Extended Nielsen-Ninomiya theorem for Floquet and non-Hermitian systems

[1] H. B. Nielsen and M. Ninomiya, “Absence of neutrinos on a lattice: (I). Proof by homo- topy theory”, Nuclear Physics B 185, 20 (1981).

[2] H. B. Nielsen and M. Ninomiya, “Absence of neutrinos on a lattice: (II). Intuitive topo- logical proof”, Nuclear Physics B 193, 173 (1981).

[3] S. Higashikawa, M. Nakagawa, and M. Ueda, “Floquet chiral magnetic effect”, Phys. Rev. Lett. 123, 066403 (2019).

[4] X.-Q. Sun, M. Xiao, T. Bzdušek, S.-C. Zhang, and S. Fan, “Three-dimensional chiral lattice fermion in floquet systems”, Phys. Rev. Lett. 121, 196401 (2018).

[5] J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, “Topological correspondence between hermitian and non-hermitian systems: anomalous dynamics”, Phys. Rev. Lett. 123, 206404 (2019).

[6] T. Bessho and M. Sato, “Nielsen-ninomiya theorem with bulk topology: duality in floquet and non-hermitian systems”, Phys. Rev. Lett. 127, 196404 (2021).

[7] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, “Chiral magnetic effect”, Phys. Rev. D 78, 074033 (2008).

[8] T. Bessho, K. Mochizuki, H. Obuse, and M. Sato, “Extrinsic topology of floquet anoma- lous boundary states in quantum walks”, arXiv preprint arXiv:2112.03167 (2021).

[9] K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance”, Phys. Rev. Lett. 45, 494 (1980).

[10] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized hall con- ductance in a two-dimensional periodic potential”, Phys. Rev. Lett. 49, 405 (1982).

[11] M. Kohmoto, “Topological invariant and the quantization of the hall conductance”, An- nals of Physics 160, 343 (1985).

[12] C. L. Kane and E. J. Mele, “Quantum spin hall effect in graphene”, Phys. Rev. Lett. 95, 226801 (2005).

[13] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions”, Phys. Rev. B 78, 195125 (2008).

[14] A. Kitaev, “Periodic table for topological insulators and superconductors”, AIP Confer- ence Proceedings 1134, 22 (2009).

[15] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy”, New Journal of Physics 12, 065010 (2010).

[16] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, “Classification of topological quantum matter with symmetries”, Rev. Mod. Phys. 88, 035005 (2016).

[17] T. Morimoto and A. Furusaki, “Topological classification with additional symmetries from clifford algebras”, Phys. Rev. B 88, 125129 (2013).

[18] K. Shiozaki and M. Sato, “Topology of crystalline insulators and superconductors”, Phys. Rev. B 90, 165114 (2014).

[19] H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an os- cillating field”, Phys. Rev. A 7, 2203 (1973).

[20] J. H. Shirley, “Solution of the schrödinger equation with a hamiltonian periodic in time”, Phys. Rev. 138, B979 (1965).

[21] T. Oka and H. Aoki, “Photovoltaic hall effect in graphene”, Phys. Rev. B 79, 081406 (2009).

[22] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, “Transport properties of nonequi- librium systems under the application of light: photoinduced quantum hall insulators without landau levels”, Phys. Rev. B 84, 235108 (2011).

[23] Y. Wang, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, “Observation of floquet-bloch states on the surface of a topological insulator”, Science 342, 453 (2013).

[24] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological haldane model with ultracold fermions”, Nature 515, 237 (2014).

[25] M. Bukov, L. D’Alessio, and A. Polkovnikov, “Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to floquet engineering”, Ad- vances in Physics 64, 139 (2015).

[26] T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of peri- odically driven quantum systems”, Physical Review B 82, 235114 (2010).

[27] M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems”, Phys. Rev. X 3, 031005 (2013).

[28] L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires”, Phys. Rev. Lett. 106, 220402 (2011).

[29] D. Carpentier, P. Delplace, M. Fruchart, and K. Gawe˛dzki, “Topological index for peri- odically driven time-reversal invariant 2d systems”, Phys. Rev. Lett. 114, 106806 (2015).

