+1}f_{i}(x)^{2}.$

Integrating with respect to $x$, we obtain

$4\,\pi\,|F_{x}|_{L^{2}(S)}^{2}=2\,\ell+1.$

It follows that

$|f|_{C^{0}(S)}\leq\sqrt{\frac{2\,\ell+1}{4\,\pi}}\,|f|_{L^{2}(S)}=O(1)\,\ell^{1/2}\,|f|_{L^{2}(S)}$

for all $f\in\Lambda_{\ell}$.
Using also Lemma 17, the assertion follows. ∎

######  Lemma 20.

There holds, as $\ell\to\infty$,

$|\nabla v_{\ell}|_{C^{0}(S)}=O(1)\,\ell.$

###### Proof.

Given $x\in S$, consider the linear map $\Lambda_{\ell}\to\mathbb{R}^{3}$
given by $f\mapsto(\nabla f)(x)$. By the Riesz representation theorem, there
is a unique

$\vec{F}_{x}=(F_{x}^{1},\,F_{x}^{2},\,F_{x}^{3})\in\Lambda_{\ell}\times\Lambda_{\ell}\times\Lambda_{\ell}$

such that, for all $f\in\Lambda_{\ell}$,

$\nabla f(x)=\int_{S}f\,\vec{F}_{x}$

As in the proof of Lemma 19, we may argue by symmetry that
$|\vec{F}_{x}|_{L^{2}(S)}$ is independent of $x$.

Given an orthonormal basis $\\{f_{i}\\}_{i=1}^{2\,\ell+1}$ of
$\Lambda_{\ell},$ we write

$\vec{F}_{x}=\sum_{i=1}^{2\,\ell+1}\left(\int_{S}f_{i}\,\vec{F}_{x}\right)\,f_{i}=\sum_{i=1}^{2\,\ell+1}(\nabla
f_{i})(x)\,f_{i}.$

Note that

$|\vec{F}_{x}|_{L^{2}(S)}^{2}=\sum_{i=1}^{2\,\ell+1}|\nabla f_{i}(x)|^{2}$

Integrating with respect to $x$, we obtain that

$4\,\pi\,|\vec{F}_{x}|_{L^{2}(S)}^{2}=\sum_{i=1}^{2\,\ell+1}|\nabla
f_{i}|_{L^{2}(S)}^{2}=\ell\,(\ell+1)(2\,\ell+1).$

Thus,

$|\nabla
f|_{C^{0}(S)}\leq\sqrt{\frac{\ell\,(\ell+1)\,(2\,\ell+1)}{4\,\pi}}\,|f|_{L^{2}(S)}=O(1)\,\ell^{3/2}\,|f|_{L^{2}(S)}$

for every $f\in\Lambda_{\ell}$.
In conjunction with Lemma 17, the assertion follows. ∎

##  Appendix B The $3j$-symbol

Recall from Appendix A the definition of $v_{\ell}\in C^{\infty}(S)$ where
$\ell\geq 0$ is an integer.
Given two integers $\ell_{1},\,\ell_{2}\geq 0$, let

$\Gamma_{\ell_{1}\ell_{2}}=\\{\ell\in\mathbb{Z}:|\ell_{1}-\ell_{2}|\leq\ell\leq\ell_{1}+\ell_{2}\\};$

cp. [Edmonds, §3.7]. Given $\ell_{3}\in\Gamma_{\ell_{1}\ell_{2}}$, recall
from, e.g., [Edmonds, §3.7], E. Wigner’s $3j$-symbol given by

(6) $\displaystyle\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}=0$

if $\ell_{1}+\ell_{2}+\ell_{3}$ is odd and

(7) $\displaystyle\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}$
$\displaystyle=(-1)^{(\ell_{1}+\ell_{2}+\ell_{3})/2}\,\sqrt{\frac{(\ell_{1}+\ell_{2}-\ell_{3})!\,(\ell_{1}+\ell_{3}-\ell_{2})!\,(\ell_{2}+\ell_{3}-\ell_{1})!}{(\ell_{1}+\ell_{2}+\ell_{3}+1)!}}\,$
$\displaystyle\qquad\times\frac{((\ell_{1}+\ell_{2}+\ell_{3})/2)!}{((\ell_{1}+\ell_{2}-\ell_{3})/2)!\,((\ell_{1}+\ell_{3}-\ell_{2})/2)!\,((\ell_{2}+\ell_{3}-\ell_{1})/2)!}$

if $\ell_{1}+\ell_{2}+\ell_{3}$ is even; see [Edmonds, (3.7.17)].

