Emergence of magnetic order in kagomé antiferromagnets

Graphical abstract

Exotic quantum spin liquid (QSL) states [1] and fractionalized quasiparticles [2, 3] in frustrated magnets are of much current interest in theoretical and experimental studies of quantum magnetism. The kagomé-lattice Heisenberg antiferromagnet (KAFM) provides a possible realization of just such novel topological states of matter. The kagomé lattice shown in Fig. 1 is one of eleven Archimedean lattices [4, 5] in two spatial dimensions, where the word kagomé itself means “weave pattern” in Japanese. The Hamiltonian for the KAFM model is given by

\begin{array}{}\text{(1)}& H=J{\displaystyle \sum _{\langle i\mathrm{,}j\rangle }}{s}_i\cdot s_j\mathrm{,}\end{array}

where s_i^2=s(s+1) and the summation \langle i\mathrm{,}j\rangle runs over all nearest-neighbor (NN) bonds (counting each bond once only) and where J(>0) is the bond strength. The KAFM is geometrically frustrated in the sense that not all pairs of nearest-neighbor spins can be simultaneously antiparallel, as is otherwise favored by the Heisenberg antiferromagnetic exchange interaction between pairs of spins. Even the classical behavior of this model at zero temperature is complicated because there are an infinite number of possible classical ground states to choose from potentially. Each potential classical ground state has nearest-neighboring spins on the “triangles” of these lattices that form angles of 120° to each other. Thermal fluctuations [6-10] favor the \sqrt{3}\times \sqrt{3} (coplanar) state (see Fig. 1). Harmonic quantum fluctuations favor coplanar over non-coplanar states [11-13], whereas anharmonic quantum fluctuations suggest that it is the \sqrt{3}\times \sqrt{3} state that is favored for s>1/2.

Results for the ground-state energy for the spin-half, KAFM are E_g/(NJ{s}^2)=-1.7544(0.002) (density matrix renormalization group (DMRG) [14]; see also Refs. [15, 16]), -1.72884 (entanglement renormalization [17]), -1.732 (series expansions around a dimer limit [18]), -1.754488 (coupled cluster method (CCM) [5, 19]), -1.75008(0.00024) (tensor network states [20]), -1.75257 (large-scale exact diagonalizations (ED) for N=42 [21, 22, 24]), and -1.75482 (ED for N=48 [23]). All of the evidence from these approximate methods (see, e.g., Refs. [5, 14, 19-21]) indicates that the long-range magnetic order parameter is zero. Candidates for the ground state of the spin-half KAFM system are a gapped spin liquid [14-16, 25, 26] (gap ~0.055(5) [16] to 0.13(1) [14]; see also recent experimental evidence of Ref. [27]), a gapless spin liquid [20, 22, 28, 29], and a valence-bond state [18, 30-33]. Although long-range magnetic order does not occur, the exact nature of the ground state is therefore still a topic of debate.

The simplest and most direct route that order may “emerge from magnetic disorder” for the KAFM is to increase the spin quantum number, s. Results for the ground-state energy for the spin-one KAFM are E_g/(NJ{s}^2)=-1.3950 (series expansions) [34], -1.40315 (CCM) [19, 35], -1.4109(2) (tensor network) [36], -1.410(2) (DMRG) [37], and -1.41095 (also DMRG) [38]. Results for the s=3/2 KAFM are E_g/(NJ{s}^2)=-1.253022 (series expansions) [34], -1.26798 (CCM) [19, 35], and -1.265(2) (tensor network) [39]. CCM calculations [19, 35] also indicate that the ground-state energy scales with s (for s\ge 3/2) via the following equation,

\begin{array}{}\text{(2)}& \frac{E_g}{NJ{s}^2}=-1-\frac{0.4140}{s}+\frac{0.0180}{s^2}\mathrm{.}\end{array}

