# Indoor Localization Algorithm For Smartphones

Kaiqing Zhang∗, Hong Hu∗, Wenhan Dai†, Yuan Shen‡ and Moe Z. Win†
∗Department of Automation, Tsinghua University, Beijing 100084, China
†Laboratory for Information and Decision Systems,
Massachusetts Institute of Technology, Cambridge, MA 02139
‡Department of Electronic Engineering, Tsinghua University, Beijing 100084,
China
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>∗ The first two authors
contribute equally to this work.

###### Abstract

Increasing sources of sensor measurements and prior knowledge have become
available for indoor localization on smartphones. How to effectively utilize
these sources for enhancing localization accuracy is an important yet
challenging problem. In this paper, we present an area state-aided
localization algorithm that exploits various sources of information.
Specifically, we introduce the concept of _area state_ to indicate the area
where the user is on an indoor map. The position of the user is then estimated
using inertial measurement unit (IMU) measurements with the aid of area
states. The area states are in turn updated based on the position estimates.
To avoid accumulated errors of IMU measurements, our algorithm uses WiFi
received signal strength indicator (RSSI) to indicate the vicinity of the user
to the routers. The experiment results show that our system can achieve
satisfactory localization accuracy in a typical indoor environment.

##  I Introduction

GPS-based localization systems can provide satisfactory performance in outdoor
scenarios, while indoor localization systems still confront some challenges in
terms of robustness, accuracy and responsiveness. The reason for these
challenges lies in the difficulty to obtain reliable measurements at low cost
and to integrate multiple information sources effectively. Many efforts have
been made to study the indoor localization problem from different perspectives
[1, 2, 3, 4, 5, 6, 7, 8]. Most of them can be categorized into two main
categories, i.e., infrastructure-based and infrastructure-free approaches.
Infrastructure-based approaches require to deploy wireless access points in
the indoor environment and are performed on the basis of wireless measurements
such as WiFi, UWB, and ultrasound. Among all these approaches, WiFi-based ones
attract much attention since WiFi routers are ubiquitous and no additional
infrastructure is needed for localization. The WiFi-based approaches can be
subdivided into two types: fingerprinting [1] and ranging-based methods [2],
both using RSSI to estimate users’ positions. However, range-based methods
often cause large localization errors due to the instability of WiFi signals
in indoor environments and fingerprinting requires laborious off-line training
and is not robust to environment changes [3]. Infrastructure-free approaches
rely primarily on IMUs to locate a user by continuously estimating the
displacement of positions from a known location, referred to as the dead-
reckoning approach. A typical example is the pedestrian navigation system
(PNS) [5, 4, 6], which performs localization by detecting steps and estimating
headings. While most of these existing PNSs depend on high-quality IMU sensors
[5, 6], achieving satisfactory performance using the IMUs on smartphones is a
challenging task. In addition to the measurements from WiFi routers and IMUs,
an indoor map is also a valuable source of information for localization. Since
walking patterns are partially determined by the geographical structure of the
environments, the indoor map serves as the prior knowledge of space
constraints in localization. A particle filter-based algorithm that combines
map information and WiFi RSSI was proposed in [10]. In [11], the algorithm
improves the localization accuracy by matching the walking trajectories onto
the map. To the best of the authors’ knowledge, few studies have fully
exploited map information in multi-sensor indoor localization. In this paper,
we develop a localization algorithm fusing the information from IMUs, indoor
maps as well as WiFi RSSI and implement a real-time system on Android
smartphones. We introduce the concept of _area state_ in the indoor map
representation and develop different localization approaches according to the
_area states_. System implementation on Android platforms demonstrates the
performance improvement achieved by exploiting map information.

##  II System Design

Figure 1: Framework of the system.

The framework of our system is illustrated in Fig. 1. In our system, the
states of the user are categorized into two types: the positional state and
the area state. The positional state refers to the coordinates of the user’s
position on a 2D map and the heading direction of the current step. The area
state refers to a particular area the user locates in, e.g., a certain
corridor. At each step, the positional states are estimated based on IMU
measurements with the aid of area states, and the area states are then updated
by new positional states based on indoor map information. We also conduct
experiments about WiFi RSSI in indoor environments and utilize the WiFi
routers as _vicinity indicators_. More details about the algorithm are
described as below.

###  II-A Dead-Reckoning using IMU Measurements

Although the accelerometer and gyroscope measurements on the phone are
accurate according to our experiments, severe drift can hardly be avoided if
the trajectories are obtained through double integration of the acceleration
measurements. Therefore, we resort to the dead-reckoning approach based on
detecting steps and step headings where the initial location and the step
length of the user are assumed to be available. Several regular patterns
emerge in the accelerometer measurements when the user is walking with the
smartphone in hand. We use the step detection method as in [6] to detect the
sharp drop of acceleration when the user is balancing out his/her steps. We
estimate heading through integration of filtered angular velocity, which is
not affected by the severe interferences from indoor electrical appliances and
magnetic materials. The initial heading and posture of the phone is assumed to
be available at the beginning of the localization. The turning can be detected
by the abrupt change of the azimuth angle if the phone remains fixed relative
to the user.

###  II-B Indoor Map Representation

To the best of the authors’ knowledge, there are few open-source geographical
data of indoor maps. The incompleteness of geographical data makes it
challenging to generalize map-based open-source projects to various sites.
Another challenge exists in the lack of standard representation rules to
define all the elements in an indoor map. Hence, we establish some intuitive
and facile representation rules for our prototype, and verify the validity of
our algorithm under these rules.

TABLE I: Basic Elements in Formulated Indoor Maps Type | Area State | Element | Description
---|---|---|---
Node | Intersection | Turning Point | Conjunction between corridors
Door | Entrance of an open area
Open Area | Room & Lobby | Relatively empty space
| Indicator | WiFi Router
Way | Corridor | Corridor | Long and narrow passage
| Wall | Constraints enclosing a room
| DoorCorridor | Connection between door & room
| WallCorridor | Connection between wall & room
Figure 2: The schematic diagram of area state representation.

We establish our indoor maps from OpenStreetMap (OSM)[12], an open source
geographical database. Map data from OSM can be accessed as an eXtensible
Markup Language (XML) file consisting of nodes, ways and relations, which can
be annotated and labeled with key-value-pairs named as tags. These tags allow
us to set the rules by labeling the nodes as intersections and open areas, and
the ways as corridors and walls. All the basic elements in an indoor map are
shown in Table I. According to the map structure, we divide the indoor
environment into three main types of _area states_ : open areas, intersections
and corridors. The schematic diagram of area state representation on the map
is presented in Fig. 2.

Figure 3: Bayesian network representation of the variables in our algorithm.

We model the time evolution of the joint positional and area states as well as
the measurements as a hidden Markov model (HMM) shown in Fig. 3. The
positional state at step $k$ is denoted as
$\left(p_{x,k},p_{y,k},\mathbf{h}_{k}^{\text{T}}\right)^{\text{T}}$, which
contains the 2D coordinates on the map and the heading at step $k$.
Specifically, the heading of the user is represented by a unit direction
vector
$\mathbf{h}_{k}={\left(\cos\theta_{k},\sin\theta_{k}\right)}^{\text{T}}$,
where $\theta_{k}$ denotes the heading angle. We denote the area state of the
user at step $k$ as $\alpha_{k}$. Specifically, $\alpha_{k}$ determines the
way of positional state transition from step $k$ to $k+1$, e.g., it is a 1D
movement in the corridor or a 2D movement in an open area. The joint state
$\mathbf{x}_{k}=\left(p_{x,k},p_{y,k},\mathbf{h}_{k}^{\text{T}},\alpha_{k}\right)^{\text{T}}$
represents the combination of positional and area state at step $k$.
Meanwhile, the transition of area state results from the transition of
positional state. Accordingly, the observation of positional state at step $k$
is denoted as
$\mathbf{y}_{k}=(p^{o}_{x,k},p^{o}_{y,k},{\mathbf{h}^{o}_{k}}^{\text{T}})^{\text{T}}$,
where $\mathbf{h}^{o}_{k}$ is obtained by integrating the angular velocity
measurements and $p^{o}_{x,k}$ and $p^{o}_{y,k}$ are obtained by moving a step
length $s$ in the direction of $\mathbf{h}^{o}_{k}$, i.e.,
$p^{o}_{x,k}=s\cos\theta_{k}$, $p^{o}_{y,k}=s\sin\theta_{k}$. Based on
$\mathbf{y}_{k}$, the position estimate is further refined by the map
information if the user is aware of what area he/she is in. This is actually
how we incorporate map information to mitigate noises and uncertainties of
position estimate based on dead-reckoning. We elaborate the map-aided
localization process as follows.

####  II-B1 Walking in Corridor

Since the width of a corridor is normally several meters and much shorter than
its length, we assume that the user moves in one dimension when walking in the
corridor. We set a threshold $\eta_{\text{str}}$ for the heading angle to
determine whether the user is still walking in a corridor. If so, the area
state $\alpha_{k}$ remains to be _Corridor_ and the estimated heading is
calibrated by the orientation information of the corridor.

####  II-B2 Turning

If greater changes of heading angle than $\eta_{\text{str}}$ are detected and
the estimated position is close to any turning point on the map, it is likely
that the user is at a turning point. The location information of the turning
point can be used to calibrate the position estimate of the user. Moreover, we
determine which corridor the user is turning to by comparing the calculated
heading and the bearing of the other $m$ corridors $c_{1},c_{2},\cdots,c_{m}$,
connected with the turning point, whose bearings are
$\mathbf{b}_{1},\mathbf{b}_{2},\cdots,\mathbf{b}_{m}$ accordingly, i.e.

$\hat{c}=\operatorname*{argmax}\limits_{c_{i},i=1,2,\cdots,m}{\mathbf{h}^{o}_{k}}^{\text{T}}\mathbf{b}_{i}$

The area state $\alpha_{k}$ is then updated as _Intersection_.

####  II-B3 Verifying the Turning

If the next corridor detected is another long passage on the map, we introduce
a Maximum a Posteriori (MAP) detection algorithm to verify the turning. We do
not align the current heading with the bearing of the next corridor instantly
because chances are that the corridor selected is not the right track.

We model $\mathbf{h}^{o}_{j}$, the heading observation of the $j$-th step
detected on a corridor, as a Gaussian distribution with the mean of
$\mathbf{b}_{i}$ when the next corridor $c_{i}$ is given. The variance of the
Gaussian is set according to a number of experiments on walking patterns. We
use
$l_{j}(c_{i})=L({c_{i}|\mathbf{h}^{o}_{j-k+1},\mathbf{h}^{o}_{j-k+2},\cdots,\mathbf{h}^{o}_{j}})$
to represent the likelihood function of $c_{i}$ given observations from $k$
steps prior to $j$,
$\mathbf{h}^{o}_{j-k+1},\mathbf{h}^{o}_{j-k+2},\cdots,\mathbf{h}^{o}_{j}$.
Furthermore, we assume
$\mathbf{h}^{o}_{j-k+1},\mathbf{h}^{o}_{j-k+2},\cdots,\mathbf{h}^{o}_{j}$ are
conditionally independent on $c_{i}$. Then we obtain the likelihood function
as

$\displaystyle l_{j}(c_{i})$
$\displaystyle=L({c_{i}|\mathbf{h}^{o}_{j-k+1},\cdots,\mathbf{h}^{o}_{j}})$
$\displaystyle\propto
p(\mathbf{h}^{o}_{j-k+1},\mathbf{h}^{o}_{j-k+2},\cdots,\mathbf{h}^{o}_{j}|c_{i})$
$\displaystyle=p(\mathbf{h}^{o}_{j-k+1}|c_{i})\cdots
p(\mathbf{h}^{o}_{j}|c_{i})$

As the likelihood evolves from step $j-k+1$ to step $j$, likelihood values of
different hypotheses in $l_{j}$ will probably diverge, from which we can cut
off the hypotheses with lower likelihood values and keep those with higher
ones. Thus, we are able to determine the next corridor $\hat{c}^{\prime}$ by
evaluating the likelihood function of consecutive $k$ step headings
$\mathbf{h}^{o}_{j-k+1},\mathbf{h}^{o}_{j-k+2},\cdots,\mathbf{h}^{o}_{j}$
happen and make the hypothesis with the maximal likelihood the best guess,
where we assume the prior probability of each corridor is equal, i.e.,

$\hat{c}^{\prime}=\operatorname*{argmax}\limits_{c_{i},i=1,2,\cdots,m}l_{j}(c_{i})$

If $\hat{c}^{\prime}$ is the same as the corridor estimated $\hat{c}$ in the
first check, we align the current heading with the corridor’s bearing.
Otherwise, we change the current corridor to $\hat{c}^{\prime}$ and make $k$
steps’ length compensation in distance. Moreover, we will make another
verification after $k$ steps until a right track is confirmed. Then the area
state $\alpha_{k}$ is updated as _Corridor_.

####  II-B4 Walking in Open Area

If the next corridor is a _Door Corridor_ that connects a door and a room or a
lobby, the user is considered walking in the open area and so is the area
state updated. The user walks in a 2D area in this case with the space
constraints imposed by the walls, i.e., the trajectories are not allowed to
traverse the walls and the user can only walk along them if his/her
trajectories cross the walls. The entire process is summarized in Algorithm 1.

Algorithm 1 Map Information Aided Algorithm

1: Initialization:

2: Given the initial phone posture, positional and area states, proper
thresholds and step length.

3: Walk in corridor:

4: Walk in 1D along the corridor.

5: First Check:

6: Detect the turning and calibrate the location with the turning points.