######  Lemma 21 ([Edmonds, (4.6.5)]).

Let $\ell_{1},\,\ell_{2}\geq 0$ be integers. There holds

$v_{\ell_{1}}\,v_{\ell_{2}}=\sum_{\ell\in\Gamma_{\ell_{1}\ell_{2}}}\,(2\,\ell+1)\,\begin{pmatrix}\ell_{1}&\ell_{2}&\ell\\\
0&0&0\end{pmatrix}^{2}\,v_{\ell}.$

Let $m\geq 3$ be an integer. Given integers $\ell_{1},\ldots,\ell_{m-1}\geq
0$, we let

(8)
$\displaystyle\Gamma(\ell_{1},\ldots,\ell_{m-1})=\\{(z_{1},\ldots,z_{m-1})\in\mathbb{Z}^{m-1}:z_{1}=\ell_{1}\text{
and }z_{i}\in\Gamma_{z_{i-1}\ell_{i}}\text{ for }i=2,\ldots,m-1\\}.$

Note that
$\Gamma(\ell_{1},\ell_{2})=\\{(\ell_{1},\ell):\ell\in\Gamma_{\ell_{1}\ell_{2}}\\}$.

######  Lemma 22.

Let $\ell_{1},\,\ldots,\,\ell_{m}\geq 0$ be integers. There holds

$\frac{1}{4\,\pi}\,\int_{S}\,\prod_{i=1}^{m}v_{\ell_{i}}=\sum_{z\in\Gamma(\ell_{1},\ldots,\ell_{m-1})}\delta_{z_{m-1}\ell_{m}}\,(2\,\ell_{m}+1)^{-1}\,\prod_{i=1}^{m-2}(2\,z_{i+1}+1)\,\begin{pmatrix}z_{i}&\ell_{i+1}&z_{i+1}\\\
0&0&0\end{pmatrix}^{2}.$

###### Proof.

Note that

$\Gamma(\ell_{1},\ldots,\ell_{m-1})=\\{(\ell_{1},z_{1},\ldots,z_{m-1}):z_{2}\in\Gamma_{\ell_{1}\ell_{2}}\text{
and }(z_{2},\ldots,z_{m-1})\in\Gamma(z_{2},\ell_{3},\ldots,\ell_{m-1})\\}.$

The assertion now follows by induction, using Lemma 21 and Lemma 17. ∎

######  Lemma 23.

Let $\delta>0$. There is an integer $\ell>1$ with the following property. Let
$\ell_{1},\,\ell_{2}\geq 0$ be integers and
$\ell_{3}\in\Gamma_{\ell_{1}\ell_{2}}$ such that

  * $\circ$

$\ell_{1}+\ell_{2}+\ell_{3}=0\,\operatorname{mod}\,4$ and

  * $\circ$

$\ell_{1},\,\ell_{2},\,\ell_{3}\geq\ell.$

Then

$\displaystyle(\ell_{1}+\ell_{2}+\ell_{3})\,\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}\geq\sqrt{\frac{2}{\pi}}-\delta.$

###### Proof.

Consider the sequence $\\{a_{k}\\}_{k=1}^{\infty}$ given by

$a_{k}=\frac{((2\,k)!)^{1/2}}{k!}\,\frac{(2\,k)^{1/4}}{2^{k}}.$

By Stirling’s formula, $\lim_{k\to\infty}a_{k}=(2/\pi)^{1/4}.$ In particular,

(9) $\displaystyle 0<\inf_{k\geq 1}a_{k}<\sup_{k\geq 1}a_{k}<\infty.$

Using (7) and that $(-1)^{\ell_{1}+\ell_{2}+\ell_{3}}=1$, we see that

(10) $\displaystyle\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}\,\bigg{[}(\ell_{1}+\ell_{2}-\ell_{3})\,(\ell_{1}+\ell_{3}-\ell_{2})\,(\ell_{2}+\ell_{3}-\ell_{1})\,(\ell_{1}+\ell_{2}+\ell_{3}+1)\,\bigg{]}^{1/4}$
$\displaystyle\qquad=\frac{(\ell_{1}+\ell_{2}+\ell_{3})^{1/4}}{(\ell_{1}+\ell_{2}+\ell_{3}+1)^{1/4}}\,\frac{a_{(\ell_{1}+\ell_{2}-\ell_{3})/2}\,a_{(\ell_{1}+\ell_{3}-\ell_{2})/2}\,a_{(\ell_{2}+\ell_{3}-\ell_{1})/2}}{a_{(\ell_{1}+\ell_{2}+\ell_{3})/2}}.$

If, as $\ell\to\infty$,

$\ell_{1}+\ell_{2}-\ell_{3}=O(1),\qquad\ell_{1}+\ell_{3}-\ell_{2}=O(1),\qquad\text{or}\qquad\ell_{2}+\ell_{3}-\ell_{1}=O(1),$

then

$\liminf_{\ell\to\infty}(\ell_{1}+\ell_{2}+\ell_{3})\,\big{[}(\ell_{1}+\ell_{2}-\ell_{3})\,(\ell_{1}+\ell_{3}-\ell_{2})\,(\ell_{2}+\ell_{3}-\ell_{1})\,(\ell_{1}+\ell_{2}+\ell_{3}+1)\,\big{]}^{-1/4}=\infty.$

