††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Engineering magnetic domain wall energies in multiferroic BiFeO3 via
epitaxial strain

Sebastian Meyer<EMAIL_ADDRESS>Nanomat/Q-mat/CESAM, Université de Liège,
B-4000 Sart Tilman, Belgium  TOM/Q-mat/CESAM, Université de Liège, B-4000 Sart
Tilman, Belgium  Fonds de la Recherche Scientifique (FRS-FNRS), Bruxelles,
Belgium  Bin Xu  Jiangsu Key Laboratory of Thin Films, School of Physical
Science and Technology, Soochow University, Suzhou 215006, China  Physics
Department and Institute for Nanoscience and Engineering, University of
Arkansas, Fayetteville, Arkansas 72701, USA  Laurent Bellaiche  Physics
Department and Institute for Nanoscience and Engineering, University of
Arkansas, Fayetteville, Arkansas 72701, USA  Bertrand Dupé
Nanomat/Q-mat/CESAM, Université de Liège, B-4000 Sart Tilman, Belgium
TOM/Q-mat/CESAM, Université de Liège, B-4000 Sart Tilman, Belgium  Fonds de la
Recherche Scientifique (FRS-FNRS), Bruxelles, Belgium

(September 1, 2024)

###### Abstract

Epitaxial strain has emerged as a powerful tool to tune magnetic and
ferroelectric properties in functional materials such as in multiferroic
perovskite oxides. Here, we use first-principles calculations to explore the
evolution of magnetic interactions in the antiferromagnetic multiferroic
BiFeO3 (BFO), one of the most promising multiferroics for future technology.
The epitaxial strain in BFO(001) oriented film is varied between
$\varepsilon_{xx,yy}\in[-2\,\%,+2\,\%]$. We find that both strengths of the
exchange interaction and Dzyaloshinskii-Moriya interaction (DMI) decrease
linearly from compressive to tensile strain whereas the uniaxial
magnetocrystalline anisotropy follows a parabolic behavior which lifts the
energy degeneracy of the (111) easy plane of bulk BFO. From the trends of the
magnetic interactions we can explain the destruction of cycloidal order in
compressive strain as observed in experiments due to the increasing anisotropy
energy. For tensile strain, we predict that the ground state remains unchanged
as a function of strain. By using the domain wall (DW) energy, we envision the
region where isolated chiral magnetic texture might occur as function of
strain i.e. where the DW and the spin spiral energy are equal. This transition
between $-1.5\,\%$ and $-0.5\,\%$ of strain should allow topologically stable
magnetic states such as antiferromagnetic skyrmions and merons to occur.
Hence, our work should trigger experimental and theoretical investigations in
this range of strain.

##  I Introduction

For over a decade, topologically stable magnetic structures, such as domain
walls (DWs) and skyrmions have been studied extensively based on their
theoretical prediction in the early 90’s Bogdanov and Yablonskii (1989);
Bogdanov and Hubert (1994) and their first experimental observation 20 years
later Mühlbauer  _et al._ (2009); Yu  _et al._ (2010). Their potential in low-
power, high-density information storage, spintronic applications and their
unique stability, resistance to external magnetic fields, and efficient
dynamics make them a fascinating area of study Kiselev  _et al._ (2011); Fert
_et al._ (2013); Parkin and Yang (2015); Back  _et al._ (2020). The formation
and control of these skyrmions have primarily been explored in ferromagnetic
materials, but their antiferromagnetic counterparts are predicted to exceed
their potential Jungwirth  _et al._ (2016); Zhang  _et al._ (2016); Barker and
Tretiakov (2016); Jungwirth  _et al._ (2018). While the stabilization of
skyrmions in FMs can be achieved e.g. by the application of an external
magnetic fields Romming  _et al._ (2013); Moreau-Luchaire  _et al._ (2016);
Soumyanarayanan  _et al._ (2017); Boulle  _et al._ (2016); Hervé  _et al._
(2018); Pollard  _et al._ (2017), antiferromagnets are unaffected by those.
Therefore, the stabilization of chiral magnetic textures in antiferromagnets
remains a challenge and depends on the precise control of magnetic
interactions.

Epitaxial strain allows for precise modification of a material’s lattice
structure through heteroepitaxial growth and has emerged as a powerful tool to
engineering magnetic properties in materials. For example, it offers the means
to tailor crystal symmetries Hatt  _et al._ (2010); Dupé  _et al._ (2011),
magnetic anisotropies, and exchange interactions and even to control the
magnetic ground state of a material Sando  _et al._ (2013). In this way,
skyrmions could be stabilized in ferromagnetic (FM) systems Hu  _et al._
(2018); Wang  _et al._ (2018); Budhathoki  _et al._ (2020); Zhang  _et al._
(2021). Such a technique can pave the way to the stabilization and control of
magnetic structures such as skyrmions in antiferromagnets. Strain engineering
has already been used in multiferroic oxides, where several ferroic orders,
such as the polarization and the magnetization, can be tuned Govinden  _et
al._ (2023); Qiang  _et al._ (2020); Hemme  _et al._ (2023); Dufour  _et al._
(2023). More interestingly, their coupling called the magneto-electric
coupling is affected by strain. Since they do not only posses a magneto-
elastic, but also a magneto-electric and piezo-electric coupling, the
magnetism can be tuned in various ways. To that end, one of the most promising
multiferroic material is BiFeO3 (BFO) where ferroelectricity and
antiferromagnetism can coexist far beyond room temperature Roginskaya  _et
al._ (1966); Neaton  _et al._ (2005); Ravindran  _et al._ (2006); Kornev  _et
al._ (2007); Haumont  _et al._ (2008).

Besides being extensively studied in the bulk, the interest in strained BFO
has also risen up due to the variety of non-collinear states that it can host.
Bulk BFO has an antiferromagnetic (AFM) spin spiral ground state - the so-
called type-I cycloid of 62 nm pitch Sosnowska  _et al._ (1982, 1996). This
spin spiral can be destroyed via biaxial strain when BFO thin film is grown on
various substrates Bai  _et al._ (2005); Béa  _et al._ (2007); Zhao  _et al._
(2006); Ratcliff  _et al._ (2011); Sando  _et al._ (2013) or by uniaxial
strain Hemme  _et al._ (2023) which suggests a high magneto-elastic response.
This response can affect all magnetic interactions in BFO e.g. the magnetic
exchange, the anisotropy and the Dzyaloshinskii-Moriya interaction (DMI) whom
ratios have yet to be quantified via density functional theory (DFT). A region
of particular importance is the transition between collinear and non-collinear
magnetic ground states as function of strain because more complex isolated
magnetic textures, such as skyrmions, antiskyrmions or merons/anti-merons can
form.

In this paper, we explore the different magnetic interactions of BiFeO3 under
the influence of bi-axial strain based on DFT. Varying the nominal strain
$\varepsilon_{xx,yy}\in[-2\,\%,+2\,\%]$ from compression to elongation, we
determine the magnetic exchange interaction, Dzyaloshinskii-Moriya interaction
and the anisotropy energy. We study the influence of epitaxial strain on the
magnetism in the $Cc$ phase of BFO to identify the region of strain, where the
transition from non-collinear to collinear order takes place. We find that
both exchange and DMI decrease linearly with strain whereas the anisotropy
follows a parabolic behavior, preferring the $[11\overline{2}]$ direction for
compressive strain and the $[1\overline{1}0]$ direction for tensile strain.
Following the ratios of magnetic interactions, we conclude that with
compressive strain, collinear G-type antiferromagnetic order is enhanced due
to an increase of the anisotropy and magnetic exchange over the DMI. In the
tensile regime, the results are ambiguous. From our calculations, we expect
that the ground state spin spiral of the bulk $R3c$ phase in BFO is preserved;
however, since the energies are extremely close, small distortions or
impurities can also favor collinear order. At last we analyze the evolution of
the domain wall energy to locate the region of strain where chiral non-
collinear magnetic textures such as antiferromagnetic skyrmions and merons
could be stabilized in BiFeO3. We identify the transition between a non-
collinear and a collinear ground state in BFO in between $-1.5\,\%$ and
$-0.5\,\%$ of compressive strain where such states should occur. The presented
work should therefore trigger experimental and theoretical investigations in
that range of strain.

This paper is structured as follows: In Sec. II, we describe our procedure for
the DFT calculations. In Sec. III, the results of the DFT calculations are
presented which are discussed in Sec. IV. A conclusive statement is given in
Sec. V.

##  II Methods

###  II.1 Structural Relaxation

We have relaxed all strained structures of BiFeO3 using the Abinit package ABI
; Gonze  _et al._ (2020) and the projector augmented wave (PAW) method Blöchl
(1994). Therefore, strain
$\varepsilon=\frac{a^{\prime}-a_{\textrm{BFO}}}{a_{\textrm{BFO}}}$ is applied
from $-2\%$ to $2\%$ for the in-plane axes of the $Cc$ phase of BFO. Here,
$a^{\prime}$ is the varied in-plane lattice constant and $a_{\textrm{BFO}}$
denotes the BFO bulk lattice constant of the $R3c$ phase. Note that the $Cc$
phase is imposed at $\varepsilon=0\,\%$ with the lattice constant of $R3c$
BFO. While $\varepsilon$ is varied from compressive strain of $-2\,\%$ to
tensile strain of $+2\,\%$, the other structural parameters are relaxed
[indicated by blurred colors of the atoms and bonds, cf. sketch in Fig. 1
(a)].