7: if the user turns to a open space at a door then

8: go to: 19

9: end if

10: Verify:

11: Perform MAP detection for the next corridor given consecutive $k$
headings.

12: if the right track is confirmed then

13: Calibrate the heading with the corridor orientation.

14: go to: 3

15: else

16: Change the current corridor and the last $k$ positional states.

17: go to: 10

18: end if

19: Walk in open space:

20: Walk freely in 2D area with walls and obstructions as space constraints.

21: if the user turns to a corridor at a door then

22: go to: 3

23: end if

###  II-C WiFi Module

In our system, WiFi module is used to refine the estimates of positional and
area states. We first conduct experimental studies on the propagation of WiFi
signals and based on the experiment results, we show that it is suitable to
use WiFi routers as vicinity indicators in our localization system.

####  II-C1 Variation of RSSI with distance

We conducted two experiments to investigate how RSSI varies with distance,
when RSSI is measured on one Android smartphone. The experiments are conducted
on 6th floor of Dreyfoos Tower, Stata Center, MIT. The floor plan are shown in
Fig. 4. The WiFi routers we use work on 802.11ac standard. Meanwhile, we also
consider the influence of tester’s orientation on RSSI. In the first
experiment, we measure the RSSI at a 0.3 m interval within a range of 4.5 m
when the tester faces towards the WiFi router. In the second experiment, the
tester is back against the router and all the other conditions are kept the
same. The maximum distance is set to be 4.5 m, since we will use WiFi routers
as indicators and this only requires the knowledge of WiFi propagation
property within a short range. One hundred RSSI measurements are collected at
each place in both experiments and they are obtained consecutively in about 30
s.

Fig. 5 depicts the samples and average values of WiFi RSSI measured at 15
different distances. When the tester faces towards the router, the RSSI is
relatively sensitive to the distance change within a certain distance $\rho$ :
a 35% decrease from 0.3 m to 1.5 m, while insensitive if the distance
increases beyond $\rho\approx 2\text{m}$. In addition, RSSI sample values
within and out of $\rho$ can be separated by a certain threshold $\phi$ ($-40$
dBm typically). This threshold is time-invariant and can be the same for
routers of the same kind. However, if the tester is back against the router
and then the human body blocks the WiFi signal propagation, the RSSIs will be
insensitive to distances in this case. The experiment results are consistent
with the well-known two-slope model [13].

Figure 4: Floor plan of 6-th floor of Dreyfoos Tower, Stata Center, MIT.
Figure 5: Variation of RSSI as a function of the distance for AP 2.

####  II-C2 Human body effects on WiFi signals

We also investigate human body effects on WiFi signals. The RSSI is measured
and collected under four different scenarios: (1) mobile phone left alone on
the cart, (2) mobile phone placed on the cart and the user stands by, (3)
mobile phone held in hand and the user walks towards the router, (4) mobile
phone held in hand and the tester walks away from the router. The distance
between mobile phone and the router is around 18 m.

The statistics of RSSI under each scenario are presented in Fig. 6 and Table
II. First, by comparing scenario 1 and scenario 2-4, we can find that the
variance of RSSI is mainly caused by human body and other interferences, e.g.
interferences from other devices, are much less significant. Second, by
comparing scenario 3 and 4, we see that human body can cause great RSSI loss
if obstructing the direct path of signal propagation. We conclude that human
body effects make the WiFi signals unstable. In fact, unstable WiFi signals
may lead to large errors on WiFi-based localization, especially when the user
and the WiFi router are not close enough. Therefore, it is hard to obtain
accurate position estimates if we use the WiFi signals to directly estimate
the distance between the user and the router.

Figure 6: Statistical distribution of the RSSI values under 4 scenarios. TABLE II: Mean and variance of RSSI in 4 different scenarios. Scenario | 1 | 2 | 3 | 4
---|---|---|---|---
Mean of RSSI/dBm | -59.05 | -59.22 | -57.83 | -65.71
$\text{Variance of RSSI/dBm}^{2}$ | 0.46 | 1.95 | 4.22 | 9.45

####  II-C3 WiFi router as a vicinity indicator

The experiment results in Fig. 5 show that RSSI is above a certain threshold
only if the user is within a short range of a WiFi router. This implies that a
high RSSI indicates vicinity to a certain router. Therefore, our system uses
the RSSI in the way that if multiple RSSI sample values from certain router
are above $\phi$, we consider that the phone is within a distance of $\rho$
from that router and the positional states is calibrated using the router’s
location. This can avoid the deterioration of localization accuracy caused by
unstable WiFi signals and also reduce the accumulative errors of IMU
measurements.

##  III Implementation and Evaluation

In this section, we elaborate more details about the implementation and
evaluate the performance of the on-line localization system.

###  III-A Experiments

All the experiments are conducted on the smartphone SamSung Galaxy S4,
GT-I9500 running the Android operating system 4.4.3. Our algorithm is
performed and evaluated on the 6-th floor of Stata Center Dreyfoos Tower at
MIT.

####  III-A1 IMU Measurements

Figure 7: Step detection (a) and heading estimation (b).

For IMU experiments, we assume the user is holding the phone in hand with the
Y-axis roughly aligning with the heading and Z-axis pointing up to the sky
while walking. Small jolting or shaking can be well-compensated by our area
state-aided algorithm. We also exploit Butterworth low pass filter to smooth
the raw data and subtract the average of the first measurements to remove the
bias.

In our experiments, the smartphone is first held in hand. The variations of
the acceleration magnitude during walking are exploited to detect the steps.
Fig. 7 (a) implies the effectiveness of step detection. Then, the smartphone
is fixed flat on a cart. We push the cart along the _Lane 4, 3, 2_ , and _1_.
It is shown in Fig. 7 (b) that the variations of azimuth estimates after a
circle are way below $360$ degrees due to the bias. The estimation errors can
be reduced to less than $2$ degrees after bias removal.

####  III-A2 Map Representation Process

We employ JOSM, an open-source map editing software, and the floor plan, to
generate the map data in XML format. The floor map is required to be
calibrated both in orientation and scale at first. Then some significant
feature points, such as turning points and doors are labeledand the relations
of the map elements pairs, e.g., doors and rooms, are represented by the links
between them. The absolute positions and orientations of the elements can be
further calculated from the longitude and latitude of each node. In general,
the process of aligning and labeling the map is very easy to operate and the
wiki-style editing makes it possible to construct map collaboratively.

(a) Anchors and Reference Points

Figure 8: Anchors, reference points and experiment trajectories. Figure 9:
Cumulative distribution function of localization errors.

###  III-B Performance Analysis

To evaluate the performance of our localization system, a user walks around
the 6th floor of Stata Center with the phone in hand. Three WiFi routers are
deployed around the lobby and their locations are marked on the map as shown
in Fig. 4.

As illustrated in Fig. 8, the real-time estimation of trajectories are
compared with the ground truth to justify the validity of our algorithm. It
shows that our system presents robust performance in corridors, which
attributes primarily to map constraints. The position estimates are calibrated
at turning points despite of some step misdetections or step length estimation
errors. Moreover, constant corrections of headings with the assistance of
corridors can well-compensate for the small attitude variations of the phone,
which allows _painless_ gestures of users during localization. In open area,
the trajectories are constrained by the walls and obstructions and refined by
the WiFi indicators. WiFi routers deployed at doors can also be exploited to
detect the position transitions of different map elements, e.g., from the
corridor to the lobby or vice verse. The localization errors of our algorithm
at the reference points, shown in Fig. 4, are recorded after ten repetitions
of experiments. Fig. 9 plots the cumulative distribution function (CDF) of
localization errors using different algorithms. It can be seen that the
proposed algorithm outperforms those algorithms not using either the area
states or the WiFi indicators. In particular, the average error is reduced
from 4.2 meters to 1.6 meters. Moreover, map information and WiFi indicators
provide different types of performance gains in terms of localization errors.
Map information can effectively mitigate small errors due to measurements
calibration while cause large errors when no absolute positional information,
e.g., WiFi indicators, is exploited. Nonetheless, the position refinement from
WiFi indicators is susceptible to small localization errors if the threshold
$\phi$ is not properly selected.

##  IV Conclusions

We designed and implemented a real-time indoor localization algorithm by
leveraging measurements from IMUs and WiFi RSSI with map information. The
proposed algorithm can effectively exploit map information and achieve
desirable localization accuracy with the aid of area states. Our localization
algorithm also uses WiFi RSSI to indicate the vicinity of the user to the
router. This system can serve as a prototype for the design of advanced indoor
localization system in the future.

## References

  * [1] P. Bahl and V. N. Padmanabhan, “RADAR: An in-building RF-based user location and tracking system,” in _IEEE Annu. Joint Conf. on Comput. and Commun. Soc._ , vol. 4, 2000, pp. 775–84.
  * [2] P. Bahl and V. N. Padmanabhan, “A comparative study on indoor localization based on RSSI measurement in wireless sensor network,” in _IEEE Int. Joint Conf. on Comput. Sci. and Software Eng._ , Nakhon, Pathom, May 2011, pp. 1–6.
  * [3] Z. Farid, R. Nordin, and M. Ismail, “Recent advances in wireless indoor localization techniques and system,” _J. Comput. Netw. Commun._ ,vol. 2013, pp. 1–10, Aug. 2013.
  * [4] O. Woodman and R. Harle, “Pedestrian localisation for indoor environments,” in _IEEE Int. Conf. on Ubiquitous Computing_ , New York, USA, Sept. 2008, pp. 114–123.
  * [5] B. Krach and P. Robertson, “Integration of foot-mounted inertial sensors into a Bayesian location estimation framework,” in _IEEE Workshop on Positioning, Navigation and Commun._ , Hannover, Germany, Mar. 2008, pp. 55–61.
  * [6] I. Skog, P. Handel, J. O. Nilsson, and J. Rantakokko, “Zero-velocity detection–An algorithm evaluation,” _IEEE Trans. Biomed. Eng._ , vol. 57, no. 11, pp. 2657–2666, Nov. 2010.
  * [7] M. Z. Win, A. Conti, S. Mazuelas, Y. Shen, W. M. Gifford, D. Dardari, and M. Chiani, “Network localization and navigation via cooperation,” IEEE Commun. Mag., vol. 49, no. 5, pp. 56–62, May 2011.
  * [8] Y. Shen and M. Z. Win, “Fundamental limits of wideband localization Part I: A general framework,” IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 4956–4980, Oct. 2010.
  * [9] D. Titterton and J. L. Weston, “Strapdown inertial navigation technology,” _IEEE RADAR, Sonar, Navigation and Avionics Series 5_ , Peter Peregrinus Ltd. UK 1997.
  * [10] F. Evennou, F. Marx, and E. Novakov, “Map-aided indoor mobile positioning system using particle filter,” in _IEEE Wireless Commun. and Networking Conf._ , vol. 4, Mar. 2005, pp. 2490–2494.
  * [11] J. A. B. Link, P. Smith, N. Viol, and K. Wehrle, “Footpath: Accurate map-based indoor navigation using smartphones,” in _IEEE Int. Conf. on Indoor Positioning and Indoor Navigation_ , Guimaraes, Sept. 2011, pp. 1–8.
  * [12] ——, “Indoor Mapping – OpenStreetMap Wiki,” Sept. 2014. [Online]. Available: http://wiki.openstreetmap.org/wiki/Indoor Mapping
  * [13] V. Erceg, L. J. Greenstein, S. Y. Tjandra, et al, “An empirically based path loss model for wireless channels in suburban environments,” _IEEE J. on Selected Areas in Commun.,_ vol. 17, no. 7, pp. 1205–1211, July 1999.


# NMR determination of van Hove singularity and Lifshitz transitions in nodal-
line semimetal ZrSiTe

Yefan Tian<EMAIL_ADDRESS>Present address: LASSP & Department of
Physics, Cornell University, Ithaca, NY 14853, USA  Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA  Yanglin Zhu
Department of Physics, Pennsylvania State University, University Park, PA
16802, USA  Rui Li  Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA  Zhiqiang Mao  Department of Physics,
Pennsylvania State University, University Park, PA 16802, USA  Joseph H. Ross,
Jr<EMAIL_ADDRESS>Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA

###### Abstract

We have applied nuclear magnetic resonance spectroscopy to study the
distinctive network of nodal lines in the Dirac semimetal ZrSiTe. The low-$T$
behavior is dominated by a symmetry-protected nodal line, with NMR providing a
sensitive probe of the diamagnetic response of the associated carriers. A
sharp low-$T$ minimum in NMR shift and $(T_{1}T)^{-1}$ provides a quantitative
measure of the dispersionless, quasi-2D behavior of this nodal line. We also
identify a van Hove singularity closely connected to this nodal line, and an
associated $T$-induced Lifshitz transition. A disconnect in the NMR shift and
line width at this temperature indicates the change in electronic behavior
associated with this topological change. These features have an orientation-
dependent behavior indicating a field-dependent scaling of the associated band
energies.

The recently discovered nodal-line semimetals (NLSMs) have been widely
investigated due to distinctive topological features including a high density
of bulk topological states and drumhead surface states. These present
intriguing possibilities for applications [1, 2, 3, 4, 5]. Among NLSMs, the
Zr$XY$ system ($X$ = Si, Ge, Sn; $Y$ = S, Se, Te) is the most studied, because
of their nodal-line network close to the Fermi energy [6]. This can lead to
enhanced correlation effects, recently indicated to be important in ZrSiSe and
ZriSiS [7, 8, 9] and associated with an unconventional mass enhancement and
predicted semimetal-exciton insulator transition [10, 11].

