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  • Letter
  • Open Access

Absence of gapless Majorana edge modes in few-leg bosonic flux ladders

F. A. Palm1,2,3,*, C. Repellin4, N. Goldman3,5, and F. Grusdt1,2

  • *Contact author: felix.palm@ulb.be
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Phys. Rev. Research 7, L012001 – Published 6 January, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L012001

Abstract

The search for Majorana excitations has seen tremendous efforts in recent years, ultimately aiming for their individual controllability in future topological quantum computers. A promising framework to realize such exotic Majorana fermions are topologically ordered non-Abelian phases of matter, such as certain fractional quantum Hall states. Quantum simulators provide unprecedented controllability and versatility to investigate such states, and developing experimentally feasible schemes to realize and identify them is of immediate relevance. Motivated by recent experiments, we consider bosons on coupled chains, subjected to a magnetic flux and experiencing Hubbard repulsion. At magnetic filling factor ν=1, similar systems on cylinders have been found to host the non-Abelian Moore-Read Pfaffian state in the bulk. Here, we address the question of whether more realistic few-leg ladders can host this exotic state and its chiral Majorana edge states. To this end, we perform extensive density-matrix renormalization-group simulations and determine the central charge of the ground state. While we do not find any evidence of gapless Majorana edge modes in systems of up to six legs, exact diagonalization of small systems reveals evidence for the Pfaffian state in the entanglement structure. By systematically varying the number of legs and monitoring the appearance and disappearance of this signal, our work highlights the importance of finite-size effects for the realization of exotic states in experimentally realistic systems.

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References (62)

