Fractal Surface States in Three-Dimensional Topological Quasicrystals
Abstract: We study topological states of matter in quasicrystals, which do not rely on crystalline orders. In the absence of a bandstructure description and spin-orbit coupling, we show that a three-dimensional quasicrystal can nevertheless form a topological insulator. It relies on a combination of noncrystallographic rotational symmetry of quasicrystals and electronic orbital space symmetry, which is the quasicrystalline counterpart of topological crystalline insulator. The resulting topological state obeys a non-trivial twisted bulk-boundary correspondence and lacks a good metallic surface. The topological surface states, localized on the top and bottom planes respecting the quasicrystalline symmetry, exhibit a new kind of multifractality with probability density concentrates mostly on high symmetry patches. They form a near-degenerate manifold of 'immobile' states whose number scales proportionally with the macroscopic sample size. This can open the door to a novel platform for topological surface physics distinct from the crystalline counterpart.
- M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
- X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
- R. Moessner and J. E. Moore, Topological Phases of Matter (Cambridge University Press, 2021).
- I. C. Fulga, D. I. Pikulin, and T. A. Loring, Aperiodic weak topological superconductors, Phys. Rev. Lett. 116, 257002 (2016).
- M. A. Bandres, M. C. Rechtsman, and M. Segev, Topological photonic quasicrystals: Fractal topological spectrum and protected transport, Phys. Rev. X 6, 011016 (2016).
- H. Huang and F. Liu, Quantum spin Hall effect and spin Bott index in a quasicrystal lattice, Phys. Rev. Lett. 121, 126401 (2018).
- A. Agarwala and V. B. Shenoy, Topological insulators in amorphous systems, Phys. Rev. Lett. 118, 236402 (2017).
- M. Xiao and S. Fan, Photonic Chern insulator through homogenization of an array of particles, Phys. Rev. B 96, 100202(R) (2017).
- S. Mansha and Y. D. Chong, Robust edge states in amorphous gyromagnetic photonic lattices, Phys. Rev. B 96, 121405(R) (2017).
- Q. Marsal, D. Varjas, and A. G. Grushin, Topological Weaire–Thorpe models of amorphous matter, Proc. Natl. Acad. Sci. U.S.A. 117, 30260 (2020).
- A. Agarwala, V. Juričić, and B. Roy, Higher-order topological insulators in amorphous solids, Phys. Rev. Res. 2, 012067 (2020).
- A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
- R. Bianco and R. Resta, Mapping topological order in coordinate space, Phys. Rev. B 84, 241106 (2011).
- T. A. Loring and M. B. Hastings, Disordered topological insulators via C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras, Europhys. Lett. 92, 67004 (2011).
- E. Prodan, Disordered topological insulators: a non-commutative geometry perspective, J. Phys. A 44, 113001 (2011).
- W. Chen, Universal topological marker, Phys. Rev. B 107, 045111 (2023).
- C. Wang, F. Liu, and H. Huang, Effective model for fractional topological corner modes in quasicrystals, Phys. Rev. Lett. 129, 056403 (2022b).
- M. Senechal, Quasicrystals and geometry (CUP Archive, 1996).
- M. Baake and U. Grimm, Aperiodic order. Volume 1. A mathematical invitation (Cambridge University Press, 2013).
- L. Fu, Topological crystalline insulators, Phys. Rev. Lett. 106, 106802 (2011).
- H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and Wannier obstructions, Phys. Rev. Lett. 121, 126402 (2018).
- Z.-D. Song, L. Elcoro, and B. A. Bernevig, Twisted bulk-boundary correspondence of fragile topology, Science 367, 794 (2020).
- M. Kohmoto, B. Sutherland, and C. Tang, Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model, Phys. Rev. B 35, 1020 (1987).
- P. Repetowicz, U. Grimm, and M. Schreiber, Exact eigenstates of tight-binding Hamiltonians on the Penrose tiling, Phys. Rev. B 58, 13482 (1998).
- M. O. Oktel, Strictly localized states in the octagonal Ammann-Beenker quasicrystal, Phys. Rev. B 104, 014204 (2021).
- F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
- A. Jagannathan, The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality, Rev. Mod. Phys. 93, 045001 (2021).
- Z. Song, Z. Fang, and C. Fang, (d−2)𝑑2(d-2)( italic_d - 2 )-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119, 246402 (2017).
- See Supplemental Material.
- A. Alexandradinata and B. A. Bernevig, Berry-phase description of topological crystalline insulators, Phys. Rev. B 93, 205104 (2016).
- B. I. Halperin, Possible states for a three-dimensional electron gas in a strong magnetic field, Jpn. J. Appl. Phys. 26, 1913 (1987).
- J. Bellissard, A. Bovier, and J.-M. Ghez, Gap labelling theorems for one dimensional discrete Schrödinger operators, Rev. Math. Phys. 4, 1 (1992).
- B. Lv, T. Qian, and H. Ding, Angle-resolved photoemission spectroscopy and its application to topological materials, Nat. Rev. Phys. 1, 608 (2019).
- E. de Prunelé, Penrose structures: Gap labeling and geometry, Phys. Rev. B 66, 094202 (2002).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.