# Velocity-aware Antenna Selection in
Predictor Antenna Systems

Hao Guo  Department of Electrical Engineering
Chalmers University of Technology
Gothenburg, Sweden
<EMAIL_ADDRESS>Behrooz Makki  Department of Ericsson Research
Ericsson Research
Gothenburg, Sweden
<EMAIL_ADDRESS>Tommy Svensson  Department of Electrical
Engineering
Chalmers University of Technology
Gothenburg, Sweden
<EMAIL_ADDRESS>
###### Abstract

Moving relay (MR), which is a candidate solution for supporting in-vehicle
users, has been investigated in different studies. Due to the mobile nature of
the MR, acquiring channel state information at the transmitter side (CSIT) is
challenging because of the fast-changing environment around the vehicle. On
top of an MR, one can use predictor antenna (PA), i.e., an additional antenna
in front of the receive antenna (RA), to obtain CSIT, and recent works have
investigated the benefits of such a set up. PA-aided CSIT acquisition normally
works with the help of different content information such as the location and
the velocity of the MR. In this paper, we study the effect of velocity
awareness on the PA system, and develop adaptive antenna selection schemes in
PA-assisted MRs. Results show that, compared to no-CSIT schemes, a velocity-
aware antenna selection-based PA system can improve the end-to-end throughput
by an order of magnitude.

###### Index Terms:

6G, channel estimation, context-aware communication, integrated access and
backhaul (IAB), moving relay, predictor antenna, throughput, vehicle-to-
everything (V2X), wireless backhaul.

##  I Introduction

As a complement to fixed relays, moving relay (MR) has been discovered to be a
candidate method to serve mobile users [1, 2]. MRs can improve the quality of
service (QoS) of in-vehicle users by reducing the vehicle penetration loss
(VPL) and performing group handover. One main challenge for MR is, however,
how to obtain accurate channel state information at the transmitter side
(CSIT) because the relay itself is naturally moving. To overcome this issue,
the concept of predictor antenna (PA) has been first proposed in [3] in which
two sets of antennas are deployed on top of a vehicle. The PA positioned in
the front is used for predicting the CSIT for the antennas behind it, referred
to as receive antenna (RA), who will later encounter a similar channel as the
PA encountered several time slots before.

Various testbed-based studies have validated the PA concept together with
advanced technologies such as massive multiple input multiple output (MIMO)
[4] and low-pass filtering [5] at high velocity. It is shown in [6] that the
PA system can be beneficial in both time division duplex (TDD) and FDD (F:
Frequency) setups with line-of-sight (LoS) and NLoS (N: non) channels.
Moreover, using Kalman filtering to interpolate for the TDD uplink estimation
is studied in [7].

The PA system, however, may face spatial mismatch, i.e., the RA does not reach
the same spatial point as the PA, due to varying vehicle velocity or the
processing delay at the base station (BS) [2, 8]. That is, for the acquired
CSIT to be accurate, the RA needs to be as close as possible to the point
where the PA has estimated the channel. Such an alignment scheme, and/or
alternative methods (e.g., rate adaptation in [8]) require knowledge of
vehicle velocity, control loop time, as well as antenna spacing. [8] focuses
on SISO (S: Single) setup and proposes analytical models of the PA system with
spatial mismatch, while the effect of multiple antennas at both transceiver
sides has not been fully investigated.

In this paper, we first review the PA concept and give a comprehensive
literature study of the PA. Then, we study the data transmission efficiency
and reliability of PA systems with multiple antennas at the MRs. Specifically,
we investigate the effect of knowing the velocity on the system performance.
Also, we design an adaptive scheme to choose the best RA based on the vehicle
velocity, such that spatial mismatch is minimized. Finally, we study the
effect of different parameters such as the antennas distance and the vehicle
velocity on the performance of PA systems. The results show that, using
adaptive antenna selection and rate adaptation, there is potential for
improving the performance of the PA system.

##  II PA Concept and Literature Review

Thanks to the flexible deployment and the VPL elimination, MR has the
potential to become to one important component in mobile networks [1, 2, 9].
The benefit of deploying MR can be mainly divided into two aspects:

  * •

Eliminating VPL for in-vehicle users: The users inside the vehicles such as
commuters normally expect similar, uninterrupted mobile service as they have
in home or office. However, vehicle itself forms a natural blockage to the
radio signal propagation which leads to high VPL [10]. Thus, VPL can
drastically degrade the signal strength and as a result, affect the received
QoS remarkably. Results in [10] reveal that the VPL is around 25 dB at sub-6
GHz and it could definitely be worse at higher frequencies such as millimeter
wave (mmw) bands, due to the fact that mmw signals are more sensitive to
propagation blockers. Using MR with both outside-vehicle and inside-vehicle
modules, the VPL can almost be eliminated and the users inside the vehicle
could experience a nearly pure LoS condition [1].

  * •

Increasing the coverage of cellular networks for out-vehicle users: As one
type of relay with relatively easy deployment, MR can also be used for
extending the coverage of macro-cell network by cooperative relying [11, 12].
By joint transmission with macro-cell BSs, the MR-aided system can remarkably
improve the coverage probability [11]. Indeed, compared to serving the in-
vehicle user equipments, data transmission to out-vehicle UEs is of lower
priority, unless for, e.g., public safety use-cases where the MR is used for
coverage extension.

In both applications of the MR, outside-vehicle backhaul links have attracted
more attention from researchers than inside-vehicle access links since the
later one is normally LoS with low design complexity. Also, wireless backhaul
is more suitable to deploy in outdoor MRs because of its flexibility and low
cost. Naturally, the design of the wireless backhaul link becomes the
bottleneck of the system capacity in order to serve many MR-assisted users
[1].

Using advanced transmission schemes at the BS side such as MIMO, Coordinated
Multi-Point (CoMP) joint transmission, and different rate allocation
algorithms can improve the performance of the MR backhaul link. Among all
these techniques CSIT plays an important role since the quality of CSIT
affects the system performance directly. On the other hand, with moving
vehicle as MR, the environment around changes quickly and the channel
estimation delay makes the CSIT at the BS side outdated quickly. The situation
becomes more severe at higher moving speed and frequency bands.

One can use diversity-based transmission schemes at the cost of extra
resources. Also, using location information can help with CSIT estimation but
the location uncertainty needs to be handled carefully [13]. Alternatively,
using channel predictors such as Kalman or Wiener can obtain the prediction
range of 0.1-0.3 times the wavelength, which is sufficient for low/moderate
speed vehicles or pedestrians at cm-wave carrier frequencies [3].
Nevertheless, these predictors become inefficient when speed and frequency
increases. Compensate by increasing the number of pilots and perform channel
interpolation would introduce unrealistic overhead.

As one promising setup aiming at acquiring better CSIT condition for MRs, the
PA setup was firstly proposed in 2012 [3], where two groups of antennas are
mounted on the top of a vehicle. The front one w.r.t. moving direction is
dedicated as PA(s) which is(are) used for channel sensing and estimation and
the obtained channel information is sent back to the BS. Then, in the next
time slot(s), the vehicle moves forward and the antennas behind PA, denoted as
RAs, would finally reach the same spatial point as the PAs. In this way, if
the processing delay at the BS is designed properly based on, e.g.,
wavelength, antenna separations and vehicle speeds, the RAs could receive the
signal at the point where the PA used to estimate the channel. By doing so the
CSIT for the RAs can achieve high accuracy and as a result, the system is
capable of utilizing various advanced transmission schemes to improve the QoS.

In [3], compared with Kalman filter-based prediction, it is shown that a PA-
assisted system can give decent CSIT accuracy in both LoS and non-line-of-
sight (NLoS) conditions with prediction range being around $\lambda$ and
$\lambda/2$, respectively, with $\lambda$ being the wavelength. Following [3],
different testbed- and simulation-based studies have investigated and revealed
the potential of the PA. [1] introduces dedicated MR links with deployment of
the PA, and studies the effect of VPL by system-level simulations. Using
multiple antennas at the BS, [14] design a downlink multiple input single
output (MISO) system with PA interpolation, in order to mitigate residual
beamforming (BF) mis-pointing. From antenna design perspective, [15] focus on
the PA-RA correction and performance coupling compensation to enhance the
performance. Using antenna coupling compensation, at least three times
$\lambda$ can be predicted with accuracy above 96% cross-correlation between
the PA and RA channels [15]. With early development of 5G, [16] investigates
optimal required number of PAs/RAs on the top of a vehicle in terms of
spectrum efficiency and power, at both 1.2 and 3.5 GHz.

Besides modeling and simulations, testbed-based research studies have been
performed to validate the feasibility of the PA concept. Part of [15] is
dedicated to the demonstration of the proposed coupling compensation in
downtown Dresden, Germany. Results in [15] indicate the superior performance
of using quarter-wavelength monopoles rather than half-wavelength dipoles in
the PA system. Then, in 2017, a testbed study was performed at 2.53 GHz in
Dresden, verifying that the prediction horizon of the PA system can be at
least up to three times the wavelength [5]. As an extension of [5], [6]
proposes interpolation-based PA estimation scheme and verifies it with
practical measurements. Finally, with massive MIMO, [4] evaluates the
performance of the PA system at 2.18 GHz, and the large gains on the signal-
to-interference ratio is a promising indication of the feasibility of
utilizing the PA in multiple antenna systems.

Given that 1) the PA system is highly sensitive to spatial mismatch, i.e., the
RAs could not reach the same point as the PAs; 2) The PA system does not have
analytical model to obtain tractable evaluations, recent studies [17, 8, 18,
19, 20, 2, 21] address these problem and further clarify the potential of the
PA system. [17] investigates the spatial mismatch problem in the PA system,
and proposes a tractable channel model to evaluate the effect of spatial
variation with rate adaptation. Then, the model in [17] has been further
evaluated in [8] with temporal evolution of the channel. With two sets of
antennas on top of a vehicle, the potential of involving PA partially into the
transmission have been studied in [18, 19], where hybrid automatic repeat
request (HARQ) is combined with system and optimal power [18] and average rate
[19] of the HARQ-based PA system are investigated respectively. Considering
finite block-length transmission, [20] studies the effect of the codeword
length on the average throughput and error probability. Also, [2] reviews the
recent progress on the PA study and proposes adaptive/non-adaptive delay
methods to be applied in the PA system with spatial mismatch. The role of the
PA in the development of MR and integrated access and backhaul (IAB) are also
summarised. Finally, [21] extends the PA concept to multiple vehicles and
investigates the potential of using PA for dynamic blockage avoidance in
cooperative internet-of-vehicle networks.

In [17, 8, 18, 19, 20, 2, 21], the speed of vehicle is assumed to be known at
the BS, which may not be practical in real applications. Alternatively, using
the speed information at the vehicle side and perform antenna selection could
potentially enhance the PA system at the presence of spatial mismatch.

##  III Model and Method

Figure 1: The illustration of the PA system with spatial mismatch.

