The Bimeron Clusters in Chiral Antiferromagnets

Magnetic bimeron is an in-plane topological counterpart of magnetic skyrmion. Despite the topological equivalence, their statics and dynamics could be distinct, making them attractive from the respective of both physics and spintronic applications. In this work, we investigate the antiferromagnetic (AFM) thin film with isotropic Dzyaloshinskii-Moriya interaction (DMI), and introduce AFM bimeron cluster as a new form of topological quasi-particles. Such exotic spin textures are subject to topological protection, and can absorb other bimeron solitons or clusters along translational direction to acquire a wide range of Neel topological number. The formation of these AFM clusters involves rearrangement of topological structures, and gives rise to remarkable topological static and dynamical properties. Moreover, the bimerons demonstrate high current-driven mobility as general AFM quasi-particles. The merits of AFM bimeron cluster reveal a potential path to unify multi-bits data creation, transmission, storage and even topology-based computation within the same material system, and may stimulate innovative spintronic devices enabling new paradigm of data manipulations.

The last decade witnessed a rapid gaining of our understanding about magnetic skyrmions [1][2][3][4][5][6][7] . These spin textures are topologically-protected, and can be effectively manipulated by spin currents or electric fields, thus they may serve as ideal information carriers. In recent years, the in-plane analogue of magnetic skyrmion, named magnetic bimeron, is gaining a lot of attention [8][9][10][11][12][13][14][15][16][17][18][19] . The topologically non-trivial square meron lattice has been experimentally observed in chiral magnet Co 8 Zn 9 Mn 3 10 . More recently, isolated meronantimeron pair and bimeron have been stabilized in Py film by magnetic imprinting 11 . Despite the topological equivalence between skyrmion and bimeron, their magnetic static and dynamic properties are distinct, making the magnetic bimeron attractive from the perspectives of both fundamental physics and practical applications in spintronic materials with in-plane anisotropy.
On the other hand, intensive efforts have been devoted to exploit topological spin textures in novel magnetic systems, such as two-dimensional (2D) materials 20 , frustrated materials 14,21 , liquid crystal 22,23 and antiferromagnets [24][25][26][27][28][29][30][31][32] . Among them, the antiferromagnetic (AFM) materials show great potential to bring topological spin textures closer to real applications. In AFM systems, the magnetic moments of coupled sub-lattices cancel out, which leads to nearly zero dipolar field and enhances the stability of nanoscale topological structures. The topological numbers of spin texture in sub-lattice space also cancel out, and thus making them free from the skyrmion Hall effect [33][34][35] . Moreover, the canting of sub-lattices magnetic momentum leads to extra torques facilitating ultrafast dynamics of AFM system. As a result, the mobility of AFM topological structures is much higher than their ferromagnetic (FM) counterparts [27][28][29] . Consequently, the manipulation of AFM topological structures requires much lower power consumption, which is highly attractive from the point of view of practical applications.
In this work, we demonstrate the stabilization of bimeron soliton and clusters in AFM thin film with isotropic DMI 36 . We combine analytical and numerical approaches, and systematically investigate the statics and current-driven dynamics of AFM bimeron soliton and their clusters. We believe the study of AFM bimeron is important to understand the topological magnetism related physics. In addition, the AFM bimeron clusters show great potential in innovative spintronic devices enabling new paradigm of data manipulations.

Results
Theoretical model. We consider the G-type cubic antiferromagnet with the in-plane uniaxial anisotropy. The AFM total energy E can be written as where J (> 0 for antiferromagnets), K a and D kl are the AFM exchange constant, the magnetic anisotropy constant and the DMI vector, respectively, and we take |D kl | = D I . n e = e X is the direction of the in-plane magnetic easy axis. By linearly combining the reduced local mangnetization m k and the neighboring magnetization m l , the net magnetization m = (m l + m k )/2 and the staggered magnetization (or Néel vector) n = (m l −m k )/2 are defined. Using m l = (m+n) and m k = (m − n), the AFM energy in continuous form can be written as: 37 where λ = 16J/a∆ 2 , A = 2J/a and L = 4J/a∆ are the homogeneous exchange constant, inhomogeneous exchange constant and parity-breaking constant, respectively. Here a is the lattice constant, and ∆ = √ 2a. w D = (D/2)[n z (∇ · n) − (n · ∇)n z ] is the DMI energy density, with the D being the DMI constant 36 . The detailed derivation procedure of Eq. (2) is given in Supplementary Note 1. For the later simulations, we adopt the parameter settings with J = 1.6475 × 10 −21 J (about 10meV), D I /J = 0.0088 and K a /J = 0.0546, damping constant α = 0.01, saturation magnetization M s = 3.76 × 10 5 A/m, and lattic constant a = 0.5nm. We also note that the generality of the results presented here has been tested by varied parameter settings.
