⇦Chinese Physics Letters, 2017, Vol. 34, No. 9, Article code 090101Views & Comments Discovery of Fractionalized Neutral Spin-1/2 Excitation of Topological Order Xiao-Gang Wen(文小刚)** Affiliations Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 31 July 2017 **Corresponding author. Email: xgwen@mit.edu
Citation Text: Wen X G 2017 Chin. Phys. Lett. 34 090101 Abstract DOI:10.1088/0256-307X/34/9/090101 PACS:01.10.-m © 2017 Chinese Physics Society Article Text After the discovery of fraction quantum Hall states in the 1980s,[1] it became more and more clear that Landau symmetry breaking theory does not describe all possible quantum phases of matter. The new quantum phases of matter were called topologically ordered phases[2,3] (for gapped cases) or quantum ordered phases[4] (for gapless cases), which correspond to patterns of many-body entanglement.[5-7] One may wonder: besides quantum Hall systems, are there other systems that realize the new topological/quantum order? In the 1980s and 1990s, it was shown theoretically that topological orders can be realized in spin liquids, such as the chiral spin liquids[8,9] and $Z_2$-spin liquids.[10,11] Also, stable quantum ordered phases can be realized in algebraic spin liquids.[12-15] The topological/quantum ordered states are not easy to detect since they are not characterized by local order parameters. On the other hand, the absence of local order parameters lead to a strange way to discover topological/quantum ordered states: one tries to detect any kind of order parameters and phases transitions as the temperature is lower to zero. If one finds nothing, then one can declare that a certain topological/quantum ordered state is discovered (if the trivial ground state can be ruled out). In fact, such a strategy was used by Y. Lee, which led to a discovery of herbertsmithite as a possible spin liquid candidate on Kagome lattice.[16] A few years earlier, another spin liquid candidate was discovered in organic Mott insulator of triangular lattice.[17] The above two are 2-dimensional spin liquids. A 3-dimensional spin liquid candidate was found in hyperkagome antiferromagnet.[18] Recently, a very promising spin liquid was discovered in honeycomb lattice $\alpha$-RuCl$_3$ with strong spin-orbital coupling.[19-25] One of the most important properties of a spin liquid is whether the spin liquid is gapped or gapless. If the spin liquid is gapped, then the next important question is whether the spin liquid has fractionalized spin-1/2 quasiparticles or not. The appearance of spin-1/2 excitations implies a non-trivial topological order in the spin liquid. However, one challenge to study herbertsmithite in more detail is to reduce the influence of magnetic impurities. The 5–10% magnetic impurities in herbertsmithite make it difficult to determine if the spin liquid is gapped or gapless.[26] In a recent work, Ref. [27], published by Chinese Physics Letters, a new kind of Kagome spin liquid was found in a new material Cu$_3$Zn(OH)$_6$FBr. The new material allows one to measure Knight shift via $^{19}$F NMR measurements (with $I = 1/2$ nuclear spin). The intrinsic Cu-spin magnetic susceptibility from Knight shift reveals a small spin gap of 8 K (compared to the spin coupling of 200 K). The small spin gap is consistent with a recent numerical calculation which found a long correlation length in the Heisenberg model on Kagome lattice.[28] Furthermore, the magnetic field dependence of spin gap indicates that the thermally excited spin excitations carry fractionalized spin-1/2. Just like the direct discovery of fractional charge via noise measurement,[29] the discovery of a totally new fractionalized neutral spin-1/2 excitation is a very exciting result. This result suggests that the Kagome spin liquid is the $Z_2$-spin liquid with a $Z_2$ topological order.[10,11] The $SO(3)$ symmetric $Z_2$ topological order features emergent spin-1/2, emergent fermions, etc.