مقالۀ پژوهشی: پلاتوها و جهش‌های مغناطش در نقطه‌های کوانتومیِ شبکه‌کاگومه مثلثی بورون

نوع مقاله : مقاله پژوهشی

نویسندگان

1 استادیار، گروه فیزیک، دانشگاه یاسوج، یاسوج، ایران.

2 استادیار، گروه فیزیک، دانشگاه یاسوج، یاسوج، ایران

چکیده

وجود پدیده­های نوینی چون پلاتوهای مغناطش (ناحیه­های با مغناطش ثابت در منحنی مغناطش) و جهش مغناطش (ناپیوستگی در منحنی مغناطش) که به صورت غیرعادی در پذیرفتاری اسپینی در دمای صفر قابل مشاهده هستند، به صورت نظری در یک شبکۀ صفر بعدیِ (نقطه کوانتومیِ) کاگومۀ مثلثیِ بورون که تحت اثر یک پتانسیل خارجی زیرشبکه قرار گرفته است، مطالعه می­شود. با تجزیه و تحلیل برهم­کنش رودرمن- کیتل- کاسویا- یوشیدا (RKKY)، حالت پایه مغناطیسی شبکۀ کاگومۀ مثلثی در حضور دو اتم ناخالصی مغناطیسی بررسی می­شود. ویژگی بارز شبکۀ کاگومۀ مثلثی بور صفر بعدی، شکل­گیری پلاتوهای مغناطش در منحنی RKKY بر حسب انرژی فرمی می­باشد. پیکربندی­های مکانی ناخالصی­های مغناطیسی، به­طور چشمگیری مکان و شدّت پلاتوها را تغییر می­دهد. بر اساس اطلاعات موجود، این نوع پلاتوهایِ مبتنی بر برهم­کنش  RKKY پیش از این گزارش نشده است. نتایج برای ساختارهایی با اندازۀ محدود (ابعاد محدود) به روشنی تأیید می­کنند؛ که هم پهنا و هم مکان پلاتو­های مغناطش با استفاده از یک پتانسیل خارجی و انرژی فرمی، قابل تنظیم و مدیریت هستند. یکی دیگر از نتایج قابل توجه در این محاسبات، رفتار غیرعادی دیگری بنام جهش مغناطش است که همراه با ناپیوستگی در منحنی­های پذیرفتاری اسپینی می­باشد. ساختار الکترونی یک شبکۀ کاگومۀ مثلثی بورون می­تواند به شدّت تحت تأثیر یک پتانسیل زیر­شبکۀ مدوله شدۀ فضایی قرار گیرد؛ چراکه با مدیریت و تغییر پتانسیل خارجی، شکاف نواری و چگالی حالت­های موضعی  قابل کنترل می­باشند. نتایج به­دست آمده، دانش قابل توجهی در زمینه طراحی آزمایش و ابزار برای ایجاد و شرح فازهای نوین مغناطیسی در علم اسپینترونیک و ابزار مگنتوالکترونیکیِ مبتنی بر شبکه­های کاگومۀ مثلثی بورون، فراهم می­کنند.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Research Paper: Magnetization Plateaus and Magnetization Jumps in Quantum Dots of Boron Triangular Kagome Lattices

نویسندگان [English]

  • Moslem Zare 1
  • Ali Modabberasl 2
1 Assistant Professor, Department of Physics, Yasouj University, Yasouj, Iran
2 Assistant Professor, Department of Physics, Yasouj University, Yasouj, Iran.
چکیده [English]

An investigation of the modern phenomena of condensed matter physics, called, magnetization plateau and magnetization jump, visible as anomalies in spin susceptibility at zero temperature, have been carried out theoretically in a zero-dimensional boron triangular Kagome lattice (0D-BTKL), namely quantum dots of BTKL, subjected to a staggering sublattice potential. By analyzing the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction, the magnetic ground state of the 0D-TKL in the presence of two magnetic adatoms, in the presence of a staggered sublattice potential is evaluated. The important salient feature of the 0D-BTKLs is the emergence of the RKKY plateaus versus the Fermi energy. The spatial configurations of the magnetic impurities dramatically change the quality and quantity of the RKKY plateaus. These RKKY plateaus have not been reported before, to the best of our knowledge. Our finite-size results successfully confirm that both the width and location of the RKKY plateaus are tunable using an external potential and Fermi energy. Another remarkable observation is the nontrivial behavior, namely the magnetization jump, which accompanies the discontinuity in the spin susceptibility curves versus the staggering potential in our calculations. We believe that our results provide significant insights towards designing further experiments to search for the realization of the magnetization plateau phases and magnetization jumps in spintronics and pseudospin electronics devices based on BTKLs.

