⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 067503 Anisotropy Engineering Edge Magnetism in Zigzag Honeycomb Nanoribbons * Baoyue Li (李宝玥)1, Yifeng Cao (曹逸锋)2, Lin Xu (徐琳)2, Guang Yang (杨光)3,2**, Zhi Ma (马治)1, Miao Ye (叶苗)4, Tianxing Ma (马天星)2 Affiliations 1School of Physics and Electronic-Electrical Engineering, Ningxia University, Yinchuan 750021 2Department of Physics, Beijing Normal University, Beijing 100875 3School of Science, Hebei University of Science and Technology, Shijiazhuang 050018 4College of Information Science and Engineering, Guilin University of Technology, Guilin 541004 Received 20 February 2019, online 18 May 2019 *Supported by the National Natural Science Foundation of China under Grant No 11774033, and the Beijing Natural Science Foundation under Grant No 1192011.
**Corresponding author. Email: yangguang@mail.bnu.edu.cn Citation Text: Li B Y, Cao Y F, Xu L, Yang G and Ma Z et al 2019 Chin. Phys. Lett. 36 067503    Abstract It has been demonstrated that the zigzag honeycomb nanoribbons exhibit an intriguing edge magnetism. Here the effect of the anisotropy on the edge magnetism in zigzag honeycomb nanoribbons is investigated using two kinds of large-scale quantum Monte Carlo simulations. The anisotropy in zigzag honeycomb nanoribbons is characterized by the ratios of nearest-neighbor hopping integrals $t_{1}$ in one direction and $t_{2}$ in another direction. Considering the electron-electron correlation, it is shown that the edge ferromagnetism could be enhanced greatly as $t_{2}/|t_{1}|$ increases from 1 to 3, which not only presents an avenue for the control of this magnetism but is also useful for exploring further novel magnetism in new nano-scale materials. DOI:10.1088/0256-307X/36/6/067503 PACS:75.75.-c, 75.50.Pp © 2019 Chinese Physics Society Article Text Since the discovery of graphene, extensive attention from the research community has been attracted by the emerging honeycomb and honeycomb-like two-dimensional (2D) materials due to their exotic electronic, optical and magnetic properties.[1] The family of these materials includes hexagonal boron nitride, transition-metal dichalcogenides,[2,3] silicene,[4–6] germanene,[7] hafnium monolayer,[8] phosphorene[9–15] as well as their allotropes,[16–23] and so forth. As a crucial prerequisite for their practical applications, various methods have been proposed to tailor and generate their properties. Among them, nanopatterning is a fruitful approach because quantum confinement realized in nanostructures often induces strikingly evident quantum phenomena.[24] Extensive studies have demonstrated that local magnetic moments appear on the edge of zigzag graphene nanoribbons (ZGNRs),[25–30] and the shape of the zigzag edge is shown in Fig. 1, where the top and the bottom of the lattice structure both show a sketch of the zigzag edge. Such quantum phenomenon in ZGNRs instigates further exploration of the edge magnetism in honeycomb and honeycomb-like nanoribbons such as molybdenum disulfide (MoS$_2$)[31] and phosphorene,[32–34] which may open an avenue to their possible applications in spintronics. For spintronics, it is required that Curie temperatures of the targeted materials should be higher than the ambient temperature, which is supposed to be approximately room temperature.[35] To solve this challenging problem, further theoretical and experimental investigations are necessary. It has been discovered that pristine graphene is nonmagnetic due to the vanishing density of states (DOS) at the Dirac point.[36] Strikingly, the appearance of edges in a honeycomb-lattice nanostructure gives rise to additional electronic states along the edges at Fermi level which form a quasi-flat band taking up one-third of the one dimensional Brillouin zone in ZGNRs.[37] These striking edge states induce novel magnetic[25,29] and optical properties.[38] As a well-controlled route, strain engineering is often utilized to modulate the magnetic properties of two dimensional materials and the corresponding nanostructures.[15,39–41] For ZGNRs, applying strain along the zigzag direction has been theoretically proposed to reinforce the edge magnetism.[27,28,42] The anisotropy induced by strain leads to the displacement of the Dirac points. Thus the electronic correlation effect is enhanced by the higher DOS in the extended flat band, which catalyses the enhancement of edge magnetism. The proper strain could even trigger room-temperature edge magnetism under a suitable Coulomb interaction.[42] Distinct from graphene, the puckered structure of phosphorene with a honeycomb lattice endows this material with strong anisotropy.[43] Consequently, the quasi-flat band of zigzag phosphorene nanoribbons (ZPNRs) expands across the entire one-dimensional Brillouin zone and it is completely detached from the bulk band.[44,45] First-principles and quantum Monte Carlo studies have shown the existence of edge ferromagnetism in ZPNRs, which is much stronger than that in ZGNRs.[32–34] Considering the relatively weak Coulomb interaction, it is predicted that the Curie temperature could be even higher, possibly even up to room temperature.[34] No matter whether the anisotropy is induced by intentionally introduced strains in the targeted materials or it is an inborn quality of the materials of interest, the study of the anisotropic effect on the ferromagnetism along the zigzag edges of honeycomb nanoribbons has great academic significance and may advance the development of spintronics.
cpl-36-6-067503-fig1.png
Fig. 1. The top view sketch of zigzag honeycomb nanoribbons. The atoms on A (B) sublattices are represented by the blue (red) circles, respectively. The black lines indicate $t_{1}$, and the pink lines indicate $t_{2}$. We adopt the periodic boundary condition in the $x$-direction and the finite size in the $y$-direction. The zigzag chains are denoted by index $R$. A unit cell is marked by the dotted line.