  1. H. L. Stormer, A. Chang, D. C. Tsui, J. C. M. Hwang, A. C. Gossard, and W. Wiegmann, Fractional quantization of the Hall effect, Phys. Rev. Lett. 50, 1953 (1983).
  2. R. B. Laughlin, Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50, 1395 (1983).
  3. B. I. Halperin, Statistics of quasiparticles and the hierarchy of fractional quantized Hall states, Phys. Rev. Lett. 52, 1583 (1984).
  4. D. Arovas, J. R. Schrieffer, and F. Wilczek, Fractional statistics and the quantum Hall effect, Phys. Rev. Lett. 53, 722 (1984).
  5. G. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360, 362 (1991).
  6. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  7. M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Direct measurement of the Zak phase in topological Bloch bands, Nat. Phys. 9, 795 (2013).
  8. L. Duca, T. Li, M. Reitter, I. Bloch, M. Schleier-Smith, and U. Schneider, An Aharonov-Bohm interferometer for determining Bloch band topology, Science 347, 288 (2015).
  9. F. Grusdt, N. Y. Yao, D. Abanin, M. Fleischhauer, and E. Demler, Interferometric measurements of many-body topological invariants using mobile impurities, Nat. Commun. 7, 11994 (2016).
  10. F. A. Palm, N. Mostaan, N. Goldman, and F. Grusdt (unpublished).
  11. M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett. 111, 185301 (2013).
  12. H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices, Phys. Rev. Lett. 111, 185302 (2013).
  13. M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, T. Menke, D. Borgnia, P. M. Preiss, F. Grusdt, A. M. Kaufman, and M. Greiner, Microscopy of the interacting Harper–Hofstadter model in the two-body limit, Nature (London) 546, 519 (2017).
  14. J. Léonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Realization of a fractional quantum Hall state with ultracold atoms, Nature (London) 619, 495 (2023).
  15. C. Wang, F.-M. Liu, M.-C. Chen, H. Chen, X.-H. Zhao, C. Ying, Z.-X. Shang, J.-W. Wang, Y.-H. Huo, C.-Z. Peng, X. Zhu, C.-Y. Lu, and J.-W. Pan, Realization of fractional quantum Hall state with interacting photons, Science 384, 579 (2024).
  16. A. Sterdyniak, N. Regnault, and G. Möller, Particle entanglement spectra for quantum Hall states on lattices, Phys. Rev. B 86, 165314 (2012).
  17. F. A. Palm, M. Buser, J. Léonard, M. Aidelsburger, U. Schollwöck, and F. Grusdt, Bosonic Pfaffian state in the Hofstadter-Bose-Hubbard model, Phys. Rev. B 103, L161101 (2021).
  18. J. Boesl, R. Dilip, F. Pollmann, and M. Knap, Characterizing fractional topological phases of lattice bosons near the first Mott lobe, Phys. Rev. B 105, 075135 (2022).
  19. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
  20. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
  21. H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-Abelian fractional quantum Hall effect states, Phys. Rev. Lett. 101, 010504 (2008).
  22. A. Sterdyniak, N. Regnault, and B. A. Bernevig, Extracting excitations from model state entanglement, Phys. Rev. Lett. 106, 100405 (2011).
  23. C. Repellin, J. Léonard, and N. Goldman, Fractional Chern insulators of few bosons in a box: Hall plateaus from center-of-mass drifts and density profiles, Phys. Rev. A 102, 063316 (2020).
  24. J. C. Y. Teo and C. L. Kane, From Luttinger liquid to non-Abelian quantum Hall states, Phys. Rev. B 89, 085101 (2014).
  25. P. Fendley, M. P. A. Fisher, and C. Nayak, Edge states and tunneling of non-Abelian quasiparticles in the ν=5/2 quantum Hall state and p+ip superconductors, Phys. Rev. B 75, 045317 (2007).
  26. Z. Liu, B. Mera, M. Fujimoto, T. Ozawa, and J. Wang, Theory of generalized Landau levels and implication for non-Abelian states, arXiv:2405.14479.
  27. C. L. Kane, R. Mukhopadhyay, and T. C. Lubensky, Fractional quantum Hall effect in an array of quantum wires, Phys. Rev. Lett. 88, 036401 (2002).
  28. A. S. Sørensen, E. Demler, and M. D. Lukin, Fractional quantum Hall states of atoms in optical lattices, Phys. Rev. Lett. 94, 086803 (2005).
  29. M. Hafezi, A. S. Sørensen, E. Demler, and M. D. Lukin, Fractional quantum Hall effect in optical lattices, Phys. Rev. A 76, 023613 (2007).
  30. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  31. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  32. C. Hubig, I. P. McCulloch, U. Schollwöck, and F. A. Wolf, Strictly single-site DMRG algorithm with subspace expansion, Phys. Rev. B 91, 155115 (2015).
  33. C. Hubig, F. Lachenmaier, N.-O. Linden, T. Reinhard, L. Stenzel, A. Swoboda, M. Grundner, S. Mardazad, F. Pauw, and S. Paeckel, The SyTen toolkit (2023), http://www.syten.eu.
  34. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.7.L012001 for details on the methods used and additional data.
  35. P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.: Theory Exp. (2004) P06002.
