[1]

# Rashba-induced spin texture and spin-layer-locking effects in the
antiferromagnetic CrI3 bilayer

Sukanya Ghosh***Present affiliation: Department of Physics and Astronomy,
Uppsala University, PO Box 516, SE-751 20 Uppsala, Sweden
*email<EMAIL_ADDRESS>Nataša Stojić  Nadia Binggeli  Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy

###### Abstract

The antiferromagnetic (AFM) CrI3 bilayer is a particularly interesting
representative of van der Waals 2D semiconductors, which are currently being
studied for their magnetism and for their potential in spintronics. Using ab
initio density-functional theory calculations, we investigate the spin texture
in momentum space of the states of the (doubly degenerate) highest valence
band of the AFM CrI3 bilayer with Cr-spin moments perpendicular to the layers.
We find the existence, in the main central part of the Brillouin zone, of a
Rashba in-plane spin texture of opposite signs on the two layers, resulting
from the intrinsic local electric fields acting on each layer. To study the
layer segregation of the wavefunctions, we apply a small electric field that
splits the degenerate states according to their layer occupancy. We find that
the wavefunctions of the highest valence band are layer-segregated, belonging
to only one of the two layers with opposite in-plane spin textures, and the
segregation occurs over nearly the whole Brillouin zone. The corresponding
layer locking of the in-plane-canted spin is related to the separation in
energy of the highest AFM band from the rest of the valence bands. We explain
how the band interactions destroy the layer locking at the $K$, $K^{\prime}$,
and $\Gamma$ points. Furthermore, we compare the layer locking of the in-
plane-canted spin in our AFM bilayer system with the hidden spin polarization
in centrosymmetric nonmagnetic materials, pointing out the differences in
segregation mechanisms and their consequences for the layer locking. We note
that a similar Rashba effect with layer locking of in-plane-canted spin could
occur in other van der Waals AFM bilayers with strong spin-orbit coupling and
an isolated energy band.

###### keywords:

Spin-layer locking Electric field First-principles calculations Reciprocal-
space spin texture 2D materials

##  1 Introduction

The recent discovery of intrinsic magnetic order in atomically thin
semiconducting van der Waals crystals [1, 2] is opening avenues for
fundamental research into 2D magnetism [3] and for the creation of low-power
spintronic devices [4]. A particularly interesting material in this context is
CrI3, in its monolayer and bilayer forms. Monolayer CrI3 is ferromagnetic (FM)
with an out-of-plane magnetization easy axis and a Curie temperature of 45 K
[5]. In a bilayer, the weak van der Waals forces bind the two monolayers,
whose intralayer couplings remain FM. The CrI3 bilayer is experimentally
observed to be a layered antiferromagnet [2, 6, 7, 8, 5], with a Néel
temperature similar to the monolayer’s Curie temperature [8, 5].

In the last decade, antiferromagnets have become increasingly more appealing
for spintronics, owing to their advantageous properties with respect to
ferromagnets: antiferromagnetic (AFM) materials do not produce stray fields,
permitting high-density memory integration, and are much less sensitive to
magnetic field perturbations, providing the necessary stability for data
storage [9, 10]. In addition, antiferromagnets have much faster spin dynamics
than ferromagnets, which is necessary for ultrafast data processing [11].

The applications of the CrI3 bilayer are conceived on the basis of its layered
AFM ground state and the ease of its transformation to the FM phase (low
critical magnetic field of 0.6–0.7 T) [2]. The switching and tuning of the
interlayer exchange from AFM to FM has been heavily investigated and obtained
experimentally by various external perturbations [8, 5, 12, 13]. For example,
the CrI3 bilayer and its multilayers are unique in their ability to
simultaneously act as a tunneling barrier and a spin-filtering layer, thanks
to their semiconducting nature and switchable magnetic state. Consequently,
they display exceptional magnetoresistance properties upon the application of
a magnetic field in both experimental studies [14, 15, 16] and theoretical
studies [17, 18]. Recently, proximity spin-orbit torque on the CrI3 bilayer in
a van der Waals heterostructure was shown theoretically [19]. At the same
time, giant second-harmonic generation was observed in the AFM CrI3 bilayer
and was explained by the simultaneous absence of both inversion symmetry and
time-reversal symmetry in its layered antiferromagnet structure [20], which
are also found to activate and control Raman divergent optical selection rules
[21]. The influence of symmetry operations for the magneto-optical Kerr effect
was recently shown for AFM and FM CrI3-CrBr3 and CrI3 bilayers [22]. However,
despite the great interest in the AFM ground state of the bilayer and newly
discovered properties and phenomena stemming from its layered magnetic order,
little is known about its k-space spin texture. Spin texture can be of vital
importance as it determines how spin-polarized currents can be manipulated for
spintronic devices. It has been heavily investigated in nonmagnetic materials,
especially after the discovery of hidden spin polarization [23, 24, 25, 26]
and spin-layer locking [27, 28] in inversion-symmetric nonmagnetic
semiconductors and semimetals. In contrast, the influence of the AFM order and
the underlying symmetries on the spin texture in general is not well
understood and is currently being studied in other systems, such as in the
spin textures in hybrid organic–inorganic perovskites [29].

In this work, we study the k-space spin texture of the CrI3 bilayer upper
valence states by using density-functional theory. We find that the spin
texture of the bilayer is essentially tangential and characterized by layer
locking of the in-plane-canted spin. In addition, the Rashba effect is not
accompanied by the typical energy splitting. To be able to evaluate the layer-
dependent properties, we apply a small electric field that splits the AFM
degenerate states according to their layer localization. We find that the
wavefunctions of the two highest valence states are spatially segregated
almost throughout the Brillouin zone (BZ) on one of the two layers featuring
opposite in-plane spin textures. We point out that the wavefunction layer
segregation is made possible by the AFM order and energy separation of the
band. We show that the layer segregation is broken essentially only at the $K$
and $K^{\prime}$ points and somewhat at the $\Gamma$ point, which can be
understood on the basis of the properties of monolayer band structure. We also
compare the Rashba effect in the AFM bilayer with the hidden spin polarization
in centrosymmetric nonmagnetic 2D materials.

##  2 System and computational method

Our calculations are performed using density-functional theory as implemented
in the Quantum ESPRESSO package with the plane-wave basis set and
pseudopotentials [30]. The local spin density approximation is used as the
exchange-correlation functional to treat interactions between the electrons
[31]. The coupling between electron spin and its orbital angular momentum is
considered by our including the fully relativistic effect. The electron-ion
interactions are described by the fully relativistic projector augmented wave
pseudopotentials [32]. The plane-wave cutoffs for the kinetic energy and
charge density for all our calculations are 60 Ry and 650 Ry, respectively. We
use the $\Gamma$-centered Monkhorst-Pack $k$-point grid of $24\times 24\times
1$. The periodic images of the CrI3 bilayer are separated by a vacuum region
with a thickness of 30 Å. In the structural relaxations, all coordinates are
relaxed until the force on each atom becomes smaller than 0.1 mRy/bohr.

We concentrate on the CrI3 bilayer with monoclinic layer stacking and $C_{2h}$
space-group symmetry. This is the common experimental form of the bilayer
obtained by exfoliation from bulk CrI3 at room temperature, and which remains
in that form when cooled to temperatures below the Néel temperature [20, 6].
The bilayer in that structure has been the focus of enormous attention because
of its layered AFM ground state, which can be conveniently switched
experimentally to the FM state (by a weak magnetic field as well as by
electrical means) [2, 5, 8, 12].

Our optimized equilibrium local density approximation lattice constant for the
CrI3 bilayer is 6.69 Å, and the innermost distance between the two layers is
3.36 Å [33], in good agreement with the findings of a previous local density
approximation study [34]. The corresponding AFM configuration is energetically
more stable than the FM configuration [33], consistent with the experimental
situation [2, 6, 7, 8, 5]. Our calculated value of the Cr atomic spin moment
of 2.79 $\mu_{\mathrm{\mathrm{B}}}$ for bilayer and monolayer CrI3 also
closely corresponds to the experimental values [35, 36], and the monolayer
magnetic anisotropy energy $E_{\parallel}-E_{\perp}$ of 0.77 meV/Cr places the
magnetization in the perpendicular direction, as found experimentally [35,
37], and in good agreement with the findings of previous theoretical studies
(see Ref. [34] and references therein).

The band structure and the spin texture of the AFM bilayer are studied in the
presence of an external electric field applied perpendicularly to the bilayer
plane (positive $z$ direction). The electric field is modeled with a sawlike
potential along the $z$ direction of the bilayer supercell. Dipole correction
is applied to avoid spurious interactions between the periodic images of CrI3
along the direction of the external electric field [38, 39]. The layer-
projected band structure and the corresponding weights of the states on each
layer are obtained from the atomic pseudo-orbital projections of the states
for the atoms in the layer.