We have used a $\sqrt{2}\times\sqrt{2}\times 2$ unit cell hosting 20 atoms.
The magnetic configuration is chosen as the collinear G-type antiferromagnetic
order. We have converged the forces on each atom down to $1\times
10^{-5}\,\textrm{hartree/bohr}$. The exchange and correlation functional is
treated with local spin density approximation $+U$ (LSDA$+U$ ), with a Hubbard
$U=4.0\,\textrm{eV}$ parameter and $J=0.4\,\textrm{eV}$ on the Fe atoms, which
are typical values for first-principles calculations on BFO Kornev  _et al._
(2007); Paillard  _et al._ (2016); Xu  _et al._ (2021). The wave-functions are
expanded with a plane-wave basis set using a kinetic energy cutoff of
$E_{\textrm{cut}}=30\,\textrm{hartree}$. By using a $k$-point-mesh for the
structural relaxation of $8\times 8\times 6$, we obtain a magnetic moment of
the Fe atoms of $|m_{\textrm{Fe}}|=4.0\,\mu_{B}$ for all relaxed structures.
The results of the relaxation are presented in Fig. 1 (b,c) and are in
accordance with previous studies Hatt  _et al._ (2010); Dupé  _et al._ (2011).

Figure 1: Structural properties of strained BiFeO3. (a) Sketch of strained
$Cc$ BFO in cubic representation: Bi-axial strain $\varepsilon$ is applied in
the $xy$ plane with an angle of $90^{\circ}$ between the $a$ and $b$ axis,
whereupon the other parameters ($c$ axis, monoclinic angle and all atomic
positions) are relaxed - indicated by blurred colors (for more details, see
text). (b) Relaxed lattice constant ($a$, $b$ and $c$ axes, squares on left
ordinate) vs. value of strain. Right ordinate: Unit cell volume (blue filled
circles). Note that the relaxation has been done in a
$\sqrt{2}\times\sqrt{2}\times 2$ unit cell - the values are presented within
this notation. (c) Left ordinate: Monoclinic angle of the $c$ axis (black
squares) and components of the oxygen octahedral tilts $\omega_{i}$ around the
$i$ axis (red circles for the second ordinate). Note that due to the imposed
$Cc$ phase at $0\,\%$ strain, the tilt angles around the three axes are not
exactly on top of each other due to shear strain in the $xy$ plane.

###  II.2 Determination of Magnetic Interactions

We use the relaxed structures as described above to determine the magnetic
interactions, such as the Heisenberg exchange interaction, the Dzyaloshinskii-
Moriya (DM) interaction and the uniaxial anisotropy energy for each value of
strain. The first two interactions are determined by calculating energy
dispersions of flat homogeneous spin spiral states without (for the exchange)
and with (for the DMI) spin-orbit coupling. For both interactions, we apply
the pseudo-cubic approximation to compare the results we have obtained for the
relaxed $R3c$ structure Xu  _et al._ (2021); Meyer  _et al._ (2023).

Spin spirals are the general solution of the Heisenberg model on a periodic
lattice and can be characterized by the spin spiral vector $\mathbf{q}$. This
vector determines the propagation direction of the spin spiral as well as the
canting angle between two neighboring spins. A magnetic moment
$\mathbf{m}_{i}$ at an atom position $\mathbf{r}_{i}$ is given by

$\mathbf{m}_{i}=m\Big{(}\cos(\mathbf{q}\cdot\mathbf{r}_{i}),\sin(\mathbf{q}\cdot\mathbf{r}_{i}),0\Big{)}$
(1)

where $m$ is the magnitude of the magnetic moment.

The energy dispersions $E(\mathbf{q})$ are obtained applying the full-
potential linearized augmented plane wave (FLAPW) approach Krakauer  _et al._
(1979); Wimmer  _et al._ (1981); Weinert  _et al._ (1982), as implemented in
the fleur code FLE . For all these calculations, we have used the LSDA+$U$
Vosko  _et al._ (1980) with a Hubbard $U=4.0\,\textrm{eV}$ parameter and
$J=0.4\,\textrm{eV}$ on the Fe atoms. Muffin-tin radii were set to 2.80 bohrs,
2.29 bohrs, and 1.29 bohrs for Bi, Fe, and O atoms, respectively and a large
plane-wave cutoff $k_{\textbf{max}}$ of 4.6 $\textrm{bohr}^{-1}$. These
parameters result in a magnetic moment of $m=4.0\,\mu_{\mathrm{B}}$ for all
phases in agreement with experiments Sosnowska  _et al._ (1996). Calculations
along the full paths of the Brillouin zone (BZ) without SOC have been
performed self-consistently using the generalized Bloch theorem Sandratskii
(1991) and a $k$-point mesh of $10\times 10\times 10$. To accurately determine
the energies around the magnetic ground state (R points of the BZ) at
$|\mathbf{q}|\rightarrow\textrm{R}$, the magnetic force theorem Mackintosh and
Andersen (1980); Oswald  _et al._ (1985) has been applied using a dense
$k$-point set of 64000 $k$ points (i.e., $40\times 40\times 40$). The energy
dispersion without SOC is interpreted using the Heisenberg exchange
interaction. For the energy contribution due to SOC, $\Delta
E_{\textrm{SOC}}$, we add SOC in first-order perturbation theory Heide  _et
al._ (2009); Meyer  _et al._ (2017) for every previously calculated point. The
resulting curve has been interpreted with the Dzyaloshinskii-Moriya
interaction Dzyaloshinsky (1958); Moriya (1960).

We determine the Heisenberg exchange interaction constants $J_{ij}$ beyond
nearest neighbors by mapping the Heisenberg Hamiltonian

$\mathcal{H}_{\textrm{ex}}=-\sum_{ij}J_{ij}(\mathbf{m}_{i}\cdot\mathbf{m}_{j})$
(2)

onto the energy dispersion $E(\mathbf{q})$ of flat spin spiral states
neglecting spin-orbit coupling (SOC). We include seven neighbors for the
exchange interaction using the pseudo-cubic approximation for all values of
strain. To simplify our model, we approximate the curvature of the combined
exchange interactions $J_{1,...,7}$ at $\mathbf{q}\rightarrow
R,\,|\mathbf{q}|\in[0,0.05]$ with an effective nearest neighbor exchange
interaction $J_{\textrm{eff}}$.

For the Dzyaloshinskii-Moriya (DM) interaction, we use the calculated data of
the energy contribution due to SOC $\Delta E_{\textrm{SOC}}$. Therefore, we
use the converse spin-current (sc) model,

$\mathcal{H}_{\textrm{SC}}=-\sum_{ij}C_{ij}\left(\mathbf{u}\times\mathbf{e}_{ij}\right)\cdot\left(\mathbf{m}_{i}\times\mathbf{m}_{j}\right)$
(3)

where $C_{ij}$ determines the strength of the DMI, $\mathbf{u}$ is the unit
vector along the polarization direction and $\mathbf{e}_{ij}$ is the unit
vector between magnetic sites $i,j$. We include three neighbors for the sc DMI
in pseudo-cubic approximation. For the effective DMI, $C_{\textrm{eff}}$, we
linearly map Eq (3) to the SOC contribution at $\mathbf{q}\rightarrow R$.

We also have calculated the magnetic anisotropy in BiFeO3. For that, we have
assumed the collinear G-type AFM state for each structure, where we have
applied SOC in different directions of the respective BZ. We have used
$20\times 20\times 20$ $k$-point-mesh in the whole Brillouin zone and applied
the second quantization with the Force Theorem Mackintosh and Andersen (1980)
to determine the energy differences between the hard axis ($[111]$ axis) and
other directions as shown in the results. For the description of the domain
wall energy, we will use an effective anisotropy $K_{\textrm{eff}}$ that is
the lowest energy of the respective direction with strain.

This method for calculating magnetic interaction in BFO has also been
described in detail in Refs. Meyer  _et al._ , 2023; Xu  _et al._ , 2021.

###  II.3 Error Estimation

In the following, we will analyze small energy differences that qualitatively
affect the ground state of BiFeO3. Especially for the DFT calculations, we
reach the limit of accuracy below $\sim 0.1\,\textrm{meV}$. It is impossible
to estimate the error in DFT calculations itself stemming mostly from the
exchange correlation functional simply because the amount of necessary
calculations for that is beyond the scope of any DFT study that is not
addressed to that specific problem. Hence, the absolute values in our
calculations may vary depending on the functional, we can only conclude on the
trends. The LSDA+$U$ approximation with the functional of Ref. Vosko  _et al._
, 1980 has been used previoulsy in Refs. Meyer  _et al._ , 2023; Xu  _et al._
, 2021.

For the effective magnetic interactions including a fitting procedure in the
first step, namely the exchange and the DMI, we will address the error based
on the error of the fits. For the DMI, the DFT data is perfectly linear and
the value $C_{\textrm{eff}}$ is determined with a negligible error of less
than 0.01 %. The values for the exchange interactions $J_{i}$ on the other
hand are fitted with an error $\Delta J_{i}$. This error affects our estimate
on $J_{\textrm{eff}}$ because it describes this energy dispersion curve within
$\mathbf{q}\rightarrow R$. Since the energy is the sum for all $J_{i}$, we
apply the sum rule to estimate the error on $J_{\textrm{eff}}$ as

$\Delta J_{\textrm{eff}}=\frac{1}{6}\sqrt{\sum_{i=1}^{7}(\Delta
J_{i})^{2}N_{i}}$ (4)

where $N_{i}$ are the number of neighbors in the $i$-th shell, which for
$J_{1}$ are 6 neighbors in the cubic lattice. The fraction
$\nicefrac{{1}}{{6}}$ in front of the square root accounts for the number of
the nearest neighbors.

Using multiplication or division, we will use the error estimate according to

$\Delta Q=|Q|\sqrt{\left(\frac{\Delta
J_{\textrm{eff}}}{J_{\textrm{eff}}}\right)^{2}}=|Q|\frac{\Delta
J_{\textrm{eff}}}{J_{\textrm{eff}}}$ (5)

where $Q$ is the quantity we determine.

##  III Results

Figure 2 shows the trends of magnetic interactions in BiFeO3 (BFO) under the
application of bi-axial strain.