Considering a setup illustrated in Fig. 1, a moving vehicle is deployed with
one PA and multiple RAs on its top. The separation between the PA and $i$-th
RA is $d_{\text{a},i}$. First, let’s consider the case with one RA to
introduce the model, i.e., the separation of the PA and the RA is
$d_{\text{a}}$, as used in, e.g., [8]. At time $t_{1}$, the PA estimates the
channel and sends the channel information back to the BS. Then, at $t_{2}$,
depending on, e.g., $d_{\text{a}}$, speed $v$, and the processing time
$t_{\text{BS}}$ at the BS, the RAs may end up in different places. In this
way, the mismatch distance $d$ between

  * •

the place where the PA estimates the channel, and

  * •

the point where the RA actually reaches at $t_{2}$

can be expressed by

$\displaystyle d=|vT-d_{\text{a}}|,$ (1)

where $T=t_{2}-t_{1}$. Denote $y$ as the received signal at the RA, and it is
given by

$\displaystyle y=\sqrt{p}hx+z.$ (2)

Here, $P$ is transmit power, and $x$ represents transmit signal with unit
energy. $z\sim\mathcal{CN}(0,1)$ is the independent and identically
distributed complex Gaussian noise added at the receiver side. Using the same
method as in [8]

$\displaystyle h=\sqrt{1-\sigma^{2}}\hat{h}+\sigma q$ (3)

to model the BS-RA channel $h$ as a function of the BS-PA channel $\hat{h}$
and $q\sim\mathcal{CN}(0,1)$. Here, $\sigma$ is a parameter which is a
function of $d$ in (1), and larger $\sigma$ represents bigger spatial
mismatch. Defining $\hat{g}\doteq(1-\sigma^{2})|\hat{h}|^{2}$ with
$h\sim\mathcal{CN}(0,1)$ and $g\doteq|{h}|^{2}$, the CDF of $g$ is given by

$\displaystyle
F_{g|\hat{g}}(x)=1-Q\left(\sqrt{\frac{2\hat{g}}{\sigma^{2}}},\sqrt{\frac{2x}{\sigma^{2}}}\right).$
(4)

Here, $Q(,)$ represents Marcum-Q function.

Algorithm 1 Velocity-aware antenna selection scheme in PA systems with spatial
mismatch

In each time slot with known velocity at the receiver side, do the followings:

  1. I.

Calculate mismatch distance $d_{i}$ for each RA using (1), with known
information of $v$, $T$ and $d_{\text{a},i}$.

  2. II.

Evaluate the expected throughput (5) using proper methods for each RA.

  3. III.

Pick the best RA with the highest expected throughput.

The goal is to maximize the outage/error-limited throughput which is defined
as

$\displaystyle\eta=\mathbb{E}\left\\{\eta|{\hat{g}}(r_{|\hat{g}}^{\text{opt}})\right\\},$
(5)

with

$\displaystyle r_{|\hat{g}}^{\text{opt}}$
$\displaystyle=\operatorname*{argmax}_{r_{|\hat{g}}\geq
0}\left\\{\left(1-\Pr\left(\log(1+gP)<r_{|\hat{g}}\right)\right)r_{|\hat{g}}\right\\}.$
(6)

In [17, 8], different analytical methods of solving (6) are presented, and
results indicate that while rate adaptation can mitigate spatial mismatch and
provide notable throughput gain compared to no CSIT, the throughput is still
very sensitive to the speed variation with one RA.

Given that the speed is known at the receiver side, it is possible to add
additional RAs and perform antenna selection to obtain better performance. The
proposed velocity-aware antenna selection scheme is summarized in Algorithm 1.
Here, we assume that in each time slot the velocity is known at the receiver
side. In this way, from (1) the mismatch distance for each RA $d_{i}$ can be
calculated. Then, we evaluate the expected throughput (5) for each RA.
Finally, the best RA can be picked with the highest throughput, which can be
expressed as

$\displaystyle\eta^{\text{opt}}=\max_{i}\eta_{i},$ (7)

where $\eta$ is given by (5).

Figure 2: Required transmit SNR as a function of vehicle speed. Here, the
required throughput is set to 5 npcu.
Figure 3: Throughput as a function of the codeword length considering FBL.
$\sigma$ is a function of the mismatch distance $d$ and larger $\sigma$
represents higher mismatch.

##  IV Simulation Results

In the simulations, we study the performance of the PA system with a focus on
the effect of speed variation on the system performance. We set moving time
$T$ = 5 ms and carrier frequency 2.68 GHz. In Figs. 2-4 we study the
performance loss of one RA system due to spatial/speed mismatch in terms of
required SNR for given average throughput, throughput with FBL codewords [20],
and average error probability under FBL [20], respectively, with antenna
separation set to 1.5 times the wavelength. Then, in Figs. 5-7, the detailed
performance of the proposed antenna selection scheme is presented.

Fig. 2 shows the required transmit SNR for given throughput threshold as a
function of different speed. Here, the throughput threshold is set to 5 npcu.
The case with PA and rate adaptation is compared to the performance with full
CSIT and no CSIT. Also, in Fig. 3, considering FBL codewords [20], throughput
is calculated for both cases with/without the PA for different spatial
mismatches. Here, $\sigma$ is a function of the mismatch distance $d$ and
larger $\sigma$ represents higher mismatch. Similarly, focusing on average
error probability, Fig. 4 shows how it changes with transmit SNR with codeword
length set to 300.

Then, applying Algorithm 1, Fig. 5 studies the throughput as a function of
vehicle speed for one, three, and five RA(s), respectively. In the case with
one RA, $d_{\text{a}}=1.5\lambda$, and for three RAs $d_{\text{a, 1}}$,
$d_{\text{a, 2}}$, and $d_{\text{a, 3}}$ are set to 1.6, 1.5 and 1.4
$\lambda$, respectively. For five RAs the antenna separations are 1.62, 1.56,
1.5, 1.44, and 1.38 $\lambda$. Using similar settings, i.e., the distance
between RAs are the same for three and five RAs, Fig. 6 indicates the best RA
index for different $d_{\text{am}}$, which is the distance between the PA and
the RA in the middle. Finally, in Fig. 7, the average throughput for the
considered speed range (100-140 km/h) are plotted for the cases with different
number of RAs and no CSIT. The antenna separations are the same as in Figs.
5-6.

Figure 4: Average error probability as a function of transmit SNR. Codeword
length is set to 300. According to (1), $v$ = 124 km/h is one case with low
mismatch while $v$ = 110 km/h has higher mismatch.
Figure 5: Throughput as a function of the number of RAs. Algorithm 1 is used
for optimal antenna selection. Here, the transmit SNR is set to 20 dB. In the
case with one RA, $d_{\text{a}}=1.5\lambda$, and for three RAs $d_{\text{a,
1}}$, $d_{\text{a, 2}}$, and $d_{\text{a, 3}}$ are set to 1.6, 1.5 and 1.4
$\lambda$, respectively. For five RAs the antenna separations are 1.62, 1.56,
1.5, 1.44, and 1.38 $\lambda$.
Figure 6: The best RA index using Algorithm 1 for different values of
$d_{\text{am}}$, which is the distance between the PA and the RA in the
middle. The parameter settings are the same as in Fig. 5.
Figure 7: Average throughput of the considered speed range (100-140 km/h) for
different cases. The parameter settings are the same as in Fig. 5.

According to the simulation figures, the following conclusions can be drawn:

  * •

With spatial mismatch, considerable performance loss can be observed. For
instance, the required SNR increases with larger mismatch and eventually
becomes equal to the case without CSIT (Fig. 2).

  * •

As shown in Fig. 3, the PA system is robust to short/moderate length packages.
However, both throughput and average error probability in Fig. 4 are
drastically affected by the speed/spatial mismatch.

  * •

Using Algorithm 1 and perform antenna selection based on vehicle speed, the
throughput becomes much smoother with increasing number of RAs (Fig. 5). As
can be seen in Fig. 6, the system can be easily extended to wider speed ranges
if the position of the RA in the middle is shifted.

  * •

Finally, the average throughput increases with number of RAs and using
Algorithm 1.

##  V Conclusions

We reviewed the PA concept and recent testbed- and theoretical- oriented
studies on the PA. With promising progress on the PA studies, it has emerges
as one promising enabler for MR and moving IAB nodes in next generation of
mobile networks. Dealing with the speed/spatial mismatch problem, we proposed
a novel antenna selection scheme to fully utilize the velocity information and
make the system more robust to speed variations. The simulation results
further clarified the effect of mismatch from different perspectives, and our
proposed velocity-aware antenna selection scheme shows notable performance
gains with multiple RA deployed at the vehicle side.

## Acknowledgement

This work was supported in part by VINNOVA (Swedish Government Agency for
Innovation Systems) within the VINN Excellence Center ChaseOn and in part by
the European Commission through the H2020 project Hexa-X (Grant Agreement no.
101015956).

