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The application of topology in condensed matter physics began with the study of the quantum Hall effect and has gradually become the main theme of modern condensed matter physics. Its importance lies in capturing the universal properties of physical systems. In particular, fractional quantum Hall liquids are the most strongly correlated systems and exhibit topological order. Its most important and universal feature is the quasiparticle (quasi-hole) elementary excitations with fractional charge and statistics, which are captured by topological field theories. However, such a macroscopic description of fractional quantum Hall liquids is not complete, because it misses an important geometric aspect that is important for both universal and non-universal properties of the system. In particular, the nature of its electrically neutral elementary excitations has not been fully understood until recently. Finite-wavelength electrically neutral elementary excitations can be viewed as charge density waves or bound states of quasi-particles-quasi-holes. However, such pictures are not applicable in the long-wave limit, so a new theoretical framework is needed. In this theoretical framework, one of the most basic degrees of freedom is the metric tensor that describes the electron correlation. Figuratively speaking, it describes the geometric shape of the correlation hole around the electron. Therefore, this theory is called the geometric theory of the fractional quantum Hall effect. Since the metric tensor is also the basic degree of freedom of the theory of gravity, this theoretical framework can be regarded as a certain type of quantum theory of gravity. Its basic elementary excitation is a spin-two graviton. This perspective discusses the geometric degrees of freedom and its quantum dynamics in quantum Hall liquids from a microscopic perspective, revealing that its basic elementary excitations are spin-two graviton-like particles with specific chirality, and focuses on the experimental detection of this chiral graviton-like particle. The figure illustrates graviton-like excitation and its chirality in the 1/3 Laughlin state using Xiao-Gang Wen’s dancing pattern analogy [Wen X G 2004 Quantum Field Theory of Many-body Systems: From the Origin of Sound to An Origin of Light and Electrons (Oxford: Oxford University Press)], with left panel showing that in the Laughlin ground state (or dancing pattern), the minimum relative angular momentum of a pair of dancers is three, ensuring sufficient separation between them, and with right panel displaying that a graviton-like excitation corresponding to a pair whose relative angular momentum changes from three to one (antisymmetry of fermion wave function only allows for odd relative angular momenta). This is not allowed in the Laughlin state, as a result, it corresponds to an excitation which is the “graviton” detected by Liang et al. [Liang J H, Liu Z Y, Yang Z H, et al. 2024 Nature 628 78 ]. In other words, the Raman process creates a “graviton” by turning a pair with relative angular momentum three (left panel) into a pair with relative angular momentum one (right panel). The angular momentum of this excitation is$1- 3 =-2 $ , corresponding to a graviton with chirality –2. For hole states like 2/3, because the chirality is reversed for holes, graviton chirality becomes +2. This figure is adopted from Yang [Yang K2024 The Innovation 5 100641 ].
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Keywords:
- topology /
- geometry /
- gravitons /
- strong correlation
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图 1 Laughlin态的两体关联函数 (a)各向异性的Laughlin态的一个代表; (b)各向同性的Laughlin 波函数. 两者关联空穴面积相同, 引自文献[4]
Fig. 1. Two-body correlation function of the Laughlin state: (a) A representative of the anisotropic Laughlin state; (b) the isotropic Laughlin wave function. The areas of the two correlation holes are the same. From Ref. [4].
