DOI: 10.5061/dryad.9p8cz8ws0
title: Data from: Local probe of bulk and edge states in a fractional Chern insulator

description: Fractional quantum Hall effect (FQHE) is a prime example of topological
 quantum many-body phe- nomena, arising from the interplay between strong
 electron correlation, topological order, and time reversal symmetry
 breaking. Recently, a lattice analog of FQHE at zero magnetic field has
 been observed, confirming the existence of a zero-field fractional Chern
 insulator (FCI). Despite this, the bulk-edge correspondence — a hallmark
 of FCI featuring an insulating bulk with conductive edges — has not been
 directly observed. In fact, this correspondence has not been visualized in
 any system for fractional states due to experimental challenges. Here we
 report the imaging of FCI edge states in twisted MoTe2 using microwave
 impedance microscopy. By tuning the carrier density, we observe the system
 evolving between metallic and FCI states, the latter of which exhibits
 insulating bulk and conductive edges as expected from bulk-boundary
 correspondence. Further analysis suggests the composite nature of the FCI
 edge states. We also observe the evolution of edge states across the
 topological phase transition as a function of interlayer electric field,
 and reveal tantalizing prospects of neighboring domains with different
 fractional orders. These findings pave the way for research into
 topologically protected 1D interfaces between various anyonic states at
 zero magnetic field, such as gapped 1d symmetry-protected phases with
 non-zero topological entanglement entropy, Halperin-Laughlin interfaces,
 and the creation of non-abelian anyons.
description: MIM measurements were performed in a 3-He cryostat with a 12 T
 superconducting magnet and homebuilt scanning setup. The MIM probe, an
 etched tungsten wire, was attached to a quartz tuning fork for topographic
 sensing and scans were taken with the tip held approximately 40 nm above
 the sample’s surface. MIM measures changes in admittance between the tip
 and sample at GHz frequencies which can be related to changes in local
 conductivity and permittivity in the sample. The input power is
 ∼0.2 <em>µ</em>W. The measurements reported here were carried
 out at 6.5 GHz at zero magnetic field.
description: # Data from: Local probe of bulk and edge states in a fractional Chern
 insulator This README.txt file was generated on 2024-10-02 by Zhurun Ji
 GENERAL INFORMATION 1\. Title of Dataset: Local probe of bulk and edge
 states in a fractional Chern insulator 2\. Contact Information A.
 Principal Investigator Contact Information Name: Zhixun Shen Institution:
 Stanford University Address: 476 Lomita Mall, Stanford, CA, 94305 Email:
 [zxshen@stanford.edu](mailto:zxshen@stanford.edu) B. Alternate Contact
 Information Name: Zhurun Ji Institution: Stanford University Address: 476
 Lomita Mall, Stanford, CA, 94305 Email:
 [zhurun@stanford.edu](mailto:zhurun@stanford.edu) 3\. Date of data
 collection (single date, range, approximate date): 2024-1-1 to 2024-4-1
 4\. Geographic location of data collection: Stanford, CA, US 5\.
 Information about funding sources that supported the collection of the
 data: This work at the Stanford Institute for Materials and Energy
 Sciences (SIMES) is supported by the QSQM, an Energy Frontier Research
 Center funded by the U.S. Department of Energy (DOE), Office of Science,
 Basic Energy Sciences (BES), under Award #DE-SC0021238. Previous
 development of MIM technique at Stanford University was funded in part by
 the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant
 GBMF4546 to ZX.S. Z. J. acknowledges support from the Stanford Science
 fellowship and the Urbanek-Chodorow fellowship. The work at University of
 Washington is mainly supported by DoE BES under award DE-SC0018171. Bulk
 MoTe2 crystal growth is supported by the Center on Programmable Quantum
 Materials, an Energy Frontier Research Center funded by DOE BES under
 award DE-SC0019443. Device fabrication used the facilities and
 instrumentation supported by NSF MRSEC DMR-230879. K.W. and T.T.
 acknowledge support from the Elemental Strategy Initiative conducted by
 the MEXT, Japan (Grant Number JPMXP0112101001) and JSPS KAKENHI (Grant
 Numbers 19H05790, 20H00354 and 21H05233). SHARING/ACCESS INFORMATION 1\.
 Links to publications that cite or use the data:
 [https://arxiv.org/abs/2404.07157](https://arxiv.org/abs/2404.07157) 2\.