Fig. 1 shows the side and top views of the CrI3 bilayer in the AFM
configuration; the arrows denote the magnetic ordering on the two layers,
$l_{1}$ and $l_{2}$. The corresponding magnetic point group is $C_{2h}$
($C_{2}$), with the following four symmetry operations: (i) identity E; (ii)
180∘ rotation about the $y$ axis Ry[180∘]; (iii) inversion combined with time
reversal I$\cdot$T; and $x$-$z$ mirror-plane reflection combined with time
reversal M${}_{xz}\cdot$T. The presence of the I$\cdot$T symmetry implies that
all bands of the pristine AFM bilayer are doubly degenerate (or multiples of
that at some k points); the pristine band structure is presented in Ref.
[33]). The presence of the electric field breaks the I$\cdot$T symmetry and
lifts that degeneracy (and also breaks the Ry[180∘] symmetry).

Figure 1:  (a) Side view of the atomic structure of the antiferromagnetic CrI3
bilayer with monoclinic layer stacking. In each CrI3 monolayer, the plane of
Cr atoms is sandwiched between two I atomic planes. The turquoise and magenta
spheres represent the Cr and I atoms, respectively. The magnetic moments on
the Cr atoms are denoted by the red arrows, while $l_{1}$ and $l_{2}$ stand
for the two layers. (b) Top view of the CrI3 bilayer, where the dashed black
lines show the unit cell and a is the lattice parameter.

We obtain the spin texture of the band states in k-space by calculating the
expectation values of the spin operators $\frac{1}{2}\hat{\sigma_{\alpha}}$,
$\alpha=x,y,z$, where $\hat{\sigma_{\alpha}}$ are the Pauli matrices. The spin
components corresponding to the $i$th energy spinor eigenfunction,
$\Psi_{i,{\bf k}}({\bf r})$, are obtained as

$S_{\alpha}({\bf k})=\frac{1}{2}\frac{<\Psi_{i,{\bf
k}}|\hat{\sigma}_{\alpha}|\Psi_{i,{\bf k}}>}{<\Psi_{i,{\bf k}}|\Psi_{i,{\bf
k}}>}$.

The particular calculation procedure to obtain the spin textures is given in
the supplementary material.

We also examine the magnetization density, m(r), of some of the band states at
selected k points. The magnetization density components are evaluated as
$m_{\alpha}({\bf r})=\mu_{\mathrm{B}}\Psi^{+}_{i,{\bf k}}({\bf
r})\hat{\sigma}_{\alpha}\Psi_{i,{\bf k}}({\bf r})$, where $\mu_{\mathrm{B}}$
is the Bohr magneton. When the states in the band are degenerate, as is the
case in the AFM bilayer without an electric field, we evaluate the
magnetization density components by summing over the degenerate states:

$m_{\alpha}({\bf r})=\mu_{\mathrm{B}}\displaystyle\sum_{i\in{\rm
band}\,n}\Psi^{+}_{i,{\bf k}}({\bf r})\hat{\sigma}_{\alpha}\Psi_{i,{\bf
k}}({\bf r})$.

##  3 Results and discussion

In Fig. 2, we show the planar average of the magnetization density components
$m_{x}(z),m_{y}(z)$, and $m_{z}(z)$ for the highest valence band of the
pristine AFM bilayer at two k points, chosen at similar distances from the BZ
center along the $k_{y}$ axis [Fig. 2(a)] and along the $k_{x}$ axis [Fig.
2(b)]. The magnetization density was obtained as the sum of the magnetization
densities of the two degenerate states of the band at those k points.

Figure 2:  Planar average of magnetization-density components $m_{x}$,
$m_{y}$, and $m_{z}$ summed over the first and second (the two highest)
valence states of the pristine CrI3 bilayer in the antiferromagnetic
configuration at (a) ${\bf k}=(0,0.15)\times 2\pi/a$ and (b) ${\bf
k}=(0.13,0)\times 2\pi/a$. The turquoise and magenta circles show the
positions of Cr and I atoms, respectively, along the $z$ direction of the
supercell, and $l_{1}$ and $l_{2}$ denote the CrI3 layers in the bilayer.
These two states contribute to form the highest valence band of the
antiferromagnetic bilayer. $m_{i}^{\rm int}(l_{1})$ and $m_{i}^{\rm
int}(l_{2})$, $i=x,y,z$, are obtained by integrating $m_{i}$ on $l_{1}$ and
$l_{2}$, respectively, and are given in the unit of $\mu_{\mathrm{B}}/$bohr2.

Apart from the $m_{z}$ component parallel to the Cr-layer spin, which is
virtually the same for the two k points, we observe the presence of an in-
plane magnetization-density component that strongly depends on the direction
and length of the k wavevector. The amplitude of this in-plane magnetization-
density component is about one tenth that of the $m_{z}$ component. For the k
point along the $k_{y}$ axis, ${\bf k}=(0,0.15)\times 2\pi/a$, in Fig. 2(a),
$m_{x}$ is significant and displays identical magnetization of opposite sign
on the two layers of the bilayer, $l_{1}$ and $l_{2}$, while $m_{y}$ remains
zero on both layers. The $m_{z}$ component also changes sign on the two
layers, following the sign of the Cr spins on the two layers. For the k point
along the $k_{x}$ axis, in Fig. 2(b), $m_{y}$ is also significant and inverts
its sign from $l_{1}$ to $l_{2}$, while keeping the same amplitude. In this
case, the $m_{x}$ component is not vanishing, but is of much smaller magnitude
than the $m_{y}$ component.

Thus, in both cases the magnetization-density component parallel to the ${\bf
k}$ vector is either zero ($m_{y}$ in Fig. 2(a)) or much smaller than the in-
plane tangential component ($m_{x}$ in Fig. 2(b) is one order of magnitude
smaller than $m_{y}$). The behavior of the in-plane tangential magnetization-
density components for both wavevectors is fully consistent with the
tangential spin texture produced by the Rashba effect for electric field ${\bf
E}$ perpendicular to the layer, and corresponding effective magnetic field
$\mathbf{B}_{\rm eff}\sim{\bf k}\times{\bf E}$. In the AFM CrI3 bilayer, each
layer feels an electric field originating from the presence of the other
layer. The fields acting on the two layers are oppositely oriented (we
estimate the strength of this intrinsic electric field to be on the order of
0.1 V/Å at each layer; see the supplementary material). This creates
tangential Rashba in-plane spin textures of opposite sign on the two layers.

For ${\bf k}$ along the $k_{x}$ axis, there is also a very small radial
$m_{x}$ magnetization-density component present on each layer. This originates
in the local chirality of the FM monolayer (i.e., its lack of horizontal
mirror-reflection symmetry), an effect discussed in detail in Ref. [40] for
the FM bilayer. In the latter case, apart from this local chirality, the FM
bilayer is also globally chiral (unlike the AFM bilayer), which results in
dominant radial components along the $k_{x}$ axis and equivalent directions of
the FM bilayer [40]. For the AFM bilayer, the simultaneous tangential behavior
and opposite signs of the magnetization-density components for the two layers
at the two k points (and also observed at other k points) suggest a layer
locking and spatial segregation of the degenerate states of the upper valence
band with opposite helicities of the in-plane spin texture in the two layers.

To confirm the layer segregation of the states with opposite in-plane-canted
spins, we applied a small electric field, $E_{\mathrm{ext}}=0.05$ V/Å, along
the $z$ direction of the unit cell, and evaluated the corresponding layer-
projected band structure and corresponding layer-weighted spin texture. The
electric field places each of the bilayer’s layers at a different potential
and thus splits the AFM degenerate energy levels according to the state’s
localization on $l_{1}$ and $l_{2}$. The effect of this electric field on the
CrI3 bilayer band structure is shown in Fig. 3. There, we also display the
weight of every state on $l_{1}$. Importantly, the highest valence band is
uniformly split nearly throughout the BZ, with a weight of the lower (upper)
split state of more than 90% (less than 8%) on $l_{1}$. In other words, the
two states belonging to the highest band are layer-separated: one is
segregated on $l_{1}$ and the other is segregated on $l_{2}$. It is important
to underline that the electric field separates the states only in energy,
while the spatial layer segregation of the states is a property of the
pristine system. The only exceptions from the spatial segregation of the two
highest states are around the special k points $K$ and $K^{\prime}$, at which
each state has a noticeable weight on both $l_{1}$ and $l_{2}$. The highest
valence states at the $K$ and $K^{\prime}$ points are rather special already
in the monolayer (where the states are degenerate; see the Appendix), and upon
creation of the AFM bilayer, they interact, hybridizing and creating an energy
splitting. To a smaller extent, also around $\Gamma$, the two highest valence
states (especially the second-highest valence state), in Fig. 3, are not
completely layer-locked. This can be similarly understood on the basis of the
monolayer band structure (Fig. A1 in the Appendix); there, at the $\Gamma$
point, the states of the two highest valence bands are particularly close in
energy and tend to exhibit increased hybridization upon formation of the AFM
bilayer. In Fig. 3, we can see that the other bands, both conduction and
valence bands, are not uniformly split in general and that many are
characterized by similar weights on the two layers. The upper valence band of
the pristine AFM CrI3 bilayer [33] has the particular property of being
separated in energy from the other bands. This separation somewhat decreases
when the band splits as a result of the application of $E_{\rm ext}$. In Fig.
3, we can see that states that are closer in energy at a given k point tend to
display increased shared weight on the two layers of the bilayer, as is the
case for the second and third valence states at $K$, $K^{\prime}$, and
$\Gamma$.

