najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2015-6-5-637-643najo-984Research ArticlePHYSICSФИЗИКАPeriodic chain of disks in a magnetic field: bulk states and edge statesGrishanovE. N.

Bolshevistskaya Str. 68, Saransk

EreminD. A.

Bolshevistskaya Str. 68, Saransk

IvanovD. A.

Bolshevistskaya Str. 68, Saransk

PopovI. Yu.

Kroverkskiy pr. 49, St. Petersburg, 197101

popov1955@gmail.comSmirnovP. I.

Kroverkskiy pr. 49, St. Petersburg, 197101

Department of Mathematics and IT, Ogarev Mordovia State UniversityRussian FederationDepartment of Higher Mathematics, ITMO UniversityRussian Federation20151508202565637643Copyright © Grishanov E.N., Eremin D.A., Ivanov D.A., Popov I.Y., Smirnov P.I., 20252025Grishanov E.N., Eremin D.A., Ivanov D.A., Popov I.Y., Smirnov P.I.Grishanov E.N., Eremin D.A., Ivanov D.A., Popov I.Y., Smirnov P.I.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/984

An explicitly solvable model for periodic chain of coupled disks in orthogonal magnetic field is considered. The spectrum for the Hamiltonian is compared with the spectrum for the corresponding chain of circles. These models are used for the comparison of the bulk and edge states. It is found that for some range of the magnetic field values the lowest band for the circles system lies below the spectrum for the corresponding disks system, i.e. the edge band is below and is separated from the lowest bulk band. 