The order parameter is given by

\begin{array}{}\text{(3)}& M=\frac{1}{N}{\displaystyle \sum _i^N}\langle \text{Ψ}|s_i^z|\text{Ψ}\rangle \mathrm{,}\end{array}

where s_i^z is defined with respect to local axes of one of the classical ground states at site i. We expect that the order parameter M/s will tend to M/s\to 1 in the limit s\to \infty. The effect of quantum fluctuations is to reduce the amount of magnetic order and so we expect for finite values of s. Results of most approximate methods for the spin-one KAFM suggest that there is no magnetic long-range order, although Ref. [40] indicated \sqrt{3}\times \sqrt{3} ground-state long-range order for integer spin quantum numbers, including s=1. Series expansion calculations [34] indicated that M/s=0.14\pm 0.03 for the s=3/2 KAFM and tensor network calculations [39] also indicate that the s=3/2 system is \sqrt{3}\times \sqrt{3} long-range ordered. CCM calculations [19, 35] indicate that the system is \sqrt{3}\times \sqrt{3} ordered for s=3/2 and that M/s is in the range 0.074 to 0.417. The KAFM demonstrates \sqrt{3}\times \sqrt{3} ground-state long-range order for s\ge 3/2 [19, 35, 40, 41]. CCM calculations [19, 35] indicated that the order parameter (with respect to the \sqrt{3}\times \sqrt{3} state) scales with s (for s\ge 3/2) via the following equation,

\begin{array}{}\text{(4)}& \frac{M}{s}=1-\frac{1.0676}{s^{0.5}}+\frac{0.0810}{s}\mathrm{,}\end{array}

whereas self-consistent spin-wave theory [11] suggested that M/s scales with s^{-1/3} to first order.

The next route to the emergence of magnetic order in the KAFM model is to introduce easy-plane anisotropy into the Hamiltonian [34, 41-45], which we shall refer to as the XXZ model, where

\begin{array}{}\text{(5)}& H={\displaystyle \sum _{\langle i\mathrm{,}j\rangle }}\{\text{Δ}{s}_i^z{s}_j^z+s_i^y{s}_j^y+s_i^x{s}_j^x\}\mathrm{.}\end{array}

The summation \langle i\mathrm{,}j\rangle again runs over all NN bonds (counting each bond once only). The XXZ model on the kagomé lattice is predicted to be magnetically ordered for all values of \text{Δ}\ge 0 for s\ge 3/2. In the limit s\to \infty, CCM calculations [42] indicate that a phase boundary between \sqrt{3}\times \sqrt{3} order at \text{Δ}=1 (KAFM) and q=0 order (see Fig. 1) at \text{Δ}=0 (kagomé XY model) occurs at the point {\text{Δ}}_c (s\to \infty )=0.727 [42], whereas results of non-linear spin-wave theory (NLSWT) [41] place this boundary at {\text{Δ}}_c (s\to \infty )=0.72235. CCM results also suggest that the boundary {\text{Δ}}_c(s) between these two phases at finite s increases with increasing s, whereas NLSWT suggest (tentatively) that the boundary {\text{Δ}}_c(s) between these two phases decreases with increasing s. (Note that CCM results for this boundary are based on the direct comparison of numerical evidence for the two states, whereas statements via NLSWT for this boundary are more speculative.) However, both NLSWT and the CCM predict also that the spin-one, NN KAFM is disordered, which agrees with the conclusions presented above for this model (see also [43]). However, CCM results also predict that a reduction in Δ from \text{Δ}=1 leads to the onset of q=0 order, whereas NLSWT predicts that a reduction in Δ from \text{Δ}=1 leads to the onset of \sqrt{3}\times \sqrt{3} order. Finally, a range of approximate methods (namely, CCM [42], NLSWT [41], DMRG [44], and variational Monte Carlo [45]) predict that the spin-half XXZ model on the kagomé lattice is disordered for all values of \text{Δ}\ge 0 (i.e., including the XY model).

Another extension of the spin-half, NN KAFM that leads to magnetic order is via the introduction of inter-layer coupling [46-48], where

\begin{array}{}\text{(6)}& H=\underset{\text{Within}\text{layer}}{\underbrace{J{\displaystyle \sum _n}{\displaystyle \sum _{\langle i\mathrm{,}j\rangle }}{s}_{i\mathrm{,}n}\cdot s_{j\mathrm{,}n}}}+\underset{\text{Between}\text{layers}}{\underbrace{J_{\perp }{\displaystyle \sum _{i\mathrm{,}n}}{s}_{i\mathrm{,}n}\cdot s_{i\mathrm{,}n+1}}}\mathrm{.}\end{array}

The summation \langle i\mathrm{,}j\rangle indicates NN bonds of strength J(>0) within a given layer (indicated by n). J_{\perp } therefore indicates the bond strength between layers n and n+1. The underlying spin model is of two spatial dimensions for J_{\perp }=0 and is of three spatial dimensions model for J_{\perp }\ne 0. Quantum magnetic systems of three spatial dimensions might well have a stronger propensity towards magnetic order than systems of lower spatial dimension and so it is not unnatural to suppose that magnetic order (of some) sort might occur. Indeed, it was observed in Refs. [47, 48] that \sqrt{3}\times \sqrt{3} magnetic order is observed for both J_{\perp }/J=1 (antiferromagnetic bonds) and J_{\perp }/J=-1 (ferromagnetic bonds). CCM results [48] then suggest that q=0 magnetic order occurs between and . A region of magnetic disorder [48] is then observed in the region , which agrees yet again with the predicted behavior of the spin-half, NN KAFM at J_{\perp }/J=0.

