# Eavesdropping on competing condensates by the edge supercurrent in a Weyl
superconductor

Stephan Kim1,2 Shiming Lei3 Leslie M. Schoop3 R. J. Cava3 N. P. Ong1,§
Departments of Physics1, Electrical and Computer Engineering2 and Chemistry3,
Princeton University, Princeton, NJ 08544, USA

In a topological insulator the metallic surface states are easily
distinguished from the insulating bulk states [1]. By contrast, in a
topological superconductor [2, 3, 4, 5], much less is known about the
relationship between an edge supercurrent and the bulk pair condensate. Can we
force their pairing symmetries to be incompatible? In the superconducting
state of the Weyl semimetal MoTe2, an edge supercurrent is observed as
oscillations in the current-voltage (_I-V_) curves induced by fluxoid
quantization [6]. We have found that the $s$-wave pairing potential of
supercurrent injected from niobium contacts is incompatible with the intrinsic
pair condensate in MoTe2. The incompatibility leads to strong stochasticity in
the switching current $I_{c}$ as well as other anomalous properties such as an
unusual antihysteretic behavior of the “wrong” sign. Under supercurrent
injection, the fluxoid-induced edge oscillations survive to much higher
magnetic fields _H_. Interestingly, the oscillations are either very noisy or
noise-free depending on the pair potential that ends up dictating the edge
pairing. Using the phase noise as a sensitive probe that eavesdrops on the
competiting bulk states, we uncover an underlying blockade mechanism whereby
the intrinsic condensate can pre-emptively block proximitization by the Nb
pair potential depending on the history.

_Preliminaries_
As reported in Ref. [6], the edge supercurrent in MoTe2 realizes a toroidal
topology despite the absence of holes in the exfoliated crystals. As $H$ is
slowly increased, fluxoid quantization leads to a sawtooth field profile for
the edge superfluid velocity ${\bf v}_{s}$, which translates to oscillations
of $I_{\rm c}$ that are clearly seen in a color-map plot of $dV/dI$ vs. $H$ (a
fluxoid is a flux quantum $\phi_{0}$ plus the superfluid circulation; Sec. VI
in Supplementary Materials SM). The large number of oscillation cycles ($\sim
90$) allow its phase noise to be analyzed. The analysis reveals that a large
phase noise appears whenever edge-state pairing, driven by Nb, is incompatible
with bulk-state pairing in MoTe2. Incompatibility between the injected
$s$-wave supercurrent and paired states intrinsic to MoTe2 has observable
consequences in both the bulk and edge states. By varying the contact geometry
across 8 devices, we can distinguish edge-state features from those in the
bulk.

Because the interfaces between Nb and MoTe2 have high transmittivity at 20 mK,
our experiment lies in the strong-coupling proximity-effect regime, well
beyond the Josephson effect regime (Sec. V in SM). In strong coupling
junctions, the system (Nb plus MoTe2) is treated as a single superconductor
with a unique $T_{c}$ and a gap function $\hat{\Psi}({\bf r})$ that is
inhomogeneous [7, 8, 9, 10, 11, 12]. In device $SA$, $T_{c}$ ($\simeq 850$ mK)
lies well above the unperturbed $T_{c}$ of pristine MoTe2 (100 mK) although
still far below that of Nb ($\sim 8$ K). Similarly, the critical field and
critical current are greatly enhanced over the values in pristine MoTe2.

To discuss the proximity effect (Sec. VII of SM) [7, 8, 9, 10, 11, 12], we
denote the Gor’kov pair amplitude in Nb by $\eta({\bf
r})=\langle\hat{\psi}_{\downarrow}({\bf r})\hat{\psi}_{\uparrow}({\bf
r})\rangle$ where $\hat{\psi}_{\alpha}({\bf r})$ annihilates an electron in
spin state $\alpha=\uparrow,\downarrow$ at $\bf r$. We call the pair amplitude
in pristine MoTe2 (of unknown symmetry) $F_{\alpha\beta}({\bf
r})=\langle\hat{\psi}_{\alpha}({\bf r})\hat{\psi}_{\beta}({\bf r})\rangle$.
Aside from MoTe2, evidence for edge supercurrents has been reported in Bi
nanowires [13] and in a Kagome superconductor [14].

_Anti-hysteretic behavior of central peak_
Figure 1a shows the color map of the differential resistance $dV/dI$ versus
$H$ and current $I$ measured in device $SA$ at temperature $T=18$ mK. In weak
$H$, supercurrent injection from Nb leads to a large critical current $I_{\rm
c}$ within a narrow central peak (shown shaded black) that reaches values of
$\sim 80\;\mu$A (20$\times$ larger than seen in pristine crystals). An
immediate surprise is the anomalous position of the central peak. In
conventional superconductors, we expect $H$ to cross zero before the flux
density $B$ does. Instead, the central peak here emerges before $H$ reaches
zero for either scan (which seems oddly anti-causal). We dub this curious
pattern “anti-hysteretic.”

When $|H|$ exceeds 10 mT, we observe a dense array of periodic peaks in the
color map (Figs. 1b and 1c). These are the fluxoid-induced oscillations [6],
now persisting to 90 mT (compared with 3 mT in pristine crystals). As shown in
Figs. 1d and 1e, the oscillations appear as narrow peaks in the differential
resistance $(dV/dI)_{0}$ measured at $I=0$. At each peak, ${\bf v}_{s}$
reverses direction to admit a fluxoid [6].

_Stochastic Switching_
The incompatibility between the intrinsic pair condensation amplitude
$F_{\alpha\beta}$ and $\eta$ injected from Nb is most evident in $I$-$V$
curves measured when $H$ lies within the central peak in Fig. 1a. The
switching transition is strongly stochastic (Sec. II in SM). In Fig. 2a, we
plot 100 curves of $dV/dI$ vs. $I$ measured at 18 mK in device $SC$ with $H$
fixed at $-2.5$ mT. The critical currents $\\{I_{\rm c}\\}$ obey a
distribution function $P(I_{\rm c},H)$ that is distinctly bimodal, with a
histogram that shows a major peak at $10.5\;\mu$A and a minor one at 8.5
$\mu$A (see Fig. 2b and Sec. II in SM). In the color map of $dV/dI(H,I)$ (Fig.
1a), the stochastic nature of the distribution is also apparent as random
spikes within the central peak. By contrast, outside the central peak
($H=-7.5$ mT), the distribution $P(I_{\rm c},H)$ becomes very narrow (Fig.
2c). The standard deviation $s(H)$ (obtained from 100 scans at each $H$) is
plotted in Fig. 2d. The profile $s(H)$ displays a narrow peak at the central
peak with peak value 100$\times$ larger than the baseline (red curve). At 311
mK, however, the peak is absent (blue curve). The stochastic switching
reflects the competition between bulk-date pairing with symmetry $\eta$ or
$F_{\alpha\beta}$.

_Phase Noise and Anti-hysteretic $(dV/dI)_{0}$ Curves_
The competition between $F_{\alpha\beta}$ and $\eta$ has a strong effect on
the phase noise of the oscillations in $(dV/dI)_{0}$. Field scans are called
inbound if $|H|$ decreases with time, and outbound if $|H|$ increases. On
inbound branches (Figs. 1b and 1d), the noise content of the oscillations is
invariably very large, whereas, on outbound branches (Figs. 1c and 1e), the
phase noise is conspicuously suppressed, especially in the interval $5<H<45$
mT, which we call the “quiet” zone.

In Fig. 3 we emphasize the differences by displaying the color map of the
oscillations over the entire field range. On the inbound branch (Fig. 3a), the
noise content is much higher than on the outbound (Fig. 3d). To quantify the
phase noise, we express the fluctuations in the frequency as an $H$-dependent
phase $\theta(H)$ (Sec. III in SM). It is expedient to subtract from
$\theta(H)$ a background $\theta_{0}(H)$ derived from an $H$-linear fit to
$\theta(H)$ in the quiet zone (5, 45) mT (Sec. III in SM). The deviation
$\Delta\theta(H)=\theta(H)-\theta_{0}(H)$ then highlights field intervals with
large phase noise. In Fig. 3b, $\Delta\theta$ on the inbound branch (blue
curve) is seen to be much larger than on the outbound curve (red). In the
quiet zone (5, 45) mT, $\Delta\theta$ is flat by design. We also employ the
smoothed derivative $\langle d\Delta\theta/dH\rangle$ which is more sensitive
to local changes in the frequency. As seen in Fig. 3c, $\langle
d\Delta\theta/dH\rangle$ is much larger on the inbound branch than in the
quiet zone on the outbound.

Aside from the oscillations, $(dV/dI)_{0}$ also displays a dissipationless
regime bracketed by the two fields $B^{\rm in}_{0}$ and $B^{\rm out}_{0}$ (red
arrows in Figs. 1d and 1e). An expanded view of these transitions is shown in
Figs. 4a – 4h for 4 devices. In device $SA$, the transition on the inbound
branch occurs at $-B^{\rm in}_{0}=-54$ mT, larger in magnitude than the field
$B^{\rm out}_{0}$ (= 28 mT) on the outbound branch (blue curves in Fig. 4e).
The reverse sweep (red curves) is the mirror reflection of the blue curve. The
resulting hysteresis is again anti-hysteretic.

_Distinguishing Edge from Bulk Supercurrents_
The observed supercurrents segregate into two groups. In group I (central peak
in Fig. 1a), $I_{\rm c}$ attains very large values ($80-200\;\mu$A) but is
easily suppressed in a weak $H$ ($3-5$ mT). The second group (II) are the weak
supercurrents ($1-2\;\mu$A) seen in the oscillations in $(dV/dI)_{0}$ which
survive to fields up to 90 mT (Figs. 1 and 3).

By tailoring the contact designs in 8 devices, we have found that the group II
features are specific to edge states. The oscillation period corresponds to a
fluxoid area ${\cal A}_{\phi}$ bracketed by the Nb contacts (Sec. I in SM).
Its perimeter is comprised of two segments (length $w$) lying under Nb
contacts, and two segments called “links” (of length $\ell_{1}$ and
$\ell_{2}$) at the uncovered physical edges (inset in Fig. 1a).

In five devices ($SA,SK,SI,SJ,SB$) the Nb contacts are laid down as parallel
strips with spacing $d$ (Table S1, Fig. 4 and Fig. S3 in SM). In the remaining
three ($SF,SH,SD$), the Nb contacts are progressively reduced in volume while
$\ell_{1}$ and $\ell_{2}$ are greatly lengthened. These changes strongly
attenuate group II features.