## References

  * [1] S. Yutao, J. Vihriala, A. Papadogiannis, M. Sternad, W. Yang, and T. Svensson, “Moving cells: a promising solution to boost performance for vehicular users,” _IEEE Commun. Mag._ , vol. 51, no. 6, pp. 62–68, Jun. 2013.
  * [2] H. Guo, B. Makki, D.-T. Phan-Huy, E. Dahlman, M.-S. Alouini, and T. Svensson, “Predictor antenna: A technique to boost the performance of moving relays,” _IEEE Commun. Mag._ , vol. 59, no. 7, pp. 80–86, Jul. 2021.
  * [3] M. Sternad, M. Grieger, R. Apelfröjd, T. Svensson, D. Aronsson, and A. B. Martinez, “Using predictor antennas for long-range prediction of fast fading for moving relays,” in _Proc. IEEE WCNCW_ , Paris, France, Apr. 2012, pp. 253–257.
  * [4] D.-T. Phan-Huy, S. Wesemann, J. Bjoersell, and M. Sternad, “Adaptive massive MIMO for fast moving connected vehicles: It will work with predictor antennas!” in _Proc. IEEE WSA_ , Bochum, Germany, Mar. 2018, pp. 1–8.
  * [5] J. Björsell, M. Sternad, and M. Grieger, “Using predictor antennas for the prediction of small-scale fading provides an order-of-magnitude improvement of prediction horizons,” in _Proc. IEEE ICCW_ , Paris, France, May 2017, pp. 54–60.
  * [6] ——, “Predictor antennas in action,” in _Proc. IEEE PIMRC_ , Montreal, Quebec, Canada, Oct. 2017, pp. 1–7.
  * [7] R. Apelfröjd, J. Björsell, M. Sternad, and D. Phan-Huy, “Kalman smoothing for irregular pilot patterns; a case study for predictor antennas in TDD systems,” in _Proc. IEEE PIMRC_ , Bologna, Italy, Sep. 2018, pp. 1–7.
  * [8] H. Guo, B. Makki, M.-S. Alouini, and T. Svensson, “A semi-linear approximation of the first-order Marcum $Q$-function with application to predictor antenna systems,” _IEEE Open Journal of the Communications Society_ , vol. 2, pp. 273–286, Feb. 2021.
  * [9] S. Jaffry, R. Hussain, X. Gui, and S. F. Hasan, “A comprehensive survey on moving networks,” _IEEE Commun. Surv. Tutor._ , vol. 23, no. 1, pp. 110–136, Oct. 2020.
  * [10] E. Tanghe, W. Joseph, L. Verloock, and L. Martens, “Evaluation of vehicle penetration loss at wireless communication frequencies,” _IEEE Trans. Veh. Technol._ , vol. 57, no. 4, p. 2036–2041, Jul. 2008.
  * [11] Q. Li, R. Q. Hu, Y. Qian, and G. Wu, “Cooperative communications for wireless networks: Techniques and applications in LTE-advanced systems,” _IEEE Wireless Commun._ , vol. 19, no. 2, pp. 22–29, Apr. 2012.
  * [12] A. Khan and A. Jamalipour, “An outage performance analysis with moving relays on suburban trains for uplink,” _IEEE Trans. Veh. Technol._ , vol. 66, no. 5, pp. 3966–3975, May 2017.
  * [13] L. S. Muppirisetty, T. Svensson, and H. Wymeersch, “Spatial wireless channel prediction under location uncertainty,” _IEEE Trans. Wirel. Commun._ , vol. 15, no. 2, pp. 1031–1044, Feb. 2016.
  * [14] D. T. Phan-Huy, M. Sternad, and T. Svensson, “Adaptive large MISO downlink with predictor antenna array for very fast moving vehicles,” in _Proc. IEEE ICCVE_ , Las Vegas, Nevada, USA, Dec. 2013, pp. 331–336.
  * [15] N. Jamaly, R. Apelfröjd, A. Belen Martinez, M. Grieger, T. Svensson, M. Sternad, and G. Fettweis, “Analysis and measurement of multiple antenna systems for fading channel prediction in moving relays,” in _Proc. IEEE EuCAP_ , The Hague, The Netherlands, Apr. 2014, pp. 2015–2019.
  * [16] D. T. Phan-Huy, M. Sternad, T. Svensson, W. Zirwas, B. Villeforceix, F. Karim, and S. E. El-Ayoubi, “5G on board: How many antennas do we need on connected cars?” in _Proc. IEEE Globecom Workshops_ , Washington, DC, USA, Dec. 2016.
  * [17] H. Guo, B. Makki, and T. Svensson, “Rate adaptation in predictor antenna systems,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 4, pp. 448–451, Apr. 2020.
  * [18] H. Guo, B. Makki, M.-S. Alouini, and T. Svensson, “Power allocation in HARQ-based predictor antenna systems,” _IEEE Wirel. Commun. Lett._ , vol. 9, no. 12, pp. 2025–2029, Dec. 2020.
  * [19] ——, “On delay-limited average rate of HARQ-based predictor antenna systems,” _IEEE Wirel. Commun. Lett._ , vol. 10, no. 8, pp. 1628–1632, Aug. 2021.
  * [20] H. Guo, B. Makki, and T. Svensson, “Predictor antennas for moving relays: Finite block-length analysis,” in _2020 3rd International Conference on Advanced Communication Technologies and Networking (CommNet)_ , Marrakech, Morocco, 2020, pp. 1–8.
  * [21] H. Guo, B. Makki, M.-S. Alouini, and T. Svensson, “High-rate uninterrupted Internet-of-vehicle communications in highways: Dynamic blockage avoidance and CSIT acquisition,” Sept. 2021, [Online] Available at: https://arxiv.org/abs/2109.12878.

  *[MR]: moving relay
  *[MRs]: moving relay
  *[QoS]: quality of service
  *[VPL]: vehicle penetration loss
  *[CSIT]: channel state information at the transmitter side
  *[PA]: predictor antenna
  *[RA]: receive antenna
  *[MIMO]: multiple input multiple output
  *[TDD]: time division duplex
  *[LoS]: line-of-sight
  *[BS]: base station
  *[mmw]: millimeter wave
  *[BSs]: base station
  *[UEs]: user equipment
  *[CoMP]: Coordinated Multi-Point
  *[RAs]: receive antenna
  *[PAs]: predictor antenna
  *[NLoS]: non-line-of-sight
  *[MISO]: multiple input single output
  *[BF]: beamforming
  *[HARQ]: hybrid automatic repeat request
  *[IAB]: integrated access and backhaul
  *[SNR]: signal-to-noise ratio
  *[npcu]: nats-per-channel-use
  *[FBL]: finite block-length


# Interplay of altermagnetism and weak ferromagnetism
in two-dimensional RuF4

Marko Milivojević  0000-0002-9583-3640 Institute of Informatics, Slovak
Academy of Sciences, 84507 Bratislava, Slovakia  Faculty of Physics,
University of Belgrade, 11001 Belgrade, Serbia  Marko Orozović
0000-0002-3018-5830 Vinča Institute of Nuclear Sciences - National Institute
of the Republic of Serbia, University of Belgrade, P. O. Box 522, RS-11001
Belgrade, Serbia  Silvia Picozzi  Consiglio Nazionale delle Ricerche CNR-SPIN,
c/o Universitá degli Studi ”G. D’Annunzio”, 66100 Chieti, Italy  Martin Gmitra
0000-0003-1118-3028 Institute of Physics, Pavol Jozef Šafárik University in
Košice, 04001 Košice, Slovakia  Institute of Experimental Physics, Slovak
Academy of Sciences, 04001 Košice, Slovakia  Srdjan Stavrić
0000-0003-2097-0955<EMAIL_ADDRESS>Vinča Institute of Nuclear Sciences -
National Institute of the Republic of Serbia, University of Belgrade, P. O.
Box 522, RS-11001 Belgrade, Serbia  Consiglio Nazionale delle Ricerche CNR-
SPIN, c/o Universitá degli Studi ”G. D’Annunzio”, 66100 Chieti, Italy

(August 28, 2024)

###### Abstract

Gaining growing attention in spintronics is a class of magnets displaying zero
net magnetization and spin-split electronic bands called altermagnets. Here,
by combining density functional theory and symmetry analysis, we show that
RuF4 monolayer is a two-dimensional $d$-wave altermagnet. Spin-orbit coupling
leads to pronounced spin splitting of the electronic bands at the $\Gamma$
point by $\sim 100\,{\rm meV}$ and turns the RuF4 into a weak ferromagnet due
to non trivial spin-momentum locking that cants the Ru magnetic moments. The
net magnetic moment scales linearly with the spin-orbit coupling strength.
Using group theory we derive an effective spin Hamiltonian capturing the spin-
splitting and spin-momentum locking of the electronic bands. Disentanglement
of the altermagnetic and spin-orbit coupling induced spin splitting uncovers
to which extent the altermagnetic properties are affected by the spin-orbit
coupling. Our results move the spotlight to the non trivial spin-momentum
locking and weak ferromagnetism in the two-dimensional altermagnets relevant
for novel venues in this emerging field of material science research.

##  I Introduction

Recent discovery of non-relativistic spin splitting in electronic bands of
materials with compensated collinear magnetic order opened new venue in
antiferromagnetic spintronicsŽutić et al. (2004); Jungwirth et al. (2016);
Šmejkal et al. (2018). These intriguing materials constitute a separate
magnetic phase, besides ferromagnets (FM) and antiferromagnets (AF), dubbed
the name altermagnets (AM) Šmejkal et al. (2022a, b) or, alternatively,
antiferromagnets with non-interconvertible spin-structure motif pairYuan et
al. (2020, 2021); Yuan and Zunger (2023). Altermagnets display a profusion of
diverse physical phenomena, including anomalous Hall effect Tschirner et al.
(2023), piezoelectricity Guo et al. (2023a), and chiral magnonsŠmejkal et al.
(2023); they can be used for efficient spin-to-charge conversion Bai et al.
(2023), to generate orientation-dependent spin-pumping currentsSun and Linder
(2023) and to proximitize superconductor materials Ouassou et al. (2023);
Beenakker and Vakhtel (2023); Sun et al. (2023), just to scratch the manifold
of their possible applications.

Many altermagnetic compounds have been classified so farGuo et al. (2023b);
Biniskos et al. (2023), including MnTe Mazin (2023); Junior et al. (2023),
MnF2Yuan et al. (2020), RuO2 Šmejkal et al. (2022a); Feng et al. (2022);
Tschirner et al. (2023) and CrSb Reimers et al. (2023). In MnTe the spin
splitting of $380\,{\rm meV}$ is experimentally determined by angle-resolved
photoemission spectroscopy (ARPES) measurements, thus confirming its
altermagnetic status previously predicted by density functional theory (DFT)
calculations. For possible spin-transport applications, apart from the
magnitude of the splitting energy, the positioning of the spin-split bands is
highly important. In this respect CrSb may be a promising candidate for
producing spin-polarized currents as a giant spin splitting of $\sim 600\,{\rm
meV}$ occurs just below the Fermi energyReimers et al. (2023).

Regarding their dimensionality, all the previously mentioned altermagnets are
three-dimensional (3D) compounds, and as of now, reports on two-dimensional
(2D) altermagnets are limited. In the work of Brekke et al Brekke et al.
(2023) it is proposed that magnon-mediated superconductivity in a 2D
altermagnet could result with a superconductor with substantially enhanced
critical temperature. However, this is accomplished using a microscopic model,
and no specific material has been suggested to test this result. Very
recently, monolayer Cr2Te2O is predicted as a promising platform for practical
realization of the spin Seebeck and spin Nernst effects of magnonsCui et al.
(2023). Further, it is shown that monolayer MnPS3, which is a conventional
antiferromagnet, can be converted into altermagnet if one of its S layers is
replaced by SeMazin et al. (2023). However, in thus obtained MnP(S,Se)3 Janus
monolayer, the spin splitting of the relevant top valence band is modest, not
exceeding $\approx 10\,{\rm meV}$ in the best case. Considering this, the
endeavor to fabricate the Janus monolayer might be deemed excessively
demanding in comparison to the anticipated benefits. Hence, for prospective
applications in miniature and highly efficient spintronic devices, it would be
beneficial to have 2D materials that inherently exhibit altermagnetic
properties.

There is yet an important peculiarity concerning 2D magnets in general.
Namely, in the absence of an external magnetic field the long-range order of
spins on a 2D lattice can sustain above $T=0\,{\rm K}$ temperature only if the
system possesses the magnetic anisotropy energy (MAE) Mermin and Wagner
(1966). MAE is crucial here as it gaps the magnon spectra. Otherwise, the low-
energy magnon excitations would destroy the long-range order at arbitrary non-
zero temperature. In this respect, altermagnets are not any different from
ferromagnets and antiferromagnets. MAE, in turn, requires the presence of a
sizable spin-orbit coupling (SOC) in the system. Therefore, for a realistic
description of a 2D altermagnet the SOC must be accounted for. This raises a
question – to which extent its non-relativistic properties, e.g. the hallmark
spin splitting of bands, are affected by SOC?

Here, we show that in the non-relativistic limit monolayer RuF4 is a 2D
altermagnet and we describe the spin splitting of band structure and spin
textures of the altermagnetic phase. Subsequently, we analyze the influence of
SOC on the altermagnetic properties. The structure of monolayer RuF4, its
crystallographic and magnetic symmetries, are described in Subsection II.1.
The altermagnetic phase is analyzed in Subsection II.2. Weak ferromagnetism
and SOC-induced spin splitting of bands are exposed in Subsections II.3 and
II.4. Finally, we summarize the paper in Section III with an outlook on the
link between altermagnetism and weak ferromagnetism.