The final route by which magnetic order may emerge from magnetic disorder is to include interactions between spins over greater distances than nearest-neighboring spins via the J₁–J₂ model [2, 20, 49-52] and/or the J₁–J₂–J₃ model [53-57], which is given by

\begin{array}{}\text{(7)}& H=J_1{\displaystyle \sum _{\langle i\mathrm{,}j\rangle }}{s}_i\cdot s_j+J_2{\displaystyle \sum _{\langle \langle i\mathrm{,}k\rangle \rangle }}{s}_i\cdot s_k+J_3{\displaystyle \sum _{\langle \langle \langle i\mathrm{,}l\rangle \rangle \rangle }}{s}_i\cdot s_l\mathrm{.}\end{array}

The summation \langle i\mathrm{,}j\rangle runs over all NN bonds, \langle \langle i\mathrm{,}k\rangle \rangle runs over all next-nearest-neighbor (NNN) bonds, and \langle \langle \langle i\mathrm{,}l\rangle \rangle \rangle runs over all next-next-nearest-neighbor (NNNN) bonds. (In each case, bonds are counted once and once only.) For the spin-half model with J_3=0, \sqrt{3}\times \sqrt{3} order is stabilized by ferromagnetic NNN interactions and q=0 order is stabilized by antiferromagnetic NNN interactions (J_2>0). Initial CCM results [52] suggest however that the magnetically disordered regime survives for a finite range of J₂ centered around the NN KAFM, namely, -0.070\lesssim J_2/J_1\lesssim 0.127; a result that is also supported by other approximate methods [49, 51]. The addition of antiferromagnetic NNNN bonds (J_3>0) (with ferromagnetic NN and NNN bonds J₂, ) was explored in Ref. [55]. The existence of ferromagnetic ordering, \sqrt{3}\times \sqrt{3} ordering, as well as non-coplanar states of magnetic order (CUBOC-1 and CUBOC-2), and finally a large magnetically disordered regime, were posited in Refs. [55, 56]. A complete picture of the ground-state phase diagram of the spin-half J₁–J₂–J₃ model is presented in Ref. [55].

Generalizations of the spin-half NN KAFM are realized physically by magnetic materials such as herbertsmithite ZnCu₃(OH)₆Cl₂ [3, 58], haydeeite Cu₃Mg(OH)₆Cl₂ [59], vesignieite BaCu₃V₂O₈(OH)₂ [60], kapellasite Cu₃Zn(OH)₆Cl₂ [61], volborthite Cu₃V₂O₇(OH)₂·2H₂O [58, 62], and francisite Cu₃Bi(SeO₃)₂ [63]. Hence, the study of such models where magnetic order may “emerge” from the disorder of the pure spin-half NN KAFM are crucial in understanding these materials. Furthermore, we have seen that these generalizations of the spin-half NN model lead to fascinating behavior from a theoretical point-of-view also. These generalized models also provide a useful tool in examining the disordered regimes in the spin-half, NN KAFM because one may observe those limiting cases where magnetic order is seen to vanish as a function of model parameters within the Hamiltonian. The behavior of the ground and excited states can be examined up to and beyond any order-to-disorder phase transitions. Finally, the NN KAFM model and its generalizations provide a very important, interesting, and challenging set of systems by which approximate methods of quantum many-body theory may be compared and contrasted.