 Was data derived from another source? no 3\. Recommended citation for this
 dataset: Ji, Zhurun et al. (2024), Local probe of bulk and edge states in
 a fractional Chern insulator, Dryad, Dataset,
 [https://doi.org/10.5061/dryad.9p8cz8ws0](https://doi.org/10.5061/dryad.9p8cz8ws0) DATA & FILE OVERVIEW 1\. File List: (1) fig1.xlsx Description: Data for Fig. 1b, 1c (2) Fig. 2/fig2b-c.xlsx Description: Data for Fig. 2b, 2c (3) Fig. 2/fig2a_nu_-0.73.xlsx, fig2a_nu_-0.78.xlsx, ..., fig2a_nu_-1.28.xlsx Description: Data for Fig. 2a (4) Fig. 3/fig3b-d.xlsx Description: Data for Fig. 3b, 3c, 3d (5) Fig. 3/Fig. 3a_nu_-0.566.xlsx, Fig. 3a_nu_-0.575.xlsx, Fig. 3a_nu_-0.583.xlsx, Fig. 3a_nu_-0.592.xlsx, Fig. 3a_nu_-0.601.xlsx, ... Fig. 3a_nu_-0.800.xlsx Description: Data for Fig. 3a (6) fig4.xlsx Description: Data for Fig. 4a, 4b 4\. Are there multiple versions of the dataset? no METHODOLOGICAL INFORMATION 1\. Description of methods used for collection/generation of data: [https://arxiv.org/abs/2404.07157](https://arxiv.org/abs/2404.07157) 2\. Instrument- or software-specific information needed to interpret the data: All .xlsx files are generated with Microsoft Excel Version 2407 running on Microsoft Windows 11 Pro. 3\. People involved with sample collection, processing, analysis and/or submission: Zhurun Ji, Heonjoon Park, Mark E. Barber, Chaowei Hu, Kenji Watanabe, Takashi Taniguchi,Jiun-Haw Chu, Xiaodong Xu and Zhi-Xun Shen DATA-SPECIFIC INFORMATION Note: We use "latex format of variable name [latex format of unit]" to annotate the variables defined in [https://arxiv.org/abs/2404.07157](https://arxiv.org/abs/2404.07157). For example, MIM stands for microwave impedance microscopy. The "[latex format of unit]" part is ignored if the variable is dimensionless or of arbitrary unit. (1) File: fig1.xlsx Sheet(s): Fig. 1b_n, Fig. 1b_D, Fig. 1b_MIM Description: 2-dimensional data of Fig. 1b, the MIM-Im signal as a function of displacement field \Tilde{D}/\epsilon_0 and carrier density \Tilde{n} (calculated using geometric capacitance models, in region 1) at 1.5 K. Fig. 1b_n: Grid of \Tilde{n} [10^{12} cm^{-2}], the local carrier density Fig. 1b_D: Grid of \Tilde{D}/\epsilon_0 [mV/nm], the local displacement field Fig. 1b_MIM: MIM-Im [mV], the imaginary part of the microwave impedance microscopy signal Sheet(s): Fig. 1c_upper Description: Data for Fig. 1c upper panel, MIM-Im signal as a function of filling factor \nu measured at T = 1.5 K Column A: \nu, the filling factor Column B: MIM-Im [mV], the imaginary part of the microwave impedance microscopy signal Sheet(s): Fig. 1c_lower Description: Data for Fig. 1c lower panel, MIM-Im signal as a function of filling factor \nu measured at T = 500 mK Column A: \nu, the filling factor Column B: MIM-Im [mV], the imaginary part of the microwave impedance microscopy signal (2) File: fig2b-c.xlsx Sheet(s): Fig. 2b_n, Fig. 2b_D, Fig. 2b_MIM Description: 2-dimensional data of Fig. 2b; The MIM-Im signal as a function of displacement field \Tilde{D}/\epsilon_0 and carrier density \Tilde{n} at 1.5 K, measured at the location denoted by the white star in panel a. Fig. 1b_n: Grid of \Tilde{n} [10^{12} cm^{-2}], the local carrier density Fig. 1b_D: Grid of \Tilde{D}/\epsilon_0 [mV/nm], the local displacement field Fig. 1b_MIM: MIM-Im [mV], the imaginary part of the microwave impedance microscopy signal Sheet(s): Fig. 2c Description: Data for Fig. 2c, linecuts averaged over 0.7 µm width around the dashed white line in panel a of MIM-Im signals (with vertical offsets) at