⇦Chinese Physics Letters, 2016, Vol. 33, No. 7, Article code 077503 Self-Consistent Spin-Wave Analysis of the 1/3 Magnetization Plateau in the Kagome Antiferromagnet * Zhong-Chao Wei(魏忠超)1, Hai-Jun Liao(廖海军)1**, Jing Chen(陈靖)1, Hai-Dong Xie(谢海东)1, Zhi-Yuan Liu(刘志远)1, Zhi-Yuan Xie(谢志远)2, Wei Li(李伟)3,4, B. Normand2, Tao Xiang(向涛)1,5 Affiliations 1Institute of Physics, Chinese Academy of Sciences, Beijing 100190 2Department of Physics, Renmin University of China, Beijing 100872 3Department of Physics, Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), Beihang University, Beijing 100191 4International Research Institute of Multidisciplinary Science, Beihang University, Beijing 100191 5Collaborative Innovation Center of Quantum Matter, Beijing 100190 Received 15 May 2016 *Supported by the National Natural Science Foundation of China under Grant Nos 10934008, 10874215 and 11174365, and the National Basic Research Program of China under Grant Nos 2012CB921704 and 2011CB309703.
**Corresponding author. Email: navyphysics@iphy.ac.cn Citation Text: Wei Z C, Liao H J, Chen J, Xie H D and Liu Z Y et al 2016 Chin. Phys. Lett. 33 077503 Abstract We propose a modified spin-wave theory to study the 1/3 magnetization plateau of the antiferromagnetic Heisenberg model on the kagome lattice. By the self-consistent inclusion of quantum corrections, the 1/3 plateau is stabilized over a broad range of magnetic fields for all spin quantum numbers $S$. The values of the critical magnetic fields and the widths of the magnetization plateaus are fully consistent with the recent numerical results from exact diagonalization and infinite projected entangled paired states. DOI:10.1088/0256-307X/33/7/077503 PACS:75.10.Jm, 75.30.Ds, 75.30.Kz © 2016 Chinese Physics Society Article Text The investigation of low-dimensional frustrated magnetism has become one of the most active frontiers in condensed matter physics. Current frontiers in the field include obtaining full insight into the entanglement structure of quantum many-body wavefunctions for different types of quantum spin liquid[1] and simplex-solid state.[2] Among many different frustrated systems, quantum antiferromagnets on the kagome lattice may be the most intriguing, due to the fact that the combination of strong geometric frustration and weak constraints maximizes quantum fluctuation effects. The kagome antiferromagnet has attracted ever-increasing attention over the last two decades, with many different methods applied and resulting in proposals for the nature of the ground state.[3-6] Realizations of the kagome geometry have now been discovered for many materials, including volborthite (Cu$_3$V$_2$O$_7$(OH)$_2\cdot$2H$_2$O),[7] herbertsmithite (ZnCu$_3$(OH)$_6$Cl$_2$),[8] vesignieite (BaCu$_3$(OH)$_6$Cl$_2$),[9] BaNi$_3$(OH)$_2$(VO$_4$)$_2$,[10] KV$_3$Ge$_2$O$_9$,[11] and jarosites of several different metal ions including chromium (KCr$_3$(OH)$_6$(SO$_4$)$_2$).[12] One of the characteristic features of kagome antiferromagnets is the appearance of magnetization plateaus in the presence of an external magnetic field. Irrespective of the method applied and the prediction for the zero-field ground state, all theoretical approaches agree that there exists a robust magnetization plateau at $m=1/3$ for all values of the spin quantum number $S$. The 1/3 plateau has been investigated theoretically by the real-space perturbation theory (RSPT),[14] exact diagonalization (ED),[15,16] density-matrix renormalization-group methods (DMRG),[17] and infinite projected entangled paired states (iPEPS).