##  II Results

###  II.1 The structure and crystal symmetries of RuF4 monolayer

Monolayer RuF4 is composed of ruthenium atoms sitting in centers of fluorine
octahedra. It is a van der Waals material that can be exfoliated from the bulk
compound which was experimentally synthesized in monoclinic $P2_{1}/c$
crystallographic space group (group No. 14) Casteel et al. (1992). We will
refer to an Ru atom with its surrounding F ligands as the Ru cluster. In each
Ru cluster two out of six F atoms belong only to that cluster, whereas four of
them are shared with four nearest neighbors. Crystal lattice of monolayer RuF4
has two sublattices with symmetrically distinct Ru clusters, RuA and RuB, that
are arranged as depicted in Fig. 1.

Figure 1:  Structure of monolayer RuF4 and a schematic depiction of
transformations $g_{1}=\mathcal{I}$ and $g_{2}=(U_{x}|t_{\frac{1}{2}})$ that
are generators of its crystallographic group. Distinct Ru clusters are
depicted by red (RuA) and blue (RuB) octahedra.

Two perpendicular lattice vectors that define the RuF4 plane, ${\bf a}_{1}$
and ${\bf a}_{2}$, have different norm, $a_{1}=5.131\,{\rm\AA}$ and
$a_{2}=5.477\,{\rm\AA}$, as obtained from our DFT calculations. Therefore, we
will refer to the two directions defined by these vectors as the shorter and
the longer in-plane directions. The structure of monolayer RuF4 belongs to the
$p2_{1}/b11$ layer group (layer group No. 17)Litvin and Wike (1991) which has
two generators: the space inversion $g_{1}=\mathcal{I}$ and the non-symmorphic
group element $g_{2}=(U_{x}|t_{\frac{1}{2}})$ that is the combination of two
operations each transforming RuA into RuB and vice versa: a rotation $U_{x}$
by $180^{\circ}$ around the shorter in-plane direction ${\bf a}_{1}$ and a
fractional lattice translation $t_{\frac{1}{2}}=\frac{1}{2}({\bf a}_{1}+{\bf
a}_{2})$. Therefore, the group symmetry consists of four elements, ${\bf
G}=\\{e,g_{1},g_{2},g_{1}g_{2}\\}$, where $e$ is the identity element and
$g_{1}g_{2}$ represents a joint action of the group generators. Importantly,
the crystal structure is centrosymmetric, but the Ru-Ru bond does not have the
point of inversion, meaning that the Dzyaloshinskii-Moriya interaction can
occur between the Ru magnetic momentsMoriya (1960). We will discuss this in
detail in the remainder of the manuscript.

###  II.2 Non-relativistic description of magnetic properties: 2D altermagnet

RuF4 has a compensated net magnetic moment due to the opposite magnetic
moments on the two Ru sublattices. Octahedral crystal field of F- ions acts on
Ru4+ ion, leading to the $t_{2g}-e_{g}$ splitting of its $4d$ orbitals and
leaving two spin-unpaired electrons in the $t_{2g}$ manifold. Therefore, from
the crystal field theory, magnetic moments of $2\,\mu_{\rm B}$ are expected on
Ru atoms. Our DFT calculations give $1.464\,\mu_{\rm B}$, in agreement with
the value reported in Ref. Wang et al. (2022). The discrepancy between the
expected and the DFT-calculated magnetic moments is due to the hybridization
between the Ru $4d$ and the F $2p$ orbitals. Concretely, the F atoms that
belong to a single Ru cluster exhibit moderate magnetic moments of 0.214
$\mu_{\rm B}$, induced by the nearest Ru magnetic moment, that are coupled
ferromagnetically to that moment. On the other hand, the F atoms shared
between two clusters are bonded to the two opposite-spin Ru atoms and are thus
non-magnetic. In summary, the magnitude of the magnetic moment on each Ru
cluster is $|{\bf m}|=|{\bf m}({\rm Ru})+2{\bf m}({\rm F})|=1.9\,\mu_{\rm B}$.

In the absence of an external magnetic field and without SOC, the direction of
magnetic moments and thus of the Néel vector $({\bf N})$ is arbitrary. To
perform symmetry analysis, we assume that ${\bf N}$ points along the longer
in-plane lattice direction ${\bf a}_{2}$, which coincides with the $y$-axis in
our choice of the coordinate frame (see Fig. 1). Space inversion $g_{1}$ has a
trivial effect on magnetic moments as they are pseudovectors, whereas $g_{2}$
leaves the magnetic order invariant because the rotation $U_{x}$ and the
fractional lattice translation $t_{\frac{1}{2}}$ each reverse the magnetic
order. Therefore, all the symmetry operations from the $p2_{1}/b11$ layer
group leave the magnetic order unaltered and are thus the symmetries of the
magnetic system as well, meaning that the magnetic space group is equal to its
parent crystallographic group (type-I). Given that (1) the magnetic space
group is of type-IBradley and Davies (1968) and (2) the combination
$\theta\mathcal{I}$ of time reversal $\theta$ and space inversion symmetry
$\mathcal{I}$ is broken due to the magnetic order, the non-relativistic spin
splitting in monolayer RuF4 is allowed Yuan et al. (2021); Guo et al. (2023b).

In Fig. 2a we plot the band structure of monolayer RuF4 calculated with spin-
polarized DFT calculations. The system is a semiconductor with an indirect
band gap of $0.83\,{\rm eV}$ between the valence band maximum, located on the
XY path, and the conduction band minimum located at S.

Figure 2:  Non-relativistic RuF4 band structure calculated with spin-polarized
DFT calculations. In (a), the $\Gamma$XS$\Gamma$YX path is assumed and the
spin splitting energies of the top valence (V1/2) and bottom conduction (C1/2)
bands are given in the inset (yellow rectangle). In (b) the top valence band
V1/2 is illustrated once again along the S${}^{\prime}\Gamma$S path, and the
$d$-wave nature of its spin-momentum locking in the full Brillouin zone (green
rectangle) at $-0.15\,{\rm eV}$ below the Fermi energy is illustrated in (c).

Along the chosen $k$-path in the Brillouin zone (BZ), the spin-up and spin-
down bands are degenerate along the $\Gamma$X, XS, and $\Gamma$Y lines,
whereas a sizable spin splitting $\Delta E_{n}({\bf k})=E^{\uparrow}_{n}({\bf
k})-E^{\downarrow}_{n}(\bf k)$ ($n$ is the band index) is observable along the
$\Gamma$S and XY lines. Notably, of all the bands in the energy window from
$-1.5\,{\rm eV}$ below to $1.5\,{\rm eV}$ above $E_{\rm F}$, the spin
splitting is the largest for the top valence (V1/2 shown in Fig. 2b) and the
bottom conduction bands (C1/2), reaching $163\,{\rm meV}$ and $117\,{\rm
meV}$, respectively (yellow rectangle in Fig. 2a). The spin-momentum locking
of the bands V1/2 presented in Fig. 2c clearly shows that monolayer RuF4 is a
$d$-wave altermagnetŠmejkal et al. (2022a).

To model the spin splitting of these bands around the $\Gamma$ point we employ
a simple time-reversal breaking Hamiltonian, compatible with the system’s
symmetry,

$\mathcal{H}_{\rm ss}=\frac{1}{2}\alpha k_{x}k_{y}\sigma_{y}.$ (1)

Here $\sigma_{y}=\pm 1$ and $\alpha$ is the band-dependent parameter
determined by fitting the model to the DFT bands. For the V1/2 and C1/2, up to
the 25% of the $\Gamma$X line, the spin splitting in an arbitrary
$k$-direction can be described using $\alpha_{V}=3450\,{\rm meV\,\AA^{2}}$ and
$\alpha_{C}=2368\,{\rm meV\,\AA^{2}}$.

As DFT calculations pointed out, the spin degeneracy of bands is not lifted
along the $\Gamma$X and $\Gamma$Y lines. This can be explained by analyzing
how the group elements connect different points of the energy surface
$E_{n}(k_{x},k_{y},s)$. Keeping the assumption that magnetic moments are in
the $y$-direction, the action of group elements reflects on the electronic
bands as

$\displaystyle g_{1}\,E_{n}(k_{x},k_{y},s)$ $\displaystyle=$ $\displaystyle
E_{n}(-k_{x},-k_{y},s),$ $\displaystyle g_{2}\,E_{n}(k_{x},k_{y},s)$
$\displaystyle=$ $\displaystyle E_{n}(k_{x},-k_{y},-s),$ $\displaystyle
g_{1}g_{2}\,E_{n}(k_{x},k_{y},s)$ $\displaystyle=$ $\displaystyle
E_{n}(-k_{x},k_{y},-s).$ (2)

Based on the equations (2), one concludes that along $\Gamma$X, where ${\bf
k}=(k_{x},0)$, and $\Gamma$Y, where ${\bf k}=(0,k_{y})$, the relations

$\displaystyle g_{2}\,E_{n}(k_{x},0,s)=E_{n}(k_{x},0,-s),$ $\displaystyle
g_{1}g_{2}\,E_{n}(0,k_{y},s)=E_{n}(0,k_{y},-s)$ (3)

express the spin degeneracy. On the other hand, these constraints are not
present along an arbitrary $k$-path and the spin splitting of bands is not
forbidden. This is exactly the case of the $\Gamma$S and XY lines, where we
observe a noticeable non-relativistic spin splitting, which is the hallmark of
the altermagnetic phase.

###  II.3 SOC turning altermagnet into a weak ferromagnet

Once the SOC is included, the spin space couples to the real space and the
Néel vector ${\bf N}$ obtains a defined crystallographic direction. Our DFT
calculations reveal that the shorter in-plane lattice direction (the $y$-axis)
is the preferential direction for ${\bf N}$, with MAE separating it from the
longer in-plane direction ($x$-axis) and from the out-of-plane direction
($z$-axis) by $1.12\,{\rm meV}/{\rm Ru}$ and $7.04\,{\rm meV}/{\rm Ru}$,
respectively.

Due to SOC, the magnetic moments are getting canted from the $y$-axis towards
the $x$-axis by $3.2^{\circ}$, as depicted in Fig. 3a. Canting lowers the
total energy only slightly ($0.08\,{\rm meV}/{\rm Ru}$), but it induces a net
magnetic moment of $|{\bf M}|=|{\bf m}_{A}+{\bf m}_{B}|=0.22\,\mu_{\rm B}$
which points in the $+x$ direction. Therefore, upon the inclusion of SOC, the
monolayer RuF4 changes its magnetic phase from the altermagnetic to the weak
ferromagnetic (WF) phase.

Figure 3:  (a) SOC-induced canting of magnetic moments on two sublattices of
RuF4. (b) Spin textures of the top valence bands V1 and V2 at energy -0.15 eV
below the Fermi level. (c) Dependence of induced net magnetic moment on the
SOC strength.

