Papers
Topics
Authors
Recent
2000 character limit reached

Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes (2310.05656v2)

Published 9 Oct 2023 in cond-mat.mes-hall and cond-mat.other

Abstract: We propose a theory, that we call the \textit{mode-shell correspondence}, which relates the topological zero-modes localised in phase space to a \textit{shell} invariant defined on the surface forming a shell enclosing these zero-modes. We show that the mode-shell formalism provides a general framework unifying important results of topological physics, such as the bulk-edge correspondence, higher-order topological insulators, but also the Atiyah-Singer and the Callias index theories. In this paper, we discuss the already rich phenomenology of chiral symmetric Hamiltonians where the topological quantity is the chiral number of zero-dimensionial zero-energy modes. We explain how, in a lot of cases, the shell-invariant has a semi-classical limit expressed as a generalised winding number on the shell, which makes it accessible to analytical computations.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (138)
  1. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett., 45:494–497, Aug 1980.
  2. Yasuhiro Hatsugai. Chern number and edge states in the integer quantum hall effect. Phys. Rev. Lett., 71:3697–3700, Nov 1993.
  3. Yasuhiro Hatsugai. Edge states in the integer quantum hall effect and the riemann surface of the bloch function. Phys. Rev. B, 48:11851–11862, Oct 1993.
  4. Quantized hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49:405–408, Aug 1982.
  5. B. I. Halperin. 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, Feb 1982.
  6. M. Büttiker. Four-terminal phase-coherent conductance. Phys. Rev. Lett., 57:1761–1764, Oct 1986.
  7. M. Büttiker. Absence of backscattering in the quantum hall effect in multiprobe conductors. Phys. Rev. B, 38:9375–9389, Nov 1988.
  8. S. Raghu and F. D. M. Haldane. Analogs of quantum-hall-effect edge states in photonic crystals. Phys. Rev. A, 78:033834, Sep 2008.
  9. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett., 100:013905, Jan 2008.
  10. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature, 461(7265):772–775, 2009.
  11. Topological wave insulators: a review. Comptes Rendus. Physique, 21(4-5):467–499, 2020.
  12. A short introduction to topological quantum computation. SciPost Physics, 3(3), sep 2017.
  13. C.W.J. Beenakker. Search for majorana fermions in superconductors. Annual Review of Condensed Matter Physics, 4(1):113–136, apr 2013.
  14. A topological Dirac insulator in a quantum spin Hall phase. Nature, 452(7190):970–974, April 2008.
  15. Alexei Kitaev. Periodic table for topological insulators and superconductors. AIP Conference Proceedings, 1134(1):22–30, 2009.
  16. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New Journal of Physics, 12(6):065010, jun 2010.
  17. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B, 78:195125, Nov 2008.
  18. The noncommutative index theorem and the periodic table for disordered topological insulators and superconductors. Journal of Mathematical Physics, 59(3):031903, mar 2018.
  19. Topological insulators in three dimensions. Phys. Rev. Lett., 98:106803, Mar 2007.
  20. Quantum spin hall effect in graphene. Phys. Rev. Lett., 95:226801, Nov 2005.
  21. Quantum spin hall effect. Phys. Rev. Lett., 96:106802, Mar 2006.
  22. Quantum spin-hall effect and topologically invariant chern numbers. Phys. Rev. Lett., 97:036808, Jul 2006.
  23. Quantized electric multipole insulators. Science, 357(6346):61–66, jul 2017.
  24. Higher-order topological insulators. Science Advances, 4(6), jun 2018.
  25. Xun-Jiang Luo. The generalization of benalcazar-bernevig-hughes model to arbitrary dimensions. arXiv preprint arXiv:2304.07714, 2023.
  26. Topological origin of equatorial waves. Science, 358(6366):1075–1077, oct 2017.
  27. Topological langmuir-cyclotron wave. Science Advances, 9(13):eadd8041, 2023.
  28. Topological modes in stellar oscillations. The Astrophysical Journal, 940(1):84, nov 2022.
  29. Jeffrey B. Parker. Topological phase in plasma physics. Journal of Plasma Physics, 87(2):835870202, 2021.