According to the literature,[46] honeycomb lattice is bipartite, which can be divided into two sets of sublattices represented by A (blue circle) and B (red circle) in Fig. 1. As shown in Fig. 1, $t_{1}$ and $t_{2}$ represent two nearest-neighbor hoping integrals and their ratio, namely, $t_{2}/|t_{1}|$, denotes the strength of anisotropy, and $t_{2}/|t_{1}|=1.0$ corresponds to the isotropic case of graphene, while for phosphorene, the value of $t_{2}/|t_{1}|$ is near 3.0. It is interesting to explore the detailed picture of the anisotropy engineering edge magnetism in zigzag honeycomb nanoribbons in the region of $t_{2}/|t_{1}|=1.0$–3.0, which may not only shed more light on some other materials, but also provide useful information on synthesizing new materials. In this work, we use two kinds of large-scale quantum Monte Carlo simulations to explore the anisotropic effect on the edge ferromagnetism of zigzag honeycomb nanoribbons. The edge ferromagnetism is found to be enhanced with increasing the value of $t_{2}/|t_{1}|$ from 1.0 to 3.0 under proper interaction because the enhanced interaction effect is caused by the higher DOS located in the extended flat band. Through the picture of the tight-binding model, we find that a band gap is shown to be higher and becomes broader as $t_{2}/|t_{1}|$ increases. The enhancement of Coulomb interaction and the doping effect on the edge magnetism are also displayed. As a prototype of the honeycomb lattice endowed with strong anisotropic nature, phosphorene can be described by a tight-binding model containing five hopping integrals $t_i$ ($i=1$, 2, 3, 4 and 5), where $t_{1}=-1.220$ eV, $t_{2}=3.665$ eV, $t_{3}=-0.205$ eV, $t_{4}=-0.105$ eV, and $t_{5}=-0.055$ eV.[47] For ZPNRs, we adopt the periodic boundary along the $x$-direction and finite lattice size in the $y$-direction. It has been verified that the anisotropic effect of ZPNRs on edge magnetism is mainly reflected by the nearest hopping terms $t_{1,2}$ due to their much higher values than those of $t_{3,4,5}$.[47] Therefore, the present study focuses on the correlation of edge magnetism and $t_{2}$ to $|t_{1}|$ ratios with the vanishing $t_{3,4,5}$ under Coulomb interaction in the honeycomb nanoribbons. The single-band Hubbard model is employed to describe the honeycomb nanoribbons and the Hamiltonian is given as $$ H=\sum_{\langle ij\rangle}{t_{ij}}c_{i\sigma }^†c_{j\sigma }+U\sum_{i}n_{i\uparrow}n_{i\downarrow}-\mu\sum_{\langle i\rangle}c_{i\sigma }^†c_{i\sigma },~~ \tag {1} $$ where $t_{ij}$ represents the hopping integral between the $i$th and $j$th sites and we consider $t_{2}/|t_{1}|=1.0$, 2.0 and 3.0 to explore the anisotropic effect on the edge magnetism, $c_{i\sigma}$ ($c_{i\sigma}^†$) denotes the annihilation (creation) operator of electron at the $i$th site, $n_{i\sigma}=c_{i\sigma}^†c_{i\sigma}$ is the occupation number operator, $\mu$ is the chemical potential, and $U$ is the on-site Coulomb repulsion. As powerful tools for treating the strong correlated systems, the determinant quantum Monte Carlo (DQMC)[48–50] and the constrained path quantum Monte Carlo (CPQMC) methods[51] are utilized to simulate magnetic correlation in the presence of Coulomb interaction.[52–57] The results of DQMC can exhibit the properties of the related systems at finite temperature, while the CPQMC is designed to explore the ground-state properties. The DQMC is free from the notorious sign problem in the half filled cases due to the particle-hole symmetry, it is mainly regarded in this study, and thus the corresponding results are guaranteed to be reliable.[34] To explore the effect of electron fillings, we present some results which are very near to the half filling using the CPQMC, and CPQMC is a method inborn to avoid the sign problem. To explore the thermodynamic properties of the edge magnetism in honeycomb nanoribbons, the uniform magnetic susceptibility $\chi$ along each edge at finite temperatures is calculated using the DQMC. The uniform magnetic susceptibility is defined as the zero-frequency spin susceptibility in the $z$ direction as $$ \chi=\int_{0}^{\beta }d\tau \sum_{ij}\langle S_{i}(\tau)\cdot S_{j}(0)\rangle,~~ \tag {2} $$ where $S_i(\tau)=e^{H\tau}S_i(0)e^{-H\tau}(\hbar=1)$ with $S_{i}=c_{i\uparrow}^†c_{i\uparrow}-c_{i\downarrow}^†c_{i\downarrow}$. First, summation runs over the sites along each edge, and then the edge magnetic susceptibility is obtained by averaging the results of the top edge and the bottom edge. Furthermore, the spatial distribution of the magnetic correlations is elucidated utilizing the CPQMC method to calculate the equal-time magnetic structure factor for each zigzag chain, which is defined as $$ M_{\rm R}=\frac{1}{L_{x}^{2}}\sum_{i,j\in {\rm Row}}S_{i,j},~~ \tag {3} $$ where $S_{i,j}= \langle S_{i}\cdot S_{j}\rangle$, $R$ is the index of the zigzag chain, $i$ and $j$ are the indices of the sites along the $R$th zigzag chain, $L_x$ represents the number of sites in each zigzag chain, and $M_{\rm R}$ is calculated along the zigzag chain from the bottom to the top as shown in Fig. 1. Through the values of spin structure factor $M_{\rm R}$, the spatial distribution of spin correlations could be clearly presented.