کلیدواژه‌ها [English]

  • Magnetization plateau
  • magnetization jump
  • Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction
  • boron triangular Kagome lattice
[1] Garanin D. and Canals B., "Classical spin liquid: Exact solution for the infinite-component antiferromagnetic model on the kagomé lattice," Physical Review B, vol. 59, no. 1, p. 443, 1999.
[2] Duan L.-M., Demler E., and Lukin M. D., "Controlling spin exchange interactions of ultracold atoms in optical lattices," Physical review letters, vol. 91, no. 9, p. 090402, 2003.
[3] Soltan-Panahi P., Struck J., Hauke P., Bick A., Plenkers W., Meineke G., Becker C., Windpassinger P., Lewenstein M. and Sengstock K., "Multi-component quantum gases in spin-dependent hexagonal lattices," Nature Physics, vol. 7, no. 5, pp. 434-440, 2011.
[4] Gemelke N., Zhang X., Hung C.-L., and Chin C., "In situ observation of incompressible Mott-insulating domains in ultracold atomic gases," Nature, vol. 460, no. 7258, pp. 995-998, 2009.
[5] Chen Y.-H., Wu W., Tao H.-S., and Liu W.-M., "Cold atoms in a two-dimensional triangular optical lattice as an artificial frustrated system," Physical Review A, vol. 82, no. 4, p. 043625, 2010.
[6] Jo G.-B., Guzman J., Thomas C. K., Hosur P., Vishwanath A., and Stamper-Kurn D. M., "Ultracold atoms in a tunable optical kagome lattice," Physical review letters, vol. 108, no. 4, p. 045305, 2012.
[7] Maruti S. and ter Haar L. W., "Magnetic properties of the two‐dimensional ‘‘triangles‐in‐triangles’’Kagomé lattice Cu9X2(cpa)6 (X= F, Cl, Br)," Journal of Applied Physics, vol. 75, no. 10, pp. 5949-5951, 1994.
[8] Mekata M., Abdulla M., Asano T., Kikuchi H., Goto T., Morishita T., Hori H., "Magnetic ordering in triangulated kagomélattice compound, Cu9Cl2(cpa)6· nH2O," Journal of magnetism and magnetic materials, vol. 177, pp. 731-732, 1998.
[9] Xie S.-Y., Li X.-B., Tian W. Q., Chen N.-K., Wang Y., Zhang S., Sun H.-B., "A novel two-dimensional MgB6 crystal: metal-layer stabilized boron kagome lattice," Physical Chemistry Chemical Physics, vol. 17, no. 2, pp. 1093-1098, 2015.
[10] Mao J., Zhang H., Jiang Y., Pan Y., Gao M., Xiao W., and Gao H.-J., "Tunability of supramolecular kagome lattices of magnetic phthalocyanines using graphene-based moiré patterns as templates," Journal of the American Chemical Society, vol. 131, no. 40, pp. 14136-14137, 2009.
[11] Gardner J. S., Gingras M. J., and Greedan J. E., "Magnetic pyrochlore oxides," Reviews of Modern Physics, vol. 82, no. 1, p. 53, 2010.
[12] Waldtmann C., Everts H.-U., Bernu B., Sindzingre P., Lhuillier C., Lecheminant P., Pierre L., "First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice," The European Physical Journal B-Condensed Matter and Complex Systems, vol. 2, no. 4, pp. 501-507, 1998.
[13] Balents L., "Spin liquids in frustrated magnets," Nature, vol. 464, no. 7286, pp. 199-208, 2010.
[14] Nisoli C., Moessner R., and Schiffer P., "Colloquium: Artificial spin ice: Designing and imaging magnetic frustration," Reviews of Modern Physics, vol. 85, no. 4, p. 1473, 2013.
[15] Norman M., "Colloquium: Herbertsmithite and the search for the quantum spin liquid," Reviews of Modern Physics, vol. 88, no. 4, p. 041002, 2016.
[16] Mielke A., "Exact ground states for the Hubbard model on the Kagome lattice," Journal of Physics A: Mathematical and General, vol. 25, no. 16, p. 4335, 1992.
[17] Tanaka A. and Ueda H., "Stability of ferromagnetism in the Hubbard model on the Kagome lattice," Physical review letters, vol. 90, no. 6, p. 067204, 2003.
