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  • Letter
  • Open Access

Interacting local topological markers: A one-particle density matrix approach for characterizing the topology of interacting and disordered states

Julia D. Hannukainen*, Miguel F. Martínez*, Jens H. Bardarson, and Thomas Klein Kvorning

  • *These authors contributed equally to this work.
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Phys. Rev. Research 6, L032045 – Published 28 August, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L032045

Abstract

While topology is a property of a quantum state itself, most existing methods for characterizing the topology of interacting phases of matter require direct knowledge of the underlying Hamiltonian. We offer an alternative by utilizing the one-particle density matrix formalism to extend the concept of the Chern, chiral, and Chern-Simons markers to include interactions. The one-particle density matrix of a free-fermion state is a projector onto the occupied bands, defining a Brillouin zone bundle of the given topological class. This is no longer the case in the interacting limit, but as long as the one-particle density matrix is gapped, its spectrum can be adiabatically flattened, connecting it to a topologically equivalent projector. The corresponding topological markers thus characterize the topology of the interacting phase. Importantly, the one-particle density matrix is defined in terms of a given state alone, making the local markers numerically favorable, and providing a valuable tool for characterizing topology of interacting systems when only the state itself is available. To demonstrate the practical use of the markers we use the chiral marker to identify the topology of midspectrum eigenstates of the Ising-Majorana chain across the transition between the ergodic and many-body localized phases. We also apply the chiral marker to random states with a known topology, and compare it with the entanglement spectrum degeneracy.

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References (88)

