Abstract
Many topological phases host gapless boundary modes that can be markedly modified by electronic interactions. Even for the long-studied edge modes of quantum Hall phases1,2, forming at the boundaries of two-dimensional electron systems, the nature of such interaction-induced changes has been elusive. Despite advances made using local probes3,4,5,6,7,8,9,10,11,12,13, key experimental challenges persist: the lack of direct information about the internal structure of edge states on microscopic scales, and complications from edge disorder. Here we use scanning tunnelling microscopy to image pristine electrostatically defined quantum Hall edge states in graphene with high spatial resolution and demonstrate how correlations dictate the structures of edge channels on both magnetic and atomic length scales. For integer quantum Hall states in the zeroth Landau level, we show that interactions renormalize the edge velocity, dictate the spatial profile for co-propagating modes and induce unexpected edge valley polarization, which differs from the bulk. Although some of our findings can be understood by mean-field theory, others show breakdown of this picture, highlighting the roles of edge fluctuations and inter-channel couplings. We also extend our measurements to spatially resolve the edge state of fractional quantum Hall phases and detect spectroscopic signatures of interactions in this chiral Luttinger liquid. Our study establishes scanning tunnelling microscopy as a promising tool for exploring the edge physics of the rapidly expanding group of two-dimensional topological phases, including recently realized fractional Chern insulators.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$32.99 / 30 days
cancel any time
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Buy this article
- Purchase on SpringerLink
- Instant access to the full article PDF.
USD 39.95
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
The data that support the findings of this study are available at figshare (https://doi.org/10.6084/m9.figshare.30456452).
References
Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982).
Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993).
Yacoby, A., Hess, H. F., Fulton, T. A., Pfeiffer, L. N. & West, K. W. Electrical imaging of the quantum Hall state. Solid State Commun. 111, 1–13 (1999).
Paradiso, N. et al. Imaging fractional incompressible stripes in integer quantum Hall systems. Phys. Rev. Lett. 108, 246801 (2012).
Pascher, N. et al. Imaging the conductance of integer and fractional quantum Hall edge states. Phys. Rev. X 4, 011014 (2014).
Suddards, M. E., Baumgartner, A., Henini, M. & Mellor, C. J. Scanning capacitance imaging of compressible and incompressible quantum Hall effect edge strips. New J. Phys. 14, 083015 (2012).
Cui, Y.-T. et al. Unconventional correlation between quantum Hall transport quantization and bulk state filling in gated graphene devices. Phys. Rev. Lett. 117, 186601 (2016).
Uri, A. et al. Nanoscale imaging of equilibrium quantum Hall edge currents and of the magnetic monopole response in graphene. Nat. Phys. 16, 164–170 (2020).
Li, G., Luican-Mayer, A., Abanin, D., Levitov, L. & Andrei, E. Y. Evolution of Landau levels into edge states in graphene. Nat. Commun. 4, 1744 (2013).
Coissard, A. et al. Absence of edge reconstruction for quantum Hall edge channels in graphene devices. Sci. Adv. 9, eadf7220 (2023).
Johnsen, T. et al. Mapping quantum Hall edge states in graphene by scanning tunneling microscopy. Phys. Rev. B 107, 115426 (2023).
Randeria, M. T. et al. Interacting multi-channel topological boundary modes in a quantum Hall valley system. Nature 566, 363–367 (2019).
Kim, S. et al. Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy. Nat. Commun. 12, 2852 (2021).
Chklovskii, D. B., Shklovskii, B. I. & Glazman, L. I. Electrostatics of edge channels. Phys. Rev. B 46, 4026–4034 (1992).
Chamon, C. D. C. & Wen, X. G. Sharp and smooth boundaries of quantum Hall liquids. Phys. Rev. B 49, 8227–8241 (1994).
Dempsey, J., Gelfand, B. Y. & Halperin, B. I. Electron-electron interactions and spontaneous spin polarization in quantum Hall edge states. Phys. Rev. Lett. 70, 3639–3642 (1993).