[10,11] However, at moment, it is not clear whether the observed spin-1/2 excitation is a boson or a fermion. Hopefully, more detailed future experiments can resolve this issue. I also like to remark that the spin liquid in $\alpha$-RuCl$_3$ does not have the $SO(3)$ spin rotation symmetry. In this case, it is harder to directly detect the fractionalization of topological order. References Two-Dimensional Magnetotransport in the Extreme Quantum LimitVacuum degeneracy of chiral spin states in compactified spaceTOPOLOGICAL ORDERS IN RIGID STATESQuantum orders and symmetric spin liquidsLocal unitary transformation, long-range quantum entanglement, wave function renormalization, and topological orderGapped quantum liquids and topological order, stochastic local transformations and emergence of unitarityRenormalization group constructions of topological quantum liquids and beyondEquivalence of the resonating-valence-bond and fractional quantum Hall statesChiral spin states and superconductivityLarge- N expansion for frustrated quantum antiferromagnetsMean-field theory of spin-liquid states with finite energy gap and topological ordersLarge- n limit of the Heisenberg-Hubbard model: Implications for high- T c superconductorsElectron Spectral Function and Algebraic Spin Liquid for the Normal State of Underdoped High T c SuperconductorsStability of U ( 1 ) spin liquids in two dimensionsProjected-Wave-Function Study of the Spin- 1 / 2 Heisenberg Model on the Kagomé LatticeSpin Dynamics of the Spin- 1 / 2 Kagome Lattice Antiferromagnet ZnCu 3 ( OH ) 6 Cl 2 Spin Liquid State in an Organic Mott Insulator with a Triangular LatticeSpin-Liquid State in the S = 1 / 2 Hyperkagome Antiferromagnet Na 4 Ir 3 O 8 Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnetEvidence for a Field-Induced Quantum Spin Liquid in α - RuCl 3 Gapless Spin Excitations in the Field-Induced Quantum Spin Liquid Phase of alpha-RuCl3Large field-induced gap of Kitaev-Heisenberg paramagnons in $\alpha$-RuCl$_{3}$Field-induced quantum criticality in the Kitaev system α − RuCl 3 Magnetic Excitations and Continuum of a Field-Induced Quantum Spin Liquid in $\alpha$-RuCl$_3$Observation of gapped anyons in the Kitaev honeycomb magnet under a magnetic fieldLocal spin susceptibility of the S = 1 2 kagome lattice in ZnCu 3 (OD) 6 Cl 2 Gapped Spin-1/2 Spinon Excitations in a New Kagome Quantum Spin Liquid Compound Cu 3 Zn(OH) 6 FBrGapped spin liquid with Z 2 topological order for the kagome Heisenberg model
[1] Tsui D C, Stormer H L and Gossard A C 1982 Phys. Rev. Lett. 48 1559
[2] Wen X G 1989 Phys. Rev. B 40 7387
[3] Wen X G 1990 Int. J. Mod. Phys. B 4 239
[4] Wen X G 2002 Phys. Rev. B 65 165113
[5] Chen X, Gu Z C and Wen X G 2010 Phys. Rev. B 82 155138
[6] Zeng B and Wen X G 2015 Phys. Rev. B 91 125121
[7] Swingle B and McGreevy J 2016 Phys. Rev. B 93 045127
[8] Kalmeyer V and Laughlin R B 1987 Phys. Rev. Lett. 59 2095
[9] Wen X G, Wilczek F and Zee A 1989 Phys. Rev. B 39 11413
[10] Read N and Sachdev S 1991 Phys. Rev. Lett. 66 1773
[11] Wen X G 1991 Phys. Rev. B 44 2664
[12] Affleck I and Marston J B 1988 Phys. Rev. B 37 3774
[13] Rantner W and Wen X G 2001 Phys. Rev. Lett. 86 3871
[14] Hermele M, Senthil T, Fisher M P A, Lee P A, Nagaosa N and Wen X G 2004 Phys. Rev. B 70 214437
[15] Ran Y, Hermele M, Lee P A and Wen X G 2007 Phys. Rev. Lett. 98 117205
[16] Helton J S, Matan K, Shores M P, Nytko E A, Bartlett B M, Yoshida Y, Takano Y, Suslov A, Qiu Y, Chung J H et al 2007 Phys. Rev. Lett. 98 107204
[17] Shimizu Y, Miyagawa K, Kanoda K, Maesato M and Saito G 2003 Phys. Rev. Lett. 91 107001
[18] Okamoto Y, Nohara M, Aruga-Katori H and Takagi H 2007 Phys. Rev. Lett. 99 137207
[19] Banerjee A, Bridges C A, Yan J Q, Aczel A A, Li L, Stone M B, Granroth G E, Lumsden M D, Yiu Y, Knolle J et al 2016 Nat. Mater. 15 733
[20] Baek S H, Do S H, Choi K Y, Kwon Y S, Wolter A U B, Nishimoto S, van den Brink J and Büchner B 2017 Phys. Rev. Lett. 119 037201
[21] Zheng J, Ran K, Li T, Wang J, Wang P, Liu B, Liu Z, Normand B, Wen J and Yu W 2017 arXiv:1703.08474 [cond-mat.str-el]
[22] Hentrich R, Wolter A U B, Zotos X, Brenig W, Nowak D, Isaeva A, Doert T, Banerjee A, Lampen-Kelley P, Mandrus D G et al 2017 arXiv:1703.08623 [cond-mat.str-el]
[23] Wolter A U B, Corredor L T, Janssen L, Nenkov K, Schönecker S, Do S H, Choi K Y, Albrecht R, Hunger J, Doert T et al 2017 Phys. Rev. B 96 041405
[24] Wang Z, Reschke S, Hüvonen D, Do S H, Choi K Y, Gensch M, Nage U, Rõõm T and Loidl A 2017 arXiv:1706.06157 [cond-mat.str-el]
[25] Jansa N, Zorko A, Gomilsek M, Pregelj M, Krämer K W, Biner D, Biffin A, Rüegg C and Klanjsek M 2017 arXiv:1706.08455 [cond-mat.str-el]
[26] Imai T, Fu M, Han T H and Lee Y S 2011 Phys. Rev. B 84 020411
[27] Feng Z, Li Z, Meng X, Yi W, Wei Y, Zhang J, Wang Y C, Jiang W, Liu Z, Li S, Liu F, Luo J, Li S, Zheng G Q, Meng Z Y, Mei J W and Shi Y 2017 Chin. Phys. Lett. 34 077502
[28] Mei J W, Chen J Y, He H and Wen X G 2017 Phys. Rev. B 95 235107
[29] de Picciotto R, Reznikov M, Heiblum M, Umansky V, Bunin G and Mahalu D 1997 Nature 389 162