The stabilization of AFM bimeron soliton and the currentdriven dynamics. Similar to the AFM skyrmions, the magnetic topology of the AFM bimeron is defined by Néel topological number Q n = (1/4π) dxdy[n · (∂ x n × ∂ y n)]. Figure 1a shows the real-space spin texture of the AFM bimeron soliton with Q n = +1, which is formed by a circular AFM meron and a crescent-shaped AFM anti-meron. The corresponding Néel vector components are shown in Figs. 1b-d, and the structure features are clearly demonstrated by n Z , which indicate a shape-defined magnetic topological dipole. We assume that the vector pointing from the perpendicular sub-lattices located in the circular meron part to that in the crescent-shaped anti-meron part as the bimeron polarity p BM , which is parallel to the in-plane magnetic easy axis.
Compared with the FM skyrmion, the AFM skyrmion can be manipulated by spin current with higher mobility and no skyrmion Hall effect [33][34][35] . The AFM bimeron shares the similar merits, while exhibits obvious anisotropic dynamics on the other hand. In this part, we derive the velocity expression based on the Thiele's approach 38,39 , and systematically investigate the dynamics driven by spin-orbit torque (SOT), originating from, e.g., the spin Hall effect in antiferromagnet/heavy metal heterostructure, or spin-transfer torque, originating from the spin current polarized by a fixed magnetic layer with the current-perpendicular-to-plane (CPP) configuration.
Taking the damping-like spin torques into account, the dynamics of the net magnetization m and the Néel vector n obey the following two coupled equations 40,41 n = (γf m − αṁ) × n + γH d m × p × n, (3a) m = (γf n − αṅ) × n + T nl + γH d n × p × n, (3b) where γ and α are the gyromagnetic ratio and the damping constant. f m = −δE/µ 0 M s δm and f n = −δE/µ 0 M s δn are the effective fields. T nl = (γf m − αṁ) × m is the higher-order nonlinear term 40 . p is the polarization vector and H d = j P/(2µ 0 eM s t Z ) is the equivalent field of spin torque, with j being the charge current density, the reduced Planck constant, P the spin polarization efficiency, µ 0 the vacuum permeability constant, e the elementary charge, and t Z the AFM layer thickness.
Based on Eqs. (3a) and (3b), the steady motion speed of AFM bimeron soliton can be semi-analytically expressed as where v x and v y are the velocity components in the continuous AFM coordinate system. d ij = dxdy(∂ i n · ∂ j n) is the component of the dissipative tensor, and u i = dxdy[(n × p) · ∂ i n] relates to the force induced by the damping-like spin torque 38 . The detailed derivation procedure of the above equation is given in Supplementary Note 2. Equation (4) applies for not only bimeron solitons, but also bimeron clusters with higher Q n , which will be discussed later.
One of the key features of the AFM bimeron under investigation is the anisotropic dynamics caused by its asymmetric structure. In order to simulate the dynamics excited by SOT, we assume a positive spin Hall angle of θ SH = P = 0.1. Figure 2a shows the SOT-driven speed of the bimeron soliton with Q n = +1 as a function of θ j , the angle between the direction of the charge current and +X. Equation (4) predicts that the speed reaches the highest value when θ j = 0 and 180 degree, corresponding to the motion along the in-plane magnetic easy axis, as indicated by the solid curves. For a moderate charge current with density j = 5 × 10 10 A/m 2 , the numerical results are consistent with those obtained by the analytical approach, indicated by the black squares. As the current density increases to 1×10 11 A/m 2 , and further to 2×10 11 A/m 2 , the SOT deforms the bimeron soliton, and Eq. (4) no longer applies, as indicated by the blue and red squares. In particular, the spin textures of AFM bimeron when j = 2×10 11 A/m 2 with varied θ j are shown in Fig. 2b to demonstrate the SOT-induced deformation. Significant constriction/expansion can be observed when θ j = 0/180 degree, which leads to lower/higher motion speed than that obtained by Eq. (4). On the other hand, only a slight tilt of spin texture is observed when θ j = 90 and 270 degree, for which the discrepancy can be safely disregarded.