References

[1] L. Balents, Spin liquids in frustrated magnets, Nature 464(7286), 199 (2010)

[2] L. Balents, M. P. Fisher, and S. M. Girvin, Fractionalization in an easy-axis kagomé antiferromagnet, Phys. Rev. B 65(22), 224412 (2002)

[3] T. H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Fractionalized excitations in the spin-liquid state of a kagomé-lattice antiferromagnet, Nature 492, 406 (2012)

[4] J. Richter, J. Schulenburg, and A. Honecker, Quantum magnetism in two dimensions: From semi-classical Néel order to magnetic disorder, Lect. Notes Phys. 645, pp 85-153, Heidelberg Berlin: Springer, 2004

[5] D. J. J. Farnell, O. Götze, J. Richter, R. F. Bishop, and P. H. Y. Li, Quantum s=1/2 antiferromagnets on Archimedean lattices: The route from semiclassical magnetic order to nonmagnetic quantum states, Phys. Rev. B 89(18), 184407 (2014)

[6] J. T. Chalker, P. C. W. Holdsworth, and E. F. Shender, Hidden order in a frustrated system: Properties of the Heisenberg kagomé antiferromagnet, Phys. Rev. Lett. 68(6), 855 (1992)

[7] J. T. Chalker and J. F. G. Eastmond, Ground-state disorder in the spin-1/2 kagomé Heisenberg antiferromagnet, Phys. Rev. B 46(21), 14201 (1992)

[8] I. Ritchey, P. Chandra, and P. Coleman, Spin folding in the two-dimensional Heisenberg kagomé antiferromagnet, Phys. Rev. B 47(22), 15342 (1993)

[9] D. A. Huse and A. D. Rutenberg, Classical antiferromagnets on the kagomé lattice, Phys. Rev. B 45(13), 7536 (1992)

[10] P. Müller, A. Zander, and J. Richter, Thermodynamics of the kagomé-lattice Heisenberg antiferromagnet with arbitrary spin S, Phys. Rev. B 98(2), 024414 (2018)

[11] A. Chubukov, Order from disorder in a kagomé antiferromagnet, Phys. Rev. Lett. 69(5), 832 (1992)

[12] S. Sachdev, Kagomé- and triangular-lattice Heisenberg antiferromagnets: Ordering from quantum fluctuations and quantum-disordered ground states with unconfined bosonic spinons, Phys. Rev. B 45(21), 12377 (1992)

[13] C. L. Henley, Long-range order in the classical kagomé antiferromagnet: Effective Hamiltonian approach, Phys. Rev. B 80, 180401 (2009)

[14] S. Depenbrock, I. P. McCulloch, and U. Schollwöck, Nature of the spin-liquid ground state of the S=1/2 Heisenberg model on the kagomé lattice, Phys. Rev. Lett. 109(6), 067201 (2012)

[15] S. Yan, D. A. Huse, and S. R. White, Spin-liquid ground state of the S=1/2 kagomé Heisenberg antiferromagnet, Science 332(6034), 1173 (2011)

[16] H. C. Jiang, Z. Y. Weng, and D. N. Sheng, Density matrix renormalization group numerical study of the kagomé antiferromagnet, Phys. Rev. Lett. 101(11), 117203 (2008)

[17] G. Evenbly and G. Vidal, Frustrated antiferromagnets with entanglement renormalization: Ground state of the spin-1/2 Heisenberg model on a kagomé lattice, Phys. Rev. Lett. 104(18), 187203 (2010)

[18] R. R. P. Singh and D. A. Huse, Ground state of the spin-1/2 kagomé-lattice Heisenberg antiferromagnet, Phys. Rev. B 76(18), 180407 (2007)

[19] O. Götze, D. J. J. Farnell, R. F. Bishop, P. H. Y. Li, and J. Richter, Heisenberg antiferromagnet on the kagomé lattice with arbitrary spin: A higher-order coupled cluster treatment, Phys. Rev. B 84(22), 224428 (2011)

[20] H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, Gapless spin-liquid ground state in the S=1/2 kagomé antiferromagnet, Phys. Rev. Lett. 118(13), 137202 (2017)

[21] A. M. Läuchli, J. Sudan, and E. S. Sorensen, Ground-state energy and spin gap of spin-1/2 kagomé-Heisenberg antiferromagnetic clusters: Large-scale exact diagonalization results, Phys. Rev. B 83(21), 212401 (2011)

[22] H. Nakano and T. Sakai, Numerical-diagonalization study of spin gap issue of the kagomé lattice Heisenberg antiferromagnet, J. Phys. Soc. Jpn. 80(5), 053704 (2011)

[23] P. W. Leung and V. Elser, Numerical studies of a 36-site kagomé antiferromagnet, Phys. Rev. B 47(9), 5459 (1993)

[24] A. M. Läuchli, J. Sudan, and R. Moessner, The S=1/2 kagomé Heisenberg antiferromagnet revisited, arXiv: 1611.06990 (2016)

[25] F. Mila, Low-energy sector of the S=1/2 kagomé antiferromagnet, Phys. Rev. Lett. 81(11), 2356 (1998)