Karlhede, A., Kivelson, S. A., Lejnell, K. & Sondhi, S. L. Textured edges in quantum Hall systems. Phys. Rev. Lett. 77, 2061–2064 (1996).
Zhang, Y. & Yang, K. Edge spin excitations and reconstructions of integer quantum Hall liquids. Phys. Rev. B 87, 125140 (2013).
Khanna, U., Goldstein, M. & Gefen, Y. Fractional edge reconstruction in integer quantum Hall phases. Phys. Rev. B 103, L121302 (2021).
Meir, Y. Composite edge states in the ν=2/3 fractional quantum Hall regime. Phys. Rev. Lett. 72, 2624–2627 (1994).
Kane, C. L., Fisher, M. P. A. & Polchinski, J. Randomness at the edge: theory of quantum Hall transport at filling ν=2/3. Phys. Rev. Lett. 72, 4129–4132 (1994).
Wan, X., Yang, K. & Rezayi, E. H. Reconstruction of fractional quantum Hall edges. Phys. Rev. Lett. 88, 056802 (2002).
Sabo, R. et al. Edge reconstruction in fractional quantum Hall states. Nat. Phys. 13, 491–496 (2017).
Bid, A. et al. Observation of neutral modes in the fractional quantum Hall regime. Nature 466, 585–590 (2010).
Venkatachalam, V., Hart, S., Pfeiffer, L., West, K. & Yacoby, A. Local thermometry of neutral modes on the quantum Hall edge. Nat. Phys. 8, 676–681 (2012).
Liu, X. et al. Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet. Science 375, 321–326 (2022).
Farahi, G. et al. Broken symmetries and excitation spectra of interacting electrons in partially filled Landau levels. Nat. Phys. 19, 1482–1488 (2023).
Hu, Y. et al. High-resolution tunnelling spectroscopy of fractional quantum Hall states. Nat. Phys. 21, 716–723 (2025).
Tsui, Y.-C. et al. Direct observation of a magnetic-field-induced Wigner crystal. Nature 628, 287–292 (2024).
Coissard, A. et al. Imaging tunable quantum Hall broken-symmetry orders in graphene. Nature 605, 51–56 (2022).
Young, A. F. et al. Spin and valley quantum Hall ferromagnetism in graphene. Nat. Phys. 8, 550–556 (2012).
Kharitonov, M. Edge excitations of the canted antiferromagnetic phase of the ν = 0 quantum Hall state in graphene: a simplified analysis. Phys. Rev. B 86, 075450 (2012).
Young, A. F. et al. Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state. Nature 505, 528–532 (2014).
Knothe, A. & Jolicoeur, T. Edge structure of graphene monolayers in the ν = 0 quantum Hall state. Phys. Rev. B 92, 165110 (2015).
Nakamura, J., Liang, S., Gardner, G. C. & Manfra, M. J. Direct observation of anyonic braiding statistics. Nat. Phys. 16, 931–936 (2020).
Werkmeister, T. et al. Anyon braiding and telegraph noise in a graphene interferometer. Science 388, 730–735 (2025).
Samuelson, N. L. et al. Anyonic statistics and slow quasiparticle dynamics in a graphene fractional quantum Hall interferometer. Preprint at https://arxiv.org/abs/2403.19628(2024).
Kundu, H. K., Biswas, S., Ofek, N., Umansky, V. & Heiblum, M. Anyonic interference and braiding phase in a Mach-Zehnder interferometer. Nat. Phys. 19, 515–521 (2023).
Veillon, A. et al. Observation of the scaling dimension of fractional quantum Hall anyons. Nature 632, 517–521 (2024).
Ruelle, M. et al. Time-domain braiding of anyons. Science 389, eadm7695 (2025).
Chiu, C.-L. et al. High spatial resolution charge sensing of quantum Hall states. Proc. Natl Acad. Sci. USA 122, e2424781122 (2025).