###  III.1 Exchange Interaction

For all applied values of strain, the nearest neighbor Heisenberg exchange
interaction $J_{1}$ [black points in Fig. 2 (a)] contains the dominant
exchange contribution and prefers collinear antiferromagnetic order. Its
strength decreases linearly about $10\,\%$ from $-28\,\textrm{meV/Fe atom}$ at
$\varepsilon=-2\,\%$ to $-26\,\textrm{meV/Fe atom}$ at $\varepsilon=+2\,\%$.
The second largest contribution stems from the second neighbor
$J_{2}\approx-1.5\,\textrm{meV/Fe atom}$ (light green squares). Its strength
does not vary drastically with strain, but it leads to exchange frustration
with strain as the ratio ${J_{2}}/{J_{1}}$ increases. The largest variation
can be seen for the third neighbor exchange interaction $J_{3}$ [blue
triangles in Fig. 2 (a)]. For compressive strain, it prefers ferromagnetic
(FM) alignment whereas for tensile strain, it prefers AFM alignment. It
linearly decreases from $J_{3}(\varepsilon=-2\,\%)=1\,\textrm{meV/Fe atom}$ to
$J_{3}(\varepsilon=+2\,\%)=-1\,\textrm{meV/Fe atom}$. Neighbors beyond the
third shell in the system hold a minor contribution
($|J_{4,5,6,7}|/|J_{1}|\leq 0.03$) and do not change significantly with
strain. Even though $J_{5}$ undergoes a sign change it only appears as a
slight correction in our method.

To simplify the model, we evaluate the effective nearest neighbor exchange
$J_{\textrm{eff}}$ as shown in black circles of Fig. 2 (a). It reduces its
strength linearly with strain (from $\sim-20\,\textrm{meV/Fe atom}$ to
$\sim-15\,\textrm{meV/Fe atom}$), with a rate of $\Delta
J_{\textrm{eff}}(\varepsilon)=1.33\,\frac{\textrm{meV/Fe
atom}}{\%\textrm{strain}}$. That means that starting from BFO with no strain,
collinear order is preferred by the exchange interaction with compressive
strain, whereas it is weakened with tensile strain. This trend is in agreement
with Ref. Walden  _et al._ (2021), where the authors also find a linear trend
in the largest exchange contribution in our range of strain Not .

From Fig. 2 (a) it is evident $J_{\textrm{eff}}$ deviating from $J_{1}$. The
ratio $\mathcal{F}=J_{\textrm{eff}}/J_{1}$ is a good indicator for exchange
frustration in magnetic systems Meyer (2020), where $\mathcal{F}\sim 1$
describes a slightly exchange frustrated system and ratios for
$0<\mathcal{F}<<1$ a strongly frustrated system. For BFO, $\mathcal{F}$
decreases with strain from $\mathcal{F}(\varepsilon=-2\%)=0.75$ to
$\mathcal{F}(\varepsilon=+2\%)=0.60$. That means, that a slight exchange
frustration is induced from compressive to tensile strain. Note that for
$\mathcal{F}\sim 0.2$ Meyer  _et al._ (2019) and $\mathcal{F}\sim 0.3$ Meyer
(2020), ferromagnetic skyrmions in the absence of magnetic field have been
observed and predicted, respectively. These ratios are much smaller than in
BiFeO3 and hence, magnetic skyrmions driven by exchange frustration are not
expected to occur in BFO. In order to be stabilized, skyrmions and other
topological magnetic structures rely on the interplay of magnetic exchange,
DMI and anisotropy.

Figure 2: Trend of magnetic interactions in BiFeO3 under bi-axial strain. (a)
Magnetic Heisenberg exchange interaction parameters $J_{1\dots 7}$ from
pseudo-cubic approximation for different values of strain. Note that the
nearest neighbor exchange $J_{1}$ is dominant and the effective nearest
neighbor exchange $J_{\textrm{eff}}$ matches the curvature of a energy
dispersion at $\mathbf{q}\rightarrow R$. The error on $J_{\textrm{eff}}$ is
stemming from $\Delta J_{i}$ as described in the text. (b) Evaluated spin-
current driven Dzyaloshinskii-Moriya interaction ($C_{1\dots 3}$ for the first
three nearest neighbors and in effective nearest-neighbor approximation
$C_{\textrm{eff}}$), (c) Energy differences for uniaxial anisotropy
calculations of different magnetization directions with respect to the [111]
direction - the inset shows a sketch with all directions in pseudo-cubic
approximation. Note that in the absence of strain, magnetization directions
within the (111) plane are energetically degenerate. In all graphs, dashed
lines serve as guide to the eye, bold solid lines
$J_{\textrm{eff}}(\varepsilon)$, $C_{\textrm{eff}}(\varepsilon)$ and
$K_{[uvw]}(\varepsilon)$ are linear fits for $J_{\textrm{eff}}$,
$C_{\textrm{eff}}$ and quadratic fits for the preferred anisotropy directions
respectively.

###  III.2 Dzyaloshinskii-Moriya Interaction

We investigate the behavior of the DMI under strain in Fig. 2 (b). As we have
shown for bulk BFO Meyer  _et al._ (2023), we have used the converse spin-
current model Katsura  _et al._ (2005); Rahmedov  _et al._ (2012) to describe
our DFT results for the DMI [Eq. (3)]. $C_{1}$ (black points) is decreasing
almost linearly from compressive to tensile strain, the variation of $C_{2}$
(green squares) and $C_{3}$ (blue triangles) is rather small; however, they
have opposite signs at almost the same magnitude for each value of strain.
Effectively, the DMI (black circles) decreases with strain from
$C_{\textrm{eff}}=0.33\,\textrm{meV/Fe atom}$ at $\varepsilon=-2\,\%$ to
$C_{\textrm{eff}}=0.25\,\textrm{meV/Fe atom}$ at $\varepsilon=+2\,\%$. This is
a change rate of about $\Delta
C_{\textrm{eff}}(\varepsilon)=-0.02\,\frac{\textrm{meV/Fe
atom}}{\%\textrm{strain}}$. That means that compared to $R3c$ BFO, the DMI
increases with compression preferring non-collinear order and weakens with
tensile strain which lowers the energy for non-collinear order. This is
opposite to the trend of the exchange interaction, hence it is important to
compare their ratios.

###  III.3 Magnetocrystalline Anisotropy

As third magnetic interaction, we focus on the magnetocrystalline anisotropy
[Fig. 2 (c)]. We determine the energy of different crystallographic directions
with respect to the hard [111] axis. In Ref. Meyer  _et al._ , 2023, we have
shown that in $R3c$ bulk BFO, the magnetocrystalline anisotropy energy is
lowest and degenerate within the full (111) plane. Here, in the absence of
strain ($\varepsilon=0\,\%$), we see the same phenomenon: The calculated
directions $[1\overline{1}0]$ (black points), $[11\overline{2}]$ (light green
squares), $10\overline{1}$ (purple pentagons) and $[1\overline{2}1]$ (orange
upside-down triangles) have the same energy at $-0.07\,\textrm{meV/Fe atom}$
(Note that a tiny deviation is caused by the imposed $Cc$ phase at 0% of
strain). The $[100]$ and the $[001]$ directions (dark green rhombus and
lighter green x, respectively) are slightly higher in energy.

Applying strain changes this behavior and the energy degeneracy of the
different directions is lifted. For tensile strain $\varepsilon>0$, the
$[1\overline{1}0]$ direction is almost constant in energy while the other
directions become unfavorable, most of all the $[11\overline{2}]$ direction.
For compressive strain, the effect is reversed. The $[1\overline{1}0]$
direction gains the least energy, whereas the $[11\overline{2}]$ direction
becomes the preferred magnetization direction by the anisotropy. The value
increases from $-0.07\,\textrm{meV/Fe atom}$ without strain by $40\,\%$ to
$-0.1\,\textrm{meV/Fe atom}$ at $\varepsilon=-2\,\%$. We focus on the two
directions with lowest energies, $[11\overline{2}]$ and $[1\overline{1}0]$.
The behavior of their energies with respect to applied strain differs
qualitatively from those of the exchange and DMI of panel (a,b). Here, we can
apply a quadratic fit to describe the trend of the anisotropy with strain
$K_{[uvw]}(\varepsilon)$. Following the trend for the two interactions, for
larger compressive strains, the $[1\overline{1}0]$ direction will be lower in
energy than the $[11\overline{2}]$ direction - in accordance with DFT
calculations of Ref. Chen  _et al._ (2018) where the authors also find an easy
(111) plane for 0 % of strain and the easy axis for large compression.
However, in the low strain regime, our results slightly differ.

With respect to the $R3c$ phase of BFO, compression leads to an increased MAE,
which favors the collinear order of the G-type AFM state. For tensile strain,
the MAE remains similar to that of bulk BFO, so a change of the magnetic
ground state is not expected from the MAE. In the following, we will discuss
the magnetic ground states we derive from our calculations, compare them to
current experimental results and show which regime of strain could stabilize
potential topological magnetic structures.

##  IV Discussion

For over a decade, the different magnetic textures in strained BiFeO3 have
been intensively investigated in experiments. However, while several studies
show a destruction of the cycloidal order with large compressive strain of BFO
on SrTiO3 substrates (estimated strain of $-1.5\,\%<\varepsilon<-0.5\,\%$
depending on the thickness of BFO) Bai  _et al._ (2005); Béa  _et al._ (2007);
Zhao  _et al._ (2006); Infante  _et al._ (2010); Ratcliff  _et al._ (2011);
Sando  _et al._ (2013, 2014), others report the persistence of cycloidal order
in the same system Ke  _et al._ (2010); Haykal  _et al._ (2020). For tensile
strain, the literature is not equivalently vast, but the destruction of the
cycloid with large tensile strain has also been observed Sando  _et al._
(2013); Haykal  _et al._ (2020). There, the amount of applied strain to
stabilize collinear order differs by $\sim 1\,\%$. These results show that the
magnetic ground state in strained BFO is very sensitive to the thickness and
growth conditions. Different cycloids, such as the type-I and type-II cycloids
in $[1\overline{1}0]$ and $[11\overline{2}]$ directions, respectively as well
as the collinear G-type AFM states are extremely close in energy. Hence, small
distortions have a large impact on the magnetic ground state. As we will show
in the following, this behaviour is in agreement with our calculations because
different magnetic states are within energy ranges down to 10$\mu$eV/atom.