In conjunction with (9) and (10), we see that

$\displaystyle(\ell_{1}+\ell_{2}+\ell_{3})\,\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}\to\infty.$

If, as $\ell\to\infty$,

$\ell_{1}+\ell_{2}-\ell_{3}\to\infty,\qquad\ell_{1}+\ell_{3}-\ell_{2}\to\infty,\qquad\text{and}\qquad\ell_{2}+\ell_{3}-\ell_{1}\to\infty,$

then

$\liminf_{\ell\to\infty}(\ell_{1}+\ell_{2}+\ell_{3})\,\big{[}(\ell_{1}+\ell_{2}-\ell_{3})\,(\ell_{1}+\ell_{3}-\ell_{2})\,(\ell_{2}+\ell_{3}-\ell_{1})\,(\ell_{1}+\ell_{2}+\ell_{3}+1)\,\big{]}^{-1/4}\geq
1$

and

$a_{(\ell_{1}+\ell_{2}-\ell_{3})/2},\,a_{(\ell_{1}+\ell_{3}-\ell_{2})/2},\,a_{(\ell_{2}+\ell_{3}-\ell_{1})/2},\,a_{(\ell_{1}+\ell_{2}+\ell_{3})/2}\to\left(\frac{2}{\pi}\right)^{1/4}.$

In conjunction with (10),

$\displaystyle\liminf_{\ell\to\infty}(\ell_{1}+\ell_{2}+\ell_{3})\,\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}\geq\sqrt{\frac{2}{\pi}}.$

The assertion follows. ∎

######  Lemma 24.

There holds

$\displaystyle\sup_{\ell_{1},\,\ell_{2},\,\ell_{3}}$
$\displaystyle\bigg{|}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}\bigg{|}\,\big{[}(\ell_{1}+\ell_{2}-\ell_{3}+1)\,(\ell_{1}+\ell_{3}-\ell_{2}+1)\,(\ell_{2}+\ell_{3}-\ell_{1}+1)\,(\ell_{1}+\ell_{2}+\ell_{3}+1)\big{]}^{1/4}$
$\displaystyle\qquad<\infty.$

where the supremum is taken over all integers $\ell_{1},\,\ell_{2}\geq 0$ and
all $\ell_{3}\in\Gamma_{\ell_{1}\ell_{2}}$.

###### Proof.

This follows as in the proof of Lemma 23. ∎

######  Lemma 25.

There holds

$\sup_{\ell_{1},\,\ell_{2}}\sum_{\ell\in\Gamma_{\ell_{1}\ell_{2}}}\ell\,\begin{pmatrix}\ell_{1}&\ell_{2}&\ell\\\
0&0&0\end{pmatrix}^{2}<\infty$

where the supremum is taken over all integers $\ell_{1},\,\ell_{2}\geq 0$.

###### Proof.

Assume that $\ell_{1}\geq\ell_{2}.$ By Lemma 24, we have

$\displaystyle\sum_{\ell\in\Gamma_{\ell_{1}\ell_{2}}}\ell\,\begin{pmatrix}\ell_{1}&\ell_{2}&\ell\\\
0&0&0\end{pmatrix}^{2}$
$\displaystyle\qquad=O(1)\,\sum_{\ell=\ell_{1}-\ell_{2}}^{\ell_{1}+\ell_{2}}\ell\,\left[(\ell_{1}+\ell_{2}-\ell+1)\,(\ell_{1}+\ell-\ell_{2}+1)\,(\ell_{2}+\ell-\ell_{1}+1)\,(\ell_{1}+\ell_{2}+\ell+1)\right]^{-1/2}.$

Note that

$\ell\,(\ell_{1}+\ell-\ell_{2}+1)^{-1/2}\,(\ell_{1}+\ell_{2}+\ell+1)^{-1/2}\leq
1.$

Moreover,

$\displaystyle\sum_{\ell=\ell_{1}-\ell_{2}}^{\ell_{1}+\ell_{2}}(\ell_{1}+\ell_{2}-\ell+1)^{-1/2}\,(\ell_{2}+\ell-\ell_{1}+1)^{-1/2}$
$\displaystyle=\sum_{i=0}^{2\,\ell_{2}}(2\,\ell_{2}-i+1)^{-1/2}\,(i+1)^{-1/2}$
$\displaystyle=O(1)\,\int_{0}^{2\,\ell_{2}}(2\,\ell_{2}+1-x)^{-1/2}\,x^{-1/2}\,\mathrm{d}x$
$\displaystyle=O(1).$