Figure 3:  Band structure of the antiferromagnetic CrI3 bilayer in the
presence of an external electric field, $E_{\rm ext}=0.05$ V/Å, applied along
the out-of-plane direction of the CrI3 bilayer. Also shown is the color-coded
weight of the states on layer 1 ($l_{\mathrm{1}}$) of the antiferromagnetic
bilayer. The energy window includes the highest valence bands and the lowest
conduction bands. The inset presents a schematic side view of the bilayer unit
cell with the two monolayers $l_{\mathrm{1}}$ and $l_{\mathrm{2}}$, where the
light blue arrows denote the local internal electric fields $E_{\mathrm{12}}$
and $E_{\mathrm{21}}$ present in the bilayer and the dark blue arrow denotes
the external electric field $E_{\mathrm{ext}}$.

In Fig. 4, we display the calculated in-plane spin texture for the two highest
valence states, split upon the action of $E_{\mathrm{ext}}$. We also show in
Fig. 4 the isovalue color map of the perpendicular spin component $S_{z}$. As
discussed above and based on Fig. 3, the spin textures in Fig. 4 also
correspond to the layer-resolved spin textures (save for the states at $K$,
$K^{\prime}$, and $\Gamma$); the first split state represents the spin texture
on $l_{2}$ and the second split state represents the spin texture on $l_{1}$.
For the first state $S_{z}$ is always negative, and for the other state it is
mostly positive. It can be seen immediately that the helicities of the in-
plane spin texture on the two layers are opposite, as we expected on the basis
of Fig. 2, and each of the layers is fully covered by only one of the two
highest states. This confirms our deduction from Fig. 2 that the AFM CrI3
bilayer is indeed characterized by a Rashba-type in-plane spin texture in
which the two highest valence states with opposite in-plane helicity are
layer-locked with nearly full spatial segregation of the wavefunctions on the
two layers. The in-plane spins have the largest magnitude at a radius of
approximately $0.15\times 2\pi/a$ around the $\Gamma$ point. They also have a
relatively large amplitude around the $M$ point along the $K$-$M$-$K^{\prime}$
line and around the $M^{\prime}$ point along the $M^{\prime}$-$K^{\prime}$
line. On the basis of the amplitude of the in-plane spin around $\Gamma$ in
Fig. 4, the Rashba coefficient for the upper valence band of the AFM bilayer
is estimated to be 0.6 eV Å (see the supplementary material).

It can be seen in Fig. 4 that in the parts of the BZ where the in-plane
components are the strongest (in particular the ring region around the
$\Gamma$ point with radius of approximately $0.15\times 2\pi/a$, and the
regions around the $M$ and $M^{\prime}$ points) the $S_{z}$ component also has
rather large values. The largest values of $S_{z}$ are at $\Gamma$ and at
$M^{\prime}$ along the line $M^{\prime}$-$K$. At the same time, at the $K$ and
$K^{\prime}$ points, the $S_{z}$ component virtually vanishes and even
reverses sign in Fig. 4(b)—these are the same points where the states interact
and lose the layer locking present everywhere else. We assign the appearance
of $S_{z}$ with reverse sign at the $K$ and $K^{\prime}$ points in the second
state to the presence of a very small amount of Cr $d$ and I $p$ states with
opposite spin, observable already in the monolayer degenerate bands at $K$
(see the Appendix). Thus, upon formation of the AFM bilayer, the monolayer
states, each having both positive and negative $S_{z}$ at $K$ and
$K^{\prime}$, will interact between the layers and result in the mixed $S_{z}$
character at each layer observed in Fig. 4 at $K$ and $K^{\prime}$. More
details are given in the Appendix, also for the $\Gamma$ point.

Figure 4:  Spin-texture plots for (a) the highest and (b) the second-highest
valence states in the Brillouin zone of the antiferromagnetic CrI3 bilayer in
the presence of an electric field of 0.05 V/Å applied perpendicularly to the
bilayer. Each of the states is layer-locked on one layer of the bilayer (see
also Fig. 3), the first state on $l_{2}$ and the second on $l_{1}$. The in-
plane spin components are represented by vectors and the perpendicular spin
component is represented by the color scheme at the right of the plots. The
reference value of the magnitude of the in-plane spin component is shown at
the bottom of the plots.

$E_{\mathrm{ext}}=0.05$ V/Å is small enough to induce a linear splitting of
the two states in the highest valence band and a first-order effect in
$E_{\mathrm{ext}}$ on the spin texture. In the limit of vanishing
$E_{\mathrm{ext}}$, the in-plane spin textures of the two states have exactly
the same magnitudes and opposite chiralities. The application of the electric
field along the positive $z$ direction slightly increases the amplitude of the
in-plane spin component for the second valence state, while there is a small
reduction of the in-plane spin component for the first valence state. For
example, with $E_{\mathrm{ext}}=0.05$ V/Å, the largest amplitude of the in-
plane spin increases (decreases) by $0.004$ for the second (first) state with
respect to its average of 0.038 for the two states. The difference in the
strength of the in-plane spin components increases as $E_{\mathrm{ext}}$
increases. It could be envisioned that the electric field could be used to
tune the relative strength of the in-plane spin texture of the two
layers/states for the bands well separated in energy. The in-plane spin
textures with $E_{\mathrm{ext}}=0.1$ V/Å and the corresponding band structure
are reported in the supplementary material. For zero and small
$E_{\mathrm{ext}}$, the contribution to the spin texture of the two layers
cancels in the band, while in the presence of a large external electric field,
there is a nonvanishing overall spin texture.

We have shown that there is a pronounced layer locking of the in-plane-canted
spin for the two highest valence states in AFM bilayer CrI3. This layer
locking without the typical Rashba energy splitting of the degenerate states
has some resemblance to the mechanism of the “hidden Rashba spin-orbit
splitting” (R-2) in centrosymmetric nonmagnetic crystals [24, 23], although in
our case there is no (hidden) Rashba energy splitting between the two doubly
degenerate bands interacting via the Rashba term (see the supplementary
material). In nonmagnetic centrosymmetric crystals, all bands must be at least
doubly degenerate, and global inversion symmetry (I) does not allow Rashba
splitting into singly degenerate bands of opposite spin polarization. However,
the presence of local asymmetry in the individual spatial I-partner sectors
does permit the R-2 effect (with splitting into doubly degenerate bands) when
the doubly degenerate states with opposite spin on the different sectors are
prevented from mixing by a mechanism such as symmetry-enforced wavefunction
segregation in nonsymmorphic crystals [24]. In the AFM bilayer, neither time-
reversal symmetry nor inversion symmetry is present. All bands must, however,
be doubly degenerate because of the presence of the symmetry operation
I$\cdot$T, and (although separated in energy) such bands interact via the
Rashba term, as local asymmetry is also present in the individual
I$\cdot$T-partner sectors. The resulting two degenerate wavefunctions of the
highest valence band are almost completely spatially segregated on the two
layers (except near the BZ edge around the $K$ and $K^{\prime}$ points, and at
$\Gamma$)—see Fig. S3 in the supplementary material. In the R-2 effect, the
segregation is not always complete, and when it is complete (in nonsymmorphic
crystals), it occurs only along the high-symmetry direction in the BZ, where
the symmetry enforcing the wavefunction segregation is present [24]. In the
AFM bilayer, an effect similar to (but not prohibitive for the mixing) that of
the nonsymmorphic symmetries is produced by the layered AFM order with a
perpendicular easy axis that naturally segregates the wavefunctions on the two
layers. However, in our case, at the k points where already in the FM
monolayer two states are degenerate or quasi-degenerate, a complete spatial
segregation in the AFM bilayer does not occur (see Figs. S5 and S6 in the
supplementary material). Thus, we conclude that for a successful spatial
wavefunction segregation of the two degenerate states of a van der Waals AFM
bilayer band, it is crucial that the relevant monolayer states as well as the
AFM band in question remain well separated in energy from the other bands.