[18] Ohgushi K., Murakami S., and Nagaosa N., "Spin anisotropy and quantum Hall effect in the kagomé lattice: Chiral spin state based on a ferromagnet," Physical Review B, vol. 62, no. 10, p. R6065, 2000.
[19] Guo H.-M. and Franz M., "Topological insulator on the kagome lattice," Physical Review B, vol. 80, no. 11, p. 113102, 2009.
[20] Tang E., Mei J.-W., and Wen X.-G., "High-temperature fractional quantum Hall states," Physical review letters, vol. 106, no. 23, p. 236802, 2011.
[21] Guterding D., Jeschke H. O., and Valentí R., "Prospect of quantum anomalous Hall and quantum spin Hall effect in doped kagome lattice Mott insulators," Scientific reports, vol. 6, no. 1, pp. 1-8, 2016.
[22] Zhang L. and Tong P., "Staggered potential and magnetic field tunable electronic switch in a kagome nanoribbon junction," Journal of Physics: Condensed Matter, vol. 31, no. 30, p. 305302, 2019.
[23] Lauchli A., Sudan J., and Sorensen E., "Ground-state energy and spin gap of spin-1/2 Kagome-Heisenberg antiferromagnetic clusters: Large-scale exact diagonalization results," Physical Review B, vol. 83, no. 21, 2011.
[24] Götze O., Farnell D. J., Bishop R., Li P., and Richter J., "Heisenberg antiferromagnet on the kagome lattice with arbitrary spin: A higher-order coupled cluster treatment," Physical Review B, vol. 84, no. 22, p. 224428, 2011.
[25] Hermele M., Ran Y., Lee P. A., and Wen X.-G., "Properties of an algebraic spin liquid on the kagome lattice," Physical Review B, vol. 77, no. 22, p. 224413, 2008.
[26] Iqbal Y., Becca F., and Poilblanc D., "Projected wave function study of Z2 spin liquids on the kagome lattice for the spin-1/2 quantum Heisenberg antiferromagnet," Physical Review B, vol. 84, no. 2, p. 020407, 2011.
[27] Ruderman M. A. and Kittel C., "Indirect exchange coupling of nuclear magnetic moments by conduction electrons," Physical Review, vol. 96, no. 1, p. 99, 1954.
[28] Kasuya T., "A theory of metallic ferro-and antiferromagnetism on Zener's model," Progress of theoretical physics, vol. 16, no. 1, pp. 45-57, 1956.
[29] Yosida K., "Magnetic properties of Cu-Mn alloys," Physical Review, vol. 106, no. 5, p. 893, 1957.
[30] Szwacki N. G., Sadrzadeh A., and Yakobson B. I., "B80 fullerene: an ab initio prediction of geometry, stability, and electronic structure," Physical review letters, vol. 98, no. 16, p. 166804, 2007.
[31] Gonzalez Szwacki N., "Boron fullerenes: a first-principles study," Nanoscale Research Letters, vol. 3, no. 2, pp. 49-54, 2008.
[32] Mannix A. J., Zhou X.-F., Kiraly B., Wood J. D., Alducin D., Myers B. D., Liu X., Fisher B. L., Santiago U., Guest J. R., Yacaman M. J., Ponce A., Oganov A. R., Hersam M. C., Guisinger N. P., "Synthesis of borophenes: Anisotropic, two-dimensional boron polymorphs," Science, vol. 350, no. 6267, pp. 1513-1516, 2015.
[33] Otten C. J., Lourie O. R., Yu M.-F., Cowley J. M., Dyer M. J., Ruoff R. S., and Buhro W. E., "Crystalline boron nanowires," Journal of the American Chemical Society, vol. 124, no. 17, pp. 4564-4565, 2002.
[34] Liu F., Tang D.-M., Gan H., Mo X., Chen J., Deng S., Xu N., Bando Y., and Golberg D., "Individual boron nanowire has ultra-high specific young’s modulus and fracture strength as revealed by in situ transmission electron microscopy" ACS nano, vol. 7, no. 11, pp. 10112-10120, 2013.
[35] Ciuparu D., Klie R. F., Zhu Y., and Pfefferle L., "Synthesis of pure boron single-wall nanotubes," The Journal of Physical Chemistry B, vol. 108, no. 13, pp. 3967-3969, 2004.