cpl-36-6-067503-fig2.png
Fig. 2. The edge magnetic susceptibility dependent on the temperature with different $t_{2}/|t_{1}|$ at half filling, $U=3.0$ and $N=4\times 6\times 6$. Inset: the edge magnetism as a function of $t_{2}/|t_{1}|$ with the certain temperature $T=1/6$ at half filling, $U=3.0$ and $N=4\times 6\times 6$.
To shed light on the anisotropic effect on the edge magnetism in the zigzag honeycomb nanoribbons, Fig. 2 is plotted to exhibit the magnetic susceptibility along the zigzag edge as a function of temperature with different ratios of $t_{2}$ to $|t_{1}|$ at half filling, Coulomb interaction $U=3.0$ and lattice size $4\times 6\times 6$. In the following we take $|t_1|$ as the unit if there is no special illustration. For graphene-based materials, $|t_1|$ is around 2.7 eV, and for phosphorene, $|t_1|$ is around 1.220 eV. The value of the on-site repulsion $U$ can be taken from its estimation in polyacetylene[58–60] $U\cong6.0$–17 eV, which clearly spans a large range of values for graphene based materials, and later the Peierls–Feynman–Bogoliubov variational principle shows that $U\simeq4$ eV is reasonable for graphene, silicene and benzene.[61] Therefore, to explore the importance of interactions on the magnetism of the nanoribbons under study, we study the model Hamiltonian in the range of $U/|t_1|=1$–5, and this is also feasible for phosphorene.[33,34] Apparently, the correlations of edge magnetic susceptibility and temperature display the Curie–Weiss behavior $\chi=A/(T-T_{\rm c})$, which describes the magnetic susceptibility $\chi$ dependent on the temperature above the Curie temperature $T_{\rm c}$. According to the reference line $y=1/x$, all the lines for $t_{2}/|t_{1}|=1.0$, 2.0 and 3.0 diverge at the finite low temperature with $U=3.0$ suggesting that the zigzag honeycomb nanoribbons have ferromagnetic behavior. Moreover, $\chi$ increases with $t_{2}/|t_{1}|$ at low temperature which presents the enhancement of the anisotropy for the edge magnetism in zigzag honeycomb nanoribbons. To provide a clearer diagram for the correlation between $\chi$ and $t_{2}/|t_{1}|$, an inset is added in Fig. 2. When the absolute value of $t_{2}/|t_{1}|$ is larger than 1.0 up to 4.0, the edge magnetic susceptibility almost linearly increases with $t_{2}/|t_{1}|$ as shown in the inset of Fig. 2. However, for the absolute value of $t_{2}/|t_{1}|$ smaller than 1 down to 0, the magnetic susceptibility slightly increases, which is similar to that in zigzag graphene nanoribbons.[42] Therefore, we may assert that stronger anisotropy can induce stronger edge magnetism in zigzag honeycomb nanoribbons.
cpl-36-6-067503-fig3.png
Fig. 3. Temperature dependence of magnetic susceptibility $\chi$ for different Coulomb interactions with the certain $t_{2}/|t_{1}|=2.0$ at half filling and $N=4\times 6\times 6$.
To understand the physical scenarios induced by the Coulomb interaction $U$, the magnetic susceptibility $\chi$ of zigzag honeycomb nanoribbons with different Coulomb interactions $U$ is computed at the same $t_{2}/|t_{1}|$ as illustrated by Fig. 3. Clearly, $\chi$ is enhanced by the interaction $U$ at the same temperature and $t_{2}/|t_{1}|$. In addition, the system is dominated by the ferromagnetic fluctuation at $U\geq 2.0$ and $t_{2}/|t_{1}|=2.0$. Hence, Figs. 2 and 3 show that both anisotropy and interaction can make the edge ferromagnetism robust in the zigzag honeycomb nanoribbons.
cpl-36-6-067503-fig4.png
Fig. 4. The magnetic structure factor for each row with different $t_{2}/|t_{1}|$ at half filling, $U=3.0$ and $N=4\times 6\times 6$.
