Abstract
Noncollinear spin textures in lowdimensional magnetic systems have been studied for decades because of their extraordinary properties and promising applications derived from the chirality and topological nature. However, material realizations of topological spin states are still limited. Employing firstprinciples and Monte Carlo simulations, we propose that monolayer chromium trichloride (CrCl_{3}) can be a promising candidate for observing the vortex/antivortex type of topological defects, socalled merons. The numbers of vortices and antivortices are found to be the same, maintaining an overall integer topological unit. By perturbing with external magnetic fields, we show the robustness of these meron pairs and reveal a rich phase space to tune the hybridization between the ferromagnetic order and meronlike defects. The signatures of topological excitations under external magnetic field also provide crucial information for experimental justifications. Our study predicts that twodimensional magnets with weak spinorbit coupling can be a promising family for realizing meronlike spin textures.
Introduction
The quest for lowdimensional (\(d \le 2\)) magnetic materials and their unique topological spin textures can be traced back to the last century. Particularly, for twodimensional (2D) magnetic systems, different ways to break the continuous rotational spin symmetry may induce different spontaneous magnetic orders by violating the Mermin–Wagner theorem^{1,2,3,4,5,6,7,8,9}. For example, with the presence of a specific easy axis in 2D structures, the longrange ferromagnetic (FM) order can sustain at finite temperature via opening a magnon gap to resist thermal agitations^{3,5,6,7,8,9,10}. This is evidenced by recent realizations of Isinglike 2D magnets, such as monolayer (ML) FM insulators CrI_{3} and Cr_{2}Ge_{2}Te_{6}, and multilayer magnetic topological insulator MnBi_{2}Te_{4}, etc.^{6,7,11,12,13,14}, which have ignited tremendous research interests to date.
Unlike the easyaxis anisotropy, an easyplane anisotropy yields a residual SO(2) symmetry that usually prohibits the formation of a spontaneous longrange magnetic order at finite temperature^{1,2,4}. On the other hand, such a residual inplane symmetry provides ample space for realizing the quasiordered states that evolve with various topological defects. A known example is the topological vortex/antivortex pairs with an algebraically decaying correlation, which can be described by the Berezinskii–Kosterlitz–Thouless (BKT) theory based on the 2D XY model^{4,15}. However, spins of realistic materials own a threedimensional (3D) degree of freedom, making them different from the ideal XY model, in which the spin is confined within the 2D easy plane. Such an extra degree of freedom gives hope to many nontrivial topological spin states, such as skyrmions, magnetic bubbles, and merons^{5,16,17,18,19,20,21}.
These noncollinear spin textures can be induced by many mechanisms, such as antisymmetric exchange interactions (DzyaloshinskiiMoriya (DM)), magnetic anisotropy, and magnetic dipolar couplings^{5}. Particularly, isotropic easyplane magnetized 2D materials can host magnetic vortices and antivortices. This type of topological defects were predicted to have curling magnetizations lying within the easy plane but pointing out of the plane around the core regions of vortices/antivortices^{5}. In condensed matter physics, such a topological structure is referred to as a meron, which corresponds to one half of a skyrmion and is stabilized via pairs^{22,23,24}. Merons have already been observed in various coupled magnetic discs and magnetic interfaces^{18,23,24,25,26,27}. For example, meron textures have been observed in thin plates of chirallattice magnet Co_{8}Zn_{9}Mn_{3} and αFe_{2}O_{3}/Co heterostructure film at room temperature^{25,27}. Recently, there was also a report about creating and stabilizing meron pairs in a continuous Py film, which inherits the inplane anisotropy by local vortex imprinting from a Co disk^{28}. However, besides magnetic discs and interfaces, a pristine singleatomic thin material that exhibits such topological defects is still absent. More importantly, the understanding of these topological defects, their interactions, and the profound relations with fundamental quantum phenomena, such as superfluidity and superconductivity, are yet unclear and forging ahead^{29,30}.
In this work, we find that, because of the weak spinorbit coupling (SOC), the magnetic dipolar interaction induced magnetic shape anisotropy (MSA) can overcome the magnetocrystalline anisotropy (MCA) to evince an easyplane, isotropic magnetic polarization in ML CrCl3. By employing an anisotropic Heisenberg model with magnetic dipoledipole (DD) interactions and exchange interactions extracted from firstprinciples simulations, we predict the existence of meronlike topological defects in ML CrCl3. Moreover, beyond pairs of vortecies and antivortecies, higherorder states involved with more than two merons are also observed in our simulations, forming complex topological excitations. Finally, to guide experiments, we show that merons are robust against external fields, and the rich hybridization between merons and the FM order can be tuned via external magnetic fields.