Due to the asymmetry of the spin texture, the bimeron soliton driven by SOT possesses a varied skyrmion Hall angle θ sk with respect to θ j . Based on Eq. (4), the steady motion velocity along the direction of the charge current v j , and the transverse velocity,v trans , can be analytically derived, and the calculated θ sk = arctan(v trans /v j ) is shown in Fig. 2c together with the numerical results. It is found that this spin structureinduced skyrmion Hall angle tends to deflect the motion of the bimeron soliton to the direction of magnetic easy axis.
Another mechanism to drive the AFM bimeron is to use the spin current with out-of-plane polarization, which is realized by injecting the charge current through a perpendicularly fixed layer. With the configuration, the effective torque originating from the spin-transfer effect will exert on the in-plane topological core of AFM bimeron soliton, and thus lead to a steady motion along the direction perpendicular to the inplane magnetic easy axis. Figure 2d shows the steady motion speed obtained by Eq. (4) and the numerical simulations with P = 0.1. The mobility of bimeron soliton is comparable to that driven by SOT, while the current-induced deformation is well alleviated, and no skyrmion Hall effect is observed. We note that symmetric magnetic skyrmions are inactive to perpendicular polarized spin currents 42 .
The formation of bimeron clusters. The bond state of skyrmions have been predicted and observed in chiral ferromagnets [43][44][45][46] , frustrated materials 14,21 and liquid crystals 22 . Despite the intriguing physics, it is important to find effective methods to manipulate these aggregated topological structures. In this part, we introduce the AFM bimeron clusters with high Q n in chiral antiferromagnets. The formation of these AFM clusters involves rearrangement of topological structures, and leads to remarkable changes in both static and dynamic characteristics. Moreover, they have high mobility as general AFM quasi-particles, making them ideal building blocks for AFM spintronic devices.
As a first step, we demonstrate the anisotropic interactions between two bimeron solitons, as shown by Fig. 3a. For the corresponding investigation, two bimeron solitons with Q n = +1 are pinned by setting the fixed in-plane spins and the distance between them are adjusted to eliminate their interactions. Then we slide one of them along the direction defined by the angle ϕ, as indicated by the inset of Fig. 3a, and the effective interaction between them can be indicated by the variation of discrete energy described by Eq. (1) 47 . For ϕ = 0 degree, the system energy rapidly increases as the bimerons approaching each other, demonstrating a repulsive interaction. And the bimeron finally collapses at the separation distance d p = 36a due to the constriction, as shown by the orange line. As ϕ increases, the interaction gradually changes from repulsive to attractive. For ϕ = 90 degree, corresponding to the alignment of bimerons in the direction orthogonal to the magnetic easy axis, a sallow potential well can be observed at d p = 44a, and further approaching of bimeron solitons leads to the merging to bimeron dimer. This process is indicated by a sudden drop of the energy at d p = 38a, as shown by the red line. To further investigate the formation of the bimeron dimer, we set ϕ = 90 degree and vary the strength of DMI, and its influence on the bimeron interactions is shown in Fig. 3b. For lower D I = 8.5 × 10 −23 J (D I /J = 0.0516), the interaction between the bimeron solitons turns to repulsive. In this case the energy barrier prohibits the merge of bimeron solitons, and both of them collapse at d p = 26a. As D I increases, the energy of bimeron dimer drops faster than the isolated solitons, and finally leads to the spontaneous formation of the former, as indicated by the pink (D I /J = 0.0561) and green (D I /J = 0.0577) lines. In the meanwhile, the merge distance and the bonding energy also increases with D I . However, it should be noted that for higher D I , the bimeron soliton tends to strip out in the direction perpendicular to the magnetic easy axis, and relaxes to a domain wall pair, as demonstrated in Supplementary Note 3. This phenomenon is similar to the case for skyrmion in the perpendicular AFM system, which narrows the stability region of bimeron solitons 48 . Figure 3c and 3d show the spatial distribution of normalized topological charge q n = n · (∂ x n × ∂ y n) of the AFM bimeron soliton and dimer, respectively. We note that the formation of dimer involves the merge of topological cores as shown in Fig. 3d, which indicates essential changes of the spin textures. In contrast, the topological cores remain protected for skyrmions in bond state 14 . Figures 3e and 3f compare the profiles of Néel vector components and q n along the translational center (n Y = 0) of the bimeron soliton and dimer. The magnetic topology of bimeron soliton is represented by the in-plane sublattice spins (n X = −1), on the other hand, that of bimeron dimer is represented by the out-of-plane ones (n Z = −1). The reorientation of topology-representative spins is another feature identifying the formation of bimeron dimer.