To further study the spatial distribution of magnetic correlations, CPQMC is used to calculate the equal-time magnetic structure factor $M_{\rm R}$ along each zigzag chain. Figure 4 presents $M_{\rm R}$ with different cases of $t_{2}/|t_{1}|=1.0$, 2.0 and 3.0 at $U=3.0$, half filling and $N=4\times 6\times 6$. For a half-filled Hubbard model on a perfect honeycomb lattice, the system shows antiferromagnetic correlations.[53] As the structure of the honeycomb lattice can be described by two interpenetrating sublattices, the spin correlation between the nearest-neighbor sites (or sites on different sublattices) is negative, due to antiferromagnetic correlations, while the spin correlation between the sites belonging to the same sublattice, for example, between the next nearest-neighbor sites, has to be positive. The value of $M_{\rm R}$ defined here is an average of the spin correlation between sites belonging to the same sublattice, thus it is positive and acts like ferromagnetic behavior.[27]
cpl-36-6-067503-fig5.png
Fig. 5. The magnetic structure factor for each row with different Coulomb interactions at half filling, $t_{2}/|t_{1}|=2.0$ and $N=4\times 6\times 6$.
The value of $M_{\rm R}$ is dramatically larger along each edge than that along each chain in the bulk so that the magnetic correlations are mainly distributed along each edge. Meanwhile, we can see that the edge magnetic correlations become larger with increasing $t_{2}/|t_{1}|$. Thus the enhancement of the anisotropy for edge magnetism is further verified by the results of the CPQMC in agreement with the conclusion obtained from the DQMC. In Fig. 5, the results of the CPQMC illustrate that $M_{\rm R}$ at each chain is dependent on the Coulomb interactions at the same $t_{2}/|t_{1}|$. It is clear that the larger interaction leads to the stronger edge magnetism which is also consistent with the results of DQMC. Even the magnetic structure factor has a finite positive value at $U=1.0$, which does not mean the exact presence of observed magnetism, we have to make careful finite size scaling analysis to explore the properties at thermodynamical limits. This costs a huge amount of CPU time and, therefore, restricts us. However, the results shown in Fig. 5 at least demonstrate that the magnetic structure factor is enhanced greatly as the interaction strength increases. The variation of the topology of the band structure caused by $t_{2}/|t_{1}|$ reveals the nature of the enhanced edge magnetism induced by the anisotropy in such systems as presented in Fig. 6. For the cases of $t_{1}=-1.0$ and $t_{2}=1.0$ in Fig. 6(b), the band structure corresponds to that of zigzag graphene nanoribbons with two Dirac cones at $K$ and $K^{\prime}$. A flat band consisting of the edge states connects these two Dirac points. The flat band takes up one-third of the one dimensional Brillouin zone. We take $t_{1}$ as the unit and increase $t_{2}$. As $t_{2}=2.0$ corresponds to $t_{2}/|t_{1}|=2.0$ in Fig. 6(c), we can see that two Dirac cones approach to ${\it \Gamma}(k=0)$ and then the flat band extends dramatically. In Fig. 6(d), we set $t_{2}=3.0$ and $t_{1}$ as the unit, and $t_{2}/|t_{1}|$ is equivalent to 3.0, which approximately corresponds to zigzag phosphorene nanoribbons according to Ref.  [47]. Under this condition, Fig. 6(d) shows a flat band occupying the entire one dimensional Brillouin zone. Meanwhile, a band gap opens up in the bulk with increasing the anisotropy. The extended flat band derived from the increasing $t_{2}/|t_{1}|$ leads to the higher density of states at Fermi level, which enhances the interaction effect. Thereby, the stronger ferromagnetism is induced by the stronger anisotropy.
cpl-36-6-067503-fig6.png
Fig. 6. The band structure of the zigzag honeycomb nanoribbons with (a) $t_2/|t_{1}|=0.0$, (b)$t_2/|t_{1}|=1.0$, (c)$t_2/|t_{1}|=2.0$, and (d)$t_2/|t_{1}|=3.0$.
cpl-36-6-067503-fig7.png
Fig. 7. The magnetic structure factor for each row at different electron fillings with $U=3.0$, $t_{2}/|t_{1}|=2.0$ and $N=4\times 6 \times 6$.
Finally, the doping effect on the edge magnetism is explored using the CPQMC. The relation between the magnetic structure factor and the electron filling $\langle n\rangle$ is illustrated in Fig. 7. It is clear that the edge ferromagnetism is sharply weakened as the electron filling moves away from the half filling and the doped charge mostly locates along the edge. Therefore, we may give a possible way to manipulate the edge magnetism in the honeycomb nanoribbons. The doping level presented in Fig. 7 is $\delta=1-\langle n\rangle=0.014$ and 0.042 respectively, namely, 1.4% or 4.2% doping ratio, which are within the current experimental capacity, as in graphene and other honeycomb-like 2D materials, doping achievable by gate voltage or chemical doping is usually on the order of $10^{12}\sim10^{13}$ cm$^{-2}$.[1] In summary, we have used both the DQMC and CPQMC methods to explore the effect of the anisotropy, the interaction and the doping on the edge ferromagnetism in the honeycomb nanoribbons. At a fixed Coulomb interaction, for example $U=3.0$, which is a reasonable interaction strength for various two dimensional materials with honeycomb-like structure, our intensive numerical results show that the edge magnetism could be enhanced remarkably as $t_{2}/|t_{1}|$ increases from 1 to 3. For a fix $t_{2}/|t_{1}|=2.0$, a ferromagnetic-like behavior is predicted as $U\geq 2.0$, and the ferromagnetic correlation is reduced greatly with a finite doping. These results provide a route for tailoring the magnetic properties of honeycomb 2D materials and searching for new materials with the honeycomb lattice. We acknowledge computational support from HSCC of Beijing Normal University. 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⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 067501 Classical Ground State Spin Ordering of the Antiferromagnetic $J_1$–$J_2$ Model * Ren-Gui Zhu (朱仁贵)** Affiliations College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000 Received 11 December 2018, online 18 May 2019 *Supported by the National Natural Science Foundation of China under Grant No 11774002.