##  4 Conclusions

We have studied the spin texture in momentum space of the highest valence band
of the AFM CrI3 bilayer characterized by the perpendicular easy axis. We found
that it displays a Rashba spin texture with opposite sign of the in-plane
spins on the two layers. In this case, the Rashba effect is not accompanied by
an energy splitting of the degenerate bands. We applied a small electric field
perpendicular to the bilayer that splits the degenerate states according to
their layer localization. We observed a layer segregation of the states over
nearly the whole BZ, each state belonging to only one of the two layers. At
the high-symmetry points $K$ and $K^{\prime}$, and to a lesser extent at
$\Gamma$, the segregation is destroyed by band interactions originating from
degeneracies or near-degeneracies present in the CrI3-monolayer band structure
at those points. We compared the Rashba effect and layer locking of the in-
plane-canted spin in the AFM CrI3 bilayer with the R-2 effect and the hidden
spin polarization in nonmagnetic centrosymmetric crystals. Although in both
cases the bands must remain doubly degenerate and the Rasbha effect is enabled
by the local asymmetry in the individual spatial inversion-related sectors,
the segregation mechanisms differ in the two cases. We conclude that, while
the layered AFM coupling makes the spatial segregation possible, the necessary
condition is the separation in energy of the band in question.

We expect that a similar effect should be observed in other van der Waals AFM
bilayers featuring a perpendicular easy axis, strong spin-orbit coupling, and
a band isolated in energy. In all such systems, the relative strength of the
in-plane canting of the spins of the two states/layers could also be tuned by
the application of an external electric field.

##  A Appendix

In Figs. 3 and 4 we see that the layer locking of the two highest valence
states shows some anomalies at the $K$ and $K^{\prime}$ points, and to some
extent also at $\Gamma$. This can be understood on the basis of the band
structure of the pristine CrI3 monolayer in the FM configuration, which is
given in Fig. A1. There, we also show the $S_{z}$ projection on the energy
bands. We see that the highest and second-highest valence states are
degenerate at $K$ and that they are very close in energy at $\Gamma$. $S_{z}$
has the same sign for the first and second valence states at $K$, while at
$\Gamma$ the two states have opposite signs. The whole highest valence band is
of roughly the same $S_{z}$ (although it is somewhat reduced near $K$).

The band structure of the weakly interacting bilayer with the AFM arrangement
of layers can be viewed as a superposition of the $S_{z}$-projected band
structure (in Fig. 3) and its time reversal ($-S_{z}$-projected band
structure) with some further modulation of the band-structure dispersion due
to the weak interlayer interactions. Indeed, the AFM CrI3 bilayer [33] has
dispersions in the BZ for the highest valence states rather similar to those
of the FM monolayer from Fig. A1. The main difference is the increase in
energy of the first band around the $\Gamma$ point and removal of the band
crossing at $K$ and $K^{\prime}$ in the bilayer case due to the interaction
between degenerate states of the two layers.

It is thus expected that the character of the FM monolayer states at a given k
point will determine to a good measure the character of the bilayer states
formed from those monolayers. The highest valence state of the CrI3 monolayer
will form the first valence band of the AFM bilayer, with twofold degeneracy
coming from the two layers with opposite spins. Interestingly, the band
crossing at $K$ of the two highest valence bands in the monolayer is
nontrivial, as the highest valence band has a Chern number of 2 and the
second-highest valence band near $K$ is characterized by a Chern number of
$-2$ [41]. The 3D probability density of the monolayer’s two highest states at
$K$ (see Fig. S6 in the supplementary material) shows that these states have
localized density on different Cr atoms. The mean $S_{z}$ for the degenerate
states at $K$ in the monolayer is only approximately $0.34$. This makes it
possible for the states of the two layers with opposite magnetization to
hybridize upon formation of the bilayer, creating an energy splitting in the
band structure of the bilayer.

At the $\Gamma$ point, the two highest valence states of the monolayer are of
opposite spin and rather close in energy. This allows some interactions
between the two states on the different layers, upon formation of the AFM
bilayer. In this case, however, because of the energy difference, each state
of the monolayer only weakly hybridizes with the state of the other layer.

Figure A1:  Spin-projected band structure of the ferromagnetic CrI3 monolayer.
The projection of the mean $S_{z}$ of the states is color-coded. The energy
window includes the highest valence bands and lowest conduction bands.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this article.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Predictions of the hybrid mesons with exotic quantum numbers $J^{PC}=2^{+-}$

Qi-Nan Wang1 Ding-Kun Lian1 Wei Chen1<EMAIL_ADDRESS>1School of
Physics, Sun Yat-Sen University, Guangzhou 510275, China

###### Abstract

We study the non-strange and strangeonium light hybrid mesons with
$J^{PC}=2^{+-}$ by using the method of QCD sum rules. The local hybrid
interpolating currents with three Lorentz indices are constructed to couple to
such exotic quantum numbers. We calculate the correlation functions up to
dimension eight condensates at the leading order of $\alpha_{s}$. In our
results, the masses of the non-strange $b_{2}$ and $h_{2}$ hybrids are about
$2.65~{}\mathrm{GeV}$, while that of the strangeonium $h_{2}^{\prime}$ hybrid
is about $2.74~{}\mathrm{GeV}$. Such exotic $2^{+-}$ hybrids can be generated
through both the two-gluon and three-gluon emission processes in the radiative
decays of $\chi_{cJ}$. Moreover, these hybrid mesons may be detectable due to
their peculiar decay behaviors and small decay widths. Using the high-
statistics data samples of $\psi(3686)$ in BESIII and BelleII, it is possible
to hunt for such hybrid states through the partial wave analyses in the
$b_{2}\to\omega/a_{1}/h_{1}/a_{2}\pi\to 4\pi$, $h_{2}\to\rho\pi\to 3\pi$ and
$h_{2}\to b_{1}\pi\to 5\pi$ processes.

Hybrid meson, Exotic quantum numbers, QCD sum rules

###### pacs:

12.39.Mk, 12.38.Lg, 14.40.Ev, 14.40.Rt

Introduction — As the most successful theory of strong interaction, the
quantum chromodynamics (QCD) has predicted the existence of exotic hadron
states beyond the conventional quark model, in which hadrons are $q\bar{q}$
mesons and $qqq$ baryons. Among the exotic hadrons, a hybrid meson
($\bar{q}gq$) is composed of a pair of quark-antiquark and a valence gluon.
Due to the existence of gluon degrees of freedom, hybrid mesons can produce
quantum numbers forbidden by the quark model, such as
$J^{PC}=0^{\pm-},1^{-+},2^{+-}$. In the past several decades, the states with
these exotic quantum numbers have received extensive research interest in both
theoretical and experimental aspects Chen:2016qju ; Esposito:2016noz ;
Guo:2017jvc ; Liu:2019zoy ; Brambilla:2019esw ; Chen:2022asf ; Meng:2022ozq .

There is no solid signal for the existence of hybrid mesons to date. The most
fascinating candidates of light hybrid mesons are three exotic quantum number
states $\pi_{1}(1400)$ IHEP-Brussels-LosAlamos-AnnecyLAPP:1988iqi ,
$\pi_{1}(1600)$ E852:2001ikk  and $\pi_{1}(2015)$ E852:2004gpn  with
$I^{G}J^{PC}=1^{-}1^{-+}$, among which the $\pi_{1}(1400)$ and $\pi_{1}(1600)$
were considered to be the same state by the coupled-channel amplitude analysis
JPAC:2018zyd  according to the COMPASS data of the $\eta^{(\prime)}\pi$ system
COMPASS:2014vkj . Recently, the BESIII Collaboration reported an isoscalar
state $\eta_{1}(1855)$ with $I^{G}J^{PC}=0^{+}1^{-+}$ in the decay process
$J/\psi\to\gamma\eta_{1}(1855)\to\gamma\eta\eta^{\prime}$, with the mass and
decay width $m=1855\pm 9_{-1}^{+6}~{}\mathrm{MeV}$ and $\Gamma=188\pm
18_{-8}^{+3}~{}\mathrm{MeV}$ respectively BESIII:2022riz ; BESIII:2022iwi .
Since its observation, the $\eta_{1}(1855)$ has been considered as a candidate
for hybrid mesonChen:2022qpd ; Chen:2023ukh ; Chen:2022isv ; Qiu:2022ktc ;
Shastry:2022upd ; Shastry:2023ths ; Shastry:2022mhk , although other
interpretations can not be excludedHuang:2022tpq ; Yan:2023vbh ; Yang:2022rck
. Furthermore, the $\eta_{1}(1855)$ was assigned as the isoscalar partner of
the $\pi_{1}(1600)$ state in a $J^{PC}=1^{-+}$ hybrid nonet Chen:2023ukh ;
Qiu:2022ktc ; Shastry:2022upd ; Shastry:2022mhk .