different filling factors traversing the physical edge. Column A: Spatial coordinate on the y axis, y [um] Column B, C, D, E, F: MIM-Im [V] at different fillings \nu of -1, -1.06,-1.11,-1.17 and -1.22, respectively (each with a 0.027 V vertical offset, added for Fig. 2c display) (3) File(s): Fig. 2a_nu_-0.73, Fig. 2a_nu_-0.78, Fig. 2a_nu_-0.84, Fig. 2a_nu_-0.89, Fig. 2a_nu_-0.95, Fig. 2a_nu_-1.00, Fig. 2a_nu_-1.06, Fig. 2a_nu_-1.11, Fig. 2a_nu_-1.17, Fig. 2a_nu_-1.22, Fig. 2a_nu_-1.28 Description: 2-dimensional scanning MIM-Im [V] data of Fig. 2a, at different fillings \nu of -0.73 - -1.28, respectively. Scan range: 3.515579 [\mu m] * 2.430399 [\mu m]. (4) File: Fig. 3/fig3b-d.xlsx Sheet(s): Fig. 3b Description: Data for Fig. 3b, linecuts of MIM-Im signal averaged over 0.3 µm width around the dashed white line in Fig. 3a (in region 1), at a series of different filling factors between −0.661 and −0.713. Column A: Spatial coordinate on the y axis, y [\mu m] Column B, C, D, E: MIM-Im [V] at different fillings \nu of -0.661, -0.679,-0.696 and -0.713, respectively Sheet(s): Fig. 3c Description: Data for Fig. 3c, linecuts of MIM-Im signal traversing the edge at filling factors ν = −3/5, −2/3 and −1, averaged over 0.5 µm width. Column A: Spatial coordinate on the y axis, y [\mu m] Column B, C, D: MIM-Im [V] at different fillings \nu of -1, -2/3 and -3/5, respectively Sheet(s): Fig. 3d_upper Description: Data for Fig. 3d's upper panel, MIM-Im signal versus \Tilde{V_g} at two representative locations in region 1 of the sample at T = 1.5 K. Column A: \Tilde{V_g} [V] (a linear combination of top and back gate voltages defined in the methods section in [https://arxiv.org/abs/2404.07157](https://arxiv.org/abs/2404.07157)) Column B: MIM-Im [V], the imaginary part of the microwave impedance microscopy signal Sheet(s): Fig. 3d_lower Description: Data for Fig. 3d's lower panel, MIM-Im signal versus \Tilde{V_g} (a linear combination of the top and back voltages defined in the methods section in [https://arxiv.org/abs/2404.07157](https://arxiv.org/abs/2404.07157)) at two representative locations in region 2 of the sample at T = 1.5 K. Column A: \Tilde{V_g} [V] Column B: MIM-Im [V], the imaginary part of the microwave impedance microscopy signal (5) File(s): Fig. 3/Fig. 3a_nu_-0.566.xlsx, Fig. 3a_nu_-0.575.xlsx, Fig. 3a_nu_-0.583.xlsx, Fig. 3a_nu_-0.592.xlsx, Fig. 3a_nu_-0.601.xlsx, ... Fig. 3a_nu_-0.800.xlsx Description: 2-dimensional real space mapping of MIM-Im [V] scanning data shown in Fig. 3a, at different fillings \nu of -0.566, -0.575, ..., -0.800, respectively, with a scan range 3.476902 [\mu m]*2.141872 [\mu m]. (6) File: Fig. 4.xlsx Sheet(s): Fig. 4a Description: Data for Fig. 4a, MIM-Im signal as a function of displacement field \Tilde{D}/\epsilon_0 at T = 1.5 K with filling factors −3/5, −2/3 and −1 states, respectively. Column A,C,E: \Tilde{D}/\epsilon_0 [mV/nm], the local displacement field Column B,D,F: MIM-Im [mV] measured at \nu= -1, -2/3 and -3/5, respectively. Fig. 4a has an vertical offset of \nu=-2/3 of 10mV, and \nu=-3/5 of 25mV for display. Sheet(s): Fig. 4b_D_-93.0, Fig. 4b_D_-67.4, Fig. 4b_D_-41.8, Fig. 4b_D_-16.2, Fig. 4b_D_-3.4, Fig. 4b_D_9.4, Fig. 4b_D_22.2, Fig. 4b_D_35.0, Fig. 4b_D_60.6, Fig. 4b_D_86.2 Description: 2-dimensional MIM-Im [V] scanning data of Fig. 4b, measured at same filling -2/3 but different displacement fields D/\epsilon_0 [mV/nm] of -93.0,-67.4, -41.8, -16.2, -3.4, 9.4, 22.2, 35.0, 60.6, 86.2, respectively, scan range 3.574977 [\mu m]*2.646694 [\mu m]