# EXPERIMENTAL RESULTS OF CRYSTAL-ASSISTED SLOW EXTRACTION AT THE SPS

M.A. Fraser<EMAIL_ADDRESS>S. Gilardoni  B. Goddard  V. Kain  D. Mirarchi S.
Montesano  S. Petrucci  S. Redaelli
R. Rossi  W. Scandale  L. Stoel1 F.M. Velotti  CERN  Geneva  Switzerland
F. Murtas  INFN LNF  Frascati  Italy
F. Galluccio  INFN Sezione di Napoli  Naples  Italy
F. Addesa  G. Cavoto2 F. Iacoangeli  INFN Sezione di Roma  Rome  Italy
1also at Vienna University of Technology  Vienna  Austria
2also at Sapienza University Roma  Rome  Italy

###### Abstract

The possibility of extracting highly energetic particles from the Super Proton
Synchrotron (SPS) by means of silicon bent crystals has been explored since
the 1990’s. The channelling effect of a bent crystal can be used to strongly
deflect primary protons and hence direct them onto an internal absorber or,
with additional deflection elements, eject them from the synchrotron. Many
studies and experiments have been carried out to investigate crystal
channelling effects. As summarised in [1, 2, 3, 4], extraction of 120 and 270
GeV proton beams has already been demonstrated in the SPS with dedicated
experiments located in the ring. At present in the SPS, the UA9 experiment is
performing studies to evaluate the possibility to use bent silicon crystals to
steer particle beams in high energy accelerators [5, 6, 7, 8, 9]. Recent
studies on the feasibility of extraction from the SPS have been made using the
UA9 infrastructure with a longer-term view of using crystals to help mitigate
slow extraction induced activation of the SPS. In this paper, the possibility
to eject particles from the SPS and into the extraction channel in Long
Straight Section (LSS) 2 using the bent crystals already installed in the SPS
is presented. Details of the concept, simulations and measurements carried out
with beam are presented, before the outlook for the future is discussed.

##  1 INTRODUCTION

In view of the proposed Beam Dump Facility (BDF) [10] and continuing requests
for higher beam intensities for Fixed Target physics, the application of bent
crystals used in different configurations and modes of operation for slow
extraction are being studied [11, 12]. Bent crystals offer promising solutions
for reducing the activation of the SPS LSS2 extraction region that is induced
by the small fraction of beam that unavoidably impinges the electrostatic
septum (ZS) during the conventional resonant slow extraction process. As a
first step in 2016, the slow extraction of a 270 GeV proton beam into the TT20
extraction line towards the North Experimental Area of the SPS was
demonstrated in dedicated Machine Development (MD) sessions using the
extraction septa in LSS2 and a bent crystal, provided by the UA9 collaboration
as part of their experimental installation in LSS5.

##  2 EXTRACTION CONCEPT

The location of the bent crystals in LSS5 is 3.5 km from the slow extraction
channel in LSS2, which makes the extraction process highly sensitive to the
working point of the machine. The crystals are positioned on the inside of the
ring and deflect inwards. Detailed studies [13] failed to identify a suitable
working point that provides the required phase advance for the channelled beam
to jump the wires of the electrostatic septum, on the outside of the ring, on
the first turn. However, they did identify the potential of the operational
Fixed Target working point ($Q_{x}=26.62$, $Q_{y}=26.58$) to extract the beam
at the ZS on the second turn, with a fractional phase advance of
$252^{\circ}$. The extraction scheme chosen is shown schematically in Fig. 1,
where the electrostatic and magnetic septa (ZS, MST and MSE) are shown, along
with a dedicated Cherenkov for Proton Flux Measurement (CpFM) detector
installed upstream of the extraction dump (TED) in the TT20 transfer line. The
channelled beam performs almost 41 betatron oscillations before reaching the
ZS.

Figure 1: Schematic of the crystal-assisted extraction scheme.

With the machine configured to store a single bunch at constant energy, the
transverse halo can be slowly and non-resonantly extracted as it diffuses into
the crystal, is channelled and deflected into the extraction septa. The
advantage of such an extraction concept for these first development tests is
that the channelled beam passes the UA9 experimental area a second time,
allowing the exploitation of the specialised beam diagnostics systems to
verify the phase advance of the channelled beam in the absence of suitable
systems in LSS2.

In order to simulate the dynamics of the extraction, the SPS was implemented
in MADX and particle tracking carried out in combination with the pycollimate
[14] scattering routine, where the crystal interaction was modelled using
single-pass UA9 measurement data [15]. More details of how the simulations
were implemented are found in [13]. The presentation of the beam at the
entrance to the ZS is shown in Fig. 2 after being tracked through the crystal
and the SPS, where the distribution is approximated as a hollow halo beam to
save computation time. In Fig.2 the crystal is aligned at $-6\sigma$ with a
channelling angle of $-160~{}\mu$rad, where positive denotes a direction
towards the outside of the ring. The simulations were used to design the LSS2
extraction bump, ensuring its closure, and to set the strengths of the
extraction septa such that the channelled beam enters TT20 on the nominal
trajectory.

Figure 2: Beam distribution (hollow halo approximation) at ZS: normalised
phase space (left) and horizontal projection (right) [13].

##  3 EXPERIMENTAL SETUP

A schematic view of the UA9 installation is shown in Fig. 3 comprising two
goniometers used to precisely align the crystals to the beam, several
detectors used as beam diagnostics and to measure different observables, as
well as absorbers and scrapers to intercept the channelled beam as well as to
test the population of the halo at relevant phase advances with respect to the
crystal.

Figure 3: Schematic of the UA9 experiment’s installation.

The collimators and absorbers are named TCXHW, TCSM, TACW and TAL, and all of
them are equipped with LHC Beam Loss Monitors (BLMs), which are significantly
more sensitive than the standard SPS BLMs. The TCSM is an LHC prototype
collimator with two horizontal jaws, composed of 1 m long blocks of graphite,
and equipped with a Beam Position Monitor (BPM) at its entrance. The TACW is
an old SPS collimator, which was equipped with a single 60 cm tungsten jaw to
suit the needs of the UA9 experiment, used to stop the crystal-channelled beam
during data taking. The experimental installation starts with the TCXHW, a 10
cm long double-sided tungsten scraper, and terminates with a station in the
high dispersive area, which includes the TAL (10 cm long double-sided tungsten
scraper) and a Roman Pot containing two Timepix high-precision pixel detectors
[16, 17]. In conjunction with UA9’s absorbers and scrapers, the channelled
beam could be stopped from circulating after a given number of turns and its
position and transverse size measured on its return to LSS5, turn-by-turn. To
directly detect the presence of channelled beam in the TT20 extraction line a
dedicated CpFM detector [18, 19] was installed that is capable of measuring
single-particle events and therefore very low extraction rates.

##  4 EXPERIMENTAL RESULTS

A low intensity LHC-type single-bunch of $1.6\times 10^{10}$ protons was used
throughout the tests in order to guarantee the protection of the fragile wires
of the ZS from damage. The effect of the imperfect closure of the LSS2
extraction bump was tested and shown not to significantly perturb the
channelling efficiency. The BPM at the TCSM observed a 0.1 mm movement as the
extraction bump was powered to its nominal value of 57 mm at the ZS, inducing
a spike in the channelling rate and increasing BLM readings in LSS5. To avoid
this type of dynamic effect it was decided to turn on the extraction bump
before aligning the crystal with the beam. The alignment of the UA9 equipment
was carried out as usual [20].

Figure 4: TT20 CpFM signal rate (red) vs. TACW position (blue) [19].

The crystal was positioned at $6\sigma$ and aligned by scanning the angle of
its goniometer and using scintillators and BLM loss levels as observables to
determine the optimal orientation of the crystal. The bending angle of the
crystal used was measured in a previous MD in 2016 as $175\pm 2~{}\mu$rad,
consistent with [5]. Once the TACW absorber was retracted and the channelled
beam was free to circulate, it was immediately detected by the CpFM in the
TT20 extraction line. After small steering corrections the beam could be
centred on the nominal beam axis at the CpFM. The extraction could be ceased
by reinserting the TACW and stopping the channelled beam in LSS5 as shown in
Figs. 4 and 5.

(a) TACW IN. (b) TACW OUT.

Figure 5: LSS5 absorber configuration for extraction.

The horizontal beam size of the extracted beam at the CpFM was also measured
at $\sigma=0.6$ mm by scanning the quartz bar of the CpFM through the
extracted beam in the horizontal plane, as shown in Fig. 6. The signal reaches
its maximum value when the quartz bar samples the entire beam. The calibration
of the CpFM is on-going in order to quantify the exact extraction rate in
terms of protons extracted per second.

Figure 6: TT20 CpFM signal (red) and position (blue) vs. time as it is moved
out of the extracted beam [19].

Further tests were made to guarantee that the extracted beam was indeed
channelled by the crystal. These involved (i) inserting the outer TCSM jaw and
TAL inner jaw to stop the channelled beam on the second turn in LSS5, also
allowing beam profiles to be reconstructed, (ii) changing the angular
alignment of crystal and (iii) adjusting the diffusion rate by exciting the
beam with the transverse damper and checking the measured extraction rate on
the CpFM. The aforementioned direct checks using the CpFM behaved as expected
with the most elegant validation being an indirect measurement carried out
with the Timepix detector. With the extraction bump turned off, and the TCXHW
inserted to intercept the channelled beam from circulating after its fourth
pass of LSS5, the channeled beam was imaged on the Timepix detector on its
third pass of LSS5, as shown in Fig. 7(a). When the extraction bump was turned
on the channelled beam disappeared from the image as it was pushed over the
septum wires of the ZS and extracted into TT20, as shown in Fig. 7(b). The
extraction was again confirmed by the CpFM.

(a) Extraction bump OFF. (b) Extraction bump ON.

Figure 7: Timepix images showing the disappearance of the channelled beam in
LSS5 when extracted in LSS2.

##  5 CONCLUSION AND OUTLOOK

The non-resonant crystal-assisted slow extraction of a 270 GeV proton beam
from the SPS towards the North Experimental Area has been demonstrated during
dedicated MD tests in 2016. The low intensity beam was extracted from the halo
of a circulating LHC-type single-bunch of $1.6\times 10^{10}$ protons and its
presence validated and characteristics probed with a dedicated CpFM detector
in the extraction line. This is the first time in the SPS that a bent crystal
has been used in conjunction with the extraction systems to bring the beam
into a transfer line towards an experimental area. In light of future
experimental requests for Fixed Target physics at 400 GeV, this is an
important step in the application of bent crystals at the SPS for the
mitigation of slow extraction induced activation.

The MD programme will continue in 2017 to increase the extraction rate,
characterise the extracted beam, quantify and compare the losses for different
extraction techniques, and calibrate the CpFM with a beam synchronous trigger
to improve the signal-to-noise ratio. The application of bent crystals to
shadow the wires of the ZS during a conventional resonant slow extraction is
being actively studied [21] and the installation of a dedicated crystal to
test this proposal in MD sessions is presently being discussed.

##  6 ACKNOWLEDGEMENTS

We are grateful to the other members of the UA9 collaboration for their
support and their contribution to the development of the experimental
apparatus used in this study. This work has been partially supported by the
ERC CoG CRYSBEAM G.A. 615089. We would also like to thank the SPS operation,
beam instrumentation, septa and RF teams for their support, as well as the
Experimental Areas and Physics co-ordinators for their flexibility and
understanding in the scheduling of the dedicated MD sessions.

## References

  * [1] W. Herr, “Diffusion of particles induced by transverse noise and its application to crystal extraction experiments”, CERN, Geneva, Switzerland, Rep. CERN-SL-92-53-AP, Nov. 1992.
  * [2] H. Akbari et al., “First results on proton extraction from the CERN-SPS with a bent crystal”, Phys. Lett. B, vol. 313, pp. 491 - 497, Sep. 1993.
  * [3] K. Elsener et al., “What did we learn from the extraction experiments with bent crystals at the CERN SPS?”, in Proc. EPAC’98, Stockholm, Sweden, June 1998, pp. 239 - 241.
  * [4] K. Elsener et al., “Proton extraction from the CERN SPS using a bent crystal”, Nucl. Instrum. Meth. B, vol. 119, pp. 215 - 230, Oct. 1996.
  * [5] W. Scandale et al., “First results on the SPS beam collimation with bent crystals”, Phys. Lett. B, vol. 692, pp. 78 - 82, Aug. 2010.
  * [6] W. Scandale et al., “Comparative results on collimation of the SPS beam of protons and Pb ions with bent crystals”, Phys. Lett. B, vol. 703, pp. 547 - 551, Aug. 2011.
  * [7] W. Scandale et al., “Strong reduction of the off-momentum halo in crystal assisted collimation of the SPS beam”, Phys. Lett. B, vol. 714, pp. 231 - 236, Aug 2012.
  * [8] W. Scandale et al., “Optimization of the crystal assisted collimation of the SPS beam”, Phys. Lett. B, vol. 726, pp. 182 - 186, Oct. 2013.
  * [9] W. Scandale et al., “Observation of strong leakage reduction in crystal assisted collimation of the SPS beam”, Phys. Lett. B, 748, pp. 451 - 454, Sept. 2015.
  * [10] A. Golutvin et al., “A Facility to Search for Hidden Particles (SHiP) at the CERN SPS”, CERN, Geneva, Switzerland, Rep. CERN-SPSC-2015-016 (SPSC-P-350), Apr. 2015.
  * [11] M.A. Fraser et al., “SPS Slow Extraction Losses and Activation: Challenges and Possibilities for Improvement”, in Proc. IPAC’17, Copenhagen, Denmark, May 2017, paper MOPIK045, this conference.
  * [12] W. Scandale, “Bent crystals in particle accelerators”, presented at the 19th ABT Engineering Forum, CERN, Geneva, Switzerland, 28 Nov. 2014, https://indico.cern.ch/event/345193/.
  * [13] F.M. Velotti, “Higher brightness beams from the SPS for the HL-LHC era”, Ph.D. thesis, EPFL, Lausanne, Switzerland, 2017.
  * [14] F.M. Velotti et al., “SPS-To-LHC transfer lines loss map generation using pycollimate”, in Proc. of IPAC’15, Richmond, VA, USA, May 2015, paper THPF09, pp. 3934 - 3937.
  * [15] R. Rossi, G. Cavoto, D. Mirarchi, S. Redaelli, W. Scandale, “Measurements of coherent interactions of 400 GeV protons in silicon bent crystals”, Nucl. Instrum. Meth. B, vol. 355, pp. 369 - 373, July 2015.
  * [16] D. Turecek, T. Holy, J. Jakubek, S. Pospisil, Z. Vykydal, “Pixelman: a multi-platform data acquisition and processing software package for Medipix2, Timepix and Medipix3 detectors”, J. Instrum., vol. 6 pp. C01046, Jan. 2011.
  * [17] V. Kraus, M. Holik, J. Jakubek, M. Kroupa, P. Soukup, Z. Vykydal, “FITPix - Fast Interface for Timepix Pixel Detectors”, J. Instrum., vol. 6, pp. C01079, Jan 2011.
  * [18] S. Montesano et al., “The Cherenkov detector for proton flux measurement (CpFM) In the UA9 experiment”, in Proc. IBIC’16, Barcelona, Spain, Sept. 2016, paper TUPG40, pp. 430 - 433.
  * [19] F.M Addesa, “Cherenkov light in vacuum beam monitor for crystal-assisted beam manipulations”, Ph.D thesis, La Sapienza Universit di Roma, Roma, Italy (in preparation).
  * [20] S. Montesano and W. Scandale, “Apparatus and experimental procedures to test crystal collimation”, in Proc. of IPAC’12, New Orleans, Louisiana, USA, May 2012, paper THEPPB011, pp. 3254 - 3256.
  * [21] F.M. Velotti et al., “Reduction Of Resonant Slow Extraction Losses With Shadowing Of Septum Wires By A Bent Crystal”, in Proc. of IPAC’17, Copenhagen, Denmark, May 2017, paper MOPIK050, this conference.