In Fig. 4, we show two parallel-strip devices, $SA$ and $SI$ (Panels a and b,
respectively), and two devices, $SH$ and $SD$, in the second series (c and d,
respectively). The corresponding $(dV/dI)_{0}$ curves are displayed in Figs.
4e-4h, together with the color maps in Figs. 4i-4j. Group II features are
prominently seen in $SA$ and $SI$ where we have $2w\gg\ell_{1}=\ell_{2}=d$. As
we increase $d$, the widths of the antihysteretic loops decrease. In devices
$SH$ (with $\ell_{1}$ = 5,420 and $\ell_{2}$ = 5,710 nm) and $SD$ ($\ell_{1}$
= 8,030 nm and $\ell_{2}$ = 17,670 nm), the edge oscillations and
antihystereses are completely suppressed (aside from residual wings in $SH$).
Data from 4 other samples are shown in Fig. S3 in SM.

In devices $SA$ and $SI$, supercurrent in the bulk flows in parallel with
supercurrent in the links if $|H|$ is weak. When $|H|$ exceeds $\sim$4 mT, the
bulk supercurrent is suppressed but the link supercurrents survive to large
$H$ because they are one-dimensional. The robustness enables fluxoid
oscillations to be observed up to 90 mT. To realize this, however, we need
$\ell_{1}$ and $\ell_{2}$ to be shorter than the decay length $\lambda$ ($\sim
800$ nm) that governs the decay of the pair amplitude $\eta$ along a link
(inset in Fig. 1a). If $\ell_{1}$, $\ell_{2}$ $>\lambda$ (the case in SH and
SD), $\eta$ decays to zero before the loop is closed, and the oscillations
vanish (Fig. 4g,h). The vanishing of the closed-loop supercurrent is also
evident in $(dV/dI)_{0}$ vs. $H$, which stays finite except when $|H|<$ 2 mT.
These findings establish that the group II features arise from edge
supercurrents at the links.

By comparison, the much larger $I_{c}$ values of the group I supercurrents
(central peak in Fig. 1a) suggest a bulk origin. The 20-fold decrease in the
peak value $I_{\rm c,max}$ as $d$ increases from 156 to 700 nm also implies
bulk states proximitized by $\eta$ (Table S1 in SM). The decrease is
consistent with the short coherence length of Nb ($\xi_{0}\sim$ 40 nm). By
contrast, at the links, $\lambda\gg\xi_{0}$.

_Proximity Effect Between Competing Symmetries_
In de Gennes’ treatment of the proximity effect (valid in the linear-gap
regime) the amplitude for a Cooper pair to propagate between points ${\bf
r}^{\prime}$ and ${\bf r}$ is given by the kernel $K({\bf r,r}^{\prime})$ [7,
9, 12] (Sec. VII in SM). Whenever a propagator segment lies in Nb, $K({\bf
r,r}^{\prime})$ gains a large enhancement from the strong $s$-channel
attraction. Hence the gap function $\hat{\Psi}({\bf r})$ in MoTe2 can adopt
either the symmetry of $F_{\alpha\beta}$ or that of $\eta$ depending on the
weighted contributions of propagators going from all points $\bf
r^{\prime}\to\bf r$ (Eq. S23 in SM). An applied field $H$ can tip the balance
by selectively weakening one or the other condensation amplitude. Calculations
beyond the linear-gap regime show how the gap function changes symmetry in a
junction between an $s$\- and a $p$-wave superconductor [10].

At $H=0$, the weighted average favors $F_{\alpha\beta}$. Vortices inserted by
$H$ initially form a vortex solid. With increasing $H$, melting of the vortex
solid leads to a dissipative vortex liquid in which the gap modulus
$|\hat{\Psi}|$ remains finite but its phase fluctuates strongly due to vortex
motion. In the vortex liquid, which survives to 80 mT, the loss of long-range
phase coherence substantially weakens the pair amplitude $F_{\alpha\beta}$
relative to $\eta$. Conversely, solidification of the vortex system strongly
tilts the competition towards the intrinsic $F_{\alpha\beta}$.

_Correlating central peak, phase noise and antihysteresis_
In each device, edge-state pairing at the links may be driven either by
$F_{\alpha\beta}$ or $\eta$ from the nearest Nb contact. From the phase noise
analysis, we infer that compatibility between edge-pairing and bulk-pairing
produces oscillations that are nearly noise-free. By contrast, incompatibility
between $s$-wave pairing at the edge and $F_{\alpha\beta}$ in the bulk
generates substantial phase noise.

Combining the group I and II features with the phase noise measured by
$\Delta\theta$ and $d\langle\Delta\theta/dH\rangle$ in device $SA$, we arrive
at a consistent picture of the antihysteretic behavior. The color map of
$dV/dI$ and the trace of $(dV/dI)_{0}$ at 18 mK are displayed in Figs. 5a and
b, respectively. Starting at $-100$ mT, the phase noise is initially small, as
seen in $\langle d\Delta\theta/dH\rangle$ (blue curve in Fig. 5c). This
implies that edge pairing is driven by the Nb contacts while all pairing of
bulk states is strongly suppressed. When $H$ reaches $-80$ mT, we observe a
surge onset in phase noise which we take as evidence for incompatibility with
the vortex liquid that appears at $-80$ mT. The phase noise remains large over
the entire interval ($-80$, $-12$) mT (red curve in 5c).

When $H$ crosses $-5$ mT, proximitization of the bulk states driven by $\eta$
becomes strongly favored ($H$ becomes too weak to depair the bulk $s$-wave
condensate). The abrupt emergence at $-4$ mT of a large bulk supercurrent (the
central peak) signals when $\hat{\Psi}$ switches to $s$-wave pairing (Figs.
5a), but this favorable interval is brief. When $H$ reaches $-2$ mT,
solidification of the vortex liquid (and the ensuing long-range phase
coherence) cause $\hat{\Psi}({\bf r})$ to switch back to the intrinsic pairing
symmetry $F_{\alpha\beta}$. Suppression of all $s$-wave paired regions
throughout the crystal collapses the central peak _before_ $H$ reaches 0-.
Hence the placement of the central peak is antihysteretic as observed.

_Blockade_
When the field crosses zero to the outbound branch ($H>0$), we should expect
to see the re-appearance of the central peak in the interval (2, 4) mT.
However, this is not observed in any of the 8 devices studied.

Together, the absence of the central peak for $H>0$ and the noise suppression
in the interval $(5,45)$ mT suggest a “blockade” mechanism. Once $\hat{\Psi}$
switches to the intrinsic symmetry $F_{\alpha\beta}$ in the limit $H\to
0^{-}$, the intrinsic condensate appears to block re-proximitization of the
bulk by $\eta$ on the outbound branch throughout the quiet zone which extends
to a field that we call $B^{\flat}\sim$ 45 mT (shaded lilac in Figs. 5a-c).
Edge pairing by Nb is blocked within the quiet zone.

The identification of $(dV/dI)_{0}$ with supercurrents at the edge clarifies
considerably the anti-hysteretic loops in Figs. 4e and 4f. As shown by the
green curve in Fig. 5b, the dissipationless interval for the edge current
spans the interval $(-B^{\rm in}_{0},B^{\rm out}_{0})=(-54,28)$ mT for a left-
to-right field scan. The interval $(-B^{\rm in}_{0},B^{\rm out}_{0})$ is
shifted left because, on the inbound branch, the edges are proximitized by the
strong Nb pair potential, whereas, on the outbound, edge pairing is driven by
the much weaker $F_{\alpha\beta}$ while $\eta$-pairing is excluded by the
blockade. Likewise, in a right-to-left scan, $(-B^{\rm out}_{0},B^{\rm
in}_{0})$ is shifted right. This accounts for the anti-hysteretic sign of the
loops in $(dV/dI)_{0}$.

_Raising the temperature_
Figure 6a shows the blockade region (shaded maroon) inferred from left-to-
right field scans taken at elevated temperatures 378, 532 and 722 mK (Fig. 6b
is for the opposite field scan). The corresponding color maps of $dV/dI$ are
displayed in Figs. 6c, d and e. For the results at 522 mK, we also show the
trace of $(dV/dI)_{0}$ (Fig. 6f) and the phase noise measured by
$\Delta\theta$ and $\langle d\Delta\theta/dH\rangle$ (Fig. 6g). The overall
patterns are similar to those taken at 18 mK (Fig. 5) except that raising $T$
decreases the field scales. The asymmetry of the phase noise, including its
suppression in the blockade interval, is apparent in Figs. 6f and 6g. We find
that the widths of the antihysteretic loops in $(dV/dI)_{0}$, the edge
dissipationless interval $(-B^{\rm in}_{0},B^{\rm out}_{0})$ and $B^{\flat}$
all decrease systematically as $T$ increases, reaching zero at the device
$T_{c}\simeq 850$ mK. The linear decrease of $B^{\flat}(T)$ as $T\to T_{c}$ is
shown in Fig. S9. These trends are consistent with the key role played by
$F_{\alpha\beta}$ in generating the antihysteresis, phase noise and the
blockade.

_Hysteretic solid-to-liquid transition_
If we inspect the blockade region in the $H$-$T$ plane (shaded maroon in Figs.
6a,b), we find that its placement mimics that of a vortex solid phase that is
field-shifted by hysteresis. We conjecture that the blockade mechanism and
quiet zone are features intrinsic to the vortex solid. If $H$ is swept from
left to right (Fig. 6a), the liquid-to-solid transition is “delayed” on the
inbound branch until $H$ reaches $-2$ mT, analogous to supercooling of a
liquid (regarding $|H|$ as an effective temperature). On the outbound branch,
the solid-to-liquid transition is also delayed and shifted to 45 mT. Hence the
observed vortex solid phase is displaced to the right as shown in Fig. 6a. For
the opposite scan, the shift is to the left. We note that the supercooling
hysteresis has the conventional sign. However, the blockade mechanism and
incompatibility between condensates together displace the bulk and edge
supercurrent responses in the opposite direction, which leads to the
antihysteresis.

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§Corresponding author email<EMAIL_ADDRESS>

Data availability
The data in the plots will be uploaded to Harvard DataVerse.

Acknowledgements
We have benefitted from discussions with Wudi Wang and Zheyi Zhu. N.P.O. and
S.K. were supported by the U.S. Department of Energy (DE-SC0017863). The
crystal growth effort was supported by a MRSEC grant from the U.S. National
Science Foundation (NSF DMR-2011750) which supported S.L, L.M.S., R.J.C and
N.P.O. The Gordon and Betty Moore Foundation’s EPiQS initiative provided
generous support via grants GBMF9466 (N.P.O.) and GBMF9064 (L.M.S.).

Author contributions
S.K. and N.P.O. conceptualized and designed the experiment. S.K. performed all
device fabrication and measurements. The crystals were grown by S.L., L.M.S.
and R.J.C. Analysis of the data were carried out by S.K. and N.P.O. who
jointly wrote the manuscript.

Competing financial interests
The authors declare no competing financial interests.

Additional Information
Supplementary Materials is available in the online version of the paper.

Correspondence and requests for materials should be addressed to N.P.O.