In the MIT bag model Barnes:1982tx ; Chanowitz:1982qj , the hybrid mesons with
$J^{PC}=1^{--},0^{-+},1^{-+}$ and $2^{-+}$ were established to be the lightest
supermultiplet states, in which only $1^{-+}$ is the exotic quantum number.
The exotic quantum numbers $J^{PC}=0^{+-}$ and $2^{+-}$ were assigned to the
higher supermultiplet, while the $0^{--}$ hybrid was much heavier with
different gluon excitation HadronSpectrum:2012gic ; Chen:2013eha ;
Meyer:2015eta . Such hybrid supermultiplet structures were supported by the
Coulomb gauge QCD Guo:2008yz  and LQCD HadronSpectrum:2012gic  calculations.
Besides, the hybrid mesons have been extensively studied by various methods,
such as the LQCD Lacock1997 ; Lacock1996 ; HadronSpectrum:2012gic ; Ma:2020bex
; Dudek2011 ; Dudek:2009kk ; Dudek2013 , the flux tube modelIsgur:1984bm ;
Isgur:1985vy ; Burns2006 , Bethe-Salpeter equationBurden2002 ; Burden:1996nh
and QCD sum rulesBalitsky:1982ps ; Balitsky:1986hf ; Barsbay:2022gtu ;
Chen:2013eha ; Chen:2021smz ; Chen:2022qpd ; Govaerts:1986pp ; Govaerts:1984hc
; Govaerts:1983ka ; Govaerts:1984bk ; Chen:2013zia ; Ho:2018cat ; Ho:2019org ;
Latorre:1984kc ; Palameta:2018yce .

Comparing to the $1^{-+}$ channel, the hybrid states with exotic
$J^{PC}=0^{+-}$ and $2^{+-}$ have received much less theoretical interest,
partly because of the absence of experimental signal. However, it is
especially interesting to note that the exotic $2^{+-}$ hybrids were predicted
to be surprisingly narrow with the decay widths less than 10 MeV in the PSS
model Page:1998gz . If this is true, such narrow states would be detectable in
the definite hybrid decay modes.

The hybrid mass is also an important parameter to determine its decay
properties. In LQCD Dudek2011 ; Dudek2013 , the masses of the light $2^{+-}$
hybrids were predicted to be about $2.4-2.8~{}\mathrm{GeV}$, which are much
heavier than those from the flux tube model Isgur:1985vy . More theoretical
investigations would be useful to diminish such a disagreement.

In this work, we investigate the light hybrid mesons with exotic quantum
numbers $J^{PC}=2^{+-}$ within the method of QCD sum rules. We shall construct
the local hybrid interpolating currents without the covariant derivative
operators. The masses of the $2^{+-}$ non-strange $\bar{q}gq$ and strangeonium
$\bar{s}gs$ hybrids can be investigated by using these three Lorentz indices
currents. Hereafter, we adopt the notation for hybrid mesons following Ref.
Meyer:2010ku  and the Particle Data Group (PDG) ParticleDataGroup:2022pth , so
that the isovector hybrid with $J^{PC}=2^{+-}$ is $b_{2}$, the isoscalar non-
strange one is $h_{2}$ while the strangeonium one is $h_{2}^{\prime}$.
Finally, according to our predicted masses, some experimental suggestions are
made. We propose to search for these light hybrid states from the decays of
$\psi(3686)$ in future experiments such as BESIII and BelleII.

Currents and projectors — A hybrid current is usually built out of a
$\bar{q}q$ pair and an excited gluon field. As in Ref. Govaerts:1986pp , the
hybrid currents with two Lorentz indices can couple to the quantum numbers
$J^{PC}=2^{-+}$ and $2^{++}$. To construct a hybrid current with
$J^{PC}=2^{+-}$ or $2^{--}$, one can consider the local operators with three
Lorentz indices or nonlocal operators with derivatives. In this work, we shall
adopt the former strategy to construct the local interpolating hybrid currents
with three Lorentz indices.

Using the gluon field strength $G_{\alpha\beta}(x)$ and Dirac matrix, the
possible hybrid currents with three Lorentz indices are

$\displaystyle J_{\alpha\beta\gamma}^{(1)}$
$\displaystyle=\bar{q}g_{s}\gamma_{\alpha}\gamma_{5}G_{\beta\gamma}q,~{}~{}~{}~{}J^{PC}=(0\,,1\,,2)^{\pm-}\,,$
(1) $\displaystyle J_{\alpha\beta\gamma}^{(2)}$
$\displaystyle=\bar{q}g_{s}\gamma_{\alpha}\gamma_{5}\tilde{G}_{\beta\gamma}q,~{}~{}~{}~{}J^{PC}=(0\,,1\,,2)^{\pm-}\,,$
$\displaystyle J_{\alpha\beta\gamma}^{(3)}$
$\displaystyle=\bar{q}g_{s}\gamma_{\alpha}G_{\beta\gamma}q,~{}~{}~{}~{}J^{PC}=(0\,,1\,,2)^{\pm+}\,,$
$\displaystyle J_{\alpha\beta\gamma}^{(4)}$
$\displaystyle=\bar{q}g_{s}\gamma_{\alpha}\tilde{G}_{\beta\gamma}q,~{}~{}~{}~{}J^{PC}=(0\,,1\,,2)^{\pm+}\,,$

where $q$ is a quark field ($u,d$ or $s$), $g_{s}$ is the strong coupling
constant, and
$\tilde{G}_{\alpha\beta}=\frac{1}{2}\varepsilon_{\alpha\beta\mu\nu}G^{\mu\nu}$
is the dual gluon field strength. Thus the interpolating currents
$J_{\alpha\beta\gamma}^{(2)}$ and $J_{\alpha\beta\gamma}^{(4)}$ have the
opposite parity with $J_{\alpha\beta\gamma}^{(1)}$ and
$J_{\alpha\beta\gamma}^{(3)}$, respectively. In general, a current with three
Lorentz indices can couple to various hadron states via the following
relations

$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}\right|0^{(-P)C}(p)\right\rangle$
$\displaystyle=Z_{1}^{0}p_{\alpha}g_{\beta\gamma}+Z_{2}^{0}p_{\beta}g_{\alpha\gamma}+Z_{3}^{0}p_{\gamma}g_{\alpha\beta}$
(2) $\displaystyle+Z_{4}^{0}p_{\alpha}p_{\beta}p_{\gamma}\,,$
$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}\right|0^{PC}(p)\right\rangle$
$\displaystyle=Z_{5}^{0}\varepsilon_{\alpha\beta\gamma\tau}p^{\tau}\,,$
$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}\right|1^{PC}(p)\right\rangle$
$\displaystyle=Z_{1}^{1}\epsilon_{\alpha}g_{\beta\gamma}+Z_{2}^{1}\epsilon_{\beta}g_{\alpha\gamma}+Z_{3}^{1}\epsilon_{\gamma}g_{\alpha\beta}$
$\displaystyle+Z_{4}^{1}\epsilon_{\alpha}p_{\beta}p_{\gamma}+Z_{5}^{1}\epsilon_{\beta}p_{\alpha}p_{\gamma}$
$\displaystyle+Z_{6}^{1}\epsilon_{\gamma}p_{\alpha}p_{\beta}\,,$
$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}\right|1^{(-P)C}(p)\right\rangle$
$\displaystyle=Z_{7}^{1}\varepsilon_{\alpha\beta\gamma\tau}\epsilon^{\tau}+Z_{8}^{1}\varepsilon_{\alpha\beta\tau\lambda}\epsilon^{\tau}p^{\lambda}p_{\gamma}$
$\displaystyle+Z_{9}^{1}\varepsilon_{\alpha\gamma\tau\lambda}\epsilon^{\tau}p^{\lambda}p_{\beta}\,,$
$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}\right|2^{(-P)C}(p)\right\rangle$
$\displaystyle=Z_{1}^{2}\epsilon_{\alpha\beta}p_{\gamma}+Z_{2}^{2}\epsilon_{\alpha\gamma}p_{\beta}+Z_{3}^{2}\epsilon_{\beta\gamma}p_{\alpha}\,,$
$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}\right|2^{PC}(p)\right\rangle$
$\displaystyle=Z_{4}^{2}\varepsilon_{\alpha\beta\tau\theta}\epsilon_{\gamma}^{~{}\tau}p^{\theta}+Z_{5}^{2}\varepsilon_{\alpha\gamma\tau\theta}\epsilon_{\beta}^{~{}\tau}p^{\theta}\,,$
$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}\right|3^{PC}(p)\right\rangle$
$\displaystyle=Z_{1}^{3}\epsilon_{\alpha\beta\gamma}\,,$

where $\epsilon_{\alpha},\epsilon_{\alpha\beta},\epsilon_{\alpha\beta\gamma}$
are the spin-1, spin-2 and spin-3 polarization tensors respectively. The
parity in Eq. (2) is $P=+$ for $J_{\alpha\beta\gamma}^{(1)}$ and
$J_{\alpha\beta\gamma}^{(4)}$ while $P=-$ for $J_{\alpha\beta\gamma}^{(2)}$
and $J_{\alpha\beta\gamma}^{(3)}$.