# Orthogonality of bilinear forms and application to matrices

Saikat Roy, Tanusri Senapati, Debmalya Sain  Department of Mathematics
National Institute of Technology Durgapur
Durgapur 713209
West Bengal
INDIA<EMAIL_ADDRESS>Department of Mathematics
Gushkara Mahavidyalaya
West Bengal
India<EMAIL_ADDRESS>(Sain) Department of Mathematics, Indian
Institute of Science, Bengaluru 560012, Karnataka, India
<EMAIL_ADDRESS>
###### Abstract.

We characterize Birkhoff-James orthogonality of continuous vector-valued
functions on a compact topological space. As an application of our
investigation, Birkhoff-James orthogonality of real bilinear forms are
studied. This allows us to present an elementary proof of the well-known
Bhatia-Šemrl Theorem in the real case.

###### Key words and phrases:

Birkhoff-James orthogonality; Bilinear forms; Bhatia-$\breve{S}$emrl Theorem.

###### 2010 Mathematics Subject Classification:

Primary 46A32, Secondary 15A63

Dr. Debmalya Sain feels elated to acknowledge the heartening friendship Mr.
Sujoy Koner. The research of Mr. Saikat Roy is supported by CSIR MHRD in the
form of Junior Research Fellowship under the supervision of Prof. Satya
Bagchi.

##  1\. Introduction

Throughout the text, the letter $\mathbb{X}$ denotes a normed linear space and
the symbols $\mathbb{H}_{1},~{}\mathbb{H}_{2}$ are reserved for finite-
dimensional Hilbert spaces. In this article, we work only with real normed
linear spaces. Let $S_{\mathbb{X}}$ denote the unit sphere of $\mathbb{X},$
i.e., $S_{\mathbb{X}}=\\{x\in\mathbb{X}:\|x\|=1\\}.$ Let $\mathcal{U}$ be a
compact topological space and let $\mathcal{C}(\mathcal{U},\mathbb{X})$ denote
the real normed linear space of all vector-valued continuous functions from
$\mathcal{U}$ to $\mathbb{X}$, equipped with the norm
$\|\cdot\|_{\mathcal{U}}$, where

$\displaystyle\|f\|_{\mathcal{U}}=\underset{u\in\mathcal{U}}{\sup}\|f(u)\|~{}~{}\forall~{}~{}f\in\mathcal{C}(\mathcal{U},\mathbb{X}).$

For any $f\in\mathcal{C}(\mathcal{U},\mathbb{X})$, we denote the norm
attainment set of $f$ by $\mathcal{M}_{f}$, i.e.,

$\displaystyle\mathcal{M}_{f}=\\{u\in\mathcal{U}:\|f(u)\|=\|f\|_{\mathcal{U}}\\}.$

We note that $\mathcal{M}_{f}\neq\emptyset$ for any
$f\in\mathcal{C}(\mathcal{U},\mathbb{X})$. Let $\mathbb{L}(\mathbb{X})$ denote
the normed linear space of all bounded linear operators defined on
$\mathbb{X}$, endowed with the usual operator norm. For any
$T\in\mathbb{L}(\mathbb{X})$, the norm attainment set of $T$ is defined as
$M_{T}=\\{x\in S_{\mathbb{X}}:\|Tx\|=\|T\|\\}.$ Observe that for any
$T\in\mathbb{L}(\mathbb{X})$, where $\mathbb{X}$ is a finite-dimensional
Banach space, the restriction of $T$ on $S_{\mathbb{X}}$ (which is again
denoted by $T$ for the sake of simplicity) is a member of
$\mathcal{C}(S_{\mathbb{X}},\mathbb{X})$ and $\|T\|=\|T\|_{S_{\mathbb{X}}}$.

Let $\mathcal{B}(\mathbb{H}_{1}\times\mathbb{H}_{2},\mathbb{R})$ denote the
Banach space of all real bilinear forms on
$\mathbb{H}_{1}\times\mathbb{H}_{2},$ endowed with the supremum norm. Let
$\langle\cdot,\cdot\rangle$ denote the usual inner product in
$\mathbb{H}_{1}$, i.e., $\langle x_{1},x_{2}\rangle=x_{1}^{T}x_{2}$ for all
$x_{1},x_{2}\in\mathbb{H}_{1}$, where $x_{1}^{T}$ stands for the transpose of
$x_{1}$. Assuming that dim $\mathbb{H}_{1}=m$ and dim $\mathbb{H}_{2}=n$, it
is well-known that each $m\times n$ matrix $A$ can be identified with a unique
linear operator in $\mathbb{L}(\mathbb{H}_{2},\mathbb{H}_{1})$. Moreover, each
$m\times n$ matrix $A$ induces a unique bilinear form $\mathcal{T}_{A}$,
defined by,

(1) $\displaystyle\mathcal{T}_{A}(x,y)=\langle
x,Ay\rangle~{}~{}\forall~{}(x,y)\in\mathbb{H}_{1}\times\mathbb{H}_{2}.$

On the other hand, every bilinear form in
$\mathcal{B}(\mathbb{H}_{1}\times\mathbb{H}_{2},\mathbb{R})$ corresponds to a
unique $m\times n$ matrix. Throughout the article, we use the symbol
$\mathcal{T}_{A}$ to denote a member of
$\mathcal{B}(\mathbb{H}_{1}\times\mathbb{H}_{2},\mathbb{R})$ of the form (1).
For any
$\mathcal{T}_{A}\in\mathcal{B}(\mathbb{H}_{1}\times\mathbb{H}_{2},\mathbb{R})$,
the norm of $\mathcal{T}_{A}$ is defined by

$\displaystyle\|\mathcal{T}_{A}\|=\underset{(x,y)\in S_{\mathbb{H}_{1}}\times
S_{\mathbb{H}_{2}}}{\sup}|\mathcal{T}_{A}(x,y)|.$

The norm attainment set of $\mathcal{T}_{A}$ is defined as
$M_{\mathcal{T}_{A}}=\\{(x,y)\in S_{\mathbb{H}_{1}}\times
S_{\mathbb{H}_{2}}:\|\mathcal{T}_{A}(x,y)\|=\|\mathcal{T}_{A}\|\\}.$ Let us
also observe that
$\mathcal{T}_{A}\in\mathcal{B}(\mathbb{H}_{1}\times\mathbb{H}_{2},\mathbb{R})$
implies that $\mathcal{T}_{A}\in\mathcal{C}(S_{\mathbb{H}_{1}}\times
S_{\mathbb{H}_{2}},\mathbb{R})$ and
$\|\mathcal{T}_{A}\|_{S_{\mathbb{H}_{1}}\times
S_{\mathbb{H}_{2}}}=\|\mathcal{T}_{A}\|$.

Given any two elements $x,y\in\mathbb{X}$, $x$ is said to be Birkhoff-James
orthogonal to $y,$ written as $x\perp_{B}y,$ if $\|x+\lambda y\|\geq\|x\|$ for
all $\lambda\in\mathbb{R};$ see [1, 4, 5]. Similarly, for any two operators
$T,~{}A\in\mathbb{L}(\mathbb{X})$, $T\perp_{B}A$, if $\|T+\lambda
A\|\geq\|T\|$ for all $\lambda\in\mathbb{R}.$ In [3], Bhatia and
$\breve{S}$emrl completely characterized Birkhoff-James orthogonality of
matrices identified as linear operators defined on a finite-dimensional
Hilbert space. Later on, Sain [7] extended the said result to characterize
Birkhoff-James orthogonality of linear operators on a finite-dimensional
Banach space $\mathbb{X}$. To obtain the desired result, the author introduced
the notions of $x^{+}$ and $x^{-}$ for any non-zero element $x\in\mathbb{X}$,
where $x^{+}=\\{y\in\mathbb{X}:\|x+\lambda y\|\geq\|x\|~{}\forall\lambda\geq
0\\}$ and $x^{-}=\\{y\in\mathbb{X}:\|x+\lambda
y\|\geq\|x\|~{}\forall\lambda\leq 0\\}$. We refer the readers to [7] for more
information in this regard.

The aim of the present article is to assimilate the above mentioned works in
the more general setting of $\mathcal{C}(\mathcal{U},\mathbb{X})$. As a
consequence of this, we completely characterize Birkhoff-James orthogonality
of real bilinear forms. As an important application, we provide an elementary
proof of the classical Bhatia-Šemrl Theorem in the real case, using Birkhoff-
James orthogonality of real bilinear forms. Moreover, we observe that some
well-known characterizations of Birkhoff-James orthogonality [7, 8] of linear
operators defined between finite-dimensional real Banach spaces follow
immediately from our obtained results.

##  2\. main result

We begin the section with a complete characterization of Birkhoff-James
orthogonality in $\mathcal{C}(\mathcal{U},\mathbb{X})$.

######  Theorem 2.1.

Let $\mathcal{U}$ be a compact topological space and let $\mathbb{X}$ be a
normed linear space. Let $f,g\in\mathcal{C}(\mathcal{U},\mathbb{X})$ be non-
zero. Then $f\perp_{B}g$ if and only if there exist
$u_{1},u_{2}\in\mathcal{M}_{f}$ such that $g(u_{1})\in f(u_{1})^{+}$ and
$g(u_{2})\in f(u_{2})^{-}$.

###### Proof.

We only prove the necessary part of the theorem as the sufficient part follows
directly. Suppose on the contrary that $g(u)\in f(u)^{-}$ for all
$u\in\mathcal{M}_{f}$. Let $u_{0}\in\mathcal{M}_{f}$ be arbitrary. Then there
exists $\lambda>0$ such that $\|f(u_{0})+\lambda
g(u_{0})\|<\|f\|_{\mathcal{U}}$. By the convexity of the norm, we obtain that

(2) $\displaystyle\|f(u_{0})+t\lambda
g(u_{0})\|<\|f\|_{\mathcal{U}}~{}\forall~{}t\in(0,1).$

Now, we consider the continuous function
$\psi:\mathcal{U}\times[-1,1]\rightarrow\mathbb{R}$ defined by

$\displaystyle\psi(u,\mu)=\|f(u)+\mu
g(u)\|~{}\forall~{}(u,\mu)\in\mathcal{U}\times[-1,1].$

Then from (2) it follows that there exists $\mu_{0}\in(0,1)$ such that
$\psi(u_{0},\mu_{0})<\|f\|_{\mathcal{U}}$. Since $\psi$ is continuous, we can
find $\delta_{0}>0$ and an open set $\mathcal{G}_{0}$ containing $u_{0}$ in
$\mathcal{U}$ such that

$\displaystyle\psi(u,\mu)<\|f\|_{\mathcal{U}}~{}\forall~{}(u,\mu)\in\mathcal{G}_{0}\times(\mu_{0}-\delta_{0},\mu_{0}+\delta_{0}).$

Once again, applying (2) in the above expression, we obtain

$\displaystyle\psi(u,\mu)<\|f\|_{\mathcal{U}}~{}\forall~{}(u,\mu)\in\mathcal{G}_{0}\times(0,\mu_{0}).$

Next, let $v_{0}\in\mathcal{U}\setminus\mathcal{M}_{f}$ be arbitrary. Then we
have $\psi(v_{0},0)<\|f\|_{\mathcal{U}}.$ Again, using the continuity of
$\psi$, we can find $\sigma_{0}>0$ and an open set $\mathcal{\hat{G}}_{0}$
containing $v_{0}$ in $\mathcal{U}$ such that

$\displaystyle\psi(v,\sigma)<\|f\|_{\mathcal{U}}~{}\forall~{}(v,\sigma)\in\mathcal{\hat{G}}_{0}\times(-\sigma_{0},\sigma_{0}).$

Clearly, the collection
$\mathcal{G}=\left\\{\mathcal{G}_{0}:u_{0}\in\mathcal{M}_{f}\right\\}\cup\left\\{\mathcal{\hat{G}}_{0}:v_{0}\in\mathcal{U}\setminus\mathcal{M}_{f}\right\\}$
forms an open cover for $\mathcal{U}$. Since $\mathcal{U}$ is compact, there
exist natural numbers $k_{1},k_{2}$ such that

$\displaystyle\mathcal{U}\subseteq\left(\bigcup\limits_{i=1}^{k_{1}}\left\\{\mathcal{G}_{i}:u_{i}\in
M_{f}\right\\}\right)\cup\left(\bigcup\limits_{j=1}^{k_{2}}\left\\{\mathcal{\hat{G}}_{j}:v_{j}\in\mathcal{U}\setminus\mathcal{M}_{f}\right\\}\right).$

Now, fix some
$\lambda_{0}\in\left(\bigcap\limits_{i=1}^{k_{1}}(0,\mu_{i})\right)\cap\left(\bigcap\limits_{j=1}^{k_{2}}(-\sigma_{j},\sigma_{j})\right).$
Clearly, $f+\lambda_{0}g\in\mathcal{C}(\mathcal{U},\mathbb{X})$. Let
$w_{0}\in\mathcal{M}_{f+\lambda_{0}g}$. However, it follows from the choice of
$\lambda_{0}$ that

$\displaystyle\|f+\lambda_{0}g\|_{\mathcal{U}}=\|f(w_{0})+\lambda
g(w_{0})\|<\|f\|_{\mathcal{U}},$

a contradiction. Similarly, if we consider $g(u)\in f(u)^{+}$ for all
$u\in\mathcal{M}_{f}$, then we once again arrive at a contradiction. This
completes the proof of the theorem. ∎

The above characterization takes a special form whenever we assume that
$\mathcal{M}_{f}$ is a connected subset of $\mathcal{U}$ for any
$f\in\mathcal{C}(\mathcal{U},\mathbb{X})$.