ZrSiTe is distinct among this family since it exhibits a non-symmorphic
symmetry-protected nodal line very close to the Fermi energy [12]. Its large
region of topologically protected drumhead surface states [13] as well as very
large spin Berry curvature and associated spin Hall and Nernst response [14]
suggest spintronic and related applications. Suggested changes of Fermi
surface topology in this family include a temperature-induced Lifshitz
transition in ZrSiSe [15], and a pressure-induced Lifshitz transition in
ZrSiTe [16] as well as indications of photo-induced phonon driven
transformation [17].

In this Letter, we have utilized nuclear magnetic resonance (NMR) spectroscopy
as a highly sensitive probe of states close to the Fermi energy. We show that
temperature-induced changes lead the chemical potential to coincide with the
protected node at low temperatures. In addition, based on density functional
theory (DFT) calculations, we demonstrate that the arrangement of parallel
Dirac nodes leads to a 2D van Hove singularity (VHS) near $E_{F}$. This leads
to temperature-induced changes, including a temperature-induced Lifshitz
transition.

Figure 1:  (a) Tetragonal crystal structure of ZrSiTe (space group: $P4/nmm$).
(b) Square-net structure of Si layer and Zr-Te layer. Figure 2:  (a) Angular
dependence of ZrSiTe NMR with a fit to $K=K_{\mathrm{iso}}+\Delta
K\cdot(3\cos^{2}\theta-1)/2$. Inset: representative crystal from growth batch.
Shift vs temperature for (b) $B\parallel a$ and (c) $B\parallel c$, with
spectra inset. Linear fits at high $T$ shown for both cases. Low-$T$ solid
curve in (c): orbital diamagnetism model. (d) $B\parallel c$ line width (open
circles) plotted with the shift (solid circles). Curves: guides to the eye.

For the experiments, the ZrSiTe single crystals were prepared using chemical
vapor transport method. The stoichiometric mixture of Zr, Si and Te powder was
sealed in a quartz tube with iodine being used as transport agent (2 mg/cm3).
Plate-like single crystals with metallic luster can be obtained via the vapor
transport growth with a temperature gradient from 950 ∘C to 850 ∘C. NMR
experiments utilized a custom spectrometer at fixed field $B=9$ T, with a
sample of six crystals from the same batch assembled edge to edge with axes
aligned, and with the field parallel to $c$ ($B\parallel c$) and $a$
($B\parallel a$). 125Te relative frequency shifts [$K=(f-f_{0})/f_{0}$] were
calibrated by aqueous Te(OH)6 and adjusted for its 707 ppm paramagnetic shift
to the dimethyltelluride standard ($f_{0}$) [18].

DFT calculations were also carried out using the PBE potential [19] as
implemented in the WIEN2k code [20]. Experimental structure parameters
$a=3.71$ $\mathrm{\SIUnitSymbolAngstrom}$, $c=9.51$
$\mathrm{\SIUnitSymbolAngstrom}$, with Zr $z=0.2238$ and Te $z=0.6399$, were
used [21, 22, 16] with cutoff parameter $k_{\mathrm{max}}=7/R_{\mathrm{MT}}$
inside the interstitial region. A mesh of 3000 $k$-points was employed for
initial calculations. Fermi surfaces and densities of states were also
calculated on a mesh of 10000 irreducible $k$-points and rendered using
XCrySDen [23].

125Te NMR spectra [insets, Fig. 2(b) and 2(c)] exhibit a peak corresponding to
the single Te site in ZrSiTe. The angular dependence of the shifts are shown
in Fig. 2(a) with $\theta$ defined in the inset. The results were fitted [red
curve in Fig. 2(a)] to $K=K_{\mathrm{iso}}+\Delta
K\cdot(3\cos^{2}\theta-1)/2$, where $K_{\mathrm{iso}}=547\pm 14$ ppm is the
isotropic shift and $\Delta K=1834\pm 23$ ppm.

The temperature-dependent shifts for $B\parallel a$ (denoted $K_{\parallel
a}$) and $B\parallel c$ ($K_{\parallel c}$) are shown in Figs. 2(b) and 2(c).
These exhibit V-shaped minima at $T_{\mathrm{min}}=20$ K, with a much larger
slope for $B\parallel c$. With ZrSiTe exhibiting a one-dimensional protected
Dirac nodal line very close to the Fermi energy [12], the large shift changes
for $K_{\parallel c}$ can be attributed to the diamagnetism associated with
this nodal line. At higher temperatures $K$ for both orientations continues to
increase, with the exception of the small drop in $K_{\parallel c}$ at 140 K.

As shown in Fig. 2(d), the $B\parallel c$ full width at half maximum (FWHM)
also exhibits a V-shaped anomaly with a maximum at 20 K (with the FWHM scale
inverted to properly match the shift). For $B\parallel a$ the width is smaller
and nearly constant. The $B\parallel c$ widths scale with $K_{\parallel c}$
below 140 K as shown in Fig. 2(d), but the change in slope above 140 K
indicates a change of broadening mechanism at high $T$. The width is scaled to
$-0.7$ times the shift in the figure, consistent with an origin for the
low-$T$ broadening due to a distribution of local fields associated with the
diagnetism of the plate-shaped crystals, with the maximum occurring where the
V-shaped diamagnetic shift has the largest magnitude.

Figure 3:  $(T_{1}T)^{-1}$ vs $T$ for $B\parallel a$ and $B\parallel c$ in (a)
linear and (b) log scale. Curves show orbital-mechanism calculation based on
protected Dirac nodal line. (c) Comparison between scaled $(T_{1}T)^{-1}$
values for $B\parallel c$ and unscaled $B\parallel a$ results.

Spin-lattice relaxation measured by inversion recovery was well-fitted to a
single exponential $M(t)=(1-Ce^{-t/T_{1}})M(\infty)$, giving $(T_{1}T)^{-1}$
values shown in Figs. 3(a) and 3(b). While in simple metals $(T_{1}T)^{-1}$ is
constant, we find sharp $(T_{1}T)^{-1}$ minima at $T_{\mathrm{min}}=20$ K for
both orientations, with an approximately linear increase on both sides. These
coincide with the observed shift minima.

At higher temperatures, there is a large orientation-dependence.
$(T_{1}T)^{-1}_{\parallel a}$ reaches a maximum at 100 K, and goes through a
minimum near 150 K, while for $B\parallel c$, these are shifted to 140 K and
225 K, respectively. The maximum at 140 K coincides with the feature in
$K_{\parallel c}$ and the break in the line width scaling [Fig. 2(d)], which
collectively indicates a change in electronic behavior at this temperature.

The band structure is shown in Fig. 4(a). In agreement with previous reports
[12], a gapless Dirac nodal line extends from R to X [$c^{*}$ direction, Fig.
4(c)], very close to $E_{F}$, and protected against spin-orbit induced gap
opening by non-symmorphic symmetry. From the band dispersions, we calculate
mean Fermi velocities ($v_{F}$) $5.6\times 10^{5}$ m/s at X, and $3.1\times
10^{5}$ m/s at R. We use the average value, $v_{F}=4.3\times 10^{5}$ m/s, in
the analysis below.

In addition to the protected nodal line, there is a cage-like network of Dirac
nodal lines [24, 22] with a finite gap due to spin-orbit coupling. The nodal
network includes segments enclosed by the large electron and hole-like Fermi
surfaces in Fig. 4(d), connected by loops in the $\Gamma$-X-M and Z-R-A planes
as sketched in Fig. 4(e) (modeled after [13]). The density of states $g(E)$
[Fig. 4(b)] is dominated near $E_{F}$ by this nodal network, where similar to
ZrSiS the excitations are essentially all Dirac fermions [25].

The protected nodal line runs parallel to segments of the nodal network, as
sketched in Fig. 4(g). Constant-energy slices from the Z-R-A plane [Figs. 4(e)
and 4(f)] further illustrate this: 35 meV below $E_{F}$ two disconnected
regions enclose only the nodal-network segments, while for $-25$ meV and above
these regions have merged, with the protected nodal line enclosed in the
disconnected region in the center. Similar features can be seen in the
$\Gamma$-X-M plane. A topology change occurs at $-27$ meV where the inner and
outer regions come into contact along R-A.

The contact points at $-27$ meV are saddle points, representing local minima
along R-A and transverse maxima. To a good approximation, because of the
nearly vanishing dispersion along R-X, this is a 2D van Hove singularity
(VHS), with an associated log energy dependence. The corresponding peak in
$g(E)$ is seen in Fig. 4(b). Such features near $E_{F}$ have been proposed to
be important for superconductivity in bulk KFe2As2 [26] and PdTe2 [27]. Here,
the origin of the VHS is similar to twisted bilayer graphene, where an avoided
crossing between closely-spaced Dirac nodes leads to a saddle point
contributing to its anomalous electronic behavior [28, 29, 30]. In the present
case, symmetry forbids the avoided crossing along the R-X centerline producing
the protected nodal line.

Regarding the $K$ and $(T_{1}T)^{-1}$ minima near 20 K, the well-known
Korringa relation [31] can connect $K$ and $(T_{1}T)^{-1}$, such as occurs due
to electron spins in simple metals. This relation is
$K^{2}T_{1}T=\hbar\gamma_{e}^{2}/(4\pi k_{B}\gamma_{n}^{2})=2.6\times 10^{-6}$
s K, where $\gamma_{e(n)}$ is the electron (nuclear) gyromagnetic ratio, and
the numerical value is for 125Te. $K_{\parallel c}$ drops by 30 ppm from 4 K
to 20 K while $(T_{1}T)^{-1}_{\parallel c}$ is reduced by 0.03 s-1 K-1 [Figs.
2(c) and 3(b)], giving $K^{2}T_{1}T=3\times 10^{-8}$ s K, much smaller than
expected due to the disappearance of a spin-hyperfine contribution. Similarly,
for $B\parallel a$ the changes in $(T_{1}T)^{-1}$ are much larger than
expected can be explained by spin mechanism, given the magnitude of
$K_{\parallel a}$.

Figure 4:  (a) ZrSiTe band structure with spin-orbit coupling. (b) Density of
states near $E_{F}$ (defined as $E=0$). (c) Tetragonal Brillouin zone with
high-symmetry points. (d) Fermi surface with hole surfaces purple (darker) and
electron surfaces green. (e) Fermi surface slice in Z-A-R plane with sketched
nodal network. (f) Slices near R at energies shown. (g) Sketch of parallel
nodal lines near R and X, with intersecting Dirac cones and effects of spin-
orbit coupling illustrated.

There is an extended-orbital mechanism for Dirac carriers that instead
provides a good explanation for the large $(T_{1}T)^{-1}$ [32, 33]. This
mechanism includes a significant contribution from carriers far from the
nucleus without relying on local hyperfine fields. Evidence for this mechanism
has been seen recently in other zirconium tellurides [34, 35]. For a quasi-2D
nodal line, one obtains $(T_{1}T)_{\parallel a}^{-1}=(3/2)(T_{1}T)_{\parallel
c}^{-1}$ [33], close to what is observed at low $T$ (Fig. 3). The
corresponding orbital mechanism for $K$ connects to bulk diamagnetism of the
Dirac electrons; $K_{\parallel c}$ scales with the line width, helping to
confirm that the bulk diamagnetism dominates. These mechanisms are not
expected to follow a Korringa relation [32]; however, the V-shaped minima in
both the shift and $(T_{1}T)^{-1}$ can be explained on the basis of orbital
coupling to the protected nodal line, as shown below.

We first consider the diamagnetic shift. The protected node will respond in
much the same way as the 2D Dirac system in graphene, assuming the node is a
linear two-band crossing with no dispersion along $c$. With the 9 T NMR field
applied along $c$ and $v_{F}=4.3\times 10^{5}$ m/s, the Landau level energies
are $\varepsilon_{\mathrm{LL}}=\pm\sqrt{2e\hbar v_{F}^{2}B|N|}=\pm
47\,[\mathrm{meV}]\,\sqrt{|N|}$, neglecting Zeeman splitting. Each level holds
a density $n_{\mathrm{LL}}=4B/(c\Phi_{0})=9.2\times 10^{18}$ cm-3 [35], with
$\Phi_{0}=4.1\times 10^{-15}$ T m2 the flux quantum, $c=9.51$
$\mathrm{\SIUnitSymbolAngstrom}$, and 4 representing the level degeneracy (2
for spin, 2 for the R-X valley degeneracy). As described in Ref. [36], the
magnetization per volume is $M=-\partial(\Omega/V)/\partial B$, with the grand
potential density
$\Omega/V=-k_{B}Tn_{\mathrm{LL}}\sum_{N}\ln[1+e^{(\varepsilon_{\mathrm{LL}}-\mu)/k_{B}T}]$,
where $\mu$ is the chemical potential. With all occupied levels in the sum, it
can be seen that this differs from the Landau diamagnetism, which depends on
carriers only at $E_{F}$. This yields an NMR shift contribution
$K_{\mathrm{dia}}=\mu_{0}M/B$, the same as the SI dimensionless susceptibility
to the extent that $M/B=\partial M/\partial B$, although in large fields $K$
becomes linear in $\mu$ and $T$ rather than exhibiting a $\mu=T=0$ singularity
as found in the low field limit [37]. The result is also distinct from the log
singularity obtained in the 3D case [33]. For the present case, demagnetizing
effects need also to be included. With crystal dimensions ($a\times b\times
c$) close to $2\times 2\times 0.5$ mm3, the mean demagnetizing factor from a
cuboid estimation [38] is $N\approx 4ab/[4ab+3c(a+b)]=0.7$, giving an observed
$K$ that is scaled by a factor of about 0.3.

