Halperin States of Particles and Holes in Ideal Time Reversal Invariant Pairs of Chern Bands and The Fractional Quantum Spin Hall Effect in Moiré MoTe$_2$
Abstract: An experiment in moir\'e MoTe$_2$ bilayers reported the first observation of a topologically ordered state with zero Hall conductivity and half of the edge conductance of a standard time-reversal invariant quantum spin Hall insulator. This state is believed to emerge at total filling one of a pair of bands with Chern numbers $C=\pm1$ related by time reversal symmetry. By viewing these bands as a pair of Landau levels with opposite magnetic fields, and starting from a parent magnet with one filled band, we demonstrate that a class of Halperin states constructed by adding particles to the empty Chern band and holes to the occupied Chern band have all the properties observed in MoTe$_2$. Remarkably, these states break time-reversal symmetry but have exactly zero Hall conductivity and helical edge conductance of $e2/2h$. These states also feature a spinless composite fermion with the same charge as the electron but split equally between both valleys. In a standard Halperin 331 state, this particle would be a neutral Bogoliubov composite fermion. However, in our context this composite fermion is charged but remains itinerant because it is split into the two valleys that effectively experience opposite magnetic fields. The existence of such charged itinerant particles is a key difference between Landau levels with opposite magnetic fields and standard multi-components Landau levels, where all the itinerant particles are charge neutral, such as the magneto-roton of the Laughlin state or the Bogoliubov composite fermion of the Moore-Read state. When the electron density changes away from the ideal filling and these itinerant charged particles are added to the parent state, the disorder potential is less efficient at localizing them as compared to standard Lanadau levels. This can explain why the state in MoTe$_2$ did not display a robust Hall plateau upon changing the electron density.
- C. Repellin and T. Senthil, Physical Review Research 2, 023238 (2020).
- Y. Sheffer and A. Stern, Physical Review B 104, L121405 (2021).
- V. Crépel and L. Fu, Physical Review B 107, L201109 (2023).
- A. P. Reddy and L. Fu, Physical Review B 108, 245159 (2023).
- C. L. Kane and E. J. Mele, Physical review letters 95, 226801 (2005a).
- C. L. Kane and E. J. Mele, Physical review letters 95, 146802 (2005b).
- B. A. Bernevig and S.-C. Zhang, Physical review letters 96, 106802 (2006).
- Y.-H. Zhang, arXiv preprint arXiv:2402.05112 (2024).
- C.-M. Jian and C. Xu, arXiv preprint arXiv:2403.07054 (2024).
- B. I. Halperin, helv. phys. acta 56, 75 (1983).
- We use units of electron charge qe=1subscript𝑞𝑒1q_{e}=1italic_q start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = 1 and ℏ=1Planck-constant-over-2-pi1\hbar=1roman_ℏ = 1, but will restore explicit units at the end in a few selected formulas.
- N. Stefanidis and I. Sodemann, Physical Review B 102, 035158 (2020).
- K. Yang, Physical Review B 58, R4246 (1998).
- X.-G. Wen, Physical Review B 43, 11025 (1991a).
- X.-G. Wen, Physical review letters 64, 2206 (1990).
- X.-G. Wen, Physical Review B 44, 5708 (1991b).
- A. MacDonald, Physical review letters 64, 220 (1990).
- R. Landauer, Philosophical magazine 21, 863 (1970).
- M. Büttiker, Physical review letters 57, 1761 (1986).
- X.-G. Wen, Quantum field theory of many-body systems: From the origin of sound to an origin of light and electrons (OUP Oxford, 2004).
- N. Read and D. Green, Physical Review B 61, 10267 (2000).
- F. Haldane and E. Rezayi, Physical review letters 60, 956 (1988).
- T.-L. Ho, Physical review letters 75, 1186 (1995).
- N. Read and E. Rezayi, Physical Review B 54, 16864 (1996).
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