Figure 1:  Anti-hysteretic central peak and edge oscillations observed in
MoTe2 measured by supercurrent injection from Nb pads in device $SA$ at 18 mK.
Panel (a) shows the color map of the differential resistance $dV/dI$ plotted
versus magnetic field $H$ and current $I$. Dissipationless regions are shown
in black (color scale at left). The central peak occurs at -3 or +3 mT when
$H$ is scanned from left to right or right to left, respectively. The inset
defines the width $w$ and spacing $d$ of the device. The decays of the Nb
pairing amplitudes in the bulk and along an edge are sketched. Panels (b) and
(c) display the fluxoid-induced edge oscillations on the inbound ($d|H|/dt<0$)
and outbound branches ($d|H|/dt>0$), respectively. Panels (d) and (e) plot the
zero-bias differential resistance $(dV/dI)_{0}$ vs. $H$ on the inbound and
outbound branches, respectively. The phase noise is much larger in (d) than in
(e). The transition to dissipationless behavior ($(dV/dI)_{0}\to 0$) occurs at
$-B^{\rm in}_{0}=-54$ mT and $B^{\rm out}_{0}=+28$ mT in (d) and (e),
respectively (red arrows). In each panel, the green arrow indicates field scan
direction.  Figure 2:  Stochastic switching in bulk states measured in device
SC. When $H$ lies within the central peak ($H$ = $-2.5$ mT, Panel a), the
switch from dissipationless to dissipative behavior is stochastic. The
critical current in 100 curves of $dV/dI$ vs. $I$ shows bunching of $I_{c}$
suggestive of a bimodal distribution $P(I_{\rm c},H)$. In Panel b, the
histogram plot of $P(I_{\rm c},H)$ confirms the bimodal distribution. However,
when $H$ lies outside the central peak ($-7.5$ mT), the switching transitions
are non-stochastic. In all 100 scans the transition occurs at $1.27\pm
0.01\;\mu$A (7 are shown in Panel c). Panel d plots the standard deviation $s$
vs. $H$ of the distribution $P(I_{\rm c},H)$ at 18 mK (Sec. II in SM). At the
peak, $s$ is a 100$\times$ larger than its value outside the peak. At 311 mK,
the peak is unresolved.  Figure 3:  Nature of the phase noise in the fluxoid-
induced oscillations over the full field interval in Sample $SA$ at 18 mK. The
four strips in Panel (a) show the observed color map of $dV/dI$ in the $H$-$I$
plane as $H$ is scanned from -90 mT to 0 mT (inbound branch). Initially,
($-90<H<-80$ mT) the phase noise is quite small. Between -80 mT and -12 mT,
the emergence of large phase noise strongly distorts the oscillations. The
noise is dominated by 2$\pi$ jumps in the phase $\theta(H)$ which lead to
random compressions and dilations of the period. We attribute the noise to
incompatibility between edge pairing induced by $\eta$ (Nb) and the intrinisic
pair amplitude $F_{\alpha\beta}$ in the vortex-liquid state of MoTe2. Panel
(b) compares the curves of the phase deviation $|\Delta\theta(H)|$ in the
inbound branch (blue) with the outbound (red).
$\Delta\theta(H)=\theta(H)-\theta_{0}(H)$ measures the phase deviation caused
by random phase jumps accumulated over the entire field interval (see text).
Panel (c) compares the smoothed derivative $\langle d\Delta\theta/dH\rangle$
between the inbound (blue) and outbound branches (red). The large phase noise
in the inbound branch causes $\langle d\Delta\theta/dH\rangle$ to lie well
above the outbound, especially between 5 and 45 mT. The four strips in Panel
(d) show the color map of $dV/dI$ as $H$ is swept from 0 to 90 mT (outbound
branch). Over the “quiet” zone (5 to 45 mT), the phase noise is negligible
except for isolated 2$\pi$ jumps (at 13.4, 35 and 36.5 mT). In all panels the
green arrow indicates field scan direction.  Figure 4:  Systematic suppression
of edge oscillations and anti-hysteresis in four devices ($SA,SI,SH$ and $SD$
in Panels a,…, d, respectively). $SA$ and $SI$ (with spacing $d$ = 156 and 285
nm, respectively) are examples from a series of 5 parallel-strip devices. $SH$
and $SD$ are from a second series in which the Nb contact volumes are
decreased while $\ell_{1}$ and $\ell_{2}$ are progressively increased. The
middle row (Panels e,…,h) shows the corresponding antihystereses in
$(dV/dI)_{0}$ vs. $H$ (blue and red arrows indicate field scan directions).
The corresponding color maps of $dV/dI$ vs. $H$ and $I$ are in the bottom row
(i,…, l). In devices $SA$ and $SI$, prominent antihysteresis and edge
oscillations are seen, both in $(dV/dI)_{0}$ (Panels e, f) and their color
maps (i, j). In devices $SH$ and $SD$, these group II features are suppressed
or absent in $(dV/dI)_{0}$ (Panels g and h) and in the color maps (k and l).
See Table S1 for parameters in the 8 devices and Fig. S3 for results from
devices $SK,SJ,SB,SF$.  Figure 5:  Correlating the central peak, phase noise
and antihysteresis measured in device $SA$ as $H$ is swept from $-100$ to
$+100$ mT at 18 mK. Panel (a) shows the color map of $dV/dI$, highlighting the
low-$I$ region. Panel (b) plots the oscillations in the zero-bias
$(dV/dI)_{0}$ (grey curve) together with its floor value (thick green curve).
The dissipationless interval for $(dV/dI)_{0}$ lies between $-B^{\rm
in}_{0}=-54$ mT and $B^{\rm out}_{0}=28$ mT (red arrows). Panel (c) plots the
phase noise measured by $|\Delta\theta|$ (red curve) and $\langle
d\Delta\theta/dH\rangle$ (blue). Within the quiet zone extending to
$B^{\flat}$ on the outbound branch (shaded lilac), the phase noise is
suppressed. We infer the existence of a blockade mechanism in this interval.
In each panel the green arrow indicates the field-scan direction.  Figure 6:
Color maps, zero-bias $(dV/dI)_{0}$ and phase noise at elevated temperatures
in device $SA$. Panels (a) and (b) display the inferred metastable states in
the $H$-$T$ plane, for left-to-right and right-to-left field scans,
respectively. The regions where the blockade mechanism operates are shaded
maroon. Thin blue stripes indicate regions in which the central peak emerges
(note the antihysteretic placement). On the outbound branch in (a), the maroon
region extends to $B^{\flat}$ ($\sim 45$ mT) at 18 mK. The dissipative vortex
liquid (orange and yellow regions) extend to $\pm 80$ mT. The device $T_{c}$
is 850 mK. Panels (c), (d) and (e) display color maps of $dV/dI$ measured at
722, 378 and 532 mK, respectively. Panel (f) plots the oscillations in the
zero-bias $(dV/dI)_{0}$ (grey curve) and its floor value (green curve). Panel
(g) display the phase noise measured by $\Delta\theta$ (red curve) and
$\langle d\Delta\theta/dH\rangle$ (blue). The lilac regions in Panels (f) and
(g) represent the quiet zone in which phase noise is minimal. As $T\to T_{c}$,
the field scales $B^{\rm in}_{0}$, $B^{\rm out}_{0}$ and $B^{\flat}$ decrease
to zero. In all panels, green arrows indicate the field-scan direction.

Methods

##  I Device fabrication

The MoTe2 devices were fabricated following standard nanofabrication
procedures. Substrates with pre-patterned alignment makers were prepared in
advance. The alignment markers were made of 5 nm of Ti and 50 nm Au. MoTe2
microflakes were mechanically exfoliated on to the substrates using Nitto
Denko SPV 5057A5 Surface Protection Tape. Microflakes with adequate sizes with
sharp edges were chosen using an optical microscope.

Niobium electrodes were made following the standard PMMA-copolymer bilayer
recipe. After the resists were spun on top of substrates, contacts were
patterned using EBPG 5150 from Raith. The devices were developed with MIBK-IPA
(1:3) solution for 60 seconds and rinsed with IPA solution. The devices were
then transferred to an EvoVac Sputtering System (Angstrom Engineering). An
_in-situ_ Ar plasma beam bombarded the surfaces of the devices for 120 seconds
to remove the top few layers of MoTe2 that were oxidized. A thin layer (3 nm)
of Ti layer was sputtered at a rate $0.1-0.2$ nm/s to form a sticking layer,
followed by 100 nm of Nb deposited at a rate 1 nm/s. Finally, a 5 nm of Au was
sputtered on top of Nb layer to protect it from oxidization. All sputtering
procedures were performed at a high vacuum of approximately $10^{-8}$ mbar.

##  II Measurement techniques

Measurements were carried out in a top-loading dilution refrigerator (Oxford
Instruments Kelvinox TLM). Once loaded to the fridge, devices were immersed in
the circulating 3He-4He mixture in the mixing chamber. The base temperature of
the fridge was $T\sim 20$ mK. Three filters were used to reduce noises during
measurements. An $LC$ $\pi$-filter, located at the top of the probe, provided
80-dB suppression for frequencies $f>100$ MHz. The two other filters were
located in the mixing chamber. The first one was a sintered metal powder
filter that consisted of Cu particles of 30-60 $\mu m$ in diameter. It
suppressed any stray electromagnetic radiation for frequencies $f>1$ GHz. The
second filter was a low-pass $RC$ filter with the cutoff frequency of $f=300$
kHz. A NbTi superconducting magnet was used to apply magnetic fields to
devices. A Keithley 2400 and a Keithley 6221 provided the current to
superconducting magnet. The smallest field-step size was as small as a few
$\mu$T.

Differential resistances of all devices were measured using a SR830 lock-in
amplifier. A DC bias from a Keithley 2400 voltage source was superposed with a
small AC excitation voltage from the lock-in amplifier through a home-made
voltage adder. The resulting voltage was converted to current through a buffer
resistor that was orders of magnitude larger in resistance than the device of
interest. The voltage signal $V$ across the device first passed through a pre-
amplifier to improve the signal-to-noise ratio. The gain was usually in the
order of $G\sim 1000$. The amplified signal then reached the lock-in amplifier
for detection. Measurements were done in quasi-4-point fashion. Although both
voltage and current contacts shared the same electrodes, the electrodes were
superconducting at 20 mK.

The differential resistance plots in the main text were obtained through the
following procedures: first, the magnetic field was set to the desired value.
Then the DC bias was ramped from zero to the desired bias value with a small
step size. After reaching the desired bias, the current was ramped back to
zero with a much larger step size. Such procedures were repeated for desired
field ranges. The collected differential resistance curves were plotted
together. The differential resistance versus field at zero bias $(dV/dI)_{0}$
plots were prepared as follows: starting with the field at zero, we swept $H$
to its maximum negative value then to the maximum positive value before
terminating at zero.


# Relative equilibria in curved restricted 4-body problems

###### Abstract.

We consider the curved 4-body problems on spheres and hyperbolic spheres.
After obtaining a criterion for the existence of quadrilateral configurations
on the equator of the sphere, we study two restricted 4-body problems, one in
which two masses are negligible, and another in which only one mass is
negligible. In the former we prove the evidence square-like relative
equilibria, whereas in the latter we discuss the existence of kite-shaped
relative equilibria.