The assertion follows. ∎

## References


# Magnetostriction of $\alpha$-RuCl3 flakes in the zigzag phase

Yun-Yi Pai<EMAIL_ADDRESS>Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831  Quantum Science Center, Oak
Ridge, Tennessee 37831  Claire E. Marvinney  Materials Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831  Quantum Science
Center, Oak Ridge, Tennessee 37831  Matthew A. Feldman  Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
Department of Physics and Astronomy, Vanderbilt University, Nashville, TN
37203  Brian Lerner  Materials Science and Technology Division, Oak Ridge
National Laboratory, Oak Ridge, TN 37831  Yoong Sheng Phang  Department of
Physics and Astronomy, University of Georgia, Athens, GA 30602  Kai Xiao
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak
Ridge, TN 37831  Jiaqiang Yan  Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831  Quantum Science Center, Oak
Ridge, Tennessee 37831  Liangbo Liang  Center for Nanophase Materials
Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831  Matthew Brahlek
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831  Benjamin J. Lawrie<EMAIL_ADDRESS>This manuscript has
been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the
US Department of Energy (DOE). The US government retains and the publisher, by
accepting the article for publication, acknowledges that the US government
retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or
reproduce the published form of this manuscript, or allow others to do so, for
US government purposes. DOE will provide public access to these results of
federally sponsored research in accordance with the DOE Public Access Plan
(http://energy.gov/downloads/doe-public-access-plan).  Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
Quantum Science Center, Oak Ridge, Tennessee 37831

###### Abstract

Motivated by the possibility of an intermediate U(1) quantum spin liquid phase
in out-of-plane magnetic fields and enhanced magnetic fluctuations in
exfoliated $\alpha$-RuCl3 flakes, we study magneto-Raman spectra of exfoliated
multilayer $\alpha$-RuCl3 in out-of-plane magnetic fields of -6 T to 6 T at
temperatures of 670 mK - 4 K. While the literature currently suggests that
bulk $\alpha$-RuCl3 is in an antiferromagnetic zigzag phase with
R$\overline{3}$ symmetry at low temperature, we do not observe R$\overline{3}$
symmetry in exfoliated $\alpha$-RuCl3 at low temperatures. While we saw no
magnetic field driven transitions, the Raman modes exhibit unexpected
stochastic shifts in response to applied magnetic field that are above the
uncertainties inferred from Bayesian analysis. These stochastic shifts are
consistent with the emergence of magnetostrictive interactions in exfoliated
$\alpha$-RuCl3.

##  I Introduction

The quantum spin liquid (QSL) is a long sought-after non-classical phase
characterized by a topological order parameter Anderson (1973); Knolle and
Moessner (2019); Savary and Balents (2016). QSLs may be critical to the
development of topologically protected quantum computing platforms because
they may host non-local excitations with Anyonic statistics. Amongst possible
candidates that could host such a topologically protected phase,
$\alpha$-RuCl3 has been extensively studied over the past several years
because it can be approximately described by the analytically solvable Kitaev
honeycomb modelKitaev (2006). Several experimental efforts have reported
features consistent with a QSL phase, including half-quantized thermal Hall
conductance plateaus Kasahara  _et al._ (2018), a scattering continuum in
Raman spectroscopy Sandilands  _et al._ (2015); Wang  _et al._ (2020a);
Wulferding  _et al._ (2020), neutron scattering Banerjee  _et al._ (2016,
2017), and nuclear magnetic resonance Baek  _et al._ (2017); Janša  _et al._
(2018).

Despite numerous reports of possible QSL signatures, many fundamental
questions remained unanswered, both theoretically and experimentally. For
example, while the room temperature structure is now accepted as C2/mLebert
_et al._ (2020); Mai  _et al._ (2019), the low temperature symmetry of
$\alpha$-RuCl3 remains in question. Early reports suggested trigonal P3112
Johnson  _et al._ (2015) and monoclinic C2/m Johnson  _et al._ (2015); Cao
_et al._ (2016); Ziatdinov  _et al._ (2016) symmetries, but more recent
reports have suggested the presence of rhombohedral R$\overline{3}$ Park  _et
al._ (2016); Glamazda  _et al._ (2017) symmetry. Further, it remains unclear
whether the Kitaev model exchange parameters should be antiferromagnetic or
ferromagnetic. Additionally, a recent density functional renormalization group
(DMRG) calculation suggested that commonly reported QSL phases induced by in-
plane magnetic fields are missing in the DMRG result, but a U(1) QSL phase can
be stabilized by out-of-plane magnetic fields Jiang, Devereaux, and Jiang
(2019). The number of antiferromagnetic zigzag phases that exist before the
onset of the QSL phase also remains in question, as does the effect of sample-
to-sample variabilityKim and Kim (2021) between ostensibly identical flakes
and between exfoliated flakes and bulk crystals.

