Abstract
In this chapter, we will theoretically establish that amorphous systems can host topologically insulating phases. We will provide a demonstration of this by constructing models (using familiar ingredients) on random lattices where fermions hop between sites within a finite range. By tuning parameters (such as the density of sites), we show that the system undergoes a quantum phase transition from a trivial to a topological phase. We characterize the topological nature by obtaining the topological invariant and associated quantized transport signatures. We also address interesting features of such quantum phase transitions. This is achieved through a detailed study of all nontrivial symmetry classes (A, AII, D, DIII and C) in two dimensions. We will also provide a demonstration of a topological insulator in three dimensions. This work opens a new direction in the experimental search for topological quantum matter, by demonstrating their possibility in, as yet unexplored, amorphous systems. We discuss several examples including glassy systems and other engineered random systems.
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References
Klitzing KV, Dorda G, Pepper M (1980) New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys Rev Lett 45:494–497
Laughlin RB (1981) Quantized hall conductivity in two dimensions. Phys Rev B 23:5632–5633
Thouless DJ, Kohmoto M, Nightingale MP, den Nijs M (1982) Quantized hall conductance in a two-dimensional periodic potential. Phys Rev Lett 49:405–408
Haldane FDM (1988) Model for a quantum hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys Rev Lett 61:2015–2018
Murakami S, Nagaosa N, Zhang S-C (2004) Spin-hall insulator. Phys Rev Lett 93:156804
Kane CL, Mele EJ (2005) \(Z_2\) topological order and the quantum spin hall effect. Phys Rev Lett 95:146802
Kane CL, Mele EJ (2005) Quantum spin hall effect in graphene. Phys Rev Lett 95:226801
Bernevig BA, Zhang S-C (2006) Quantum spin hall effect. Phys Rev Lett 96:106802
Bernevig BA, Hughes TL, Zhang S-C (2006) Quantum spin hall effect and topological phase transition in HgTe quantum wells. Science 314(5806):1757–1761
König M, Wiedmann S, Brne C, Roth A, Buhmann H, Molenkamp LW, Qi X-L, Zhang S-C (2007) Quantum spin hall insulator state in HgTe quantum wells. Science 318(5851):766–770
Fu L, Kane CL, Mele EJ (2007) Topological insulators in three dimensions. Phys Rev Lett 98:106803
Moore JE, Balents L (2007) Topological invariants of time-reversal-invariant band structures. Phys Rev B 75:121306
Roy R (2009) Topological phases and the quantum spin hall effect in three dimensions. Phys Rev B 79:195322
Hsieh D, Qian D, Wray L, Xia Y, Hor YS, Cava RJ, Hasan MZ (2008) A topological dirac insulator in a quantum spin hall phase. Nature 452:970–974
Hasan MZ, Kane CL (2010) Colloquium: topological insulators. Rev Mod Phys 82:3045–3067
Qi X-L, Zhang S-C (2011) Topological insulators and superconductors. Rev Mod Phys 83(4):1057
Ando Y (2013) Topological insulator materials. J Phys Soc Jpn 82(10):102001
Qi X-L, Hughes TL, Zhang S-C (2008) Topological field theory of time-reversal invariant insulators. Phys Rev B 78:195424
Schnyder AP, Ryu S, Furusaki A, Ludwig AWW (2008) Classification of topological insulators and superconductors in three spatial dimensions. Phys Rev B 78:195125
Ryu S, Schnyder AP, Furusaki A, Ludwig AWW (2010) Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J Phys 12(6):065010
Kitaev A (2009) Periodic table for topological insulators and superconductors. AIP Conf Proc 1134(1):22–30. http://aip.scitation.org/doi/pdf/10.1063/1.3149495
Altland A, Zirnbauer MR (1997) Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys Rev B 55:1142–1161
Kitaev AY (2001) Unpaired majorana fermions in quantum wires. Phys Uspekhi 44(10S):131
Chadov S, Qi X, Kübler J, Fecher GH, Felser C, Zhang SC (2010) Tunable multi-functional topological insulators in ternary heusler compounds. Nat Mater 9(7):541–545
Das A, Ronen Y, Most Y, Oreg Y, Heiblum M, Shtrikman H (2012) Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of majorana fermions. Nat Phys 8(12):887–895
Chang C-Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L-L et al (2013) Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator. Science 340(6129):167–170
Nadj-Perge S, Drozdov IK, Li J, Chen H, Jeon S, Seo J, MacDonald AH, Bernevig BA, Yazdani A (2014) Observation of majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346(6209):602–607
Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T (2014) Experimental realization of the topological haldane model with ultracold fermions. Nature 515(7526):237–240
Kobayashi K, Ohtsuki T, Imura K-I (2013) Disordered weak and strong topological insulators. Phys Rev Lett 110:236803
Diez M, Fulga IC, Pikulin DI, TworzydÅo J, Beenakker CWJ (2014) Bimodal conductance distribution of Kitaev edge modes in topological superconductors. New J Phys 16(6):063049
Li J, Chu R-L, Jain JK, Shen S-Q (2009) Topological Anderson insulator. Phys Rev Lett 102:136806
Fulga IC, van Heck B, Edge JM, Akhmerov AR (2014) Statistical topological insulators. Phys Rev B 89:155424
Ringel Z, Kraus YE, Stern A (2012) Strong side of weak topological insulators. Phys Rev B 86:045102
Kraus YE, Lahini Y, Ringel Z, Verbin M, Zilberberg O (2012) Topological states and adiabatic pumping in quasicrystals. Phys Rev Lett 109:106402
Fulga IC, Pikulin DI, Loring TA (2016) Aperiodic weak topological superconductors. Phys Rev Lett 116:257002
Bandres MA, Rechtsman MC, Segev M (2016) Topological photonic quasicrystals: fractal topological spectrum and protected transport. Phys Rev X 6:011016
Christ N, Friedberg R, Lee T (1982) Random lattice field theory: general formulation. Nucl Phys B 202(1):89–125
Loring TA, Hastings MB (2010) Disordered topological insulators via C*-algebras. Eur Phys Lett 92(6):67004
Bernevig BA, Hughes TL (2013) Topological insulators and topological superconductors. Princeton University Press, Princeton
Roy R (2006) Topological invariants of time reversal invariant superconductors. arXiv:cond-mat/0608064
Qi X-L, Hughes TL, Raghu S, Zhang S-C (2009) Time-reversal-invariant topological superconductors and superfluids in two and three dimensions. Phys Rev Lett 102:187001
Senthil T, Marston JB, Fisher MPA (1999) Spin quantum hall effect in unconventional superconductors. Phys Rev B 60:4245–4254
Chern T (2016) \(d + id\) and \(d\) wave topological superconductors and new mechanisms for bulk boundary correspondence. AIP Advances 6(8)
Datta S (1997) Electronic transport in mesoscopic systems. Cambridge University Press, Cambridge
Medhi A, Shenoy VB (2012) Continuum theory of edge states of topological insulators: variational principle and boundary conditions. J Phys Condens Matter 24(35):355001
Fradkin E (2013) Field theories of condensed matter physics. Cambridge University Press, Cambridge
Scappucci G, Capellini G, Lee WCT, Simmons MY (2009) Ultradense phosphorus in germanium delta-doped layers. Appl Phys Lett 94(16):162106
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Agarwala, A. (2019). Topological Insulators in Amorphous Systems. In: Excursions in Ill-Condensed Quantum Matter. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-21511-8_3
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DOI: https://doi.org/10.1007/978-3-030-21511-8_3
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