Florin Diacu1,2,3 and Sawsan Alhowaity3,4

1Yangtze Center of Mathematics, Sichuan University, Chengdu, China

2Yale-NUS College, National University of Singapore, Singapore

3Department of Mathematics and Statistics, University of Victoria, Canada

4Department of Mathematics, University of Shaqra, Saudi Arabia

<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1\. Introduction

The classical $N$-body problem has a long history. Isaac Newton first proposed
it in 1687 in his first edition of Principia in the context of the Moon’s
motion. He assumed that universal gravitation acts between celestial bodies
(reduced to point masses) in direct proportion with the product of the masses
and in inverse proportion with the square of the distance. The study of the
$N$-body problem was further advanced by the Bernoulis, Lagrange, Laplace,
Euler, Cauchy, Jacobi, Dirichlet, Poincaré and many others.

The idea of extending the gravitational force between point masses to spaces
of constant curvature occurred soon after the discovery of hyperbolic
geometry. In the 1830s, independently of each other, Bolyai and Lobachevsky
realized that there must be an intimate connection between the laws of physics
and the geometry of the universe, [2], [29], [25]. A few years earlier, Gauss
had interpreted Newton’s gravitational law as stating that the attracting
force between bodies is inversely proportional with the area of the sphere of
radius equal to the distance between the point masses (i.e. proportional to
$1/r^{2}$, where $r$ is the distance). Using this idea, Bolyai and Lobachevsky
suggested that, should space be hyperbolic, the attracting force between
bodies must be inversely proportional to the hyperbolic area of the
corresponding hyperbolic sphere (i.e. proportional to
$1/\sinh^{2}(|\kappa|^{1/2}r)$, where $r$ is the distance and $\kappa<0$ the
curvature of the hyperbolic space). This is equivalent to saying that, in
hyperbolic space, the potential that describes the gravitational force is
proportional to $\coth(|\kappa|^{1/2}r)$.

The above analytic expression of the potential was first introduced by
Schering, [33], [34], and then extended to elliptic space by Killing, [22–24].
But with no physical ways of checking the validity of this generalization of
the gravitational force, it was unclear whether the cotangent potential had
any physical meaning, the more so since Lipschitz had proposed a different
extension of the law, which turned out to be short lived, [28]. The
breakthrough came at the dawn of the 20th century when Liebmann made two
important discoveries, [26], [27]. He showed that two basic properties of the
Newtonian potential are also satisfied by the cotangent potential: (1) in the
Kepler problem, which studies the motion of one body around a fixed centre,
the potential is a harmonic function (i.e. a solution of the Laplace equation
in the Euclidean case, but of the Laplace-Beltrami equation in the non-flat
case); (2) in both the flat and the non-flat case, all bounded orbits of the
Kepler problem are closed, a property discovered by Bertrand for the Newtonian
law, [1]. These similarities between the flat and the curved problem convinced
the scientific community that the cotangent potential was the natural way to
express gravity in spaces of constant curvature.

The curved $N$-body problem became somewhat neglected after the birth of
general relativity, but was revived after the discretization of Einstein’s
equation showed that an $N$-body problem in spaces of variable curvature is
too complicated to be treated with analytical tools. In the 1990s, the Russian
school of celestial mechanics considered both the curved Kepler and the curved
2-body problem, [24], [35]. After understanding that, unlike in the Euclidean
case, these problems are not equivalent, the latter failing to be integrable,
[35], the 2-body case was intensively studied by several researchers of this
school. More recently, the work of Diacu, Santoprete, and Pérez-Chavela
considered the curved $N$-body problem for $N>2$ in a new framework, leading
to many interesting results, [3–20], [32]. Other researchers developed these
ideas further, [20] [30], [31], [37–40], and the problem is growing in
popularity.

In this short note we prove three results. The first is a criterion for the
existence of quadrilateral relative equilibria on the equator of the sphere.
The second shows that if two masses are negligible and the other two are
equal, then square-like relative equilibria exists on spheres,
but—surprisingly—not on hyperbolic spheres. The element of surprise arises
from the fact that, in the general problem, square-like equilibria exist both
on the hyperbolic sphere and on the sphere (except for the case when they are
on the equator), [5]. Finally we prove that if only one mass is negligible and
the other three are equal, some kite-shaped relative equilibria exist on
spheres, but not on hyperbolic spheres.

##  2\. Equations of motion

We consider the motion of four bodies on 2-dimensional surfaces of constant
curvature $\kappa$, namely spheres $\mathbb{S}_{\kappa}^{2}$ for $\kappa>0$,
the Euclidean plane $\mathbb{R}^{2}$ for $\kappa=0$, and hyperbolic spheres
$\mathbb{H}_{\kappa}^{2}$ for $\kappa<0$. We will arrange these objects in
$\mathbb{R}^{3}$ such that they all have a common point at which lie all the
north poles of the spheres and the vertices of the hyperbolic spheres, to all
of which the plane $\mathbb{R}^{2}$ is tangent. If we fix the origin of a
coordinate system at this point, then we can write

$\mathbb{S}_{\kappa}^{2}:=\\{(x,y,z)\ \\!|\
\\!\kappa(x^{2}+y^{2}+z^{2})+2\kappa^{\frac{1}{2}}z=0\\}\ \ {\rm for}\ \
\kappa>0,$ $\mathbb{H}_{\kappa}^{2}:=\\{(x,y,z)\ \\!|\
\\!\kappa(x^{2}+y^{2}-z^{2})+2|\kappa|^{\frac{1}{2}}z=0\hskip 5.69046ptz\geq
0\\}\ \ {\rm for}\ \ \kappa<0.$

Consider now four point masses, $m_{i}>0,\ i=1,2,3,4$, whose position vectors,
velocities, and accelerations are given by

${\bf r}_{i}=(x_{i},y_{i},z_{i}),\ \dot{\bf
r}_{i}=(\dot{x}_{i},\dot{y}_{i},\dot{z}_{i}),\ \ddot{\bf
r}_{i}=(\ddot{x}_{i},\ddot{y}_{i},\ddot{z}_{i}),\ i=1,2,3,4.$

Then, as shown in [9], the equations of motion take the form

$\begin{cases}\ddot{x}_{i}=\sum_{j=1,j\neq
i}^{N}\frac{m_{j}\Big{[}{x}_{j}-\Big{(}1-\frac{\kappa
r_{ij}^{2}}{2}\Big{)}{x}_{i}\Big{]}}{\Big{(}1-\frac{\kappa
r_{ij}^{2}}{4}\Big{)}^{3/2}r_{ij}^{3}}-\kappa(\dot{\bf r}_{i}\cdot\dot{\bf
r}_{i})x_{i}\\\ \ddot{y}_{i}=\sum_{j=1,j\neq
i}^{N}\frac{m_{j}\Big{[}{y}_{j}-\Big{(}1-\frac{\kappa
r_{ij}^{2}}{2}\Big{)}{y}_{i}\Big{]}}{\Big{(}1-\frac{\kappa
r_{ij}^{2}}{4}\Big{)}^{3/2}r_{ij}^{3}}-\kappa(\dot{\bf r}_{i}\cdot\dot{\bf
r}_{i})y_{i}\\\ \ddot{z}_{i}=\sum_{j=1,j\neq
i}^{N}\frac{m_{j}\Big{[}{z}_{j}-\Big{(}1-\frac{\kappa
r_{ij}^{2}}{2}\Big{)}{z}_{i}\Big{]}}{\Big{(}1-\frac{\kappa
r_{ij}^{2}}{4}\Big{)}^{3/2}r_{ij}^{3}}-(\dot{\bf r}_{i}\cdot\dot{\bf
r}_{i})(\kappa z_{i}+\sigma|\kappa|^{1/2}),\ i=1,2,3,4,\end{cases}$

where $\sigma=1$ for $\kappa\geq 0$, $\sigma=-1$ for $\kappa<0$, and

$r_{ij}:=\begin{cases}[(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}]^{1/2}\
\ {\rm for}\ \ \kappa\geq
0\cr[(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}-(z_{i}-z_{j})^{2}]^{1/2}\ \ {\rm
for}\ \ \kappa<0\cr\end{cases}$

for $i,j\in\\{1,2,3,4\\}$. The above system has eight constraints, namely

$\kappa(x_{1}^{2}+y_{i}^{2}+\sigma z_{i}^{2})+2|\kappa|^{1/2}z_{i}=0,$
$\kappa{\bf r}_{i}\cdot\dot{\bf r}_{i}+|\kappa|^{1/2}\dot{z}_{i}=0,\hskip
8.5359pti=1,2,3,4.$

If satisfied at an initial instant, these constraints are satisfied for all
time because the sets $\mathbb{S}_{\kappa}^{2},\mathbb{R}^{2}$, and
$\mathbb{H}_{\kappa}^{2}$ are invariant for the equations of motion, [5].
Notice that for $\kappa=0$ we recover the classical Newtonian equations of the
$4$-body problem on the Euclidean plane, namely

$\ddot{\bf r}_{i}=\sum_{j=1,j\neq i}^{N}\frac{m_{j}({\bf r}_{j}-{\bf
r}_{i})}{r_{ij}^{3}},$

where ${\bf r}_{i}=(x_{i},y_{i},0),\ i=1,2,3,4.$

##  3\. Relative equilibria

Relative equilibria are solutions for which the relative distances remain
constant during the motion. We first introduce some coordinates
$(\varphi,\omega)$, which were originally used in [9] for the case $N=3$, to
detect relative equilibria on and near the equator of
$\mathbb{S}_{\kappa}^{2}$, where $\varphi$ measures the angle from the
$x$-axis in the $xy$-plane, while $\omega$ is the height on the vertical
$z$-axis. In these new coordinates, the constraints become

$x_{i}^{2}+y_{i}^{2}+\omega_{i}^{2}+2\kappa^{-1/2}\omega_{i}=0,\hskip
5.69046pti=1,2,3,4.$

With the notation,

$\Omega_{i}=x_{i}^{2}+y_{i}^{2}=-\kappa^{-1/2}\omega_{i}(\kappa^{1/2}\omega_{i}+2)\geq
0,\ \omega_{i}\in[-2\kappa^{-1/2},0],\ i=1,2,3,4,$

where equality occurs when the body is at the North or the South Pole of the
sphere, the $(\varphi,\omega)$-coordinates are given by the transformations