The resulting shift is displayed as a heat map in Fig. 5(a), with the line-
node energy defined as $E=0$. We find that the V-shaped minimum (and
corresponding line width maximum) can be obtained only if $\mu(T)$ crosses the
node at $T_{\mathrm{min}}$. Assuming $\mu=-Ak_{B}(T-20\,[\mathrm{K}])$, Fig.
5(c) shows curves calculated for various multipliers $A$. As expected [36],
there is a linear change in $K_{\mathrm{dia}}$ away from the node crossing as
long as $\mu$ is between the $N=\pm 1$ levels. Larger values of $A$ produce a
relatively sharp minimum, similar to the measured results.

In Fig. 2(c), this model is compared to the data using $A=6$, with good
agreement below the 20 K node crossing. $\mu(T)$ in this case is shown by the
white arrow in Fig. 5(a). Together with the line width results, this is a good
indication that indeed the bulk Dirac diamagnetism is the dominant
contribution to the low-$T$ shift. There are previous reports [39, 40, 41] of
diamagnetic shifts in point-node Dirac/Weyl systems; here we provide a
quantitative model for this behavior based on the quasi-2D protected node in
ZrSiTe. Above 20 K, the data fall below the calculated curve, likely caused by
a change in Landau level configuration as $\mu$ approaches the VHS [Fig.
4(f)].

Ref. [33] gives $(T_{1}T)^{-1}$ for the quasi-2D extended-orbital case. With
zero Dirac gap for the protected nodal line, the result is

$\bigg{(}\frac{1}{T_{1}T}\bigg{)}_{\parallel
a}=\frac{3}{2}\bigg{(}\frac{1}{T_{1}T}\bigg{)}_{\parallel c}=\\\
4\beta\frac{\mu_{0}^{2}\gamma_{n}^{2}e^{2}k_{B}}{(4\pi)^{2}}\int
dE\bigg{[}-\frac{\partial f(E)}{\partial
E}\bigg{]}\frac{E}{\hbar^{2}cv_{F}}\ln\frac{2|E|}{\hbar\omega_{0}},$ (1)

with $E=\pm v_{F}k\hbar$, $c$ the lattice constant, and $f(E)$ the $\mu$\- and
$T$-dependent Fermi function. $\beta$ is a multiplier relating to the overall
density of Dirac carriers within the unit cell [32], while the leading factor
of 4 is due to the valley degeneracy since the Fermi golden rule mechanism
introduces a product of source and final state multiplicities.

Fig. 5(c) shows a heat map for $(T_{1}T)_{\parallel c}^{-1}$ computed with
$c=9.51$ $\mathrm{\SIUnitSymbolAngstrom}$, $v_{F}=4.3\times 10^{5}$ m/s and
$\beta=1.2\times 10^{-4}$. The white arrow represents
$\mu=-Ak_{B}(T-20\,[\mathrm{K}])$, with $A=6$ from the shift analysis, while
Fig. 5(b) shows results for other multipliers. The linear behavior away from
the node is characteristic of the quasi-2D case [33], with a point node
instead exhibiting quadratic behavior. Larger values of $A$ generate the
relatively sharp minimum seen in the data. The curves in Fig. 3(b) were
generated from this model using $A=6$, including the factor $3/2$ for
$B\parallel c$. Both curves include an additive constant 0.175 (s K)-1, due to
nodal-network carriers at $E_{F}$ not belonging to the protected node, assumed
to have little temperature dependence at these temperatures. While the
agreement is particularly good below 20 K, the curves fall above the data for
$T>20$ K. Similar to the shift results, this is likely due to the VHS.

While the shift analysis has only the overall multiplier $A$ as adjustable
parameter, for $(T_{1}T)^{-1}$ the orbital mechanism also includes the
multiplier $\beta$. It is difficult to know this parameter a priori; however,
comparison shows that the results are in the expected range: ZrTe5 with its
point-node exhibits a nearly-quadratic $T$\- and $\mu$-dependent
$(T_{1}T)^{-1}$ [42] fitted to $(T_{1}T)^{-1}\cong 40\mu^{2}$ (s K eV2)-1, or
a 0.004 (s K)-1 difference as $\mu$ increases from zero to 10 meV. This is the
same as the 20 K to 4 K change in $\mu$ modeled here [Fig. 5(a)], for which
$(T_{1}T)_{\parallel a}^{-1}$ changes by 0.055 (s K)-1, 14 times larger than
for ZrTe5 for the same $\Delta\mu$. The valley degeneracy plus the additional
phase space for electron scattering in the line node case leads to [35] a
ratio $(T_{1}T)_{\parallel
a}^{-1}/(T_{1}T)_{\mathrm{point}}^{-1}=\frac{3v_{F}}{2\pi\hbar\mu c}$ for the
line node vs the point-node, where the small differences in $v_{F}$ between
materials has been neglected. This ratio is 13, consistent with the scaled
comparison of relaxation rates.

Figure 5:  (a) Heat map showing ($\mu,T$)-dependent $K_{\parallel c}$
calculated from nodal-line diamagnetism. (b) $K_{\parallel c}$ vs $T$ assuming
$\mu(T)$ as shown. (c) Heat map showing calculated ($\mu,T$)-dependent
$(T_{1}T)_{\parallel c}^{-1}$. (d) $(T_{1}T)_{\parallel c}^{-1}$ vs $T$
assuming $\mu(T)$ as shown.

The model in which the protected line is a quasi-2D nodal line with very
little dispersion agrees well with the data, for example even 10 meV
dispersion along this line will lead to considerable broadening of the
V-shaped shift minima, which is not observed. Also the factor 3/2 in Eq. (1)
requires $\Delta k_{z}\neq 0$ scattering among degenerate states along the
line. These factors together suggest that the protected nodal line is indeed
very close to the quasi-2D case.

For $B\parallel a$, there will be no Landau quantization and thus the Dirac
diamagnetism vanishes. The $K_{\parallel a}$ minimum must therefore be due to
a spin contribution which presumably also underlies the larger $K_{\parallel
c}$ minimum. The spin shift is proportional to $g(E_{F})$ [31]. Since
$g(E_{F})\propto|E|$ for the quasi-2D case, this will also give a V-shaped
shift as $\mu$ crosses the node energy. However, the spin $(T_{1}T)^{-1}$
contribution is quadratic and as noted above based on the Korringa product,
this term will be negligible compared to the observed $(T_{1}T)^{-1}$
magnitude.

The sharp VHS adjacent to the protected nodal line implies that additional
electronic features relying upon the enhanced $g(E)$ could be engineered into
this system [26, 27, 28, 29, 30], by tuning $E_{F}$ to the VHS. Tuning could
be accomplished by chemical pressure [43], physical pressure or strain [16,
44], and light activation [17] with the required energy difference being quite
small. This could be another avenue for switching the topological behavior for
quantum computing applications. Note that a VHS may also be present due to
extrema of the nodal network [8]; however, such 3D saddle points would not
give the sharp features we observe.

At 140 K, there is a disconnect for $B\parallel c$, seen in the shift vs $T$
[Fig. 2(c)] as well as the line width scaling [Fig. 2(d)]. Based on the model
described above, we infer that this occurs when $\mu$ crosses the VHS as it
continues to decrease relative to the line-node energies, thus this change
corresponds to a $T$-induced Lifshitz transition. We label this transition
temperature $T_{\mathrm{VHS}}$. There is also a broad maximum in
$(T_{1}T)^{-1}$ at this temperature, although the maximum occurs at a reduced
temperature of about 100 K for $B\parallel a$.

To show that the behaviors for the two orientations are closely related, Fig.
3(c) shows $(T_{1}T)^{-1}$ with the temperature scale reduced by a factor of
0.7 for $B\parallel c$. For the additional scaled $B\parallel c$ plot, a
common baseline of 0.11 (s K)-1 was subtracted, and then the remainder was
scaled by the $3/2$ factor appearing in Eq. (1). The broad minimum at
temperatures above $T_{\mathrm{VHS}}$ for $B\parallel c$ follows the same
trend as for the $B\parallel a$ data, indicating that only the relative energy
scale is modified by the field.

Examination of the band-structure shows that the obvious $g(E)$ minimum near
$-130$ meV [Fig. 4(b)] is associated with the nodal lines parallel to R-X that
contribute to the VHS, so it is reasonable to associate the $(T_{1}T)^{-1}$
minima above $T_{\mathrm{VHS}}$ with $\mu$ crossing this feature. However, the
roughly quadratic shape of the minimum contrasts with the V-shape for the
crossing near 20 K. This points to a more conventional local-orbital mechanism
for the high-$T$ $(T_{1}T)^{-1}$, since [45] normally
$(T_{1}T)^{-1}\propto\int
d\varepsilon\frac{df}{d\varepsilon}g^{2}(\varepsilon)$, and with
$g(\varepsilon)\propto\varepsilon$ for the line node, this will give
$(T_{1}T)^{-1}\propto\mu^{2}$.

The relatively large change of $\mu$ at low $T$ cannot be explained on the
basis of charge conservation alone. Indeed, from the calculated $g(E)$ at
$E_{F}$ (Fig. 4), standard considerations [46] lead to a change of $\mu(T)$ by
only a few meV up to room temperature within a rigid-band picture. On the
other hand, based on the consistent picture provided by the shifts, the
$(T_{1}T)^{-1}$, and line width it is clear that $\mu(T)$ has a large
$T$-dependence and crosses the protected node at low $T$. In ZrSiSe, transport
results were recently proposed to give evidence for even larger $T$-induced
changes of $\mu$ [15]. For the present case, we assume it is relative shifts
of band energies that contribute to this process causing the node to shift
relative to $\mu$, perhaps induced by the promotion of carriers from donor
states, in a manner akin to the changes that have been proposed [17] to
accompany the optical promotion of carriers in ZrSiTe.

With the $T_{1}$ minima at high $T$ understood as due to $\mu$ crossing the
vertical line-node parts of the nodal network, the absence of a corresponding
feature in $K$ can be understood in light of the relatively large gap in the
nodal-network system. With a gap, the diamagnetic shift is greatly reduced,
even though the orbital mechanism can still induce a significant
$(T_{1}T)^{-1}$ contribution [33]. The origin of the apparent temperature
scaling above $T_{\mathrm{VHS}}$ we assume to be the orientation-dependent
field-induced rearrangement of the Dirac density of states due to Landau level
formation, although renormalization of the Dirac energy dispersion such as
indicated in ZrSiSe [8] might also be important.

In conclusion, we present NMR results that delineate the effects of the
symmetry-protected topological nodal line in ZrSiTe, and we also show that
there is a van Hove singularity (VHS) close in energy which plays a
significant role, and leads to a $T$-induced Lifshitz transition. DFT
calculations confirm the presence of the VHS just below $E_{F}$. Sharp minima
at 20 K in NMR shifts and $(T_{1}T)^{-1}$ are matched to a theoretical model
indicating that the chemical potential crosses the protected nodal line along
R-X. The results provide an indication of the 2D nature of this nodal line.
Well-defined changes in the NMR results at higher temperatures indicate the
influence of the 2D VHS, leading to a Lifshitz transition which occurs at 140
K with the field oriented perpendicular to the layers.

###### Acknowledgements.

This work was supported by the Robert A. Welch Foundation, Grant No. A-1526
and Texas A&M University. The financial support for sample preparation was
provided by the National Science Foundation through the Penn State 2D Crystal
Consortium-Materials Innovation Platform (2DCC-MIP) under National Science
Foundation cooperative agreement DMR-1539916.