The layered Van der Waals nature of $\alpha$-RuCl3 enables the possibility of
heterostructure assembly, proximity effect engineering, and strain engineering
Geim and Grigorieva (2013); Das  _et al._ (2015); Huang  _et al._ (2020);
Rizzo  _et al._ (2020); Wang  _et al._ (2020b). Exfoliated flakes a few layers
thick are air-stableZhou  _et al._ (2019); Du  _et al._ (2018), greatly
increasing the flexibility of the workflow. On the other hand, reduced
dimensionality yields stronger order-parameter fluctuations and eventually
suppression of long-range orderMermin and Wagner (1966). Stacking faults may
also open up additional hopping pathways to stabilize the QSL phaseZhou  _et
al._ (2019). Magnetic fluctuations have been reported to persist Lin  _et al._
(2020) or be enhanced Zhou  _et al._ (2019); Du  _et al._ (2018) in the few-
layer to monolayer limit. Furthermore, strain gradients may induce synthetic
gauge fields that locally tune topological phasesRachel, Fritz, and Vojta
(2016); Perreault  _et al._ (2017). A recent first principle calculation
suggests that 2% strain is enough to drive monolayer $\alpha$-RuCl3 from the
AFM zigzag phase to the spin-polarized phase. Topological devices with well-
patterned QSL puddles may be possible with appropriate gauge field
engineering.

Raman spectroscopy is a flexible and powerful tool for resolving sample
symmetry as well as microscopic electron-phonon and phonon-magnon
interactions. In addition to the above described characterization of C2/m and
R$\overline{3}$ symmetry, Raman spectroscopy has been used to characterize
temperature dependent hysteresis Glamazda  _et al._ (2017), several magnon
modes, and a possible Majorana mode Wulferding  _et al._ (2020); Hig . Here,
motivated by a possible intermediate U(1) QSL phase in out-of-plane magnetic
fields Jiang, Devereaux, and Jiang (2019) and larger tunability in exfoliated
$\alpha$-RuCl3, we study the Raman spectra of exfoliated $\alpha$-RuCl3 in
out-of-plane magnetic fields in a Faraday geometryWulferding  _et al._ (2020)
at temperatures as low as $T=$ 670 mK.

##  II Sample Details and Experimental Setup

Bulk $\alpha$-RuCl3 single crystals were grown by vapor transport May, Yan,
and McGuire (2020). The $\alpha$-RuCl3 flakes were mechanically exfoliated
onto a 300 nm SiO2 film on a Si substrate. Due to signal-to-noise-ratio (SNR)
constraints and large parameter sweeps, we focused on thick flakes with
lateral dimensions of tens of microns. Variable temperature Raman spectra were
acquired for temperatures of 3 K - 300 K in a Montana Instruments closed-cycle
cryostat with a Princeton Instruments Isoplane SCT-320 spectrograph, a Pixis
400BR Excelon camera, and a 2400 line/mm grating. The sample was probed in a
backscattering configuration (beam path $\|$ c*, where c* is the outer product
of the crystallographic a and b axes illustrated in Figure 1 (a)) with a
532.03 nm excitation at 2.0 mW power and 45 sec integration time. The laser
excitation was removed with Semrock 532 nm RazorEdge ultrasteep dichroic and
long-pass edge filters with cutoff at 90 cm-1.

The $T=$ 4 K and 670 mK magnetic field dependent Raman spectra were acquired
in a customized Leiden dilution refrigerator with free space optics access
that is described in further detail elsewhereLawrie  _et al._ (2021). Raman
spectra were recorded with an Andor Kymera 193 spectrograph with a Newton
EMCCD DU970P-BV camera and a 2400 line/mm grating. The exposure time was 1800
seconds per spectrum. The sample was probed in a backscattering configuration
with a 2 mW (200 uW), 532.2096 nm laser excitation at 4K (670 mK). The
excitation was filtered by a set of 3 volume Bragg gratings (Optigrate, 1
volume Bragg beam splitter and 2 volume Bragg notch filters). The Faraday
geometry (B $\|$ c* and beam path) induces a $\theta_{F}(B)=-25.60^{\circ}$/T
polarization rotation due to the beam propagation through the objective.

Figure 1: (a) $\alpha$-RuCl3 lattice structure. (b) optical image of flake 1.
(c) Raman spectra of flake 1 from $T=$ 3 K to 300 K. (d) intensity plot of the
Raman spectra in (c).

Initial temperature dependent Raman spectra acquired as flake 1 (illustrated
in Figure 1 (b)) warmed from $T=$ 3 K to 270 K are shown in Figure 1 (c) and
(d). All the Raman modes here are consistent with those reported in the
literature. Using peak assignments from the R$\overline{3}$ space groupLi  _et
al._ (2019) for low temperature, we identify E${}^{1}_{g}$ at 116 cm-1,
E${}^{2}_{g}$ at 164 cm-1, E${}^{3}_{g}$ at 272 cm-1, E${}^{4}_{g}$ at 296
cm-1, and A${}^{1}_{1}g$ at 313 cm-1. We note that the low energy tail below
125 cm-1 is not negligible and results in distortion of the lineshape of the
E${}^{1}_{g}$ Fano peak at 116 cm-1 in this dataset (this is not the case for
the other datasets taken with 3 volume Bragg gratings).