######  Theorem 2.2.

Let $\mathcal{U}$ be a compact topological space and let $\mathbb{X}$ be a
normed linear space. Let $f\in\mathcal{C}(\mathcal{U},\mathbb{X})$ be such
that $f$ is non-zero and $\mathcal{M}_{f}$ is connected. Then for any
$g\in\mathcal{C}(\mathcal{U},\mathbb{X})$, $f\perp_{B}g$ if and only if there
exists $u_{0}\in\mathcal{M}_{f}$ such that $f(u_{0})\perp_{B}g(u_{0})$.

###### Proof.

We only prove the necessary part of the theorem as the sufficient part of the
theorem follows trivially. Suppose on the contrary that
$f(u_{0})\not\perp_{B}g(u_{0})$ for any $u_{0}\in\mathcal{M}_{f}$. Therefore,
for each $u\in\mathcal{M}_{f}$, either $g(u)\in f(u)^{+}$ or $g(u)\in
f(u)^{-}$. Now, we consider the following collections:

$\displaystyle C_{1}=\\{u\in\mathcal{M}_{f}:\|f(u)+\lambda
g(u)\|>\|f\|_{\mathcal{U}},~{}\lambda>0\\},$ $\displaystyle
C_{2}=\\{u\in\mathcal{M}_{f}:\|f(u)+\lambda
g(u)\|>\|f\|_{\mathcal{U}},~{}\lambda<0\\}.$

We claim that $C_{1}\cup C_{2}=\mathcal{M}_{f}$. Let $v\in\mathcal{M}_{f}$ be
arbitrary. Since $f(v)\not\perp_{B}g(v)$, there exists $\mu\in\mathbb{R}$ such
that $\|f(v)+\mu g(v)\|<\|f\|_{\mathcal{U}}$. Without loss of generality, let
$\mu<0$, i.e., $v\not\in C_{2}$. Let $\lambda>0$ be arbitrary. Then there
exists $t\in(0,1)$ such that $f(v)=t(f(v)+\mu g(v))+(1-t)(f(v)+\lambda g(v)).$
This proves that $\|f\|_{\mathcal{U}}<\|f(v)+\lambda g(v)\|$, i.e., $v\in
C_{1}$. Similarly, assuming $\mu>0$, it can be shown that $v\in C_{2}$.
Therefore, $C_{1}\cup C_{2}=\mathcal{M}_{f}$, as desired.

Next, we claim that both $C_{1},C_{2}$ are non-empty. Indeed, following the
same arguments used in the proof of Theorem 2.1, one can easily verify that
$C_{1},C_{2}\neq\emptyset$. Finally, we claim that $C_{1},C_{2}$ form a
separation of $\mathcal{M}_{f}$. To establish our claim, we observe that
$\bar{C_{1}}\cap C_{2}=\emptyset$ and $C_{1}\cap\bar{C_{2}}=\emptyset$, as
otherwise we can find $u_{0}\in\mathcal{M}_{f}$ such that
$f(u_{0})\perp_{B}g(u_{0})$. However, this is a contradiction to the fact that
$\mathcal{M}_{f}$ is connected. This completes the proof of the theorem. ∎

The above theorem may not be true if $\mathcal{M}_{f}$ is not connected. We
furnish the following example in support of this fact.

######  Example 2.2.1.

Let $\mathcal{U}=[0,2]\subset\mathbb{R}$ and $\mathbb{X}=\mathbb{R}$. We
define $f,g\in\mathcal{C}(\mathcal{U},\mathbb{X})$ by $f(u)=sin~{}\pi u$ and
$g(u)=1$ for all $u\in[0,2]$. Clearly,
$\mathcal{M}_{f}=\\{\frac{1}{2},\frac{3}{2}\\}$. Now,
$f(\frac{1}{2})g(\frac{1}{2})>0$, i.e., $g(\frac{1}{2})\in f(\frac{1}{2})^{+}$
and $f(\frac{3}{2})g(\frac{3}{2})<0$, i.e., $g(\frac{3}{2})\in
f(\frac{3}{2})^{-}$. Then from Theorem 2.1, it follows that $f\perp_{B}g$,
however, $g(v)\neq 0$ for any $v\in\mathcal{M}_{f}$, i.e.,
$f(v)\not\perp_{B}g(v)$ for any $v\in\mathcal{M}_{f}$.

Our next theorem, in some sense, assimilates the above two theorems. This
turns out to be an important tool in our further developments. We omit the
proof as it is trivial in view of Theorem 2.1 and Theorem 2.2.

######  Theorem 2.3.

Let $\mathcal{U}$ be a compact topological space and let $\mathbb{X}$ be a
normed linear space. Let $f,g\in\mathcal{C}(\mathcal{U},\mathbb{X})$ be non-
zero. Let $\mathcal{D}_{f}\subseteq\mathcal{M}_{f}$ be such that
$\mathcal{D}_{f}$ is connected. Then the following conditions are equivalent:
$(i)$ There exist $u_{1},u_{2}\in\mathcal{D}_{f}$ such that $g(u_{1})\in
f(u_{1})^{+}$ and $g(u_{2})\in f(u_{2})^{-}$.
$(ii)$ There exists $u_{0}\in\mathcal{D}_{f}$ such that
$f(u_{0})\perp_{B}g(u_{0})$.

As mentioned in the introduction, we obtain some of the earlier results on the
Birkhoff-James orthogonality of linear operators as simple corollaries to
Theorem 2.1 and Theorem 2.2.

######  Corollary 2.3.1.

[7, Theorem 2.2] Let $\mathbb{X}$ be a finite-dimensional real Banach space.
Let $T,A\in\mathbb{L}(\mathbb{X})$. Then $T\perp_{B}A$ if and only if there
exist $x,y\in M_{T}$ such that $Ax\in Tx^{+}$ and $Ay\in Ty^{-}$.

###### Proof.

The proof of the sufficient part is trivial. To prove the necessary part, we
first observe that

$\displaystyle\|T+\lambda A\|_{S_{\mathbb{X}}}=\|T+\lambda
A\|\geq\|T\|=\|T\|_{S_{\mathbb{X}}},$

for all $\lambda\in\mathbb{R}$. In other words,
$T,A\in\mathcal{C}(S_{\mathbb{X}},\mathbb{X})$ (by considering the respective
restriction operators on $S_{\mathbb{X}}$) with $T\perp_{B}A$. Therefore, it
follows from Theorem 2.1 that there exist $x,y\in\mathcal{M}_{T}$ such that
$Ax\in Tx^{+}$ and $Ay\in Ty^{-}$. This completes the proof of the necessary
part as $M_{T}=\mathcal{M}_{T}$. ∎

######  Corollary 2.3.2.

[8, Theorem 2.1] Let $\mathbb{X}$ be a finite-dimensional real Banach space.
Let $T\in\mathbb{L}(\mathbb{X})$ be such that $T$ attains its norm at only
$\pm D$, where $D$ is a connected subset of $S_{\mathbb{X}}$. Then for
$A\in\mathbb{L}(\mathbb{X})$ with $T\perp_{B}A$ there exists $x\in D$ such
that $Tx\perp_{B}Ax$.

###### Proof.

Considering the restriction on $S_{\mathbb{X}},$ it follows that
$T,A\in\mathcal{C}(S_{\mathbb{X}},\mathbb{X})$ with $T\perp_{B}A$. In view of
Theorem 2.1, there exists $x,y\in\mathcal{M}_{T}$ such that $Ax\in Tx^{+}$ and
$Ay\in Ty^{-}$. Without loss of generality, let $x\in D$. We claim that there
exists $y_{0}\in D$ such that $Ay_{0}\in Ty_{0}^{-}$. Indeed, otherwise the
linearity of $T$ would imply $Az\in Tz^{+}$ for all $z\in\mathcal{M}_{T}$,
which contradicts Theorem 2.1. Now, it follows from Theorem 2.3 that there
exists $x\in\mathcal{M}_{T}$ such that $Tx\perp_{B}Ax$. As before, note that
$\mathcal{M}_{T}=M_{T}.$ This completes the proof of the corollary. ∎

We would like to characterize Birkhoff-James orthogonality of real bilinear
forms on $\mathbb{H}_{1}\times\mathbb{H}_{2}$. To obtain the desired
characterization, let us first present some useful facts about bilinear forms
on $\mathbb{H}_{1}\times\mathbb{H}_{2}$. We omit the proofs as one can verify
them directly.

######  Proposition 2.4.

Let $\mathbb{H}_{1},\mathbb{H}_{2}$ be finite-dimensional Hilbert spaces and
let
$\mathcal{T}_{A},\mathcal{T}_{B}\in\mathcal{B}(\mathbb{H}_{1}\times\mathbb{H}_{2},\mathbb{R})$
be non-zero. Then the following hold true:
$(i)$ $\mathcal{T}_{A}+\lambda\mathcal{T}_{B}=\mathcal{T}_{A+\lambda B}$ for
all $\lambda\in\mathbb{R}$.
$(ii)$ $\|\mathcal{T}_{A}\|=\|A\|$.
$(iii)$ $M_{\mathcal{T}_{A}}=\\{(\pm\frac{Ay}{\|A\|},y):y\in M_{A}\\}.$

Now, the promised characterization:

######  Theorem 2.5.

Let $\mathbb{H}_{1},\mathbb{H}_{2}$ be finite-dimensional Hilbert spaces and
let
$\mathcal{T}_{A},\mathcal{T}_{B}\in\mathcal{B}(\mathbb{H}_{1}\times\mathbb{H}_{2},\mathbb{R})$
be non-zero. Then $\mathcal{T}_{A}\perp_{B}\mathcal{T}_{B}$ if and only if
there exists $(x_{0},y_{0})\in M_{\mathcal{T}_{A}}$ such that
$\mathcal{T}_{B}(x_{0},y_{0})=0$.

###### Proof.

We only prove the necessary part of the theorem, since the sufficient part
follows trivially. We observe that $M_{\mathcal{T}_{A}}=W_{1}\cup W_{2}$,
where

$\displaystyle W_{1}$
$\displaystyle=\left\\{\left(\frac{Ay}{\|A\|},y\right):y\in
M_{A}\right\\}~{}~{}and~{}~{}W_{2}=\left\\{\left(-\frac{Ay}{\|A\|},y\right):y\in
M_{A}\right\\}.$

If $M_{A}=\\{\pm y\\}$ for some $y\in S_{\mathbb{H}_{2}}$, then the proof
follows directly from Theorem 2.3, using the bilinearity of $\mathcal{T}_{A}$.
Otherwise, define
$f_{1},f_{2}:\mathbb{H}_{2}\rightarrow\mathbb{H}_{1}\times\mathbb{H}_{2}$ by
$f_{1}(y)=\left(\frac{Ay}{\|A\|},y\right)$ and
$f_{2}(y)=\left(-\frac{Ay}{\|A\|},y\right)$, respectively. It follows from
Theorem $2.2$ of [8] that $M_{A}$ is connected. Consequently, $W_{1}$ and
$W_{2}$ are also connected, as they are continuous images of $M_{A}$. Now,
$\mathcal{T}_{A},\mathcal{T}_{B}\in\mathcal{C}(S_{\mathbb{X}}\times
S_{\mathbb{Y}},\mathbb{R})$ with $\mathcal{T}_{A}\perp_{B}\mathcal{T}_{B}$.
Therefore in view of Theorem 2.1, there exist $(x_{1},y_{1}),(x_{2},y_{2})\in
M_{\mathcal{T}_{A}}$ such that
$\mathcal{T}_{B}(x_{1},y_{1})\in\mathcal{T}_{A}(x_{1},y_{1})^{+}$ and
$\mathcal{T}_{B}(x_{2},y_{2})\in\mathcal{T}_{A}(x_{2},y_{2})^{-}$. Since
$\mathcal{T}_{A},\mathcal{T}_{B}$ are real valued, we must have
$\mathcal{T}_{A}(x_{1},y_{1})\mathcal{T}_{B}(x_{1},y_{1})>0$ and
$\mathcal{T}_{A}(x_{2},y_{2})\mathcal{T}_{B}(x_{2},y_{2})<0.$ Without loss of
generality, suppose $(x_{1},y_{1})\in W_{1}$. Then there must be
$(x_{2},y_{2})\in W_{1}$ such that
$\mathcal{T}_{A}(x_{2},y_{2})\mathcal{T}_{B}(x_{2},y_{2})<0$. Indeed,
otherwise the bilinearity of $\mathcal{T}_{A}$ would imply
$\mathcal{T}_{A}(x,y)\mathcal{T}_{B}(x,y)>0$ for all $(x,y)\in
M_{\mathcal{T}_{A}}$, which contradicts Theorem 2.1. Now, the proof follows
from Theorem 2.3. ∎

As a natural outcome of our present study, we may obtain a completely new and
elementary proof of the classical Bhatia-Šemrl Theorem in the real case, as an
application of Theorem 2.5. Note that in the following theorem, $A$ and $B$
are $n\times n$ matrices with real entries and $\mathbb{H}$ is an
$n$-dimensional real Hilbert space for some $n>1$.

######  Theorem 2.6.

A matrix $A$ is orthogonal to $B$ if and only if there exists a unit vector
$y_{0}\in\mathbb{H}$ such that $||A||=||Ay_{0}||$ and $\langle
Ay_{0},By_{0}\rangle=0$.

###### Proof.

Without loss of generality, let $A$ be non-zero. We only prove the necessary
part of the theorem as the proof of the sufficient part follows trivially.
Consider the bilinear forms
$\mathcal{T}_{A},\mathcal{T}_{B}\in\mathcal{B}(\mathbb{H}\times\mathbb{H},\mathbb{R})$.
Now, for any $\lambda\in\mathbb{R}$, we have

$\displaystyle\|\mathcal{T}_{A}+\lambda\mathcal{T}_{B}\|=\|A+\lambda
B\|\geq\|A\|=\|\mathcal{T}_{A}\|.$

Therefore, $\mathcal{T}_{A}\perp_{B}\mathcal{T}_{B}$. In light of Theorem 2.5,
there exists $(x_{0},y_{0})\in M_{\mathcal{T}_{A}}$ such that
$\mathcal{T}_{B}(x_{0},y_{0})=0$. Also, since $(x_{0},y_{0})\in
M_{\mathcal{T}_{A}}$, we must have $x_{0}=\pm\frac{Ay_{0}}{\|A\|}$ for some
$y_{0}\in M_{A}$. This implies that
$\mathcal{T}_{B}(x_{0},y_{0})=\pm\frac{1}{\|A\|}\langle
Ay_{0},By_{0}\rangle=0$. This establishes the theorem. ∎

Let us end this article with the following closing remark:

######  Remark 2.1.

The study of Birkhoff-James orthogonality of linear operators between Hilbert
spaces and Banach spaces have been conducted in detail in [2, 3, 7, 8]. The
main purpose of this article is to illustrate that it is possible to extend
these results to the topological setting. Moreover, as a natural outcome of
such a study, Birkhoff-James orthogonality of real bilinear forms can be
characterized completely, which in turn allows us to obtain a new and
elementary proof of the Bhatia-Šemrl Theorem. Although a number of proofs of
the said theorem are available in the literature [2, 3, 6, 9], we believe that
our proof is less complicated and differs from each of those in motivation.