[18] The RSPT provides analytical results for the critical magnetic fields and the width of the plateau.[14] However, a qualitative discrepancy has arisen with recent numerical results from ED[15] and iPEPS,[18] not least in that the calculated plateau width increases with increasing $S$, whereas it decreases within RSPT. To improve qualitative and quantitative understanding of the 1/3 magnetization plateau in the kagome antiferromagnet, in this work we employ a self-consistent spin-wave theory to study its properties. The theory contains a single quantum correction parameter, determined self-consistently from the expectation values of the magnon densities. We compute these densities for all $S$, derive the spin-wave spectrum, and evaluate both the critical magnetic fields and the width of the 1/3 plateau, finding complete consistency with the recent numerical results. We study the Hamiltonian $$\begin{align} H=\sum_{\langle i,j\rangle}{\boldsymbol S}_{i}\cdot{\boldsymbol S}_{j}-h\sum_{i}S_{i}^{z},~~ \tag {1} \end{align} $$ where ${\boldsymbol S}_i$ is the spin-$S$ operator on site $i$, $\langle i,j \rangle$ denotes the sum over neighboring sites, and $J$, the nearest-neighbor antiferromagnetic exchange coupling, is set as the energy scale ($J=1$). Recent numerical studies of the kagome and Husimi lattices by the method of projected entangled simplex states (PESS)[6] have demonstrated very explicitly[13] that the origin of the 1/3-plateau phase is the creation of a semiclassical up-up-down spin configuration on every triangle, as shown in Fig. 1. We stress that this statement holds for all values of $S$, even $S=1/2$. As a consequence, it is entirely justified to employ a spin-wave description, which we implement by performing a Holstein–Primakoff transformation from spin operators to bosonic degrees of freedom, $$\begin{alignat}{1} S_{i}^{+}=\,& \sqrt{2S-d_{i}^{†ger} d_{i}} d_{i},~ S_{i}^{z}=S-d_{i}^{†ger} d_{i},~i \in {\rm A,B},~~ \tag {2} \end{alignat} $$ $$\begin{alignat}{1} S_{i}^{+}=\,&-d_{i}^{†ger} \sqrt{2S-d_{i}^{†ger} d_{i}},~ S_{i}^{z}=d_{i}^{†ger} d_{i}-S,~i \in {\rm C}.~~ \tag {3} \end{alignat} $$ Here we have assumed that spins on the A and B sublattices are oriented along ${\hat z}$, while those on the C sublattice are oriented along $-{\hat z}$. We restrict all of our considerations to zero temperature.
cpl-33-7-077503-fig1.png
Fig. 1. The up-up-down spin configuration of the 1/3 magnetization plateau on the kagome lattice. The two primitive lattice vectors are denoted as ${\boldsymbol a}_1=(1,0)a$ and ${\boldsymbol a}_2=\Big(\frac{1}{2},\frac{\sqrt 3}{2}\Big)a$, where we set $a=1$. The unit cell, denoted by the dashed blue lines, contains the three sites $(A, B, C)$. Here $\delta_{1}$, $\delta_{2}$, and $\delta_{3}$ denote the nearest-neighbor lattice vectors.
In a conventional linear spin-wave theory ($S \rightarrow \infty$), the 1/3 plateau is stable only at $h=2S$. However, the effects of quantum fluctuations at finite $S$ may be included through the effective mean-field boson densities at each site, $n_{i} \equiv \langle d_{i}^{†ger} d_{i}\rangle$, which are determined self-consistently and act to stabilize the magnetization plateau over a finite range of $h$. By introducing the approximation $$ \sqrt{2S-d_{i}^{†ger} d_{i}} \approx \sqrt{2S-\langle d_{i}^{†ger} d_{i} \rangle}=\sqrt{2S-n_{i}},~~ \tag {4} $$ the Hamiltonian (1) can be decoupled in the form $$\begin{align} \mathcal{H}_{\rm MF}=\sum_{k} \alpha_{k}^{†ger}[\mathcal{H}({\boldsymbol k})+\delta h {\it \Lambda}] \alpha_{k}+\mathcal{C},~~ \tag {5} \end{align} $$ where $\alpha_{k}\equiv(a_{k},b_{k},c_{-k}^{†ger})^{T}$, $a_{k}$, $b_{k}$, and $c_{k}$ are the Fourier transforms of the operators, $d_{i}$, for each of the A, B, and C sublattices, $\delta h=(h-2S)$, ${\it \Lambda}={\rm diag}(1,1,-1)$, and $\mathcal{C}$ is a constant. Here $\mathcal{H}({\boldsymbol k})$ specifies the quadratic Hamiltonian at $h=2S$, $$ \hat{\mathcal{H}}({\boldsymbol k})=\left(\begin{matrix} 2S & d_{k} & f_{k} \\ d_{k} & 2S & g_{k} \\ f_{k} & g_{k} & 2S \end{matrix}\right), $$ where $$\begin{align} d_{k}=\,&\sqrt{(2S-n_{\rm A})(2S-n_{\rm B})}\cos({\boldsymbol k}\cdot{\boldsymbol \delta_{1}}), \\ f_{k}=\,&-\sqrt{(2S-n_{\rm A})(2S-n_{\rm C})}\cos({\boldsymbol k}\cdot{\boldsymbol \delta_{2}}),\\ g_{k}=\,&-\sqrt{(2S-n_{\rm B})(2S-n_{\rm C})}\cos({\boldsymbol k} \cdot {\boldsymbol \delta_{3}}),~~ \tag {6} \end{align} $$ with the bond vectors $\delta_{i}$ as shown in Fig. 1. In Eq. (6), $n_{\rm A}$, $n_{\rm B}$ and $n_{\rm C}$ are respectively the expectation values of the magnon density, $\langle d_{i}^{†ger} d_{i}\rangle$, on each of the A, B, and C sublattices. From the Holstein–Primakoff transformation, the normalized longitudinal magnetizations on the three sublattices are $m_{\rm A}=1- n_{\rm A}/S$, $m_{\rm B}=1-n_{\rm B}/S$, and $m_{\rm C}=n_{\rm C}/S-1$. In the 1/3-plateau phase, by definition $(m_{\rm A}+ m_{\rm B}+m_{\rm C})/3=1/3$ and therefore $n_{\rm A}+n_{\rm B}=n_{\rm C}$. Further, in the following we employ the reflection symmetry of the system about a vertical axis through the C sites (Fig. 1), which specifies that $m_{\rm A}=m_{\rm B}$ and $n_{\rm A}=n_{\rm B}$. Thus the self-consistent spin-wave theory for the 1/3-plateau phase contains only one parameter to be determined, which we denote as $x=n_{\rm A}=n_{\rm B}=n_{\rm C}/2$. The diagonalization of a general quadratic bosonic Hamiltonian is nontrivial. Here we summarize the procedure[19] to diagonalize the mean-field Hamiltonian (5). In the generalized Bogoliubov transformation $$ \alpha_{k}=U_{k} \beta_{k},~~ \tag {7} $$ the commutation relations of the new bosons, $\beta_{k} \equiv (\beta_{1,k}, \beta_{2,k}, \beta_{3,-k}^{†ger})^{T}$, will be preserved if $U_{k}$ satisfies the condition $$ U_{k}^{†ger} {\it \Lambda} U_{k}=U_{k} {\it \Lambda} U_{k}^{†ger}={\it \Lambda}.~~ \tag {8} $$ Substituting Eq. (8) into Eq. (5) and making use of the condition (8), one will obtain $$ \mathcal{H}_{\rm MF}=\sum_{k} \beta_{k}^{†ger} [ U_{k}^{†ger} \mathcal{H}({\boldsymbol k}) U_{k}+\delta h {\it \Lambda}] \beta_{k}+\mathcal{C}.~~ \tag {9} $$ To obtain a diagonal form, $U_{k}$ must satisfy the further condition $$ U_{k}^{†ger} \mathcal{H}({\boldsymbol k}) U_{k}=D_{k}={\rm diag} (\lambda_{1}({\boldsymbol k}), \lambda_{2}({\boldsymbol k}), \lambda_{3}({\boldsymbol k})),~~ \tag {10} $$ where the eigenvalues $\{\lambda_{n}({\boldsymbol k})\}$, which are the spin-wave dispersion relations at $h=2S$, should be positive definite, i.e., $\lambda_{n}({\boldsymbol k})>0$. This requires that the matrix $\mathcal{H}({\boldsymbol k})$ should also be positively definite,[19] in which case there exists a matrix $K$ such that $\mathcal{H}({\boldsymbol k})=K^{†ger} K$ (Cholesky decomposition or eigen decomposition