The direction of the induced magnetic moment in RuF4 can be inferred from the
group theory analysis. In general, the magnetic moment ${\bf
M}=(M_{x},M_{y},M_{z})$ of any system must stay invariant under the action of
all the group symmetry elements. Since the inversion $g_{1}$ has a trivial
action on the pseudovector ${\bf M}$ through the $\mathcal{D}$ representation
111$\mathcal{D}(g_{1})=\Big{(}\begin{smallmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{smallmatrix}\Big{)}$,
$\mathcal{D}(g_{2})=\Big{(}\begin{smallmatrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{smallmatrix}\Big{)}$,
$\mathcal{D}(g_{1}g_{2})=\Big{(}\begin{smallmatrix}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{smallmatrix}\Big{)}$, $\mathcal{D}(g_{1}){\bf M}={\bf M}$, whereas
the group elements $g_{2}$ and $g_{1}g_{2}$ change the sign of the $y$ and $z$
component of ${\bf M}$, $\mathcal{D}(g_{2}){\bf M}=\mathcal{D}(g_{1}g_{2}){\bf
M}=(M_{x},-M_{y},-M_{z})\equiv(M_{x},M_{y},M_{z})$, both the $M_{y}$ and
$M_{z}$ must be zero. Therefore, the net magnetic moment of RuF4 must point
along the $x$-axis, in accordance with our DFT results.

The transition from the altermagnetic to the weak ferromagnetic phase leaves
its fingerprints on the spin texture. Spin texture is a vector field in the
reciprocal space defined through the expectation value of the spin operator
${\bf s}=(s_{x},s_{y},s_{z})$ in a given Bloch state
$|\psi_{k_{x},k_{y}}^{n}\rangle$,

$\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=\langle\psi_{k_{x},k_{y}}^{n}|s_{i}|\psi_{k_{x},k_{y}}^{n}\rangle,\quad
i=x,y,z.$ (4)

The spin texture, like the net magnetic moment, complies with the symmetry of
the system. We provide detailed derivation in Appendix A and arrive here
straight at the conclusion by expressing the action of the group elements on
the spin texture,

$\displaystyle g_{1}:$ $\displaystyle\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=\langle s_{i}\rangle^{n}_{-k_{x},-k_{y}},$
$\displaystyle g_{2}:$ $\displaystyle\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=(-1)^{1-\delta_{i,x}}\langle
s_{i}\rangle^{n}_{k_{x},-k_{y}},$ $\displaystyle g_{1}g_{2}:$
$\displaystyle\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=(-1)^{1-\delta_{i,x}}\langle
s_{i}\rangle^{n}_{-k_{x},k_{y}},$ (5)

where $\delta_{i,x}$ is the Kronecker delta, equal to 1 if $i=x$ and 0
otherwise.

The Eq. II.3 shows that the spin texture in the I quadrant of the BZ
$(k_{x}>0,k_{y}>0)$ is related to the spin texture in the II, III, and IV
through the action of the group elements $g_{1}g_{2}$, $g_{1}$, and $g_{2}$,
respectively. We can illustrate these relations with a concrete example. In
Fig. 3c we show the spin texture of the valence bands V1/2 at the constant
energy surface $-0.15\,{\rm eV}$ below the Fermi level. Now, it is clear that
the spin expectation values $\langle s_{y/z}\rangle_{k_{x},k_{y}}$ from the II
and IV quadrants cancel out the spin expectation values in the I and III
quadrants (actually $\langle s_{z}\rangle$ is zero everywhere, which is not
shown in the plot). The cancellation of the spin expectation values from
different quadrants of the BZ leads to the vanishing of $M_{y}$ and $M_{z}$ in
the real space. Indeed, each component of the net magnetic moment can be
expressed as an integral of the corresponding spin expectation values over the
BZ, $M_{i}=\sum_{n}^{{\rm occ}}\int_{BZ}\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}{\rm d}^{2}{\bf k}$, where the ”occ” denotes
that the sum runs over the occupied states. On the other hand, $\langle
s_{x}\rangle_{k_{x},k_{y}}>0$ in all the four quadrants, yielding non-zero
$M_{x}$.

In the following, we question the microscopic mechanism responsible for
canting the magnetic moments. In the work of Wang et alWang et al. (2022) it
is shown that the Dzyaloshinskii-Moriya interaction (DMI) acting between the
Ru moments is mainly responsible for their canting, with some contribution
from the single-ion anisotropy (SIA). The DMI between the opposite-spin Ru
atoms is allowed because the Ru-Ru bond does not display inversionMoriya
(1960), as we mentioned in the Subsection II.1. Given that SOC is the common
origin of both the DMI and SIA, the magnitude of the induced net magnetic
moment must depend on the SOC strength. To shed light on this relation, we
scale the SOC strength in DFT calculations by using the modified SOC constant
$\widetilde{\lambda}=\xi\lambda$, where the dimensionless parameter $\xi$
ranges from 0 (no SOC) to 1 (realistic SOC). Then, we constrain the magnetic
moments in the directions that are canted by angle $\alpha$ from the $y$-axis.
For a set of directions $\alpha$ the equilibrium canting angle $\alpha_{\xi}$
is found by minimizing the auxiliary function
$E_{\xi}(\alpha)=c_{0}+c_{2}(\alpha-\alpha_{\xi})^{2}$, where
$E_{\xi}(\alpha)$ is the total energy of the system for a particular canting
angle $\alpha$ and SOC strength $\widetilde{\lambda}=\xi\lambda$. Finally, for
each $\xi$ we perform DFT calculations for magnetic moments canted by
$\alpha_{\xi}$ and we calculate the net magnetic moment. The plot in Fig. 3c
shows that the net magnetic moment is linear in SOC strength.

The linearity of the dependence $M(\tilde{\lambda})$ can be explained by
turning to the microscopic expression for the Dzyaloshinskii-Moriya
interaction. First, we noticed that the magnitude of the Ru cluster’s magnetic
moment ${\bf m}={\bf m}({\rm Ru})+2{\bf m}({\rm F})$ is almost unaffected by
SOC, as the magnitude of the magnetic moment on Ru atom changes only slightly
from $1.464\,\mu_{\rm B}$ for $\xi=0$ to $1.453\,\mu_{\rm B}$ for $\xi=1$
while $|{\bf m}({\rm F})|$ doesn’t change at all, staying $0.214\,\mu_{\rm
B}$. Second, the magnitude of the net magnetic moment ${\bf M}={\bf
m}_{A}+{\bf m}_{B}$ and the canting angle $\alpha_{\xi}$ are related by
$M=2m\sin\alpha_{\xi}$, where $m=|{\bf m}_{A/B}|$. Now, if we assume that
canted magnetic moments stay in the $xy$-plane, the DMI energy can be
expressed as

$E_{\rm DMI}={\bf D}\cdot({\bf m}_{A}\times{\bf
m}_{B})=D\cos\theta_{D}\sin(2\alpha_{\xi}),$ (6)

where $\theta_{D}=\measuredangle({\bf D},z)$. As pointed out by MoriyaMoriya
(1960) the magnitude of the Dzyaloshinskii vector is linear in SOC constant,
$D\sim\lambda$ (here $D\sim\tilde{\lambda}$ as we are using modified SOC
constant), whereas its direction depends on the symmetry of the structure and
on the atomic positioning. Therefore, the angle $\theta_{D}$ does not depend
on $\tilde{\lambda}$ and the DMI energy can be expressed as $E_{\rm
DMI}\sim\tilde{\lambda}\sin(2\alpha_{\xi})$. As long as the canting angle is
small the approximate relation $E_{\rm DMI}\sim 2\tilde{\lambda}\alpha_{\xi}$
holds. Noticing that the DMI energy is quadratic in Dzyaloshinskii vector
magnitude222For a quantum spin-1 system described by Hamiltonian
$\mathcal{H}=J{\bf S}_{1}\cdot{\bf S}_{2}+{\bf D}\cdot({\bf S}_{1}\times{\bf
S}_{2})$, ${\bf D}=D{\bf e}_{z}$, the DMI correction of the eigenvalues is
equal to $\pm\sqrt{J^{2}+D^{2}}\sim\pm J\big{(}1+\frac{D^{2}}{2J^{2}}\big{)}$
and $\frac{J}{2}(1\pm 3\sqrt{1+\frac{8D^{2}}{9J^{2}}})\sim\frac{J}{2}(1\pm
3(1+\frac{4D^{2}}{9J^{2}}))$, where it is assumed that the DMI is much smaller
than the Heisenberg exchange. Therefore, the DMI correction to the energy is
$E_{\rm DMI}\sim D^{2}/J$., $E_{\rm DMI}\sim D^{2}$ , the last relation shows
that the canting angle is linear in the modified SOC constant,
$\alpha_{\xi}\sim\tilde{\lambda}$. Finally, for small canting angles, the
linearity of the $M(\tilde{\lambda})$ dependence stems from the relation
$M\sim 2m\alpha_{\xi}\sim m\tilde{\lambda}$.

###  II.4 SOC-induced spin splitting of bands

Spin-orbit coupling, besides inducing net magnetic moment in the 2D
altermagnet RuF4, is responsible for significant spin splitting of the
electronic bands. To shed light on this SOC effect, in Fig. 4(a)-(b) we plot
the band structure calculated with SOC along the same $k$-path and in the same
energy window as in the Fig. 2. In addition, we project the spin expectation
values $\langle{s_{x}}\rangle_{k_{x},k_{y}}$ and
$\langle{s_{y}}\rangle_{k_{x},k_{y}}$ over each band to reveal how their spin
polarization evolves along different $k$-directions.

Figure 4:  Non-collinear relativistic DFT calculations of the band structure
for RuF4 along high symmetry lines. (a) spin expectation values $\langle
s_{x}({\bf k})\rangle$ and (b) $\langle s_{y}({\bf k})\rangle$ plotted as a
band color. (c) spin splitting of the top valence V1/2 and bottom conduction
bands C1/2 is along the same $k$-path given, with the AM-induced spin
splitting plotted for comparison with dashed lines. (d) spin splitting of V1/2
and C1/2 at the $\Gamma$ point calculated for the different SOC strength
$\xi$. (e) the difference between spin splitting energies with and without SOC
of V1/2 and C1/2 bands along the $\Gamma$S path for different SOC strengths
$\xi$.

Due to SOC, the bands V1/2 and C1/2, which are degenerate in the AM phase
along some special directions, are now split in almost the entire BZ,
displaying highly nonuniform splitting (Fig. 4c). Intriguingly, the non-
relativistic (AM-induced) and relativistic (SOC-induced) spin splitting reach
their respective maxima in different regions of the BZ: while SOC-induced
splitting is maximal at $\Gamma$ and S points, where the bands were previously
degenerate in the AM phase, the AM-induced splitting in the middle of the
$\Gamma$S and XY lines is barely affected by SOC.

Apart from the splitting energies, the polarization of bands displays non-
uniform $k$-dependence. For example, the V1/2 bands are polarized in the
$x$-direction along the $\Gamma$X and $\Gamma$Y lines, with $\langle
s_{x}\rangle<0$ for V1 and $\langle s_{x}\rangle>0$ for V2 (Fig. 4a-b). The
polarization in the $x$-direction along these $k$-lines is the most evident
for the V7/8 bands. On the other hand, both $\langle s_{x}\rangle$ and
$\langle s_{y}\rangle$ are nonzero and vary substantially along the $\Gamma$S
and XY lines, meaning that V1/2 have mixed polarization along these
directions. Also, there are $k$-directions along which the bands are almost
unpolarized, such as the XS line where $\langle s_{x}\rangle\approx\langle
s_{y}\rangle\approx 0$.