Due to the translational attractive interaction between the same solitons, they can accumulate along the direction perpendicular to the magnetic easy axis, and thus form the AFM bimeron clusters with a wide range of Q n . Figure 4a compares the energy composition of bimeron clusters and the isolated bimeron solitons with the same Q n up to 20. There are two features worth noting. Firstly, DMI prefers the formation of bimeron clusters than the isolated bimeron solitons, as indicated by the green lines. As a result, the stabilization of bimeron clusters in chiral antiferromagnets is easier than bimeron solitons. Secondly, the stabilization of bimeron soliton is mainly determined by the competition between AFM exchange and DMI, while the magnetic anisotropy plays a minor role, as indicated by the lines with hollow symbols. However, the formation of bimeron clusters significantly increases the magnetic anisotropy energy, as indicated by the blue lines. As a result, we note that the stabilization of bimeron solitons and clusters may follow different paths. Or in other words, the stabilizations of bimeron solitons and clusters may not guarantee each other. Figure 4b shows the energy difference ∆E between bimeron clusters with topological number Q n Energy difference between clusters with neighboring Néel topological numbers vs. Qn. c Real-space spin textures and d Z component of Néel vectors of bimeron cluster with Qn from +2 to +6. and Q n -1. As Q n increases, ∆E quickly drops and then converges to a value smaller than the energy of an isolated bimeron soliton, which may indicate the possibility to stabilize AFM bimeron clusters with arbitrary Q n . Figure 4c and 4d show the real-space spin textures and Z component of Néel vectors of the bimeron cluster with Q n ranged from +2 to +6. Here we note that the bimeron clusters can exist in both Atype and G-type antiferromagnets with isotropic DMI and inplane uniaxial magnetic anisotropy. However, for chiral FM system, the out-of-plane components of magnetic momentum increases with Q n , which may lead to higher magneto-static energy and destabilize the clusters.
Current driven dynamics of bimeron clusters. Similar to the solitons, the AFM bimeron clusters can also be effectively manipulated by spin currents. We adopted j = 1 × 10 10 A/m 2 to avoid the deformation, and the calculated speeds of bimeron clusters driven by SOT and CPP-STT are shown in Fig. 5a, where good agreements between Eq. (4) and the numerical simulations can be observed. For the SOT-driven case, the AFM bimeron clusters have anisotropic dynamics similar to the AFM bimeron soliton, and prefer the motion along the magnetic easy axis. As Q n increases, the speed driven by SOT nonlinearly increases, as indicated by the solid lines. However, that by CPP-STT decreases, as indicated by the green dashed line. In order to understand this difference, we calculated the strength of the effective spin torque force, F st,i = −µ 0 H d M s t Z u i , as a function of Q n , as shown in Fig. 5b. For the SOT-driven cases with the charge current in all directions, F st increases almost linearly with Q n , and well explains the anisotropic dynamics observed in Fig. 5a. In contrast, for the CPP-STT-driven case, the effective force keeps the same when Q n ≥ 2. Since the accumulation of Q n will increase the dissipation and effective mass of the bimeron clusters, but not the effective driven force, the speed of AFM bimeron clusters tend to decrease with Q n .