Yang, W. et al. Evidence for correlated electron pairs and triplets in quantum Hall interferometers. Nat. Commun. 15, 10064 (2024).
Werkmeister, T. et al. Strongly coupled edge states in a graphene quantum Hall interferometer. Nat. Commun. 15, 6533 (2024).
Wen, X. G. Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states. Phys. Rev. B 41, 12838–12844 (1990).
Chang, A. M. Chiral Luttinger liquids at the fractional quantum Hall edge. Rev. Mod. Phys. 75, 1449–1505 (2003).
Grayson, M., Tsui, D. C., Pfeiffer, L. N., West, K. W. & Chang, A. M. Resonant tunneling into a biased fractional quantum Hall edge. Phys. Rev. Lett. 86, 2645–2648 (2001).
Cohen, L. A. et al. Universal chiral Luttinger liquid behavior in a graphene fractional quantum Hall point contact. Science 382, 542–547 (2023).
Assouline, A. et al. Emission and coherent control of Levitons in graphene. Science 382, 1260–1264 (2023).
Zibrov, A. A. et al. Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level. Nature 549, 360–364 (2017).
Spanton, E. M. et al. Observation of fractional Chern insulators in a van der Waals heterostructure. Science 360, 62–66 (2018).
Cai, J. et al. Signatures of fractional quantum anomalous Hall states in twisted MoTe2. Nature 622, 63–68 (2023).
Zeng, Y. et al. Thermodynamic evidence of fractional Chern insulator in moiré MoTe2. Nature 622, 69–73 (2023).
Lu, Z. et al. Fractional quantum anomalous Hall effect in multilayer graphene. Nature 626, 759–764 (2024).
Li, H. et al. Electrode-free anodic oxidation nanolithography of low-dimensional materials. Nano Lett. 18, 8011–8015 (2018).
Cohen, L. A. et al. Nanoscale electrostatic control in ultraclean van der Waals heterostructures by local anodic oxidation of graphite gates. Nat. Phys. 19, 1502–1508 (2023).
Li, G., Luican, A. & Andrei, E. Y. Self-navigation of a scanning tunneling microscope tip toward a micron-sized graphene sample. Rev. Sci. Instrum. 82, 073701 (2011).
Acknowledgements
We thank A. Stern for discussions. The work at Princeton is primary supported by the US DOE-BES grant DE-FG02-07ER46419 and by the Gordon and Betty Moore Foundation’s EPiQS initiative grant GBMF9469. Other support was provided by ONR grant N000142412471, NSF-MRSEC through the Princeton Center for Complex Materials grant NSFDMR-2011750, NSF grant DMR-2312311, NSF grant OMA-2326767, and the US Army Research Office MURI project under grant number W911NF-21-2-0147. J.Y. is supported by the Princeton Materials Science Postdoctoral Fellowship. K.G.W. is supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program. M.P.Z., R.F., A.S.M., Tianle Wang and Taige Wang are funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 (Theory of Materials program KC2301). R.F. is also supported by the Gordon and Betty Moore Foundation (Grant No. GBMF8688). Work at UCSB was supported primarily by the ONR under award N00014-23−1−2066 as well as by a Brown Investigator Award to A.F.Y. K.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Nos. 21H05233 and 23H02052) and the World Premier International Research Center Initiative, MEXT, Japan.