  1. A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321, 2 (2006).
  2. E. Prodan, Non-commutative tools for topological insulators, New J. Phys. 12, 065003 (2010).
  3. E. Prodan, Disordered topological insulators: A non-commutative geometry perspective, J. Phys. A: Math. Theor. 44, 113001 (2011).
  4. T. A. Loring and M. B. Hastings, Disordered topological insulators via C*-algebras, Europhys. Lett. 92, 67004 (2010).
  5. R. Bianco and R. Resta, Mapping topological order in coordinate space, Phys. Rev. B 84, 241106(R) (2011).
  6. T. A. Loring, K-theory and pseudospectra for topological insulators, Ann. Phys. 356, 383 (2015).
  7. A. Marrazzo and R. Resta, Locality of the anomalous Hall conductivity, Phys. Rev. B 95, 121114(R) (2017).
  8. T. A. Loring, A guide to the Bott index and localizer index, arXiv:1907.11791.
  9. I. Mondragon-Shem and T. L. Hughes, Robust topological invariants of topological crystalline phases in the presence of impurities, arXiv:1906.11847.
  10. B. Irsigler, J.-H. Zheng, and W. Hofstetter, Microscopic characteristics and tomography scheme of the local Chern marker, Phys. Rev. A 100, 023610 (2019).
  11. T. A. Loring and H. Schulz-Baldes, The spectral localizer for even index pairings, J. Noncommut. Geom. 14, 1 (2020).
  12. H. Schulz-Baldes and T. Stoiber, The spectral localizer for even index pairings, Proc. Amer. Math. Soc. 149, 134 (2021).
  13. L. Jezequel, C. Tauber, and P. Delplace, Estimating bulk and edge topological indices in finite open chiral chains, J. Math. Phys. 63, 121901 (2022).
  14. A. Cerjan and T. A. Loring, Local invariants identify topology in metals and gapless systems, Phys. Rev. B 106, 064109 (2022).
  15. P. d'Ornellas, R. Barnett, and D. K. K. Lee, Quantized bulk conductivity as a local Chern marker, Phys. Rev. B 106, 155124 (2022).
  16. J. D. Hannukainen, M. F. Martínez, J. H. Bardarson, and T. K. Kvorning, Local topological markers in odd spatial dimensions and their application to amorphous topological matter, Phys. Rev. Lett. 129, 277601 (2022).
  17. W. Chen, Universal topological marker, Phys. Rev. B 107, 045111 (2023).
  18. S. Franca and A. G. Grushin, Topological zero-modes of the spectral localizer of trivial metals, Phys. Rev. B 109, 195107 (2024).
  19. J. D. Hannukainen, Anomaly and topology: On the axial anomaly, domain wall dynamics, and local topological markers in quantum matter, Ph.D. thesis, KTH Royal Institute of Technology, 2024.
  20. O. Penrose and L. Onsager, Bose-Einstein condensation and liquid helium, Phys. Rev. 104, 576 (1956).
  21. E. Koch and S. Goedecker, Locality properties and Wannier functions for interacting systems, Solid State Commun. 119, 105 (2001).
  22. S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Many-body localization characterized from a one-particle perspective, Phys. Rev. Lett. 115, 046603 (2015).
  23. S. Bera, T. Martynec, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, One-particle density matrix characterization of many-body localization, Ann. Phys. 529, 1600356 (2017).
  24. G. Kells, N. Moran, and D. Meidan, Localization enhanced and degraded topological order in interacting p-wave wires, Phys. Rev. B 97, 085425 (2018).
  25. M. Hopjan and F. Heidrich-Meisner, Many-body localization from a one-particle perspective in the disordered one-dimensional Bose-Hubbard model, Phys. Rev. A 101, 063617 (2020).
  26. M. Hopjan, G. Orso, and F. Heidrich-Meisner, Detecting delocalization-localization transitions from full density distributions, Phys. Rev. B 104, 235112 (2021).
  27. X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82, 155138 (2010).
  28. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  29. M. Cheng and Z.-C. Gu, Topological response theory of Abelian symmetry-protected topological phases in two dimensions, Phys. Rev. Lett. 112, 141602 (2014).
  30. M. P. Zaletel, Detecting two-dimensional symmetry-protected topological order in a ground-state wave function, Phys. Rev. B 90, 235113 (2014).
  31. J. C. Wang, Z.-C. Gu, and X.-G. Wen, Field-theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology, and beyond, Phys. Rev. Lett. 114, 031601 (2015).
  32. A. Altland, D. Bagrets, and A. Kamenev, Topology versus Anderson localization: Nonperturbative solutions in one dimension, Phys. Rev. B 91, 085429 (2015).
  33. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  34. Z. Wang, X.-L. Qi, and S.-C. Zhang, Topological order parameters for interacting topological insulators, Phys. Rev. Lett. 105, 256803 (2010).
  35. V. Gurarie, Single-particle Green's functions and interacting topological insulators, Phys. Rev. B 83, 085426 (2011).
  36. L. Wang, X. Dai, and X. C. Xie, Interaction-induced topological phase transition in the Bernevig-Hughes-Zhang model, Europhys. Lett. 98, 57001 (2012).
  37. Z. Wang, X.-L. Qi, and S.-C. Zhang, Topological invariants for interacting topological insulators with inversion symmetry, Phys. Rev. B 85, 165126 (2012).
  38. L. Wang, H. Jiang, X. Dai, and X. C. Xie, Pole expansion of self-energy and interaction effect for topological insulators, Phys. Rev. B 85, 235135 (2012).
  39. S. R. Manmana, A. M. Essin, R. M. Noack, and V. Gurarie, Topological invariants and interacting one-dimensional fermionic systems, Phys. Rev. B 86, 205119 (2012).
  40. J. Zhao, P. Mai, B. Bradlyn, and P. Phillips, Failure of topological invariants in strongly correlated matter, Phys. Rev. Lett. 131, 106601 (2023).
  41. A. A. Markov and A. N. Rubtsov, Local marker for interacting topological insulators, Phys. Rev. B 104, L081105 (2021).
  42. D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Localization-protected quantum order, Phys. Rev. B 88, 014206 (2013).
  43. J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Many-body localization in a disordered quantum Ising chain, Phys. Rev. Lett. 113, 107204 (2014).
  44. D. Pekker, G. Refael, E. Altman, E. Demler, and V. Oganesyan, Hilbert-Glass transition: New universality of temperature-tuned many-body dynamical quantum criticality, Phys. Rev. X 4, 011052 (2014).