## References

  * [1] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935) 169-172.
  * [2] T. Bhattacharyya, P. Grover, Characterization of Birkhoff-James orthogonality, J. Math. Anal. Appl. 407 (2013) 350-358.
  * [3] R. Bhatia, P. $\breve{S}$emrl, Orthogonality of matrices and distance problems, Linear Algebra Appl. 287 (1999) 77-85.
  * [4] R. C. James, Inner product in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947) 559-566.
  * [5] R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265-292.
  * [6] D. J. Ke$\breve{c}$ki$\grave{c}$, Gateaux derivative of B(H) norm, Proc. Amer. Math. Soc. 133 (2005) 2061-2067.
  * [7] D. Sain, Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Anal. Appl. 447 (2017) 860-866.
  * [8] D. Sain, K. Paul, Operator norm attainment and inner product spaces, Linear Algebra Appl. 439 (2013) 2448-2452.
  * [9] A. Turn$\breve{s}$ek, A remark on orthogonality and symmetry of operators in B(H), Linear Algebra Appl. 535 (2017) 141-150.


# Incommensurate broken-helix and broken-fanlike states in axion insulator
candidate EuIn2As2

Masaki Gen<EMAIL_ADDRESS>RIKEN Center for Emergent Matter Science
(CEMS), Wako 351-0198, Japan  Yukako Fujishiro  RIKEN Center for Emergent
Matter Science (CEMS), Wako 351-0198, Japan  Kazuki Okigami  Department of
Applied Physics, University of Tokyo, Tokyo 113-8656, Japan  Satoru Hayami
Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan
Kiyohiro Adachi  Materials Characterization Support Team, RIKEN Center for
Emergent Matter Science (CEMS), Wako 351-0198, Japan  Daisuke Hashizume
Materials Characterization Support Team, RIKEN Center for Emergent Matter
Science (CEMS), Wako 351-0198, Japan  Takashi Kurumaji  Department of Advanced
Materials Science, University of Tokyo, Kashiwa 277-8561, Japan  Division of
Physics, Mathematics and Astronomy, California Institute of Technology,
Pasadena, California 91125, USA  Hajime Sagayama  Institute of Materials
Structure Science, High Energy Accelerator Research Organization, Tsukuba
305-0801, Japan  Hironori Nakao  Institute of Materials Structure Science,
High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan
Yoshinori Tokura  RIKEN Center for Emergent Matter Science (CEMS), Wako
351-0198, Japan  Department of Applied Physics, University of Tokyo, Tokyo
113-8656, Japan  Tokyo College, University of Tokyo, Tokyo 113-8656, Japan
Taka-hisa Arima  RIKEN Center for Emergent Matter Science (CEMS), Wako
351-0198, Japan  Department of Advanced Materials Science, University of
Tokyo, Kashiwa 277-8561, Japan

###### Abstract

Zintl phase EuIn2As2 has garnered growing attention as an axion insulator
candidate, triggered by the identification of a commensurate
double-${\mathbf{Q}}$ broken-helix state in previous studies, however, its
periodicity and symmetry remain subjects of debate. Here, we perform resonant
x-ray scattering experiments on EuIn2As2, revealing an incommensurate nature
of the broken-helix state, where both the wave number and the amplitude of the
helical modulation exhibit systematic sample dependence. Furthermore, the
application of an in-plane magnetic field brings about a broken-fanlike state
preserving the double-${\mathbf{Q}}$ nature, which could be attributed to
multiple-spin interactions in momentum space. We propose that the itinerant
character of EuIn2As2, possibly induced by Eu deficiency, gives rise to the
helical modulation and impedes the realization of a theoretically-predicted
axion state with the collinear antiferromagnetic order.

Europium-based compounds offer a fertile playground for exploring nontrivial
magnetotransport phenomena 2004_Wig ; 2016_Kaw ; 2020_Ros ; 2023_May . In the
presence of Fermi surfaces, the carrier-mediated Ruderman-Kittel-Kasuya-Yosida
(RKKY) interaction can stabilize a variety of modulated magnetic structures
such as helix 2009_Nan ; 2020_Tak ; 2022_Kur ; 2023_Soh_2  and skyrmions
2019_Kan ; 2022_Tak ; 2023_Gen ; 2023_Sin ; 2023_Mat . The correlation between
magnetism and electronic band topology is one other intriguing aspect, as
extensively studied in the 122 families of the Zintl phase 2006_Jia ; 2019_Ma
; 2020_Su ; 2021_Wan ; 2022_Bia ; 2023_Kre ; 2023_Cuo . First-principles
calculations predict that EuIn2As2 can host an axion insulating state (AXS)
with the layered antiferromagnetic (AFM) order 2009_Ess ; 2019_Xu , thereby
potentially exhibiting the quantized magnetoelectric effect 2019_Tok ;
2022_Ber ; 2021_Fij . Given that the AXS remains unestablished in
stoichiometric bulk materials 2021_Fij ; 2011_Wan ; 2017_Mog ; 2019_Zha ;
2019_Li ; 2020_Den , a comprehensive investigation on the physical properties
of EuIn2As2 stands as a critical issue 2008_Gof ; 2020_Zha ; 2020_Sat ;
2020_Reg ; 2020_Yu ; 2021_Rib ; 2022_Gon ; 2022_Yan ; 2023_Soh ; 2023_Wu ;
2023_Don , not only from the fundamental viewpoint but also for its
applications in next-generation devices 2019_Tok .

EuIn2As2 forms a hexagonal crystal structure of space group $P6_{3}/mmc$,
consisting of alternating stacking of Eu triangular layers and In2As2 blocks
[Fig. 1(a)] (Note that one unit cell contains two Eu layers). The magnetism is
dominated by intraplane ferromagnetic and interplane AFM interactions between
localized Eu2+ moments, with an easy-plane anisotropy becoming evident at low
temperatures 2020_Zha . Despite the theoretical prediction of a gapped
insulating state 2019_Xu , angle-resolved photoemission spectroscopies (ARPES)
revealed holelike bulk bands crossing the Fermi level 2020_Sat ; 2020_Reg .
Neutron diffraction (ND) 2021_Rib  and resonant x-ray scattering (RXS)
experiments 2023_Soh  observed a concurrent short-period magnetic modulation,
${\mathbf{Q}}_{1}=(0,0,q_{1z})$, along with the AFM component
${\mathbf{Q}}_{2}=(0,0,1)$; the reported $q_{1z}$ values are 0.303(1) 2021_Rib
and 0.3328(6) 2023_Soh , respectively, in the reciprocal-lattice unit. A phase
separation scenario was excluded by azimuthal scans in the RXS 2023_Soh . Both
studies proposed a six-layer-period double-${\mathbf{Q}}$ helical structure,
termed broken helix, under the assumption that ${\mathbf{Q}}_{1}$ represents
an exactly commensurate modulation with $q_{1z}=1/3$ 2021_Rib ; 2023_Soh .
Initially, the AXS was believed to be realized in the commensurate broken-
helix state, as the ${\mathcal{T}}{\mathcal{C}}_{2}$ symmetry (the combination
of time-reversal and two-fold rotational operations) is preserved once the
principal axis of the broken helix is aligned along a specific
crystallographic axis 2021_Rib ; 2023_Soh . However, a recent optical
birefringence study challenged the above picture in terms of the symmetry, and
proposed the unpinned nature of the broken helix owing to minimal hexagonal
anisotropy 2023_Don .

Here, we reexamine the magnetic structure of EuIn2As2. While the preceding ND
study identified an incommensurate ${\mathbf{Q}}_{1}$ modulation 2021_Rib , no
incommensurate spin configuration has been considered so far 2021_Rib ;
2023_Soh ; 2023_Don . Whether ${\mathbf{Q}}_{1}$ is commensurate or not is
pivotal, as the latter is not compatible with the AXS due to the
${\mathcal{T}}{\mathcal{C}}_{2}$ symmetry breaking. The mechanism for the
emergence of the ${\mathbf{Q}}_{1}$ modulation also remains puzzling 2023_Soh
; 2023_Don . To address these issues, we perform the RXS experiments,
revealing that the ${\mathbf{Q}}_{1}$ peak exhibits variability in the
$q_{1z}$ value (0.25–0.29) as well as the intensity across samples [Fig.
1(b)]. The complemental single-crystal structure analysis suggests that the
${\mathbf{Q}}_{1}$ modulation stems from the RKKY interaction mediated by
doped holes due to Eu deficiency. We argue that, to avoid the loss of
generality, the broken-helix state should be considered as a superposition of
incommensurate helical and collinear AFM modulations, as illustrated in Fig.
1(c). We also reveal the appearance of a double-${\mathbf{Q}}$ fanlike state
with higher-harmonic modulations in an in-plane magnetic field, giving
insights into a theoretical framework to describe the complicated magnetism in
EuIn2As2.

Figure 1: (a) Crystal structure of EuIn2As2. The black line represents a
crystallographic unit cell. (b) Ratio of the magnetic-moment amplitude of the
${\mathbf{Q}}_{1}=(0,0,q_{1z})$ and ${\mathbf{Q}}_{2}=(0,0,1)$ modulations,
$m_{{\mathbf{Q}}_{1}}/m_{{\mathbf{Q}}_{2}}$, versus the $q_{1z}$ value in the
broken-helix state for samples used in this study (5 K) and those in previous
studies by Riberrolles et al. 2021_Rib  and by Soh et al. 2023_Soh  (6 K).
$m_{{\mathbf{Q}}_{1}}/m_{{\mathbf{Q}}_{2}}$ is estimated from the observed
peak intensities. (c) Schematics of the broken-helix state and an equivalent
double-${\mathbf{Q}}$ representation as the superposition of incommensurate
helical ${\mathbf{Q}}_{1}$ and collinear AFM ${\mathbf{Q}}_{2}$ modulations.
Magnetic moments are in the $ab$-plane, and the modulation is along the $c$
axis.

Single crystals of EuIn2As2 were grown by an indium flux method. Details of
sample growth and characterization are described in the Supplemental Material
(SM) SM . We picked up two specimens (#1 and #2) from the same batch for the
RXS. RXS experiments were performed at BL-3A, Photon Factory, KEK, by using
horizontally ($\pi$) polarized incident x-rays in resonance with the Eu
$L_{2}$ absorption edge ($E=7.612$ keV) 2022_Kur ; 2023_Gen . Samples with the
as-grown (001) plane were glued on an aluminium plate and set in a cryostat
equipped with a vertical-field superconducting magnet. The scattering plane
was set to be $(H,0,L)$, and a magnetic field was applied along the
crystallographic $b$ axis [Fig. 2(a)]. We could access fundamental Bragg peaks
at $(3m,0,2n)$ and $(3m\pm 1,0,n)$ ($m,n$: integer), and their magnetic
satellite peaks in the reciprocal space [Fig. 2(b)]. Unless otherwise stated,
we performed the $(0,0,L)$ scan with $L=12$–16, and the scattered x-rays were
detected without analyzing the $\pi^{\prime}$ and $\sigma^{\prime}$
polarizations, parallel and perpendicular to the scattering plane,
respectively. For polarization analysis, the 006 reflection of a pyrolytic
graphite (PG) crystal was used, where the scattering angle was $\sim$92∘ near
the Eu $L_{2}$ edge.

Figure 2: (a) Experimental geometry of the RXS. ${\mathbf{k}}_{\rm i}$
(${\mathbf{k}}_{\rm f}$) and $\pi$ ($\pi^{\prime}$ and $\sigma^{\prime}$)
represent the propagation vector and polarization direction of incident
(scattered) x-rays, respectively. (b) Positions of the Bragg peaks in the
$(H,0,L)$ scattering plane below $T_{\rm N2}$. Gray squares are the
fundamental peaks, and red (blue) circles are the magnetic ${\mathbf{Q}}_{1}$
(${\mathbf{Q}}_{2}$) peaks. (c) RXS profiles observed in the $(0,0,L)$ scan at
5 K (red) and 30 K (gray) in zero field for sample #1. [(d)–(f)] Temperature
dependence of (d) magnetic susceptibility $M/H$, (e) $q_{1z}$, and (f)
integrated intensity of the $00L$ reflections with $L=14-q_{1z}$ and $L=13$ in
zero field.

As in Refs. 2021_Rib ; 2023_Soh , we observe two kinds of magnetic Bragg
peaks, ${\mathbf{Q}}_{1}=(0,0,q_{1z})$ and ${\mathbf{Q}}_{2}=(0,0,1)$ [Fig.
2(c)]. Figures 2(d)–2(f) show the temperature dependence of magnetic
susceptibility, $q_{1z}$, and integrated intensities of the ${\mathbf{Q}}_{1}$
and ${\mathbf{Q}}_{2}$ peaks in (nearly) zero field for sample #1. Upon
cooling, the ${\mathbf{Q}}_{1}$ and ${\mathbf{Q}}_{2}$ peaks emerge below
$T_{\rm N1}=17.6$ K and $T_{\rm N2}=16.2$ K, respectively. $q_{1z}$ gradually
decreases from 0.32 to 0.29 in the intermediate-temperature (IT) phase, while
$q_{1z}$ is almost temperature independent in the low-temperature (LT) phase.
These trends also agree with Refs. 2021_Rib ; 2023_Soh , although there is a
discrepancy in the $q_{1z}$ value [Fig. 1(b)]. Notably, a distinct $q_{1z}$
value of 0.25 is observed for sample #2, and we confirm that the way of
adhering samples to the Al plate has little influence on $q_{1z}$ (see the SM
SM ). Therefore, the observed variability in $q_{1z}$ appears to be primarily
attributed to the sample dependence rather than extrinsic strain.