For the interpolating currents in Eq. (1), they can not couple to a spin-3
state since the last two Lorentz indices in $G_{\beta\gamma}$ and
$\tilde{G}_{\beta\gamma}$ are antisymmetric while the spin-3 polarization
tensor $\epsilon_{\alpha\beta\gamma}$ is completely symmetric. According to
Eq. (2), all these currents can couple to both positive and negative parity
states with spin-0, spin-1 and spin-2 via different tensor structures. It is
explicit that the charge conjugation $C$-parities of
$J_{\alpha\beta\gamma}^{(1)}$ and $J_{\alpha\beta\gamma}^{(2)}$ are negative
while those of $J_{\alpha\beta\gamma}^{(3)}$ and $J_{\alpha\beta\gamma}^{(4)}$
are positive.

The couplings between a three Lorentz indices current and hadron states are
very complicate, as shown in Eq. (2). For the currents
$J_{\alpha\beta\gamma}^{(1)}$ and $J_{\alpha\beta\gamma}^{(2)}$, we concern
the following couplings to study the hybrid states with $J^{PC}=2^{+-}$

$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}^{(1)}\right|2^{+-}(p)\right\rangle=$
$\displaystyle
Z_{4}^{2}\varepsilon_{\alpha\beta\tau\theta}\epsilon_{\gamma}^{~{}\tau}p^{\theta}+Z_{5}^{2}\varepsilon_{\alpha\gamma\tau\theta}\epsilon_{\beta}^{~{}\tau}p^{\theta}$
(3) $\displaystyle=$ $\displaystyle
f^{-}\left(\varepsilon_{\alpha\beta\tau\theta}\epsilon_{\gamma}^{~{}\tau}p^{\theta}-\varepsilon_{\alpha\gamma\tau\theta}\epsilon_{\beta}^{~{}\tau}p^{\theta}\right)\,,$
$\displaystyle\left\langle
0\left|J_{\alpha\beta\gamma}^{(2)}\right|2^{+-}(p)\right\rangle=$
$\displaystyle
Z_{1}^{2}\epsilon_{\alpha\beta}p_{\gamma}+Z_{2}^{2}\epsilon_{\alpha\gamma}p_{\beta}+Z_{3}^{2}\epsilon_{\beta\gamma}p_{\alpha}$
(4) $\displaystyle=$ $\displaystyle
f^{-\prime}(\epsilon_{\alpha\beta}p_{\gamma}-\epsilon_{\alpha\gamma}p_{\beta})\,,$

in which $f^{-}$ and $f^{-\prime}$ are the coupling constants between the
currents and the corresponding $2^{+-}$ hybrids. In accord with the currents
in Eq. (1), the tensor structures of the above relations are rewritten as
antisymmetric for the indices $\\{\beta\gamma\\}$, so that the symmetric part
vanishes. Due to these couplings, one can prove that
$J_{\alpha\beta\gamma}^{(1)}$ and $J_{\alpha\beta\gamma}^{(2)}$ actually
couple to the same hybrid state, by multiplying a Levi-Civita tensor on both
side of Eq. (3). To study the hybrid mesons with $J^{PC}=2^{+-}$, we shall
investigate the interpolating current $J_{\alpha\beta\gamma}^{(1)}$ in this
work.

QCD sum rules — The two-point correlation function of
$J_{\alpha\beta\gamma}^{(1)}$ can be written as

$\displaystyle\Pi_{\alpha_{1}\beta_{1}\gamma_{1},\alpha_{2}\beta_{2}\gamma_{2}}(p^{2})$
$\displaystyle=i\int d^{4}xe^{ip\cdot x}\left\langle
0\left|T\left[J_{\alpha_{1}\beta_{1}\gamma_{1}}^{(1)}(x)J_{\alpha_{2}\beta_{2}\gamma_{2}}^{(1)\dagger}(0)\right]\right|0\right\rangle\,,$
(5)

which contains six Lorentz indices. To study the hybrid state with
$J^{PC}=2^{+-}$, we decompose this correlation function into invariant
function associated with such quantum numbers by constructing the normalized
projector

$\displaystyle\mathbb{P}_{\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2}}=$
$\displaystyle\frac{1}{80p^{2}}\times$ (6)
$\displaystyle\sum\left(\varepsilon_{\alpha_{1}\beta_{1}\tau_{1}\theta_{1}}\epsilon_{\gamma_{1}}^{~{}\tau_{1}}p^{\theta_{1}}-\varepsilon_{\alpha_{1}\gamma_{1}\tau_{1}\theta_{1}}\epsilon_{\beta_{1}}^{~{}\tau_{1}}p^{\theta_{1}}\right)$
$\displaystyle\times\left(\varepsilon_{\alpha_{2}\beta_{2}\tau_{2}\theta_{2}}\epsilon_{\gamma_{2}}^{~{}\tau_{2}*}p^{\theta_{2}}-\varepsilon_{\alpha_{2}\gamma_{2}\tau_{2}\theta_{2}}\epsilon_{\beta_{2}}^{~{}\tau_{2}*}p^{\theta_{2}}\right)\,.$

The above summation over the polarization tensor is

$\sum\epsilon_{\alpha_{1}\beta_{1}}\epsilon_{\alpha_{2}\beta_{2}}^{*}=\eta_{\alpha_{1}\alpha_{2}}\eta_{\beta_{1}\beta_{2}}+\eta_{\alpha_{1}\beta_{2}}\eta_{\beta_{1}\alpha_{2}}-\frac{2}{3}\eta_{\alpha_{1}\beta_{1}}\eta_{\alpha_{2}\beta_{2}}\,,$
(7)

where

$\eta_{\alpha\beta}=\frac{p_{\alpha}p_{\beta}}{p^{2}}-g_{\alpha\beta}\,.$ (8)

One may wonder whether there is the Lorentz structure
$\varepsilon_{\beta\gamma\tau\theta}\epsilon_{\alpha}^{~{}\tau}p^{\theta}$ in
Eq. (3). As a matter of fact, the projector constructed by this structure is
the same as that in Eq. (6). In other words, there are only two independent
Lorentz structures in Eq. (3).

The invariant function $\Pi_{2}\left(p^{2}\right)$ corresponding to the
$J^{PC}=2^{+-}$ can be extracted as

$\displaystyle\Pi_{2}(p^{2})$
$\displaystyle=\mathbb{P}^{\alpha_{1},\beta_{1},\gamma_{1},\alpha_{2},\beta_{2},\gamma_{2}}\Pi_{\alpha_{1}\beta_{1}\gamma_{1},\alpha_{2}\beta_{2}\gamma_{2}}(p^{2})\,.$
(9)

This correlation function is usually evaluated via the method of operator
production expansion (OPE) at the quark-gluon level, as a function of various
QCD parameters and nonperturbative condensates.

At the hadron level, the correlation function $\Pi(p^{2})$ can be described
via the dispersion relation

$\Pi(p^{2})=\frac{\left(p^{2}\right)^{N}}{\pi}\int_{0}^{\infty}\frac{\operatorname{Im}\Pi(s)}{s^{N}\left(s-p^{2}-i\epsilon\right)}ds+\sum_{n=0}^{N-1}b_{n}\left(p^{2}\right)^{n}\,,$
(10)

where $b_{n}$ is the subtraction constant. In QCD sum rules, the imaginary
part of the correlation function is defined as the spectral function

$\rho(s)=\frac{1}{\pi}\text{Im}\Pi(s)=f_{X}^{2}\delta(s-m_{X}^{2})+\cdots\,,$
(11)

in which the “one pole plus continuum” parametrization is used, and ”$\cdots$”
denotes the continuum and higher states. The parameters $f_{X}$ and $m_{X}$
are the coupling constant and mass of the lowest-lying hybrid meson
respectively.