of $\mathcal{H}({\boldsymbol k})$). The diagonalization of $\mathcal{H}({\boldsymbol k})$ therefore maps to the diagonalization of $K {\it \Lambda} K^{†ger}$, meaning to the exercise of finding a further unitary matrix, $V$, such that $V^{†ger}(K {\it \Lambda} K^{†ger})V=L$, with $L$ being diagonal. The solutions satisfying the two conditions (8) and (10) simultaneously are then $D_k= {\it \Lambda} L$ and $U_{k}=K^{-1} V D^{1/2}_{k}$. Here $U_{k}$ may be in fact obtained directly by diagonalizing the matrix ${\it \Lambda} \mathcal{H}({\boldsymbol k})$, i.e., from the equation $({\it \Lambda} \mathcal{H}({\boldsymbol k})) U_{k}=U_{k} ({\it \Lambda} D_{k})$. It is important to note that, due to Eq. (8), the generalized Bogoliubov transformation matrix $U_{k}$ is independent of the magnetic field $h$. Thus the energies of the spin-wave excitations at fields away from $h=2S$ are obtained simply by uniform shifts of the three magnon modes at $h=2S$, $$ \omega_{1,2}({\boldsymbol k})=\lambda_{1,2}({\boldsymbol k})+\delta h,~\omega_{3}({\boldsymbol k})=\lambda_{3}({\boldsymbol k})-\delta h.~~ \tag {11} $$ Thus the mean-field Hamiltonian (5) can be rewritten as $$ \mathcal{H}_{\rm MF}=\sum_{k} \sum_{j=1}^{3} \omega_{j}({\boldsymbol k}) \beta_{j,k}^{†ger} \beta_{j,k}+\mathcal{C}.~~ \tag {12} $$ To show the spin-wave dispersion relations $\omega_{j}({\boldsymbol k})$, it is necessary to solve for $x$. However, some preliminary remarks on the nature of the 1/3-plateau phase are already in order. It is clear from Eq. (11) that modes $\omega_{1,2}({\boldsymbol k})$ are pushed up by an increase of the magnetic field while the mode $\omega_{3}({\boldsymbol k})$ is pushed down, and conversely when $h$ decreases. When the lowest mode touches energy zero, the plateau phase becomes unstable. Thus the lower transition point $h_{\rm c1}$, out of the 1/3-plateau phase is determined by the lower gap of two modes $\lambda_{1,2}({\boldsymbol k})$, whereas the upper transition $h_{\rm c2}$, is determined by the gap of $\lambda_{3}({\boldsymbol k})$. Defining ${\it \Delta}_{j}={\min}(\lambda_{j}({\boldsymbol k}))$ as the energy gaps of the three spin-wave branches at field $h=2S$, $$ h_{\rm c1}=2S-{\min} ({\it \Delta}_{1},{\it \Delta}_{2}),~ h_{\rm c2}=2S+{\it \Delta}_{3},~~ \tag {13} $$ and the width of the plateau is given by $$\begin{align} {\it \Delta}_{\rm w}=h_{\rm c2}-h_{\rm c1}={\min} ({\it \Delta}_{1},{\it \Delta}_{2})+{\it \Delta}_{3}.~~ \tag {14} \end{align} $$ To determine the magnon-density parameter $x$, we use a numerical iterative method to solve the equation $$\begin{align} x=\frac{1}{N} \sum_{k}|U_{k}(1,3)|^{2},~~ \tag {15} \end{align} $$ where $U_{k}$ is the Bogoliubov transformation matrix of Eq. (7), which depends in turn on $x$. Due to the fact that $U_{k}$ does not depend on the magnetic field, $x$ is also independent of $h$ in the 1/3-plateau phase, although it does depend on $S$. As the above noted, we need therefore solve Eq. (15) only at $h=2S$ to obtain the spin-wave spectra, $\{\omega_{j}({\boldsymbol k})\}$, and hence the critical fields $h_{\rm c1}$ and $h_{\rm c2}$ in Eq. (13).
cpl-33-7-077503-fig2.png
Fig. 2. Dispersion relations of the three elementary magnon excitations, $j=1$, 2 and 3, for selected symmetry directions in the first Brillouin zone (inset), illustrated for field $h=2S$ and $S=1/2$. Here ${\it \Delta}_{1}$ and ${\it \Delta}_{3}$ denote respectively the energy gaps of modes 1 and 3.