[30] J. K. Asbóth and H. Obuse, “Bulk-boundary correspondence for chiral symmetric quan- tum walks”, Phys. Rev. B 88, 121406 (2013).

[31] T. Morimoto, H. C. Po, and A. Vishwanath, “Floquet topological phases protected by time glide symmetry”, Phys. Rev. B 95, 195155 (2017).

[32] D. J. Thouless, “Quantization of particle transport”, Phys. Rev. B 27, 6083 (1983).

[33] R. Roy and F. Harper, “Periodic table for floquet topological insulators”, Phys. Rev. B 96, 155118 (2017).

[34] L. Privitera, A. Russomanno, R. Citro, and G. E. Santoro, “Nonadiabatic breaking of topological pumping”, Phys. Rev. Lett. 120, 106601 (2018).

[35] A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal- superconducting hybrid structures”, Phys. Rev. B 55, 1142 (1997).

[36] C.-K. Chiu, H. Yao, and S. Ryu, “Classification of topological insulators and supercon- ductors in the presence of reflection symmetry”, Phys. Rev. B 88, 075142 (2013).

[37] C.-K. Chiu and A. P. Schnyder, “Classification of reflection-symmetry-protected topo- logical semimetals and nodal superconductors”, Phys. Rev. B 90, 205136 (2014).

[38] K. Mochizuki, T. Bessho, M. Sato, and H. Obuse, “Topological quantum walk with dis- crete time-glide symmetry”, Phys. Rev. B 102, 035418 (2020).

[39] T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks”, Phys. Rev. A 82, 033429 (2010).

[40] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru- Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks”, Nat. Commun. 3, 1 (2012).

[41] H. Obuse, J. K. Asbóth, Y. Nishimura, and N. Kawakami, “Unveiling hidden topological phases of a one-dimensional hadamard quantum walk”, Phys. Rev. B 92, 045424 (2015).

[42] M. Nakagawa, R.-J. Slager, S. Higashikawa, and T. Oka, “Wannier representation of floquet topological states”, Phys. Rev. B 101, 075108 (2020).

[43] M. J. Rice and E. J. Mele, “Elementary excitations of a linearly conjugated diatomic polymer”, Phys. Rev. Lett. 49, 1455 (1982).

[44] B. A. Bernevig, Topological insulators and topological superconductors (Princeton Uni- versity Press, 2013).

[45] J. K. Asbóth, L. Oroszlány, and A. Pályi, “A short course on topological insulators”, Lecture notes in physics 919, 166 (2016).

[46] M. V. Berry, “Quantal phase factors accompanying adiabatic changes”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45 (1984).

[47] A. Cerjan, S. Huang, M. Wang, K. P. Chen, Y. Chong, and M. C. Rechtsman, “Experi- mental realization of a weyl exceptional ring”, Nat. Photonics 13, 623 (2019).

[48] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -symmetry breaking in com- plex optical potentials”, Phys. Rev. Lett. 103, 093902 (2009).

[49] J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity- time symmetry breaking transitions in a dissipative floquet system of ultracold atoms”, Nat. commun. 10, 1 (2019).

[50] V. Kozii and L. Fu, “Non-hermitian topological theory of finite-lifetime quasiparticles: prediction of bulk fermi arc due to exceptional point”, arXiv preprint arXiv:1708.05841 (2017).

[51] T. Yoshida, R. Peters, and N. Kawakami, “Non-hermitian perspective of the band struc- ture in heavy-fermion systems”, Phys. Rev. B 98, 035141 (2018).

[52] T. Yoshida, R. Peters, N. Kawakami, and Y. Hatsugai, “Symmetry-protected exceptional rings in two-dimensional correlated systems with chiral symmetry”, Phys. Rev. B 99, 121101(R) (2019).

[53] M. Papaj, H. Isobe, and L. Fu, “Nodal arc of disordered dirac fermions and non-hermitian band theory”, Phys. Rev. B 99, 201107 (2019).

[54] H. Shen and L. Fu, “Quantum oscillation from in-gap states and a non-hermitian landau level problem”, Phys. Rev. Lett. 121, 026403 (2018).