Author information
Authors and Affiliations
Contributions
J.Y., H.H., K.G.W. and A.Y. conceived of the experiment. J.Y., H.H., K.G.W. and A.Y. performed the STM measurements and analysed the data. H.H. fabricated the device structure with help from J.Y., K.G.W. and L.C. L.C. and A.F.Y. provided the patterned graphite gate. R.F., A.S.M., Tianle Wang, Taige Wang and M.P.Z. carried out the theoretical calculations. K.W. and T.T. provided the hBN crystals. All authors contributed to the writing of the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 Sensing and tuning edge potential.
a, Schematic illustration of the working principle of STS potential sensing. When graphene is highly incompressible (e.g. v = 2), variation in local potential φ as felt by graphene (blue) shifts the energy of LLs (red), the latter of which is measured by the VB at which resonant tunneling to LLs occurs, i.e. peak in dI/dV(VB). b, STS measured along a linear trajectory across the gate-defined edge with νB = νP = 2. VPG and VBG are adjusted within the gap to balance ϕ in the PG- and BG- controlled bulk to extract the intrinsic potential variations. c,d, ϕ extracted from the STS mapping near a flipped L-shaped edge of PG (same as Fig. 1b) with νB = νP = 2, where VPG and VBG are adjusted within the gap to balance (c) or offset (d) φ across the edge. b is taken at y = 300 nm in c,d. Yellow line overlaid on the x-axis in b denotes the spatial extent of the scanned area in c,d.
Extended Data Fig. 2 Representative FFT images obtained from the atomic-scale STS maps of edge states.
a-e, Upper panels, modulus of the complex FFT amplitude |n(k)|; Lower panels, complex phase masked by the modulus, |n(k)|arg(n(k)) (see Methods and Supplementary Information Section 7 for details). Images are obtained from representative points on the edge channels in Fig. 2h (a), the two edge channels in Fig. 2i (left, b; right, c), and the two channels in Fig. S5a (left, d; right, e). VB = 0 except for a where a small negative offset VB = −1 mV is applied to maintain sufficient dI/dV contrast for accurate FFT analysis. The six prominent peaks in the top panel of a, and the corresponding peaks at the same (kx, ky) in b-e are Bragg peaks of the graphene lattice. In b-e a new set of peaks at the Kekulé vectors are more prominent than the Bragg peaks, indicating strong intervalley coherence27.
Extended Data Fig. 3 Inverse Fast Fourier Transform (iFFT) analysis of valley polarization.
a, Fourier filter for retaining only the Bragg peaks in the FFT data. b-e, Representative examples of iFFT generated real-space intensity maps, for a strongly valley polarized state (Z ~ 1, b); graphene lattice (Z = 0, c); left (d) and right (e) panels in Fig. 3b. Partial valley polarization is visible in d,e as larger iFFT amplitude on one sublattices. Yellow hexagons denote expected graphene atomic positions. Weaker contrast near the boundaries of b-e is a result of the Blackman window function applied to obtain the FFT peaks.
Extended Data Fig. 4 Tuning the reconstruction of co-propagating edge states.
a, STS imaging of edge states with (νB, νP) = (−2, 2), but with a sharper edge potential compared to Fig. 2j. The four edge channels in Fig. 2j are ‘squeezed’ into two pairs of spatially overlapping edge channels. b,c, Valley polarization Z = cosθ (b) and inter-valley coherence phase φ (c) measured along the two edge channels in a. Inset in b illustrates valley Bloch sphere with θ, φ specified. d, cartoon illustration of the potential-tuned edge spin transition inferred from isospin measurements. In a smooth potential as in Fig. 2j, four channels acquire four distinct flavors (top panel). The spin anti-alignment within the left (cyan) and right (orange) pairs of edge states is inferred from the same isospin orientation within each pair and Pauli exclusion (middle panel). By increasing the steepness of the potential (bottom panel), each pair merges into a single channel, and retains its isospin orientation as they merge, therefore resulting in two spin-unpolarized channels. This demonstrates a potential-tuned spin phase transition at the edge16,17,18,19.
Supplementary information
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yu, J., Han, H., Wolinski, K.G. et al. Visualizing interaction-driven restructuring of quantum Hall edge states. Nature 648, 585–590 (2025). https://doi.org/10.1038/s41586-025-09858-3
Received:
Accepted:
Published:
Version of record:
Issue date:
DOI: https://doi.org/10.1038/s41586-025-09858-3