In general, the magnetic ground state is driven by the interplay between the
magnetic exchange $J$, the DMI $C$ and the anisotropy $K$. For example, when
$K$ is large as compared to $J$ and $C$ a FM or AFM ground state is favored
depending on the sign of $J$. When $C$ is large, a non-collinear ground state
is favored since the DM favors 90∘ spin spirals. The ratio between the
different contributions is therefore important. We compare these interactions
in Figure 3. In panel (a), the ratio of
$|J_{\textrm{eff}}|/|C_{\textrm{eff}}|$ is shown in grey (squares show the
ratios of DFT determined magnetic interactions - the bold line is the ratio of
the fitted interactions with strain). Increasing the ratios means that
collinear order is enhanced, decreasing ratios destabilize collinear magnetic
ground states.

Applying compressive strain, the exchange interaction slightly increases with
respect to the DMI - collinear G-type AFM order is more favorable than in the
absence of strain. With tensile strain, the ratio lowers - hence, non-
collinear order gains energy. The anisotropy [panel (b)] increases for
compressive strain from about 24 % to almost 29 % of the DMI due to the
increasing preference of the $[11\overline{2}]$ direction shown in Fig. 2 (c).
This enhances the stability of collinear order with a quantization axis along
$[11\overline{2}]$. With tensile strain, the preferred magnetization direction
changes to $[1\overline{1}0]$ with a constant value of the anisotropy. As the
DMI decreases, the ratio $\nicefrac{{K}}{{C}}$ increases and will also
stabilize collinear magnetic order. Note that the increase of anisotropy with
respect to DMI starting from 0 % of strain differs for compressive and tensile
strain - e.g. a ratio of 0.26 is reached at $\sim+1.5\,\%$ within the tensile
regime, but at around $\sim-0.75\,\%$ in the compressive regime. Due to both
increasing $\nicefrac{{$J$}}{{$C$}}$ and $\nicefrac{{$K$}}{{$C$}}$ with
compression, we expect collinear order to be favored even in low strain
regimes compared to the $R3c$ bulk BFO phase. For tensile strain, the
conclusion is not that unambiguous: while the exchange loses energy with
respect to the DMI, the anisotropy gains energy. The magnetic behavior
crucially depends on both trends. In Fig. 2, the effective exchange and DMI
depend linearly on the nominal strain, but the anisotropy can be described
with a quadratic behavior.

Figure 3: Interplay between the magnetic interactions for different values of
strain. (a) Ratio of magnetic exchange $J_{\textrm{eff}}$ with respect to the
Dzyaloshinskii-Moriya interaction $C_{\textrm{eff}}$. Note that the large
error estimate stems solely from the small error $\Delta J_{\textrm{eff}}$.
The squares show the values, determined from DFT whereas the solid line shows
the ratio
$\nicefrac{{$J_{\textrm{eff}}(\varepsilon)$}}{{$C_{\textrm{eff}}(\varepsilon)$}}$
from the fits of panel (a) and (b) of Fig. 2. (b) Ratio between anisotropy
energy $K_{\textrm{eff}}$ and $C_{\textrm{eff}}$. The circles show the values,
determined from DFT whereas the solid line shows the ratio
$\nicefrac{{$K_{\textrm{eff}}(\varepsilon)$}}{{$C_{\textrm{eff}}(\varepsilon)$}}$
from the fits of panel (c) and (b) of Fig. 2, where $K_{\textrm{eff}}$ denotes
the direction of lowest energy. (c) Domain wall energy $E_{\textrm{DW}}$ [Eq.
(6)] as a function of applied strain. The blue points are derived from the DFT
values and the blue solid line shows the trend from the fits of
$J_{\textrm{eff}}(\varepsilon)$, $C_{\textrm{eff}}(\varepsilon)$ and
$K_{\textrm{eff}}(\varepsilon)$. The red point shows the determined value for
the $R3c$ phase of BFO according to Ref. Xu  _et al._ (2021) where the red
line denotes the trend which is renormalized to that point.

To quantify where isolated chiral magnetic textures may be stable, we have
used the domain wall energy, defined as Craik (2003); de Lacheisserie  _et
al._ (2005)

$E_{\textrm{DW}}(\varepsilon)=\frac{4}{a^{2}(\varepsilon)}\sqrt{J_{\textrm{eff}}(\varepsilon)\cdot
K_{\textrm{eff}}(\varepsilon)}-\frac{2\pi}{a^{2}(\varepsilon)}C_{\textrm{eff}}(\varepsilon),$
(6)

with $a(\varepsilon)$ being the pseudo-cubic lattice constant for each value
of strain, taken from the diagonal [111] direction to address for the changes
in the $c$ axis due to relaxation. We also renormalized $a$ into the
$[1\overline{1}0]$ direction to account for the known type-I cycloid. As
compared to other quantities such as the isolated skyrmion energies or vortex
energies, the DW energy is independent of the size of the magnetic textures
and therefore does not require the expensive spin dynamics energy minimization
in a magnetic supercell of at least 100$\times$100 nm at least 111Note that
compared to this manuscript, in Li  _et al._ (2023), the authors assume a
[111] directed anisotropy which exceeds the bulk value by a factor of 3 to
stabilize magnetic skyrmions in BFO. Here, we have calculated the anisotropy
based on DFT and only find an easy axis within (111) plane.. The DW energy is
directly quantifying the energy cost or gain of non collinear textures based
on the effective exchange, the anisotropy and the DMI. Furthermore, a domain
wall can be approximated as half of a cycloid, where for positive values of
energy, collinear magnetic order is preferred and for negative values of
$E_{\textrm{DW}}$, a spin spiral ground state is expected 222Note that
contrary to the spin spiral pitch $\lambda$, the DW width
$\Delta\propto\sqrt{J/K}$ is independent of the DMI, hence for spin spirals
with a certain pitch length the quantitative description differs as well as
its energy. However, due to the simplicity of Eq. (6), the trends for the
magnetic ground state energies can be recognized directly and give
qualitatively the same results as spin spiral energies..

In Fig. 3 (c), the DW energy with respect to the applied strain is shown. For
all values of strain, the DWs are higher in energy than the collinear state
(blue points with error bars); however, the energies are extremely small
(below 0.1 meV). As described above, compressing BFO prefers collinear order,
i.e. an increased DW energy. For tensile strain, the energy differences are
almost zero, meaning that we predict a similar ground state compared to the
magnetic state at 0 % strain. The trend coincides with experimental
observations: For compressive strain, the collinear G-type AFM state is
preferred, driven by the large increase of anisotropy in the
$[11\overline{2}]$ direction and the increase in exchange with respect to the
DMI. This result is robust against small errors within our calculated
interactions. For tensile strain, our results show the same difficulties as
for experimental observations: The energies of the states are so close that
slightly differing strains, the magnetic ground state can differ (cf. Refs.
Sando  _et al._ (2013) and Haykal  _et al._ (2020)). This is indicated by the
error bars from our calculations.

As a reference point in red, we plot the estimated DW energy from Ref. Xu  _et
al._ (2021), where we investigated the $R3c$ phase of BFO and slightly adapted
the DFT parameters to obtain a spin cycloid with a pitch of about $\sim
62\,\textrm{nm}$. With these parameters (corresponding to
$C_{\textrm{eff}}=0.66\,\textrm{meV/Fe atom}$,
$K_{\textrm{eff}}=0.069\,\textrm{meV/Fe atom}$ and
$J_{\textrm{eff}}=-11.78\,\textrm{meV/Fe atom}$), the DW energy is slightly
negative ($-0.017\,\nicefrac{{\textrm{meV}}}{{\AA}}$). Following our estimated
trend for the magnetic interactions by renormalizing the DW energy from fits
(red curve), we can expect a collinear G-type AFM from $\sim-1.5\,\%$ of
compressive strain and a cycloidal ground state for low compressive strains.
In between these regimes, at compressive strain
$-1.5\,\%<\varepsilon_{xx,yy}<-0.5\,\%$, we expect the transition between non-
collinear and collinear order for BFO. This is the region where topologically
stable magnetic structures, such as isolated skyrmions or merons and anti-
merons could be stabilized in BFO. An investigation of those state is beyond
the scope of this paper and part of a different study.

##  V Conclusion

We have investigated the evolution of magnetic interactions in BiFeO3 under
bi-axial strain between $-2\,\%<\varepsilon_{xx,yy}<+2\,\%$ using first-
principles calculations. Going from compressive to tensile strain, both
exchange and Dzyaloshinskii-Moriya interaction decrease linearly whereas the
uniaxial anisotropy prefers a $[11\overline{2}]$ magnetization direction for
the calculated range of compressive strain and a $[1\overline{1}0]$ axis for
tensile strain. Comparing the ratios of the three interactions, collinear
G-type antiferromagnetic order in BiFeO3 is strengthened for compressive
strain due to a large increase of anisotropy by about 40 % in accordance with
experimental observations. For tensile strain our results show a similar
magnetic ground state as for bulk BFO, where for a slight change in the
ratios, also the collinear ground state can be favored. At the transition
between the cycloid and the collinear AFM state for medium compressive strain
more complex magnetic structures such as skyrmions or merons might occur which
is why we believe this regime to be interesting for further investigation.