Indeed, our single-crystal structural analyses reveal a Eu deficiency of
1.6(5)% for sample #1 and 1.1(4)% for sample #2 (see the SM SM ), indicating
that a larger Eu deficiency leads to a larger $q_{1z}$ value. Recalling the
first-principles calculations predicting a gapped insulating state with the
AFM order (${\mathbf{Q}}_{2}$) 2019_Xu , the additional ${\mathbf{Q}}_{1}$
modulation is expected to be induced by doped hole carriers which contribute
to the RKKY interaction between the Eu2+ moments. We also find that the
intensity ratio of the ${\mathbf{Q}}_{1}$ to the ${\mathbf{Q}}_{2}$ peak is
higher for samples with larger $q_{1z}$ [Fig. 1(b)], suggesting that the
${\mathbf{Q}}_{1}$ component is enhanced by a larger number of carriers. The
above scenarios agree with the 3D character of a hole pocket observed in a
previous ARPES 2020_Sat , although the details of the Fermi-surface shape need
to be clarified to comprehend the quantitative correlation between the carrier
number (related to the off-stoichiometry) and the magnetic modulation period.

Figure 3: [(a)–(c)] Magnetic-field dependence of (a) magnetization $M$ (left
axis) and its field derivative $dM/dH$ (right axis), (b) $q_{1z}$, and (c)
integrated intensity $I(H)$ of the $00L$ reflections with $L=16-q_{1z}$ and
$L=15$. Dark and light colors represent data in field-increasing and
decreasing processes, respectively. (d) Logarithmic contour plot of the RXS
intensity in the $(0,0,L)$ scan near $H_{\rm c1}$ in a field-increasing
process. Black arrows on the right represent peak widths for
$3{\mathbf{Q}}_{1}$ and ${\mathbf{Q}}_{2}-{\mathbf{Q}}_{1}$. (e) Magnetic-
field dependence of the integrated intensity of the $3{\mathbf{Q}}_{1}$ (cyan)
and ${\mathbf{Q}}_{2}-{\mathbf{Q}}_{1}$ (pink) peaks. The absolute value is
normalized by the intensity of the 0013 reflection at 0 T. All the data are
taken at 5 K with $H\parallel b$.

Next, we investigate the magnetic structure changes with the application of an
in-plane magnetic field for sample #1, which undergoes a metamagnetic
transition at $\mu_{0}H_{\rm c1}=0.2$ T at 5 K 2020_Zha ; 2020_Yu ; 2023_Soh .
Figures 3(a)–(c) show the field dependence of magnetization, $q_{1z}$, and
integrated intensities of the ${\mathbf{Q}}_{1}$ and ${\mathbf{Q}}_{2}$ peaks.
With increasing a magnetic field, both ${\mathbf{Q}}_{1}$ and
${\mathbf{Q}}_{2}$ peaks survive until the saturation field of 1.0 T. Here,
$q_{1z}$ exhibits only a small change of at most $\pm 0.01$ even across
$H_{\rm c1}$. With subsequently decreasing a magnetic field, a hysteresis
appears around $H_{\rm c1}$ in RXS, in line with the magnetization process.
Ultimately, neither the intensities of the ${\mathbf{Q}}_{1}$ and
${\mathbf{Q}}_{2}$ peaks nor $q_{1z}$ reverts to the original values at zero
field.

We find remarkable features in the RXS profiles near $H_{\rm c1}$. Figure 3(d)
shows a logarithmic contour plot of the RXS intensity against a magnetic
field, observed in the $(0,0,L)$ scan in the $L$ range between 12.9 and 13.8.
Apart from the strong ${\mathbf{Q}}_{1}$ and ${\mathbf{Q}}_{2}$ peaks at
$L=13.71$ and 13, respectively, peaks around $L=13.29$ and 13.13 are
discernible in certain field ranges. These additional peaks likely correspond
to higher-harmonic ${\mathbf{Q}}_{2}-{\mathbf{Q}}_{1}$ and $3{\mathbf{Q}}_{1}$
modulations, respectively. The field evolution of integrated intensities of
these peaks are displayed in Fig. 3(e). The weak
${\mathbf{Q}}_{2}-{\mathbf{Q}}_{1}$ peak appears above 0.1 T and persists
until at least 0.4 T, except immediately after $H_{\rm c1}$ where the
intensity of the $3{\mathbf{Q}}_{1}$ peak becomes prominent. Notably, a
$2{\mathbf{Q}}_{1}$ peak at $L=13.42$, expected in the conventional fanlike
structure 2003_Kos ; 2020_Ghi , is not observed in the entire field range
[Fig. 3(d)]. These results indicate that the high-field (HF) phase above
$H_{\rm c1}$ is characterized by a superposition of ${\mathbf{Q}}_{1}$ and
${\mathbf{Q}}_{2}$, ruling out a phase separation scenario.

Figure 4: [(a)(b)] Schematic geometical configuration of the RXS focusing on
the ${\mathbf{Q}}_{1}$ and ${\mathbf{Q}}_{2}$ peaks around the magnetic Bragg
spots $(-2,0,11.71)$ and $(0,0,9)$, respectively. See also Figs. 2(a) and (b).
[(c)–(h)] RXS profiles of the polarization analysis at 5 K at 0 T [(c)(f)],
0.1 T [(d)(g)], and 0.3 T [(e)(h)] with $H\parallel b$. Panels (c)–(e) and
(f)–(h) show the data of the $(-2,0,L)$ scan around $L=11.71$ and $(0,0,L)$
scan around $L=9$, respectively.

To get information on the orientation of magnetic moments in each phase at 5
K, we performed polarization analysis with $H\parallel b$. In general, the
magnetic scattering intensity $I$ is given by $I\propto({\mathbf{e}}_{\rm
i}\times{\mathbf{e}}_{\rm f})\cdot{\mathbf{m}}_{\mathbf{Q}}$, where
${\mathbf{e}}_{\rm i}$ (${\mathbf{e}}_{\rm f}$) is the polarization vector of
the incident (scattered) beam, and ${\mathbf{m}}_{\mathbf{Q}}$ is a magnetic
moment of the ${\mathbf{Q}}$ modulation. The $\pi$–$\pi^{\prime}$ channel
($I_{\pi-\pi^{\prime}}$) always detects the modulated component along $b$
($m_{b}$), whereas the $\pi$–$\sigma^{\prime}$ channel
($I_{\pi-\sigma^{\prime}}$) contains those along $a^{*}$ ($m_{a^{*}}$) and $c$
($m_{c}$) in the ratio of $\cos^{2}\omega:\sin^{2}\omega$, where $\omega$ is
the angle between the propagation vector of the incident beam
(${\mathbf{k}}_{\rm i}$) and the $a^{*}$ axis. Figures 4(a) and 4(b) show the
experimental configurations of polarization analysis focusing on the Bragg
spots at $(-2,0,11.71)$ and $(0,0,9)$, in which $I_{\pi-\sigma^{\prime}}$
mainly reflects $m_{a^{*}}$ ($\sim$100% and 83%) of the ${\mathbf{Q}}_{1}$ and
${\mathbf{Q}}_{2}$ modulations, respectively. Additional measurements focusing
on the Bragg spots at $(2,0,11.71)$ and $(0,0,17)$ confirm the absence of
$m_{c}$ (see the SM SM ), i.e., all the spins lie within the $ab$ plane.

At zero field, both $I_{\pi-\pi^{\prime}}$ and $I_{\pi-\sigma^{\prime}}$ have
intensities at $(-2,0,11.71)$ and $(0,0,9)$ [Figs. 4(c) and 4(f)]. At 0.1 T
($<H_{\rm c1}$), a drastic enhancement is observed in
$I_{\pi-\sigma^{\prime}}$ at $(0,0,9)$, accompanied by a suppression of
$I_{\pi-\pi^{\prime}}$ [Fig. 4(g)], suggesting the reorientation of magnetic
domains so as to lie the AFM component perpendicular to the field direction.
In contrast, $I_{\pi-\pi^{\prime}}$ and $I_{\pi-\sigma^{\prime}}$ still have
comparable intensities at $(-2,0,11.71)$ [Fig. 4(d)], suggesting that the
${\mathbf{Q}}_{1}$ modulation originates from a helical component, rather than
sinusoidal. Followed by the metamagnetic transition, the presence of only
$m_{a^{*}}$ is confirmed for both the ${\mathbf{Q}}_{1}$ and
${\mathbf{Q}}_{2}$ modulations, as evidenced by the disappearance of
$I_{\pi-\pi^{\prime}}$ at the both Bragg positions at 0.3 T ($>H_{\rm c1}$)
[Figs. 4(e) and 4(h)]. Accordingly, the HF phase can be ascribed to a
double-${\mathbf{Q}}$ fanlike state.

Let us here update the understanding of the magnetic structure in EuIn2As2. We
show that the LT phase is a broken-helix state with a superposition of
incommensurate helical ${\mathbf{Q}}_{1}$ and AFM ${\mathbf{Q}}_{2}$
modulations, which can be approximately described as
${\mathbf{m}}({\mathbf{r}})\propto(1,i,0)\sum_{\eta=1,2}{m_{{\mathbf{Q}}_{\eta}}}\exp(i{\mathbf{Q}}_{\eta}\cdot{\mathbf{r}})+{\rm
c.c.}$ in zero field. This expression encompasses the commensurate broken
helix proposed in Refs. 2021_Rib ; 2023_Soh  as a special case. By comparing
the observed intensities of the ${\mathbf{Q}}_{1}$ and ${\mathbf{Q}}_{2}$
peaks with the calculated ones based on the above formula assuming an equal
moment-size condition, we estimate $m_{{\mathbf{Q}}_{1}}/m_{{\mathbf{Q}}_{2}}$
at the lowest temperature for each sample used in this study and in Refs.
2021_Rib ; 2023_Soh , as shown in Fig. 1(b) (see the SM SM ); e.g.,
$m_{{\mathbf{Q}}_{1}}/m_{{\mathbf{Q}}_{2}}\approx 0.75$ and 0.22 for samples
#1 ($q_{1z}=0.29$) and #2 ($q_{1z}=0.25$), respectively. A positive
correlation between $q_{1z}$ and $m_{{\mathbf{Q}}_{1}}/m_{{\mathbf{Q}}_{2}}$
can be clearly seen, which agrees with the RKKY picture for the emergence of
helimagnetism by hole doping, as mentioned above. We also note that the IT
phase was identified as a sinusoidal state by Mössbauer 2021_Rib  and optical
birefringence measurements 2023_Don . As the hexagonal anisotropy is
negligibly weak 2020_Zha ; 2023_Don , it is natural to believe that the
sinusoidal modulation is stabilized by thermal fluctuations and gradually
transforms into helix with decreasing temperature. We thus infer that there is
a slight elliptical distortion of the ${\mathbf{Q}}_{1}$ component in the LT
phase at 5 K.

Our RXS also reveals a helix-fan transition while keeping the
double-${\mathbf{Q}}$ nature, as evidenced by the observation of higher
harmonics ${\mathbf{Q}}_{2}-{\mathbf{Q}}_{1}$. Since the real-space picture of
the double-${\mathbf{Q}}$ fanlike state is rather nontrivial, a theoretical
description based on the microscopic model is necessary. We first investigate
ground states of a spin Hamiltonian composed of long-range Heisenberg exchange
and biquadratic exchange interactions in real space on a 1D chain system, as
proposed in Ref. 2023_Don . Our variational calculations reveal that this
model can stabilize an incommensurate broken helix with a dominant
${\mathbf{Q}}_{2}$ component in zero field, while it fails to preserve the
double-${\mathbf{Q}}$ nature and instead stabilizes a canted AFM state in high
magnetic fields (see the SM SM ).

Figure 5: Magnetic-field dependence of (a) magnetization $m$, and (b)(c) spin
structure factor $S({\mathbf{q}})$ per a spin at several ${\mathbf{Q}}$’s for
$h\parallel x$, calculated by simulated annealing for Eq. (1) with $K=0.9$,
$\alpha=0.48$, and $\alpha^{\prime}=0.9$. Local spin configurations at
$h=0.1$, 0.3, and 0.5 are schematically illustrated by purple arrows in panels
(d), (e), and (f), respectively. The light blue (red) arrow indicates the $x$
($y$) component of each spin.

In order to reproduce the double-${\mathbf{Q}}$ fanlike state in the HF phase,
we employ an effective spin Hamiltonian composed of exchange interactions in
momentum space:

$\begin{split}{\mathcal{H}}=&-J{\mathbf{S}}_{{\mathbf{Q}}_{1}}\cdot{\mathbf{S}}_{-{\mathbf{Q}}_{1}}-J^{\prime}{\mathbf{S}}_{{\mathbf{Q}}_{2}}\cdot{\mathbf{S}}_{-{\mathbf{Q}}_{2}}-J^{\prime\prime}{\mathbf{S}}_{{\mathbf{Q}}_{3}}\cdot{\mathbf{S}}_{-{\mathbf{Q}}_{3}}\\\
&+K({\mathbf{S}}_{{\mathbf{Q}}_{1}}\cdot{\mathbf{S}}_{-{\mathbf{Q}}_{1}})^{2}+K^{\prime}({\mathbf{S}}_{{\mathbf{Q}}_{2}}\cdot{\mathbf{S}}_{-{\mathbf{Q}}_{2}})^{2}+K^{\prime\prime}({\mathbf{S}}_{{\mathbf{Q}}_{3}}\cdot{\mathbf{S}}_{-{\mathbf{Q}}_{3}})^{2}\\\
&-K_{2}({\mathbf{S}}_{{\mathbf{Q}}_{1}}\cdot{\mathbf{S}}_{{\mathbf{Q}}_{2}})({\mathbf{S}}_{-{\mathbf{Q}}_{1}}\cdot{\mathbf{S}}_{-{\mathbf{Q}}_{2}})-h\sum_{i}S_{i}^{x},\end{split}$
(1)

where we take into account dominant interactions with
${\mathbf{Q}}_{1}=0.29\pi$ and ${\mathbf{Q}}_{2}=\pi$. In addition to the
conventional bilinear and biquadratic terms for ${\mathbf{Q}}_{1}$ and
${\mathbf{Q}}_{2}$ 2017_Hay , Eq. (1) includes an intertwined coupling term
($K_{2}$) with a negative sign, which plays a crucial role in stabilizing a
double-${\mathbf{Q}}$ fanlike state in high magnetic fields. We also introduce
higher-harmonic terms ($J^{\prime\prime}$ and $K^{\prime\prime}$) with
${\mathbf{Q}}_{3}=3{\mathbf{Q}}_{1}$ to reproduce the enhancement of the
$3{\mathbf{Q}}_{1}$ modulation immediately after the metamagnetic transition.
The last term in Eq. (1) represents the Zeeman coupling to an in-plane
magnetic field $h\parallel x$. For simplicity, we set $J^{\prime}=\alpha J$,
$J^{\prime\prime}=\alpha^{\prime}J$, $K^{\prime}=\alpha^{2}K$,
$K^{\prime\prime}=\alpha^{\prime 2}K$, $K_{2}=K/2$, $J=1$ as the energy unit,
and consider $XY$ spins with $|{\mathbf{S}}_{i}|=1$. We performed simulated
annealing with $N=200$ spins for several parameter sets, and calculated the
magnetization $m$ and spin structure factor, defined as
$S({\mathbf{q}})=(1/N)\sum_{i,j}\langle{\mathbf{S}}_{i}\cdot{\mathbf{S}}_{j}\rangle
e^{i{\mathbf{q}}\cdot({\mathbf{r}}_{i}-{\mathbf{r}}_{j})}$. Figures 5(a)–5(c)
show one typical result, which qualitatively reproduces our RXS data (Fig. 3).
The system undergoes metamagnetic transitions, accompanied by the enhancement
of $S(3{\mathbf{Q}}_{1})$ and $S({\mathbf{Q}}_{2}-{\mathbf{Q}}_{1})$. We note
that other parameter sets can bring about qualitatively distinct behaviors in
higher harmonics, such as the enhancement of
$S({{\mathbf{Q}}_{2}-2{\mathbf{Q}}_{1}})$ instead of $S(3{\mathbf{Q}}_{1})$.