Results
Easyplane anisotropy in ML CrCl_{3}
The easyplane anisotropy is crucial for the formation of topological defects^{5,31}. Usually, the anisotropy is described by the magnetic anisotropy energy (MAE), which characterizes the dependence of energy on the orientation of magnetization. There are two origins of MAE owing to the relativistic effect. The first is MCA that is mainly determined by the SOC. The second is the shape anisotropy related to MSA that is originated from the Breit modification of the relativistic twoelectron energy. MSA is ascribed to the classical magnetic DD interactions^{32,33}:
where \(\mu _0\) is the vacuum permeability, and \({\mathbf{m}}_i\), \({\mathbf{m}}_j\) are the local magnetic moments with a spatial separation of \({\mathbf{r}}_{ij}\). Such a DD interaction usually favors magnetization along the elongated direction of materials. For thin films, this provides an origin of inplane anisotropy^{5,32}. Typically, the shape anisotropy can be added as a posterior term after the firstprinciples electronic calculations^{33}. In most materials, the magnetic DD interaction is small compared with the MCA interaction^{34}. However, it may play an important role in weakMCA magnets with negligible SOC. This leads us to study ML CrCl_{3}, which is known for its weak SOC^{33}. 2D CrCl_{3} has been recently fabricated^{35,36,37}, and the XY physics was discussed in a recent experiment on ML CrCl_{3}^{38}.
According to Eq. (1), the MSA energy is calculated by \(E_{{\mathrm{MSA}}} \,=\, E_\parallel ^{{\mathrm{D}}  {\mathrm{D}}}  E_ \bot ^{{\mathrm{D}}  {\mathrm{D}}}\) with the magnetization rotating from the inplane direction (\(\parallel\)) to the outofplane direction (\(\bot\)). The MCA energy is calculated by \(E_{{\mathrm{MCA}}} \,=\, E_\parallel ^{{\mathrm{SOC}}}  E_ \bot ^{{\mathrm{SOC}}}\), which is obtained from firstprinciples density functional theory (DFT) + U calculations. The calculation setups and convergence tests are included in the Method section. The schematic plot of the FM anisotropy energy (taking the outofplane direction as a reference) is presented in Fig. 1(a). Because of the competition between MCA and MSA, the overall MAE of ML CrCl_{3} is about −34 μeV (the negative sign means a preferred inplane polarization). Thus, the zerotemperature groundstate magnetization in ML CrCl_{3} is inplane (Fig. 1(c)). Besides, the inplane MAE is nearly perfectly isotropic, and the variation is <0.1 μeV. In this work, we choose the Hubbard U and Hund exchange J as 2.7 eV and 0.7 eV, respectively, which were used in published literatures and provided good agreements with available measurements^{3,39,40,41}. Moreover, we have checked other choices of U and J within a reasonable range and confirmed that they do not qualitatively affect the preferred inplane MAE and subsequent topological spin textures of ML CrCl_{3} (see the Supplementary Note 1 and Supplementary Table 1). These results also agree with recent calculations and measurements^{33,35,36,37}.
For the purpose of comparison, we have done calculations for ML CrI_{3}, which has a large SOC. Its MAE is around 300 μeV, resulting in a known Isinglike outofplane magnetic order^{3,6,42,43}. The corresponding anisotropy energy and preferred zerotemperature outofplane magnetization are schematically plotted in Fig. 1(b, d). The calculated anisotropy energies and magnetic coupling constants are listed in Table 1 for these widely studied chromium trihalides, and ML CrCl_{3} is the most promising structure to realize the inplane groundstate polarization. Moreover, the relationship between the MCA energy and SOC strength is discussed in the Supplementary Note 2 to show that SOC is the main mechanism in deciding the MCA energy.
Lowtemperature magnetic phase
To explore the magnetic phase at low temperature, we have employed a classical Monte Carlo (MC) simulation based on an anisotropic Heisenberg model with parameters from firstprinciples simulations. This Heisenberg Hamiltonian is essentially an approximation to the manybody electronic Hamiltonian in a localized basis, which only contains the spin degree of freedom^{44}. This method provides good agreements of the Curie temperature (T_{c}) for ML CrI_{3} and CrBr_{3} with experiments^{3,9,41,44,45}. More specifically, we add the magnetic DD interaction (\(E_{{\mathrm{D}}  {\mathrm{D}}}\)) from Eq. (1):
where A describes the easyaxis singleion anisotropy, \(\lambda _{1,2}\) and \(J_{1,2}\) represent the anisotropic and isotropic exchange couplings up to the next nearest neighbors, and \({\mathbf{B}}\) is the external magnetic field.