**Corresponding author. Email: rgzhu@mail.ahnu.edu.cn Citation Text: Zhu R G 2019 Chin. Phys. Lett. 36 067501    Abstract The classical frustrated antiferromagnetic $J_1$–$J_2$ model is considered in a description of the classical spin wave for a vector spin system. Its ground state (GS) spin ordering is analyzed by minimizing its energy. Our analytical derivations show that all the spins in the GS phase must lie in planes that are parallel to each other. When applying the derived formulations to concrete lattices such as the square and simple cubic lattices, we find that in the large $J_2$ region, a large continuous GS degeneracy concluded by a qualitative analysis is lifted, and collinear striped ordering is selected as the GS phase. DOI:10.1088/0256-307X/36/6/067501 PACS:75.10.Jm, 75.30.Et, 75.10.Hk © 2019 Chinese Physics Society Article Text In modern condensed matter physics, frustrated spin systems with competing interactions have been intensively studied in the last 30 years,[1,2] providing us a vast and intriguing world of physics with many novel and exotic magnetic phenomena, even at the classical level. In this research area, of great interest is the frustrated antiferromagnetic (AF) $J_1$–$J_2$ spin model with the Hamiltonian $$ H=\frac{1}{2}J_1\sum_{ < ij>}{\boldsymbol S}_i\cdot{\boldsymbol S}_j+\frac{1}{2}J_2\sum_{[ij]}{\boldsymbol S}_i\cdot{\boldsymbol S}_j,~~ \tag {1} $$ where $J_1$ and $J_2$ are the nearest-neighboring (NN) and the next-nearest-neighboring (NNN) exchange couplings, respectively, and both of them are considered to be AF, i.e., $J_1, J_2>0$. In research of the ground state (GS) phases of this frustrated spin model, the subtle interplay between frustration and fluctuations is a fascinating subject. Frustration often tends to promote the degeneracy of the classical GS phases, while fluctuations can act strongly among the degenerate GS manifold, lift the degeneracy, and select certain spin ordering as the true and stable GS phase. This phenomenon has been called 'order by disorder'.[3–5] A typical example is the discovery of collinear striped ordering in the region of large $J_2$ (or $J_1\to 0$ and $J_2\ne 0$). For the classical model, in this region, because $J_2\gg J_1$, the system is often qualitatively and approximately decoupled into two interpenetrating sublattices with only NNN AF bonds. Thus the two sublattices for every order antiferromagnetically with their spin directions are independent of each other, and the energy of the system is independent of the NN exchange coupling $J_1$, which leads to a highly degenerate classical GS manifold including collinear and noncollinear spin orderings.[6] However, for the quantum model, when quantum fluctuations are considered, this degeneracy will be lifted, and the GS becomes magnetically ordered collinearly as a stripe phase.[6–10] Studying the corresponding classical model is often the first step in studying the quantum spin model, so that influences of the quantum effect can be exhibited more distinctly by comparison. Furthermore, some research methods for the quantum spin model, such as the coupled cluster method,[9] the spin-wave theory[11] and the double-time Green's function method,[12] often recur to the GSs of the corresponding classical model as their reference state at the beginning of the research. Thus it is essential to make the properties of the corresponding classical model clear for the quantum model. However, for the frustrated AF $J_1$–$J_2$ model, in the large $J_2$ region, although the collinear striped ordering as the GS phase has been well accepted at the quantum level, it is still not obvious at the classical level.[13] In this Letter, we restudy the GS of the frustrated AF $J_1$–$J_2$ model still at the classical level, but go beyond the qualitative analysis mentioned above. Resorting to a classical description of the spin wave for a vector spin system, we present detailed analytical derivations of the GS energy and its corresponding spin ordering. Our results show more information about the classical GS than the qualitative analysis did. Especially for the square and simple cubic lattices,[14–16] our results show that in the large $J_2$ region, the collinear striped ordering is obviously the GS phase at the classical level. Here, at the classical level, the lattice spins are treated as classical vectors with length $S$, and their directions can continuously change in real space. According to the