Additionally, less reported $\alpha$-RuCl3 Raman modes at 222 cm-1 and 345
cm-1 are observed. The 222 cm-1 peak has been attributed to stacking faultsLin
_et al._ (2020) or defectsLi  _et al._ (2019). It has been reported that thin
flakes are more prone to stacking faults than single crystalsKim and Kim
(2021), which is consistent with the fact that the $\alpha$-RuCl3 flakes
studied here are in the thin crystal to thick flake limit. The 345 cm-1 peak
has been attributed to A${}^{2}_{1g}$ due to its $XX$ polarization (parallel
polarization) natureLin  _et al._ (2020) and to defectsLi  _et al._ (2019).
Notably, Li et al. Li  _et al._ (2019) reported that both the 222 cm-1 peak
and 345 cm-1 peak only showed up for blue (488 nm) excitation but not for red
(633 nm) excitation.

Figure 2: Raman spectra as a function of out-of-plane magnetic field at (a)
$T=$ 4 K and (d) $T\sim$ 670 mK. (b) The 164 cm-1 and (c) the 313 cm-1 Raman
modes exhibit a clear fluctuation in intensity and energy with magnetic field.
At 670 mK, the 164 cm-1 (e) and 313 cm-1 (f) modes also exhibit a magnetic
field dependence. The signal-to-noise ratio for the spectra at $T=$ 670 mK is
reduced by $\sim\sqrt{10}$ because of the reduced laser power and unchanged
integration time. (g) angular dependence of the 164 cm-1 mode at $B=0$ T with
$XX$ (parallel) polarization configuration. (h) Intensity of the Ag and Bg
modes as a function of the angle $\alpha$ between polarization and the sample
orientation from (g). (i) Raman spectra as a function of field with $XX$
(parallel) polarization configuration and a constant $\alpha$ at $T=4$ K.

##  III Magneto-Raman spectroscopy

Figure 2 (a) shows the measured Raman spectra of flake 2 as a function of out-
of-plane magnetic field ($B$ $\|$ $\textbf{c}^{*}$) at $T=$ 4 K with the
magnetic field swept from $B=0$ T, $+$6 T, $-$6 T, to $0$ T in 1 T increments.
Figure 2 (d) shows the results of the same measurement repeated at $T\sim$ 670
mK with 0.5 T steps. Two clear observations can be made:

  1. 1.

The absolute intensity of the 313 cm-1 peak has a clear asymmetric behavior as
a function of magnetic field; it has local minima at $B\sim-$ 6 T and $B\sim+$
1 T and a local maximum at $B\sim+$ 5 T.

  2. 2.

Almost all the peaks appear to fluctuate as a function of magnetic field, an
effect that is clear in the magnified 164 cm-1 and 313 cm-1 modes shown in
Figure 2 (b-c) and (e-f).

The first observation can be simply explained by the fact that the angle
$\alpha$ between the polarization and the crystal a, b axes was not fixed
during the magnetic field sweep. The angle rotates at
$\theta_{F}(B)=-25.60^{\circ}$/T due to Faraday rotation in the objective.
Since the 313 cm-1 mode is sensitive to XX polarization only, it is known to
have a cos(2$\alpha$) dependenceMai  _et al._ (2019), where $\alpha$ is the
angle between the polarization and a axis. The observed magnetic-field
dependence of the 313 cm-1 peak intensity is consistent with this
polarization-rotation effect. The same explanation could, at first glance, be
true for the second observation: as reported in Mai et al. Mai  _et al._
(2019), the spectral weights of the modes from the $A_{g}$ series and $B_{g}$
series are a function of $\alpha$. It is natural to consider the possibility
that a similar effect may be present here. However, the measurement reported
by Mai et al. Mai  _et al._ (2019) was performed at room temperature, and the
irreducible representation of the space group C2/m was assumed. For the low
temperature spectra reported here, the likely space group R$\overline{3}$
yields phonon modes at similar energies, and the decomposed $E_{g}$ modes,
though doubly degenerate, are not broken. Fluctuations as a function of
magnetic field have not previously been observed for low temperature
polarization resolved Raman spectroscopy of single crystal $\alpha$-RuCl3Mai
and Aguilar (2021).

We subsequently performed a polarization sweep at zero magnetic field on flake
3. Figure 2 (g) illustrates the normalized 164 cm-1 mode as a function of the
polarization angle. A clear 2-fold oscillation is observed. Fitting the 164
cm-1 peak to a pair of closely spaced peaks yields Figure 2 (h). This angular
dependence is different from the expected R$\overline{3}$ angular independence
Mai and Aguilar (2021). It is also different from the reported angular
dependence for C2/m, which has a 4-fold symmetryMai  _et al._ (2019). However,
it is worth pointing out that fits of the peak position may be affected by the
baseline, which could, in principle, be affected by the angle $\alpha$.