nanostructuremagnetic fieldexplicitly solvable modelThis work was partially financially supported by the Government of the Russian Federation (grant 074-U01), by the Ministry of Science and Education of the Russian Federation (GOSZADANIE 2014/190, Project 14.Z50.31.0031), by grants of the President of Russia (state contracts 14.124.13.2045-MK and 14.124.13.1493-MK).ReferencesC. Hermosa, J.V. Alvarez, et al. Intrinsic electrical conductivity of nanostructured metal-organic polymer chains. Nature Comm., 2013, 4, P. 1709.C. Hermosa, J.V. Alvarez, et al. Intrinsic electrical conductivity of nanostructured metal-organic polymer chains. Nature Comm., 2013, 4, P. 1709.A. Medhi, V.B. Shenoy. Continuum theory of edge states of topological insulators: variational principle and boundary conditions. J. Phys.: Condens. Matter, 2012, 24 (35), P. 355001.A. Medhi, V.B. Shenoy. Continuum theory of edge states of topological insulators: variational principle and boundary conditions. J. Phys.: Condens. Matter, 2012, 24 (35), P. 355001.M.Z. Hasan, C.L. Kane. Topological insulators. Rev. Mod. Phys., 2010, 82 P. 3045-3067.M.Z. Hasan, C.L. Kane. Topological insulators. Rev. Mod. Phys., 2010, 82 P. 3045-3067.S. Mao, Y. Kuramoto, K.-I. Imura, A. Yamakage. Analytic theory of edge modes in topological insulators. J. Phys. Soc. Japan, 2010, 79, P. 124709.S. Mao, Y. Kuramoto, K.-I. Imura, A. Yamakage. Analytic theory of edge modes in topological insulators. J. Phys. Soc. Japan, 2010, 79, P. 124709.J.J. Cha, K.J. Koski, Yi. Cui. Topological insulator nanostructures. Phys. Status Solidi RRL, 2013, 7 (1-2), P. 15–25.J.J. Cha, K.J. Koski, Yi. Cui. Topological insulator nanostructures. Phys. Status Solidi RRL, 2013, 7 (1-2), P. 15–25.N.I. Gerasimenko, B.S. Pavlov. Scattering problems on noncompact graphs. Theoret. Math. Phys., 1988, 74, P. 230–240N.I. Gerasimenko, B.S. Pavlov. Scattering problems on noncompact graphs. Theoret. Math. Phys., 1988, 74, P. 230–240P. Exner, P. Seba. Free quantum motion on a branching graph. Rep. Math. Phys., 1989, 28, P. 7–26.P. Exner, P. Seba. Free quantum motion on a branching graph. Rep. Math. Phys., 1989, 28, P. 7–26.P. Exner, et al. (Eds.) Analysis on Graphs and Its Applications. Proc. Symp. Pure Math., (AMS, Providence, Rhode Island, 2008).P. Exner, et al. (Eds.) Analysis on Graphs and Its Applications. Proc. Symp. Pure Math., (AMS, Providence, Rhode Island, 2008).O. Post. Spectral Analysis on Graph-like Spaces. Springer, Berlin, 2010.O. Post. Spectral Analysis on Graph-like Spaces. Springer, Berlin, 2010.I.S. Lobanov, I.Yu. Popov. Scattering by zigzag-armchair nanotube junction. Nanosystems: Physics, Chemistry, Mathematics, 2012, 3 (2), P. 6–28.I.S. Lobanov, I.Yu. Popov. Scattering by zigzag-armchair nanotube junction. Nanosystems: Physics, Chemistry, Mathematics, 2012, 3 (2), P. 6–28.I.Yu. Popov, A.N. Skorynina, I.V. Blinova. On the existence of point spectrum for branching strips quantum graph. J. Math. Phys., 2014, 55, P. 033504/1–20.I.Yu. Popov, A.N. Skorynina, I.V. Blinova. On the existence of point spectrum for branching strips quantum graph. J. Math. Phys., 2014, 55, P. 033504/1–20.V. Kostrykin, R. Schrader. Kirchhoff’s rule for quantum wires. J. Physics A: Math. Gen., 1999, 32, P. 595–630.V. Kostrykin, R. Schrader. Kirchhoff’s rule for quantum wires. J. Physics A: Math. Gen., 1999, 32, P. 595–630.K. Pankrashkin. On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon. Nanosystems: Phys., Chem., Math., 2015, 6 (1), P. 46–56.K. Pankrashkin. On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon. Nanosystems: Phys., Chem., Math., 2015, 6 (1), P. 46–56.P. Duclos, P. Exner, O. Turek. On the spectrum of a bent chain graph. J. Phys. A: Math. Theor., 2008, 41, P. 415206.P. Duclos, P. Exner, O. Turek. On the spectrum of a bent chain graph. J. Phys. A: Math. Theor., 2008, 41, P. 415206.I.Yu. Popov, P.I. Smirnov. Spectral problem for branching chain quantum graph. Phys. Lett. A, 2013, 377, P. 439–442.I.Yu. Popov, P.I. Smirnov. Spectral problem for branching chain quantum graph. Phys. Lett. A, 2013, 377, P. 439–442.I.S. Lobanov, E.S. Trifanova. Direct and inverse problems in the model of quantum graphs. Nanosystems: Phys., Chem., Math., 2012, 3 (5), P. 6–32.I.S. Lobanov, E.S. Trifanova. Direct and inverse problems in the model of quantum graphs. Nanosystems: Phys., Chem., Math., 2012, 3 (5), P. 6–32.I.S. Lobanov, A.I. Trifanov, E.S. Trifanova. Genetic algorithm for constructing graphene nanoribbon with given electronic transport properties. Nanosystems: Phys., Chem., Math., 2013, 4, P. 512–523.I.S. Lobanov, A.I. Trifanov, E.S. Trifanova. Genetic algorithm for constructing graphene nanoribbon with given electronic transport properties. Nanosystems: Phys., Chem., Math., 2013, 4, P. 512–523.D.A. Eremin, I.Yu. Popov. Quantum ring with wire: a model of two-particles problem. Nanosystems: Phys., Chem., Math., 2012, 2 (2), P. 6–32.D.A. Eremin, I.Yu. Popov. Quantum ring with wire: a model of two-particles problem. Nanosystems: Phys., Chem., Math., 2012, 2 (2), P. 6–32.M.A. Kokoreva, V.A. Margulis, M.A. Pyataev. Electron transport in a two-terminal AharonovBohm ring with impurities. Physica E, 2011, 43, P. 1610.M.A. Kokoreva, V.A. Margulis, M.A. Pyataev. Electron transport in a two-terminal AharonovBohm ring with impurities. Physica E, 2011, 43, P. 1610.B.S. Pavlov, I.Yu. Popov, V.A. Geyler, O.S. Pershenko. Possible construction of a quantum multiplexer. Europhys. Lett., 2000, 52, P. 196.B.S. Pavlov, I.Yu. Popov, V.A. Geyler, O.S. Pershenko. Possible construction of a quantum multiplexer. Europhys. Lett., 2000, 52, P. 196.I.Yu. Popov. The resonator with narrow slit and the model based on the operator extensions theory. J.Math. Phys., 1992, 33 (11), P. 3794–3801.I.Yu. Popov. The resonator with narrow slit and the model based on the operator extensions theory. J.Math. Phys., 1992, 33 (11), P. 3794–3801.I.Yu. Popov. Model of point-like window for electromagnetic Helmholtz resonator. Zeitschrift fur Analysis und ihre Anwendungen. 2013, 32 (2), P. 155–162.I.Yu. Popov. Model of point-like window for electromagnetic Helmholtz resonator. Zeitschrift fur Analysis und ihre Anwendungen. 2013, 32 (2), P. 155–162.

The authors declare that there are no conflicts of interest present.