[26] L. Messio, B. Bernu, and C. Lhuillier, Kagomé antiferromagnet: A chiral topological spin liquid? Phys. Rev. Lett. 108(20), 207204 (2012)

[27] M. Fu, T. Imai, T. H. Han, and Y. S. Lee, Evidence for a gapped spin-liquid ground state in a kagomé Heisenberg antiferromagnet, Science 350(6261), 655 (2015)

[28] Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Gapless spin-liquid phase in the kagomé spin-1/2 Heisenberg antiferromagnet, Phys. Rev. B 87, 060405(R) (2013)

[29] M. Hermele, Y. Ran, P. A. Lee, and X. G. Wen, Properties of an algebraic spin liquid on the kagomé lattice, Phys. Rev. B 77(22), 224413 (2008)

[30] J. B. Marston and C. Zeng, Spin-Peierls and spin-liquid phases of kagomé quantum antiferromagnets, J. Appl. Phys. 69(8), 5962 (1991)

[31] P. Nikolic and T. Senthil, Physics of low-energy singlet states of the kagomé lattice quantum Heisenberg antiferromagnet, Phys. Rev. B 68(21), 214415 (2003)

[32] D. Schwandt, M. Mambrini, and D. Poilblanc, Generalized hard-core dimer model approach to low-energy Heisenberg frustrated antiferromagnets: General properties and application to the kagomé antiferromagnet, Phys. Rev. B 81(21), 214413 (2010)

[33] D. Poilblanc, M. Mambrini, and D. Schwandt, Effective quantum dimer model for the kagomé Heisenberg antiferromagnet: Nearby quantum critical point and hidden degeneracy, Phys. Rev. B 81(18), 180402 (2010)

[34] J. Oitmaa and R. R. P. Singh, Competing orders in spin-1 and spin-3/2 XXZ kagomé antiferromagnets: A series expansion study, Phys. Rev. B 93(1), 014424 (2016)

[35] D. J. J. Farnell, O. Götze, J. Richter, R. Zinke, R. F. Bishop, and P. H. Y. Li, The interplay between lattice topology, frustration, and spin quantum number in quantum antiferromagnets on Archimedean Lattices, Phys. Rev. B 98(22), 224402 (2018)

[36] T. Liu, W. Li, A. Weichselbaum, J. von Delft, and G. Su, Simplex valence-bond crystal in the spin-1 kagomé Heisenberg antiferromagnet. Phys. Rev. B 91, 060403(R) (2015)

[37] H. Changlani and A. M. Läuchli. Trimerized ground state of the spin-1 Heisenberg antiferromagnet on the kagomé lattice, Phys. Rev. B 91, 100407(R) (2015)

[38] S. Nishimoto and M. Nakamura, Non-symmetry-breaking ground state of the S=1 Heisenberg model on the kagomé lattice, Phys. Rev. B 92, 140412(R) (2015)

[39] T. Liu, W. Li, and G. Su, Spin-ordered ground state and thermodynamic behaviors of the spin-3/2 kagomé Heisenberg antiferromagnet, Phys. Rev. E 94(3), 032114 (2016)

[40] O. Cépas and A. Ralko, Resonating color state and emergent chromodynamics in the kagomé antiferromagnet, Phys. Rev. B 84(2), 020413 (2011)

[41] A. L. Chernyshev and M. E. Zhitomirsky, Quantum selection of order in an XXZ antiferromagnet on a kagomé lattice, Phys. Rev. Lett. 113(23), 237202 (2014)

[42] O. Götze and J. Richter, Ground-state phase diagram of the XXZ spin-s kagomé antiferromagnet: A coupled-cluster study, Phys. Rev. B 91(10), 104402 (2015)

[43] C. Xu and J. E. Moore, Geometric criticality for transitions between plaquette phases in integer-spin kagomé XXZ antiferromagnets, Phys. Rev. B 72(6), 064455 (2005)

[44] Y. C. He and Y. Chen, Distinct spin liquids and their transitions in spin-1/2 XXZ kagomé antiferromagnets, Phys. Rev. Lett. 114(3), 037201 (2015)

[45] W. J. Hu, S. S. Gong, F. Becca, and D. N. Sheng, Variational Monte Carlo study of a gapless spin liquid in the spin-1/2 XXZ antiferromagnetic model on the kagomé lattice, Phys. Rev. B 92(20), 201105 (2015)