To improve the convergence of the OPE series and suppress the contributions
from continuum and higher states, the Borel transformation can be performed to
the correlation function in both hadron and quark-gluon levels. The QCD sum
rules are then obtained as

$\Pi\left(s_{0},M_{B}^{2}\right)=f_{X}^{2}e^{-m_{X}^{2}/M_{B}^{2}}=\int_{0}^{s_{0}}dse^{-s/M_{B}^{2}}\rho(s)\,,$
(12)

where $M_{B}$ is the Borel mass introduced via the Borel transformation and
$s_{0}$ is the continuum threshold. Then the mass and coupling of the lowest-
lying hadron can be extracted as

$\displaystyle m_{X}\left(s_{0},M_{B}^{2}\right)$
$\displaystyle=\sqrt{\frac{\frac{\partial}{\partial\left(-1/M_{B}^{2}\right)}\Pi\left(s_{0},M_{B}^{2}\right)}{\Pi\left(s_{0},M_{B}^{2}\right)}}\,,$
(13) $\displaystyle f_{X}\left(s_{0},M_{B}^{2}\right)$
$\displaystyle=\sqrt{\Pi\left(s_{0},M_{B}^{2}\right)e^{m_{X}^{2}/M_{B}^{2}}}\,.$

In this work, we calculate the correlation function and spectral density at
the leading order of $\alpha_{s}$ up to dimension eight condensates. For the
current $J_{\alpha\beta\gamma}^{(1)}$, the correlation function for the hybrid
meson with $J^{PC}=2^{+-}$ is obtained as

$\displaystyle\Pi_{2}(s_{0},M_{B}^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}ds~{}e^{-\frac{s}{M_{B}^{2}}}\left\\{\frac{\alpha_{s}s^{3}}{2^{8}\times
3^{2}~{}\pi^{3}}+\frac{11\alpha_{s}m_{q}s}{2^{4}\times
3^{2}~{}\pi}\Big{\langle}\bar{q}q\Big{\rangle}\right.$ (14)
$\displaystyle\left.+\frac{s}{2^{4}\times
3^{2}~{}\pi}\Big{\langle}\alpha_{s}GG\Big{\rangle}+\frac{1}{2^{8}~{}\pi^{2}}\Big{\langle}g_{s}^{3}G^{3}\Big{\rangle}\right\\}$
$\displaystyle+\frac{19\pi\alpha_{s}}{2^{4}\times
3^{2}}\Big{\langle}\bar{q}q\Big{\rangle}\Big{\langle}g_{s}\bar{q}\sigma
Gq\Big{\rangle}-\frac{1}{2^{7}\times
3^{2}}\Big{\langle}\alpha_{s}GG\Big{\rangle}^{2}$
$\displaystyle-\frac{5}{2^{8}\times
3^{4}}(\mathrm{ln}(M_{B}^{2}/\mu^{2})-\gamma_{E})\Big{\langle}\alpha_{s}GG\Big{\rangle}^{2}\,.$

At the leading order of $\alpha_{s}$, the four-quark condensate gives no
contribution to the correlation function since the corresponding feynman
diagram has no loop Ho:2018cat . For the light hybrid $\bar{u}gu$ and
$\bar{d}gd$ systems, we don’t consider the contributions from the quark
condensate and quark-gluon mixed condensate due to the negligible light quark
mass. For the strangeonium hybrid $\bar{s}gs$ system, we consider the quark
condensate contribution but neglect the mixed condensate since it will be much
more suppressed as compared to the quark condensate Ho:2018cat .

Numerical analysis — To perform the numerical analyses, we use the following
values for various QCD parameters at the renormalization scale
$\mu=2~{}\mathrm{GeV}$ and the QCD scale $\Lambda_{QCD}=300~{}\mathrm{GeV}$
ParticleDataGroup:2022pth ; Narison:2011xe ; Narison:2018dcr ; Jamin:2002ev :

$\displaystyle\alpha_{s}\left(\mu\right)$
$\displaystyle=\frac{4\pi}{9\ln\left(\mu^{2}/\Lambda_{\mathrm{QCD}}^{2}\right)}\,,$
$\displaystyle m_{s}$ $\displaystyle=93_{-5}^{+11}~{}\mathrm{MeV}\,,$
$\displaystyle\Big{\langle}\bar{q}q\Big{\rangle}$ $\displaystyle=-(0.24\pm
0.01)^{3}~{}\mathrm{GeV}^{3}\,,$
$\displaystyle\Big{\langle}\bar{s}s\Big{\rangle}$ $\displaystyle=(0.8\pm
0.1)\times\Big{\langle}\bar{q}q\Big{\rangle}\,,$
$\displaystyle\Big{\langle}g_{s}\bar{q}\sigma Gq\Big{\rangle}$
$\displaystyle=-(0.8\pm
0.2)\times\Big{\langle}\bar{q}q\Big{\rangle}~{}\mathrm{GeV}^{2}\,,$
$\displaystyle\Big{\langle}g_{s}\bar{s}\sigma Gs\Big{\rangle}$
$\displaystyle=(0.8\pm 0.2)\times\Big{\langle}g_{s}\bar{q}\sigma
Gq\Big{\rangle}\,,$ (15) $\displaystyle\Big{\langle}\alpha_{s}GG\Big{\rangle}$
$\displaystyle=(6.35\pm 0.35)\times 10^{-2}~{}\mathrm{GeV}^{4}\,,$
$\displaystyle\Big{\langle}g_{s}^{3}G^{3}\Big{\rangle}$
$\displaystyle=-(8.2\pm
1.0)\times\Big{\langle}\alpha_{s}GG\Big{\rangle}~{}\mathrm{GeV}^{2}\,.$

As shown in Eq. (13), the hadron mass and coupling are the function of $M_{B}$
and $s_{0}$. The working regions of these two parameters can be determined by
requiring suitable OPE convergence and pole contribution. To guarantee the
good OPE convergence, we require that the contributions from the $D>6$
condensates be less than 2%, i.e

$R_{D>6}=\left|\frac{\Pi_{2}^{D>6}\left(M_{B},\infty\right)}{\Pi_{2}^{tot}\left(M_{B},\infty\right)}\right|<2\%\,.$
(16)

For the $\bar{q}gq$ light hybrid meson with $J^{PC}=2^{+-}$, this requirement
leads to the lower bound of the Borel mass $M_{B}^{2}\geq
2.39~{}\mathrm{GeV}^{2}$. To get the upper bound of $M_{B}^{2}$, we need to
fix the value of $s_{0}$ at first. In Fig. 1, we show the variations of the
hadron mass $m_{X}$ with the threshold $s_{0}$ for various Borel mass
$M_{B}^{2}$. It is shown that the variation of $m_{X}$ with $M_{B}^{2}$ is
minimized around $s_{0}\sim 10.0~{}\mathrm{GeV}^{2}$, which will result in the
working region $9.0\leq s_{0}\leq 11.0~{}\mathrm{GeV}^{2}$. Using this value
of $s_{0}$, the upper bound of $M_{B}^{2}$ can be obtained by requiring the
pole contribution be larger than 50%, i.e

$PC=\frac{\Pi\left(M_{B},s_{0}\right)}{\Pi\left(M_{B},\infty\right)}>50\%\,.$
(17)

Figure 1: Variations of hadron mass with $s_{0}$ and $M_{B}^{2}$ for the
$\bar{q}gq$ light hybrid meson.

Finally, the working region of the Borel parameter can be determined to be
$2.39\leq M_{B}^{2}\leq 2.94~{}\mathrm{GeV}^{2}$. We show the Borel curves in
these regions in Fig. 1, from which we can see that the sum rule is stable in
the above parameter regions. We can extract the hadron mass as

$\displaystyle m_{b_{2}/h_{2}}$ $\displaystyle=2.65\pm 0.09~{}\mathrm{GeV}\,,$
(18)

and the coupling as

$\displaystyle f_{b_{2}/h_{2}}$ $\displaystyle=(1.25\pm 0.18)\times
10^{-1}~{}\mathrm{GeV}^{4}\,.$ (19)

The errors come from the continuum threshold $s_{0}$, condensates
$\Big{\langle}\alpha_{s}GG\Big{\rangle}$, $\Big{\langle}g_{s}\bar{q}\sigma
Gq\Big{\rangle}$ and $\Big{\langle}g_{s}^{3}G^{3}\Big{\rangle}$. The
uncertainty from the Borel mass is small enough to be neglected. The hadron
mass extracted in Eq. (18) is degenerate for the isoscalar and isovector
$\bar{q}gq$ light hybrid states with $J^{PC}=2^{+-}$, since we don’t
distinguish them in our calculations.