## References

  * Burkov  _et al._ [2011] A. A. Burkov, M. D. Hook, and L. Balents, Topological nodal semimetals, Phys. Rev. B 84, 235126 (2011).
  * Fang  _et al._ [2015a] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Topological nodal line semimetals with and without spin-orbital coupling, Phys. Rev. B 92, 081201(R) (2015a).
  * Lv  _et al._ [2016] Y.-Y. Lv, B.-B. Zhang, X. Li, S.-H. Yao, Y. B. Chen, J. Zhou, S.-T. Zhang, M.-H. Lu, and Y.-F. Chen, Extremely large and significantly anisotropic magnetoresistance in ZrSiS single crystals, Appl. Phys. Lett. 108, 244101 (2016).
  * Sankar  _et al._ [2017] R. Sankar, G. Peramaiyan, I. P. Muthuselvam, C. J. Butler, K. Dimitri, M. Neupane, G. N. Rao, M.-T. Lin, and F. C. Chou, Crystal growth of Dirac semimetal ZrSiS with high magnetoresistance and mobility, Sci. Rep. 7, 40603 (2017).
  * Hu  _et al._ [2017] J. Hu, Z. Tang, J. Liu, Y. Zhu, J. Wei, and Z. Mao, Nearly massless Dirac fermions and strong Zeeman splitting in the nodal-line semimetal ZrSiS probed by de Haas–van Alphen quantum oscillations, Phys. Rev. B 96, 045127 (2017).
  * Klemenz  _et al._ [2019] S. Klemenz, S. Lei, and L. M. Schoop, Topological semimetals in square-net materials, Annu. Rev. Mater. Res. 49, 185 (2019).
  * Rudenko  _et al._ [2018] A. N. Rudenko, E. A. Stepanov, A. I. Lichtenstein, and M. I. Katsnelson, Excitonic instability and pseudogap formation in nodal line semimetal ZrSiS, Phys. Rev. Lett. 120, 216401 (2018).
  * Shao  _et al._ [2020] Y. Shao, A. N. Rudenko, J. Hu, Z. Sun, Y. Zhu, S. Moon, A. J. Millis, S. Yuan, A. I. Lichtenstein, D. Smirnov, Z. Q. Mao, M. I. Katsnelson, and D. N. Basov, Electronic correlations in nodal-line semimetals, Nat. Phys. 16, 636 (2020).
  * Gatti  _et al._ [2020] G. Gatti, A. Crepaldi, M. Puppin, N. Tancogne-Dejean, L. Xian, U. De Giovannini, S. Roth, S. Polishchuk, P. Bugnon, A. Magrez, H. Berger, F. Frassetto, L. Poletto, L. Moreschini, S. Moser, A. Bostwick, E. Rotenberg, A. Rubio, M. Chergui, and M. Grioni, Light-induced renormalization of the Dirac quasiparticles in the nodal-line semimetal ZrSiSe, Phys. Rev. Lett. 125, 076401 (2020).
  * Pezzini  _et al._ [2018] S. Pezzini, M. R. Van Delft, L. M. Schoop, B. V. Lotsch, A. Carrington, M. I. Katsnelson, N. E. Hussey, and S. Wiedmann, Unconventional mass enhancement around the Dirac nodal loop in ZrSiS, Nat. Phys. 14, 178 (2018).
  * Wang  _et al._ [2020a] J.-R. Wang, G.-Z. Liu, X. Wan, and C. Zhang, Quantum criticality of the excitonic insulating transition in the nodal-line semimetal ZrSiS, Phys. Rev. B 101, 245151 (2020a).
  * Topp  _et al._ [2016] A. Topp, J. M. Lippmann, A. Varykhalov, V. Duppel, B. V. Lotsch, C. R. Ast, and L. M. Schoop, Non-symmorphic band degeneracy at the Fermi level in ZrSiTe, New J. Phys. 18, 125014 (2016).
  * Muechler  _et al._ [2020] L. Muechler, A. Topp, R. Queiroz, M. Krivenkov, A. Varykhalov, J. Cano, C. R. Ast, and L. M. Schoop, Modular arithmetic with nodal lines: Drumhead surface states in ZrSiTe, Phys. Rev. X 10, 011026 (2020).
  * Yen and Guo [2020] Y. Yen and G.-Y. Guo, Tunable large spin Hall and spin Nernst effects in the Dirac semimetals $\mathrm{Zr}XY(X=\mathrm{Si},\mathrm{Ge};Y=\mathrm{S},\mathrm{Se},\mathrm{Te})$, Phys. Rev. B 101, 064430 (2020).
  * Chen  _et al._ [2020] F. C. Chen, Y. Fei, S. J. Li, Q. Wang, X. Luo, J. Yan, W. J. Lu, P. Tong, W. H. Song, X. B. Zhu, L. Zhang, H. B. Zhou, F. W. Zheng, P. Zhang, A. L. Lichtenstein, M. I. Katsnelson, Y. Yin, N. Hao, and Y. P. Sun, Temperature-Induced Lifshitz Transition and Possible Excitonic Instability in ZrSiSe, Phys. Rev. Lett. 124, 236601 (2020).
  * Krottenmüller  _et al._ [2020] M. Krottenmüller, M. Vöst, N. Unglert, J. Ebad-Allah, G. Eickerling, D. Volkmer, J. Hu, Y. L. Zhu, Z. Q. Mao, W. Scherer, and C. A. Kuntscher, Indications for Lifshitz transitions in the nodal-line semimetal ZrSiTe induced by interlayer interaction, Phys. Rev. B 101, 081108(R) (2020).
  * Kirby  _et al._ [2020] R. J. Kirby, L. Muechler, S. Klemenz, C. Weinberg, A. Ferrenti, M. Oudah, D. Fausti, G. D. Scholes, and L. M. Schoop, Signature of an Ultrafast Photo-Induced Lifshitz Transition in the Nodal-Line Semimetal ZrSiTe, arXiv:2011.04646 (2020).
  * Inamo [1996] M. Inamo, 125Te NMR Evidence for the Existence of Trinuclear Tellurate Ion in Aqueous Solution, Chem. Lett. 25, 17 (1996).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Blaha  _et al._ [2020] P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H. Madsen, and L. D. Marks, WIEN2k: An APW+ lo program for calculating the properties of solids, J. Chem. Phys. 152, 074101 (2020).
  * Bensch and Dürichen [1994] W. Bensch and P. Dürichen, Structure of the layer compound ZrSiTe, Acta Crystallogr. C Struct. Commun. 50, 346 (1994).
  * Hu  _et al._ [2016] J. Hu, Z. Tang, J. Liu, X. Liu, Y. Zhu, D. Graf, K. Myhro, S. Tran, C. N. Lau, J. Wei, and Z. Mao, Evidence of topological nodal-line fermions in ZrSiSe and ZrSiTe, Phys. Rev. Lett. 117, 016602 (2016).
  * Kokalj [1999] A. Kokalj, XCrySDen–a new program for displaying crystalline structures and electron densities, J. Mol. Graph. Model. 17, 176 (1999).
  * Schoop  _et al._ [2016] L. M. Schoop, M. N. Ali, C. Straßer, A. Topp, A. Varykhalov, D. Marchenko, V. Duppel, S. S. P. Parkin, B. V. Lotsch, and C. R. Ast, Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS, Nat. Commun. 7, 11696 (2016).
  * Fu  _et al._ [2019] B.-B. Fu, C.-J. Yi, T.-T. Zhang, M. Caputo, J.-Z. Ma, X. Gao, B. Q. Lv, L.-Y. Kong, Y.-B. Huang, P. Richard, M. Shi, V. N. Strocov, C. Fang, H.-M. Weng, Y.-G. Shi, T. Qian, and H. Ding, Dirac nodal surfaces and nodal lines in ZrSiS, Sci. Adv. 5, eaau6459 (2019).
  * Fang  _et al._ [2015b] D. Fang, X. Shi, Z. Du, P. Richard, H. Yang, X. X. Wu, P. Zhang, T. Qian, X. Ding, Z. Wang, T. K. Kim, M. Hoesch, A. Wang, X. Chen, J. Hu, H. Ding, and H.-H. Wen, Observation of a Van Hove singularity and implication for strong-coupling induced Cooper pairing in KFe2As2, Phys. Rev. B 92, 144513 (2015b).
  * Kim  _et al._ [2018] K. Kim, S. Kim, J. S. Kim, H. Kim, J.-H. Park, and B. I. Min, Importance of the van Hove singularity in superconducting PdTe2, Phys. Rev. B 97, 165102 (2018).
  * Luican  _et al._ [2011] A. Luican, G. Li, A. Reina, J. Kong, R. R. Nair, K. S. Novoselov, A. K. Geim, and E. Y. Andrei, Single-layer behavior and its breakdown in twisted graphene layers, Phys. Rev. Lett. 106, 126802 (2011).
  * Brihuega  _et al._ [2012] I. Brihuega, P. Mallet, H. González-Herrero, G. Trambly de Laissardière, M. M. Ugeda, L. Magaud, J. M. Gómez-Rodríguez, F. Ynduráin, and J.-Y. Veuillen, Unraveling the intrinsic and robust nature of van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis, Phys. Rev. Lett. 109, 196802 (2012).
  * Andrei and MacDonald [2020] E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  * Carter  _et al._ [1977] G. C. Carter, L. H. Bennett, and D. J. Kahan,  _Metallic shifts in NMR: a review of the theory and comprehensive critical data compilation of metallic materials_ (Pergamon, New York, 1977).
  * Okvátovity  _et al._ [2019] Z. Okvátovity, H. Yasuoka, M. Baenitz, F. Simon, and B. Dóra, Nuclear spin-lattice relaxation time in TaP and the Knight shift of Weyl semimetals, Phys. Rev. B 99, 115107 (2019).
  * Maebashi  _et al._ [2019] H. Maebashi, T. Hirosawa, M. Ogata, and H. Fukuyama, Nuclear magnetic relaxation and Knight shift due to orbital interaction in Dirac electron systems, J. Phys. Chem. Solids 128, 138 (2019).
  * Tian  _et al._ [2019] Y. Tian, N. Ghassemi, and J. H. Ross, Jr., Dirac electron behavior and NMR evidence for topological band inversion in ZrTe5, Phys. Rev. B 100, 165149 (2019).
  * Tian  _et al._ [2020] Y. Tian, N. Ghassemi, and J. H. Ross, Jr., Topological nodal line in ZrTe2 demonstrated by nuclear magnetic resonance, Phys. Rev. B 102, 165149 (2020).
  * Li  _et al._ [2015] Z. Li, L. Chen, S. Meng, L. Guo, J. Huang, Y. Liu, W. Wang, and X. Chen, Field and temperature dependence of intrinsic diamagnetism in graphene: Theory and experiment, Phys. Rev. B 91, 094429 (2015).
  * Raoux  _et al._ [2015] A. Raoux, F. Piéchon, J.-N. Fuchs, and G. Montambaux, Orbital magnetism in coupled-bands models, Phys. Rev. B 91, 085120 (2015).
  * Prozorov and Kogan [2018] R. Prozorov and V. G. Kogan, Effective demagnetizing factors of diamagnetic samples of various shapes, Phys. Rev. Appl. 10, 014030 (2018).
  * Wang  _et al._ [2020b] C. G. Wang, Y. Honjo, L. X. Zhao, G. F. Chen, K. Matano, R. Zhou, and G.-q. Zheng, Landau diamagnetism and Weyl-fermion excitations in TaAs revealed by ${}^{75}\mathrm{As}$ NMR and NQR, Phys. Rev. B 101, 241110(R) (2020b).
  * Tay  _et al._ [2020] D. Tay, T. Shang, P. Puphal, E. Pomjakushina, H.-R. Ott, and T. Shiroka, Unusual ${}^{27}\mathrm{Al}$ NMR shift in the Weyl-fermion systems LaAlGe and PrAlGe, Phys. Rev. B 102, 241109(R) (2020).
  * Suetsugu  _et al._ [2021] S. Suetsugu, K. Kitagawa, T. Kariyado, A. W. Rost, J. Nuss, C. Mühle, M. Ogata, and H. Takagi, Giant orbital diamagnetism of three-dimensional Dirac electrons in ${\mathrm{Sr}}_{3}\mathrm{PbO}$ antiperovskite, Phys. Rev. B 103, 115117 (2021).
  * Tian  _et al._ [2021] Y. Tian, N. Ghassemi, and J. H. Ross, Jr., Gap-opening transition in Dirac semimetal ZrTe5, arXiv:2101.07387 (2021).
  * Hosen  _et al._ [2017] M. M. Hosen, K. Dimitri, I. Belopolski, P. Maldonado, R. Sankar, N. Dhakal, G. Dhakal, T. Cole, P. M. Oppeneer, D. Kaczorowski, F. Chou, M. Z. Hasan, T. Durakiewicz, and M. Neupane, Tunability of the topological nodal-line semimetal phase in $\mathrm{ZrSi}X$-type materials ($X=\mathrm{S},\mathrm{Se},\mathrm{Te}$), Phys. Rev. B 95, 161101(R) (2017).
  * Zhou  _et al._ [2020] W. Zhou, A. N. Rudenko, and S. Yuan, Effect of Mechanical Strain on the Optical Properties of Nodal-Line Semimetal ZrSiS, Adv. Electron. Mater. 6, 1900860 (2020).
  * Slichter [1990] C. P. Slichter,  _Principles of magnetic resonance_ (Springer, New York, 1990).
  * Ashcroft and Mermin [1976] N. W. Ashcroft and N. D. Mermin,  _Solid state physics_ (Holt, Rinehart, and Winston, New York, 1976).


# Theorem on the photon sphere of a general asymptotically-flat, static, and
spherical symmetrical wormhole with Z-2 symmetry

Naoki Tsukamoto1<EMAIL_ADDRESS>1Department of Physics, Faculty of
Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo
162-8601, Japan

###### Abstract

A photon sphere or an antiphoton sphere on the throat of static and spherical
symmetrical wormholes with Z-2 symmetry against the throat is often
overlooked. On this paper, we prove a theorem that a general asymptotically-
flat, static, spherical symmetrical, and Z-2 symmetrical wormhole without a
thin shell must have at least a photon sphere. We confirm our result in a case
of a Reissner-Nordström black-hole-like wormhole spacetime or a charged
Damour-Solodukhin wormhole spacetime. We should keep in mind that the theorem
does not depend on gravitational theories and energy conditions and that we
can apply it for neither Z-2 asymmetric wormholes nor thin-shell wormholes.

##  I Introduction

In general relativity, a static and spherically symmetric compact object has a
photon (antiphoton) sphere formed by unstable (stable) circular photon orbits
Claudel:2000yi ; Perlick_2004_Living_Rev  and rays can be deflected strongly
near the photon sphere Hagihara_1931 ; Darwin_1959 ; Atkinson_1965 ;
Luminet_1979 ; Ohanian_1987 ; Nemiroff_1993 ; Frittelli_Kling_Newman_2000 ;
Virbhadra_Ellis_2000 ; Bozza_Capozziello_Iovane_Scarpetta_2001 ; Bozza:2002zj
; Eiroa:2002mk ; Perlick:2003vg ; Bozza_2010 ; Tsukamoto:2016jzh ;
Shaikh:2019itn . The existence of the photon sphere has a strong influence on
the image of a collapsing star Ames_1968 ; Synge:1966okc ; Yoshino:2019qsh ,
the high-frequency behavior of the photon absorption cross section
Sanchez:1977si ; Decanini:2010fz , the high-frequency spectrum of quasinormal
modes of compact objects Press:1971wr ; Goebel_1972 ; Raffaelli:2014ola ,
centrifugal force and gyroscopic precession Abramowicz_Prasanna_1990 ;
Abramowicz:1990cb ; Allen:1990ci ; Hasse_Perlick_2002 , and Bondi’s sonic
horizon of a radial fluid Mach:2013gia ; Chaverra:2015bya ; Cvetic:2016bxi ;
Koga:2016jjq ; Koga:2018ybs ; Koga:2019teu . It is concerned that stable
circular photon orbits might cause instability of compact objects with a
trivial topology Keir:2014oka ; Cardoso:2014sna ; Cunha:2017qtt ;
Cunha:2017eoe ; Cunha:2022gde ; Zhong:2022jke .