$x_{i}=\Omega_{i}^{1/2}\cos\varphi_{i},\
y_{i}=\Omega_{i}^{1/2}\sin\varphi_{i}.$

Thus the equations of motion take the form

$\begin{cases}\ddot{\varphi_{i}}=\Omega_{i}^{-1/2}\sum_{j=1,j\neq
i}^{N}\frac{m_{j}\Omega_{j}^{1/2}\sin(\varphi_{j}-\varphi_{i})}{\rho_{ij}^{3}(1-\frac{\kappa\rho_{ij}^{2}}{4})^{3/2}}-\frac{\dot{\varphi_{i}}\dot{\Omega_{i}}}{\Omega_{i}}\\\
\ddot{\omega_{i}}=\Omega_{i}^{-1/2}\sum_{j=1,j\neq
i}^{N}\frac{m_{j}\Big{[}\omega_{j}+\omega_{i}+\frac{\kappa\rho_{ij}^{2}}{2}(\omega_{i}+\kappa^{-1/2})\Big{]}}{\rho_{ij}^{3}\Big{(}1-\frac{\kappa\rho_{ij}^{2}}{4}\Big{)}^{3/2}}-(\kappa\omega_{i}+\kappa^{1/2})(\frac{\dot{\Omega_{i}^{2}}}{4\Omega_{i}}+\dot{\varphi_{i}}^{2}\Omega_{i}+\dot{\omega_{i}^{2}}),\end{cases}$

where

$\dot{\Omega}_{i}=-2\kappa^{-1/2}\dot{\omega_{i}}(\kappa^{1/2}\omega_{i}+1)$
$\rho_{ij}^{2}=\Omega_{i}+\Omega_{j}-2\Omega_{i}^{1/2}\Omega_{j}^{1/2}\cos(\varphi_{i}-\varphi_{j})+(\omega_{i}-\omega_{j})^{2},\hskip
5.69046pti,j=1,2,3,4,\hskip 5.69046pti\neq j.$

##  4\. Relative equilibria on the equator

If we restrict the motion of the four bodies to the equator of
$\mathbb{S}_{\kappa}^{2}$, then

$\omega_{i}=-\kappa^{-1/2},\hskip 8.5359pt\dot{\omega_{i}}=0,\hskip
8.5359pt\Omega_{i}=\kappa^{-1},\ i=1,2,3,4,$

and the equations of motion take the form

$\ddot{\varphi}_{i}=\kappa^{3/2}\sum_{j=1,j\neq
i}^{4}\frac{m_{j}\sin(\varphi_{j}-\varphi_{i})}{|\sin(\varphi_{j}-\varphi_{i})|^{3}},\
\ i=1,2,3,4.$

For the relative equilibria, the angular velocity is the same constant for all
masses, so we denote this velocity by $\alpha\neq 0$ and take

$\varphi_{1}=\alpha t+a_{1},\hskip 8.5359pt\varphi_{2}=\alpha t+a_{2},\hskip
8.5359pt\varphi_{3}=\alpha t+a_{3},\hskip 8.5359pt\varphi_{4}=\alpha t+a_{4},$

where $a_{1},a_{2},a_{3},a_{4}$ are real constants, so

$\ddot{\varphi}_{i}=0,\ i=1,2,3,4.$

Using the notation

$s_{1}:=\frac{\kappa^{3/2}\sin(\varphi_{1}-\varphi_{2})}{|\sin(\varphi_{1}-\varphi_{2})|^{3}},\hskip
8.5359pts_{2}:=\frac{\kappa^{3/2}\sin(\varphi_{2}-\varphi_{3})}{|\sin(\varphi_{2}-\varphi_{3})|^{3}},\hskip
8.5359pts_{3}:=\frac{\kappa^{3/2}\sin(\varphi_{3}-\varphi_{1})}{|\sin(\varphi_{3}-\varphi_{1})|^{3}},$
$s_{4}:=\frac{\kappa^{3/2}\sin(\varphi_{4}-\varphi_{1})}{|\sin(\varphi_{4}-\varphi_{1})|^{3}},\hskip
8.5359pts_{5}:=\frac{\kappa^{3/2}\sin(\varphi_{2}-\varphi_{4})}{|\sin(\varphi_{2}-\varphi_{4})|^{3}},\hskip
8.5359pts_{6}:=\frac{\kappa^{3/2}\sin(\varphi_{3}-\varphi_{4})}{|\sin(\varphi_{3}-\varphi_{4})|^{3}},$

we obtain from the equations of motion that

$\begin{cases}-m_{2}s_{1}+m_{3}s_{3}+m_{4}s_{4}=0\cr
m_{1}s_{1}-m_{3}s_{2}-m_{4}s_{5}=0\cr-m_{1}s_{3}+m_{2}s_{2}-m_{4}s_{6}=0\cr-
m_{1}s_{4}+m_{2}s_{5}+m_{3}s_{6}=0.\end{cases}$

To have other solutions of the masses than $m_{1}=m_{2}=m_{3}=m_{4}=0$, the
determinant of the above system must vanish, which is equivalent to

$s_{1}s_{6}+s_{3}s_{5}=s_{2}s_{4}.$

We have thus proved the following result.

######  Theorem 1.

A necessary condition that the quadrilateral inscribed in the equator of
$\mathbb{S}_{\kappa}^{2}$, with the four masses $m_{1},m_{2},m_{3},m_{4}>0$ at
its vertices, forms a relative equilibrium is that

$s_{1}s_{6}+s_{3}s_{5}=s_{2}s_{4}.$

##  5\. Equivalent equations of motion

Let us now introduce some equivalent equations of motion that are suitable for
the kind of solutions we are seeking. First, rewriting the above constraints
as

$\kappa(x_{i}^{2}+y_{i}^{2})+(|\kappa|^{1/2}z_{i}+1)^{2}=1,$

and solving explicitly for $z_{i}$, we obtain

$z_{i}=|\kappa|^{-1/2}[\sqrt{1-\kappa(x_{i}^{2}+y_{i}^{2})}-1].$

The idea here is to eliminate the four equations involving
$z_{1},z_{2},z_{3},z_{4}$, but they still appear in the terms $r_{ij}^{2}$ in
the form $\sigma(z_{i}-z_{j})^{2}$ as

$\sigma(z_{i}-z_{j})^{2}=\frac{\kappa(x_{i}^{2}+y_{i}^{2}-x_{j}^{2}-y_{j}^{2})^{2}}{\left[\sqrt{1-\kappa(x_{i}^{2}+y_{i}^{2})}+\sqrt{1-\kappa(x_{j}^{2}+y_{j}^{2})}\right]^{2}}.$

The case of physical interest is when $\kappa$ is not far from zero, so the
above expression exist even for $\kappa>0$ under this assumption. Then the
equations of motion become

$\begin{cases}\ddot{x_{i}}=\sum_{j=1,j\neq
i}^{N}\frac{m_{j}\Big{[}{x}_{j}-\Big{(}1-\frac{\kappa\rho_{ij}^{2}}{2}\Big{)}{x}_{i}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{ij}^{2}}{4}\Big{)}^{3/2}\rho_{ij}^{3}}-\kappa(\dot{x_{i}}^{2}+\dot{y_{i}}^{2}+\kappa
B_{i})x_{i}\\\ \ddot{y_{i}}=\sum_{j=1,j\neq
i}^{N}\frac{m_{j}\Big{[}{y}_{j}-\Big{(}1-\frac{\kappa\rho_{ij}^{2}}{2}\Big{)}{y}_{i}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{ij}^{2}}{4}\Big{)}^{3/2}\rho_{ij}^{3}}-\kappa(\dot{x_{i}}^{2}+\dot{y_{i}}^{2}+\kappa
B_{i})y_{i},\end{cases}$

where

$\rho_{ij}^{2}=(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+\frac{\kappa(A_{i}-A_{j})^{2}}{(\sqrt{1-\kappa
A_{i}}+\sqrt{1-\kappa A_{j}})^{2}},$ $A_{i}=x_{i}^{2}+y_{i}^{2},$
$B_{i}=\frac{(x_{i}\dot{x_{i}}+y_{i}\dot{y_{i}})^{2}}{1-\kappa(x_{i}^{2}+y_{i}^{2})},\hskip
11.38092pti=1,2,3,4.$

It is obvious that for $\kappa=0$ we recover the classical Newtonian equations
of motion of the planar 4-body problem.

##  6\. The case of two negligible masses

We now consider the case when two out of the four given masses are negligible,
$m_{3}=m_{4}=0$. Then the equations of motion become

$\begin{cases}\ddot{x}_{1}=\frac{m_{2}\Big{[}{x}_{2}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{x}_{1}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}^{3/2}\rho_{12}^{3}}-\kappa(\dot{x_{1}}^{2}+\dot{y_{1}}^{2}+\kappa
B_{1})x_{1}\\\
\ddot{y}_{1}=\frac{m_{2}\Big{[}{y}_{2}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{y}_{1}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{4}\Big{)}^{3/2}\rho_{12}^{3}}-\kappa(\dot{x_{1}}^{2}+\dot{y_{1}}^{2}+\kappa
B_{1})y_{1}\\\ \end{cases}$
$\begin{cases}\ddot{x}_{2}=\frac{m_{1}\Big{[}{x}_{1}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{x}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{4}\Big{)}^{3/2}\rho_{12}^{3}}-\kappa(\dot{x_{2}}^{2}+\dot{y_{2}}^{2}+\kappa
B_{2})x_{2}\\\
\ddot{y}_{2}=\frac{m_{1}\Big{[}{y}_{1}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{y}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{4}\Big{)}^{3/2}\rho_{12}^{3}}-\kappa(\dot{x_{2}}^{2}+\dot{y_{2}}^{2}+\kappa
B_{2})y_{2}\\\ \end{cases}$
$\begin{cases}\ddot{x}_{3}=\frac{m_{1}\Big{[}{x}_{1}-\Big{(}1-\frac{\kappa\rho_{13}^{2}}{2}\Big{)}{x}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{13}^{2}}{4}\Big{)}^{3/2}\rho_{13}^{3}}+\frac{m_{2}\Big{[}{x}_{2}-\Big{(}1-\frac{\kappa\rho_{32}^{2}}{2}\Big{)}{x}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{23}^{2}}{4}\Big{)}^{3/2}\rho_{23}^{3}}-\kappa(\dot{x_{3}}^{2}+\dot{y_{3}}^{2}+\kappa
B_{3})x_{3}\\\
\ddot{y}_{3}=\frac{m_{1}\Big{[}{y}_{1}-\Big{(}1-\frac{\kappa\rho_{13}^{2}}{2}\Big{)}{y}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{13}^{2}}{4}\Big{)}^{3/2}\rho_{13}^{3}}+\frac{m_{2}\Big{[}{y}_{2}-\Big{(}1-\frac{\kappa\rho_{32}^{2}}{2}\Big{)}{y}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{32}^{2}}{2}\Big{)}^{3/2}\rho_{32}^{3}}-\kappa(\dot{x_{3}}^{2}+\dot{y_{3}}^{2}+\kappa
B_{3})y_{3}\\\ \end{cases}$
$\begin{cases}\ddot{x}_{4}=\frac{m_{1}\Big{[}{x}_{1}-\Big{(}1-\frac{\kappa\rho_{14}^{2}}{2}\Big{)}{x}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{14}^{2}}{4}\Big{)}^{3/2}\rho_{14}^{3}}+\frac{m_{4}\Big{[}{x}_{4}-\Big{(}1-\frac{\kappa\rho_{42}^{2}}{2}\Big{)}{x}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{42}^{2}}{4}\Big{)}^{3/2}\rho_{42}^{3}}-\kappa(\dot{x_{4}}^{2}+\dot{y_{4}}^{2}+\kappa
B_{4})x_{4}\\\
\ddot{y}_{4}=\frac{m_{1}\Big{[}{y}_{1}-\Big{(}1-\frac{\kappa\rho_{14}^{2}}{2}\Big{)}{y}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{14}^{2}}{4}\Big{)}^{3/2}\rho_{14}^{3}}+\frac{m_{4}\Big{[}{y}_{4}-\Big{(}1-\frac{\kappa\rho_{42}^{2}}{2}\Big{)}{y}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{42}^{2}}{4}\Big{)}^{3/2}\rho_{42}^{3}}-\kappa(\dot{x_{4}}^{2}+\dot{y_{4}}^{2}+\kappa
B_{4})y_{4},\end{cases}$

where $\rho_{ij}^{2}=\rho_{ji}^{2},\ i\neq j$,

$\rho_{ij}^{2}=(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+\frac{\kappa(x_{i}^{2}+y_{i}^{2}-x_{j}^{2}-y_{j}^{2})^{2}}{[\sqrt{1-\kappa(x_{i}^{2}+y_{i}^{2})}+\sqrt{1-\kappa(x_{j}^{2}+y_{j}^{2})}]^{2}}.$