Figure 2 (i) illustrates a set of Raman spectra on flake 3 as a function of
magnetic field at $T=4$ K with parallel $XX$ polarization configuration and a
constant polarization angle $\alpha$ with respect to the flake. This was done
by compensating half of the $\theta_{F}(B)=-25.60^{\circ}$/T on both the
excitation and collection path. We note that the 345 cm-1 peak on this flake
appears at 359 cm-1, consistent with previous reports that this peak is
related to stacking faults or defects Li  _et al._ (2019)) that may vary
significantly from flake to flake.

Figure 3: (a) Peak assignment from R$\overline{3}$ and C2/m space groups. The
R$\overline{3}$ has doubly degenerate $E_{g}$s and singlet $A_{g}$s. The peaks
in C2/m consist of $A_{g}$ series and $B_{g}$ series. (the solid line is the
resulting spectrum.) (b) An example spectrum taken at $T=670$ mK fit with
R$\overline{3}$ peaks. (c) The same example spectrum taken at $T=670$ mK fit
with C2/m peaks.  Figure 4: Posterior distributions from Bayesian inference.
(a) Selected inferred peak positions using R$\overline{3}$ peak assignment for
spectra taken at $T=4$ K. (b) Selected inferred peak positions using C2/m peak
assignment for spectra taken at $T=4$ K. (c) Selected inferred peak positions
using R$\overline{3}$ peak assignment for spectra taken at $T=670$ mK. (d)
Selected inferred peak positions using C2/m peak assignment for spectra taken
at $T=670$ mK. (e) Selected inferred peak positions using R$\overline{3}$ peak
assignment for spectra taken at $T=4$ K with parallel ($XX$) polarization.

##  IV Bayesian Inference

In order to better understand the Raman spectra shown above, we employed
Bayesian inference techniques, considering both the modes from R$\overline{3}$
with unsplit degeneracies and C2/m. For R$\overline{3}$, as shown in Figure 3
(a), the model consists of 2 Fano lineshapes ($E_{1g}$, $E_{2g}$), 5 Gaussian
peaks ($E_{3g}$, $E_{4g}$, $A^{1}_{1g}$, $A_{3g}$, and $R_{1}$). To capture
the baseline, we use a linear background with a broad Gaussian centered at
$\sim$ 300 cm-1. For C2/m, peaks consist of the $A_{g}$ series and $B_{g}$
series. The A1g, B1g, A2g, B2g are fitted to Fano lineshapes and the rest to
Gaussian. We fit the Raman spectra from 90 cm-1 to 400 cm-1 in a single step
with Scipy. We then used the fitting result as the starting center of the
distribution of our priors. The priors of the parameters are assumed to be
Gaussian. The Bayesian inference was done using Hamiltonian Monte Carlo
PyMC3Salvatier, Wiecki, and Fonnesbeck (2016). For fast convergence we use a
No U-Turns (NUTS) sampler. We use 4 chains with 3,000 samples per chain. It
takes between 3 and 100 minutes for a spectrum to converge, depending on the
complexity of the models.

Figure 3 (b) shows the resulting reconstructed spectra from the posterior
predictives using R$\overline{3}$ peak assignments plotted with an original
spectrum taken at $T=670$ mK. Figure 3 (c) shows the resulting reconstructed
spectra from the posterior predictives using C2/m peak assignments plotted
with an original spectrum taken at $T=670$ mK. Figure 4 shows the posterior
distribution. Note that these histograms are the distributions of the
posteriors at each field. Since the magnetic field was swept from $B=0$ T to
$B=+$6 T to $B=-$6 T to $B=0$, the extrema are singly visited, $B=0$ T is
triply visited, and all the other $B$ field values are doubly visited. No
manual normalization was applied. We see:

  1. 1.

While the uncertainty for each posterior peak position is different, most of
the peaks shown here move above their highest density interval (HDI) Hyndman
(1996).

  2. 2.

While the models R$\overline{3}$ and C2/m are very different, the posterior
peak positions for 222 cm-1, 313 cm-1, and 341 cm-1 are quantitatively similar
within a dataset. This means that these peak positions are robust against bias
from the details of the other peaks. The joint scatter plot of peak position
against, for example, the baseline, are essentially orthogonal. Hence the
change of the peak position as a function of the magnetic field is not likely
to be introduced by a polarization dependent baseline.

  3. 3.

While polarization dependent frequency shifts exist for some of the modes such
as the 163 cm-1 mode for C2/m symmetry, the 222 cm-1 and 313 cm-1 modes are
known to not exhibit frequency shifts as a function of the polarization angle.
Furthermore, the polarization induced frequency shift has clear rotational
symmetry that is missing here.

  4. 4.

