×

zbMATH — the first resource for mathematics

Bulk-edge correspondence for two-dimensional topological insulators. (English) Zbl 1291.82120
The paper proves the bulk-edge correspondence for 2D topological insulators and independent particles. With this aim, the authors introduce a \(\mathbb Z^2\) bulk topological invariant equal to the number of pairs of edge states modulo 2. They claim that there is a room for a strict mathematical approach just as for quantum Hall systems, where Laughlin’s argument has gained in precision and detail by the subsequent mathematical discussion. First, the class of bulk, edge, single-particle discrete Schrödinger operators are introduced which describe insulators, topological or otherwise. A general class of 2D single-particle lattice Hamiltonians is considered with and without edge. They are symmetric under fermionic time reversal and potentially describe topological insulators. By this, the periodicity is postulated only in the direction parallel to the edge. The duality is formulated in its most basic version, in a precise though still preliminary form. Then the Hamiltonian is obtained from the Schrödinger operator on the honeycomb lattice modeling graphene in the single-particle approximation and considering two types of boundary conditions (zigzag and armchair). As a result, the authors give a new proof of the absence of edge states for armchair boundary conditions. In order to define the index of bundles on the 2-torus, as well as some auxiliary indices, their sections and transition matrices are considered and they are classified in terms of a \(\mathbb Z^2\)-invariant. As a result, the bulk index is defined and the basic version of duality is stated in a whole. The bulk index is formulated for the specific and more familiar case, where the lattice Hamiltonian is doubly periodic. In this case, the relevant 2-torus is the Brillouin zone. The authors discuss how that index arises from the general one. All the obtained results have a counterpart in the case of quantum Hall (QH) systems. Then, an independent, alternate version of the duality is included which is based on scattering theory and on Levinson’s theorem, as well as on a comparison between the two versions. Then the authors conjecture an analogous alternate version for quantum spin Hall (QHS) systems. Finally, the paper presents some results on the obtained indices, including their comparison with references.

MSC:
82D20 Statistical mechanical studies of solids
81V70 Many-body theory; quantum Hall effect
81U05 \(2\)-body potential quantum scattering theory
82D80 Statistical mechanical studies of nanostructures and nanoparticles
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Avila, J.C., Schulz-Baldes, H., Villegas-Blas, C.: Topological invariants of edge states for periodic two-dimensional models. http://arXiv.org/abs/1202.0537v1 [math ph], 2012, to appear in Math. Phys., Anal. Geom · Zbl 1271.81210
[2] Bernevig, B.A.; Hughes, T.L.; Zhang, S.-C., Quantum spin Hall effect and topological phase transition in hgte quantum wells, Science, 314, 1757-1761, (2006)
[3] Bräunlich, G.; Graf, G.M.; Ortelli, G., Equivalence of topological and scattering approaches to quantum pumping, Commun. Math. Phys., 295, 243-259, (2010) · Zbl 1227.34089
[4] Essin, A.M.; Gurarie, V., Bulk-boundary correspondence of topological insulators from their green’s functions, Phys. Rev. B, 84, 125132, (2011)
[5] Fröhlich, J.; Kerler, T., Universality in quantum Hall systems, Nucl. Phys. B, 354, 369-417, (1991)
[6] Fröhlich, J.; Studer, U.M., Gauge invariance and current algebra in nonrelativistic many-body theory, Rev. Mod. Phys, 65, 733, (1993)
[7] Fröhlich, J., Studer, U.M., Thiran, E.: Quantum theory of large systems of non-relativistic matter. Les Houches Lectures 1994, London, New York: Elsevier (1995) available at http://arXiv.org/abs/cond-mat/9508062v1, 1995
[8] Fröhlich, J.; Zee, A., Large scale physics of the quantum Hall fluid, Nucl. Phys. B, 364, 517-540, (1991)
[9] Fu, L.; Kane, C.L., Time reversal polarization and a \(Z\)_{2} adiabatic spin pump, Phys. Rev. B, 74, 195312, (2006)
[10] Fujita, M.; Wakabayashi, K.; Nakada, K.; Kusakabe, K., Peculiar localized state at zigzag graphite edge, J. Phys. Soc. Jpn., 65, 1920-1923, (1996)
[11] Haldane, F.D.M, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett., 61, 2015-2018, (1988)
[12] Hasan, M.Z.; Kane, C.L., Topological insulators, Rev. Mod. Phys., 82, 3045-3067, (2010)
[13] Hatsugai, Y., Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett., 71, 3697, (1993) · Zbl 0972.81712
[14] Hatsugai, Y.; Ryu, S., Topological origin of zero-energy edge states in particle-hole symmetric systems, Phys. Rev. Lett., 89, 077002, (2002)
[15] Hsieh, D.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y.S.; Cava, R.J.; Hasan, M.Z., A topological Dirac insulator in a quantum spin Hall phase, Nature, 452, 970, (2008)
[16] Kane, C.L.; Mele, E.J., \(Z\)_{2} topological order and the quantum spin Hall effect, Phys. Rev. Lett., 95, 146802, (2005)
[17] Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer-Verlag, 1980 · Zbl 0435.47001
[18] Kohn, W., Analytic properties of Bloch waves and Wannier functions, Phys. Rev., 115, 809-821, (1959) · Zbl 0086.45101
[19] König, M.; Wiedmann, S.; Brüne, C.; Roth, A.; Buhmann, H.; Molenkamp, L.W.; Qi, X.-L.; Zhang, S.-C., Quantum spin Hall insulator state in hgte quantum wells, Science, 318, 766, (2007)
[20] Moore, J.E.; Balents, L., Topological invariants of time-reversal-invariant band structures, Phys. Rev. B, 75, 121306(r), (2007)
[21] Nakada, K.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M.S., Edge state in graphene ribbons: nanometer size effect and edge shape dependence, Phys. Rev. B., 54, 17954, (1996)
[22] Nakahara, M.: Geometry, Topology and Physics. Graduate Student Series in Physics, London: Institute of Physics Publishing, 1990 · Zbl 0764.53001
[23] Pfeffer, W.F., More on involutions of a circle, Amer. Math. Monthly, 81, 613, (1974) · Zbl 0281.54025
[24] Prodan, E., Robustness of the spin-Chern number, Phys. Rev. B, 80, 125327, (2009)
[25] Qi, X.-L.; Wu, Y.-S.; Zhang, S.-C., Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors, Phys. Rev. B, 74, 085308, (2006)
[26] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III. Scattering Theory. New York: Academic Press, 1979 · Zbl 0405.47007
[27] Roy, R., \(Z\)_{2} classification of quantum spin Hall systems: an approach using time-reversal invariance, Phys. Rev. B, 79, 195321, (2009)
[28] Schulz-Baldes, H.; Kellendonk, J.; Richter, T., Simultaneous quantization of edge and bulk Hall conductivity, J. Phys. A: Math. Gen., 33, l27, (2000) · Zbl 0985.81137
[29] Sheng, D.N.; Weng, Z.Y.; Sheng, L.; Haldane, F.D.M., Quantum spin-Hall effect and topologically invariant Chern numbers, Phys. Rev. Lett., 97, 036808, (2006)
[30] Thouless, D.J., Quantisation of particle transport, Phys. Rev. B, 27, 6083-6087, (1983)
[31] Wen, X.G., Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states, Phys. Rev. B, 41, 12838-12844, (1990)
[32] Zhang, S.-C., The Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect, Int. J. Mod. Phys. B, 6, 25-58, (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.