The spin-wave spectra for the $S=1/2$ case at field $h=2S$ are shown in Fig. 2. The minima of the two lower modes, $j=1$ and 3, are both located at the ${\it \Gamma}$ point (${\boldsymbol k}=0$). There are two trivial level-crossings between these modes, occurring at the points marked $P_1$ and $P_2$ in Fig. 2 while actually forming a circle in the Brillouin zone, which changes size with $\delta h$. However, the crossing between modes $j=1$ and 2, which have the same $\delta h$-dependence, occurs at a single point, $P_3$, along the line ${\it \Gamma}$–$X$. This nontrivial exact crossing is a consequence of the reflection symmetry through the ${\hat y}$-axis and leads to a Dirac-type spectrum between the eigenmodes $j=1$ and 2. In more detail, the magnon spectra along ${\it \Gamma}$–$X$ can be found analytically by diagonalizing the matrix ${\it \Lambda} \mathcal{H} ({\boldsymbol k}=(k_x,0))$ to obtain $$\begin{align} \lambda_{1}(k_x)=\,& \begin{cases} 2S-(2S-x)\cos (k_x/2),& k_x \le k_{\rm c},\\ \frac{1}{2} [\sqrt{\gamma_{k}}+(2S - x) \cos (k_x/2)], & k_x > k_{\rm c}, \\ \end{cases} \\ \lambda_{2}(k_x)=\,& \begin{cases} \frac{1}{2} [\sqrt{\gamma_{k}}+(2S - x) \cos (k_x/2)], & k_x \le k_{\rm c}, \\ 2S-(2S - x)\cos (k_x/2), & k_x > k_{\rm c},\\\end{cases}\\ \lambda_{3}(k_x)=\,&\frac{1}{2}[\sqrt{\gamma_{k}}-(2S - x) \cos (k_x/2)],~~ \tag {16} \end{align} $$ where $\gamma_{k}=[(2S-x) \cos (k_x/2)+4x]^{2}+24 x (S-x)$. The crossing point is therefore located at momentum $k_x=k_{\rm c}=2 \cos^{-1} \Big(\frac{3+x/S-\sqrt{1+ 18(x/S)-3(x/S)^2}}{4-2(x/S)}\Big)$. In the classical limit, $S \rightarrow \infty$, the position of crossing is $k_{\rm c}=2\pi/3$. The gaps of modes $1$ and $3$ are $$ {\it \Delta}_{1}=x,~{\it \Delta}_{3}=\frac{x}{2}-S+\sqrt{S^{2}+9 S x-\frac{15}{4}x^{2}},~~ \tag {17} $$ and hence the normalized critical magnetic fields and the width of the 1/3 plateau are $$\begin{alignat}{1} \!\!\!\!\!\!\frac{h_{\rm c1}}{S}=\,& 2-\frac{x}{S},~~ \tag {18} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!\frac{h_{\rm c2}}{S}=\,& 2-(1-\frac{x}{2S})+\sqrt{1+9\Big(\frac{x}{S}\Big)-\frac{15}{4} \Big(\frac{x}{S}\Big)^{2}},~~ \tag {19} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!\frac{{\it \Delta}_{\rm w}}{S}=\,&\frac{3}{2}\Big(\frac{x}{S}\Big)-1+\sqrt{1+9\Big(\frac{x}{S}\Big) -\frac{15}{4}\Big(\frac{x}{S}\Big)^{2}}.~~ \tag {20} \end{alignat} $$
cpl-33-7-077503-fig3.png
Fig. 3. Self-consistent magnon-density parameter $x/S$, shown as a function of $1/S$. The red line indicates a two-parameter power-law fit of the form $x/S=\mu (1/S)^{\eta} / [1+\mu(1/S)^{\eta}]$ (Eq. (22)). The inset shows the magnitudes of the normalized magnetization on the A and B sublattices, with the result $m_{\rm A}=m_{\rm B}=1-x/S$ from the self-consistent spin-wave analysis shown as the solid red line and the corresponding results from iPEPS calculations with bond dimension $D=10$[18] shown as square blue points.