Figures 5(d)-5(f) show schematics of local spin configurations in three
different phases at $h=0.1$, 0.3, and 0.5, respectively. The spin
configuration at $h=0.3$ is far from the conventional fanlike state in that
some spins still possess a negative $x$ component. Notably, a square-wave-
shaped modulation appears in the $y$ component (perpendicular to $h$),
reflecting the effect of $3{\mathbf{Q}}_{1}$ along with the ${\mathbf{Q}}_{1}$
modulation. At $h=0.5$, the spin configuration is like a conventional fanlike
state, while the remaining ${\mathbf{Q}}_{2}$ component contributes to a
complex manner of the spin flipping, resulting in a broken fanlike state.
Though our magnetization data [Fig. 3(a)] shows no clear signs of a transition
between these field-induced phases, the above-mentioned pictures would
qualitatively describe the field evolution of a magnetic structure above
$H_{\rm c1}$ in EuIn2As2. There have been reports of other Eu-based itinerant
magnets exhibiting the coexistence of two magnetic modulations along the $c$
axis, such as EuRh2As2 2009_Nan , EuCuSb 2020_Tak , and EuZnGe 2022_Kur ,
although their in-field magnetic structures have yet to be understood. The
present study establishes benchmark experimental and theoretical approaches
for a better understanding of the complex helimagnetism characterized by
interplane magnetic modulations.

In summary, we elucidate the incommensurate nature of the broken-helix state
in EuIn2As2 through RXS experiments, contradicting the previously proposed
commensurate broken-helix 2021_Rib ; 2023_Soh . Furthermore, we identify the
emergence of a broken fanlike state with higher-harmonic modulations in an in-
plane magnetic field. The double-${\mathbf{Q}}$ nature likely arises from the
RKKY mechanism via hole carriers introduced by Eu deficiency. We propose that
electron doping in off-stoichiometric EuIn2As2 through chemical substitution
can be a promising pathway to shift the Fermi energy to the band gap and
suppress the helical modulation, realizing the theoretically predicted AXS
with the collinear AFM order 2019_Xu .

This work was financially supported by the JSPS KAKENHI Grants-In-Aid for
Scientific Research (No. 22H00101, No. 22K14011, No. 23H04869, No. 23H05431,
and No. 23K13068), JST SPRING (No. JPMJSP2108), and JST PRESTO (No. MJPR20L8).
The resonant x-ray scattering experiments were performed with the approval of
the Photon Factory Program Advisory Committee (Proposal No. 2022G551 and No.
2023G093). The authors appreciate A. Kikkawa and M. Kriener for the invaluable
support in the synthesis of the single crystals used in this study. The
authors appreciate H. Takeda, S. Kitou, M. Hirschberger, and K. Kakurai for
fruitful discussions.

## References

  * (1) G. A. Wigger, C. Beeli, E. Felder, H. R. Ott, A. D. Bianchi, and Z. Fisk, Percolation and the Colossal Magnetoresistance of Eu-Based Hexaboride, Phys. Rev. Lett. 93, 147203 (2004).
  * (2) K. Kawashima, T. Kinjo, T. Nishio, S. Ishida, H. Fujihisa, Y. Gotoh, K. Kihou, H. Eisaki, Y. Yoshida, and A. Iyo, Superconductivity in Fe-Based Compound EuAFe4As4 (A = Rb and Cs), J. Phys. Soc. Jpn. 85, 064710 (2016).
  * (3) P. Rosa, Y. Xu, M. Rahn, J. Souza, S. Kushwaha, L. Veiga, A. Bombardi, S. Thomas, M. Janoschek, E. Bauer, M. Chan, Z. Wang, J. Thompson, N. Harrison, P. Pagliuso, A. Bernevig, and F. Ronning, Colossal magnetoresistance in a nonsymmorphic antiferromagnetic insulator, npj Quantum Mater. 5, 52 (2020).
  * (4) A. H. Mayo, H. Takahashi, M. S. Bahramy, A. Nomoto, H. Sakai, and S. Ishiwata, Magnetic Generation and Switching of Topological Quantum Phases in a Trivial Semimetal ${\alpha}$-EuP3, Phys. Rev. X 12, 011033 (2023).
  * (5) S. Nandi, A. Kreyssig, Y. Lee, Y. Singh, J. W. Kim, D. C. Johnston, B. N. Harmon, and A. I. Goldman, Magnetic ordering in EuRh2As2 studied by x-ray resonant magnetic scattering, Phys. Rev. B 79, 100407(R) (2009).
  * (6) H. Takahashi, K. Aono, Y. Nambu, R. Kiyanagi, T. Nomoto, M. Sakano, K. Ishizaka, R. Arita, and S. Ishiwata, Competing spin modulations in the magnetically frustrated semimetal EuCuSb, Phys. Rev. B 102, 174425 (2020).
  * (7) T. Kurumaji, M. Gen, S. Kitou, H. Sagayama, A. Ikeda, and T. Arima, Anisotropic magnetotransport properties coupled with spiral spin modulation in a magnetic semimetal EuZnGe, Phys. Rev. Mater. 6, 094410 (2022).
  * (8) J.-R. Soh, I. S.-Ramírez, X. Yang, J. Sun, I. Zivkovic, J. A. R.-Velamazán, O. Fabelo, A. Stunault, A. Bombardi, C. Balz, M. D. Le, H. C. Walker, J. H. Dil, D. Prabhakaran, H. M. Rønnow, F. de Juan, M. G. Vergniory, and A. T. Boothroyd, Weyl metallic state induced by helical magnetic order, npj Quantum Mater. 9, 7 (2024).
  * (9) K. Kaneko, M. D. Frontzek, M. Matsuda, A. Nakao, K. Munakata, T. Ohhara, M. Kakihana, Y. Haga, M. Hedo, T. Nakama, and Y. Ōnuki, Unique Helical Magnetic Order and Field-Induced Phase in Trillium Lattice Antiferromagnet EuPtSi, J. Phys. Soc. Jpn. 88, 013702 (2019).
  * (10) R. Takagi, N. Matsuyama, V. Ukleev, L. Yu, J. S. White, S. Francoual, J. R. L. Mardegan, S. Hayami, H. Saito, K. Kaneko, K. Ohishi, Y. Ōnuki, T. Arima, Y. Tokura, T. Nakajima, and S. Seki, Nat. Commun. 13, 1472 (2022).
  * (11) M. Gen, R. Takagi, Y. Watanabe, S. Kitou, H. Sagayama, N. Matsuyama, Y. Kohama, A. Ikeda, Y. Ōnuki, T. Kurumaji, T. Arima, and S. Seki, Rhombic skyrmion lattice coupled with orthorhombic structural distortion in EuAl4, Phys. Rev. B 107, L020410 (2023).
  * (12) D. Singh, Y. Fujishiro, S. Hayami, S. H. Moody, T. Nomoto, P. R. Baral, V. Ukleev, R. Cubitt, N.-J. Steinke, D. J. Gawryluk, E. Pomjakushina, Y. Ōnuki, R. Arita, Y. Tokura, N. Kanazawa, and J. S. White, Transition between distinct hybrid skyrmion textures through their hexagonal-to-square crystal transformation in a polar magnet, Nat. Commun. 14, 8050 (2023).
  * (13) T. Matsumura, K. Kurauchi, M. Tsukagoshi, N. Higa, H. Nakao, M. Kakihana, M. Hedo, T. Nakama, Y. Ōnuki, Distorted triangular skyrmion lattice in a noncentrosymmetric tetragonal magnet, arXiv:2306.14767.
  * (14) J. Jiang and S. M. Kauzlarich, Colossal Magnetoresistance in a Rare Earth Zintl Compound with a New Structure Type: EuIn2P2, Chem. Mater. 18, 435 (2006).
  * (15) J.-Z. Ma, S. M. Nie, C. J. Yi, J. Jandke, T. Shang, M. Y. Yao, M. Naamneh, L. Q. Yan, Y. Sun, A. Chikina, et al., Spin Fluctuation Induced Weyl Semimetal State in the Paramagnetic Phase of EuCd2As2, Sci. Adv. 5, eaaw4718 (2019).
  * (16) H. Su, B. Gong, W. Shi, H. Yang, H. Wang, W. Xia, Z. Yu. P.-J. Guo, J. Wang, L. Ding, L. Xu, X. Li, X. Wang, Z. Zou, N. Yu, Z. Zhu, Y. Chen, Z. Liu, K. Liu, G. Li, and Y. Guo, Magnetic exchange induced Weyl state in a semimetal EuCd2Sb2, APL Mater. 8, 011109 (2020).
  * (17) Z.-C. Wang, J. D. Rogers, X. Yao, R. Nichols, K. Atay, B. Xu, J. Franklin, I. Sochnikov, P. J. Ryan, D. Haskel, and F. Taft, Colossal Magnetoresistance without Mixed Valence in a Layered Phosphide Crystal, Adv. Mater. 33, 2005755 (2021).
  * (18) J. Blawat, M. Marshall, J. Singleton, E. Feng, H. Cao, W. Xie, and R. Jin, Unusual Electrical and Magnetic Properties in Layered EuZn2As2, Adv. Quantum Tech. 5, 2200012 (2022).
  * (19) S. Krebber, M. Kopp, C. Garg, K. Kummer, J. Sichelschmidt, S. Schulz, G. Poelchen, M. Mende, A. V. Virovets, K. Warawa, M. D. Thomson, A. V. Tarasov, D. Yu. Usachov, D. V. Vyalikh, H. G. Roskos, J. Müller, C. Krellner, and K. Kliemt, Colossal magnetoresistance in EuZn2P2 and its electronic and magnetic structure, Phys. Rev. B 108, 045116 (2023).
  * (20) G. Cuono, R. M. Sattigeri, C. Autieri, and T. Dietl, Ab initio overestimation of the topological region in Eu-based compounds, Phys. Rev. B 108, 075150 (2023).
  * (21) A. M. Essin, J. E. Moore, and D. Vanderbilt, Magnetoelectric polarizability and axion electrodynamics in crystalline insulators, Phys. Rev. Lett. 102, 146805 (2009).
  * (22) Y. Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, Higher-Order Topology of the Axion Insulator EuIn2As2, Phys. Rev. Lett. 122, 256402 (2019).
  * (23) Y. Tokura, K. Yasuda, and A. Tsukazaki, Magnetic topological insulators, Nat. Rev. Phys. 1, 126 (2019).
  * (24) B. A. Bernevig, C. Felser, and H. Beidenkopf, Progress and prospects in magnetic topological materials, Nature 603, 41 (2022).
  * (25) K. M. Fijalkowski, N. Liu, M. Hartl, M. Winnerlein, P. Mandal, A. Coschizza, A. Fothergill, S. Grauer, S. Schreyeck, K. Brunner, M. Greiter, R. Thomale, C. Gould, and L. W. Molenkamp, Any axion insulator must be a bulk three-dimensional topological insulator, Phys. Rev. B 103, 235111 (2021).
  * (26) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B 83, 205101 (2011).
  * (27) M. Mogi, M. Kawamura, R. Yoshimi, A. Tsukazaki, Y. Kozuka, N. Shirakawa, K. S. Takahashi, M. Kawasaki, and Y. Tokura, A magnetic heterostructure of topological insulators as a candidate for an axion insulator, Nat. Mater. 16, 516 (2017).
  * (28) D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Topological Axion States in the Magnetic Insulator MnBi2Te4 with the Quantized Magnetoelectric Effect, Intrinsic magnetic topological insulators in van der Waals layered MnBi2Te4-family materials, Phys. Rev. Lett. 122, 206401 (2019).
  * (29) J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan, and Y. Xu, Sci. Adv, 5, eaaw5685 (2019).
  * (30) Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, Y. Zhang, Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4, Science 367, 895 (2020).
  * (31) A. M. Goforth, P. Klavins, J. C. Fettinger, and S. M. Kauzlarich, Magnetic Properties and Negative Colossal Magnetoresistance of the Rare Earth Zintl phase EuIn2As2, Inorg. Chem. 47, 11048 (2008).
  * (32) Y. Zhang, K. Deng, X. Zhang, M. Wang, Y. Wang, C. Liu, J.-W. Mei, S. Kumar, E. F. Schwier, K. Shimada, C. Chen, and B. Shen, In-plane antiferromagnetic moments and magnetic polaron in the axion topological insulator candidate EuIn2As2, Phys. Rev. B 101, 205126 (2020).
  * (33) T. Sato, Z. Wang, D. Takane, S. Souma, C. Cui, Y. Li, K. Nakayama, T. Kawakami, Y. Kubota, C. Cacho, T. K. Kim, A. Arab, V. N. Strocov, Y. Yao, and T. Takahashi, Signature of band inversion in the antiferromagnetic phase of axion insulator candidate EuIn2As2, Phys. Rev. Res. 2, 033342 (2020).
  * (34) S. Regmi, M. M. Hosen, B. Ghosh, B. Singh, G. Dhakal, C. Sims, B. Wang, F. Kabir, K. Dimitri, Y. Liu, A. Agarwal, H. Lin, D. Kaczorowski, A. Bansil, and M. Neupane, Temperature-dependent electronic structure in a higher-order topological insulator candidate EuIn2As2, Phys. Rev. B 102, 165153 (2020).
  * (35) F. H. Yu, H. M. Mu, W. Z. Zhuo, Z. Y. Wang, Z. F. Wang, J. J. Ying, and X. H. Chen, Elevating the magnetic exchange coupling in the compressed antiferromagnetic axion insulator candidate EuIn2As2, Phys. Rev. B 102, 180404(R) (2020).
  * (36) M. Gong, D. Sar, J. Friedman, D. Kaczorowski, S. A. Razek, W.-C. Lee, and P. Aynajian, Surface state evolution induced by magnetic order in axion insulator candidate EuIn2As2, Phys. Rev. B 106, 125156 (2022).
  * (37) J. Yan, Z. Z. Jiang, R. C. Xiao, W. J. Lu, W. H. Song, X. B. Zhu, X. Luo, Y. P. Sun, and M. Yamashita, Field-induced topological Hall effect in antiferromagnetic axion insulator candidate EuIn2As2, Phys. Rev. Research 4, 013163 (2022).
  * (38) Q. Wu, T. Hu, D. Wu, R. Li, L. Yue, S. Zhang, S. Xu, Q. Liu, T. Dong, and N. Wang, Spin dynamics in the axion insulator candidate EuIn2As2, Phys. Rev. B 107, 174411 (2023).
  * (39) S. X. M. Riberolles, T. V. Trevisan, B. Kuthanazhi, T. W. Heitmann, F. Ye, D. C. Johnston, S. L. Bud’ko, D. H. Ryan, P. C. Canfield, A. Kreyssig, A. Vishwanath, R. J. McQueeney, L.-L. Wang, P. P. Orth, and B. G. Ueland, Magnetic crystalline-symmetry-protected axion electrodynamics and field-tunable unpinned Dirac cones in EuIn2As2, Nat. Commun. 12, 999 (2021).
  * (40) J.-R. Soh, A. Bombardi, F. Mila, M. C. Rahn, D. Prabhakaran, S. Francoual, H. M. Rønnow, and A. T. Boothroyd, Understanding unconventional magnetic order in a candidate axion insulator by resonant elastic x-ray scattering, Nat. Commun. 14, 3387 (2023).
  * (41) E. Donoway, T. V. Trevisan, A. Liebman - Peláez, R. P. Day, K. Yamakawa, Y. Sun, J. R. Soh, D. Prabhakaran, A. Boothroyd, R. M. Fernandes, J. G. Analytis, J. E. Moore, J. Orenstein, and V. Sunko, Symmetry-breaking pathway towards the unpinned broken helix, arXiv:2310.16018.
  * (42) See Supplemental Material at xxxx for details of the experiments and analyses, which includes Refs. RIETAN ; CrysAlis ; Jana .
  * (43) F. Izumi and K. Momma, Three-Dimensional Visualization in Powder Diffraction, Solid State Phenom. 130, 15 (2007).
  * (44) CrysAlisPro (Agilent Technologies Ltd, Yarnton, 2014).
  * (45) V. Petříček, M. Dušek, and L. Palatinus, Crystallographic computing system JANA2006: General features, Z. Kristallogr. Cryst. Mater. 229, 345 (2014).
  * (46) T. Kosugia, S. Kawanob, N. Achiwac, A. Onoderad, Y. Nakaie, and N. Yamamoto, Direct evidence of helifan structures in holmium by single crystal neutron diffraction, Physica B 334, 365 (2003).
  * (47) N. J. Ghimire, R. L. Dally, L. Poudel, D. C. Jones, D. Michel, N. Thapa Magar, M. Bleuel, M. A. McGuire, J. S. Jiang, J. F. Mitchell, J. W. Lynn, I. I. Mazin, Competing magnetic phases and fluctuation-driven scalar spin chirality in the kagome metal YMn6Sn6, Sci. Adv. 6, eabe2680 (2020).
  * (48) In the case of $q_{1z}=1/3$, the high-harmonic ${\mathbf{Q}}_{2}-2{\mathbf{Q}}_{1}$ and $3{\mathbf{Q}}_{1}$ positions accidentally overlap with the ${\mathbf{Q}}_{1}$ and ${\mathbf{Q}}_{2}$ positions, respectively.
  * (49) S. Hayami, R. Ozawa, and Y. Motome, Effective bilinear-biquadratic model for noncoplanar ordering in itinerant magnets, Phys. Rev. B 95, 224424 (2017).