  45. N. Laflorencie, G. Lemarié, and N. Macé, Topological order in random interacting Ising-Majorana chains stabilized by many-body localization, Phys. Rev. Res. 4, L032016 (2022).
  46. S. Roy, J. T. Chalker, I. V. Gornyi, and Y. Gefen, Measurement-induced steering of quantum systems, Phys. Rev. Res. 2, 033347 (2020).
  47. N. Lang and H. P. Büchler, Entanglement transition in the projective transverse field Ising model, Phys. Rev. B 102, 094204 (2020).
  48. S. Sang and T. H. Hsieh, Measurement-protected quantum phases, Phys. Rev. Res. 3, 023200 (2021).
  49. A. Lavasani, Y. Alavirad, and M. Barkeshli, Measurement-induced topological entanglement transitions in symmetric random quantum circuits, Nat. Phys. 17, 342 (2021).
  50. C. Fleckenstein, A. Zorzato, D. Varjas, E. J. Bergholtz, J. H. Bardarson, and A. Tiwari, Non-Hermitian topology in monitored quantum circuits, Phys. Rev. Res. 4, L032026 (2022).
  51. G. Kells, D. Meidan, and A. Romito, Topological transitions in weakly monitored free fermions, SciPost Phys. 14, 031 (2023).
  52. A. G. Grushin, in Low-Temperature Thermal and Vibrational Properties of Disordered Solids, edited by M. A. Ramos (World Scientific, London, 2022), Chap. 11.
  53. P. Corbae, J. D. Hannukainen, Q. Marsal, D. Muñoz-Segovia, and A. G. Grushin, Amorphous topological matter: Theory and experiment, Europhys. Lett. 142, 16001 (2023).
  54. F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Phys. 19, 498 (2018).
  55. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  56. P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, arXiv:2403.07111.
  57. J. Venderley, V. Khemani, and E.-A. Kim, Machine learning out-of-equilibrium phases of matter, Phys. Rev. Lett. 120, 257204 (2018).
  58. S. Moudgalya, D. A. Huse, and V. Khemani, Perturbative instability towards delocalization at phase transitions between MBL phases, arXiv:2008.09113.
  59. R. Sahay, F. Machado, B. Ye, C. R. Laumann, and N. Y. Yao, Emergent ergodicity at the transition between many-body localized phases, Phys. Rev. Lett. 126, 100604 (2021).
  60. T. B. Wahl, F. Venn, and B. Béri, Local integrals of motion detection of localization-protected topological order, Phys. Rev. B 105, 144205 (2022).
  61. 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).
  62. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  63. L. Fidkowski, Entanglement spectrum of topological insulators and superconductors, Phys. Rev. Lett. 104, 130502 (2010).
  64. T. L. M. Lezama, S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, One-particle density matrix occupation spectrum of many-body localized states after a global quench, Phys. Rev. B 96, 060202(R) (2017).
  65. M. Nakahara, Geometry, Topology and Physics (CRC Press, Boca Raton, FL, 2018).
  66. M. R. Zirnbauer, Riemannian symmetric superspaces and their origin in random matrix theory, J. Math. Phys. 37, 4986 (1996).
  67. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  68. A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009).
  69. S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  70. M. Serbyn, Z. Papić, and D. A. Abanin, Local cnservation laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013).
  71. V. Ros, M. Müller, and A. Scardicchio, Integrals of motion in the many-body localized phase, Nucl. Phys. B 891, 420 (2015).
  72. L. Fidkowski and A. Kitaev, Effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81, 134509 (2010).
  73. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L032045 for a derivation of the coefficient of the chiral marker in D dimensions, and a demonstration of how the chiral marker works in practice by characterizing the topology of the XYZ-Majorana model, which includes Refs. [16, 19, 72, 86, 87, 88].
  74. The particle hole constraint is CρC=1−ρ, where C=σxK and K represents complex conjugation.
  75. For each disorder realization we have calculated the energy eigenstate in the even fermion parity sector with energy closest to the infinite-temperature value E=0.
  76. If ɛα−ɛα+2<ε, ε=10−14, we instead take the average of λα=(ɛα−ɛα+3)/(ɛα−ɛα+4). If ɛα−ɛα+4<ε we use λα=(ɛα−ɛα+5)/(ɛα−ɛα+6). We repeat this process until the denominator is greater than ε.
  77. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  78. T. Klein Kvorning, C. Spånslätt, AtMa P. O. Chan, and S. Ryu, Nonlocal order parameters for states with topological electromagnetic response, Phys. Rev. B 101, 205101 (2020).
  79. K. Inamura, R. Kobayashi, and S. Ryu, Non-local order parameters and quantum entanglement for fermionic topological field theories, J. High Energy Phys. 01 (2020) 121.
  80. Z.-C. Gu and X.-G. Wen, Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory, Phys. Rev. B 90, 115141 (2014).
  81. J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature (London) 467, 68 (2010).
  82. L. A. Peña Ardila, M. Heyl, and A. Eckardt, Measuring the single-particle density matrix for fermions and hard-core bosons in an optical lattice, Phys. Rev. Lett. 121, 260401 (2018).
  83. 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).
  84. M. Leeuwenhoek, S. Gröblacher, M. P. Allan, and Y. M. Blanter, Modeling Green's function measurements with two-tip scanning tunneling microscopy, Phys. Rev. B 102, 115416 (2020).
  85. M. Hopjan, F. Heidrich-Meisner, and V. Alba, Scaling properties of a spatial one-particle density-matrix entropy in many-body localized systems, Phys. Rev. B 104, 035129 (2021).
  86. A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys.-Usp. 44, 131 (2001).
  87. D. Meidan, A. Romito, and P. W. Brouwer, Scattering matrix formulation of the topological index of interacting fermions in one-dimensional superconductors, Phys. Rev. Lett. 113, 057003 (2014).
  88. I. C. Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, Scattering formula for the topological quantum number of a disordered multimode wire, Phys. Rev. B 83, 155429 (2011).

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Phys. Rev. Research 6, L032045– Published 28 August, 2024
  • Received 12 July 2023
  • Revised 12 June 2024
  • Accepted 8 August 2024
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