Concerning the form of the magnon-density function $x(S)$ in Fig. 3 we show the quantity $x/S$ determined numerically by solving the self-consistent equation (15). It is clear that $x/S$ grows sub-linearly from zero, and similarly that $m_{\rm A}=m_{\rm B}=1-x/S$ (inset of Fig. 3) falls monotonically from full polarization in the classical limit. We propose the power-law form $$ m_{\rm A}=m_{\rm B}=\frac{1}{1+\mu (1/S)^\eta},~~ \tag {21} $$ for the sublattice magnetization and thus $$ \frac{x}{S}=\frac{\mu(1/S)^{\eta}}{1+\mu (1/S)^{\eta}} \,=\, \frac{1}{1+\mu^{-1} S^\eta},~~ \tag {22} $$ for the magnon density. We find that this two-parameter fit, with prefactor $\mu= 0.206(1)$ and exponent $\eta=0.690(1)$, offers an excellent account of the data over the entire range of $S$, i.e., not only where $1/S$ is small while even when $S=1/2$ (last point). We comment that significantly better statistics still can be obtained by generalizing this type of power-law fit to two exponents. Returning to Eqs. (18)-(20), the nonzero boson-density expectation values on all sites ($n_{\rm A}=n_{\rm B}=n_{\rm C}/2=x$) lead to finite energy gaps (Eq. (17)), which stabilize the 1/3 plateau over a broad range of magnetic fields, as shown in Fig. 4 for all values of $S$. In the self-consistent spin-wave treatment, all of these quantities increase with the spin quantum number $S$, although their normalized values decrease towards the expected limits (Figs. 3 and 4). This type of behavior is completely consistent with the numerical results obtained from ED[15] and iPEPS,[18] which are shown for comparison in Fig. 4. The $1/S$-dependence of both critical fields $h_{\rm c1}$ and $h_{\rm c2}$, and of the plateau width ${\it \Delta}_{\rm w}$, predicted by the self-consistent spin-wave theory have the correct qualitative trends, and in fact close quantitative agreement with the numerical calculations. It is striking that our analytical results are accurate at a semi-quantitative level even for the extreme quantum case $S=1/2$. These results demonstrate that the self-consistent spin-wave theory captures properly the nature of the 1/3 magnetization plateau in the kagome antiferromagnet.
cpl-33-7-077503-fig4.png
Fig. 4. (a) Normalized critical magnetic fields $h_{\rm c1}/S$ and $h_{\rm c2}/S$ of the 1/3-plateau phase, shown as a function of $1/S$. Data from the self-consistent spin-wave theory (solid red lines) are compared with the results from ED (green),[15] iPEPS with bond dimension $D=10$ (blue),[18] and RSPT (dashed black).[14] (b) Width ${\it \Delta}_{\rm w}=(h_{\rm c2}-h_{\rm c1})$ of the 1/3 plateau, shown as a function of $1/S$; all data labels as in panel (a). The inset shows the normalized width ${\it \Delta}_{\rm w}/S$.