classical description of the spin wave,[17–19] we parameterize them by spherical coordinates $$\begin{align} &S_n^x=S\sin\theta\cos\varphi_n,\\ &S_n^y=S\sin\theta\sin\varphi_n,\\ &S_n^z=S\cos\theta,~~ \tag {2} \end{align} $$ where the orientation of the $z$-axis is arbitrarily chosen in real space, and the azimuthal angle $\varphi_n$ is dependent on the lattice site, but the polar angle $\theta$ is not. Taking Eq. (2) into Eq. (1), and rewriting the azimuthal angle by introducing a wave vector ${\boldsymbol q}$, $$ \varphi_n={\boldsymbol q}\cdot{\boldsymbol r}_n,~~ \tag {3} $$ we can finally obtain the energy of the model in a parameterized form $$ E_{\rm cl}=\frac{1}{2}Z_1J_1NS^2[\cos^2\theta(1+p)+\sin^2\theta(\gamma_{1{\boldsymbol q}}+p\gamma_{2{\boldsymbol q}})],~~ \tag {4} $$ with the structure factors $$ \gamma_{i{\boldsymbol q}}=\frac{1}{Z_i}\sum_{{\boldsymbol\delta}_i}\cos({\boldsymbol q}\cdot{\boldsymbol\delta}_i), i=1,2,~~ \tag {5} $$ where $N$ is the total number of lattice sites, $p=(Z_2 J_2)/(Z_1J_1)$ is the frustration parameter, $Z_1$ ($Z_2$) is the coordination number of the nearest (next-nearest) neighbors, and ${\boldsymbol\delta}_1$ and ${\boldsymbol\delta}_2$ are the lattice vectors connecting the NN and NNN sites, respectively. In the expression of the classical energy, i.e., Eq. (4), there is a function $f(\theta)=A\cos^2\theta+B\sin^2\theta$ with $A=1+p$ and $B=\gamma_{1{\boldsymbol q}}+p\gamma_{2{\boldsymbol q}}$ for $B\le A$ because of $\gamma_{1{\boldsymbol q}}\le1$ and $\gamma_{2{\boldsymbol q}}\le1$. The single minimum point of this function is $\theta=\pi/2$, thus we can obtain an upper limit of the classical GS energy by substituting $\theta=\pi/2$ into Eq. (4) and obtain $$ \bar{E}_{\rm cl}=\frac{1}{2}Z_1J_1NS^2(\gamma_{1{\boldsymbol q}}+p\gamma_{2{\boldsymbol q}}).~~ \tag {6} $$ Because the orientation of the $z$-axis is arbitrarily selected, the polar angle $\theta=\pi/2$ means that all the spin vectors in real space must lie in planes that are parallel to each other. This result is independent of the frustration parameter $p$ and applicable to all dimensions. Let us consider the square lattice. For convenience, without loss of generality, the $z$-axis is set perpendicular to the square plane, and the $x$- and $y$-axes are set along the two edges of the primitive square plaquette, respectively. We have shown that the polar angle $\theta$ must be $\pi/2$ for the GS spin ordering, thus here all the spin vectors lie in the square plane (or $x$–$y$ plane). Whether they are aligned collinearly or noncollinearly can be judged from the values of the azimuthal angles $\varphi_n,n=1,2,\ldots$. For a square lattice, the coordination numbers are $Z_1=4$ and $Z_2=4$. In the unit of lattice spacing ($a=1$), the lattice vectors connecting four NN sites are ${\boldsymbol a}_1=-{\boldsymbol a}_3=(1,0,0)$, ${\boldsymbol a}_2=-{\boldsymbol a}_4=(0,1,0)$, and the ones connecting four NNN sites are ${\boldsymbol b}_1=-{\boldsymbol b}_3=(1,1,0)$, ${\boldsymbol b}_2=-{\boldsymbol b}_4=(-1,1,0)$. From Eq. (5), we obtain the structure factors for the square lattice $$\begin{align} \gamma_{1{\boldsymbol q}}=\,&\frac{1}{2}(\cos q_x+\cos q_y),~~ \tag {7} \end{align} $$ $$\begin{align} \gamma_{2{\boldsymbol q}}=\,&\cos q_x\cos q_y.~~ \tag {8} \end{align} $$ In the region of small $J_2$ (or $p\ll 1$), the main contribution to the classical energy $\bar{E}_{\rm cl}$ in Eq. (6) comes from the NN structure factor $\gamma_{1{\boldsymbol q}}$. From Eq. (7), we find that the minimum of $\gamma_{1{\boldsymbol q}}$ occurs at the following wave vectors $$ {\boldsymbol q}=\pi(m,n,C), \mbox{with odd}~m,~n,~~ \tag {9} $$ where $C$ is an arbitrary constant. This form of wave vectors leads to GS spin ordering with the relative azimuthal angles between the NN spins and between the NNN spins as $$ \begin{cases} \Delta\varphi_1={\boldsymbol q}\cdot{\boldsymbol a}_i=k\pi,~\mbox{with odd}~k\\ \Delta\varphi_2={\boldsymbol q}\cdot{\boldsymbol b}_i=k\pi,~\mbox{with even}~k \end{cases},~\mbox{for}~~ i=1,2,3,4,~~ \tag {10} $$ which means that all the nearest neighbors (next-nearest neighbors) of a given spin are aligned exactly antiparallel (parallel) to it. This is the classical two-sublattice Néel ordering phase, as shown in Fig. 1(a).
cpl-36-6-067501-fig1.png
Fig. 1. Classical GS phases of the square lattice: (a) Néel ordering for small $J_2$; (b) and (c) the two degenerate collinear striped orderings for large $J_2$.