[36] Liu F., Shen C., Su Z., Ding X., Deng S., Chen J., Xu N. and Gao H., "Metal-like single crystalline boron nanotubes: synthesis and in situ study on electric transport and field emission properties," Journal of Materials Chemistry, vol. 20, no. 11, pp. 2197-2205, 2010.
[37] Kiran B., Bulusu S., Zhai H.-J., Yoo S., Zeng X. C., and Wang L.-S., "Planar-to-tubular structural transition in boron clusters: B20 as the embryo of single-walled boron nanotubes," Proceedings of the National Academy of Sciences, vol. 102, no. 4, pp. 961-964, 2005.
[38] Oger E., Crawford N. R., Kelting R., Weis P., Kappes M. M., and Ahlrichs R., "Boron cluster cations: transition from planar to cylindrical structures," Angewandte Chemie International Edition, vol. 46, no. 44, pp. 8503-8506, 2007.
[39] An W., Bulusu S., Gao Y., and Zeng X. C., "Relative stability of planar versus double-ring tubular isomers of neutral and anionic boron cluster B20 and B20-," The Journal of chemical physics, vol. 124, no. 15, p. 154310, 2006.
[40] Liu H., Gao J., and Zhao J., "From boron cluster to two-dimensional boron sheet on Cu (111) surface: growth mechanism and hole formation," Scientific reports, vol. 3, no. 1, pp. 1-9, 2013.
[41] Wu X., Dai J., Zhao Y., Zhuo Z., Yang J., and Zeng X. C., "Two-dimensional boron monolayer sheets," ACS nano, vol. 6, no. 8, pp. 7443-7453, 2012.
[42] Liu Y., Penev E. S., and Yakobson B. I., "Probing the synthesis of two‐dimensional boron by first‐principles computations," Angewandte Chemie International Edition, vol. 52, no. 11, pp. 3156-3159, 2013.
[43] Zhang Z., Yang Y., Gao G., and Yakobson B. I., "Two‐dimensional boron monolayers mediated by metal substrates," Angewandte Chemie, vol. 127, no. 44, pp. 13214-13218, 2015.
[44] Eremets M. I., Struzhkin V. V., Mao H.-k., and Hemley R. J., "Superconductivity in boron," Science, vol. 293, no. 5528, pp. 272-274, 2001.
[45] Oganov A. R., Chen J., Gatti C., Ma Y., Ma Y., Glass C. W., Liu Z., Yu T., Kurakevych O. O. and Solozhenko V. L., "Ionic high-pressure form of elemental boron," Nature, vol. 457, no. 7231, pp. 863-867, 2009.
[46] Xie S.-Y., Li X.-B., Tian W. Q., Chen N.-K., Zhang X.-L., Wang Y., Zhang S., and Sun H.-B., "First-principles calculations of a robust two-dimensional boron honeycomb sandwiching a triangular molybdenum layer," Physical Review B, vol. 90, no. 3, p. 035447, 2014.
[47] Zhang L., Wang Z., Du S., Gao H.-J., and Liu F., "Prediction of a Dirac state in monolayer TiB2," Physical Review B, vol. 90, no. 16, p. 161402, 2014.
[48] Zhang H., Li Y., Hou J., Tu K., and Chen Z., "FeB6 monolayers: the graphene-like material with hypercoordinate transition metal," Journal of the American Chemical Society, vol. 138, no. 17, pp. 5644-5651, 2016.
[49] Li J., Fan X., Wei Y., Liu J., Guo J., Li X., Wang V., Liangc Y. and Chen G., "Voltage-gated spin-filtering properties and global minimum of planar MnB6, and half-metallicity and room-temperature ferromagnetism of its oxide sheet," Journal of Materials Chemistry C, vol. 4, no. 46, pp. 10866-10875, 2016.
[50] Tai G., Hu T., Zhou Y., Wang X., Kong J., Zeng T., You Y., Wang Q., "Synthesis of atomically thin boron films on copper foils," Angewandte Chemie International Edition, vol. 54, no. 51, pp. 15473-15477, 2015.
[51] Feng B., Zhang J., Zhong Q., Li W., Li S., Li H., Cheng P., Meng S., Chen L. and Wu K., "Experimental realization of two-dimensional boron sheets," Nature chemistry, vol. 8, no. 6, pp. 563-568, 2016.