###### Acknowledgements.

B.D., S.M. thank Prof. Philippe Ghosez, Louis Bastogne, Dr. Subhadeep
Bandyopadhyay, Dr. He Xu and Prof. Matthieu J. Verstraete for helpful
discussions. This work is supported by the National Natural Science Foundation
of China under Grant No. 12074277, the startup fund from Soochow University
and the support from Priority Academic Program Development (PAPD) of Jiangsu
Higher Education Institutions. S.M., B.D., and L.B. acknowledge the DARPA
Grant No. HR0011727183-D18AP00010 (TEE Program) and the European Union’s
Horizon 2020 research and innovation programme under Grant Agreements No.
964931 (TSAR). L.B. also thanks ARO Grant No. W911NF-21-1-0113 and ARO Grant
No. W911NF-21-2-0162 (ETHOS). Computing time was provided by ARCHER and
ARCHER2 based in the United Kingdom at National Supercomputing Service with
support from the PRACE aisbl, from the Consortium d’Équipements de Calcul
Intensif (FRS-FNRS Belgium GA 2.5020.11) and the LUMI CECI/Belgium for
awarding this project access to the LUMI supercomputer, owned by the EuroHPC
Joint Undertaking, hosted by CSC (Finland) and the LUMI consortium through
LUMI CECI/Belgium, ULiege-NANOMAT-SKYRM-1. Sebastian Meyer is a Postdoctoral
Researcher [CR] of the Fonds de la Recherche Scientifique – FNRS. Bertrand
Dupé is a Research Associate [CQ] of the Fonds de la Recherche Scientifique –
FNRS.

## References

  * Bogdanov and Yablonskii (1989) A. Bogdanov and D. Yablonskii, Zh. Eksp. Teor. Fiz 95, 178 (1989).
  * Bogdanov and Hubert (1994) A. Bogdanov and A. Hubert, Journal of Magnetism and Magnetic Materials 138, 255 (1994).
  * Mühlbauer  _et al._ (2009) S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Boni, Science 323, 915 (2009).
  * Yu  _et al._ (2010) X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010).
  * Kiselev  _et al._ (2011) N. S. Kiselev, a. N. Bogdanov, R. Schäfer, and U. K. Rößler, Journal of Physics D: Applied Physics 44, 392001 (2011).
  * Fert  _et al._ (2013) A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnology 8, 152 (2013).
  * Parkin and Yang (2015) S. Parkin and S.-H. Yang, Nature Nanotechnology 10, 195 (2015).
  * Back  _et al._ (2020) C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert, M. Garst, T. Ma, S. Mankovsky, T. L. Monchesky, M. Mostovoy, N. Nagaosa, S. S. P. Parkin, C. Pfleiderer, N. Reyren, A. Rosch, Y. Taguchi, Y. Tokura, K. von Bergmann, and J. Zang, Journal of Physics D: Applied Physics 53, 363001 (2020).
  * Jungwirth  _et al._ (2016) T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016).
  * Zhang  _et al._ (2016) X. Zhang, Y. Zhou, and M. Ezawa, Scientific Reports 6, 24795 (2016).
  * Barker and Tretiakov (2016) J. Barker and O. A. Tretiakov, Physical Review Letters 116, 147203 (2016).
  * Jungwirth  _et al._ (2018) T. Jungwirth, J. Sinova, A. Manchon, X. Marti, J. Wunderlich, and C. Felser, Nature Physics 14, 200 (2018).
  * Romming  _et al._ (2013) N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Science 341, 636 (2013).
  * Moreau-Luchaire  _et al._ (2016) C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M. George, M. Weigand, J. Raabe, V. Cros, and A. Fert, Nature Nanotechnology 11, 444 (2016).
  * Soumyanarayanan  _et al._ (2017) A. Soumyanarayanan, M. Raju, A. L. Gonzalez Oyarce, A. K. C. Tan, M.-Y. Im, A. P. Petrović, P. Ho, K. H. Khoo, M. Tran, C. K. Gan, F. Ernult, and C. Panagopoulos, Nature Materials 16, 898 (2017).
  * Boulle  _et al._ (2016) O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de Souza Chaves, A. Locatelli, T. O. Menteş, A. Sala, L. D. Buda-Prejbeanu, O. Klein, M. Belmeguenai, Y. Roussigné, A. Stashkevich, S. M. Chérif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I. M. Miron, and G. Gaudin, Nature Nanotechnology 11, 449 (2016).
  * Hervé  _et al._ (2018) M. Hervé, B. Dupé, R. Lopes, M. Böttcher, M. D. Martins, T. Balashov, L. Gerhard, J. Sinova, and W. Wulfhekel, Nature Communications 9, 1015 (2018).
  * Pollard  _et al._ (2017) S. D. Pollard, J. A. Garlow, J. Yu, Z. Wang, Y. Zhu, and H. Yang, Nature Communications 8, 14761 (2017).
  * Hatt  _et al._ (2010) A. J. Hatt, N. A. Spaldin, and C. Ederer, Physical Review B: Condensed Matter and Materials Physics 81, 54109 (2010).
  * Dupé  _et al._ (2011) B. Dupé, S. Prosandeev, G. Geneste, B. Dkhil, and L. Bellaiche, Physical review letters 106, 237601 (2011).
  * Sando  _et al._ (2013) D. Sando, A. Agbelele, D. Rahmedov, J. Liu, P. Rovillain, C. Toulouse, I. C. Infante, A. P. Pyatakov, S. Fusil, E. Jacquet, C. Carrétéro, C. Deranlot, S. Lisenkov, D. Wang, J.-M. L. Breton, M. Cazayous, A. Sacuto, J. Juraszek, A. K. Zvezdin, L. Bellaiche, B. Dkhil, A. Barthélémy, and M. Bibes, Nature Materials 12, 641 (2013).
  * Hu  _et al._ (2018) J. M. Hu, T. Yang, and L. Q. Chen, npj Computational Materials 2018 4:1 4, 1 (2018).
  * Wang  _et al._ (2018) J. Wang, Y. Shi, and M. Kamlah, Phys. Rev. B 97, 024429 (2018).
  * Budhathoki  _et al._ (2020) S. Budhathoki, A. Sapkota, K. M. Law, S. Ranjit, B. Nepal, B. D. Hoskins, A. S. Thind, A. Y. Borisevich, M. E. Jamer, T. J. Anderson, A. D. Koehler, K. D. Hobart, G. M. Stephen, D. Heiman, T. Mewes, R. Mishra, J. C. Gallagher, and A. J. Hauser, Phys. Rev. B 101, 220405(R) (2020).
  * Zhang  _et al._ (2021) Y. Zhang, J. Liu, Y. Dong, S. Wu, J. Zhang, J. Wang, J. Lu, A. Rückriegel, H. Wang, R. Duine, H. Yu, Z. Luo, K. Shen, and J. Zhang, Phys. Rev. Lett. 127, 117204 (2021).
  * Govinden  _et al._ (2023) V. Govinden, P. Tong, X. Guo, Q. Zhang, S. Mantri, M. M. Seyfouri, S. Prokhorenko, Y. Nahas, Y. Wu, L. Bellaiche, T. Sun, H. Tian, Z. Hong, N. Valanoor, and D. Sando, Nature Communications 14, 4178 (2023).
  * Qiang  _et al._ (2020) B. W. Qiang, N. Togashi, S. Momose, T. Wada, T. Hajiri, M. Kuwahara, and H. Asano, Applied Physics Letters 117, 142401 (2020).
  * Hemme  _et al._ (2023) P. Hemme, J.-C. Philippe, A. Medeiros, A. Alekhin, S. Houver, Y. Gallais, A. Sacuto, A. Forget, D. Colson, S. Mantri, B. Xu, L. Bellaiche, and M. Cazayous, Phys. Rev. Lett. 131, 116801 (2023).
  * Dufour  _et al._ (2023) P. Dufour, A. Abdelsamie, J. Fischer, A. Finco, A. Haykal, M. F. Sarott, S. Varotto, C. Carrétéro, S. Collin, F. Godel, N. Jaouen, M. Viret, M. Trassin, K. Bouzehouane, V. Jacques, J.-Y. Chauleau, S. Fusil, and V. Garcia, Nano Letters 23, 9073 (2023).
  * Roginskaya  _et al._ (1966) Y. E. Roginskaya, Y. Y. Tomashpol’Skiǐ, Y. N. Venevtsev, V. M. Petrov, and G. S. Zhdanov, Soviet Physics JETP 23, 47 (1966).
  * Neaton  _et al._ (2005) J. B. Neaton, C. Ederer, U. V. Waghmare, N. A. Spaldin, and K. M. Rabe, Physical Review B: Condensed Matter and Materials Physics 71, 014113 (2005).
  * Ravindran  _et al._ (2006) P. Ravindran, R. Vidya, A. Kjekshus, H. Fjellvåg, and O. Eriksson, Physical Review B: Condensed Matter and Materials Physics 74, 224412 (2006).
  * Kornev  _et al._ (2007) I. A. Kornev, S. Lisenkov, R. Haumont, B. Dkhil, and L. Bellaiche, Phys. Rev. Lett. 99, 227602 (2007).
  * Haumont  _et al._ (2008) R. Haumont, I. Kornev, S. Lisenkov, L. Bellaiche, J. Kreisel, and B. Dkhil, Physical Review B: Condensed Matter and Materials Physics 78, 134108 (2008).
  * Sosnowska  _et al._ (1982) I. Sosnowska, T. P. Neumaier, and E. Steichele, J. Phys. C Solid State Phys. 15, 4835 (1982).
  * Sosnowska  _et al._ (1996) I. Sosnowska, R. Przeniosło, P. Fischer, and V. Murashov, Journal of Magnetism and Magnetic Materials 160, 384 (1996).
  * Bai  _et al._ (2005) F. Bai, J. Wang, M. Wuttig, J. Li, N. Wang, A. P. Pyatakov, A. K. Zvezdin, L. E. Cross, and D. Viehland, Applied Physics Letters 86, 032511 (2005).
  * Béa  _et al._ (2007) H. Béa, M. Bibes, S. Petit, J. Kreisel, and A. Barthélémy, Philosophical Magazine Letters 87, 165 (2007).
  * Zhao  _et al._ (2006) T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A. Doran, M. P. Cruz, Y. H. Chu, C. Ederer, N. A. Spaldin, R. R. Das, D. M. Kim, S. H. Baek, C. B. Eom, and R. Ramesh, Nature Materials 5, 823 (2006).
  * Ratcliff  _et al._ (2011) W. Ratcliff, D. Kan, W. Chen, S. Watson, S. Chi, R. Erwin, G. J. McIntyre, S. C. Capelli, and I. Takeuchi, Advanced Functional Materials 21, 1567 (2011).
  * (41) www.abinit.org.
  * Gonze  _et al._ (2020) X. Gonze, B. Amadon, G. Antonius, F. Arnardi, L. Baguet, J.-M. Beuken, J. Bieder, F. Bottin, J. Bouchet, E. Bousquet, N. Brouwer, F. Bruneval, G. Brunin, T. Cavignac, J.-B. Charraud, W. Chen, M. Côté, S. Cottenier, J. Denier, G. Geneste, P. Ghosez, M. Giantomassi, Y. Gillet, O. Gingras, D. R. Hamann, G. Hautier, X. He, N. Helbig, N. Holzwarth, Y. Jia, F. Jollet, W. Lafargue-Dit-Hauret, K. Lejaeghere, M. A. Marques, A. Martin, C. Martins, H. P. Miranda, F. Naccarato, K. Persson, G. Petretto, V. Planes, Y. Pouillon, S. Prokhorenko, F. Ricci, G.-M. Rignanese, A. H. Romero, M. M. Schmitt, M. Torrent, M. J. van Setten, B. Van Troeye, M. J. Verstraete, G. Zérah, and J. W. Zwanziger, Computer Physics Communications 248, 107042 (2020).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Paillard  _et al._ (2016) C. Paillard, B. Xu, B. Dkhil, G. Geneste, and L. Bellaiche, Phys. Rev. Lett. 116, 247401 (2016).
  * Xu  _et al._ (2021) B. Xu, S. Meyer, M. J. Verstraete, L. Bellaiche, and B. Dupé, Phys. Rev. B 103, 214423 (2021).
  * Meyer  _et al._ (2023) S. Meyer, B. Xu, M. J. Verstraete, L. Bellaiche, and B. Dupé, Phys. Rev. B 108, 024403 (2023).
  * Krakauer  _et al._ (1979) H. Krakauer, M. Posternak, and A. J. Freeman, Physical Review B 19, 1706 (1979).
  * Wimmer  _et al._ (1981) E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Physical Review B 24, 864 (1981).
  * Weinert  _et al._ (1982) M. Weinert, E. Wimmer, and A. J. Freeman, Physical Review B 26, 4571 (1982).
  * (50) www.flapw.de.
  * Vosko  _et al._ (1980) S. H. Vosko, L. Wilk, and M. Nusair, Canadian Journal of Physics 58, 1200 (1980).
  * Sandratskii (1991) L. M. Sandratskii, Journal of Physics: Condensed Matter 3, 8565 (1991).
  * Mackintosh and Andersen (1980) A. Mackintosh and O. Andersen, ed. M. Springford, Cambridge Univ. Press, London , 149 (1980).
  * Oswald  _et al._ (1985) A. Oswald, R. Zeller, P. J. Braspenning, and P. H. Dederichs, Journal of Physics F: Metal Physics 15, 193 (1985).
  * Heide  _et al._ (2009) M. Heide, G. Bihlmayer, and S. Blügel, Physica B: Condensed Matter 404, 2678 (2009).
  * Meyer  _et al._ (2017) S. Meyer, B. Dupé, P. Ferriani, and S. Heinze, Phys. Rev. B 96, 094408 (2017).
  * Dzyaloshinsky (1958) I. Dzyaloshinsky, Journal of Physics and Chemistry of Solids 4, 241 (1958).
  * Moriya (1960) T. Moriya, Physical Review 120, 91 (1960).
  * Walden  _et al._ (2021) M. R. Walden, C. V. Ciobanu, and G. L. Brennecka, Journal of Applied Physics 130, 104102 (2021).
  * (60) Note that in Walden  _et al._ (2021), the authors distinguish between in-plane and out-of-plane components of the exchange, whereas we use a pseudo-cubic approximation.
  * Meyer (2020) S. Meyer,  _Complex spin structures in frustrated ultrathin films_ , Ph.D. thesis (2020).
  * Meyer  _et al._ (2019) S. Meyer, M. Perini, S. von Malottki, A. Kubetzka, R. Wiesendanger, K. von Bergmann, and S. Heinze, Nat. Commun. 10, 3823 (2019).
  * Katsura  _et al._ (2005) H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005).
  * Rahmedov  _et al._ (2012) D. Rahmedov, D. Wang, J. Íñiguez, and L. Bellaiche, Phys. Rev. Lett. 109, 037207 (2012).
  * Chen  _et al._ (2018) Z. Chen, Z. Chen, C.-Y. Kuo, Y. Tang, L. R. Dedon, Q. Li, L. Zhang, C. Klewe, Y.-L. Huang, B. Prasad, A. Farhan, M. Yang, J. D. Clarkson, S. Das, S. Manipatruni, A. Tanaka, P. Shafer, E. Arenholz, A. Scholl, Y.-H. Chu, Z. Q. Qiu, Z. Hu, L.-H. Tjeng, R. Ramesh, L.-W. Wang, and L. W. Martin, Nature Communications 9, 3764 (2018).
  * Infante  _et al._ (2010) I. C. Infante, S. Lisenkov, B. Dupé, M. Bibes, S. Fusil, E. Jacquet, G. Geneste, S. Petit, A. Courtial, J. Juraszek, L. Bellaiche, A. Barthélémy, and B. Dkhil, Phys. Rev. Lett. 105, 057601 (2010).
  * Sando  _et al._ (2014) D. Sando, A. Agbelele, C. Daumont, D. Rahmedov, W. Ren, I. C. Infante, S. Lisenkov, S. Prosandeev, S. Fusil, E. Jacquet, C. Carrétéro, S. Petit, M. Cazayous, J. Juraszek, J.-M. L. Breton, L. Bellaiche, B. Dkhil, A. Barthélémy, and M. Bibes, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, 20120438 (2014).
  * Ke  _et al._ (2010) X. Ke, P. P. Zhang, S. H. Baek, J. Zarestky, W. Tian, and C. B. Eom, Physical Review B 82, 134448 (2010).
  * Haykal  _et al._ (2020) A. Haykal, J. Fischer, W. Akhtar, J.-Y. Chauleau, D. Sando, A. Finco, F. Godel, Y. A. Birkhölzer, C. Carrétéro, N. Jaouen, M. Bibes, M. Viret, S. Fusil, V. Jacques, and V. Garcia, Nature Communications 11, 1704 (2020).
  * Craik (2003) D. J. Craik,  _Magnetism: principles and applications_ (2003).
  * de Lacheisserie  _et al._ (2005) E. D. T. de Lacheisserie, D. Gignoux, and M. Schlenker,  _Magnetism_ , Vol. 1 (Springer Science & Business Media, 2005).
  * Note (1) Note that compared to this manuscript, in Li  _et al._ (2023), the authors assume a [111] directed anisotropy which exceeds the bulk value by a factor of 3 to stabilize magnetic skyrmions in BFO. Here, we have calculated the anisotropy based on DFT and only find an easy axis within (111) plane.
  * Note (2) Note that contrary to the spin spiral pitch $\lambda$, the DW width $\Delta\propto\sqrt{J/K}$ is independent of the DMI, hence for spin spirals with a certain pitch length the quantitative description differs as well as its energy. However, due to the simplicity of Eq. (6), the trends for the magnetic ground state energies can be recognized directly and give qualitatively the same results as spin spiral energies.
  * Li  _et al._ (2023) Z. Li, T. Chirac, J. Tranchida, V. Garcia, S. Fusil, V. Jacques, J.-Y. Chauleau, and M. Viret, Phys. Rev. Res. 5, 043109 (2023).