The point that deserves special attention is the lifting of spin degeneracy
along the X$\Gamma$Y path. What is particularly striking is a large spin
splitting at $\Gamma$, which is a time-reversal invariant momenta (TRIM)
point, of $\Delta_{\Gamma}^{V_{1/2}}=86\,{\rm meV}$ and
$\Delta_{\Gamma}^{C_{1/2}}=79\,{\rm meV}$ for the V1/2 and C1/2 bands,
respectively. Moreover, $\Delta_{\Gamma}$ scales linearly with the SOC
strength, as shown in Fig. 4d. At first glance, this linear splitting seems to
occur due to a nonzero net magnetic moment pointing in the $x$-direction, but
our additional DFT calculations reveal that even when the Ru magnetic moments
are strictly constrained in the $y$-direction (so that ${\bf M}=0$), the spin
splitting of bands along X$\Gamma$Y persists.

To shed light on this, let us consider a non-relativistic Hamiltonian of the
AM phase $\mathcal{H}^{\rm nrel}_{\rm AM}$ and include SOC as a perturbation.
The complete derivation is exposed in Appendix B but here we discuss the most
important conclusions. Assuming that two Bloch wavefunctions corresponding to
the same ${\bf k}=(k_{x},k_{y})$ point,
$|{\Psi_{1}}\rangle_{k_{x},k_{y}}=|{\psi_{1}}\rangle\otimes|{\uparrow}\rangle$
and
$|{\Psi_{2}}\rangle_{k_{x},k_{y}}=|{\psi_{2}}\rangle\otimes|{\downarrow}\rangle$,
are with the same energy $E$, the perturbation $\mathcal{H}_{\rm
SOC}=({\bm{\nabla}}V\times{\bf p})\cdot{\bm{\sigma}}=\lambda{\bf
L}\cdot{\bm{\sigma}}$ in the
$\\{|{\Psi_{1}}\rangle_{k_{x},k_{y}},|{\Psi_{2}}\rangle_{k_{x},k_{y}}\\}$
basis has the matrix elements of the form

$\langle\mathcal{H}_{\rm SOC}\rangle=\lambda\begin{pmatrix}\ell_{y}&{\rm
i}(\ell_{x}-\ell_{z})\\\ -{\rm i}(\ell_{x}-\ell_{z})&\ell_{y}\end{pmatrix},$
(7)

where $\ell_{i}=\ell_{i}(k_{x},k_{y})$, $i=x,y,z$, are real numbers that can
be connected to nonzero matrix elements of the pseudovector operator ${\bf
L}$, see Appendix B. From the Eq. 7, the SOC contribution to the level $E$ is
$\lambda\Big{(}\ell_{y}\pm(\ell_{x}-\ell_{z})\Big{)}$, meaning that SOC lifts
the degeneracy of $|{\Psi_{1}}\rangle_{k_{x},k_{y}}$ and
$|{\Psi_{2}}\rangle_{k_{x},k_{y}}$ inducing the spin splitting of $\Delta_{\bf
k}=2\lambda(\ell_{x}-\ell_{z})$. The derivation exposed in Appendix B applies
to any two degenerate Bloch wavefunctions as it only exploits the action of
the group element $g_{2}$ and the assumption that ${\bf N}$ points in the
$y$-direction. Therefore, the conclusions of Appendix B apply to any ${\bf k}$
point, showing that the linear-in-SOC spin splitting is a cooperative effect
of breaking the time-reversal symmetry due to the magnetic order and of the
spin-orbit coupling.

The spin degeneracy along the $\Gamma$S and XY lines is lifted already in the
AM phase, but the inclusion of SOC changes this spin splitting drastically. To
show how the SOC-induced splitting gradually develops over the AM-induced one,
we plot the difference of the SOC-induced and AM-induced splitting energies
for V1/2 and C1/2 bands along the $\Gamma$S path for different SOC strengths
$\tilde{\lambda}=\xi\lambda$ ( Fig. 4e). As $\xi$ varies from 0 to 1 the SOC-
induced splitting increases, but the fraction of the $\Gamma$S path where it
dominates over the AM-induced splitting never surpasses $40\%$. Therefore, the
AM-induced splitting is still dominant along the $\Gamma$S line despite the
sizable SOC-induced spin splitting.

##  III Discussion and conclusion

In this work we have shown that in the non-relativistic limit, the monolayer
RuF4 represents a two-dimensional $d$-wave altermagnet. Using the type-I
magnetic space group, we have revealed the symmetry properties of the non-
relativistic spin splitting in the altermagnetic phase. By applying the theory
of invariants, we have derived a simple model Hamiltonian that captures the
spin-momentum locking of the top valence and the bottom conduction bands, both
of which can be activated for spin transport by doping. In the relativistic
limit, we have shown that spin-orbit coupling induces weak ferromagnetism and
significantly changes the spin splitting and the spin texture of bands. Using
the perturbation theory, we explained the appearance of a sizable SOC-induced
spin splitting at the time-reversal invariant $\Gamma$ point, which is
reminiscent of Zeeman splitting but occurs even in the limit of vanishing net
magnetic moment. We have shown that the magnetic moment induced due to canting
of Ru spins is linear in SOC strength. This implies that in altermagnetic
compounds containing heavy atoms, which exhibit very strong SOC, the
altermagnetic phase could be significantly altered by weak ferromagnetism.

Finally, it’s worth noting that Autieri et al.Autieri et al. (2023) recently
highlighted that the emergence of weak ferromagnetism in metallic altermagnets
is a property unparalleled in conventional antiferromagnets. As such, the
appearance of weak magnetization in metallic compounds can be used to
demarcate the conventional antiferromagnets from altermagnets. Yet, as we have
shown in this work, RuF4 is a semiconductor and we argue that the appearance
of SOC-induced weak ferromagnetism may be a general property of all the
altermagnetic compounds, not just metals. This becomes especially encouraging
once one realizes that the small component of the magnetic moment, induced by
SOC and perpendicular to the Néel vector, could be employed as a knob to
control the direction of dominant magnetic arrangement in altermagnets.
Indeed, the Néel vector in weak ferromagnets can be dragged by a small
external magnetic field using the experimental technique exposed in the work
of Dmitrienko et al.Dmitrienko et al. (2014). In this way one could hopefully
control the direction of spin-polarized currents in altermagnets. Therefore,
we strongly believe that the relation between weak ferromagnetism and
altermagnetism warrants greater attention, and would stimulate interest to
delve into this intriguing topic.

##  IV Methods

DFT calculations were performed using the VASP code Kresse and Furthmüller
(1996). The effects of electronic exchange and correlation were described at
the Generalized Gradient Approximation (GGA) level using the
Perdew–Burke–Ernzerhof (PBE) functional Perdew et al. (1996). The Kohn-Sham
wavefunctions were expanded on a plane wave basis set with a cutoff of
$500\,{\rm eV}$. We used the pseudopotentials with Ru 4p,5s,4d and F 2$s$,2$p$
states as valence states. The lattice constants $a_{1}=5.131\,{\rm\AA}$ and
$a_{2}=5.477\,{\rm\AA}$ were obtained from spin-polarized non-relativistic DFT
calculations using the Birch-Murnaghan equation of state and assuming the
antiparallel arrangement of Ru magnetic moments. In the out-of-plane
direction, we used the lattice constant of $a_{3}=20\,{\rm\AA}$ to ensure
sufficient separation between the periodic replicas. The atomic positions were
relaxed until all the forces’ components dropped below $0.002\,{\rm
eV/\AA^{2}}$. The Brillouin zone was sampled with $10\times 9\times 1$ ${\bf
k}$ points mesh during the relaxation and with a finer $15\times 14\times 1$
mesh during the self-consistent field calculations. To find the canting angle
of the Ru magnetic moment we performed non-collinear DFT calculations with SOC
and for a set of directions we constrained the magnetic moments using the
penalty functional Ma and Dudarev (2015). For producing the bandstructure and
spin texture plots we used the Pyprocar packageHerath et al. (2020). For
setting up DFT calculations and for structural visualization we used the
Atomic Simulation Environment Larsen et al. (2017), XCrySdenKokalj (1999), and
VestaMomma and Izumi (2011) packages.

###### Acknowledgements.

We acknowledge support by the Italian Ministry for Research and Education
through the Nanoscience Foundries and Fine Analysis (NFFA-Trieste, Italy)
project and through the PRIN-2017 project ”TWEET: Towards ferroelectricity in
two dimensions” (IT-MIUR grant No. 2017YCTB59). M.M, M.O., and S.S.
acknowledge the financial support provided by the Ministry of Education,
Science, and Technological Development of the Republic of Serbia. This project
has received funding from the European Union’s Horizon 2020 Research and
Innovation Programme under the Programme SASPRO 2 COFUND Marie Sklodowska-
Curie grant agreement No. 945478. M.G. acknowledges financial support provided
by Slovak Research and Development Agency provided under Contract No. APVV-SK-
CZ-RD-21-0114 and by the Ministry of Education, Science, Research and Sport of
the Slovak Republic provided under Grant No. VEGA 1/0695/23 and Slovak Academy
of Sciences project IMPULZ IM-2021-42 and project FLAG ERA JTC 2021 2DSOTECH.
The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V.
for providing computational resources on the GCS Supercomputer SuperMUC-NG at
Leibniz Supercomputing Centre and the Italian supercomputing centre CINECA for
providing computational resources through the ISCRA initiative, in particular
projects ISCRA-C HP10C6WZ1O and ISCRA-B HP10BA00W3.