# Curious convergent series of integers with missing digits

Melvyn B. Nathanson  Lehman College (CUNY), Bronx, NY 10468
<EMAIL_ADDRESS>To Ron Graham

###### Abstract.

A classical theorem of Kempner states that the sum of the reciprocals of
positive integers with missing decimal digits converges. This result is
extended to much larger families of “missing digits” sets of positive integers
with both convergent and divergent harmonic series.

###### Key words and phrases:

Integers with missing digits, harmonic series

###### 2010 Mathematics Subject Classification:

11A63, 11B05, 11B75, 11K16.

##  1\. Kempner’s theorem

> “It is well known that the series
>
> $\sum_{n=1}^{\infty}\frac{1}{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots$
>
> diverges. The object of this Note is to prove that if the denominators do
> not include all natural numbers 1, 2, 3, …, but only those which do not
> contain any figure 9, the series converges. The method of proof holds
> unchanged if, instead of 9, any other figure 1, 2, …, 8 is excluded, but not
> for the figure 0.”
>
> A. J. Kempner, _Amer. Math. Monthly_ 21 (1914), 48–50

A _harmonic series_ is a series of the form $\sum_{a\in A}1/a$, where $A$ is a
set of positive integers. Mathematicians have long been interested in the
convergence or divergence of harmonic series. Let $c\in\\{1,2,\ldots,9\\}$,
and let $A_{10}(c)$ be the set of positive integers in which the digit $c$
does not occur in the usual decimal representation. Kempner [6] proved in 1914
that $\sum_{a\in A_{10}(c)}1/a$ converges. He called this “a curious
convergent series.” More generally, for every integer $g\geq 2$, every
positive integer $n$ has a unique $g$-adic representation of the form
$n=\sum_{i=0}^{k}c_{i}g^{i}$, with digits $c_{i}\in\\{0,1,2,\ldots,g-1\\}$ for
$i=0,1,\ldots,k$ and $c_{k}\neq 0$. If $A_{g}(c)$ is the set of integers whose
$g$-adic representation contains no digit $c$, then the infinite series
$\sum_{a\in A_{g}(c)}1/a$ converges. This includes the case $c=0$, which was
not discussed by Kempner.

Kempner’s theorem has been studied and extended by Baillie [1], Farhi [2],
Gordon [3], Irwin [5], Lubeck-Ponomarenko [7], Schmelzer and Baillie [10], and
Wadhwa [11, 12]. It is Theorem 144 in Hardy and Wright [4].

The $g$-adic representation is a special case of a more general method to
represent the positive integers. A _$\mathcal{G}$ -adic sequence_ is a
strictly increasing sequence of positive integers
$\mathcal{G}=(g_{i})_{i=0}^{\infty}$ such that $g_{0}=1$ and $g_{i}$ divides
$g_{i+1}$ for all $i\geq 0$. The integer quotients

$d_{i}=\frac{g_{i+1}}{g_{i}}$

satisfy $d_{i}\geq 2$ and

(1) $g_{k+1}=g_{k}d_{k}=d_{0}d_{1}d_{2}\cdots d_{k}$

for all $k\geq 0$. Every positive integer $n$ has a unique representation in
the form

(2) $n=\sum_{i=0}^{k}c_{i}g_{i}$

where $c_{i}\in\\{0,1,\ldots,d_{i}-1\\}$ for all $i\in\\{0,1,\ldots,k\\}$ and
$c_{k}\neq 0$. We call (2) the $\mathcal{G}$-adic representation of $n$. This
is equivalent to de Bruijn’s additive system (Nathanson [8, 9]).

Harmonic series constructed from sets of positive integers with missing
$\mathcal{G}$-adic digits do not necessarily converge. In Theorem 1 we
construct sets of integers with missing $\mathcal{G}$-adic digits whose
harmonic series converge, and also sets of integers with missing
$\mathcal{G}$-adic digits whose harmonic series diverge.

##  2\. $\mathcal{G}$-adic representations with bounded quotients

Define the interval of integers

$[a,b]=\\{n\in\mathbf{Z}:a\leq n\leq b\\}.$

Let $\mathcal{G}=(g_{i})_{i=0}^{\infty}$ be a $\mathcal{G}$-adic sequence with
quotients $d_{i}=g_{i+1}/g_{i}$. Let $I$ be a set of nonnegative integers,
and, for all $i\in I$, let $U_{i}$ be a nonempty proper subset of
$[0,d_{i}-1]$. For every nonnegative integer $k$, let $A_{k}$ be the set of
integers $n\in[g_{k},g_{k+1}-1]$ whose $\mathcal{G}$-adic representation
$n=\sum_{i=0}^{k}c_{i}g_{i}$ satisfies the following missing digits condition:

(3) $c_{i}\in[0,d_{i}-1]\setminus U_{i}\qquad\text{for all $i\in
I\cap[0,k]$.}$

######  Lemma 1.

The set $A_{k}$ satisfies:

  1. (a)

$A_{k}=\emptyset$ if and only if $k\in I$ and $U_{k}=[1,d_{k}-1]$.

  2. (b)

If $A_{k}\neq\emptyset$, then

(4) $|A_{k}|\leq\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\ i\notin
I\end{subarray}}^{k}d_{i}\leq 2|A_{k}|.$

###### Proof.

We use the inequality

(5) $x\leq 2(x-1)\qquad\text{for $x\geq 2$.}$

If $n\in[g_{k},g_{k+1}-1]$, then $n$ has the $\mathcal{G}$-adic representation

$n=\sum_{i=0}^{k-1}c_{i}g_{i}+c_{k}g_{k}$

with $c_{k}\neq 0$ and so $c_{k}\in[1,d_{k}-1]$. It follows that
$A_{k}=\emptyset$ if and only if $k\in I$ and $U_{k}=[1,d_{k}-1]$.

For $A_{k}\neq\emptyset$, there are three cases.

  1. (i)

If $k\in I$ and $0\in U_{k}$, then

$|A_{k}|=\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\ i\notin
I\end{subarray}}^{k}d_{i}<2|A_{k}|.$

  2. (ii)

If $k\in I$ and and $0\notin U_{k}$, then $U_{k}$ is a proper subset of
$[1,d_{k}-1]$ and so $|U_{k}|\leq d_{k}-2$. Inequality (5) gives

$d_{k}-|U_{k}|\leq 2\left(d_{k}-|U_{k}|-1\right).$

We obtain

$\displaystyle|A_{k}|$
$\displaystyle=(d_{k}-|U_{k}|-1)\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k-1}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\ i\notin
I\end{subarray}}^{k}d_{i}$ $\displaystyle<\prod_{\begin{subarray}{c}i=0\\\
i\in I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\
i\notin I\end{subarray}}^{k}d_{i}$ $\displaystyle\leq
2(d_{k}-|U_{k}|-1)\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k-1}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\ i\notin
I\end{subarray}}^{k}d_{i}$ $\displaystyle=2|A_{k}|.$

  3. (iii)

We have $d_{k}\geq 2$ and so $d_{k}\leq 2(d_{k}-1)$ from inequality (5). If
$k\notin I$, then

$\displaystyle|A_{k}|$
$\displaystyle=(d_{k}-1)\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\ i\notin
I\end{subarray}}^{k-1}d_{i}<\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\ i\notin
I\end{subarray}}^{k}d_{i}$
$\displaystyle=d_{k}\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\ i\notin
I\end{subarray}}^{k-1}d_{i}\leq 2(d_{k}-1)\prod_{\begin{subarray}{c}i=0\\\
i\in I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\
i\notin I\end{subarray}}^{k-1}d_{i}$ $\displaystyle=2|A_{k}|.$

This completes the proof. ∎

Let $A$ be a set of nonnegative integers, and let $A(n)$ be the number of
elements $a\in A$ with $a\leq n$. The upper asymptotic density of the set $A$
is $d_{U}(A)=\limsup_{n\rightarrow\infty}A(n)/n$. If
$\lim_{n\rightarrow\infty}A(n)/n$ exists, then
$d(A)=\lim_{n\rightarrow\infty}A(n)/n$ is called the asymptotic density of the
set $A$. The set $A$ has _asymptotic density zero_ if $d(A)=d_{U}(A)=0$.

######  Lemma 2.

Let $A$ be a set of positive integers. If $\sum_{a\in A}1/a<\infty$, then
$d(A)=0$.

###### Proof.

We show that $d_{U}(A)>0$ implies $\sum_{a\in A}1/a=\infty$.

If $d_{U}(A)=\limsup_{n\rightarrow\infty}A(n)/n=\alpha>0$, then, for every
$\varepsilon>0$, we have

$\frac{A(n)}{n}<\alpha+\varepsilon\qquad\text{for all integers $n\geq
N(\varepsilon)$}$

and

(6) $\frac{A(n_{i})}{n_{i}}>\alpha-\varepsilon\qquad\text{for infinitely many
integers $n_{i}$.}$

Let $\varepsilon<\alpha/3$. There is a sequence of positive integers
$(n_{i})_{i=0}^{\infty}$ satisfying inequality (6) such that $n_{0}\geq
N(\varepsilon)$ and $n_{i}>2n_{i-1}$ for all $i\geq 1$. We have

$\displaystyle A(n_{i})-A(n_{i-1})$
$\displaystyle>(\alpha-\varepsilon)n_{i}-(\alpha+\varepsilon)n_{i-1}$
$\displaystyle>(\alpha-\varepsilon)n_{i}-\frac{(\alpha+\varepsilon)n_{i}}{2}$
$\displaystyle=n_{i}\left(\frac{\alpha-3\varepsilon}{2}\right)$

and so

$\displaystyle\sum_{\begin{subarray}{c}a\in A\\\ n_{i-1}<a\leq
n_{i}\end{subarray}}\frac{1}{a}$
$\displaystyle\geq\frac{A(n_{i})-A(n_{i-1})}{n_{i}}>\frac{\alpha-3\varepsilon}{2}>0.$

It follows that

$\sum_{\begin{subarray}{c}a\in A\\\ 1\leq a\leq
n_{k}\end{subarray}}\frac{1}{a}\geq\sum_{i=1}^{k}\sum_{\begin{subarray}{c}a\in
A\\\ n_{i-1}<a\leq
n_{i}\end{subarray}}\frac{1}{a}>k\left(\frac{\alpha-3\varepsilon}{2}\right)$

and the infinite series $\sum_{a\in A}1/a$ diverges. Equivalently, convergence
of the infinite series $\sum_{a\in A}1/a$ implies $d(A)=0$. This completes the
proof. ∎

The converse of Lemma 2 is false. The set of prime numbers has asymptotic
density zero, but the sum of the reciprocals of the primes diverges.

######  Theorem 1.

Let $\mathcal{G}=(g_{i})_{i=0}^{\infty}$ be a $\mathcal{G}$-adic sequence with
bounded quotients, that is,

(7) $d_{i}=\frac{g_{i+1}}{g_{i}}\leq d$

for some integer $d\geq 2$ and all $i=0,1,2,\ldots$. Let $I$ be a set of
nonnegative integers, and, for all $i\in I$, let $U_{i}$ be a nonempty proper
subset of $[0,d_{i}-1]$.