  30. Topological active matter. Nature Reviews Physics, 4(6):380–398, may 2022.
  31. Quantum gyroelectric effect: Photon spin-1 quantization in continuum topological bosonic phases. Phys. Rev. A, 98:023842, Aug 2018.
  32. Mário G. Silveirinha. Chern invariants for continuous media. Phys. Rev. B, 92:125153, Sep 2015.
  33. G. E. Volovik. The Universe in a Helium Droplet. OUP Oxford, 2009.
  34. R Bott and R Seeley. Some remarks on the paper of callias. Communications in Mathematical Physics, 62(3):235–245, 1978.
  35. Pierre Delplace. Berry-Chern monopoles and spectral flows. SciPost Phys. Lect. Notes, page 39, 2022.
  36. Frédéric Faure. Manifestation of the topological index formula in quantum waves and geophysical waves. arXiv preprint arXiv:1901.10592, 2019.
  37. From ray tracing to topological waves in continuous media. arXiv preprint arXiv:2207.01479, 2022.
  38. Topology in shallow-water waves: A violation of bulk-edge correspondence. Communications in Mathematical Physics, 383(2):731–761, mar 2021.
  39. Guillaume Bal. Topological invariants for interface modes. Communications in Partial Differential Equations, 47(8):1636–1679, 2022.
  40. Guillaume Bal. Topological charge conservation for continuous insulators. Journal of Mathematical Physics, 64(3), 2023.
  41. Topological transition in stratified fluids. Nature Physics, 15(8):781–784, 2019.
  42. Unidirectional modes induced by nontraditional coriolis force in stratified fluids. Phys. Rev. Lett., 128:184501, May 2022.
  43. Robust propagating in-gap modes due to spin-orbit domain walls in graphene. Phys. Rev. B, 106:035139, Jul 2022.
  44. Valley chern numbers and boundary modesin gapped bilayer graphene. Proceedings of the National Academy of Sciences, 110(26):10546–10551, 2013.
  45. All-si valley-hall photonic topological insulator. New Journal of Physics, 18(2):025012, feb 2016.
  46. Valley topological phases in bilayer sonic crystals. Phys. Rev. Lett., 120:116802, Mar 2018.
  47. Quantum valley hall effect in wide-gap semiconductor sic monolayer. Scientific Reports, 10(5044), 2020.
  48. Jeffrey C. Y. Teo and C. L. Kane. Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B, 82:115120, Sep 2010.
  49. Topological characterization of periodically driven quantum systems. Phys. Rev. B, 82:235114, Dec 2010.
  50. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X, 3:031005, Jul 2013.
  51. Michel Fruchart. Complex classes of periodically driven topological lattice systems. Physical Review B, 93:115429, 2015.
  52. Floquet topological insulators for sound. Nature Communications, 7(1), jun 2016.
  53. Exploring topological phases with quantum walks. Phys. Rev. A, 82:033429, Sep 2010.
  54. Topological quantum walks: Theory and experiments. Frontiers of Physics, 14:1–6, 2019.
  55. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nature Communications, 6:8260, 10 2015.
  56. Pierre A. L. Delplace. Topological chiral modes in random scattering networks. SciPost Phys., 8:081, 2020.
  57. Optical resonator analog of a two-dimensional topological insulator. Phys. Rev. Lett., 110:203904, May 2013.
  58. Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems. Phys. Rev. B, 90:125143, Sep 2014.
  59. Topological index for periodically driven time-reversal invariant 2d systems. Phys. Rev. Lett., 114:106806, Mar 2015.
  60. Topological band theory for non-hermitian hamiltonians. Phys. Rev. Lett., 120:146402, Apr 2018.
  61. New topological invariants in non-hermitian systems. Journal of Physics: Condensed Matter, 31(26):263001, Apr 2019.
  62. Topological phases of non-hermitian systems. Phys. Rev. X, 8:031079, Sep 2018.
  63. Topologically protected bound states in photonic parity–time-symmetric crystals. Nature Materials, 16(4):433–438, April 2017.
  64. Gain- and loss-induced topological insulating phase in a non-hermitian electrical circuit. Physical Review Applied, 13(1), Jan 2020.
  65. Non-hermitian physics. Advances in Physics, 69(3):249–435, Jul 2020.
  66. Topological insulator laser: Experiments. Science, 359(6381), 2018.
  67. Nonlinear topological photonics. Applied Physics Reviews, 7(2):021306, 2020.