We can now show that when $m_{1}=m_{2}=:m>0$ and $m_{3}=m_{4}=0$, square-like
relative equilibria, i.e. equilateral equiangular quadrilaterals, always exist
on $\mathbb{S}_{\kappa}^{2}$, but not on $\mathbb{H}_{\kappa}^{2}$.

######  Theorem 2.

In the curved 4-body problem, assume that $m_{1}=m_{2}=:m>0$ and
$m_{3}=m_{4}=0$. Then, in $\mathbb{S}_{\kappa}^{2}$, there is a circle of
radius $r<\kappa^{-1/2}$, parallel with the $xy$-plane, such that a square
configuration inscribed in this circle, with $m_{1},m_{2}$ at the opposite
ends of one diagonal and $m_{3},m_{4}$ at the opposite ends of the other
diagonal, forms a relative equilibrium. But in $\mathbb{H}_{\kappa}^{2}$,
there is no such solution.

$m_{1}$$m_{2}$$m_{4}$$m_{3}$ Figure 1.  The case of 2 equal masses and 2
negligible masses.

###### Proof.

We must check the existence of a solution of the form

${\bf q}=(q_{1},q_{2},q_{3},q_{4})\in\mathbb{S}_{\kappa}^{2},\hskip
8.5359pt{\bf q_{i}}=(x_{i},y_{i}),\hskip 5.69046pti=1,2,3,4.$
$x_{1}=r\cos\alpha t,\hskip 14.22636pty_{1}=r\sin\alpha t,\hskip 25.6073pt$
$x_{2}=-r\cos\alpha t,\hskip 14.22636pty_{2}=-r\sin\alpha t,\hskip 14.22636pt$
$x_{3}=r\cos(\alpha t+\pi/2)=-r\sin\alpha t,\hskip
14.22636pty_{3}=r\sin(\alpha t+\pi/2)=r\cos\alpha t,\hskip 14.22636pt$
$x_{4}=-r\cos(\alpha t+\pi/2)=r\sin\alpha t,\hskip
22.76228pty_{4}=-r\sin(\alpha t+\pi/2)=-r\cos\alpha t,\hskip 14.22636pt$

where

$\hskip 14.22636ptx_{i}^{2}+y_{i}^{2}=r^{2},\ \
\rho^{2}=\rho_{13}^{2}=\rho_{14}^{2}=\rho_{23}^{2}=\rho_{24}^{2}=2r^{2},\ \
\rho_{12}^{2}=\rho_{34}^{2}=4r^{2}.$

Substituting these expressions into the system, the first four equations lead
us to

$\alpha^{2}=\frac{m}{4r^{3}(1-\kappa r^{2})^{3/2}},$

whereas the last four equations yield

$\alpha^{2}=\frac{2m(1-\frac{\kappa\rho^{2}}{2})}{\rho^{3}(1-\frac{\kappa\rho^{2}}{4})^{3/2}(1-\kappa
r^{2})}.$

So, to have a solution, the equation

$\frac{m}{4r^{3}(1-\kappa
r^{2})^{3/2}}=\frac{2m(1-\frac{\kappa\rho^{2}}{2})}{\rho^{3}(1-\frac{\kappa\rho^{2}}{4})^{3/2}(1-\kappa
r^{2})}$

must be satisfied. This equation is equivalent to

$\frac{1}{8r^{3}(1-\kappa r^{2})^{3/2}}=\frac{1}{2\sqrt{2}r^{3}(1-\frac{\kappa
r^{2}}{2})^{3/2}},$

which leads to

$3\kappa r^{2}=2.$

Obviously, in the case of $\mathbb{H}_{\kappa}^{2}$, we have $\kappa<0$, so
this equation has no solutions. For $\mathbb{S}_{\kappa}^{2}$, it leads to

$r=\sqrt{2/3}\kappa^{-1/2}.$

Since $r<\kappa^{-1/2}$, such a solution always exists in
$\mathbb{S}_{\kappa}^{2}$. ∎

$m_{1}$$m_{3}$$m_{2}$$m_{4}$ Figure 2. The case of two equal masses and two
negligible masses.

##  7\. The case of one negligible mass

Let $m_{1},m_{2},m_{3}>0$ and assume that $m_{4}=0$. Then the equations of
motion take the form

$\begin{cases}\ddot{x_{1}}=\frac{m_{2}\Big{[}{x}_{2}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{x}_{1}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{4}\Big{)}^{3/2}\rho_{12}^{3}}+\frac{m_{3}\Big{[}{x}_{3}-\Big{(}1-\frac{\kappa\rho_{31}^{2}}{2}\Big{)}{x}_{1}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{31}^{2}}{4}\Big{)}^{3/2}\rho_{31}^{3}}-\kappa(\dot{x_{1}}^{2}+\dot{y_{1}}^{2}+\kappa
B_{1})x_{1}\\\
\ddot{y_{1}}=\frac{m_{2}\Big{[}{y}_{2}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{y}_{1}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{4}\Big{)}^{3/2}\rho_{12}^{3}}+\frac{m_{3}\Big{[}{y}_{3}-\Big{(}1-\frac{\kappa\rho_{31}^{2}}{2}\Big{)}{y}_{1}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{31}^{2}}{4}\Big{)}^{3/2}\rho_{31}^{3}}-\kappa(\dot{x_{1}}^{2}+\dot{y_{1}}^{2}+\kappa
B_{1})y_{1}\\\ \end{cases}$
$\begin{cases}\ddot{x_{2}}=\frac{m_{1}\Big{[}{x}_{1}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{x}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{4}\Big{)}^{3/2}\rho_{12}^{3}}+\frac{m_{3}\Big{[}{x}_{3}-\Big{(}1-\frac{\kappa\rho_{32}^{2}}{2}\Big{)}{x}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{32}^{2}}{4}\Big{)}^{3/2}\rho_{32}^{3}}-\kappa(\dot{x_{2}}^{2}+\dot{y_{2}}^{2}+\kappa
B_{2})x_{2}\\\
\ddot{y_{2}}=\frac{m_{1}\Big{[}{y}_{1}-\Big{(}1-\frac{\kappa\rho_{12}^{2}}{2}\Big{)}{y}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{12}^{2}}{4}\Big{)}^{3/2}\rho_{12}^{3}}+\frac{m_{3}\Big{[}{y}_{3}-\Big{(}1-\frac{\kappa\rho_{32}^{2}}{2}\Big{)}{y}_{2}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{32}^{2}}{4}\Big{)}^{3/2}\rho_{32}^{3}}-\kappa(\dot{x_{2}}^{2}+\dot{y_{2}}^{2}+\kappa
B_{2})y_{2}\end{cases}$
$\begin{cases}\ddot{x_{3}}=\frac{m_{1}\Big{[}{x}_{1}-\Big{(}1-\frac{\kappa\rho_{13}^{2}}{2}\Big{)}{x}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{13}^{2}}{4}\Big{)}^{3/2}\rho_{13}^{3}}+\frac{m_{2}\Big{[}{x}_{2}-\Big{(}1-\frac{\kappa\rho_{32}^{2}}{2}\Big{)}{x}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{32}^{2}}{4}\Big{)}^{3/2}\rho_{32}^{3}}-\kappa(\dot{x_{3}}^{2}+\dot{y_{3}}^{2}+\kappa
B_{3})x_{3}\\\
\ddot{y_{3}}=\frac{m_{1}\Big{[}{y}_{1}-\Big{(}1-\frac{\kappa\rho_{13}^{2}}{2}\Big{)}{y}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{13}^{2}}{4}\Big{)}^{3/2}\rho_{13}^{3}}+\frac{m_{2}\Big{[}{y}_{2}-\Big{(}1-\frac{\kappa\rho_{32}^{2}}{2}\Big{)}{y}_{3}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{32}^{2}}{4}\Big{)}^{3/2}\rho_{32}^{3}}-\kappa(\dot{x_{3}}^{2}+\dot{y_{3}}^{2}+\kappa
B_{3})y_{3}\end{cases}$
$\begin{cases}\ddot{x_{4}}=\frac{m_{1}\Big{[}{x}_{1}-\Big{(}1-\frac{\kappa\rho_{14}^{2}}{2}\Big{)}{x}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{14}^{2}}{4}\Big{)}^{3/2}\rho_{14}^{3}}+\frac{m_{2}\Big{[}{x}_{2}-\Big{(}1-\frac{\kappa\rho_{42}^{2}}{2}\Big{)}{x}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{42}^{2}}{4}\Big{)}^{3/2}\rho_{42}^{3}}+\frac{m_{3}\Big{[}{x}_{3}-\Big{(}1-\frac{\kappa\rho_{43}^{2}}{2}\Big{)}{x}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{43}^{2}}{4}\Big{)}^{3/2}\rho_{43}^{3}}\\\
\hfill-\kappa(\dot{x_{4}}^{2}+\dot{y_{4}}^{2}+\kappa B_{4})x_{4}\\\
\ddot{y_{4}}=\frac{m_{1}\Big{[}{y}_{1}-\Big{(}1-\frac{\kappa\rho_{14}^{2}}{2}\Big{)}{y}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{14}^{2}}{4}\Big{)}^{3/2}\rho_{14}^{3}}+\frac{m_{2}\Big{[}{y}_{2}-\Big{(}1-\frac{\kappa\rho_{42}^{2}}{2}\Big{)}{y}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{42}^{2}}{4}\Big{)}^{3/2}\rho_{42}^{3}}+\frac{m_{3}\Big{[}{y}_{3}-\Big{(}1-\frac{\kappa\rho_{43}^{2}}{2}\Big{)}{y}_{4}\Big{]}}{\Big{(}1-\frac{\kappa\rho_{43}^{2}}{4}\Big{)}^{3/2}\rho_{43}^{3}}\\\
\hfill-\kappa(\dot{x_{4}}^{2}+\dot{y_{4}}^{2}+\kappa B_{4})y_{4}.\end{cases}$

We will next show that if the non-negligible masses are equal, then there
exist some kite-shaped relative equilibria.