For a long time, such a strong gravitational field predicted by general
relativity, which was unlikely to be observed, had been not considered
seriously in astrophysics. Recently, however, phenomena in a strong
gravitational field has been considered eagerly since the direct detection of
gravitational waves from binary black holes have been reported by LIGO
Scientific Collaboration and Virgo Collaboration Abbott:2016blz  and the black
hole shadows in the centers of a galaxy M87 and the Milky way have been
reported by Event Horizon Telescope Collaboration Akiyama:2019cqa ;
EventHorizonTelescope:2022wkp .

Theorems on the circular photon orbits such as the upper and lower bounds of
radii of circular photon orbits around a black hole Hod:2017xkz ; Hod:2020pim
; Hod:2023jmx  and the numbers of circular photon orbits surround compact
objects without an event horizon Hod:2017zpi  were investigated. The theorems
can be applied for the compact objects with a trivial topology in general
relativity under the energy conditions. On the other hand, recently, Kudo and
Asada have proved that a spacetime cannot be asymptotically flat if its
outermost circular photon orbit is stable Kudo:2022ewn . We note that Kudo and
Asada’s proof relies on only the assumptions of a metric and it depends on
neither the energy conditions, gravitational theories, nor the topology of the
spacetime.

General relativity permits wormhole spacetimes with a nontrivial topology
since the Einstein equations are local-field equations which do not determine
the topology of the spacetime. A traversable wormhole has a throat which
connects two regions of one or two universes Visser_1995 ; Morris:1988cz . The
photon sphere and the antiphoton sphere on the throat of the wormhole with Z-2
symmetry against the throat is often overlooked. In Ref. Dey:2008kn , Dey and
Sen claimed that an Ellis-Bronnikov wormhole Ellis:1973yv ; Bronnikov:1973fh
with vanishing Arnowitt-Deser-Misner(ADM) masses, the Z-2 symmetry, and a
metric

$\displaystyle\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}\rho^{2}+(\rho^{2}+a^{2})\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right),$
(1)

where $a$ is a constant, has no photon sphere on its throat but it contradicts
papers by several authors Ellis:1973yv ; Chetouani_Clement_1984 ;
Perlick:2003vg ; Perlick_2004_Living_Rev ; Muller:2004dq ; Nandi:2006ds ;
Muller:2008zza ; Bhattacharya:2010zzb ; Gibbons:2011rh ; Nakajima:2012pu ;
Tsukamoto:2012xs ; Ohgami:2015nra ; Tsukamoto:2016zdu ; Tsukamoto:2016qro ;
Tsukamoto:2017edq ; Bugaev:2023mlc . Simpson and Visser suggested a black-
bounce spacetime Simpson:2018tsi  with a metric

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{\sqrt{\rho^{2}+a^{2}}}\right)\mathrm{d}t^{2}+\frac{\mathrm{d}\rho^{2}}{1-\frac{2M}{\sqrt{\rho^{2}+a^{2}}}}$
(2)
$\displaystyle+(\rho^{2}+a^{2})\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right),$

where $M$ is a positive ADM mass, and it is a wormhole for $a>2M$. In the case
of no ADM mass $M=0$, the metric corresponds to the one of the Ellis-Bronnikov
wormhole Ellis:1973yv ; Bronnikov:1973fh  with no ADM masses. The existence of
the antiphoton sphere for $2M<a<3M$, the marginal photon sphere for $a=3M$,
and the photon sphere for $a>3M$ on the throat were found in Ref.
Tsukamoto:2020bjm  while the antiphoton and photon spheres were overlooked in
Refs. Simpson:2018tsi ; Nascimento:2020ime ; Guerrero:2021ues ;
Bronnikov:2021liv . A Damour-Solodukhin wormhole Damour:2007ap  with a metric

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}+\lambda^{2}\right)\mathrm{d}\tilde{t}^{2}$
(3)
$\displaystyle+\frac{\mathrm{d}r^{2}}{1-\frac{2M}{r}}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right),$

where $M>0$ and $\lambda\geq 0$ are assumed, has the antiphoton sphere for
$\lambda<1/\sqrt{2}$, the photon sphere for $\lambda>1/\sqrt{2}$, and the
marginal photon sphere for $\lambda=1/\sqrt{2}$ on the throat
Tsukamoto:2020uay ; Tsukamoto:2023rzd . However, the (anti)photon sphere on
the throat is often neglected Nandi:2018mzm ; Ovgun:2018fnk ;
Bhattacharya:2018leh ; Ovgun:2018swe ; Ovgun:2018oxk ; Vagnozzi:2022moj .

Recently, Shaikh et al. Shaikh:2018oul  have proved that the a general,
static, and spherically symmetric wormhole with Z-2 symmetry against the
throat has unstable or stable circular photon orbits on the throat and they
have investigated the condition when the wormhole has three photon spheres,
where one of them is on the throat and the others are off the throat, and two
antiphoton shperes off the throat. Bronnikov and Baleevskikh Bronnikov:2018nub
also have given an alternative proof on circular photon orbits on the throat
of the Z-2 symmetric wormhole and they have investigated deflection angle of a
light scatted by an asymmetric wormhole. We emphasize that their result is
caused by the geometry of the throat and by symmetry against the throat and it
depends on neither the energy conditions nor gravitational theories. They are
interested in the Z-2 symmetric wormhole with the photon sphere on the throat
Shaikh:2018oul ; Bronnikov:2018nub  while a few Z-2 symmetric wormholes with
the antiphoton sphere on the throat have been studied also in Refs.
Shaikh:2019jfr ; Tsukamoto:2021apr ; Tsukamoto:2021caq .

On this paper, by combining Kudo and Asada’s theorem Kudo:2022ewn  and the
theorem on the Z-2 symmetrical wormhole in Ref. Shaikh:2018oul  or Ref.
Bronnikov:2018nub , we prove a theorem that a general asymptotically-flat,
static, and spherically symmetrical wormhole with Z-2 symmetry against the
throat has at least one photon sphere. The theorem will helps to prevent the
overlook the photon spheres and the antiphoton spheres of the wormhole. We
confirm the theorem in a Reissner-Nordström black-hole-like wormhole spacetime
or a charged Damour-Solodukhin wormhole spacetime Lemos:2008cv ;
Tsukamoto:2019ihj .

This paper is organized as follows. In Sec. II, we give a proof for the
theorem on the photon sphere and antiphoton spheres in a general
asymptotically-flat, static, and spherically symmetrical wormhole spacetime
with Z-2 symmetry against the throat. We consider the Reissner-Nordström
black-hole-like wormhole Lemos:2008cv ; Tsukamoto:2019ihj  as an example in
Sec. III, and we summarize and discuss in Sec. IV. In this paper we use the
units in which a light speed and Newton’s constant are unity.

##  II circular photon orbits on a throat

We consider a general, static, and spherical symmetrical wormhole spacetime
with Z-2 symmetry against a throat. Its metric is given by, in coordinates
$(t,x,\theta,\varphi)$,

$\displaystyle\mathrm{d}s^{2}=-A(x)\mathrm{d}t^{2}+\frac{\mathrm{d}x^{2}}{A(x)}+r^{2}(x)\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right),$

where the radial coordinate $x$ is defined for $-\infty<x<\infty$. We assume
that $A(x)$ and $r(x)$ are positive, continuous, and finite for
$-\infty<x<\infty$. The spacetime has time-translational and axial Killing
vectors $t^{\mu}\partial_{\mu}=\partial_{t}$ and
$\varphi^{\mu}\partial_{\mu}=\partial_{\varphi}$ due to its stationality and
axisymmetry, respectively. We set $\theta=\pi/2$ without loss of generality
because of spherically symmetry. We assume that $r(x)$ has a positive minimal
value at $x=0$, i.e., there is the throat at $x=x_{\mathrm{th}}=0$. We also
assume that the wormhole has a Z-2 symmetry against the throat. From the
assumptions, we obtain
$A^{\prime}(x_{\mathrm{th}})=r^{\prime}(x_{\mathrm{th}})=0$ and
$r^{\prime\prime}(x_{\mathrm{th}})\geq 0$, where the prime denotes a
differentiation with respect to $x$.

The trajectory of a ray is expressed by

$\displaystyle-A(x)\dot{t}^{2}+\frac{\dot{x}^{2}}{A(x)}+r^{2}(x)\dot{\varphi}^{2}=0,$
(5)

where the dot denotes a differentiation with respect to an affine parameter
along the trajectory. By using the impact parameter $b\equiv E/L$ of the ray,
where $E\equiv-g_{\mu\nu}t^{\mu}\dot{x}^{\nu}$ and $L\equiv
g_{\mu\nu}\varphi^{\mu}\dot{x}^{\nu}$ are conserved energy and angular
momentum of the ray, respectively, Eq. (5) can be rewritten in

$\displaystyle\dot{x}^{2}+V_{\mathrm{eff}}(x,b)=0,$ (6)

where $V_{\mathrm{eff}}(x,b)$ is the effective potential of the ray given by

$\displaystyle V_{\mathrm{eff}}(x,b)\equiv
E^{2}A(x)\left(\frac{b^{2}}{r^{2}(x)}-\frac{1}{A(x)}\right).$ (7)

The first and second derivatives of the effective potential are obtains as

$\displaystyle
V_{\mathrm{eff}}^{\prime}(x,b)=E^{2}b^{2}\left(\frac{A^{\prime}r-2Ar^{\prime}}{r^{3}}\right)$
(8)

and

$\displaystyle
V_{\mathrm{eff}}^{\prime\prime}(x,b)=E^{2}b^{2}\left(\frac{A^{\prime\prime}r^{2}-4A^{\prime}rr^{\prime}+6Ar^{\prime
2}-2Arr^{\prime\prime}}{r^{4}}\right).$

If the spacetime permits

$\displaystyle
A^{\prime}(x_{\mathrm{c}})r(x_{\mathrm{c}})-2A(x_{\mathrm{c}})r^{\prime}(x_{\mathrm{c}})=0$
(10)

to be $V_{\mathrm{eff}}^{\prime}(x_{\mathrm{c}},b)=0$, it has a circular
photon orbit at $x=x_{\mathrm{c}}$. The ray of the circular orbit must have
with a critical impact parameter $b=b_{\mathrm{c}}$, where $b_{\mathrm{c}}$ is
given by

$\displaystyle
b_{\mathrm{c}}\equiv\pm\sqrt{\frac{r^{2}(x_{\mathrm{c}})}{A(x_{\mathrm{c}})}},$
(11)

so that $V_{\mathrm{eff}}(x_{\mathrm{c}},b_{\mathrm{c}})$ vanishes. The
circular photon orbit is unstable (stable) if
$V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{c}},b_{\mathrm{c}})$ is negative
(positive). A sphere formed by the unstable (stable) circular photon orbits is
called photon sphere (antiphoton sphere).

We find $V_{\mathrm{eff}}^{\prime}(x_{\mathrm{th}},b)=0$ and
$V_{\mathrm{eff}}(x_{\mathrm{th}},b_{\mathrm{th}})=0$, where $b_{\mathrm{th}}$
is a critical immpact parameter

$\displaystyle
b_{\mathrm{th}}=\pm\sqrt{\frac{r^{2}(x_{\mathrm{th}})}{A(x_{\mathrm{th}})}},$
(12)

on the throat $x=x_{\mathrm{th}}=0$. Thus, the rays make the photon sphere
(antiphoton sphere) on the throat if
$V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{th}},b_{\mathrm{th}})$ is negative
(positive). Here, $V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{th}},b)$ is
obtained by

$\displaystyle
V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{th}},b)=E^{2}b^{2}\left(\frac{A^{\prime\prime}(x_{\mathrm{th}})r(x_{\mathrm{th}})-2A(x_{\mathrm{th}})r^{\prime\prime}(x_{\mathrm{th}})}{r^{3}(x_{\mathrm{th}})}\right).$

If the wormhole spacetime is asymptotically flat at both regions against the
throat,

$\displaystyle A(\pm\infty)=1+O\left(x^{-1}\right)=1+O\left(r^{-1}\right),$
(14) $\displaystyle
A^{\prime}(\pm\infty)=O\left(x^{-2}\right)=O\left(r^{-2}\right),$ (15)
$\displaystyle r(\pm\infty)=\pm x+O\left(x^{0}\right)$ (16)

should hold and, hence, we obtain

$\displaystyle
V_{\mathrm{eff}}^{\prime}(\pm\infty,b)=\pm\frac{2E^{2}b^{2}}{r^{3}(\pm\infty)}.$
(17)

From $V_{\mathrm{eff}}^{\prime}(-\infty,b)>0$,
$V_{\mathrm{eff}}^{\prime}(x_{\mathrm{th}},b)=0$, and
$V_{\mathrm{eff}}^{\prime}(\infty,b)<0$, and an intermediate value theorem, we
obtain the following theorem on the asymptotically-flat wormhole;

(a)

For $V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{th}},b)<0$, the numbers of the
photon spheres and antiphoton spheres are $2n-1$ and $2n-2$, respectively,
where $n$ is a positive integer. One of the photon spheres is on the throat.