  68. Stefano Longhi. Non-hermitian tight-binding network engineering. Physical Review A, 93(2), feb 2016.
  69. Tony E. Lee. Anomalous edge state in a non-hermitian lattice. Phys. Rev. Lett., 116:133903, Apr 2016.
  70. Ye Xiong. Why does bulk boundary correspondence fail in some non-hermitian topological models. arXiv preprint arXiv:1705.06039, 2017.
  71. Edge states and topological invariants of non-hermitian systems. Phys. Rev. Lett., 121:086803, Aug 2018.
  72. Non-hermitian chern bands. Phys. Rev. Lett., 121:136802, Sep 2018.
  73. Hidden chern number in one-dimensional non-hermitian chiral-symmetric systems. Phys. Rev. B, 100:161105, Oct 2019.
  74. Non-bloch topological invariants in a non-hermitian domain wall system. Phys. Rev. B, 100:035102, Jul 2019.
  75. Non-hermitian boundary modes and topology. Phys. Rev. Lett., 124:056802, Feb 2020.
  76. The analysis of bulk boundary correspondence under the singularity of the generalized brillouin zone in non-hermitian system. arXiv preprint arXiv:2106.06384, 2021.
  77. Biorthogonal bulk-boundary correspondence in non-hermitian systems. Phys. Rev. Lett., 121:026808, Jul 2018.
  78. Non-hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence. Phys. Rev. B, 99:081302, Feb 2019.
  79. Restoration of the non-hermitian bulk-boundary correspondence via topological amplification. arXiv preprint arXiv:2207.12427, 2022.
  80. Non-hermitian topological invariants in real space. Phys. Rev. Lett., 123:246801, Dec 2019.
  81. Towards the theory of types iii and iv non-hermitian weyl fermions. arXiv preprint arXiv:2110.13714, 2021.
  82. Infernal and exceptional edge modes: Non-hermitian topology beyond the skin effect. arXiv preprint arXiv:2304.13743, 2023.
  83. Exceptional ring of the buoyancy instability in stars. Phys. Rev. Res., 6:L012055, Mar 2024.
  84. Simultaneous quantization of edge and bulk hall conductivity. Journal of Physics A: Mathematical and General, 33(2):L27–L32, dec 1999.
  85. Edge current channels and chern numbers in the integer quantum hall effect. Reviews in Mathematical Physics, 14(01):87–119, 2002.
  86. Bulk-boundary correspondence of topological insulators from their respective green’s functions. Physical Review B, 84(12), sep 2011.
  87. Bulk-edge correspondence for two-dimensional topological insulators. Communications in Mathematical Physics, 324(3):851–895, oct 2013.
  88. Jean Bellissard. Noncommutative geometry and quantum hall effect. In Proceedings of the International Congress of Mathematicians: August 3–11, 1994 Zürich, Switzerland, pages 1238–1246. Springer, 1995.
  89. Bulk and boundary invariants for complex topological insulators. K, 2016.
  90. Constantine Callias. Axial anomalies and index theorems on open spaces. Communications in Mathematical Physics, 62(3):213–234, 1978.
  91. Lars Hörmander. The weyl calculus of pseudo-differential operators. Communications on Pure and Applied Mathematics, 32(3):359–443, 1979.
  92. BV Fedosov. Analytic formulae for the index of elliptic operators. Tr. Mosk. Mat. Obs, 30:159–241, 1974.
  93. The index of elliptic operators: I. Annals of Mathematics, 87(3):484–530, 1968.
  94. Ezra Getzler. Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Communications in Mathematical Physics, 92(2):163–178, June 1983.
  95. Ezra Getzler. A short proof of the local atiyah-singer index theorem. Topology, 25(1):111–117, 1986.
  96. Heat kernels and Dirac operators. Springer Science & Business Media, 2003.
  97. Maciej Zworski. Semiclassical analysis, volume 138. American Mathematical Society, 2022.
  98. Estimating bulk and edge topological indices in finite open chiral chains. Journal of Mathematical Physics, 63(12), December 2022.
  99. Topological boundary modes in isostatic lattices. Nature Physics, 10(1):39–45, January 2014.
  100. Geometry and topology tango in ordered and amorphous chiral matter. SciPost Phys., 12:038, 2022.
  101. Model-free characterization of topological edge and corner states in mechanical networks. arXiv preprint arXiv:2304.04832, 2023.
  102. The index of elliptic operators: I. Annals of mathematics, pages 484–530, 1968.
  103. Zak phase and the existence of edge states in graphene. Phys. Rev. B, 84:195452, Nov 2011.