[55] M. Ezawa, “Non-hermitian higher-order topological states in nonreciprocal and recipro- cal systems with their electric-circuit realization”, Phys. Rev. B 99, 201411 (2019).

[56] X.-X. Zhang and M. Franz, “Non-hermitian exceptional landau quantization in electric circuits”, Phys. Rev. Lett. 124, 046401 (2020).

[57] G. Gamow, “Zur quantentheorie des atomkernes”, Zeitschrift für Physik 51, 204 (1928).

[58] E. Majorana, “Scattering of an α particle by a radioactive nucleus”, EJTP 3, 293 (2006).

[59] H. Feshbach, “Unified theory of nuclear reactions”, Ann. Phys. 5, 357 (1958).

[60] H. Feshbach, “Unified theory of nuclear reactions”, Ann. Phys. 19, 287 (1962).

[61] C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having symmetry”, Phys. Rev. Lett. 80, 5243 (1998).

[62] Y. C. Hu and T. L. Hughes, “Absence of topological insulator phases in non-hermitian PT -symmetric hamiltonians”, Phys. Rev. B 84, 153101 (2011).

[63] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, “Biorthogonal bulk- boundary correspondence in non-hermitian systems”, Phys. Rev. Lett. 121, 026808 (2018).

[64] S. Yao and Z. Wang, “Edge states and topological invariants of non-hermitian systems”, Phys. Rev. Lett. 121, 086803 (2018).

[65] S. Yao, F. Song, and Z. Wang, “Non-hermitian chern bands”, Phys. Rev. Lett. 121, 136802 (2018).

[66] K. Yokomizo and S. Murakami, “Non-bloch band theory of non-hermitian systems”, Phys. Rev. Lett. 123, 066404 (2019).

[67] N. Okuma and M. Sato, “Topological phase transition driven by infinitesimal instability: majorana fermions in non-hermitian spintronics”, Phys. Rev. Lett. 123, 097701 (2019).

[68] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, and M. Ueda, “Topolog- ical phases of non-hermitian systems”, Phys. Rev. X 8, 031079 (2018).

[69] Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Uni- directional invisibility induced by -symmetric periodic structures”, Phys. Rev. Lett. 106, 213901 (2011).

[70] A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices”, Nature 488, 167 (2012).

[71] L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V. R. Almeida, Y.-F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity- time metamaterial at optical frequencies”, Nat. Mater. 12, 108 (2013).

[72] B. Peng, S¸ . K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity–time-symmetric whispering-gallery microcavities”, Nature Physics 10, 394 (2014).

[73] J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by us- ing exceptional points: application to microcavity sensors for single-particle detection”, Phys. Rev. Lett. 112, 203901 (2014).

[74] Z.-P. Liu, J. Zhang, i. m. c. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-x. Liu, “Metrology with PT -symmetric cavities: enhanced sensitivity near the PT -phase transition”, Phys. Rev. Lett. 117, 110802 (2016).

[75] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Enhanced sensitivity at higher-order exceptional points”, Nature 548, 187 (2017).

[76] W. Chen, S¸ . K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity”, Nature 548, 192 (2017).

[77] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, “Symmetry and topology in non- hermitian physics”, Phys. Rev. X 9, 041015 (2019).

[78] H. Zhou and J. Y. Lee, “Periodic table for topological bands with non-hermitian symme- tries”, Phys. Rev. B 99, 235112 (2019).

[79] N. Hatano and D. R. Nelson, “Localization transitions in non-hermitian quantum me- chanics”, Phys. Rev. Lett. 77, 570 (1996).

[80] N. Hatano and D. R. Nelson, “Vortex pinning and non-hermitian quantum mechanics”, Phys. Rev. B 56, 8651 (1997).

[81] N. Hatano and D. R. Nelson, “Non-hermitian delocalization and eigenfunctions”, Phys. Rev. B 58, 8384 (1998).

[82] Q. Niu, D. J. Thouless, and Y.-S. Wu, “Quantized hall conductance as a topological invariant”, Physical Review B 31, 3372 (1985).