The details of extracting those coefficients from firstprinciples simulations are presented in the Method section.
We begin with the intrinsic ML CrCl_{3} without an external field. The evolution of the averaged magnetic moment with temperature for ML CrCl_{3} is presented in Fig. 2(a). We find that the outofplane magnetization (the bluedotted line M_{z}) is almost completely quenched. Surprisingly, the inplane magnetization (the red diamond M_{in}) with large fluctuations emerges even after the result is averaged over 25 ensembles. In contrast to CrCl_{3}, as shown in Fig. 2(b), ML CrI_{3} exhibits a normal Isinglike outofplane magnetism with negligible fluctuations, and the inplane magnetization is always zero. A sharp phase transition is observed around 42 K for ML CrI_{3}, which is close to the measured Curie temperature of 45 K^{6}.
The inset of Fig. 2(a) further addresses the substantial fluctuations of the inplane magnetization by comparing the statistically averaged inplane magnetic moment with a singleround simulation. Importantly, for a singleround simulation, there is no obvious trend of the evolution of the inplane polarization even between adjacent temperatures. Such a remarkable randomness strongly indicates the existence of a weak polarized state or a quasiordered phase, e.g., the vortex/antivortex states described by BKT physics^{4,15,46}. The detailed comparison of averaged magnetization for ML CrCl_{3} and CrI_{3} with and without DD interactions can be found in the Supplementary Note 3.
We have also checked the potential existence of quasiorder in ML CrCl_{3} by analyzing the spinspin correlation functions, which is an effective way to identify orders^{15,47}. In Fig. 2(c), the spin correlation of ML CrCl_{3} exhibits a quasilongrange decay to zero at a low temperature (4 K) and an exponential decay to zero at a hightemperature (12 K) that is within the paramagnetic (PM) phase. These behaviors indicate that (1) there is no longrange order in ML CrCl_{3}; (2) quasi orders may exist at low temperature because of the algebraiclike decaying correlation function. On the other hand, ML CrI_{3} exhibits a normal FM behavior in Fig. 2(d): the spin correlation exponentially decays to a nonzero value at 40 K because of the longrange FM order while it exponentially decays to zero at 48 K because of the PM phase.
In the following, by further analyzing the realspace arrangement of local magnetic moments as obtained through MC simulations, we confirm the topological vortex/antivortex type of defects within a 3D spin space. Figure 3(a–c) present the realspace magnetic moments at different temperatures. At low temperatures (Fig. 3(a, b)), we can identify the vortex/antivortex type of topological spin defects, which are marked by the ovals. At higher temperature (Fig. 3(c)), the spin structures disappear. Importantly, the numbers of vortex and antivortex defects are always the same, maintaining an overall integer topological unit. The chirality of topological pairs is indicated by the blue (vortex type) or red (antivortex type) part of the ovals. In addition, we have plotted the corresponding relative phase maps, which are used in experiments to identify spin textures^{48,49}. In Fig. 3(d–f), the phase maps clarify the chirality of topological defects at core regions.
At low temperature (0.25 K), we find that these vortex (antivortex) type spin defects can be connected with neighbor alternative antivortex (vortex) type defects via a spin flux closure as indicated by the dashed line in Fig. 3(d). The enlarged Fig. 3(g) provides a clear microstructure diagram of those topological defects. Interestingly, unlike the strict XY model where spin rotator is constrained within the 2D XY plane, the topological structures in ML CrCl_{3} contain 3D information, and this 3D spin effect is particularly significant around the core regions of the topological structures. As seen from the side view in Fig. 3(h), the magnetic moments around cores point out of the plane and form alterable spin hills^{26,28,49,50,51}. Compared with the vortex and antivortex pairs in the XYmodel, whose free energy mainly comes from the core regions^{4,15}, the benefit of forming such 3D spin texture is manifest by avoiding large swirl angles around the cores of defects and, hence, lowering the system energy. Meanwhile, our simulation does not display any correlation between the magnetic moment swirling and the direction of the central spin hill. This agrees with the experimental observations that all four possible up/down combinations of paired meroncore polarities show up with nearly equal probabilities^{28}.
Such topological defects have been referred to as merons, which have a \(\pm\, 1/2\) skyrmion charge^{5,23,52} and are similar to experimental observations^{18,23,24,25,26}. This is a material analogy to a solution of the Yang–Mills theory, in which merons as described in the context of quark confinement can only exist in pairs owing to the one half of topological charge carried^{22,23,52,53}. This character agrees with the equal numbers of vortex/antivortex defects observed in our simulations. Particularly, because of the pairing character of these topological spin defects, it is expected that doped free carriers could be intrinsically spinpolarized, and the transport of charged meron pairs might be dispersionless at low temperature^{54,55,56}. Although those dynamic behaviors and quantitative interactions between merons are beyond the scope of this study, they would be interesting topics for future studies^{57,58,59}.