By contrast, in the RSPT approach[14] one performs an expansion in powers of $1/S$ to obtain the plateau properties in the form $$\begin{align} h_{\rm c1}/S=2-\frac{1}{8S}-\frac{1}{4S^{2}},~~ \tag {23} \end{align} $$ $$\begin{align} h_{\rm c2}/S=\,& 2+\frac{3}{8S}+\frac{1}{4S^{2}},~~ \tag {24} \end{align} $$ $$\begin{align} {\it \Delta}_{\rm w}/S= \,& \frac{1}{2S}+\frac{1}{2S^{2}},~~ \tag {25} \end{align} $$ to order $(1/S)^2$. However, not only is this form fated to diverge in the most quantum systems, as $1/S \rightarrow 1$, but there is also a qualitative discrepancy with our analytical results and with the recent numerical results: the RSPT predictions for $h_{\rm c1}$ and $h_{\rm c2}$ are concave up as a function of $1/S$ where they should be concave down, and conversely, while the non-normalized plateau width trends in the wrong direction. Thus only in the extreme classical limit do the considerations of RSPT appear to be valid. In summary, we have investigated the 1/3 magnetization plateau of the kagome antiferromagnet using a straightforward modified spin-wave theory, which contains only one self-consistent parameter. We have shown that the quantum corrections contained in this magnon-density parameter open finite energy gaps, which stabilize the 1/3-plateau phase over a broad range of magnetic fields. The qualitative and quantitative behavior of the critical fields and plateau widths is in excellent agreement with the recent numerical results for the same quantities obtained by exact diagonalization[15] and by tensor-network methods.[18] These results indicate that the self-consistent spin-wave theory provides an accurate description of the properties of the magnetization plateau in the kagome antiferromagnet. We suggest that the same type of theory should also be applied to describe the properties of magnetization plateaus in a number of other frustrated systems, including the extended square and honeycomb geometries as well as the triangular,[20] checkerboard, Shastry–Sutherland, and Husimi antiferromagnets.[13] References Resonating valence bonds: A new kind of insulator?Spin liquids in frustrated magnetsSimplex solid states of SU ( N ) quantum antiferromagnetsNumerical studies of antiferromagnetism on a Kagomé netSpin-Peierls and spin-liquid phases of Kagomé quantum antiferromagnetsPhysics of low-energy singlet states of the Kagome lattice quantum Heisenberg antiferromagnetGround state of the spin-1/2 kagome-lattice Heisenberg antiferromagnetTriplet and singlet excitations in the valence bond crystal phase of the kagome lattice Heisenberg modelFrustrated Antiferromagnets with Entanglement Renormalization: Ground State of the Spin- 1 2 Heisenberg Model on a Kagome LatticeValence-bond crystal in the extended kagome spin- 1 2 quantum Heisenberg antiferromagnet: A variational Monte Carlo approachKagome´- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinonsDensity Matrix Renormalization Group Numerical Study of the Kagome AntiferromagnetSpin-Liquid Ground State of the S = 1/2 Kagome Heisenberg AntiferromagnetNature of the Spin-Liquid Ground State of the S = 1 / 2 Heisenberg Model on the Kagome LatticeIdentifying topological order by entanglement entropyProjected-Wave-Function Study of the Spin- 1 / 2 Heisenberg Model on the Kagomé LatticeGapless spin-liquid phase in the kagome spin- 1 2 Heisenberg antiferromagnetTensor Renormalization of Quantum Many-Body Systems Using Projected Entangled Simplex StatesSpin-1/2 Kagomé -Like Lattice in Volborthite Cu 3 V 2 O 7 (OH) 2 ·2H 2 OA Structurally