# Heavy Flavour Spectroscopy

Chang-Zheng Yuan
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
University of Chinese Academy of Sciences, Beijing

(17-21 July 2023)

###### Abstract

The discovery of hadronic states beyond the conventional two-quark meson and
three-quark baryon picture in the last two decades is one of the most amazing
accomplishments in fundamental physics research. We review the experimental
progress on the study of the exotic states (also known as the $\rm XYZ$
particles) beyond the conventional quark model. We give a general review and
then focus on the lineshape measurement of the $X(3872)$, observation of new
decay modes of the $Y(4230)$ and new vector charmoniumlike states $Y(4500)$
and $Y(4790)$, evidence for the neutral isospin partners of the charged
charmoniumlike $Z_{cs}$ states, discoveries of the tetraquark state candidates
with four different flavours or two-pairs of charm-anticharm quarks and the
pentaquark states.

Introduction: Hadron spectroscopy is a field of frequent discoveries and
surprises, and the theoretical difficulties in understanding the strong
interaction in the color-confinement regime make the field even more
fascinating. The tremendous data collected by the BaBar, Belle, BESIII, LHCb,
and other experiments and improved theoretical tools developed to analyze the
experimental data result in rapid progress of the field.

In the conventional quark model, mesons are composed of one quark and one
anti-quark, while baryons are composed of three quarks. However, many
quarkoniumlike states were discovered at two $B$-factories BaBar and Belle [2]
in the first decade of the 21st century. Whereas some of these are good
candidates of quarkonium states, many other states have exotic properties,
which may indicate that exotic states, such as multi-quark state, hadronic
molecule, or hybrid, have been observed [3].

BaBar and Belle experiments finished their data taking in 2008 and 2010,
respectively, and the data are still used for various physics analyses. BESIII
[4] and LHCb [5] experiments started data taking and contributed to the study
of exotic hadrons since 2008. Most of the discoveries of the such states were
made at these four experiments.

Figure 1 shows the history of the discovery of some of the new hadrons,
started from the observation of the $X(3872)$ in 2003 [6]. In this article, we
present recent experimental progress and focus on those states with exotic
properties, including the $X(3872)$, $Y(4260)$, $Z_{c}(3900)$, $P_{c}$ and
their siblings.