Beyond meron pairs, our MC simulations reveal more complicated spin textures, such as the higherorder states involved more than two merons. As indicated in Fig. 3(a, d), some pairs of vortex/antivortex type merons are found to be antiparallelly aligned with each other at low temperature, forming “quadrupolelike” topological excitations^{57,60}. The presence of these hierarchical excitations indicates that, although the classical MC treatment does not take into account the quantum nature of spins, it does include the correlations of magnetic moments from the Heisenberg model^{9}. (see Supplementary Fig. 4 for more examples of coupled meron pairs).
As temperature increases, those higherorder excitations are firstly annihilated by thermal fluctuations as indicated by the snapshot at 3.25 K in Fig. 3(b, e). On the other hand, the relatively robust pairing between vortex/antivortex type merons still persists, although thermal energy blurs the topological structures and makes them harder to be recognized in Fig. 3(e). Finally, at the higher temperature (~12 K), the paired merons totally melt, as shown in Fig. 3(c, f). The system becomes a disordered, PM phase with an exponentially decaying spinspin correlation, which agrees with Fig. 2(c).
There are a few important mechanisms that are crucial for understanding the simulation results. First, we clarify the role of DD interactions in generating meronlike spin textures. It is known that DD interactions complicate system behaviors due to its nonlocal character and weakened singularities of the longitudinal mode of spin waves^{61,62,63}. For 3D spins in ML CrCl_{3}, DD interactions mainly impact the MAE and introduce the preferred inplane groundstate polarization. This is evidenced by the similar averaged magnetic moments calculated with different truncations of DD interactions in Fig. 2(e). In the above simulations, we truncate the DD interaction with a cutoff radius around 31 Å, as adopted in previous studies^{50,64}. When the truncation radius increases from 15 Å to 52 Å, the statistically averaged value of magnetic moment at a certain temperature (T = 3 K) does not change essentially. Moreover, we find that the residual DD interaction beyond a 31 Å truncation contributes <5% to the total MSA energy. Thus, the overall DD interaction energy is, at least, an order of magnitude smaller than the energy from the isotropic exchange interaction J. As a result, except at extremely low temperature, the residual energy due to the DD interaction truncation shall not qualitatively affect the spin textures of our studied systems. It is worth mentioning that our simulation cannot give accurate asymptotical behaviors approaching zero temperature due to the exponentially increasing simulation time, but this limit does not affect the main results of this work.
The simulationsize effect also needs to be considered. As Bramwell and Holdsworth discussed, a spontaneous magnetization could show up in a finitesize XY model owing to the suppression of Goldstone modes that usually destroy ferromagnetism^{63,65}. A similar behavior could happen in 3D spin cases. As shown in Fig. 2(f) at 3 K, when the simulation size is not big enough (less than ~400 Å), the averaged magnetic moment of ML CrCl_{3} remains nearly a constant and does not exhibit large fluctuations, resembling the typical FM order. This also agrees with previous anisotropic Heisenbergmodel studies^{64,66} on planar phases of 2D lattices, in which only the planar FM type curve was observed due to small simulation sizes. On the other hand, when the simulation size is significantly larger than that of meron pairs (~ 200 Å estimated from Fig. 3(a)), quasi orders start to show up because the system is large enough to hold meronpairs. Particularly, with simulation size increasing, owing to the geometric features and growing density of merons, the averaged magnetization is expected to gradually decrease. This is what we observed in Fig. 2(f): when the system size increases, the averaged inplane magnetic moment of ML CrCl_{3} (the deep blue line with red circles) decreases with large fluctuations. In contrast, the magnetic polarization of Isinglike ML CrI_{3} (the green line with light blue circles) does not depend on the system size, showing a normal FM state. Therefore, we conclude that a finitesize simulation can capture those meron pairs if the system size is significantly larger than that of merons. In other words, larger systems can hold more meron pairs but do not change the existence of the meronlike phase.
Finally, we also check the proximity effect from substrates, such as the widely used hexagonal boron nitride (hBN). We do not find significant modifications to the meron states when ML CrCl_{3} is attached to hBN, as shown in the Supplementary Table 2, Supplementary Fig. 3, and Supplementary Note 4. This verifies the feasibility of experimental realization for our theoretical predictions.