Perfect S = 1 / 2 Kagomé AntiferromagnetVesignieite BaCu 3 V 2 O 8 (OH) 2 as a Candidate Spin-1/2 Kagome AntiferromagnetMagnetization plateaus of the spin- 1 2 kagome antiferromagnets volborthite and vesignieiteFrustrated magnetism in the S = 1 kagomé lattice BaNi3(OH)2(VO4)2Exotic Magnetism of Novel S = 1 Kagome Lattice Antiferromagnet KV 3 Ge 2 O 9Observation of 1/3 Magnetization-Plateau-Like Anomaly in S =3/2 Perfect Kagomé Lattice Antiferromagnet KCr 3 (OH) 6 (SO 4 ) 2 (Cr-jarosite)Heisenberg antiferromagnet on the Husimi latticeReal-space perturbation theory for frustrated magnets: application to magnetization plateausMagnetization Process of the Spin- S Kagome-Lattice Heisenberg AntiferromagnetNumerical study of magnetization plateaus in the spin- 1 2 kagome Heisenberg antiferromagnetControlling frustrated liquids and solids with an applied field in a kagome Heisenberg antiferromagnetSpin- S kagome quantum antiferromagnets in a field with tensor networksTime reversal symmetry breaking chiral spin liquids: Projective symmetry group approach of bosonic mean-field theoriesDiagonalization of the quadratic boson hamiltonianSelf-consistent spin wave analysis of the magnetization plateau in triangular antiferromagnet
[1] Anderson P W 1973 Mater. Res. Bull. 8 153
Balents L 2010 Nature 464 199
[2] Arovas D P 2008 Phys. Rev. B 77 104404
[3] Zeng C and Elser V 1990 Phys. Rev. B 42 8436
Marston J B and Zeng C 1991 J. Appl. Phys. 69 5962
Nikolic P and Senthil T 2003 Phys. Rev. B 68 214415
Singh R R P and Huse D A 2007 Phys. Rev. B 76 180407
Singh R R P and Huse D A 2008 Phys. Rev. B 77 144415
Evenbly G and Vidal G 2010 Phys. Rev. Lett. 104 187203
Iqbal Y, Becca F and Poilblanc D 2011 Phys. Rev. B 83 100404
[4] Sachdev S 1992 Phys. Rev. B 45 12377
Jiang H C, Weng Z Y and Sheng D N 2008 Phys. Rev. Lett. 101 117203
Yan S, Huse D A and White S R 2011 Science 332 1173
Depenbrock S, McCulloch I P and Schollwöck U 2012 Phys. Rev. Lett. 109 067201
Jiang H C, Wang Z H and Balents L 2012 Nat. Phys. 8 902
Li T 2016 arXiv:1601.02165v1[cond-mat.str-el]
[5] Ran Y, Hermele M, Lee P A and Wen X G 2007 Phys. Rev. Lett. 98 117205
Iqbal Y, Becca F, Sorella S and Poilblanc D 2013 Phys. Rev. B 87 060405
[6] Xie Z Y, Chen J, Yu J F, Kong X, Normand B and Xiang T 2014 Phys. Rev. X 4 011025
[7] Hiroi Z, Hanawa M, Kobayashi N, Nohara M, Takagi H, Kato Y and Takigawa M 2001 J. Phys. Soc. Jpn. 70 3377
[8] Shores M P, Nytko E A, Barlett B M and Nocera D G 2005 J. Am. Chem. Soc. 127 13462
[9] Okamoto Y, Yoshida H and Hiroi Z 2009 J. Phys. Soc. Jpn. 78 033701
Okamoto Y, Tokunaga M, Yoshida H, Matsuo A, Kindo K and Hiroi Z 2011 Phys. Rev. B 83 180407
[10] Freedman D E, Chisnell R, McQueen T M, Lee Y S, Payen C and Nocera D G 2012 Chem. Commun. 48 64
[11] Hara S, Sato H and Narumi Y 2012 J. Phys. Soc. Jpn. 81 073707
[12] Okuta K, Hara S, Sato H, Narumi Y and Kindo K 2011 J. Phys. Soc. Jpn. 80 063703
[13] Liao H J, Xie Z Y, Chen J, Han X J, Xie H D, Normand B and Xiang T 2016 Phys. Rev. B 93 075154
[14] Zhitomirsky M E 2015 J. Phys.: Conf. Ser. 592 012110
[15] Nakano H and Sakai T 2015 J. Phys. Soc. Jpn. 84 063705
[16] Capponi S, Derzhko O, Honecker A, Läuchli A M and Richter J 2013 Phys. Rev. B 88 144416
[17] Nishimoto S, Shibata N and Hotta C 2013 Nat. Commun. 4 2287
[18] Picot T, Ziegler M, Orús R and Poilblanc D 2016 Phys. Rev. B 93 060407(R)
[19] Messio L, Lhuillier C and Misguich G 2013 Phys. Rev. B 87 125127
Colpa J H P 1978 Physica A 93 327
[20] Takano J, Tsunetsugu H and Zhitomirsky M E 2011 J. Phys.: Conf. Ser. 320 012011