  36. F. A. Palm, S. Mardazad, A. Bohrdt, U. Schollwöck, and F. Grusdt, Snapshot-based detection of ν=12 Laughlin states: Coupled chains and central charge, Phys. Rev. B 106, L081108 (2022).
  37. M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua, and U. Schollwöck, Vortex and Meissner phases of strongly interacting bosons on a two-leg ladder, Phys. Rev. B 91, 140406(R) (2015).
  38. F. Kolley, M. Piraud, I. P. McCulloch, U. Schollwöck, and F. Heidrich-Meisner, Strongly interacting bosons on a three-leg ladder in the presence of a homogeneous flux, New J. Phys. 17, 092001 (2015).
  39. S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck, and T. Vekua, Symmetry-broken states in a system of interacting bosons on a two-leg ladder with a uniform Abelian gauge field, Phys. Rev. A 94, 063628 (2016).
  40. A. Petrescu, M. Piraud, G. Roux, I. P. McCulloch, and K. Le Hur, Precursor of the Laughlin state of hard-core bosons on a two-leg ladder, Phys. Rev. B 96, 014524 (2017).
  41. J.-D. Chen, H.-H. Tu, Y.-H. Wu, and Z.-F. Xu, Quantum phases of two-component bosons on the Harper-Hofstadter ladder, Phys. Rev. A 102, 043322 (2020).
  42. A. J. Daley, J. M. Taylor, S. Diehl, M. Baranov, and P. Zoller, Atomic three-body loss as a dynamical three-body interaction, Phys. Rev. Lett. 102, 040402 (2009).
  43. M. Hafezi, P. Adhikari, and J. M. Taylor, Engineering three-body interaction and Pfaffian states in circuit QED systems, Phys. Rev. B 90, 060503(R) (2014).
  44. C. H. Lee, W. W. Ho, B. Yang, J. Gong, and Z. Papić, Floquet mechanism for non-Abelian fractional quantum Hall states, Phys. Rev. Lett. 121, 237401 (2018).
  45. F. Petiziol, M. Sameti, S. Carretta, S. Wimberger, and F. Mintert, Quantum simulation of three-body interactions in weakly driven quantum systems, Phys. Rev. Lett. 126, 250504 (2021).
  46. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  47. B. Michen, C. Repellin, and J. C. Budich, Adiabatic preparation of fractional Chern insulators from an effective thin-torus limit, Phys. Rev. Res. 5, 023100 (2023).
  48. Y.-C. He, F. Grusdt, A. Kaufman, M. Greiner, and A. Vishwanath, Realizing and adiabatically preparing bosonic integer and fractional quantum Hall states in optical lattices, Phys. Rev. B 96, 201103(R) (2017).
  49. B. Wang, M. Aidelsburger, J. Dalibard, A. Eckardt, and N. Goldman, Cold-atom elevator: From edge-state injection to the preparation of fractional Chern insulators, Phys. Rev. Lett. 132, 163402 (2024).
  50. F. A. Palm, J. Kwan, B. Bakkali-Hassani, M. Greiner, U. Schollwöck, N. Goldman, and F. Grusdt, Growing extended Laughlin states in a quantum gas microscope: A patchwork construction, Phys. Rev. Res. 6, 013198 (2024).
  51. T. Blatz, J. Kwan, J. Léonard, and A. Bohrdt, Bayesian optimization for robust state preparation in quantum many-body systems, Quantum 8, 1388 (2024).
  52. A. Petrescu, H. F. Song, S. Rachel, Z. Ristivojevic, C. Flindt, N. Laflorencie, I. Klich, N. Regnault, and K. Le Hur, Fluctuations and entanglement spectrum in quantum Hall states, J. Stat. Mech.: Theory Exp. (2014) P10005.
  53. A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body-localized system, Science 364, 256 (2019).
  54. H. Pichler, G. Zhu, A. Seif, P. Zoller, and M. Hafezi, Measurement protocol for the entanglement spectrum of cold atoms, Phys. Rev. X 6, 041033 (2016).
  55. M. Dalmonte, B. Vermersch, and P. Zoller, Quantum simulation and spectroscopy of entanglement Hamiltonians, Nat. Phys. 14, 827 (2018).
  56. T. V. Zache, C. Kokail, B. Sundar, and P. Zoller, Entanglement spectroscopy and probing the Li-Haldane conjecture in topological quantum matter, Quantum 6, 702 (2022).
  57. A. Nardin, R. Lopes, M. Rizzi, L. Mazza, and S. Nascimbene, Bisognano-Wichmann Hamiltonian for the entanglement spectroscopy of fractional quantum Hall states, arXiv:2312.07604.
  58. M. K. Joshi, C. Kokail, R. van Bijnen, F. Kranzl, T. V. Zache, R. Blatt, C. F. Roos, and P. Zoller, Exploring large-scale entanglement in quantum simulation, Nature (London) 624, 539 (2023).
  59. Q. Redon, Q. Liu, J.-B. Bouhiron, N. Mittal, A. Fabre, R. Lopes, and S. Nascimbene, Realizing the entanglement Hamiltonian of a topological quantum Hall system, arXiv:2307.06251.
  60. N. Goldman, J. Beugnon, and F. Gerbier, Detecting chiral edge states in the Hofstadter optical lattice, Phys. Rev. Lett. 108, 255303 (2012).
  61. F. Binanti, N. Goldman, and C. Repellin, Spectroscopy of edge and bulk collective modes in fractional Chern insulators, Phys. Rev. Res. 6, L012054 (2024).
  62. B.-Y. Sun, N. Goldman, M. Aidelsburger, and M. Bukov, Engineering and probing non-Abelian chiral spin liquids using periodically driven ultracold atoms, PRX Quantum 4, 020329 (2023).

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Phys. Rev. Research 7, L012001– Published 6 January, 2025
  • Received 2 August 2024
  • Accepted 31 October 2024
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