Response to external field
It is highly motivated to further investigate the response of the topologically paired merons to an external static magnetic field, which has been widely employed to study magnetic properties of materials^{6,7,35,37}. As discussed above, the polarization direction of the local magnetic moments is crucial for determining the existence of the meron phases. As a result, a magnetic field applied along different directions is expected to induce different responses. First, we apply a small inplane magnetic field (70 Gs), and the results are shown in Fig. 4(a). At low temperature (below 2 K), the magnetic moment is close to the saturated value (3 μ_{B}/Cr), and the fluctuation is substantially quenched, indicating a welldefined FM order. This is confirmed by Fig. 4(c, g), in which both the realspace magnetic moment distribution and projected magnetic component along the externalfield direction show a FM order. This suggests that the applied inplane magnetic field breaks the inplane isotropy, forming a preferred polarization direction, which stabilizes the FM order. Interestingly, as temperature goes up, the magnetic polarization decreases, and larger fluctuation accompanies. In the realspace snapshot (Fig. 4(d)), meron pairs emerge. Those thermally excited topological defects are embedded in the aligned spin sea, which is tilted along the direction of the external field. Figure 4(d, h) provide a physics picture analogy of spin distributions in the real space and projected space. The latter can be directly compared with the experimentally observed figure format^{48,49}.
We then explore the case where the applied magnetic field is perpendicular to the material plane. If the outofplane magnetic field is below a critical value (\(B_ \bot ^{{\mathrm{crit}}} \approx 0.2{\mathrm{T}}\)), although it is not strong enough to completely govern the preferred magnetization direction, the local magnetic moments are no longer perfectly within the material plane. Consequently, we observe a net averaged magnetization along the outofplane direction. In this weak magnetic field situation, the topological meron pairs may be preserved. Take \(B_ \bot = 0.1{\mathrm{T}}\) as an example shown in Fig. 4(b). At low temperature, an averaged outofplane magnetization shows up with the help of external field (the orangedotted curve M_{z}). Meanwhile, the inplane magnetic moment (the reddotted curve M_{xy}) exhibits large fluctuations, indicating the existence of excited merons. This is evidenced by both the realspace and projected component plots of the magnetic moments in Fig. 4(e, i). The hot spots in the projected component figure indicate the core regions of merons, which correspond to the previously discussed spin hills at meron cores. When the temperature reaches 10 K, the averaged inplane magnetization approaches zero. On the other hand, the outofplane magnetization is preserved, as shown in Fig. 4(f, j). This is another type of hybridization between FM and meron states. Moreover, the reason for the abnormal elevation of outofplane FM polarization with increasing temperature at around 10 K is that the DD interaction is more sensitive to the arrangements of adjacent magnetic moments than the onsite Zeeman energy provided by the external field.
Discussion
In summary, our studies show that the overall isotropic inplane magnetized 2D material such as ML CrCl_{3} could provide the opportunity for generating merontype topological defects without involving any other interactions. This would give hope to realizing topological spin structures in a much broader range of materials with weak SOC, such as CrF_{3}. Meanwhile, these meronlike topological spin defects may shed light on the understanding of the recently observed suppressed magnetic ordering in ML NiPS_{3}, an AFM easyplane anisotropy material^{67}. In addition to the previously studied Josephson junction arrays or superfluid helium systems.^{15,29,54}, these meronlike pairs may provide an alternative way to study the exotic relations between topological spin textures and quantum critical phenomena, including the superfluidity and superconducting behaviors, in pristine 2D magnetic systems.
Methods
Density functional theory (DFT) calculations
The DFT calculations are performed within the generalized gradient approximation using the Perdew–Burke–Ernzerhof functional as implemented in the Vienna Ab initio Simulation Package^{68}. A planewave basis set with a kinetic energy cutoff of 450 eV and a 5 × 5 × 1 kpoint sampling is adopted for a 2 × 2 × 1 supercell to mimic different magnetic configurations for extracting magnetic interaction constants. The kpoint grid is 12 × 12 × 1 for a 2 × 2 × 1 supercell. The vacuum distance is set to be 20 Å between adjacent layers to avoid spurious interactions. The van der Waals interaction is included by the DFTD2 method^{69}, and SOC is considered. We choose the Hubbard U = 2.7 eV and Hund J = 0.7 eV for Cr^{3+} ions. The structure is relaxed until the force converges within 0.01 eV/Å. The optimal lattice constant is 6.01 Å, which is close to the bulk monoclinic structure lattice constant 6.06 Å^{70}.