######  Theorem 3.

Consider the curved 4-body problem with masses $m_{1}=m_{2}=m_{3}:=m>0$ and
$m_{4}=0$. Then, in $\mathbb{S}_{\kappa}^{2}$, there exists at least one kite-
shaped relative equilibrium for which the equal masses lie at the vertices of
an equilateral triangle, whereas the negligible mass is at the intersection of
the extension of one height of the triangle with the circle on which all the
bodies move. In $\mathbb{H}_{\kappa}^{2}$, however, there are no such kite-
shaped relative equilibria.

$m_{1}$$m_{2}$$m_{3}$$m_{4}$ Figure 3. A kite configuration of 3 equal masses
and one negligible mass.

###### Proof.

We will check a solution of the form

$x_{1}=r\cos\alpha t,\hskip 14.22636pty_{1}=r\sin\alpha t,\hskip 28.45274pt$
$x_{2}=r\cos\Big{(}\alpha t+\frac{2\pi}{3}\Big{)},\hskip
14.22636pty_{2}=r\sin\Big{(}\alpha t+\frac{2\pi}{3}\Big{)}$
$x_{3}=r\cos\Big{(}\alpha t+\frac{4\pi}{3}\Big{)},\hskip
14.22636pty_{3}=r\sin\Big{(}\alpha t+\frac{4\pi}{3}\Big{)},\hskip 14.22636pt$
$x_{4}=r\cos\Big{(}\alpha t-\frac{\pi}{3}\Big{)},\hskip
14.22636pty_{4}=r\sin\Big{(}\alpha t-\frac{\pi}{3}\Big{)},\hskip 14.22636pt$

where

$\rho_{12}^{2}=\rho_{13}^{2}=\rho_{23}^{2}=3r^{2},\ \
\rho_{43}^{2}=\rho_{41}^{2}=r^{2},\ \ \rho_{24}^{2}=4r^{2}.$

Substituting these expressions into the above system, we are led to the
conclusion that the following two equations must be satisfied,

$\alpha^{2}=\frac{m}{\sqrt{3}r^{3}(1-\frac{3\kappa r^{2}}{4})^{3/2}},$
$\alpha^{2}=\frac{m}{4r^{3}(1-\kappa
r^{2})^{3/2}}+\frac{m}{r^{3}(1-\frac{\kappa r^{2}}{4})^{3/2}}.$

Comparing these equations we obtain the condition for the existence of the
kite-shaped relative equilibria,

$\frac{1}{\sqrt{3}(1-\frac{3\kappa r^{2}}{4})^{3/2}}=\frac{1}{4(1-\kappa
r^{2})^{3/2}}+\frac{1}{(1-\frac{\kappa r^{2}}{4})^{3/2}}.$

Straightforward computations show that $r$ is a solution of this equation if
it is a root of the polynomial

$P(r)=a_{24}r^{24}+a_{22}r^{22}+a_{20}r^{20}+a_{18}r^{18}+a_{16}r^{16}+a_{14}r^{14}+a_{12}r^{12}+$
$a_{10}r^{10}+a_{8}r^{8}+a_{6}r^{6}+a_{4}r^{4}+a_{2}r^{2}+a_{0},$
$a_{24}=\frac{6697290145}{16777216}\kappa^{12},\ \
a_{22}=-\frac{2884257825}{524288}\kappa^{11},\ \
a_{20}=\frac{18063189465}{524288}\kappa^{10},$
$a_{18}=-\frac{4241985935}{32768}\kappa^{9},\ \ \
a_{16}=\frac{21267471735}{65536}\kappa^{8},$
$a_{14}=-\frac{584429805}{1024}\kappa^{7},\ \ \
a_{12}=\frac{737853351}{1024}\kappa^{6},\ \ \
a_{10}=-\frac{41995431}{64}\kappa^{5},$
$a_{8}=\frac{109080063}{256}\kappa^{4},\ \ \
a_{6}=-\frac{1530101}{8}\kappa^{3},$ $a_{4}=\frac{446217}{8}\kappa^{2},\ \ \
a_{2}=-9318\kappa,\ \ \ a_{0}=649$

that belongs to the interval $r\in(0,\kappa^{-1/2})$ for
$\mathbb{S}_{\kappa}^{2}$, but needs only to be positive for
$\mathbb{H}_{\kappa}^{2}$. To find out if we have such a root, we make the
substitution $x=r^{2}$, and obtain the polynomial

$Q(x)=a_{24}x^{12}+a_{22}x^{11}+a_{20}x^{10}+a_{18}x^{9}+a_{16}x^{8}+a_{14}x^{7}+a_{12}x^{6}+$
$a_{10}x^{5}+a_{8}x^{4}+a_{6}x^{3}+a_{4}x^{2}+a_{2}x+a_{0}.$

By Descartes’s rule of signs the number of positive roots depends on the
number of changes of sign of the coefficients, which in turn depends on the
sign of $\kappa$. So let us discuss the two cases separately.

In $\mathbb{S}_{\kappa}^{2}$, i.e. for $\kappa>0$, there are twelve changes of
sign, so $Q$ can have twelve, ten, eight, six, four, two, or zero positive
roots, so this does not guarantee the existence of a positive root. However,
we can notice that $Q(\frac{\kappa^{-1}}{2})=-2.4959<0$ and $Q(0)=649>0$, so a
root must exist for $x\in(0,\kappa^{-1/2})$, i.e. for $r\in(0,\kappa^{-1})$, a
remark that proves the existence of at least one kite-shaped relative
equilibrium.

In $\mathbb{H}_{\kappa}^{2}$, i.e. for $\kappa<0$, we seek a positive root of
$Q$. But for $\kappa<0$, there is no sign change in $Q(x)$, so the polynomial
has no positive roots. Therefore there are no kite solutions in
$\mathbb{H}_{\kappa}^{2}$. This remark completes the proof. ∎

Acknowledgment. Florin Diacu did most of the work on this paper while visiting
the Yangtze Center of Mathematics at Sichuan University as Distinguished
Foreign Professor in April-May 2017. He was also supported in part by a grant
from the Yale-NUS College at the National University of Singapore and an NSERC
of Canada Discovery Grant. Sawsan Alhowaity was funded by a scholarship from
the University of Shaqra, Saudia Arabia, towards the completion of her
doctoral degree at the University of Victoria in Canada.

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# Emotion-guided Cross-domain Fake News Detection using Adversarial Domain
Adaptation

Arjun Choudhry11footnotemark: 1
Biometric Research Laboratory
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>
&Inder Khatri11footnotemark: 1
Biometric Research Laboratory
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>
&Arkajyoti Chakraborty
Biometric Research Laboratory
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>
&Dinesh Kumar Vishwakarma
Biometric Research Laboratory
Delhi Technological University
New Delhi, India
<EMAIL_ADDRESS>
&Mukesh Prasad
School of Computer Science
University of Technology Sydney
Ultimo, Australia
<EMAIL_ADDRESS>

###### Abstract

Recent works on fake news detection have shown the efficacy of using emotions
as a feature or emotions-based features for improved performance. However, the
impact of these emotion-guided features for fake news detection in cross-
domain settings, where we face the problem of domain shift, is still largely
unexplored. In this work, we evaluate the impact of emotion-guided features
for cross-domain fake news detection, and further propose an emotion-guided,
domain-adaptive approach using adversarial learning. We prove the efficacy of
emotion-guided models in cross-domain settings for various combinations of
source and target datasets from FakeNewsAMT, Celeb, Politifact and Gossipcop
datasets.††*Equal Contribution

_K_ eywords Fake News Detection $\cdot$ Domain Adaptation $\cdot$ Emotion
Classification $\cdot$ Adversarial Training $\cdot$ Cross-domain Analysis

##  1 Introduction

In recent years, our reliance on social media as a source of information has
increased multi-fold, bringing along exponential increase in the spread of
_fake news_. To counter this, researchers have proposed various approaches for
fake news detection (Shu et al., 2019; Sheng et al., 2022). However, models
trained on one domain are often brittle and vulnerable to incorrect
predictions for the samples of another domain (Saikh et al., 2019; Pérez-Rosas
et al., 2018). This is primarily due to the shift between the two domains, as
depicted in Figure 1(1). To handle this, some domain-adaptive frameworks
(Zhang et al., 2020; Huang et al., 2021; Li et al., 2021) have been proposed
which help align the source and target domains in the feature space to
ameliorate domain shift across different problems. These frameworks guide the
feature extractors to extract domain-invariant features by aligning the source
and target domains in the feature space, thus generalizing well across
domains. However, due to the absence of labels in the target-domain data, the
adaptation is often prone to negative transfer, which can disturb the class-
wise distribution and affect the discriminability of the final model, as shown
in Figure 1(2).

Some recent studies have observed a correlation between the veracity of a text
and its emotions. There exists a prominent affiliation for certain emotions
with fake news, and for other emotions with real news (Vosoughi et al., 2018),
as illustrated in Figure 1(3). Further, some works have successfully utilized
emotions as features, or emotion-guided features to aid in fake news detection
(Guo et al., 2019; Zhang et al., 2021; Choudhry et al., 2022). However, we
observe that these works only consider the in-domain setting for evaluation,
without analyzing the robustness of these frameworks to domain shift in cross-
domain settings. This is another important direction that needs to be
explored.

Figure 1: (1) Cross-domain texts not aligned. (2) Domain adaptation leads to
some alignment. (3) Emotion-guided classification in one domain. (4) Emotion-
guided domain adaptation leads to improved alignment of the two domains.

In this paper, we study the efficacy of emotion-aided models in capturing
better generalizable features for cross-domain fake news detection. Table 1
shows the improvements observed in various cross-domain settings when our
emotion-guided models were evaluated in cross-domain settings. We observe that
emotion-guided frameworks show improved performance in cross-domain settings,
as compared to their baseline models without the said emotion-aided features,
thus underscoring the generalized feature extraction in emotion-aided models.
We further propose an emotion-guided unsupervised domain adaptation framework,
which utilizes emotion labels in a multi-task adversarial setting for better
feature alignment across domains. The emotion labels for emotion
classification, trained parallel to the fake news detection branch in the
multi-task learning setup, help provide additional supervision for improved
alignment during domain adaptation, mitigating the issue of incorrect
alignment of domains. This is illustrated in Figure 1(4)). This leads to
better discriminability. We experimentally prove the efficacy of our approach
across a variety of datasets in cross-domain settings for various combinations
of single-task or multi-task, domain-adaptive or non-adaptive, emotion-guided
or unguided settings on the accuracy of the models.