## References

## References

  * Žutić et al. (2004) I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
  * Jungwirth et al. (2016) T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11, 231 (2016), ISSN 1748-3395.
  * Šmejkal et al. (2018) L. Šmejkal, Y. Mokrousov, B. Yan, and A. H. MacDonald, Nat. Phys. 14, 242 (2018), ISSN 1745-2481.
  * Šmejkal et al. (2022a) L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 031042 (2022a), ISSN 2160-3308.
  * Šmejkal et al. (2022b) L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022b), ISSN 2160-3308.
  * Yuan et al. (2020) L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Phys. Rev. B 102, 014422 (2020), ISSN 2469-9969.
  * Yuan et al. (2021) L.-D. Yuan, Z. Wang, J.-W. Luo, and A. Zunger, Phys. Rev. Mater. 5, 014409 (2021), ISSN 2475-9953.
  * Yuan and Zunger (2023) L.-D. Yuan and A. Zunger, Adv. Mater. 35, 2211966 (2023), ISSN 0935-9648.
  * Tschirner et al. (2023) T. Tschirner, P. Ke${\rm\beta}$ler, R. D. G. Betancourt, T. Kotte, D. Kriegner, B. Büchner, J. Dufouleur, M. Kamp, V. Jovic, L. Smejkal, et al., APL Mater. 11, 101103 (2023).
  * Guo et al. (2023a) S.-D. Guo, X.-S. Guo, K. Cheng, K. Wang, and Y. S. Ang, Appl. Phys. Lett. 123, 082401 (2023a).
  * Šmejkal et al. (2023) L. Šmejkal, A. Marmodoro, K.-H. Ahn, R. González-Hernández, I. Turek, S. Mankovsky, H. Ebert, S. W. D’Souza, O. Šipr, J. Sinova, et al., Phys. Rev. Lett. 131, 256703 (2023), ISSN 1079-7114.
  * Bai et al. (2023) H. Bai, Y. C. Zhang, Y. J. Zhou, P. Chen, C. H. Wan, L. Han, W. X. Zhu, S. X. Liang, Y. C. Su, X. F. Han, et al., Phys. Rev. Lett. 130, 216701 (2023).
  * Sun and Linder (2023) C. Sun and J. Linder, Phys. Rev. B 108, L140408 (2023), ISSN 2469-9969.
  * Ouassou et al. (2023) J. A. Ouassou, A. Brataas, and J. Linder, Phys. Rev. Lett. 131, 076003 (2023).
  * Beenakker and Vakhtel (2023) C. W. J. Beenakker and T. Vakhtel, Phys. Rev. B 108, 075425 (2023).
  * Sun et al. (2023) C. Sun, A. Brataas, and J. Linder, Phys. Rev. B 108, 054511 (2023).
  * Guo et al. (2023b) Y. Guo, H. Liu, O. Janson, I. C. Fulga, J. van den Brink, and J. I. Facio, Materials Today Physics 32, 100991 (2023b), ISSN 2542-5293.
  * Biniskos et al. (2023) N. Biniskos, F. J. dos Santos, M. dos Santos Dias, S. Raymond, K. Schmalzl, P. Steffens, J. Persson, N. Marzari, S. Blügel, S. Lounis, et al., APL Mater. 11, 081103 (2023).
  * Mazin (2023) I. I. Mazin, Phys. Rev. B 107, L100418 (2023).
  * Junior et al. (2023) P. E. F. Junior, K. A. de Mare, K. Zollner, K.-h. Ahn, S. I. Erlingsson, M. van Schilfgaarde, and K. Výborný, Phys. Rev. B 107, L100417 (2023).
  * Feng et al. (2022) Z. Feng, X. Zhou, L. Šmejkal, L. Wu, Z. Zhu, H. Guo, R. González-Hernández, X. Wang, H. Yan, P. Qin, et al., Nat. Electron. 5, 735 (2022), ISSN 2520-1131.
  * Reimers et al. (2023) S. Reimers, L. Odenbreit, L. Smejkal, V. N. Strocov, P. Constantinou, A. B. Hellenes, R. J. Ubiergo, W. H. Campos, V. K. Bharadwaj, A. Chakraborty, et al., arXiv (2023), eprint 2310.17280.
  * Brekke et al. (2023) B. Brekke, A. Brataas, and A. Sudbø, Phys. Rev. B 108, 224421 (2023), ISSN 2469-9969.
  * Cui et al. (2023) Q. Cui, B. Zeng, P. Cui, T. Yu, and H. Yang, Phys. Rev. B 108, L180401 (2023), ISSN 2469-9969.
  * Mazin et al. (2023) I. Mazin, R. González-Hernández, and L. Šmejkal, arXiv (2023), eprint 2309.02355.
  * Mermin and Wagner (1966) N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966), ISSN 1079-7114.
  * Casteel et al. (1992) W. J. Casteel, Jr., A. P. Wilkinson, H. Borrmann, R. E. Serfass, and N. Bartlett, Inorg. Chem. 31, 3124 (1992), ISSN 0020-1669.
  * Litvin and Wike (1991) D. B. Litvin and T. R. Wike, in _Character Tables and Compatibility Relations of The Eighty Layer Groups and Seventeen Plane Groups_ (Springer, Boston, MA, Boston, MA, USA, 1991), pp. 18–26, ISBN 978-1-4613-0495-1.
  * Moriya (1960) T. Moriya, Phys. Rev. 120, 91 (1960), ISSN 1536-6065.
  * Wang et al. (2022) N. Wang, J. Chen, N. Ding, H. Zhang, S. Dong, and S.-S. Wang, Phys. Rev. B 106, 064435 (2022), ISSN 2469-9969.
  * Bradley and Davies (1968) C. J. Bradley and B. L. Davies, Rev. Mod. Phys. 40, 359 (1968), ISSN 1539-0756.
  * Autieri et al. (2023) C. Autieri, R. M. Sattigeri, G. Cuono, and A. Fakhredin, arXiv (2023), eprint 2312.07678.
  * Dmitrienko et al. (2014) V. E. Dmitrienko, E. N. Ovchinnikova, S. P. Collins, G. Nisbet, G. Beutier, Y. O. Kvashnin, V. V. Mazurenko, A. I. Lichtenstein, and M. I. Katsnelson, Nat. Phys. 10, 202 (2014), ISSN 1745-2481.
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996), ISSN 2469-9969.
  * Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996), ISSN 1079-7114.
  * Ma and Dudarev (2015) P.-W. Ma and S. L. Dudarev, Phys. Rev. B 91, 054420 (2015), ISSN 2469-9969.
  * Herath et al. (2020) U. Herath, P. Tavadze, X. He, E. Bousquet, S. Singh, F. Muñoz, and A. H. Romero, Comput. Phys. Commun. 251, 107080 (2020), ISSN 0010-4655.
  * Larsen et al. (2017) A. H. Larsen, J. J. Mortensen, J. Blomqvist, I. E. Castelli, R. Christensen, M. Dułak, J. Friis, M. N. Groves, B. Hammer, C. Hargus, et al., J. Phys.: Condens. Matter 29, 273002 (2017), ISSN 0953-8984.
  * Kokalj (1999) A. Kokalj, J. Mol. Graphics Model. 17, 176 (1999), ISSN 1093-3263.
  * Momma and Izumi (2011) K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011), ISSN 0021-8898.

##  Appendix A General constraints on spin expectation values in the
relativistic case

Let $|{\psi_{k_{x},k_{y}}^{n}}\rangle$ represents an eigenstate of the
altermagnetic Hamiltonian in a relativistic phase $\mathcal{H}_{AM}^{\rm rel}$

$\mathcal{H}_{AM}^{\rm
rel}|{\psi_{k_{x},k_{y}}^{n}}\rangle=E_{n}(k_{x},k_{y})|{\psi_{k_{x},k_{y}}^{n}}\rangle,$
(8)

with group symmetry ${\bf G}=p2_{1}/b11$ of both the relativistic and/or
altermagnetic phase, since the magnetic space group is equal to the
crystallographic group, see Section II.2. The commutation relation
$[d(g),\mathcal{H}_{AM}^{\rm rel}]=0$ holds for any element $g\in{\bf G}$,
where $d$ is the representation of the group acting in the Hilbert space of
the Hamiltonian $\mathcal{H}_{AM}^{\rm rel}$. For a given state
$|{\psi_{k_{x},k_{y}}^{n}}\rangle$, the spin expectation values are defined as

$\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=\langle\psi_{k_{x},k_{y}}^{n}|s_{i}|\psi_{k_{x},k_{y}}^{n}\rangle,\quad
i=x,y,z.$ (9)

Using the action of the group element $g\in{\bf G}$ in the Hilbert space in
which the Hamiltonian $\mathcal{H}$ acts, the previous equation can be
transformed to

$\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=\langle\psi_{k_{x},k_{y}}^{n}|d^{{\dagger}}(g)d(g)s_{i}d^{{\dagger}}(g)d(g)|\psi_{k_{x},k_{y}}^{n}\rangle$
(10)

By analyzing the action of nontrivial group elements $g_{1}=\mathcal{I}$,
$g_{2}=(U_{x}|t_{\frac{1}{2}})$ and $g_{1}g_{2}$, one obtains general
constraints on the spin expectation values

$\displaystyle g_{1}:$ $\displaystyle\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=\langle s_{i}\rangle^{n}_{-k_{x},-k_{y}},$
$\displaystyle g_{2}:$ $\displaystyle\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=(-1)^{1-\delta_{i,x}}\langle
s_{i}\rangle^{n}_{k_{x},-k_{y}},$ $\displaystyle g_{1}g_{2}:$
$\displaystyle\langle
s_{i}\rangle^{n}_{k_{x},k_{y}}=(-1)^{1-\delta_{i,x}}\langle
s_{i}\rangle^{n}_{-k_{x},k_{y}},$ (11)

where $\delta_{i,x}$ is the Kronecker delta, equal to 1 if $i=x$ and 0
otherwise. To obtain the previous relations, we have used the following
identities

$\displaystyle g_{1}:$ $\displaystyle
d(g_{1})s_{i}d^{{\dagger}}(g_{1})=s_{i},\phantom{xxxxxxxxxx}d(g_{1})|\psi_{k_{x},k_{y}}^{n}\rangle=|\psi_{-k_{x},-k_{y}}^{n}\rangle,$
$\displaystyle g_{2}:$ $\displaystyle
d(g_{2})s_{i}d^{{\dagger}}(g_{2})=(-1)^{1-\delta_{i,x}}s_{i},\phantom{xxx}d(g_{2})|\psi_{k_{x},k_{y}}^{n}\rangle=|\psi_{k_{x},-k_{y}}^{n}\rangle$
$\displaystyle g_{1}g_{2}:$ $\displaystyle
d(g_{1}g_{2})s_{i}d^{{\dagger}}(g_{1}g_{2})=(-1)^{1-\delta_{i,x}}s_{i},d(g_{1}g_{2})|\psi_{k_{x},k_{y}}^{n}\rangle=|\psi_{-k_{x},k_{y}}^{n}\rangle.$
(12)

In short, the set of equations A show that in a system with $p2_{1}/b11$
symmetry the spin texture $\langle s_{x}\rangle$ can accumulate, yielding a
finite net magnetic moment in the $x$-direction, whereas the $\langle
s_{y}\rangle$ and $\langle s_{z}\rangle$ from different quadrants of the BZ
cancel out, as explained in the main text and illustrated in Fig. 3b.

Furthermore, the equations A in the special cases of the $\Gamma$ point and
along the $\Gamma$X and $\Gamma$Y lines, suggest that spin expectation values
are nonzero in the $x$-direction solely since for $k_{y}=0$ ($\Gamma$X line)
we have

$\displaystyle\langle s_{x}\rangle^{n}_{k_{x},0}$ $\displaystyle=$
$\displaystyle\langle s_{x}\rangle^{n}_{k_{x},0},$ $\displaystyle\langle
s_{y}\rangle^{n}_{k_{x},0}$ $\displaystyle=$ $\displaystyle-\langle
s_{y}\rangle^{n}_{k_{x},0}=0,$ $\displaystyle\langle
s_{z}\rangle^{n}_{k_{x},0}$ $\displaystyle=$ $\displaystyle-\langle
s_{z}\rangle^{n}_{k_{x},0}=0,$ (13)

while along the $\Gamma$Y line ($k_{y}=0$) one gets

$\displaystyle\langle s_{x}\rangle^{n}_{0,k_{y}}$ $\displaystyle=$
$\displaystyle\langle s_{x}\rangle^{n}_{0,k_{y}},$ $\displaystyle\langle
s_{y}\rangle^{n}_{0,k_{y}}$ $\displaystyle=$ $\displaystyle-\langle
s_{y}\rangle^{n}_{0,k_{y}}=0,$ $\displaystyle\langle
s_{z}\rangle^{n}_{0,k_{y}}$ $\displaystyle=$ $\displaystyle-\langle
s_{z}\rangle^{n}_{0,k_{y}}=0.$ (14)

This conclusion is in agreement with the band structure plotted in Fig. 4
which shows that along the X$\Gamma$Y path the $x$-component of spin is
nonzero (blue and red bands Fig. 4a) while the $y$-component equals to zero
(gray bands Fig. 4b).