  104. Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett., 89:077002, Jul 2002.
  105. Lasing in topological edge states of a one-dimensional lattice. Nature Photonics, 11(10):651–656, sep 2017.
  106. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nature Communications, 6(1), apr 2015.
  107. Observation of optical shockley-like surface states in photonic superlattices. Optics letters, 34:1633–5, 07 2009.
  108. The Bulk-Edge Correspondence for Disordered Chiral Chains. Communications in Mathematical Physics, 363(3):829–846, November 2018.
  109. Solitons in polyacetylene. Phys. Rev. Lett., 42:1698–1701, Jun 1979.
  110. Shockley model description of surface states in topological insulators. Phys. Rev. B, 86:075304, Aug 2012.
  111. Orbital embedding and topology of one-dimensional two-band insulators. Phys. Rev. B, 104:235428, Dec 2021.
  112. A. Venaille and P. Delplace. Wave topology brought to the coast. Phys. Rev. Research, 3:043002, Oct 2021.
  113. Chiral chains with two valleys and disorder of finite correlation length. Phys. Rev. B, 108:075430, Aug 2023.
  114. Experimental Realization of Two-Dimensional Weak Topological Insulators. Nano Letters, 22(7):3125–3132, April 2022. Publisher: American Chemical Society.
  115. Colloquium: Topological insulators. Rev. Mod. Phys., 82:3045–3067, Nov 2010.
  116. Quantum transport and two-parameter scaling at the surface of a weak topological insulator. Phys. Rev. Lett., 108:076804, Feb 2012.
  117. Strong side of weak topological insulators. Phys. Rev. B, 86:045102, Jul 2012.
  118. Disordered weak and strong topological insulators. Phys. Rev. Lett., 110:236803, Jun 2013.
  119. Aperiodic weak topological superconductors. Phys. Rev. Lett., 116:257002, Jun 2016.
  120. Prediction of weak topological insulators in layered semiconductors. Phys. Rev. Lett., 109:116406, Sep 2012.
  121. A weak topological insulator state in quasi-one-dimensional bismuth iodide. Nature, 566(7745):518–522, February 2019.
  122. Observation and control of the weak topological insulator state in ZrTe5. Nature Communications, 12(1):406, January 2021.
  123. Z2subscript𝑍2{Z}_{2}italic_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT topological order and the quantum spin hall effect. Phys. Rev. Lett., 95:146802, Sep 2005.
  124. Topological weyl semi-metal from a lattice model. Europhysics Letters, 97(6):67004, mar 2012.
  125. Orbital edge states in a photonic honeycomb lattice. Phys. Rev. Lett., 118:107403, Mar 2017.
  126. Building crystalline topological phases from lower-dimensional states. Phys. Rev. B, 96:205106, Nov 2017.
  127. Multiplicative topological phases. Communications Physics, 5(1):262, October 2022.
  128. Multiplicative topological semimetals. arXiv preprint arXiv:2301.02404, 2023.
  129. Multiplicative majorana zero-modes. arXiv preprint arXiv:2301.02765, 2023.
  130. Second-order topological phases protected by chiral symmetry. Phys. Rev. B, 100:235302, Dec 2019.
  131. Higher-order topological phases in crystalline and non-crystalline systems: a review. arXiv preprint arXiv:2309.03688, 2023.
  132. Defects in graphene : A topological description. arXiv preprint arXiv:2304.08905, 2023.
  133. Zero modes of the vortex-fermion system. Nuclear Physics B, 190(4):681–691, 1981.
  134. (d−2)𝑑2(d-2)( italic_d - 2 ) -dimensional edge states of rotation symmetry protected topological states. Physical Review Letters, 119(24), dec 2017.
  135. Chiral-symmetric higher-order topological phases of matter. Physical Review Letters, 128(12), mar 2022.
  136. Boundary-obstructed topological phases. Physical Review Research, 3(1), mar 2021.
  137. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett., 119:246401, Dec 2017.
  138. Locality of the windowed local density of states. arXiv preprint arXiv:2101.00272, 2021.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Slide Deck Streamline Icon: https://streamlinehq.com

Whiteboard

Dice Question Streamline Icon: https://streamlinehq.com

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 tweet and received 5 likes.

Upgrade to Pro to view all of the tweets about this paper:

Start a free 7-day Pro trial