Substituting Eq. (9) into Eq. (6), we obtain the classical GS energy of the Néel phase $$ E_0^{\rm N}({\rm square})=-\frac{1}{2}Z_1J_1NS^2(1-p).~~ \tag {11} $$ In the region of large $J_2$ (or $p\gg1$), the main contribution to the classical energy $\bar{E}_{\rm cl}$ in Eq. (6) comes from the NNN structure factor $\gamma_{2{\boldsymbol q}}$. From Eq. (8), we find that the minimum of $\gamma_{2{\boldsymbol q}}$ occurs at the following wave vectors $$ {\boldsymbol q}=\pi(m,n,C), ~\mbox{with odd}~m~{\rm and even}~n~{\rm or vice versa},~~ \tag {12} $$ where $C$ is an arbitrary constant. If we choose odd $m$ and even $n$, this form of wave vectors will lead to GS spin ordering with the relative azimuthal angles between the NN spins as $$\begin{align} &\Delta\varphi_1={\boldsymbol q}\cdot{\boldsymbol a}_i=k\pi, \\ &\mbox{with odd}~k~ {\rm for}~i=1,~3, ~~\mbox{with even}~k~{\rm for}~i=2,~4,~~ \tag {13} \end{align} $$ and the ones between the NNN spins as $$ \Delta\varphi_2={\boldsymbol q}\cdot{\boldsymbol b}_i=k\pi,~\mbox{with odd}~k~{\rm for}~ i=1,~2,~3,~4.~~ \tag {14} $$ The expression of $\Delta\varphi_1$ in Eq. (13) means that the NN spins along the $x$-axis are aligned exactly antiparallel to each other, while the ones along the $y$-axis are aligned parallel to each other. The expression of $\Delta\varphi_2$ in Eq. (14) means that all the NNN spins are aligned exactly antiparallel to each other. This spin ordering is a collinear striped ordering, as shown in Fig. 1(b). Going back to Eq. (12), if we choose even $m$ and odd $n$, we can obtain nothing but a degenerate collinear striped ordering of the former, with the NN spins along the $y$-axis aligned antiparallel to each other, whereas those along the $x$-axis are aligned parallel to each other, as shown in Fig. 1(c). Substituting Eq. (12) into Eq. (6), we can obtain the classical GS energy of this collinear striped ordering phase $$ E_0^{\rm S}(\mbox{square})=-\frac{1}{2}Z_1J_1NS^2p.~~ \tag {15} $$ Interestingly, because $p=(Z_2J_2)/(Z_1J_1)$, this energy is in fact independent of the NN coupling $J_1$, the same as the energy from the qualitative analysis that neglects $J_1$ at the beginning. However, our analysis shows that the GS phase is collinear striped ordering, which is a result beyond the qualitative analysis. For a simple cubic (SC) lattice, it is convenient to set the $x,y,z$-axes along the three edges of the primitive cube, respectively. The coordination numbers are $Z_1=6$ and $Z_2=12$. In the unit of lattice spacing ($a=1$), the lattice vectors connecting six NN sites are ${\boldsymbol a}_1=-{\boldsymbol a}_4=(1,0,0)$, ${\boldsymbol a}_2=-{\boldsymbol a}_5=(0,1,0)$, ${\boldsymbol a}_3=-{\boldsymbol a}_6=(0,0,1)$, and the ones connecting twelve NNN sites are ${\boldsymbol b}_1=-{\boldsymbol b}_7=(1,1,0)$, ${\boldsymbol b}_2=-{\boldsymbol b}_8=(-1,1,0)$, ${\boldsymbol b}_3=-{\boldsymbol b}_9=(1,0,1)$, ${\boldsymbol b}_4=-{\boldsymbol b}_{10}=(-1,0,1)$, ${\boldsymbol b}_5=-{\boldsymbol b}_{11}=(0,1,1)$, ${\boldsymbol b}_6=-{\boldsymbol b}_{12}=(0,-1,1)$. Taking these lattice vectors into Eq. (5), we obtain the structure factors for the SC lattice, $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\gamma_{1{\boldsymbol q}}=\,&\frac{1}{3}[\cos q_x+\cos q_y+\cos q_z],~~ \tag {16} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\gamma_{2{\boldsymbol q}}=\,&\frac{1}{3}[\cos q_x\cos q_z\!+\!\cos q_x\cos q_y\!+\!\cos q_y\cos q_z].~~ \tag {17} \end{alignat} $$ The subsequent analyses are similar to those for the square lattice, thus we give only the results and some necessary discussions in the following. In the region of small $J_2$, the minimum of classical energy for this case occurs at the following wave vectors $$ {\boldsymbol q}=\pi(l,m,n),~\mbox{with}~l,~m,~n~{\rm all odd},~~ \tag {18} $$ and the relative pitch angles between NN spins and between NNN spins are $$\begin{align} &\Delta\varphi_1={\boldsymbol q}\cdot{\boldsymbol a}_i=k\pi,~\mbox{with odd}~k~{\rm for}~ i=1,~2,\cdots,6\\ &\Delta\varphi_2={\boldsymbol q}\cdot{\boldsymbol b}_i=k\pi,~\mbox{with even}~k~{\rm for}~ i=1,~2,~\cdots,12,~~ \tag {19} \end{align} $$ which also lead to the Néel phase, as shown in Fig. 2(a). The corresponding classical GS energy is $$ E_0^{\rm N}(\mbox{SC})=-\frac{1}{2}Z_1J_1NS^2(1-p).~~ \tag {20} $$
cpl-36-6-067501-fig2.png
Fig. 2. Classical GS phases of the SC lattice: (a) Néel ordering for small $J_2$; (b), (c) and (d) the three degenerate collinear striped orderings. The direction of the (0,0,0) site spin vector is assumed to be along the $x$-axis.