[52] Tang H. and Ismail-Beigi S., "Novel precursors for boron nanotubes: the competition of two-center and three-center bonding in boron sheets," Physical review letters, vol. 99, no. 11, p. 115501, 2007.
[53] Kim S., Han W. H., Lee I.-H., and Chang K., "Boron triangular Kagome lattice with half-metallic ferromagnetism," Scientific reports, vol. 7, no. 1, pp. 1-8, 2017.
[54] Zhao Y., Ban C., Xu Q., Wei S.-H., and Dillon A. C., "Charge-driven structural transformation and valence versatility of boron sheets in magnesium borides," Physical Review B, vol. 83, no. 3, p. 035406, 2011.
[55] Ziff R. M. and Gu H., "Universal condition for critical percolation thresholds of kagomé-like lattices," Physical Review E, vol. 79, no. 2, p. 020102, 2009.
[56] Balakrishnan J., Kok Wai Koon G., Jaiswal M., Castro Neto A., and Özyilmaz B., "Colossal enhancement of spin–orbit coupling in weakly hydrogenated graphene," Nature Physics, vol. 9, no. 5, pp. 284-287, 2013.
[57] Weeks C., Hu J., Alicea J., Franz M., and Wu R., "Engineering a robust quantum spin Hall state in graphene via adatom deposition," Physical Review X, vol. 1, no. 2, p. 021001, 2011.
[58] Calleja F., Ochoa H., Garnica M., Barja S., Navarro J. J., Black A., Otrokov M. M., Chulkov E. V., Arnau A., Vazquez de Parga A. L., Guinea F., Miranda R., "Spatial variation of a giant spin–orbit effect induces electron confinement in graphene on Pb islands," Nature Physics, vol. 11, no. 1, pp. 43-47, 2015.
[59] Zare M., "RKKY interaction in biased single-layer silicene," Physical Review B, vol. 100, no. 8, p. 085434, 2019.
[60] Zare M., Parhizgar F., and Asgari R., "Strongly anisotropic RKKY interaction in monolayer black phosphorus," Journal of Magnetism and Magnetic Materials, vol. 456, pp. 307-315, 2018.
[61] Zare M., Parhizgar F., and Asgari R., "Topological phase and edge states dependence of the RKKY interaction in zigzag silicene nanoribbon," Physical Review B, vol. 94, no. 4, p. 045443, 2016.
[62] Zare M. and Sadeghi E., "Exchange interaction of magnetic impurities in a biased bilayer phosphorene nanoribbon," Physical Review B, vol. 98, no. 20, p. 205401, 2018.
[63] Zare M., "Strain-induced modulation of exchange interaction in monolayer zigzag nanoribbons of B2S," Materials Research Express, vol. 6, no. 10, p. 105097, 2019.
[64] Siemensmeyer K., Wulf E., Mikeska H.-J., Flachbart K., Gabáni S., Mat’aš S., Priputen P., Efdokimova A., and Shitsevalova N., "Fractional magnetization plateaus and magnetic order in the Shastry-Sutherland magnet TmB4," Physical review letters, vol. 101, no. 17, p. 177201, 2008.
[65] Yoshii S., Yamamoto T., Hagiwara M., Michimura S., Shigekawa A., Iga F., Takabatake T., Kindo K., "Multistep magnetization plateaus in the Shastry-Sutherland system TbB4," Physical review letters, vol. 101, no. 8, p. 087202, 2008.
[66] Hida K., "Magnetic properties of the spin-1/2 ferromagnetic-ferromagnetic-antiferromagnetic trimerized heisenberg chain," Journal of the Physical Society of Japan, vol. 63, no. 6, pp. 2359-2364, 1994.
[67] Zare M., "Observation of flat band, RKKY plateau, and magnetization jump in quasi-one-dimensional triangular kagome lattice model," Journal of Applied Physics, vol. 128, no. 16, p. 163903, 2020.
[68] Kohno M. and Takahashi M., "Magnetization process of the spin-1/2 XXZ models on square and cubic lattices," Physical Review B, vol. 56, no. 6, p. 3212, 1997.
[69] Sakai T. and Takahashi M., "Metamagnetism of antiferromagnetic XXZ quantum spin chains," Physical Review B, vol. 60, no. 10, p. 7295, 1999.