(b)

For $V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{th}},b)>0$, the numbers of the
photon spheres and antiphoton spheres are $2n$ and $2n-1$, respectively. One
of the antiphoton spheres is on the throat.

(c)

The photon sphere and the antiphoton sphere must be alternately.

(d)

The outermost circular photon orbits are stable. Thus, the wormhole has at
least one photon sphere on or off the throat.

(e)

In marginal cases, i.e.,
$V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{th}},b)=0$, we can get the similar
theorem to (a) and (b) which being replaced
$V_{\mathrm{eff}}^{\prime\prime}(x_{\mathrm{th}},b)$ in (a) and (b) into the
higher-order derivatives of the effective potential.

##  III Reissner-Nordström black-hole-like wormhole

We confirm that the photon sphere and the antiphoton sphere in a Reissner-
Nordström black-hole-like wormhole spacetime or a charged Damour-Solodukhin
wormhole spacetime Lemos:2008cv ; Tsukamoto:2019ihj  with a metric, in
coordinates $(\tilde{t},r,\theta,\varphi)$,

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\lambda^{2}\right)\mathrm{d}\tilde{t}^{2}$
$\displaystyle+\frac{\mathrm{d}r^{2}}{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right),$

where we assume $M>0$, $\lambda\geq 0$, and $\left|Q\right|\leq M$. A tidal
force Lemos:2008cv  and a particle collision Tsukamoto:2019ihj  in the
Reissner-Nordström black-hole-like wormhole spacetime was investigated. If
$Q=0$, it becomes a Damour-Solodukhin wormhole Damour:2007ap , which was
suggested as a black hole mimicker, and gravitational lensing Nandi:2018mzm ;
Ovgun:2018fnk ; Bhattacharya:2018leh ; Ovgun:2018swe ; Ovgun:2018oxk ,
quasinormal modes and gravitational waves Bueno:2017hyj ; Volkel:2018hwb , the
emissions Karimov:2019qco  and images of accretion disks Paul:2019trt , and
its shadow Amir:2018pcu  were studied.

By transforming the time coordinate $\tilde{t}$ into
$t\equiv\sqrt{1+\lambda^{2}}\tilde{t}$, we obtain the metric in the
coordinates $(t,r,\theta,\varphi)$ as

$\displaystyle\mathrm{d}s^{2}=-A(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{B(r)}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right),$
(19)

where $A(r)$ and $B(r)$ are given by

$\displaystyle
A(r)=1-\frac{2M}{(1+\lambda^{2})r}+\frac{Q^{2}}{(1+\lambda^{2})r^{2}}$ (20)

and

$\displaystyle B(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}},$ (21)

respectively. We note that the spacetime is asymptotically flat. A throat is
at $r=r_{\mathrm{th}}\equiv M+\sqrt{M^{2}-Q^{2}}$ but the radial coordinate
$r$ does not cover the other side of the throat. We set $\theta=\pi/2$ without
loss of generality.

The trajactory of a ray can be expressed by

$\displaystyle\dot{r}^{2}+v(r,b)=0,$ (22)

where the effective potential $v(r)$ is given by

$\displaystyle v(r,b)\equiv
E^{2}B(r)\left(\frac{b^{2}}{r^{2}}-\frac{1}{A(r)}\right).$ (23)

There are two photon (antiphoton) spheres at $r=r_{\mathrm{m}}$
($r=r_{\mathrm{a}}$) in both regions against the throat, if
$\lambda<\lambda_{\mathrm{ma}}$ and $r_{\mathrm{m}}>r_{\mathrm{th}}$
($\lambda<\lambda_{\mathrm{ma}}$ and $r_{\mathrm{a}}>r_{\mathrm{th}}$) are
satisfied. Here, $r_{\mathrm{m}}$ and $r_{\mathrm{a}}$ are given by

$\displaystyle
r_{\mathrm{m}}\equiv\frac{3M+\sqrt{9M^{2}-8\left(1+\lambda^{2}\right)Q^{2}}}{2\left(1+\lambda^{2}\right)}$
(24)

and

$\displaystyle
r_{\mathrm{a}}\equiv\frac{3M-\sqrt{9M^{2}-8\left(1+\lambda^{2}\right)Q^{2}}}{2\left(1+\lambda^{2}\right)},$
(25)

respectively, and $\lambda_{\mathrm{ma}}$ is defined as

$\displaystyle\lambda_{\mathrm{ma}}\equiv\frac{\sqrt{9M^{2}-8Q^{2}}}{2\sqrt{2}\left|Q\right|}.$
(26)

We find that $r_{\mathrm{m}}=r_{\mathrm{th}}$ or
$r_{\mathrm{a}}=r_{\mathrm{th}}$ holds if $\lambda=\lambda_{\mathrm{th}}$,
where $\lambda_{\mathrm{th}}$ is defined by

$\displaystyle\lambda_{\mathrm{th}}\equiv\sqrt{1+\frac{M\left(\sqrt{M^{2}-Q^{2}}-M\right)}{Q^{2}}}$
(27)

for $0<\left|Q\right|<M$ and $\lambda_{\mathrm{th}}\equiv 1/\sqrt{2}$ for
$Q=0$. Figure 1 shows $\lambda_{\mathrm{th}}$ and $\lambda_{\mathrm{ma}}$ as
the functions of $\left|Q\right|/M$. We find that $\lambda=1/\sqrt{5}\sim
0.447$ and $\left|Q\right|/M=\sqrt{15}/4\sim 0.968$ is the only solution of
$\lambda=\lambda_{\mathrm{th}}=\lambda_{\mathrm{ma}}$.

Figure 1:  $\lambda_{\mathrm{ma}}$ and $\lambda_{\mathrm{th}}$ as the
functions of $\left|Q\right|/M$ are shown by as solid (green) and dashed (red)
curves, respectively. Given $\left|Q\right|/M$, We find that
$\lambda_{\mathrm{ma}}$ is greater than or equal to $\lambda_{\mathrm{th}}$.
We get $\lambda_{\mathrm{th}}=\lambda_{\mathrm{ma}}$ if and only if
$\left|Q\right|/M=\sqrt{15}/4\sim 0.968$.

We can confirm

$\displaystyle
v(r_{\mathrm{m}},b_{\mathrm{m}})=\frac{\mathrm{d}v(r_{\mathrm{m}},b_{\mathrm{m}})}{\mathrm{d}r}=0,$
(28) $\displaystyle
v(r_{\mathrm{a}},b_{\mathrm{a}})=\frac{\mathrm{d}v(r_{\mathrm{a}},b_{\mathrm{a}})}{\mathrm{d}r}=0$
(29)

and

$\displaystyle\frac{\mathrm{d}^{2}v(r_{\mathrm{m}},b_{\mathrm{m}})}{\mathrm{d}r^{2}}<0,$
(30)
$\displaystyle\frac{\mathrm{d}^{2}v(r_{\mathrm{a}},b_{\mathrm{a}})}{\mathrm{d}r^{2}}>0,$
(31)

where $b_{\mathrm{m}}$ and $b_{\mathrm{a}}$ are the critical impact parameters
given by

$\displaystyle
b_{\mathrm{m}}\equiv\pm\sqrt{\frac{r_{\mathrm{m}}^{2}}{A(r_{\mathrm{m}})}},$
(32) $\displaystyle
b_{\mathrm{a}}\equiv\pm\sqrt{\frac{r_{\mathrm{a}}^{2}}{A(r_{\mathrm{a}})}},$
(33)

in straight-forward calculations. We plot $r_{\mathrm{m}}/M$,
$r_{\mathrm{a}}/M$, $r_{\mathrm{th}}/M$ in Fig. 2 and we show examples of the
second derivatives of the effective potentials in Fig. 3.

Figure 2:  The specific radii of two photon spheres off the throat
$r_{\mathrm{m}}/M$, two antiphoton spheres off the throat $r_{\mathrm{a}}/M$,
and another photon sphere or another antiphoton sphere on the throat
$r_{\mathrm{th}}/M$ when $Q/M=0,0.6,0.9$, and $0.99$ are shown in left-top,
right-top, left-bottom, and right-bottom panels, respectively. Thick-dashed
(green) and thick-solid (black) curves denote the specific radii of the photon
spheres off the throat $r_{\mathrm{m}}/M$ and the antiphoton spheres off the
throat $r_{\mathrm{a}}/M$, respectively. Thin-dashed (red) and thin-solid
(blue) curves denote the specific radius of the photon sphere and the
antiphoton sphere on the throat $r_{\mathrm{th}}/M$, respectively.  Figure 3:
The second derivatives of the effective potentials for $Q/M=0.99$ are shown.
Thin-solid (blue) and thick-solid (red) curves denote the second derivatives
of effective potentials for the photon spheres and the antiphoton spheres off
the throat, respectively. A thick-dashed (green) curve denotes the one for
another photon sphere or another antiphoton sphere on the throat. We confirm
that the curves have two zero points at $\lambda=\lambda_{\mathrm{th}}$ and
$\lambda=\lambda_{\mathrm{ma}}$.

We use rather a proper length $\rho$ from the throat as new radial coordinate
to cover over the Reissner-Nordström black-hole-like wormhole spacetime than
the radial coordinate $x$ since the latter is really complicated. We obtain
the proper length as

$\displaystyle\left|\rho\right|$ $\displaystyle\equiv$
$\displaystyle\int^{r}_{r_{\mathrm{th}}}\frac{\mathrm{d}r}{\sqrt{B(r)}}$
$\displaystyle=$ $\displaystyle\sqrt{r^{2}-2Mr+Q^{2}}$
$\displaystyle+M\log\left(\frac{r-M+\sqrt{r^{2}-2Mr+Q^{2}}}{\sqrt{M^{2}-Q^{2}}}\right).$

By using the coordinates $(t,\rho,\theta,\varphi)$, the metric is written in

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\left[1-\frac{2M}{(1+\lambda^{2})r(\rho)}+\frac{Q^{2}}{(1+\lambda^{2})r^{2}(\rho)}\right]\mathrm{d}t^{2}$
$\displaystyle+\mathrm{d}\rho^{2}+r^{2}(\rho)\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right).$

Note that the throat is at $\rho=\rho_{\mathrm{th}}\equiv 0$ and we can extend
the domain of the radial coordinate $\rho$ into $-\infty<\rho<\infty$ because
of the Z-2 symmetry of the wormhole spacetime. The trajectory of the ray can
be rewritten as

$\displaystyle\dot{\rho}^{2}+V(\rho,b)=0,$ (36)

where the effective potential $V(\rho,b)$ is defined by

$\displaystyle V(\rho,b)\equiv
E^{2}\left(\frac{b^{2}}{r^{2}(\rho)}-\frac{1}{A(r(\rho))}\right).$ (37)

The ray with the critical impact parameter

$\displaystyle
b=b_{\mathrm{th}}\equiv\pm\sqrt{\frac{r_{\mathrm{th}}^{2}}{A(r_{\mathrm{th}})}}$
(38)

satisfies

$\displaystyle
V(\rho_{\mathrm{th}},b_{\mathrm{th}})=\frac{\mathrm{d}V(\rho_{\mathrm{th}},b_{\mathrm{th}})}{\mathrm{d}\rho}=0.$
(39)

The second derivative of the effective potential is obtained as

$\displaystyle\frac{\mathrm{d^{2}}V(\rho_{\mathrm{th}},b_{\mathrm{th}})}{\mathrm{d}\rho^{2}}$
$\displaystyle=E^{2}\left(-\frac{b_{\mathrm{th}}^{2}}{r_{\mathrm{th}}^{3}}+\frac{1}{2A^{2}(r_{\mathrm{th}})}\frac{\mathrm{d}A(r_{\mathrm{th}})}{\mathrm{d}r}\right)\frac{\mathrm{d}B(r_{\mathrm{th}})}{\mathrm{d}r}$
$\displaystyle=\frac{-2\left(\lambda^{2}+1\right)\left(M\sqrt{M^{2}-Q^{2}}+M^{2}-Q^{2}\right)E^{2}Z}{\lambda^{4}\left(\sqrt{M^{2}-Q^{2}}+M\right)^{4}\left(2M\sqrt{M^{2}-Q^{2}}+2M^{2}-Q^{2}\right)^{2}},$

where $Z$ is given by

$\displaystyle Z$ $\displaystyle\equiv$ $\displaystyle
5M^{2}Q^{2}+\left(3MQ^{2}-4M^{3}\right)\sqrt{M^{2}-Q^{2}}-4M^{4}-Q^{4}$ (41)
$\displaystyle+\lambda^{2}\left[8M^{4}+Q^{4}-8M^{2}Q^{2}\right.$
$\displaystyle\left.+\left(8M^{3}-4MQ^{2}\right)\sqrt{M^{2}-Q^{2}}\right].$

We confirm
$\frac{\mathrm{d^{2}}V(\rho_{\mathrm{th}},b_{\mathrm{th}})}{\mathrm{d}\rho^{2}}>0$
for $\lambda<\lambda_{\mathrm{th}}$ and
$\frac{\mathrm{d^{2}}V(\rho_{\mathrm{th}},b_{\mathrm{th}})}{\mathrm{d}\rho^{2}}<0$
for $\lambda>\lambda_{\mathrm{th}}$ in a straight calculation. Thus, there is
an antiphoton sphere on the throat for $\lambda<\lambda_{\mathrm{th}}$ and a
photon sphere on the throat for $\lambda>\lambda_{\mathrm{th}}$.