Determination of the Heisenberg Hamiltonian coefficients
We obtain the magnetic interaction coefficients of the anisotropic Heisenberg model Hamiltonian (Eq. (2)) by calculating the total energies of different magnetic configurations. We consider the FM and Néel antiferromagnetic (AFM) configurations. The corresponding energies for a unit cell are
in which two magnetic orientations (inplane and outofplane) are calculated to determine the anisotropic coupling constants. Moreover, we flip the magnetic moment of a Cr^{3+} cation in the 2 × 2 × 1 supercell to realize more magnetic configurations (the energies are normalized to one unit cell).
As a result, we obtain six equations, which are used to solve for the five magnetic interaction coefficients in Eq. (1) and the reference energy (\(E_0\)). The extracted coefficients and corresponding magnetic anisotropy energies for CrX_{3} (X = Cr, Br, I) are summarized in the Table 1.
MC simulations
Based on the Hamiltonian Eq. (2), we perform MC simulations via the Metropolis algorithm on 2D hexagonal lattices with a size of 160 × 160 unit cells (if not particularly specified), which contain 51,200 magnetic moments. The periodic boundary condition is implemented. All magnetic moments are set to along the outofplane direction at the initial state to mimic experimental cooling conditions with the help of external field. We run for \(5.12 \times 10^9\) MC steps (\(10 \times 10^5\) steps per site in average) to ensure that the thermal equilibrium is achieved.
The outofplane and inplane magnetizations are defined as:
where N represents the number of total magnetic moments in the simulated system, and <X> denotes the time average of corresponding components after the system achieves the thermal equilibrium.
In magnetic systems, the static spinspin correlation function describes the average scalar product of spins at two lattice sites with a fixed distance \(\left {{\mathbf{r}}_1  {\mathbf{r}}_2} \right = r\). It can be written as:
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one or twodimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966).
Hohenberg, P. C. Existence of longrange order in one and two dimensions. Phys. Rev. 158, 383–386 (1967).
Lado, J. L. & FernándezRossier, J. On the origin of magnetic anisotropy in two dimensional CrI3. 2D Mater. 4, 035002 (2017).
Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in twodimensional systems. J. Phys. C. Solid State Phys. 6, 1181–1203 (1973).
Hellman, F. et al. Interfaceinduced phenomena in magnetism. Rev. Mod. Phys. 89, 025006 (2017).
Huang, B. et al. Layerdependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270–273 (2017).
Gong, C. et al. Discovery of intrinsic ferromagnetism in twodimensional van der Waals crystals. Nature 546, 265–269 (2017).
Burch, K. S., Mandrus, D. & Park, J. G. Magnetism in twodimensional van der Waals materials. Nature 563, 47–52 (2018).
Olsen, T. Theory and simulations of critical temperatures in CrI3 and other 2D materials: easyaxis magnetic order and easyplane Kosterlitz–Thouless transitions. MRS Commun. 9, 1142–1150 (2019).
Chen, L. et al. Topological spin excitations in honeycomb ferromagnet CrI3. Phys. Rev. X 8, 041028 (2018).
Deng, Y. et al. Gatetunable roomtemperature ferromagnetism in twodimensional Fe3GeTe2. Nature 563, 94–99 (2018).
Fei, Z. et al. Twodimensional itinerant ferromagnetism in atomically thin Fe3GeTe2. Nat. Mater. 17, 778–782 (2018).
Gong, Y. et al. Experimental realization of an intrinsic magnetic topological insulator. Chin. Phys. Lett. 36, 076801 (2019).
Otrokov, M. M. et al. Unique thicknessdependent properties of the van der waals interlayer antiferromagnet MnBi2Te4 films. Phys. Rev. Lett. 122, 107202 (2019).
José, J. V. 40 years of BKT theory, (Workd Scientific, 2013).
Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).
Yu, X. Z. et al. Realspace observation of a twodimensional skyrmion crystal. Nature 465, 901–904 (2010).
Shinjo, T. Magnetic vortex core observation in circular dots of permalloy. Science 289, 930–932 (2000).
Tong, Q., Liu, F., Xiao, J. & Yao, W. Skyrmions in the Moiré of van der Waals 2D Magnets. Nano Lett. 18, 7194–7199 (2018).
Wang, W. et al. Spin chirality fluctuation in twodimensional ferromagnets with perpendicular magnetic anisotropy. Nat. Mater. 18, 1054–1059 (2019).
Han, M.G. et al. Topological magneticspin textures in twodimensional van der Waals Cr 2 Ge 2 Te 6. Nano Lett. 19, 7859–7865 (2019).
Ezawa, M. Compact merons and skyrmions in thin chiral magnetic films. Phys. Rev. B 83, 100408 (2011).