The coexistence of topological counterparts and currentdriven topology modification. Another distinguishable feather of bimeron clusters is that their counterparts with opposite topological number Q n can coexist within the same AFM background. In order to understand this phenomenon, we analyze the AFM energy and topological charge of bimeron cluster by group symmetry. We use (m X , m Y , m Z ) A to denote the spin vectors of the bimeron A. Next, the spin vectors are operated as follows, and then we get the bimeron B. The above operation is performed in the coordinate system of the discrete model (cf. Supplementary Figure 1), and the transformation to the co-ordinate system of the continuous model leads to The AFM Néel vector and the net magnetization of the bimeron B are obtained as On the other hand, the operation in Eq. (5) will cause the changes in the spatial derivatives of the AFM Néel vector, Based on Eqs. (7) and (8), we get (9) Combining Eqs. (2), (7) and (9), it is found that the AFM energy of bimeron A is the same as that of bimeron B, while they have the opposite Néel topological charge Q n . Thus, we prove that the bimeron clusters with opposite topological charge can coexist in the AFM film with isotropic DMI. Based on the similar approach, we also demonstrate in Supplementary Note 5 that for the clusters with opposite Q n , the in-plane polarized spin current leads to their motions in the same direction, while the out-of-plane polarized one to the opposite directions.
Compared with thin films have the perpendicular magnetic anisotropy, the form of topological quasiparticles allowed by the in-plane AFM system is significantly enriched, making the bimeron clusters ideal multi-bit data carriers. Moreover, through the merge/annihilation of similar/opposite bimeron clusters, the topological number can be easily modified. Here we adopted j = 1 × 10 10 A/m 2 , P = 0.1, and use the CPP-STT as the driving force, which leads to the most significant speed difference between cluster with different Q n . To demonstrate the merge of similar clusters, a bimeron soliton with Q n = +1 and a cluster with Q n = +3 are aligned in the direction perpendicular to the in-plane magnetic easy axis. Then the spin current is injected to the AFM thin film to drive both the particles moving in −Y direction. Due to their speed difference [about 36 m/s according to Fig. 5a], the bimeron soliton will catch up with the cluster at about 200 ps, and finally they merge to a new cluster with Q n = +4 (cf. Supplementary Movie 1). After the merge of bimeron soliton and cluster, the total energy of the system decreases about 160 K. To demonstrate the annihilation of opposite clusters, a soliton with Q n = -1 and a cluster with Q n = +2 are used, and a slight misalignment is introduced to facilitate their annihilation, as shown in Fig. 6b. With the applied spin current, the soliton and cluster move toward each other. The annihilation happens at about 200 ps, with a significant drop of system energy (about 2600 K), which leads to a quick burst of spin waves, and finally the soliton with Q n = +1 remains (cf. Supplementary Movie 2). Based on the above features, we note that AFM bimeron clusters actually provide a full set of signed integers, and the above-mentioned processes can be regarded as the analogues of summation "3+1" and subtraction "2-1". Through the process of particle creation and annihilation driven by spin current, the topological number of the bimeron cluster can be easily modified, revealing a potential path towards magnetic topology-based computing.

Discussion
The study demonstrates that the bimeron clusters with a wide range of topological numbers of different sign have the potential to be stabilized in antiferromagnetic thin film with isotropic DMI, and exhibit rich and versatile dynamics driven by spin currents. Such findings indicate that AFM bimeron may serve as an ideal candidate to investigate skyrmionrelated physics, such as particle interaction, attraction, repulsion, bonding and mutual annihilation.
From applied perspective, through the processes of particle merging and annihilation driven by spin currents, the topological number of the bimeron clusters can be easily modified, revealing an appealing path towards magnetic topology-based computing. The AFM bimeron cluster may be utilized to unify multi-bits data creation, transmission, storage and computation within the same material system, paving the way for new data manipulation paradigm.

Code availability
An open-source micromagnetic simulation program Mumax 3 is used in this study 49 . The code is available at http://mumax.github.io/.