Figure 1: Discovery of some heavy exotic states from experiments.

The lineshape of the $X(3872)$:  The $X(3872)$ was observed in 2003 by the
Belle experiment [6], and confirmed very soon by the CDF [7] and $D0$ [8]
experiments in $p\bar{p}$ collision.

The mass of the $X(3872)$ has been measured as $3871.65\pm 0.06$ MeV [9],
which is lower than the mass threshold of $\bar{D}^{0}D^{*0}$, $3871.69\pm
0.11$ MeV, by $0.04\pm 0.12$ MeV, to be compared with the bounding energy of
the deuteron of 2.2 MeV.

The width measurements are less precise and model dependent since the
$X(3872)$ is very narrow and the mass resolution of the experiments is usually
much larger than the intrinsic width. Fitting the $\pi^{+}\pi^{-}J/\psi$
invariant mass distribution with a Breit-Wigner (BW) function, LHCb reported a
width of about 1 MeV (the mass resolution is 2.4–3.0 MeV); and the fit with a
Flatté function with constraints from other measurements yields a FWHM of 0.22
MeV which depends strongly on the $X(3872)\to\bar{D}^{0}D^{*0}$ coupling [10,
11].

Although the statistics are low at BESIII experiment, the high efficiencies of
reconstructing all the $X(3872)$ decays modes and the very good mass
resolution in the $\bar{D}^{0}D^{*0}$ mode ($<1$ MeV) make it possible to
measure the lineshape of the $X(3872)$ state. BESIII determined the pole
locations of the $X(3872)$ based on a simultaneous fit to the data samples of
$X(3872)\to D^{0}\bar{D}^{0}\pi^{0}$ and $X(3872)\to\pi^{+}\pi^{-}J/\psi$,
with the $X(3872)$ produced in $e^{+}e^{-}\to\gamma X(3872)$ process [12]. The
parameterization of the $X(3872)$ lineshape, with the effect of $D^{*0}$ width
taken into account, is developed in Ref. [13]. The fit results and the
lineshape of the $X(3872)$ are shown in Fig. 2. The lineshape parameters are
determined to be $g=(0.16\pm 0.10^{+1.12}_{-0.11})$, $\Gamma_{0}=(2.67\pm
1.77^{+8.01}_{-0.82})$ MeV and $M_{X}=(3871.63\pm 0.13^{+0.06}_{-0.05})$ MeV.
Here $g$ denotes the effective coupling constant of the $X(3872)$ to neutral
and charged $D^{*}\bar{D}$; the constant $\Gamma_{0}$ represents all the
channels except $D^{*}\bar{D}$, and is separated into three parts:
$\Gamma_{0}=\Gamma_{\pi^{+}\pi^{-}{J}/\psi}+\Gamma_{\rm known}+\Gamma_{\rm
unknown}$; and $M_{X}$ is the mass of the $X(3872)$. The FWHM of the lineshape
is determined to be $(0.44^{+0.13~{}+0.38}_{-0.35~{}-0.25})$ MeV. Two poles
are found on the first and second Riemann sheets corresponding to the
$D^{*0}\bar{D}^{0}$ branch cut. The pole location on the first sheet is much
closer to the $D^{*0}\bar{D}^{0}$ threshold than the other, and is determined
to be $(7.04\pm 0.15^{+0.07}_{-0.08})$ MeV above the $D^{0}\bar{D}^{0}\pi^{0}$
threshold with an imaginary part $(-0.19\pm 0.08^{+0.14}_{-0.19})$ MeV.

Figure 2:  The fit to the $D^{0}\bar{D}^{0}\pi^{0}$ (left) and
$\pi^{+}\pi^{-}J/\psi$ (middle) invariant mass distributions. Data are taken
from Ref. [12]. The $X(3872)$ lineshape at the best estimation is shown in
right panel. The vertical dashed line indicates the position of
$D^{*0}\bar{D}^{0}$ threshold.

Belle measured the $X(3872)$ lineshape with $B\to X(3872)K\to
D^{0}\bar{D}^{*0}K$ [14]. The peak near the threshold in the
$D^{0}\bar{D}^{*0}$ invariant mass spectrum is fitted using a relativistic BW
function. Belle determined a mass of $(3873.71^{+0.56}_{-0.50}\pm 0.13)$ MeV
and a width of $(5.2^{+2.2}_{-1.5}\pm 0.4)$ MeV. The peak is also studied
using a Flatté lineshape and the lower limit on the $D\bar{D}^{*}$ coupling
constant $g$ is determined to be $0.075$ at 95% credibility. A coupled channel
analysis of the data used in this analysis and those in
$X(3872)\to\pi^{+}\pi^{-}J/\psi$ decay is highly recommended to get reliable
information about the $X(3872)$ lineshape.

Observation of new $\psi(4230)$ decays and new vector states $Y(4500)$ and
$Y(4790)$:  The $Y$ states were discovered in the initial state radiation in
the $B$-factory experiments, and they have $J^{PC}=1^{--}$. So these state can
also be produced directly in $e^{+}e^{-}$ annihilation experiment like BESIII.
Much improved measurements of the $Y(4260)$ [15], $Y(4360)$, $Y(4660)$ [16]
and so on are achieved, their new decay modes are discovered and new vector
states are observed.

The most precise measurements of the $Y(4260)$ are from the BESIII experiment
[17, 18]. By doing a high luminosity energy scan in the vicinity of the
$Y(4260)$, BESIII found the peak of the $Y(4260)$ is much lower (so it is now
named the $\psi(4230)$) than that from previous measurements and the width is
narrow, and there is a high mass shoulder with a mass of 4.32 GeV if fitted
with a BW function. Since then, more new decay modes of the $\psi(4230)$ were
observed including $\pi^{+}\pi^{-}h_{c}$[19], $\pi^{+}\pi^{-}\psi(2S)$ [20],
$\omega\chi_{c0}$ [21], $\pi\bar{D}D^{*}+c.c.$ [22], $\pi\bar{D}^{*}D^{*}$
[23], and $K\bar{K}J/\psi$ [24, 25].

The cross sections of $e^{+}e^{-}\rightarrow K^{+}K^{-}J/\psi$ at center-of-
mass energies from 4.127 to 4.600 GeV are measured based on 15.6 fb-1 data
collected by the BESIII experiment [24]. Two resonant structures are observed
in the line shape of the cross sections. The mass and width of the first
structure are measured to be $4225.3\pm 2.3\pm 21.5$ MeV and ($72.9\pm 6.1\pm
30.8$) MeV, respectively. They are consistent with those of the established
$\psi(4230)$. The second structure is observed for the first time with a
statistical significance greater than 8$\sigma$, denoted as $Y(4500)$. Its
mass and width are determined to be $4484.7\pm 13.3\pm 24.1$ MeV and $111.1\pm
30.1\pm 15.2$ MeV, respectively. This state is confirmed in
$e^{+}e^{-}\to\pi\bar{D}^{*}D^{*}$ reported in Ref. [23].

With the world’s largest $e^{+}e^{-}$ scan data sample between $4.226$ and
$4.95$ GeV accumulated by BESIII, the Born cross sections of $e^{+}e^{-}\to
D_{s}^{\ast+}D_{s}^{\ast-}$ are measured precisely [26]. Besides two
enhancements in the energy dependent cross sections at around 4.2 and 4.45 GeV
that may come from the $\psi(4160)$ or $\psi(4230)$ and the $\psi(4415)$,
respectively, a third resonance structure ($Y(4790)$) is observed at around
4.7$\sim$4.8 GeV with statistical significance greater than $6.1\sigma$. Due
to the limited number of data points around 4.79 GeV, the fitted mass of the
third structure varies from 4786 to 4793 MeV and the width from 27 to 60 MeV.
This could be the same state observed in $e^{+}e^{-}\to
K_{S}^{0}K_{S}^{0}J/\psi$ with a statistical significance of $4.0\sigma$ [25].

In the charmonium energy region between 3 and 5 GeV, we now have identified 6
well known $\psi$ peaks ($J/\psi$, $\psi(2S)$, $\psi(3770)$, $\psi(4040)$,
$\psi(4160)$, and $\psi(4415)$) and 9 new $Y$ structures ($Y(4230)$,
$Y(4320)$, $Y(4360)$, $Y(4390)$, $Y(4500)$, $Y(4630)$, $Y(4660)$, $Y(4710)$,
and $Y(4790)$). They are all vector states and they cannot be all charmonium
states. While more experimental efforts are needed to resolve the origins of
these states, theoretical efforts are also necessary to identify if the vector
charmonium hybrids and/or tetraquark states have already been observed.

Neutral partners of charged charmoniumlike $Z_{cs}$ states:  The charged
charmoniumlike state $Z_{c}(3900)$ discovered in $\pi J/\psi$ by the BESIII
[27] and Belle [28] experiments, the $Z_{c}(4020)$ discovered in $\pi h_{c}$
by the BESIII [29] experiment, and the $Z_{c}(4430)$ discovered in
$\pi\psi(2S)$ by the Belle [30] experiment are all states with minimal quark
content of $c\bar{c}u\bar{d}$.

Recent studies try to search for states with one of the four quarks replaced
by a different quark, for example, the $Z_{cs}$ states with quark content
$c\bar{c}u\bar{s}$. BESIII announced observation of a near-threshold structure
$Z_{cs}(3985)$ in the $K^{+}$ recoil-mass spectra in $e^{+}e^{-}\to
K^{+}(D^{-}_{s}D^{*0}+D^{*-}_{s}D^{0})$ [31] with a mass of 3983 MeV and a
width of about 10 MeV; and LHCb reported two resonances decaying into
$K^{\pm}J/\psi$, the $Z_{cs}(4000)$ with a mass of 4003 MeV and a width of
about 131 MeV, and the $Z_{cs}(4220)$ with a mass of 4216 MeV and a width of
about 233 MeV [32]. The widths of the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ are
quite different, maybe one of them is the strange partner of the $Z_{c}(3900)$
with the $d$ quark replaced with an $s$ quark. Both BESIII [33] and LHCb [34]
reported evidence for the neutral partners of the $Z_{cs}$ states at around 4
GeV with quark content $c\bar{c}u\bar{s}$. These indicate that these states
form isospin doublets.