Our contributions can be summarized as follows:

  * •

We suggest the use of emotion classification as an auxiliary task for improved
fake news detection in cross-domain settings, indicating the applicability of
emotion-guided features across domains.

  * •

We compare how Ekman’s and Plutchik’s base emotion classes individually affect
the performance of our multi-task domain-adaptive framework, and if there are
meaningful differences between them.

  * •

We propose an emotion-guided domain-adaptive framework for fake news detection
across domains. We show that domain-adaptive fake news detection models better
align the two domains with the help of supervised learning using emotion-aided
features.

  * •

We evaluate our approach on a variety of source and target combinations from
four datasets. Our results prove the efficacy of our approach.

##  2 Related Works

Several studies over the last few years have explored the correlation of fake
news detection with emotions. K et al. (2020) _emotionized_ text
representations using explicit emotion intensity lexicons. Guo et al. (2019)
utilized the discrepancies between publisher’s emotion and the thread’s
comments’ emotions to detect fake news. However, most of these methods relied
upon some additional inputs during evaluation. Choudhry et al. (2022) proposed
an emotion-aided multi-task learning approach, where emotion classification
was the auxiliary task implicitly aligning fake news features according to
emotion labels.

Inspired by Ganin et al. (2015), Zhang et al. (2020) proposed the first fake
news detection work to tackle domain shifts between different datasets. They
proposed a multi-modal framework with a Gradient Reversal Layer (GRL) to learn
domain-invariant features across different domains and used a joint fake news
detector on the extracted features. Huang et al. (2021) proposed a robust and
generalized fake news detection framework adaptable to a new target domain
using adversarial training to make the model robust to outliers and Maximum
Mean Difference (MMD)-based loss to align the features of source and target.
Li et al. (2021) extended the problem by treating it as a multi-source domain
adaptation task, using the labeled samples from multiple source domains to
improve the performance on unlabeled target domains. They also utilized weak
labels for weak supervision on target samples to improve performance.

However, no previous work has aligned features between different domains using
emotion-guided features and domain adaptation using adversarial training. We
show that applying both of these approaches leads to improved performance due
to better alignment of inter-domain features.

##  3 Proposed Methodology

###  3.1 Datasets, Emotion Annotation & Preprocessing

We use the FakeNewsAMT (Pérez-Rosas et al., 2018), Celeb (Pérez-Rosas et al.,
2018), Politifact111https://www.politifact.com, and
Gossipcop222https://www.gossipcop.com datasets for cross-domain fake news
detection. FakeNewsAMT is a multi-domain dataset containing samples from
technology, education, business, sports, politics, and entertainment domains.
The Celeb dataset has been derived from the web, and contains news about
celebrities. Politifact is a web-scrapped dataset containing political news,
while Gossipcop contains news extracted from the web, manually annotated via
crowd-sourcing and by experts.

Figure 2: Pictorial depiction of our emotion-guided domain-adaptive approach
for cross-domain fake news detection.

We use the Unison model (Colnerič and Demšar, 2020) to annotate all datasets
with the core emotions from Ekman’s (Ekman, 1992) (6 emotions: Joy, Surprise,
Anger, Sadness, Disgust, Fear) and Plutchik’s (Plutchik, 1982) (8 emotions:
Joy, Surprise, Trust, Anger, Anticipation, Sadness, Disgust, Fear) emotion
theories. During preprocessing, we convert text to lowercase, remove
punctuation, and de-contract verb forms (eg. “I’d” to “I would”).

###  3.2 Multi-task Learning

We use multi-task learning (MTL) to incorporate emotion classification as an
auxiliary task to our fake news detection branch. Multi-task learning enables
a model to learn the shared features between two or more correlated tasks for
improved feature extraction and performance. We use Ekman’s or Plutchik’s
emotions labels for emotion classification branch in our MTL models to see
which performs better, and compare the performance with the corresponding
single-task (STL) models in domain-adaptive and non-adaptive settings.

###  3.3 Emotion-guided Domain-adaptive Framework

We propose the cumulative use of domain adaptation and emotion-guided feature
extraction for cross-domain fake news detection. Our approach aims to improve
the feature alignment between different domains using adversarial domain
adaptation, by leveraging the correlation between the emotion and the veracity
of a text (as shown in Figure 1(4)). Figure 2 shows our proposed framework. We
use an LSTM-based (Hochreiter and Schmidhuber, 1997) feature extractor, which
is trained using the accumulated loss from fake news classifier, emotion
classifier and the discriminator (aids in learning domain-invariant features).
LSTM can be replaced with better feature extractors. We used it specifically
for easier comparison to non-adapted emotion-guided and non-adapted single-
task models. The domain classifier acts as the discriminator. In our proposed
framework, we do not use the truth labels for the target dataset for domain
adaptation. However, we utilize the target domain emotion labels in our
approach to better align the two domains using the emotion labels for
supervised learning. The fake news classification loss, emotion classification
loss, adversarial loss, and total loss are defined as in Equations 1, 2, 3,
and 4:

$L_{FND}\ =\ \min\limits_{\theta_{l},\theta_{f}}\sum_{i=1}^{n_{s}}L_{f}^{i}$
(1)

$L_{emo}\ =\ \min\limits_{\theta_{l},\theta_{e}}\sum_{i=1}^{n_{s}}L_{es}^{i}\
+\ \sum_{j=1}^{n_{t}}L_{et}^{j}))$ (2)

$L_{adv}\ =\
\min\limits_{\theta_{d}}(\max\limits_{\theta_{l}}(\sum_{i=1}^{n_{s}}L_{ds}^{i}\
+\ \sum_{j=1}^{n_{t}}L_{dt}^{j}))$ (3)

$L_{Total}\ =\ (1-\alpha-\beta)*L_{FND}\ +\ \alpha\ *\ (L_{adv})\ +\ \beta\ *\
(L_{emo})$ (4)

where $n_{s}$ and $n_{t}$ are number of samples in source and target sets;
$\theta_{d}$, $\theta_{f}$, $\theta_{e}$ and $\theta_{l}$ are parameters for
discriminator, fake news classifier, emotion classifier and LSTM feature
extractor; $L_{d_{s}}$ and $L_{d_{t}}$ are binary crossentropy loss for source
and target classification; $L_{es}$ and $L_{et}$ are crossentropy loss for
emotion classification; $L_{f}$ is binary crossentropy loss for Fake News
Classifier; $\alpha$ and $\beta$ are weight parameters in $L_{Total}$. We
optimised $\alpha$ and $\beta$ for each setting for optimal performance.

We used 300 dimension GloVe (Pennington et al., 2014) embeddings. All models
were trained for up to 50 epochs, stopped when the peak validation accuracy
for the in-domain validation set was achieved. We used a batch size of 25 for
every experiment. Each model used the Adam optimizer with learning rate
0.0025. We used an LSTM layer with 256 units for feature extraction, while
both fake news detection and emotion classification branches consisted of two
dense layers each.

##  4 Experimental Analysis & Results

We evaluated our proposed approach on various combinations of source and
target datasets. Each model was optimized on an in-domain validation set from
the source set. Table 1 illustrates our results proving the efficacy of using
emotion-guided models in non-adapted and domain-adapted cross-domain settings.
It compares non-adaptive models, domain-adaptive models, and our emotion-
guided domain-adaptive models in various settings. MTL(E) and MTL(P) refer to
emotion-guided multi-task frameworks using Ekman’s and Plutchik’s emotions
respectively. STL refers to the single-task framework. DA refers to the use of
the domain-adaptive framework, containing a discriminator. Some findings
observed are:

Source | Target | Setting | Accuracy
---|---|---|---
FakeNewsAMT | Celeb | STL | 0.420
MTL(E) | 0.520
MTL(P) | 0.530
DA STL | 0.560
DA MTL(E) | 0.540
DA MTL(P) | 0.600
Celeb | FakeNewsAMT | STL | 0.432
MTL(E) | 0.471
MTL(P) | 0.476
DA STL | 0.395
DA MTL(E) | 0.501
DA MTL(P) | 0.551
Politifact | Gossipcop | STL | 0.527
MTL(E) | 0.555
MTL(P) | 0.603
DA STL | 0.585
DA MTL(E) | 0.698
DA MTL(P) | 0.671
Celeb | Gossipcop | STL | 0.488
MTL(E) | 0.501
MTL(P) | 0.490
DA STL | 0.525
DA MTL(E) | 0.555
DA MTL(P) | 0.587
FakeNewsAMT | Gossipcop | STL | 0.451
MTL(E) | 0.652
MTL(P) | 0.620
DA STL | 0.790
DA MTL(E) | 0.805
DA MTL(P) | 0.795
FakeNewsAMT | Politifact | STL | 0.363
MTL(E) | 0.450
MTL(P) | 0.530
DA STL | 0.621
DA MTL(E) | 0.704
DA MTL(P) | 0.621
Table 1: Cross-domain evaluation of non-adaptive and adaptive models on
FakeNewsAMT, Celeb, Politifact and Gossipcop datasets. Emotion-guided domain-
adaptive models (DA MTL(E) and DA MTL(P)) outperform their corresponding STL
models in cross-domain settings. Domain-adaptive MTL models consistently
outperform baseline STL, non-adaptive MTL and domain-adaptive STL models.

MTL(E) and MTL(P) models outperform their STL counterparts in cross-domain
settings, as seen in Table 1. This indicates improved extraction of
generalizable features by the emotion-guided models, which aids in improved
fake news detection across datasets from different domains. MTL(E) and MTL(P)
further perform comparably for most settings, and each outperforms the other
in three settings respectively.

DA STL models generally outperform STL models in cross-domain settings across
multiple combinations of datasets. However, we see the STL model outperformed
the DA STL model for Celeb dataset as the source dataset and FakeNewsAMT
dataset as target, confirming that unguided adaptation can sometimes lead to
negative alignment, reducing the performance of the model.

DA MTL(E) and DA MTL(P) models improve performance in cross-domain settings.
Table 1 shows improved results obtained using the emotion-guided adversarial
DA models over their non-adaptive counterparts. This shows the scope for
improved feature extraction even after using DA, and emotion-guided models act
as a solution aiding in correct alignment of the samples and features
extracted by the adaptive framework from different domains. Emotion-guided DA
models mitigated the issue of negative alignment when Celeb dataset was the
source and FakeNewsAMT dataset the target, where STL model outperformed the DA
STL model. The emotion-guided DA models helped correctly align the two
domains, leading to significantly improved performance.

##  5 Conclusion

In this work, we showed the efficacy of emotion-guided models for improved
cross-domain fake news detection and further presented an emotion-guided
domain-adaptive fake news detection approach. We evaluated our proposed
framework against baseline STL, emotion-guided MTL, DA STL and emotion-guided
DA MTL models for various source and target combinations from four datasets.
Our proposed approach led to improved cross-domain fake news detection
accuracy, indicating that emotions are generalizable across domains and aid in
better alignment of different domains during domain adaptation.

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