Abstract
As we had seen in Table 1.1, the classification of topological systems is described by a tenfold scheme and a periodicity in spatial dimensions d. Notions of integral dimensions and bulk-boundary correspondence lies at the heart of the topological band theory [1,2,3,4]. A nontrivial invariant calculated for a periodic system signals existence of robust boundary states for the same system with a boundary. This correspondence is the progenitor of formulations of various invariants such as TKNN invariant (Chern number) [5], the Pfaffian and others (for a recent review see [3]) which lead to exotic boundary physics. However, not every system has a well defined “bulk” or “boundary”. Neither does every system have a well defined dimension. Is there a notion of a topological state in such systems? If yes, how can they be characterized?
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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 83:1057–1110
Chiu CK, Teo JCY, Schnyder AP, Ryu S (2016) Classification of topological quantum matter with symmetries. Rev Mod Phys 88:035005
Ludwig AWW (2016) Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond. Phys Scr 2016(T168):014001. http://arxiv.org/abs/1512.08882
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
Domany E, Alexander S, Bensimon D, Kadanoff LP (1983) Solutions to the Schrödinger equation on some fractal lattices. Phys Rev B 28:3110–3123
Mandelbrot BB, Pignoni R (1983) The fractal geometry of nature. WH Freeman, New York
Gefen Y, Mandelbrot BB, Aharony A (1980) Critical phenomena on fractal lattices. Phys Rev Lett 45:855–858
Gefen Y, Aharony A, Mandelbrot BB, Kirkpatrick S (1981) Solvable fractal family, and its possible relation to the backbone at percolation. Phys Rev Lett 47:1771–1774
Rammal R, Toulouse G (1982) Spectrum of the Schrödinger equation on a self-similar structure. Phys Rev Lett 49:1194–1197
Alexander S, Orbach R (1982) Density of states on fractals:fractons. Journal de Physique Lettres 43(17):625–631
van Veen E, Yuan S, Katsnelson MI, Polini M, Tomadin A (2016) Quantum transport in Sierpinski carpets. Phys Rev B 93:115428
Alexander S (1984) Some properties of the spectrum of the Sierpinski gasket in a magnetic field. Phys Rev B 29:5504–5508
Fukushima M, Shima T (1992) On a spectral analysis for the Sierpinski gasket. Potential Anal 1(1):1–35
Wang XR (1995) Localization in fractal spaces: exact results on the Sierpinski gasket. Phys Rev B 51:9310–9313
Chakrabarti A (1996) Exact results for infinite and finite Sierpinski gasket fractals: extended electron states and transmission properties. J Phys Condens Matter 8(50):10951
Pal B, Chakrabarti A (2012) Staggered and extreme localization of electron states in fractal space. Phys Rev B 85:214203
Gordon JM, Goldman AM, Maps J, Costello D, Tiberio R, Whitehead B (1986) Superconducting-normal phase boundary of a fractal network in a magnetic field. Phys Rev Lett 56:2280–2283
Shang J, Wang Y, Chen M, Dai J, Zhou X, Kuttner J, Hilt G, Shao X, Gottfried JM, Wu K (2015) Assembling molecular Sierpiński triangle fractals. Nat Chem 7(5):389–393
Song ZG, Zhang YY, Li SS (2014) The topological insulator in a fractal space. Appl Phys Lett 104(23):1–5
Bernevig BA, Hughes TL (2013) Topological insulators and topological superconductors. Princeton University Press, Princeton
Loring TA, Hastings MB (2010) Disordered topological insulators via C*-algebras. Eur Phys Lett 92(6):67004
Datta S (1997) Electronic transport in mesoscopic systems. Cambridge University Press, Cambridge
Fradkin E (2013) Field theories of condensed matter physics. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Agarwala, A. (2019). Seeking Topological Phases in Fractals. In: Excursions in Ill-Condensed Quantum Matter. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-21511-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-21511-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21510-1
Online ISBN: 978-3-030-21511-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)