[83] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, “Topological origin of non-hermitian skin effects”, Phys. Rev. Lett. 124, 086801 (2020).

[84] K. Kawabata, T. Bessho, and M. Sato, “Classification of exceptional points and non- hermitian topological semimetals”, Phys. Rev. Lett. 123, 066405 (2019).

[85] H. Shen, B. Zhen, and L. Fu, “Topological band theory for non-hermitian hamiltonians”, Phys. Rev. Lett. 120, 146402 (2018).

[86] Y. Ashida, Z. Gong, and M. Ueda, “Non-hermitian physics”, Advances in Physics 69, 249 (2020).

[87] A. A. Zyuzin and A. Y. Zyuzin, “Flat band in disorder-driven non-hermitian weyl semimet- als”, Phys. Rev. B 97, 041203(R) (2018).

[88] T. Matsushita, Y. Nagai, and S. Fujimoto, “Disorder-induced exceptional and hybrid point rings in weyl/dirac semimetals”, Phys. Rev. B 100, 245205 (2019).

[89] L. H. Karsten, “Lattice fermions in euclidean space-time”, Physics Letters B 104, 315 (1981).

[90] H. B. Nielsen and M. Ninomiya, “The adler-bell-jackiw anomaly and weyl fermions in a crystal”, Physics Letters B 130, 389 (1983).

[91] S. Murakami, “Phase transition between the quantum spin hall and insulator phases in 3d: emergence of a topological gapless phase”, New Journal of Physics 9, 356 (2007).

[92] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, “Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates”, Phys. Rev. B 83, 205101 (2011).

[93] A. A. Burkov, M. D. Hook, and L. Balents, “Topological nodal semimetals”, Phys. Rev. B 84, 235126 (2011).

[94] A. A. Zyuzin and A. A. Burkov, “Topological response in weyl semimetals and the chiral anomaly”, Phys. Rev. B 86, 115133 (2012).

[95] M. M. Vazifeh and M. Franz, “Electromagnetic response of weyl semimetals”, Phys. Rev. Lett. 111, 027201 (2013).

[96] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a weyl fermion semimetal and topological fermi arcs”, Science 349, 613 (2015).

[97] N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl and dirac semimetals in three- dimensional solids”, Rev. Mod. Phys. 90, 015001 (2018).

[98] S. Kobayashi, K. Shiozaki, Y. Tanaka, and M. Sato, “Topological blount’s theorem of odd-parity superconductors”, Phys. Rev. B 90, 024516 (2014).

[99] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, “Topological nodal line semimetals with and without spin-orbital coupling”, Phys. Rev. B 92, 081201(R) (2015).

[100] F. Nathan and M. S. Rudner, “Topological singularities and the general classification of floquet–bloch systems”, New Journal of Physics 17, 125014 (2015).

[101] L. Zhou, C. Chen, and J. Gong, “Floquet semimetal with floquet-band holonomy”, Phys. Rev. B 94, 075443 (2016).

[102] C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental observation of the topological structure of exceptional points”, Phys. Rev. Lett. 86, 787 (2001).

[103] M. S. Rudner and L. S. Levitov, “Topological transition in a non-hermitian quantum walk”, Phys. Rev. Lett. 102, 065703 (2009).

[104] M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, “Time-Reversal Symmetry in Non- Hermitian Systems”, Progress of Theoretical Physics 127, 937 (2012).

[105] Y. C. Hu and T. L. Hughes, “Absence of topological insulator phases in non-hermitian PT -symmetric hamiltonians”, Phys. Rev. B 84, 153101 (2011).

[106] K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, “Edge states and topological phases in non-hermitian systems”, Phys. Rev. B 84, 205128 (2011).

[107] H. Schomerus, “Topologically protected midgap states in complex photonic lattices”, Opt. Lett. 38, 1912 (2013).

[108] J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system”, Phys. Rev. Lett. 115, 040402 (2015).

[109] T. E. Lee, “Anomalous edge state in a non-hermitian lattice”, Phys. Rev. Lett. 116, 133903 (2016).