The $Z_{c}(3900)$ ($Z_{c}(4020)$) and $Z_{cs}$ states may form multiplets
shown in Fig. 3, the missing states can be searched for with the existing or
future data samples.

Figure 3: The possible multiplets of the $Z_{c}(3900)$ and $Z_{c}(4020)$.

New tetraquark states from LHC experiments:  LHCb experiment observed two new
resonances with four different flavors with mass of $2908\pm 11\pm
20\,\rm{MeV}$ and width of $136\pm 23\pm 13\,\rm{MeV}$, which decay to
$D^{+}_{s}\pi^{+}$ and $D^{+}_{s}\pi^{-}$, respectively, from a combined
amplitude analysis for the decays $B^{0}\to\overline{D}{}^{0}D^{+}_{s}\pi^{-}$
and $B^{+}\to D^{-}D^{+}_{s}\pi^{+}$, which are related by isospin symmetry.
The former state indicates the first observation of a doubly charged open-
charm tetraquark state with minimal quark content $c\bar{s}u\bar{d}$, and the
latter state is a neutral tetraquark composed of $c\bar{s}\bar{u}d$ quarks
($T_{c\bar{s}}$). Both states are found to have spin-parity $0^{+}$, and their
resonant parameters are consistent with each other, which suggests that they
belong to an isospin triplet [35].

Tetraquark states $T_{cs}$ with four different flavors ($cs\bar{u}\bar{d}$)
have been search for at LHCb and evidence ($3.9\sigma$) for two states
($X_{0}(2900)$ and $X_{1}(2900)$) in $D^{-}K^{+}$ system were reported from a
PWA of $B^{+}\to D^{+}D^{-}K^{+}$ events by the LHCb experiment [36]. They are
good candidates for the flavour partners of the $T_{c\bar{s}}$ states, and
more flavour partners with other quark contents and spin-parities are
expected.

The LHCb, ATLAS, and CMS experiments reported observation of states decay to
two charmonium states [37, 38, 39]. The $X(6900)$ is observed in all these
three experiments, and a new structure ($X(6600)$), with a significance above
$5\sigma$, and evidence for another new structure ($X(7300)$), with a local
significance of $4.1\sigma$, are found at CMS. The masses, widths, and
significances are obtained in model-dependent ways without considering
possible interference between the resonances. These are good candidates for
tetraquark states with two pairs of charm-anticharm quarks.

Pentaquark states:  In the decay of $\Lambda_{b}\to J/\psi pK^{-}$ analyzed by
the LHCb experiments, there are three very narrow peaks in the invariant mass
distribution of $J/\psi p$ [40]. In a simple fit to the invariant mass
spectrum, the resonance parameters of the $P_{c}(4312)^{+}$,
$P_{c}(4440)^{+}$, and $P_{c}(4457)^{+}$ are determined. They are all narrow,
and the $P_{c}(4312)^{+}$ state peaks right below the
$\Sigma_{c}^{+}\bar{D}^{0}$ threshold, the $P_{c}(4457)^{+}$ state peaks right
below the $\Sigma_{c}^{+}\bar{D}^{*0}$ threshold, while the $P_{c}(4440)^{+}$
state peaks about 20 MeV below it.

Being so close to the thresholds, they are very good candidates for the
molecules of a charmed baryon and an anti-charmed meson, and more similar
states close to the other baryon-meson thresholds are expected. In a following
amplitude analysis of $B\to J/\psi\Lambda\bar{p}$ decays, a narrow resonance
in the $J/\psi\Lambda$ system, consistent with a pentaquark candidate with
strangeness, is observed with high significance [41]. The mass and the width
of this new state are measured to be $4338.2\pm 0.7\pm 0.4$ MeV and $7.0\pm
1.2\pm 1.3$ MeV, respectively. It is very close to the $\Xi_{c}^{+}D^{-}$
threshold of 4337.4 MeV. Evidence for states at around
$\Xi_{c}^{0}\bar{D}^{*0}$ threshold are reported in Ref. [42].

Summary and Perspectives:  Many states with exotic properties were observed in
the past two decades. Some of them are quite close to the thresholds of two
heavy objects, either two heavy flavor mesons or one heavy flavor meson and
one heavy flavor baryon, like the $X(3872)$ ($\bar{D}^{0}D^{*0}$), $Y(4220)$
($D_{s}^{*+}D_{s}^{*-}$ or $\bar{D}D_{1})$, $Z_{c}(3900)^{+}$
($\bar{D}^{0}D^{*+}$), $Z_{c}(4020)^{+}$ ($\bar{D}^{*0}D^{*+}$), $Z_{cs}^{+}$
($\bar{D}^{0}D_{s}^{*+}$), $P_{c}(4312)^{+}$ ($\Sigma_{c}^{+}\bar{D}^{0}$),
$P_{c}(4440)^{+}$ and $P_{c}(4457)^{+}$ ($\Sigma_{c}^{+}\bar{D}^{*0}$); and
some other states are not close to such thresholds, such as the $Y(4360)$,
$Y(4660)$, $Z_{c}(4430)^{+}$, and $Z_{cs}(4220)^{+}$. These may suggest that
we did observe the hadronic molecules close to thresholds and we also observed
hadronic states with some other quark configurations like compact tetraquark
states and so on. It is expected that more results will be produced by the
Belle II, BESIII, LHCb, and other experiments.

Acknowledgments:  We thank the organizers for the invitation to give a talk at
the conference in such a beautiful city. This work is supported in part by
National Key Research and Development Program of China (No. 2020YFA0406300),
and National Natural Science Foundation of China (NSFC, Nos. 11961141012 and
11835012).

## References

  * [1]
  * [2] A. J. Bevan et al. [BaBar and Belle], Eur. Phys. J. C 74, 3026 (2014).
  * [3] For recent reviews, see for example H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 86, no.2, 026201 (2023); N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, Phys. Rept. 873, 1-154 (2020); F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, no. 1, 015004 (2018).
  * [4] C. Z. Yuan, Int. J. Mod. Phys. A 33, no. 21, 1830018 (2018).
  * [5] R. Aaij et al. [LHCb], Int. J. Mod. Phys. A 30, 1530022 (2015).
  * [6] S. K. Choi et al. [Belle], Phys. Rev. Lett. 91, 262001 (2003).
  * [7] D. Acosta et al. [CDF], Phys. Rev. Lett. 93, 072001 (2004).
  * [8] V. M. Abazov et al. [D0], Phys. Rev. Lett. 93, 162002 (2004).
  * [9] R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022).
  * [10] R. Aaij et al. [LHCb], Phys. Rev. D 102, no.9, 092005 (2020).
  * [11] R. Aaij et al. [LHCb], JHEP 08, 123 (2020).
  * [12] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 124, 242001 (2019).
  * [13] C. Hanhart, Y. S. Kalashnikova and A. V. Nefediev, Phys. Rev. D 81, 094028 (2010).
  * [14] H. Hirata et al. [Belle], Phys. Rev. D 107, no.11, 112011 (2023).
  * [15] B. Aubert et al. [BaBar], Phys. Rev. Lett. 95, 142001 (2005).
  * [16] X. L. Wang et al. [Belle], Phys. Rev. Lett. 99, 142002 (2007).
  * [17] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 118, no.9, 092001 (2017).
  * [18] M. Ablikim et al. [BESIII], Phys. Rev. D 106, no.7, 072001 (2022).
  * [19] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 118, no.9, 092002 (2017).
  * [20] M. Ablikim et al. [BESIII], Phys. Rev. D 104, no.5, 052012 (2021).
  * [21] M. Ablikim et al. [BESIII], Phys. Rev. D 99, no.9, 091103 (2019).
  * [22] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 122, no.10, 102002 (2019).
  * [23] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 130, no.12, 121901 (2023).
  * [24] M. Ablikim et al. [BESIII], Chin. Phys. C 46, no.11, 111002 (2022).
  * [25] M. Ablikim et al. [BESIII], Phys. Rev. D 107, no.9, 092005 (2023).
  * [26] M. Ablikim et al. [BESIII], [arXiv:2305.10789 [hep-ex]].
  * [27] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 110, 252001 (2013).
  * [28] Z. Q. Liu et al. [Belle], Phys. Rev. Lett. 110, 252002 (2013)
  * [29] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 111, no.24, 242001 (2013).
  * [30] S. K. Choi et al. [Belle], Phys. Rev. Lett. 100, 142001 (2008).
  * [31] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 126, no.10, 102001 (2021).
  * [32] R. Aaij et al. [LHCb], Phys. Rev. Lett. 127, no.8, 082001 (2021).
  * [33] M. Ablikim et al. [BESIII], Phys. Rev. Lett. 129, no.11, 112003 (2022).
  * [34] [LHCb], [arXiv:2301.04899 [hep-ex]].
  * [35] R. Aaij et al. [LHCb], Phys. Rev. Lett. 131, no.4, 041902 (2023).
  * [36] R. Aaij et al. [LHCb], Phys. Rev. Lett. 125, 242001 (2020).
  * [37] R. Aaij et al. [LHCb], Sci. Bull. 65, no.23, 1983-1993 (2020).
  * [38] G. Aad et al. [ATLAS], [arXiv:2304.08962 [hep-ex]].
  * [39] A. Hayrapetyan et al. [CMS], [arXiv:2306.07164 [hep-ex]].
  * [40] R. Aaij et al. [LHCb], Phys. Rev. Lett. 122, no.22, 222001 (2019).
  * [41] R. Aaij et al. [LHCb], Phys. Rev. Lett. 131, no.3, 031901 (2023).
  * [42] R. Aaij et al. [LHCb], Sci. Bull. 66, 1278-1287 (2021).