Also, it should be mentioned that the conclusions agree with the ones given in
Section B, where we have shown that the SOC is solely responsible for the
observed splitting along the high-symmetry lines $\Gamma$X and $\Gamma$Y, as
well as in the $\Gamma$ point.

##  Appendix B Perturbation theory of SOC-induced removal of spin degeneracy
at arbitrary $k$ point

The spin splitting at the $\Gamma$ point and along the $\Gamma$X and $\Gamma$Y
lines can be explained using the perturbation theory, where the SOC
Hamiltonian is introduced as a perturbation. To do this, we apply a first-
order degenerate perturbation theory, valid for bands not only along the
X$\Gamma$Y path, but also at other $k$ points of the Brillouin zone which host
the degenerate bands in the altermagnetic phase.

We describe the two degenerate states as
$|{\Psi_{1}}\rangle=|{\psi_{1}}\rangle\otimes|{\uparrow}\rangle$ and
$|{\Psi_{2}}\rangle=|{\psi_{2}}\rangle\otimes|{\downarrow}\rangle$, where
$|{\psi_{1/2}}\rangle$ corresponds to the orbital part of the wave function,
while $|{\uparrow/\downarrow}\rangle$ are eigenstates of the Pauli
$\sigma_{y}$ operator for the eigenvalues 1 and $-1$, respectively. We chose
$\sigma_{y}$ because we assumed that the spin quantization axis is along the
$y$-direction. Now, if the $\mathcal{H}^{\rm nrel}_{\rm AM}$ is the non-
relativistic one-electron Hamiltonian (AM emphasizes here that this is the
non-relativistic Hamiltonian of the altermagnetic phase), then the following
relation holds

$\mathcal{H}^{\rm nrel}_{\rm AM}|{\Psi_{1/2}}\rangle=E|{\Psi_{1/2}}\rangle.$
(15)

In the orbital space the action of the group element $g_{2}$ is through the
representation $d_{O}(g_{2})$, while in the magnetic space acts as a
representation $d_{M}$, $d_{M}(g_{2})=\big{(}\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\big{)}$, which flips the magnetic moments
$d_{M}(g_{2})|{\uparrow/\downarrow}\rangle=|{\downarrow/\uparrow}\rangle$.
Using the commutation relation between the Hamiltonian and the elements of the
group, $[d_{O}(g_{2})\otimes d_{M}(g_{2}),\mathcal{H}^{\rm nrel}_{\rm AM}]=0$,
one can say that

$\displaystyle\Big{(}d_{O}(g_{2})\otimes d_{M}(g_{2})\Big{)}\mathcal{H}^{\rm
nrel}_{\rm AM}|{\psi_{1}}\rangle\otimes|{\uparrow}\rangle$ $\displaystyle=$
$\displaystyle\Big{(}d_{O}(g_{2})\otimes
d_{M}(g_{2})\Big{)}E|{\psi_{1}}\rangle\otimes|{\uparrow}\rangle$
$\displaystyle\mathcal{H}^{\rm nrel}_{\rm
AM}\Big{(}d_{O}(g_{2})|{\psi_{1}}\rangle\Big{)}\otimes|{\downarrow}\rangle$
$\displaystyle=$ $\displaystyle
E\Big{(}d_{O}(g_{2})|{\psi_{1}}\rangle\Big{)}\otimes|{\downarrow}\rangle.$
(16)

The last relation is equal to $\mathcal{H}^{\rm nrel}_{\rm
AM}|{\psi_{2}}\rangle\otimes|{\downarrow}\rangle=E|{\psi_{2}}\rangle\otimes|{\downarrow}\rangle$,
which leads us to conclude that the relation
$d_{O}(g_{2})|{\psi_{1}}\rangle=|{\psi_{2}}\rangle$ must hold. Furthermore,
using the fact that $d_{O}(g_{2})d_{O}(g_{2})=d_{O}(g_{2}^{2})=I$, where $I$
is the identity matrix, one can conclude that
$d_{O}(g_{2}^{2})|{\psi_{1}}\rangle=|{\psi_{1}}\rangle$, which suggests that
the relation $d_{O}(g_{2})|{\psi_{2}}\rangle=|{\psi_{1}}\rangle$ holds.
Therefore, the group element $g_{2}$ interchanges the orbital parts of two
degenerate states. This will be useful for calculating the matrix elements of
the SOC operator in the $\\{|\Psi_{1}\rangle,|\Psi_{2}\rangle\\}$ basis.

To account for the SOC effects we use the first-order perturbation theory and
calculate the allowed matrix elements of the operator $H_{\rm
SOC}=({\bm{\nabla}}V\times{\bf p})\cdot{\bm{\sigma}}$, where ${\bm{\nabla}}V$
represents the gradient of the crystal potential, ${\bf p}$ the momentum
operator, and ${\bm{\sigma}}$ the Pauli operator. Now, the fact that
$({\bm{\nabla}}V\times{\bf p})$ transforms as a pseudovector allows us to
rewrite the SOC Hamiltonian as $H_{\rm SOC}=\lambda{\bf L}\cdot{\bm{\sigma}}$,
where $\lambda$ describes the SOC strength. Note that ${\bf L}$ is not the
angular momentum but it transforms like one. Now, the matrix elements of
$H_{\rm SOC}$ in the $\\{|\Psi_{1}\rangle,|\Psi_{2}\rangle\\}$ basis are

$\langle H_{\rm
SOC}\rangle=\lambda\begin{pmatrix}\langle\psi_{1}|L_{y}|\psi_{1}\rangle&{\rm
i}\langle\psi_{1}|L_{x}|\psi_{2}\rangle-\langle\psi_{1}|L_{z}|\psi_{2}\rangle\\\
-{\rm
i}\langle\psi_{2}|L_{x}|\psi_{1}\rangle-\langle\psi_{2}|L_{z}|\psi_{1}\rangle&-\langle\psi_{2}|L_{y}|\psi_{2}\rangle\end{pmatrix}.$
(17)

Here we have used that the Pauli operators in a given spin basis
$\\{|{\uparrow}\rangle=\frac{1}{\sqrt{2}}\big{(}-{\rm
i}|{+}\rangle+|{-}\rangle\big{)},|{\downarrow}\rangle=\frac{1}{\sqrt{2}}\big{(}{\rm
i}|{+}\rangle+|{-}\rangle\big{)}\\}$ ($|{\pm}\rangle$ are eigenvectors of the
$\sigma_{z}$ operator, $\sigma_{z}|{\pm}\rangle=\pm|{\pm}\rangle$) are equal
to $\sigma_{x}=\big{(}\begin{smallmatrix}0&{\rm i}\\\ {-\rm
i}&0\end{smallmatrix}\big{)}$, $\sigma_{y}=\big{(}\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\big{)}$, $\sigma_{z}=\big{(}\begin{smallmatrix}0&-1\\\
-1&0\end{smallmatrix}\big{)}$. The symmetry argument in this case gives us

$\displaystyle\langle\psi_{1}|L_{y}|\psi_{1}\rangle$ $\displaystyle=$
$\displaystyle\langle\psi_{1}|d_{O}^{{\dagger}}(g_{2})d_{O}(g_{2})L_{y}d_{O}^{{\dagger}}(g_{2})d_{O}(g_{2})|\psi_{1}\rangle=-\langle\psi_{2}|L_{y}|\psi_{2}\rangle=\ell_{y},$
(18)

where we have used that $d_{O}(g_{2})L_{y}d_{O}^{{\dagger}}(g_{2})=-L_{y}$ and
that $\ell_{y}$ is the real number. Also, we calculate the matrix elements
$\langle\psi_{1}|L_{x}|\psi_{2}\rangle$ and
$\langle\psi_{1}|L_{z}|\psi_{2}\rangle$,

$\displaystyle\ell_{x}^{12}=\langle\psi_{1}|L_{x}|\psi_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\psi_{1}|d_{O}^{{\dagger}}(g_{2})d_{O}(g_{2})L_{x}d_{O}^{{\dagger}}(g_{2})d_{O}(g_{2})|\psi_{2}\rangle=\langle\psi_{2}|L_{x}|\psi_{1}\rangle=(\ell_{x}^{12})^{*},$
$\displaystyle\ell_{z}^{12}=\langle\psi_{1}|L_{z}|\psi_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\psi_{1}|d_{O}^{{\dagger}}(g_{2})d_{O}(g_{2})L_{z}d_{O}^{{\dagger}}(g_{2})d_{O}(g_{2})|\psi_{2}\rangle=-\langle\psi_{2}|L_{z}|\psi_{1}\rangle=-(\ell_{z}^{12})^{*},$

and use the relations $d_{O}(g_{2})L_{x}d_{O}^{{\dagger}}(g_{2})=L_{x}$ and
$d_{O}(g_{2})L_{z}d_{O}^{{\dagger}}(g_{2})=-L_{z}$.

The conditions obtained in (B) imply that
$\ell_{x}^{12}=\ell_{x}\in\mathcal{R}$ and $\ell_{z}^{12}={\rm i}\ell_{z}$,
where $\ell_{z}\in\mathcal{R}$, allowing us to simplify $\langle H_{\rm
SOC}\rangle$ to

$\langle H_{\rm SOC}\rangle=\lambda\begin{pmatrix}\ell_{y}&{\rm
i}(\ell_{x}-\ell_{z})\\\ -{\rm i}(\ell_{x}-\ell_{z})&\ell_{y}\end{pmatrix}.$
(20)

Using the final form of $\langle H_{\rm SOC}\rangle$ given by Eq. (20), the
SOC-induced energy contribution to the degenerate level in the altermagnetic
case is equal $\lambda\Big{(}\ell_{y}\pm(\ell_{x}-\ell_{z})\Big{)}$. The spin
splitting due to the SOC is equal to $2\lambda(\ell_{x}-\ell_{z})$.
Furthermore, by calculating the eigenvectors of $E_{0}I_{2}+\langle H_{\rm
SOC}\rangle$, where $E_{0}$ represents the energy of the double degenerate
bands, while $I_{2}$ is the $2\times 2$ identity matrix in the eigenbasis
$\\{|{\uparrow}\rangle,|{\downarrow}\rangle\\}$ of the altermagnetic phase, we
find that the spin expectation values have the nonzero component in the
$x$-direction only, consistent with the general conclusion from Appendix A.
However, the first-order perturbation theory predicts $\langle
s_{x}\rangle=\pm 1$ (in the units of $\hbar$/2) at the $\Gamma$ point, which
is not consistent with the data obtained from first-principles calculations,
where $\langle s_{x}\rangle<1$. This implies that the higher-order
perturbation theory needs to be applied to account for the interaction with
the bands higher in energy, responsible for the reduction of the intensity of
the spin expectation value.