We summarize our results on the Reissner-Nordström black-hole-like wormhole.
(i)In the case of $\left|Q\right|/M<\sqrt{15}/4$: It has two photon spheres
off the throat and one antiphoton sphere on the throat for
$\lambda<\lambda_{\mathrm{th}}$ and one photon sphere on the throat only for
$\lambda>\lambda_{\mathrm{th}}$. (ii)In the case of
$\left|Q\right|/M>\sqrt{15}/4$: It has two photon spheres off the throat and
one antiphoton sphere on the throat for $\lambda<\lambda_{\mathrm{th}}$, three
photon spheres (one of them is on the throat and the others are off the
throat) and two antiphoton spheres off the throat for
$\lambda_{\mathrm{th}}<\lambda<\lambda_{\mathrm{ma}}$ and one photon sphere on
the throat only for $\lambda>\lambda_{\mathrm{ma}}$.

##  IV Summary and Discussion

A general, static, and spherical symmetrical wormhole with Z-2 symmetry
against the throat has a photon sphere or an antiphoton sphere on the throat.
We give a new theorem that a general asymptotically-flat, static, spherically
symmetrical, and Z-2 symmetrical wormhole has at least one photon sphere. We
emphasize that the theorem is due to the geometry of the throat and the Z-2
symmetry against the throat. Thus, our result does not depend on gravitational
theories and energy conditions. We should keep in mind that we can apply the
theorem for neither Z-2 asymmetrical wormholes nor thin-shell wormholes.

We have considered a Reissner-Nordström black-hole-like wormhole or a charged
Damour-Solodukhin wormhole as an example. We have concentrated on non-marginal
cases that the second derivative of the effective potential does not vanish on
the throat in Sec. III for simplicity but we can easily extend our result to
the marginal cases that it vanishes. We find a marginal photon sphere on the
throat for $\lambda=\lambda_{\mathrm{th}}$ in the case of
$\left|Q\right|/M<\sqrt{15}/4$ and the marginal photon sphere on the throat
for $\lambda=\lambda_{\mathrm{ma}}$ in the case of
$\left|Q\right|/M>\sqrt{15}/4$. In the case of the
$\left|Q\right|/M=\sqrt{15}/4$, we find the marginal photon sphere on the
throat for $\lambda=\lambda_{\mathrm{th}}=\lambda_{\mathrm{ma}}$.

We do not consider rotating wormholes on this paper but the effects of the
throat on circular photon orbits or the shadow of the rotating wormholes also
escape notice sometimes Nedkova:2013msa ; Abdujabbarov:2016efm ; Amir:2018pcu
. The throat can change the shape of the shadow of the rotating wormholes
Shaikh:2018kfv ; Kasuya:2021cpk . The extension of this work to a general
rotating wormhole is left as future work.

## References

  * (1) C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis, J. Math. Phys. 42, 818 (2001).
  * (2) V. Perlick, Living Rev. Relativity 7, 9 (2004).
  * (3) Y. Hagihara, Jpn. J. Astron. Geophys., 8, 67 (1931).
  * (4) C. Darwin, Proc. R. Soc. Lond. A 249, 180 (1959).
  * (5) R. d’ E. Atkinson, Astron. J. 70, 517 (1965).
  * (6) J.-P. Luminet, Astron. Astrophys. 75, 228 (1979).
  * (7) H. C. Ohanian, Am. J. Phys. 55, 428 (1987).
  * (8) R. J. Nemiroff, Am. J. Phys. 61, 619 (1993).
  * (9) S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D 61, 064021 (2000).
  * (10) K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D 62, 084003 (2000).
  * (11) V. Bozza, S. Capozziello, G. Iovane, and G. Scarpetta, Gen. Relativ. Gravit. 33, 1535 (2001).
  * (12) E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D 66, 024010 (2002).
  * (13) V. Bozza, Phys. Rev. D 66, 103001 (2002).
  * (14) V. Perlick, Phys. Rev. D 69, 064017 (2004).
  * (15) V. Bozza, Gen. Relativ. Gravit. 42, 2269 (2010).
  * (16) N. Tsukamoto, Phys. Rev. D 95, 064035 (2017).
  * (17) R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, Phys. Rev. D 99, 104040 (2019).
  * (18) W. L. Ames and K. S. Thorne, Astrophys. J. 151, 659 (1968).
  * (19) J. L. Synge, Mon. Not. Roy. Astron. Soc. 131, no. 3, 463 (1966).
  * (20) H. Yoshino, K. Takahashi, and K. i. Nakao, Phys. Rev. D 100, 084062 (2019).
  * (21) N. G. Sanchez, Phys. Rev. D 18, 1030 (1978).
  * (22) Y. Decanini, A. Folacci, and B. Raffaelli, Phys. Rev. D 81, 104039 (2010).
  * (23) W. H. Press, Astrophys. J. 170, L105 (1971).
  * (24) C. J. Goebel, Astrophys. J. 172, L95 (1972).
  * (25) B. Raffaelli, Gen. Rel. Grav. 48, 16 (2016).
  * (26) M. A. Abramowicz and A. R. Prasanna, Mon. Not. Roy. Astr. Soc. 245, 720 (1990).
  * (27) M. A. Abramowicz, Mon. Not. Roy. Astr. Soc. 245, 733 (1990).
  * (28) B. Allen, Nature 347, 615 (1990).
  * (29) W. Hasse and V. Perlick, Gen. Relativ. Gravit. 34, 415 (2002).
  * (30) P. Mach, E. Malec, and J. Karkowski, Phys. Rev. D 88, 084056 (2013)
  * (31) E. Chaverra and O. Sarbach, Class. Quant. Grav. 32, 155006 (2015)
  * (32) M. Cvetic, G. W. Gibbons, and C. N. Pope, Phys. Rev. D 94, 106005 (2016).
  * (33) Y. Koga and T. Harada, Phys. Rev. D 94, 044053 (2016).
  * (34) Y. Koga and T. Harada, Phys. Rev. D 98, 024018 (2018).
  * (35) Y. Koga, Phys. Rev. D 99, 064034 (2019).
  * (36) J. Keir, Class. Quant. Grav. 33, 135009 (2016).
  * (37) V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, Phys. Rev. D 90, 044069 (2014).
  * (38) P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Phys. Rev. Lett. 119, 251102 (2017).
  * (39) P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Phys. Rev. D 96, 024039 (2017).
  * (40) P. Cunha, V.P., C. Herdeiro, E. Radu, and N. Sanchis-Gual, Phys. Rev. Lett. 130, 061401 (2023).
  * (41) Z. Zhong, V. Cardoso, and E. Maggio, Phys. Rev. D 107, 044035 (2023).
  * (42) B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. 116, 061102 (2016).
  * (43) K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. 875, L1 (2019).
  * (44) K. Akiyama et al. [Event Horizon Telescope Collaboration], Astrophys. J. Lett. 930, L12 (2022).
  * (45) S. Hod, Phys. Lett. B 727, 345 (2013).
  * (46) S. Hod, Phys. Rev. D 101, 084033 (2020).
  * (47) S. Hod, JHEP 12, 178 (2023).
  * (48) S. Hod, Phys. Lett. B 776, 1 (2018).
  * (49) R. Kudo and H. Asada, Phys. Rev. D 105, 084014 (2022).
  * (50) M. Visser, Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics, Woodbury, NY, 1995).
  * (51) M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
  * (52) T. K. Dey and S. Sen, Mod. Phys. Lett. A 23, 953 (2008).
  * (53) H. G. Ellis, J. Math. Phys. 14, 104 (1973).
  * (54) K. A. Bronnikov, Acta Phys. Polon. B 4, 251 (1973).
  * (55) L. Chetouani and G. Clément, Gen. Relativ. Gravit. 16, 111 (1984).
  * (56) T. Muller, Am. J. Phys. 72, 1045 (2004).
  * (57) K. K. Nandi, Y. Z. Zhang, and A. V. Zakharov, Phys. Rev. D 74, 024020 (2006).
  * (58) T. Muller, Phys. Rev. D 77, 044043 (2008).
  * (59) A. Bhattacharya and A. A. Potapov, Mod. Phys. Lett. A 25, 2399 (2010).
  * (60) G. W. Gibbons and M. Vyska, Class. Quant. Grav. 29, 065016 (2012).
  * (61) K. Nakajima and H. Asada, Phys. Rev. D 85, 107501 (2012).
  * (62) N. Tsukamoto, T. Harada, and K. Yajima, Phys. Rev. D 86, 104062 (2012).
  * (63) T. Ohgami and N. Sakai, Phys. Rev. D 91, 124020 (2015).
  * (64) N. Tsukamoto and T. Harada, Phys. Rev. D 95, 024030 (2017).
  * (65) N. Tsukamoto, Phys. Rev. D 94, 124001 (2016).
  * (66) N. Tsukamoto, Phys. Rev. D 95, 084021 (2017).
  * (67) M. A. Bugaev, P. S. Samorodskaya, I. D. Novikov and S. V. Repin, Phys. Rev. D 108, 124059 (2023).
  * (68) A. Simpson and M. Visser, JCAP 02, 042 (2019).
  * (69) N. Tsukamoto, Phys. Rev. D 103, 024033 (2021).
  * (70) J. R. Nascimento, A. Y. Petrov, P. J. Porfirio, and A. R. Soares, Phys. Rev. D 102, 044021 (2020).
  * (71) K. A. Bronnikov, R. A. Konoplya and T. D. Pappas, Phys. Rev. D 103, 124062 (2021).
  * (72) M. Guerrero, G. J. Olmo, D. Rubiera-Garcia, and D. S. C. Gómez, JCAP 08, 036 (2021).
  * (73) T. Damour and S. N. Solodukhin, Phys. Rev. D 76, 024016 (2007)
  * (74) N. Tsukamoto, Phys. Rev. D 101, 104021 (2020) [erratum: Phys. Rev. D 106, 049901 (2022)].
  * (75) N. Tsukamoto, Class. Quant. Grav. 40, 228001 (2023).
  * (76) K. K. Nandi, R. N. Izmailov, E. R. Zhdanov, and A. Bhattacharya, JCAP 1807, 027 (2018).
  * (77) A. Ovgun, Phys. Rev. D 98, 044033 (2018).
  * (78) A. Bhattacharya and R. K. Karimov, arXiv:1811.00768 [gr-qc].
  * (79) A. Ovgun, arXiv:1811.06870 [gr-qc].
  * (80) A. Ovgun, Universe 5, 115 (2019).
  * (81) S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and P. S. Joshi, et al. Class. Quant. Grav. 40, 165007 (2023).
  * (82) R. Shaikh, P. Banerjee, S. Paul and T. Sarkar, Phys. Lett. B 789, 270 (2019) [erratum: Phys. Lett. B 791, 422 (2019)].
  * (83) K. A. Bronnikov and K. A. Baleevskikh, Grav. Cosmol. 25, 44 (2019).
  * (84) R. Shaikh, P. Banerjee, S. Paul, and T. Sarkar, JCAP 1907, 028 (2019); R. Casadio, A. Fabbri and L. Mazzacurati, Phys. Rev. D 65, 084040 (2002); N. Dadhich, S. Kar, S. Mukherji and M. Visser, Phys. Rev. D 65, 064004 (2002).
  * (85) N. Tsukamoto, Phys. Rev. D 105, 064013 (2022); K. A. Bronnikov and S. W. Kim, Phys. Rev. D 67, 064027 (2003); K. A. Bronnikov, V. N. Melnikov, and H. Dehnen, Phys. Rev. D 68, 024025 (2003).
  * (86) N. Tsukamoto, Phys. Rev. D 104, 064022 (2021); N. Tsukamoto, Phys. Rev. D 105, 084036 (2022); F. S. N. Lobo, M. E. Rodrigues, M. V. de Sousa Silva, A. Simpson and M. Visser, Phys. Rev. D 103, 084052 (2021).
  * (87) J. P. S. Lemos and O. B. Zaslavskii, Phys. Rev. D 78, 024040 (2008).
  * (88) N. Tsukamoto and T. Kokubu, Phys. Rev. D 101, 044030 (2020).
  * (89) R. K. Karimov, R. N. Izmailov, and K. K. Nandi, Eur. Phys. J. C 79, 952 (2019).
  * (90) P. Bueno, P. A. Cano, F. Goelen, T. Hertog, and B. Vercnocke, Phys. Rev. D 97, 024040 (2018).
  * (91) S. H. Volkel and K. D. Kokkotas, Class. Quant. Grav. 35, 105018 (2018).
  * (92) S. Paul, R. Shaikh, P. Banerjee, and T. Sarkar, JCAP 03, 055 (2020).
  * (93) M. Amir, K. Jusufi, A. Banerjee, and S. Hansraj, Class. Quant. Grav. 36, 215007 (2019).
  * (94) P. G. Nedkova, V. K. Tinchev and S. S. Yazadjiev, Phys. Rev. D 88, 124019 (2013).
  * (95) A. Abdujabbarov, B. Juraev, B. Ahmedov, and Z. Stuchlík, Astrophys. Space Sci. 361, 226 (2016).
  * (96) R. Shaikh, Phys. Rev. D 98, 024044 (2018).
  * (97) S. Kasuya and M. Kobayashi, Phys. Rev. D 103, 104050 (2021).