[110] D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, “Edge modes, degenera- cies, and topological numbers in non-hermitian systems”, Phys. Rev. Lett. 118, 040401 (2017).

[111] L Xiao, X Zhan, Z. Bian, K. Wang, X Zhang, X. Wang, J Li, K Mochizuki, D Kim, N Kawakami, et al., “Observation of topological edge states in parity–time-symmetric quantum walks”, Nature Physics 13, 1117 (2017).

[112] M. Chernodub, “The nielsen-ninomiya theorem, pt-invariant non-hermiticity and single 8-shaped dirac cone”, Journal of Physics A: Mathematical and Theoretical 50, 385001 (2017).

[113] H. Zhou and J. Y. Lee, “Periodic table for topological bands with non-hermitian symme- tries”, Phys. Rev. B 99, 235112 (2019).

[114] K. Kawabata, K. Shiozaki, and M. Ueda, “Anomalous helical edge states in a non- hermitian chern insulator”, Phys. Rev. B 98, 165148 (2018).

[115] H. Xu, D. Mason, L. Jiang, and J. Harris, “Topological energy transfer in an optome- chanical system with exceptional points”, Nature 537, 80 (2016).

[116] S. Longhi, “Topological phase transition in non-hermitian quasicrystals”, Phys. Rev. Lett. 122, 237601 (2019).

[117] R. Okugawa and T. Yokoyama, “Topological exceptional surfaces in non-hermitian sys- tems with parity-time and parity-particle-hole symmetries”, Phys. Rev. B 99, 041202(R) (2019).

[118] F. Song, S. Yao, and Z. Wang, “Non-hermitian skin effect and chiral damping in open quantum systems”, Phys. Rev. Lett. 123, 170401 (2019).

[119] Z.-Y. Ge, Y.-R. Zhang, T. Liu, S.-W. Li, H. Fan, and F. Nori, “Topological band theory for non-hermitian systems from the dirac equation”, Phys. Rev. B 100, 054105 (2019).

[120] F. Song, S. Yao, and Z. Wang, “Non-hermitian topological invariants in real space”, Phys. Rev. Lett. 123, 246801 (2019).

[121] K.-I. Imura and Y. Takane, “Generalized bulk-edge correspondence for non-hermitian topological systems”, Phys. Rev. B 100, 165430 (2019).

[122] K. Kimura, T. Yoshida, and N. Kawakami, “Chiral-symmetry protected exceptional torus in correlated nodal-line semimetals”, Phys. Rev. B 100, 115124 (2019).

[123] W. B. Rui, M. M. Hirschmann, and A. P. Schnyder, “ -symmetric non-hermitian dirac semimetals”, Phys. Rev. B 100, 245116 (2019).

[124] K. Moors, A. A. Zyuzin, A. Y. Zyuzin, R. P. Tiwari, and T. L. Schmidt, “Disorder-driven exceptional lines and fermi ribbons in tilted nodal-line semimetals”, Phys. Rev. B 99, 041116(R) (2019).

[125] Z. Yang and J. Hu, “Non-hermitian hopf-link exceptional line semimetals”, Phys. Rev. B 99, 081102(R) (2019).

[126] L. Jin and Z. Song, “Bulk-boundary correspondence in a non-hermitian system in one dimension with chiral inversion symmetry”, Phys. Rev. B 99, 081103(R) (2019).

[127] L. Zhou, “Non-hermitian floquet topological superconductors with multiple majorana edge modes”, Phys. Rev. B 101, 014306 (2020).

[128] T. Ohashi, S. Kobayashi, and Y. Kawaguchi, “Generalized berry phase for a bosonic bogoliubov system with exceptional points”, Phys. Rev. A 101, 013625 (2020).

[129] S. Longhi, “Non-bloch-band collapse and chiral zener tunneling”, Phys. Rev. Lett. 124, 066602 (2020).

[130] Z. Yang, C.-K. Chiu, C. Fang, and J. Hu, “Jones polynomial and knot transitions in her- mitian and non-hermitian topological semimetals”, Phys. Rev. Lett. 124, 186402 (2020).