图 2 用Wen[1] 的舞蹈规则类比来说明1/3填充Laughlin 态中的类引力子激发及其手性, 左图为在 Laughlin 基态(或舞蹈规则)中, 一对舞者的最小相对角动量为3, 以确保他们之间有足够的距离; 右图为类引力子激发对应于相对角动量从3变为1的一个电子对(费米子波函数的反对称性只允许奇数相对角动量); 这在 Laughlin 态下是不允许的, 因此, 它对应于一种激发, 即文献[12]检测到的“引力子”, 换句话说, 拉曼过程通过将相对角动量为3的电子对(左图)变成相对角动量为1的电子对(右图)来产生“引力子”; 该激发的角动量为 1 – 3 = –2, 对应引力子手性为–2. 对于2/3及3/5填充的空穴状态, 由于空穴的手性与电子相反, 引力子手性变为 +2, 引自文献[13]
Fig. 2. Illustration of graviton-like excitation and its chirality in the 1/3 Laughlin state using Wen’s[1] dancing pattern analogy. Left panel: In the Laughlin ground state (or dancing pattern), the minimum relative angular momentum of a pair of dancers is 3, ensuring sufficient separation between them. Right panel: A graviton-like excitation corresponds to a pair whose relative angular momentum changes from 3 to 1 (antisymmetry of fermion wave function only allows for odd relative angular momenta). This is not allowed in the Laughlin state, as a result, it corresponds to an excitation which is the “graviton” detected by Liang et al.[12]. In other words, the Raman process creates a “graviton” by turning a pair with relative angular momentum 3 (left panel) to a pair with relative angular momentum 1 (right panel). The angular momentum of this excitation is 1 – 3 = –2, corresponding to graviton chirality –2. For hole states like 2/3 and 3/5, because the chirality is reversed for holes, graviton chirality becomes +2. From Ref. [13].
图 3 圆极化偏振的拉曼散射, 不同入射光与出射光偏振的组合对应于系统不同角动量的元激发, 在1/3填充的Laughlin态中, 只有对应于角动量为–2的组合有共振峰(见右图绿色曲线), 与文献[10]的预言相符, 引自文献[12]
Fig. 3. Circularly polarized Raman scattering. Different combinations of incident and outgoing light polarization couple to elementary excitations of different angular momentums in the system. In the 1/3 filled Laughlin state, only the combination corresponding to the angular momentum –2 has a resonance peak (see the green curve on the right), which is consistent with the prediction of Ref. [10]. Cited from Ref. [12].
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[1] Wen X G 2004 Quantum Field Theory of Many-body Systems: From the Origin of Sound to An Origin of Light and Electrons (Oxford: Oxford University Press
[2] Girvin S M, MacDonald A H, Platzman P M 1986 Phys. Rev. B 33 2481
Google Scholar
[3] Haldane F D M 2011 Phys. Rev. Lett. 107 116801
Google Scholar
[4] Qiu R Z, Haldane F D M, Wan X, Yang K, Yi S 2012 Phys. Rev. B 85 115308
Google Scholar
[5] Yang K 2013 Phys. Rev. B 88 241105
Google Scholar
[6] Halperin B I, Lee P A, Read N 1993 Phys. Rev. B 47 7312
Google Scholar
[7] Jo I, Rosales K A V, Mueed M A, Pfeiffer L N, West K W, Baldwin K W, Winkler R, Padmanabhan M, Shayegan M 2017 Phys. Rev. Lett. 119 016402
Google Scholar
[8] Yang K 2016 Phys. Rev. B 93 161302
Google Scholar
[9] Liu Z, Gromov A, Papic Z 2018 Phys. Rev. B 98 155140
Google Scholar
[10] Liou S F, Haldane F D M, Yang K, Rezayi E H 2019 Phys. Rev. Lett. 123 146801
Google Scholar
[11] Nguyen D X, Haldane F D M, Rezayi E H, Son D T, Yang K 2022 Phys. Rev. Lett. 128 246402
Google Scholar
[12] Liang J H, Liu Z Y, Yang Z H, Huang Y L, Wurstbauer U, Dean C R, West K W, Pfeiffer L N, Du L J, Pinczuk P 2024 Nature 628 78
Google Scholar
[13] Yang K 2024 The Innovation 5 100641
Google Scholar
[14] Nguyen D X, Son D T 2021 Phys. Rev. Res. 3 023040
Google Scholar
[15] Haldane F D M, Rezayi E H, Yang K 2021 Phys. Rev. B 104 L121106
Google Scholar
[16] Ma K K W, Peterson M R, Scarola V W, Yang K 2023 Encyclopedia of Condensed Matter Physics (2nd Ed.) (Academic Press
[17] 万歆, 王正汉, 杨昆 2013 物理 42 558
Wan X, Wang Z H, Yang K 2013 Physics 42 558
[18] Ma K K W, Yang K 2022 Phys. Rev. B 105 045306
Google Scholar
[19] Ma K K W 2022 arXiv: 2209.11119
[20] Ma K K W, Yang K 2024 arXiv: 2408.00058
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