In the region of large $J_2$, the minimum of classical energy for this case occurs at the following wave vectors $$ {\boldsymbol q}=\pi(l,m,n),~\mbox{with only one of}~l,~m,~n~{\rm even}.~~ \tag {21} $$ If we choose even $l$, the relative pitch angles between NN spins and between NNN spins are $$\begin{alignat}{1} &\Delta\varphi_1= {\boldsymbol q}\cdot{\boldsymbol a}_{i}=k{\pi},\\ &\mbox{with even}~k~{\rm for}~i=1,~4,\\ &\mbox{with odd}~k~{\rm for}~i=2,~3,~5,~6,~~ \tag {22} \end{alignat} $$ $$\begin{alignat}{1} &\Delta\varphi_2={\boldsymbol q}\cdot{\boldsymbol b}_i=k{\pi},\\ &\mbox{with even}~k~{\rm for}~i=5,~6,~11,~12,\\ &\mbox{with odd}~k~{\rm for}~i=1,~2,~3,~4,~7,~8,~9,~10,~~ \tag {23} \end{alignat} $$ which lead to the following spin ordering: the NN spins along the $x$-axis are all aligned parallel to each other, while the NN spins along the $y$- and $z$-axes are all aligned antiparallel to each other. This is also a kind of collinear striped ordering, as shown in Fig. 2(b). If we choose even $m$ or even $n$, we will obtain the other two collinear striped ordering phases which are degenerate with the former, as shown in Figs. 2(c) and 2(d). Substituting Eq. (21) into Eq. (6), we can obtain the classical energy of the stripe ordered phase $$ E_0^{{\rm S}}(\mbox{SC})=-\frac{1}{6}Z_1J_1NS^2(1+p).~~ \tag {24} $$ It is noticed that this form of energy is dependent on the NN coupling $J_1$, which is different from the case of the square lattice. In summary, it has been widely accepted that collinear striped spin ordering is the GS phase of the quantum frustrated AF $J_1$–$J_2$ spin model in the large $J_2$ region. However, for the classical model, it is not obvious whether the case is the same. At the classical level, when $J_2\gg J_1$, the system is often qualitatively and approximately decoupled into two independent sublattices with NNN bonds only, which leads a highly degenerate GS manifold including collinear and noncollinear spin orderings. In this study, going beyond the qualitative and approximate treatment mentioned above, we have restudied the classical model resorting to a classical description of the spin wave for a vector spin system. From the analytical derivation for the classical GS energy and its corresponding spin ordering, we conclude firstly that in the GS phase, all the spins must lie in planes that are parallel to each other. Then for square and SC lattices, we further conclude that in the large $J_2$ region, the spin ordering in the GS phase must be collinear and striped. Thus for the classical model, our results show that in a classical spin wave description, the collinear striped ordering can also be selected as the true GS phase from the highly degenerate GS manifold. Finally, we are obliged to point out that our derivations are not suitable for the region near the critical frustration parameter denoted by $p_{\rm c}=J_{\rm 2c}/J_1$, where the frustration is so strong that it causes the system to be in a magnetic disordered phase between the Néel phase and the stripe ordered phase. If we solve the equation $E_0^{\rm N}=E_0^{\rm S}$, we can obtain $p_{\rm c}=1/2$ for the square lattice and $p_{\rm c}=1/4$ for the SC lattice, which are just the classical critical frustration parameters of the AF $J_1$–$J_2$ model on these two lattices, respectively.[6] Thus our derivations and results are effective far outside of the region $J_2/J_1\sim p_{\rm c}$ where the frustration is weak. References Order as an effect of disorderOrdering due to disorder in a frustrated vector antiferromagnetOrdering Due to Quantum Fluctuations in the Frustrated Heisenberg ModelPhase diagram of the frustrated spin-1/2 Heisenberg antiferromagnet in 2 dimensionsSpin Wave Theory for a Frustrated Antiferromagnetic Heisenberg Model on a Square LatticeGround-state ordering of the J 1 − J 2 model on the simple cubic and body-centered cubic latticesFrustrated J 1 − J 2 − J 3 Heisenberg antiferromagnet on the simple cubic latticeNon-linear spin wave theory results for the frustrated S=\frac 12 Heisenberg antiferromagnet on a body-centered cubic latticeThe J 1 - J 2 model on the face-centered-cubic latticesFrustrated two dimensional quantum magnetsMagnetic behaviors and multitransition temperatures in a nanoislandTensor-product state approach to spin- 1 2 square J 1 − J 2 antiferromagnetic Heisenberg model: Evidence for deconfined quantum criticalityMagnetic properties of a nanoribbon: An effective-field theoryMagnetic solitons in a frustrated ferromagnetic spin chainFrustrated Ferromagnetic Spin Chain near the Transition Point
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