[131] C. C. Wojcik, X.-Q. Sun, T. Bzdušek, and S. Fan, “Homotopy characterization of non- hermitian hamiltonians”, Phys. Rev. B 101, 205417 (2020).

[132] Z. Yang, A. Schnyder, J. Hu, and C.-K. Chiu, “Fermion doubling theorems in 2d non- hermitian systems”, arXiv:1912.02788.

[133] K. Kawabata and S. Ryu, “Nonunitary scaling theory of non-hermitian localization”, arXiv:2005.00604.

[134] F. Terrier and F. K. Kunst, “Dissipative analogue of four-dimensional quantum hall physics”, arXiv:2003.11042.

[135] M. N. Chernodub and A. Cortijo, “Non-hermitian chiral magnetic effect in equilibrium”, Symmetry 12, 761 (2020).

[136] D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, “Non-hermitian boundary modes and topology”, Phys. Rev. Lett. 124, 056802(R) (2020).

[137] P. Titum, E. Berg, M. S. Rudner, G. Refael, and N. H. Lindner, “Anomalous floquet- anderson insulator as a nonadiabatic quantized charge pump”, Phys. Rev. X 6, 021013 (2016).

[138] J. C. Budich, Y. Hu, and P. Zoller, “Helical floquet channels in 1d lattices”, Phys. Rev. Lett. 118, 105302 (2017).

[139] M. M. Wauters, A. Russomanno, R. Citro, G. E. Santoro, and L. Privitera, “Localization, topology, and quantized transport in disordered floquet systems”, Phys. Rev. Lett. 123, 266601 (2019).

[140] K. Zhang, Z. Yang, and C. Fang, “Correspondence between winding numbers and skin modes in non-hermitian systems”,

[141] M. Sato, “Topological properties of spin-triplet superconductors and fermi surface topol- ogy in the normal state”, Phys. Rev. B 79, 214526 (2009).

[142] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, “Topological invariants for the fermi surface of a time-reversal-invariant superconductor”, Phys. Rev. B 81, 134508 (2010).

[143] M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, “Topology of andreev bound states with flat dispersion”, Phys. Rev. B 83, 224511 (2011).

[144] A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal- superconducting hybrid structures”, Phys. Rev. B 55, 1142 (1997).

[145] D. Nakamura, T. Bessho, and M. Sato, in preparation (2021).

[146] M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, “Second-order topological insulators and superconductors with an order-two crystalline symmetry”, Phys. Rev. B 97, 205135 (2018).

[147] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Electric multipole moments, topo- logical multipole moment pumping, and chiral hinge states in crystalline insulators”, Phys. Rev. B 96, 245115 (2017).

[148] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Quantized electric multipole in- sulators”, Science 357, 61 (2017).

[149] W. A. Benalcazar, T. Li, and T. L. Hughes, “Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators”, Phys. Rev. B 99, 245151 (2019).

[150] Z. Song, Z. Fang, and C. Fang, “(d 2)-dimensional edge states of rotation symmetry protected topological states”, Phys. Rev. Lett. 119, 246402 (2017).

[151] J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, “Reflection- symmetric second-order topological insulators and superconductors”, Phys. Rev. Lett. 119, 246401 (2017).

[152] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. Parkin, B. A. Bernevig, and T. Neupert, “Higher-order topological insulators”, Science advances 4, eaat0346 (2018).

[153] M. Ezawa, “Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices”, Phys. Rev. Lett. 120, 026801 (2018).

[154] E. Khalaf, “Higher-order topological insulators and superconductors protected by inver- sion symmetry”, Phys. Rev. B 97, 205136 (2018).

[155] A. Matsugatani and H. Watanabe, “Connecting higher-order topological insulators to lower-dimensional topological insulators”, Phys. Rev. B 98, 205129 (2018).

[156] W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene”, Phys. Rev. Lett. 42, 1698 (1979).

[157] A. Eckardt, “Colloquium: atomic quantum gases in periodically driven optical lattices”, Rev. Mod. Phys. 89, 011004 (2017).

[158] T. Oka and S. Kitamura, “Floquet engineering of quantum materials”, Annual Review of Condensed Matter Physics 10, 387 (2019).