[70] Dmitriev D. and Krivnov V. Y., "Frustrated ferromagnetic spin-1/2 chain in a magnetic field," Physical Review B, vol. 73, no. 2, p. 024402, 2006.
[71] Nakano H., Hasegawa Y., and Sakai T., "Magnetization jump in the magnetization process of the spin-1/2 heisenberg antiferromagnet on a distorted square-kagome lattice," Journal of the Physical Society of Japan, vol. 84, no. 11, p. 114703, 2015.
[72] Poulis N., Van den Handel J., Ubbink J., Poulis J., and Gorter C., "On antiferromagnetism in a single crystal," Physical Review, vol. 82, no. 4, p. 552, 1951.
[73] Møller H. B., Shapiro S., and Birgeneau R., "Field-dependent magnetic phase transitions in mixed-valent tmse", Physical Review Letters, vol. 39, no. 16, p. 1021, 1977.
[74] Hardy V., Maignan A., Hébert S., Yaicle C., Martin C., Hervieu M., Lees M. R., Rowlands G., D. Mc K. Paul, and Raveau B., "Observation of spontaneous magnetization jumps in manganites", Physical Review B, vol. 68, no. 22, p. 220402, 2003.
[75] Honecker A., Schulenburg J., and Richter J., "Magnetization plateaus in frustrated antiferromagnetic quantum spin models," Journal of Physics: Condensed Matter, vol. 16, no. 11, p. S749, 2004.
[76] Depenbrock S., McCulloch I. P., and Schollwöck U., "Nature of the spin-liquid ground state of the S=1/2 Heisenberg model on the kagome lattice," Physical review letters, vol. 109, no. 6, p. 067201, 2012.
[77] Cabra D. C. and Grynberg M. D., "Ground-state magnetization of polymerized spin chains," Physical Review B, vol. 59, no. 1, p. 119, 1999.
[78] Cabra D. C., Honecker A., and Pujol P., "Magnetization curves of antiferromagnetic Heisenberg spin-1/2 ladders," Physical review letters, vol. 79, no. 25, p. 5126, 1997.
[79] Morita K., Sugimoto T., Sota S., and Tohyama T., "Magnetization plateaus in the spin-1/2 antiferromagnetic Heisenberg model on a kagome-strip chain," Physical Review B, vol. 97, no. 1, p. 014412, 2018.
[80] Totsuka K., "Magnetization plateau in the S=1/2 Heisenberg spin chain with next-nearest-neighbor and alternating nearest-neighbor interactions", Phys. Rev. B, 57, 3454, 1998.
[81] Momoi T., Totsuka K., "Magnetization plateaus as insulator-superfluid transitions in quantum spin systems", Phys. Rev. B, 61, 3231, 2000.
[82] Kubo K. and Momoi T., "Ground state of a spin system with two- and four-spin exchange interactions on the triangular lattice", Z. Phys. B, 103, 485, 1997. 
[83] Honecker A., " Lanczos study of the S = 1/2 frustrated square-lattice anti-ferromagnet in a magnetic field", Can. J. Phys. 79, 1557, 2001.
[84] Dey D., Das S., Kumar M., and Ramasesha S., “Magnetization plateaus of spin-1/2 system on a 5/7 skewed ladder “, Phys. Rev. B, 101, 195110, 2020.
[85] Schulenburg J., Honecker A., Schnack J., Richter J., and Schmidt H.-J., “Macroscopic Magnetization Jumps due to Independent Magnons in Frustrated Quantum Spin Lattices “Phys. Rev. Lett. 88, 167207, 2002.
[86] Schmidt H.-J., Richter J., and Moessner R., “Linear independence of localized magnon states “J. Phys. A 39, 10673, 2006.
[87] Nishimoto S., Shibata N., and Hotta C.,” Controlling frustrated liquids and solids with an applied field in a kagome Heisenberg antiferromagnet “Nat. Commun. 4, 2287, 2013.
[88] Capponi S., Derzhko O., Honecker A., Läuchli A. M., and Richter J.,” Numerical study of magnetization plateaus in the spin-12 kagome Heisenberg antiferromagnet “Phys. Rev. B, 88, 144416, 2013.
[89] Pal S., Mukherjee A., and Lal S.,” Topological approach to quantum liquid ground states on geometrically frustrated Heisenberg antiferromagnets “J. Phys.: Condens. Matter, 32, 165805, 2020.