[46] D. Schmalfuß, J. Richter, and D. Ihle, Absence of long-range order in a spin-half Heisenberg antiferromagnet on the stacked kagomé lattice, Phys. Rev. B 70(18), 184412 (2004)

[47] D. Guterding, R. Valentí, and H. O. Jeschke, Reduction of magnetic interlayer coupling in barlowite through isoelectronic substitution, Phys. Rev. B 94(12), 125136 (2016)

[48] O. Götze and J. Richter, The route to magnetic order in the spin-1/2 kagomé Heisenberg antiferromagnet: The role of interlayer coupling, Europhys. Lett. 114(6), 67004 (2016)

[49] T. Tay and O. I. Motrunich, Variational study of J₁–J₂ Heisenberg model on kagomé lattice using projected Schwinger-boson wave functions, Phys. Rev. B 84, 020404(R) (2011)

[50] R. Suttner, C. Platt, J. Reuther, and R. Thomale, Renormalization group analysis of competing quantum phases in the J₁–J₂ Heisenberg model on the kagomé lattice, Phys. Rev. B 89(2), 020408 (2014)

[51] F. Kolley, S. Depenbrock, I. P. McCulloch, U. Schollwöck, and V. Alba, Phase diagram of the J₁–J₂ Heisenberg model on the kagomé lattice, Phys. Rev. B 91(10), 104418 (2015)

[52] J. Richter and O. Götze, unpublished

[53] A. B. Harris, C. Kallin, and A. J. Berlinsky, Possible Néel orderings of the Kagomé antiferromagnet, Phys. Rev. B 45(6), 2899 (1992)

[54] B. Bernu, C. Lhuillier, E. Kermarrec, F. Bert, P. Mendels, R. H. Colman, and A. S. Wills, Exchange energies of kapellasite from high-temperature series analysis of the kagomé lattice J₁–J₂–J_d Heisenberg model, Phys. Rev. B 87(15), 155107 (2013)

[55] Y. Iqbal, H. O. Jeschke, J. Reuther, R. Valenti, I. I. Mazin, M. Greiter, and R. Thomale, Paramagnetism in the kagomé compounds (Zn,Mg,Cd)Cu₃(OH)₆Cl₂, Phys. Rev. B 92(22), 220404 (2015)

[56] S. Bieri, L. Messio, B. Bernu, and C. Lhuillier, Gapless chiral spin liquid in a kagomé Heisenberg model, Phys. Rev. B 92(6), 060407 (2015)

[57] A. Wietek, A. Sterdyniak, and A. M. Läuchli, Nature of chiral spin liquids on the kagomé lattice, Phys. Rev. B 92(12), 125122 (2015)

[58] Z. Hiroi, H. Yoshida, Y. Okamoto, and M. Takigawa, Spin-1/2 kagomé compounds: Volborthite vs. herbertsmithite, J. Phys. Conf. Ser. 145(1), 012002 (2009)

[59] O. Janson, J. Richter, and H. Rosner, Modified kagomé physics in the natural spin-1/2 kagomé lattice systems: Kapellasite Cu₃Zn(OH)₆Cl₂ and haydeeite Cu₃Mg(OH)₆Cl₂, Phys. Rev. Lett. 101(10), 106403 (2008)

[60] Y. Okamoto, H. Yoshida, and Z. Hiroi, Vesignieite BaCu₃V₂OV₈(OH)₂ as a candidate spin-1/2 kagomé antiferromagnet, J. Phys. Soc. Jpn. 78(3), 033701 (2009)

[61] B. Fåk, E. Kermarrec, L. Messio, B. Bernu, C. Lhuillier, F. Bert, P. Mendels, B. Koteswararao, F. Bouquet, J. Ollivier, A. D. Hillier, A. Amato, R. H. Colman, and A. S. Wills, Kapellasite: A kagomé quantum spin liquid with competing interactions, Phys. Rev. Lett. 109(3), 037208 (2012)

[62] O. Janson, J. Richter, P. Sindzingre, and H. Rosner, Coupled frustrated quantum spin-1/2 chains with orbital order in volborthite Cu₃V₂O₇(OH)₂·2H₂O, Phys. Rev. B 82(10), 104434 (2010)

[63] I. Rousochatzakis, J. Richter, R. Zinke, and A. A. Tsirlin, Frustration and Dzyaloshinsky-Moriya anisotropy in the kagomé francisites Cu₃Bi(SeO₃)₂O₂X (X=Br, Cl), Phys. Rev. B 91(2), 024416 (2015)

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