[159] M. H. Kolodrubetz, F. Nathan, S. Gazit, T. Morimoto, and J. E. Moore, “Topological floquet-thouless energy pump”, Physical review letters 120, 150601 (2018).

[160] Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks”, Phys. Rev. A 48, 1687 (1993).

[161] J Kempe, “Quantum random walks: an introductory overview”, Contemporary Physics 44, 307 (2003).

[162] H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments”, Physical Review B 84, 195139 (2011).

[163] D. Gross, V. Nesme, H. Vogts, and R. F. Werner, “Index theory of one dimensional quantum walks and cellular automata”, Communications in Mathematical Physics 310, 419 (2012).

[164] J. K. Asbóth, “Symmetries, topological phases, and bound states in the one-dimensional quantum walk”, Phys. Rev. B 86, 195414 (2012).

[165] B. Tarasinski, J. K. Asbóth, and J. P. Dahlhaus, “Scattering theory of topological phases in discrete-time quantum walks”, Phys. Rev. A 89, 042327 (2014).

[166] J. K. Asboth and J. M. Edge, “Edge-state-enhanced transport in a two-dimensional quan- tum walk”, Phys. Rev. A 91, 022324 (2015).

[167] C Cedzich, F. Grünbaum, C Stahl, L Velázquez, A. Werner, and R. Werner, “Bulk-edge correspondence of one-dimensional quantum walks”, Journal of Physics A: Mathemati- cal and Theoretical 49, 21LT01 (2016).

[168] F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. De Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions”, Nature communications 7, 1 (2016).

[169] S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Mea- suring topological invariants in disordered discrete-time quantum walks”, Phys. Rev. A 96, 033846 (2017).

[170] C Cedzich, T Geib, F. Grünbaum, C Stahl, L Velázquez, A. Werner, and R. Werner, “The topological classification of one-dimensional symmetric quantum walks”, in Annales henri poincaré, Vol. 19, 2 (Springer, 2018), pp. 325–383.

[171] C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk”, Phys. Rev. Lett. 121, 100502 (2018).

[172] C Cedzich, T Geib, C Stahl, L Velázquez, A. Werner, and R. Werner, “Complete homo- topy invariants for translation invariant symmetric quantum walks on a chain”, Quantum 2, 95 (2018).

[173] C Cedzich, T Geib, A. Werner, and R. Werner, “Chiral floquet systems and quantum walks at half-period”, in Annales henri poincaré, Vol. 22, 2 (Springer, 2021), pp. 375– 413.

[174] P. W. Anderson, “Absence of diffusion in certain random lattices”, Physical review 109, 1492 (1958).

[175] S. Matsuura, P.-Y. Chang, A. P. Schnyder, and S. Ryu, “Protected boundary states in gapless topological phases”, New Journal of Physics 15, 065001 (2013).

[176] B. I. Halperin, “Quantized hall conductance, current-carrying edge states, and the exis- tence of extended states in a two-dimensional disordered potential”, Physical Review B 25, 2185 (1982).

[177] Z. Fedorova, H. Qiu, S. Linden, and J. Kroha, “Observation of topological transport quantization by dissipation in fast thouless pumps”, Nature communications 11, 1 (2020).

[178] N. Konno, “A path integral approach for disordered quantum walks in one dimension”, arXiv preprint quant-ph/0406233 (2004).

[179] A. Joye and M. Merkli, “Dynamical localization of quantum walks in random environ- ments”, Journal of Statistical Physics 140, 1025 (2010).

[180] A. Joye, “Random time-dependent quantum walks”, Communications in mathematical physics 307, 65 (2011).

[181] D. Evensky, R. Scalettar, and P. G. Wolynes, “Localization and dephasing effects in a time-dependent anderson hamiltonian”, Journal of Physical Chemistry 94, 1149 (1990).

[182] L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, “Observation of photonic anomalous floquet topological insulators”, Nature communications 8, 1 (2017).

[183] S. Mukherjee, A. Spracklen, M. Valiente, E. Andersson, P. Öhberg, N. Goldman, and R. R. Thomson, “Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice”, Nature communications 8, 1 (2017).