Phatak, C., PetfordLong, A. K. & Heinonen, O. Direct observation of unconventional topological Spin structure in coupled magnetic discs. Phys. Rev. Lett. 108, 067205 (2012).
Tan, A. et al. Topology of spin meron pairs in coupled Ni/Fe/Co/Cu(001) disks. Phys. Rev. B 94, 014433 (2016).
Chmiel, F. P. et al. Observation of magnetic vortex pairs at room temperature in a planar αFe2O3/Co heterostructure. Nat. Mater. 17, 581–585 (2018).
Van Waeyenberge, B. et al. Magnetic vortex core reversal by excitation with short bursts of an alternating field. Nature 444, 461–464 (2006).
Yu, X. Z. et al. Transformation between meron and skyrmion topological spin textures in a chiral magnet. Nature 564, 95–98 (2018).
Gao, N. et al. Creation and annihilation of topological meron pairs in inplane magnetized films. Nat. Commun. 10, 5603 (2019).
Blatter, G., Feigel’man, M. V., Geshkenbein, V. B., Larkin, I. A. & Vinokur, V. M. Vortices in hightemperature superconductors. Rev. Mod. Phys. 66, 1125 (1994).
Zwierlein, M. W., AboShaeer, J. R., Schirotzek, A., Schunck, C. H. & Ketterle, W. Vortices and superfluidity in a strongly interacting Fermi gas. Nature 435, 1047–1051 (2005).
Irkhin, V. & Katanin, A. A. Kosterlitzthouless and magnetic transition temperatures in layered magnets with a weak easyplane anisotropy. Phys. Rev. B 60, 2990–2993 (1999).
Jansen, H. J. F. Magnetic anisotropy in densityfunctional theory. Phys. Rev. B 59, 4699 (1999).
Xue, F., Hou, Y., Wang, Z. & Wu, R. Twodimensional Ferromagnetic van der Waals CrX3 (X=Cl, Br, I) monolayers with enhanced anisotropy and curie temperature. Phys. Rev. B 100, 224429 (2019).
Griffiths, R. B. Free energy of interacting magnetic dipoles. Phys. Rev. 176, 655–659 (1968).
Cai, X. et al. Atomically Thin CrCl3: an inplane layered antiferromagnetic insulator. Nano Lett. 19, 3993–3998 (2019).
Klein, D. R. et al. Enhancement of interlayer exchange in an ultrathin twodimensional magnet. Nat. Phys. 15, 1255–1260 (2019).
Kim, H. H. et al. Evolution of interlayer and intralayer magnetism in three atomically thin chromium trihalides. Proc. Natl Acad. Sci. 166, 11131–11136 (2019).
BedoyaPinto, A. et al. Intrinsic 2DXY ferromagnetism in a van der Waals monolayer. Preprint at https://arxiv.org/abs/2006.07605 (2020).
Liu, J., Sun, Q., Kawazoe, Y. & Jena, P. Exfoliating biocompatible ferromagnetic Crtrihalide monolayers. Phys. Chem. Chem. Phys. 18, 8777–8784 (2016).
Jiang, P., Li, L., Liao, Z., Zhao, Y. X. & Zhong, Z. Spin directioncontrolled electronic band structure in twodimensional ferromagnetic CrI3. Nano Lett. 18, 3844–3849 (2018).
Lu, X., Fei, R. & Yang, L. Curie temperature of emerging twodimensional magnetic structures. Phys. Rev. B 100, 205409 (2019).
Pizzochero, M. & Yazyev, O. V. Inducing magnetic phase transitions in monolayer CrI3 via lattice deformations. J. Phys. Chem. C. 124, 7585–7590 (2020).
Sivadas, N., Okamoto, S., Xu, X., Fennie, C. J. & Xiao, D. StackingDependent Magnetism in Bilayer CrI 3. Nano Lett. 18, 7658–7664 (2018).
Pizzochero, M., Yadav, R. & Yazyev, O. V. Magnetic exchange interactions in monolayer CrI3 from manybody wavefunction calculations. 2D Mater. 7, 035005 (2020).
Torelli, D., Thygesen, K. S. & Olsen, T. High throughput computational screening for 2D ferromagnetic materials: the critical role of anisotropy and local correlations. 2D Mater. 6, 045018 (2019).
Hadzibabic, Z., Krüger, P., Cheneau, M., Battelier, B. & Dalibard, J. BerezinskiiKosterlitzThouless crossover in a trapped atomic gas. Nature 441, 1118–1121 (2006).