Let $n=\sum_{i=0}^{k}c_{i}g_{i}$ be the $\mathcal{G}$-adic representation of
the positive integer $n$. Let $A$ be the set of positive integers $n$ that
satisfy the missing digits condition (3). If

(8) $I(k)\geq\frac{(1+\delta)\log k}{\log(d/(d-1))}$

for some $\delta>0$ and all $k\geq k_{0}=k_{0}(\delta)$, then the set $A$ has
asymptotic density zero and the harmonic series $\sum_{a\in A}1/a$ converges.

If

(9) $I(k)\leq\frac{(1-\delta)\log k}{\log d}$

for some $\delta>0$ and all $k\geq k_{1}=k_{1}(\delta)$, then the harmonic
series $\sum_{a\in A}1/a$ diverges.

Kempner’s theorem is the special case $g_{i}=10^{i}$, $d_{i}=10$, and
$U_{i}=\\{9\\}$ for all $i\in I=\mathbf{N}_{0}$.

###### Proof.

For all $k\in\mathbf{N}_{0}$, the finite sets

$A_{k}=A\cap[g_{k},g_{k+1}-1]$

are pairwise disjoint and $A=\bigcup_{k=0}^{\infty}A_{k}.$

For all $i\in I$, we have

$1\leq|U_{i}|\leq d_{i}-1$

and

$\frac{1}{d}\leq\frac{1}{d_{i}}\leq\frac{d_{i}-|U_{i}|}{d_{i}}=1-\frac{|U_{i}|}{d_{i}}\leq
1-\frac{1}{d}<1.$

Let $I(k)$ satisfy inequality (8). We obtain

(10)
$\left(1-\frac{1}{d}\right)^{I(k)}\leq\left(\frac{d-1}{d}\right)^{\frac{(1+\delta)\log
k}{\log(d/(d-1))}}=\frac{1}{k^{1+\delta}}.$

If $a\in A_{k}$, then $a\geq g_{k}=d_{0}d_{1}\cdots d_{k-1}$. By Lemma 1,

$\displaystyle\sum_{\begin{subarray}{c}a\in A\\\ a\geq
g_{k_{0}}\end{subarray}}\frac{1}{a}$
$\displaystyle=\sum_{k=k_{0}}^{\infty}\sum_{a\in
A_{k}}\frac{1}{a}\leq\sum_{k=k_{0}}^{\infty}\frac{|A_{k}|}{g_{k}}$
$\displaystyle\leq\sum_{k=k_{0}}^{\infty}\frac{d_{k}}{\prod_{i=0}^{k}d_{i}}\prod_{\begin{subarray}{c}i=0\\\
i\in I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\
i\notin I\end{subarray}}^{k}d_{i}$ $\displaystyle\leq
d\sum_{k=k_{0}}^{\infty}\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k}\frac{d_{i}-|U_{i}|}{d_{i}}$ $\displaystyle\leq
d\sum_{k=k_{0}}^{\infty}\left(1-\frac{1}{d}\right)^{I(k)}$ $\displaystyle\leq
d\sum_{k=k_{0}}^{\infty}\frac{1}{k^{1+\delta}}<\infty.$

Thus, the harmonic series converges. By Lemma 2, the set $A$ has asymptotic
density zero.

Let $I(k)$ satisfy inequality (9). We obtain

(11)
$\left(\frac{1}{d}\right)^{I(k)}\geq\left(\frac{1}{d}\right)^{\frac{(1-\delta)\log
k}{\log d}}=\frac{1}{k^{1-\delta}}.$

If $a\in A_{k}$, then $a<g_{k+1}=d_{0}d_{1}\cdots d_{k-1}d_{k}$. By Lemma 1,

$\displaystyle\sum_{\begin{subarray}{c}a\in A\\\ a\geq
g_{k_{1}}\end{subarray}}\frac{1}{a}$
$\displaystyle=\sum_{k=k_{1}}^{\infty}\sum_{a\in
A_{k}}\frac{1}{a}\geq\sum_{k=k_{1}}^{\infty}\frac{|A_{k}|}{g_{k+1}}$
$\displaystyle\geq\frac{1}{2}\sum_{k=k_{1}}^{\infty}\frac{1}{\prod_{i=0}^{k}d_{i}}\prod_{\begin{subarray}{c}i=0\\\
i\in I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\
i\notin I\end{subarray}}^{k}d_{i}$
$\displaystyle=\frac{1}{2}\sum_{k=k_{1}}^{\infty}\prod_{\begin{subarray}{c}i=0\\\
i\in
I\end{subarray}}^{k}\frac{d_{i}-|U_{i}|}{d_{i}}\geq\frac{1}{2}\sum_{k=k_{1}}^{\infty}\prod_{\begin{subarray}{c}i=0\\\
i\in I\end{subarray}}^{k}\frac{1}{d_{i}}$
$\displaystyle\geq\frac{1}{2}\sum_{k=k_{1}}^{\infty}\left(\frac{1}{d}\right)^{I(k)}$
$\displaystyle\geq\frac{1}{2}\sum_{k=k_{1}}^{\infty}\frac{1}{k^{1-\delta}}$

and the harmonic series diverges. This completes the proof. ∎

######  Corollary 1.

Let $I$ be a set of nonnegative integers, and let $(v_{i})_{i\in I}$ be a
sequence of 0s and 1s. Let $A$ be the set of integers $n$ such that, if
$n\in\left[2^{k},2^{k+1}-1\right]$ has the $2$-adic representation
$n=\sum_{i=0}^{k}c_{i}2^{i}$, then $c_{i}=v_{i}$ for all $i\in I\cap[0,k]$. If

$I(k)\geq(1+\delta)\log_{2}k$

for some $\delta>0$ and all $k\geq k_{0}(\delta)$, then the harmonic series
$\sum_{a\in A}1/a$ converges. If

$I(k)\leq(1-\delta)\log_{2}k$

for some $\delta>0$ and all $k\geq k_{1}(\delta)$, then the harmonic series
$\sum_{a\in A}1/a$ diverges.

###### Proof.

For all $i\in I$, let $u_{i}=1-v_{i}$ and $U_{i}=\\{u_{i}\\}$. Apply Theorem
1. ∎

It is an open problem to determine the convergence or divergence of
$\sum_{a\in A}1/a$ if $I(k)\sim\log_{2}k$.

##  3\. $\mathcal{G}$-adic representations with unbounded quotients

Let $\mathcal{G}=(g_{i})_{i=0}^{\infty}$ be a $\mathcal{G}$-adic sequence with
quotients $d_{i}=g_{i+1}/g_{i}$. Let $I$ be an infinite set of nonnegative
integers, and, for all $i\in I$, let $U_{i}$ be a nonempty proper subset of
$[0,d_{i}-1]$. If the sequence $\mathcal{G}=(g_{i})_{i=0}^{\infty}$ has
bounded quotients $d_{i}\leq d$, then

$\frac{|U_{i}|}{d_{i}}\geq\frac{1}{d}$

for all $i\in I$ and the infinite series $\sum_{i\in I}\frac{|U_{i}|}{d_{i}}$
diverges. Equivalently, the convergence of this series implies that
$\mathcal{G}$ has unbounded quotients.

Let $n=\sum_{i=0}^{k}c_{i}g_{i}$ be the $\mathcal{G}$-adic representation of
the positive integer $n$. Let $A$ be the set of positive integers whose
$\mathcal{G}$-adic representations satisfy the missing digits condition (3).
The missing digits set $A$ is finite if and only if $I$ is a cofinite set of
nonnegative integers and $U_{i}=[1,d_{i}-1]$ for all sufficiently large $i$.
The harmonic series of a finite set of positive integers converges.

Theorem 1 shows that harmonic series constructed from infinite sets of
integers with missing $\mathcal{G}$-adic digits do not always converge. It
follows from Theorem 1 that if

$I(k)\geq(\log k)^{1+\delta}$

for some $\delta>0$ and all $k\geq k_{0}(\delta)$, and if $\sum_{a\in A}1/a$
diverges, then the sequence $\mathcal{G}$ must have _unbounded quotients_ ,
that is,

$\limsup d_{i}=\infty.$

Theorem 2 gives a sufficient condition for the divergence of harmonic series
of sets of positive integers constructed from $\mathcal{G}$-adic sequences
with unbounded quotients. We use the following inequality, which is easily
proved by induction: If $0\leq x_{i}<1$ for $i=1,\ldots,n$, then

(12) $\prod_{i=1}^{n}(1-x_{i})\geq 1-\sum_{i=1}^{n}x_{i}.$

######  Theorem 2.

Let $\mathcal{G}=(g_{i})_{i=0}^{\infty}$ be a $\mathcal{G}$-adic sequence, and
let $n=\sum_{i=0}^{k}c_{i}g_{i}$ be the $\mathcal{G}$-adic representation of
the positive integer $n$. Let $I$ be an infinite set of nonnegative integers,
and, for all $i\in I$, let $U_{i}$ be a nonempty proper subset of
$[0,d_{i}-1]$. Let $A$ be the set of positive integers whose
$\mathcal{G}$-adic representations satisfy the missing digits condition (3).
If the set $A$ is infinite and if

(13) $\sum_{i\in I}\frac{|U_{i}|}{d_{i}}<\infty$

then the sequence $\mathcal{G}=(g_{i})_{i=0}^{\infty}$ has unbounded quotients
and the harmonic series $\sum_{a\in A}1/a$ diverges.

For example, the “missing digits” set constructed from
$\mathcal{G}=(g_{i})_{i=0}^{\infty}$ with $g_{i}=2^{i(i+1)/2}$ and
$d_{i}=2^{i+1}$ and with $I=\mathbf{N}_{0}$ and $U_{i}=\\{0\\}$ for all $i\in
I$ has a divergent harmonic series.

###### Proof.

Because the infinite series (13) converges, there is an integer $i_{0}\in I$
such that

$\sum_{\begin{subarray}{c}i=i_{0}\\\ i\in I\end{subarray}}^{\infty}\
\frac{|U_{i}|}{d_{i}}<\frac{1}{2}.$

Inequality (12) implies that, for all $k\in\mathbf{N}_{0}$,

$\prod_{\begin{subarray}{c}i=i_{0}\\\ i\in I\end{subarray}}^{k}\
\left(1-\frac{|U_{i}|}{d_{i}}\right)\geq 1-\sum_{\begin{subarray}{c}i=i_{0}\\\
i\in I\end{subarray}}^{k}\frac{|U_{i}|}{d_{i}}>\frac{1}{2}$

and so

$\displaystyle\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{k}\left(1-\frac{|U_{i}|}{d_{i}}\right)$
$\displaystyle=\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{i_{0}-1}\left(1-\frac{|U_{i}|}{d_{i}}\right)\prod_{\begin{subarray}{c}i=i_{0}\\\
i\in I\end{subarray}}^{k}\left(1-\frac{|U_{i}|}{d_{i}}\right)$
$\displaystyle>\frac{1}{2}\prod_{\begin{subarray}{c}i=0\\\ i\in
I\end{subarray}}^{i_{0}-1}\left(1-\frac{|U_{i}|}{d_{i}}\right)=\delta>0.$

Let $A_{k}=A\cap[g_{k},g_{k+1}-1]$. The set $A$ is infinite if and only if
$A_{k}\neq\emptyset$ for infinitely many $k$. Applying inequality (4) of Lemma
1, we obtain

$\displaystyle\sum_{a\in A}\frac{1}{a}$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\
A_{k}\neq\emptyset\end{subarray}}^{\infty}\sum_{a\in
A_{k}}\frac{1}{a}\geq\sum_{\begin{subarray}{c}k=0\\\
A_{k}\neq\emptyset\end{subarray}}^{\infty}\frac{|A_{k}|}{g_{k+1}}$
$\displaystyle\geq\frac{1}{2}\sum_{\begin{subarray}{c}k=0\\\
A_{k}\neq\emptyset\end{subarray}}^{\infty}\frac{1}{\prod_{i=0}^{k}d_{i}}\prod_{\begin{subarray}{c}i=0\\\
i\in I\end{subarray}}^{k}(d_{i}-|U_{i}|)\prod_{\begin{subarray}{c}i=0\\\
i\notin I\end{subarray}}^{k}d_{i}$
$\displaystyle=\frac{1}{2}\sum_{\begin{subarray}{c}k=0\\\
A_{k}\neq\emptyset\end{subarray}}^{\infty}\prod_{\begin{subarray}{c}i=0\\\
i\in I\end{subarray}}^{k}\left(1-\frac{|U_{i}|}{d_{i}}\right)$

and so the harmonic series $\sum_{a\in A}\frac{1}{a}$ diverges. This completes
the proof. ∎

## References

  * [1] R. Baillie, _Sums of reciprocals of integers missing a given digit_ , Amer. Math. Monthly 86 (1979), no. 5, 372–374.
  * [2] B. Farhi, _A curious result related to Kempner’s series_ , Amer. Math. Monthly 115 (2008), no. 10, 933–938.
  * [3] R. A. Gordon, _Comments on “Subsums of the harmonic series”_ , Amer. Math. Monthly 126 (2019), no. 3, 275–279.
  * [4] G. H. Hardy and E. M. Wright, _An Introduction to the Theory of Numbers_ , 6th ed., Oxford University Press, Oxford, 2008.
  * [5] F. Irwin, _A curious convergent series_ , Amer. Math. Monthly 23 (1916), no. 5, 149–152.
  * [6] A. J. Kempner, _A curious convergent series_ , Amer. Math. Monthly 21 (1914), no. 2, 48–50.
  * [7] B. Lubeck and V. Ponomarenko, _Subsums of the harmonic series_ , Amer. Math. Monthly 125 (2018), no. 4, 351–355.
  * [8] M. B. Nathanson, _Additive systems and a theorem of de Bruijn_ , Amer. Math. Monthly 121 (2014), no. 1, 5–17.
  * [9] by same author, _Limits and decomposition of de Bruijn’s additive systems_ , Combinatorial and Additive Number Theory II (New York), Springer Proc. Math. Stat., vol. 220, Springer, 2017, pp. 255–267.
  * [10] T. Schmelzer and R. Baillie, _Summing a curious, slowly convergent series_ , Amer. Math. Monthly 115 (2008), no. 6, 525–540.
  * [11] A. D. Wadhwa, _An interesting subseries of the harmonic series_ , Amer. Math. Monthly 82 (1975), no. 9, 931–933.
  * [12] by same author, _Some convergent subseries of the harmonic series_ , Amer. Math. Monthly 85 (1978), no. 8, 661–663.