For the strangeonium $\bar{s}gs$ hybrid system, the quark condensate is taken
into account for the correlation function. However, we find that the
contribution from quark condensate is much smaller than those from the two-
and tri-gluon condensates, as shown in Fig. 2. Adopting the same criteria as
the above mentioned, the Borel parameter and continuum threshold can be
determined to be $2.18\leq M_{B}^{2}\leq 3.20~{}\mathrm{GeV}^{2}$ and
$10.0\leq s_{0}\leq 12.0~{}\mathrm{GeV}^{2}$. We show the Borel curves in Fig.
3.

Figure 2: OPE convergence for the $\bar{s}gs$ hybrid system.

The mass and coupling of this state are

$\displaystyle m_{h_{2}^{\prime}}$ $\displaystyle=2.74\pm
0.09~{}\mathrm{GeV}\,,$ (20)

and

$\displaystyle f_{h_{2}^{\prime}}$ $\displaystyle=(1.42\pm 0.18)\times
10^{-1}~{}\mathrm{GeV}^{4}\,.$ (21)

Figure 3: Variations of hadron mass with $s_{0}$ and $M_{B}^{2}$ for the
$\bar{s}gs$ strangeonium hybrid meson.

Our calculations of the $2^{+-}$ hybrid masses in Eqs. (18) and (20) are in
good agreement with the LQCD predictions Dudek2011 ; Dudek2013 . Accordingly,
we shall discuss the hybrid production and decay in the following.

Production and decay — Similar to the $1^{-+}$ hybrid candidate
$\eta_{1}(1855)$ being observed in the radiative decays of
$J/\psi$BESIII:2022riz ; Chen:2022isv , a light hybrid meson $X$ with
$J^{PC}=2^{+-}$ can be generated from the decay of charmoniums, such as
$\chi_{cJ}\rightarrow\gamma X$. For this reason,
$\psi(3686)\rightarrow\gamma\chi_{cJ}\rightarrow\gamma\gamma X$ is a possible
process to produce $2^{+-}$ light hybrid mesons. Analogous to the radiative
decay $J/\psi\rightarrow\gamma\eta_{1}(1855)$, the decay
$\chi_{cJ}\rightarrow\gamma X$ can also occur through the three-gluon emission
processes as shown in Fig. 4. In additon, the $2^{+-}$ light hybrids can be
generated through the two-gluon emission process either, as shown in Fig. 4,
since the $\chi_{c0/2}$ can decay in such a process. For this reason, the
production rates of the $2^{+-}$ hybrids in $\chi_{cJ}$ radiative decays may
be larger than that of $\eta_{1}(1855)$ in $J/\psi$ radiative decays. This
advantage can be utilized to search for these hybrid mesons from the high-
statistics samples of $\psi(3686)$ in BESIII and BelleII experiments.

A hybrid meson may decay into two conventional mesons in two ways, one of
which is that a pair of $\bar{q}q$ or $\bar{s}s$ is excited from the valence
gluon and then combines with the valence quark and antiquark respectively to
form two mesons. Another way is the so-called QCD axial anomaly Akhoury:1987ed
; Ball:1995zv ; Chao:1989yp . Both mechanisms are at the order of
$\alpha_{s}$, and thus may be on the same order of magnitude. Meanwhile, a
hybrid meson can also decay into final states containing a lighter hybrid
meson plus a conventional meson, for example, the S-wave decay processes
$b_{2}\to\pi_{1}\omega/\eta_{1}\rho$, $h_{2}\to\eta_{1}\omega/\pi_{1}\rho$.

We list some possible two-meson decay modes of $2^{+-}$ hybrids with different
$I^{G}$ in Table LABEL:tab:decay_mode, in which the S-wave, P-wave and D-wave
decay channels are considered. It is clear that all S-wave decay channels have
limited phase spaces, while P-wave channels such as $a_{1}\pi$, $b_{1}\pi$,
$h_{1}\pi$, $a_{2}\pi$, $K\bar{K}_{1}$ have larger phase spaces. It is
interesting to find that all two S-wave mesons decay channels are in D-wave,
such as $\omega\pi$, $\rho\eta^{(\prime)}$, $\rho\pi$,
$\omega\eta^{(\prime)}$, $K\bar{K}^{\ast}$, with much larger phase spaces than
those of S-wave and P-wave channels. Such decay behaviors may result in the
small total decay width for these $2^{+-}$ hybrid mesons Page:1998gz , making
it possible to isolate and detect them. Accordingly, we suggest to search for
the $b_{2}$ hybrid in the $4\pi$ final states via
$b_{2}\to\omega/a_{1}/h_{1}/a_{2}\pi\to 4\pi$ process, while the $h_{2}$
hybrid in the $3\pi$ or $5\pi$ final states via $h_{2}\to\rho\pi\to 3\pi$ or
$h_{2}\to b_{1}\pi\to 5\pi$ process. The two strange mesons channels may also
be used to reproduce these hybrid states, such as
$K\bar{K}^{\ast},K^{(\ast)}\bar{K}_{1}$ and so on.

Besides, these $2^{+-}$ hybrid mesons can also decay into the $\pi\gamma$ and
$\eta^{(\prime)}\gamma$ final states via the electromagnetic interaction.

Figure 4: Two possible production mechanisms of $2^{+-}$ light hybrids from $\chi_{cJ}$ radiative decays. Table 1: Some possible two-meson decay modes for the hybrids with the exotic quantum numbers $I^{G}J^{PC}=0^{-}2^{+-}$ ($h_{2}$ and $h_{2}^{\prime}$) and $1^{+}2^{+-}$ ($b_{2}$). The two strange mesons decay modes are appropriate for $b_{2},h_{2}$ and $h_{2}^{\prime}$ at the same time. $I^{G}(J^{PC})$ |  $0^{-}(2^{+-})$ |  $1^{+}(2^{+-})$
---|---|---
S-wave  |  $K_{0}^{*}\bar{K}_{2}^{*}$
$a_{1}b_{1}$, $f_{1}h_{1}$, $f_{2}h_{1}$, |  $a_{1}h_{1}$, $f_{1}b_{1}$, $f_{2}b_{1}$,
|  $\eta_{1}\omega$, $\pi_{1}\rho$ |  $\pi_{1}\omega$, $\eta_{1}\rho$
P-wave |  $K\bar{K}_{1}$, $K\bar{K}_{2}^{*}$, $K^{*}\bar{K}_{0}^{*}$, $K^{*}\bar{K}_{1}$, $K^{*}\bar{K}_{2}^{*}$
$h_{1}\eta$, $b_{1}\pi$, $f_{0}\omega$ , |  $b_{1}\eta$, $h_{1}\pi$, $f_{0}\rho$ ,
$f_{1}\omega$, $f_{2}\omega$, $a_{0}\rho$, $a_{1}\rho$ |  $f_{1}\rho$, $f_{2}\rho$, $a_{0}\omega$, $a_{1}\pi$, $a_{2}\pi$
D-wave |  $K\bar{K}^{\ast}$
$\rho\pi$, $\omega\eta^{(\prime)}$ |  $\omega\pi$, $\rho\eta^{(\prime)}$

Conclusion — We have investigated the masses of the light hybrid mesons with
exotic quantum number $J^{PC}=2^{+-}$ in QCD sum rule method by constructing
local interpolating currents with three Lorentz indices. To calculate the
correlation functions and spectral densities, we consider the contributions of
nonperturbative effect up to dimension eight condensates at the leading order
of $\alpha_{s}$.

After the numerical analyses, the extracted masses are about
$2.65~{}\mathrm{GeV}$ for the non-strange $\bar{q}gq$ hybrids ($b_{2}$ and
$h_{2}$) and $2.74~{}\mathrm{GeV}$ for the $\bar{s}gs$ strangeonium hybrid
($h_{2}^{\prime}$), respectively. As discussed above, their peculiar decay
behaviors may result in the surprising narrowness of these $2^{+-}$ hybrids.
We suggest to hunt for such hybrid states through the partial wave analyses in
the $b_{2}\to\omega/a_{1}/h_{1}/a_{2}\pi\to 4\pi$, $h_{2}\to\rho\pi\to 3\pi$
and $h_{2}\to b_{1}\pi\to 5\pi$ processes in BESIII and BelleII experiments by
using the high-statistics data samples of $\psi(3686)$.

## ACKNOWLEDGMENTS

This project is supported by the National Natural Science Foundation of China
under Grants No. 12175318, the National Key Research and Development Program
of China (2020YFA0406400), the Fundamental Research Funds for the Central
Universities.

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