The 163 cm-1 mode has 2 clear hysteresis loops in the R$\overline{3}$ model,
as shown in Figure 5 that is less clear in the C2/m model. The posterior for
the 163 cm-1 mode in the C2/m model has a multinomial distribution, which is
usually an indication of the model not converging well, possibly due to the
complexity of the model, which includes multiple closely spaced Raman modes.

  5. 5.

The trend is different each sweep, even for the same flake. For example, the
310 cm-1 peak has a positive slope as a function of magnetic field in the
first sweep at $T=4$ K, but not in the second sweep at $T=670$ mK.

Figure 5: Hysteretic frequency shift of the 163 cm-1 mode. Waterfall plot of
the posterior distribution of the 163 cm-1 mode position as a function of the
magnetic field (from Fig. 4) (a). As the magnetic field is swept from 0 T
$\to$ $+6$ T $\to$ 0 T $\to$ $-$ 6 T $\to$ 0 T, the peak position follows the
arrow 1 $\to$ 2 $\to$ 3 $\to$ 4\. This hysteretic response appears in Bayesian
inference as well as frequentist fits. The hysteresis is statistically
significant. For example, from the Bayesian inferred result of the first
spectrum at $B$ = 0 T, the mean = 163.74 cm-1, the 3% HDI = 163.69 cm-1, and
the 97% HDI = 167.79 cm-1, and thus the middle 94% HDI = 0.10 cm-1 with a
standard deviation = 0.027 cm-1). For this case, the size of the hysteresis of
the loop of the positive field is about 0.19 cm-1, which is 7.4 $\sigma$.

##  V Discussion and Conclusion

The variation in vibrational Raman modes as a function of magnetic field
reported here is consistent with spin-phonon interactions and magnetostrictive
effects. Recently Gass et al. Gass  _et al._ (2020) and Schonemann et al.
Schönemann  _et al._ (2020) reported contraction along the c∗ axis and along
the in-plane direction in response to applied in-plane fields. Though not
reported yet, it is natural to assume similar effects may occur in response to
out-of-plane magnetic fields as well. While Raman shifts seemed to be
negligible in a previous study of single crystals Wulferding  _et al._ (2020),
magnetostrictive effects may be stronger in exfoliated flakes due to reduced
rigidity. On the other hand, related materials like CrI3 have strong Raman
shifts for out-of-plane magnetic fields of $B=1$ TMcCreary  _et al._ (2020). A
non-reciprocal magneto-phonon interaction was also reported for CrI3 in the
form of polarization rotation asymmetric with magnetic field Liu  _et al._
(2020), but little to negligible shift of the vibrational Raman modes was
described in that report. In addition to CrI3, Raman frequency shifts in
response to magnetic field have been observed in 15R-BaMnO3Poojitha  _et al._
(2020).

The hysteretic shifts in $\alpha$-RuCl3 Raman modes reported here are
statistically significant, but highly variable. This could plausibly be due to
variation in microscopic spin arrangement and strain that can result from
substrate roughness Bunch  _et al._ (2007) or from variations in flake
geometry and thickness and may also affected by the exfoliation process. For
example, in CrI3, Raman modes depend on the spin arrangement and are sensitive
to spin flipsMcCreary  _et al._ (2020). As the magnetic field is applied, the
spins in $\alpha$-RuCl3 flakes may re-orient, and the flake itself may deform
mechanically. As the magnetic field is removed, the flake may settle in a new
spin-mechanical configuration that is more stable than the initial
configuration. This is consistent with the fact that we see a stronger Raman
shift in the 4K sweep than the mK sweep, which was performed after the 4K
sweep.

Knowing that the Raman modes move around stochastically in a manner that may
be linked to microscopic details, one may consider a device with multiple
individually addressable, non-degenerate Majorana QSLs by patterning different
local strain gradients in $\alpha$-RuCl3 on pillars of different sizes.
Hysteretic magnetostrictive effects may also be useful for neuromorphic or
quantum memory applications, though substantial additional research is
essential to corroborate such claims.

###### Acknowledgements.

The authors would like to acknowledge insightful discussion with Thuc Mai,
Rolando Valdes Aguilar, and Arnab Banerjee. This research was sponsored by the
U. S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division. Postdoctoral (CEM) research
support was provided by the Intelligence Community Postdoctoral Research
Fellowship Program at the Oak Ridge National Laboratory, administered by Oak
Ridge Institute for Science and Education through an interagency agreement
between the U.S. Department of Energy and the Office of the Director of
National Intelligence. Portions of this material (JY) are based upon work
supported by the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Quantum Science Center.
Exfoliation (KX), variable temperature photoluminescence (BL) and modeling
(LL) were performed at the Center for Nanophase Materials Sciences, which is a
U.S. Department of Energy Office of Science User Facility. Student (MAF, BEL,
YSP) research support was provided by the Department of Defense through the
National Defense Science $\&$ Engineering Graduate Fellowship (NDSEG) and by
the DOE Science Undergraduate Laboratory Internships (SULI) program.

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