McCoy, B. M. & Wu, T. T. The TwoDimensional Ising Model. Encyclopedia of Mathematical Physics, (Harvard University Press, 1973). https://doi.org/10.4159/harvard.9780674180758.
Park, J. P. & Crowell, P. A. Interactions of spin waves with a magnetic vortex. Phys. Rev. Lett. 95, 167201 (2005).
Wachowiak, A. et al. Direct observation of internal spin structure of magnetic vortex cores. Science 298, 577–580 (2002).
Lobanov, I. S., Jónsson, H. & Uzdin, V. M. Mechanism and activation energy of magnetic skyrmion annihilation obtained from minimum energy path calculations. Phys. Rev. B 94, 174418 (2016).
Hertel, R. & Schneider, C. M. Exchange explosions: magnetization dynamics during vortexantivortex annihilation. Phys. Rev. Lett. 97, 177202 (2006).
Actor, A. Classical solutions of SU(2) YangMills theories. Rev. Mod. Phys. 51, 461–525 (1979).
Callan, C. G., Dashen, R. & Gross, D. J. Toward a theory of the strong interactions. Phys. Rev. D. 17, 2717–2763 (1978).
Yuli, O. et al. Enhancement of the superconducting transition temperature of La2xSrxCuO4 bilayers: role of pairing and phase stiffness. Phys. Rev. Lett. 101, 057005 (2008).
Berciu, M. & John, S. Magnetic structure factor in cuprate superconductors: evidence for charged meron vortices. Phys. Rev. B 69, 224515 (2004).
Kim, S. K. Dynamics of bimeron skyrmions in easyplane magnets induced by a spin supercurrent. Phys. Rev. B 99, 224406 (2019).
Zhang, X. et al. Static and dynamic properties of bimerons in a frustrated ferromagnetic monolayer. Phys. Rev. B 101, 144435 (2020).
Shen, L. et al. Dynamics of ferromagnetic bimerons driven by spin currents and magnetic fields. Preprint at https://arxiv.org/abs/2005.11924 (2020).
Kim, S. K. Dynamics of bimeron skyrmions in easyplane magnets induced by a spin supercurrent. Phys. Rev. B 99, 224406 (2019).
Li, X. et al. The bimeron clusters in chiral antiferromagnets. Preprint at https://arxiv.org/abs/2002.04387 (2020).
Maleev, S. Dipole forces in twodimensional and layered ferromagnets. Sov. J. Exp. Theor. Phys. 43, 1240 (1976).
Bruno, P. Spinwave theory of twodimensional ferromagnets in the presence of dipolar interactions and magnetocrystalline anisotropy. Phys. Rev. B 43, 6015–6021 (1991).
Maier, P. G. & Schwabl, F. Ferromagnetic ordering in the twodimensional dipolar XY model. Phys. Rev. B 70, 134430 (2004).
Mól, L. A. S. & Costa, B. V. Anisotropic Heisenberg model with dipolar interactions: Monte Carlo simulations of the planartoparamagnetic phase transition in a bilayer system. Phys. Rev. B 79, 054404 (2009).
Bramwell, S. T. & Holdsworth, P. C. W. Magnetization: a characteristic of the KTB transition. Phys. Rev. B 49, 8811–8814 (1994).
Rapini, M., Dias, R. A. & Costa, B. V. Phase transition in ultrathin magnetic films with longrange interactions: Monte Carlo simulation of the anisotropic Heisenberg model. Phys. Rev. B 75, 014425 (2007).
Kim, K. et al. Suppression of magnetic ordering in XXZtype antiferromagnetic monolayer NiPS3. Nat. Commun. 10, 345 (2019).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Grimme, S. Accurate description of van der Waals complexes by density functional theory including empirical corrections. J. Comput. Chem. 25, 1463–1473 (2004).
McGuire, M. A. et al. Magnetic behavior and spinlattice coupling in cleavable van der Waals layered CrCl3 crystals. Phys. Rev. Mater. 1, 1–10 (2017).
Acknowledgements
The work is supported by the Air Force Office of Scientific Research (AFOSR) grant No. FA95501710304 and National Science Foundation (NSF) CAREER Grant No. DMR1455346. The computational resources have been provided by the Stampede of TeraGrid at the Texas Advanced Computing Center (TACC) through XSEDE.
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L.Y. supervised the project. X.L performed the DFT and MC simulations and drafted the paper. All authors discussed the results and edited the paper.
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Lu, X., Fei, R., Zhu, L. et al. Meronlike topological spin defects in monolayer CrCl_{3}. Nat Commun 11, 4724 (2020). https://doi.org/10.1038/s41467020185738
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DOI: https://doi.org/10.1038/s41467020185738
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