Abstract
We prove the discrete Bethe-Sommerfeld conjecture on the graphene lattice, on its dual lattice (the triangular lattice), and on the extended Harper lattice. For each of these lattice geometries, we analyze the behavior of small periodic potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we describe sharp arithmetic criteria on the periods that ensure that no gaps open, and we characterize those energies at which gaps may open in the perturbative regime. In all three cases, we provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero.
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A. Avila, S. Jitomirskaya and C. Marx, Spectral theory of extended Harper’s model and a question by Erdős and Szekeres, Invent. Math. 210 (2017), 283–339.
S. Becker and M. Zworski, Magnetic oscillations in a model of graphene, Comm. Math. Phys. 367 (2019), 941–989.
S. Becker, R. Han and S. Jitomirskaya, Cantor spectrum of graphene in magnetic fields, Invent. Math. 218 (2019), 979–1041.
J. Bellissard and B. Simon, Cantor spectrum for the almost Mathieu equation, J. Funct. Anal. 48 (1982), 408–419.
G. Berkolaiko and A. Comech, Symmetry and Dirac points in graphene spectrum, J. Spectr. Theory 8 (2018), 1099–1148.
A. Brouwer and W. Haemers, Spectra of Graphs, Springer, New York, 2012.
A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009), 109–162.
F. Chung, Spectral Graph Theory, American Mathematical Society, Providence, RI, 1997.
D. Cvetković, M. Doob, I. Gutman and A. Torĝasev, Recent Results in the Theory of Graph Spectra, Elsevier, Amsterdam, 1988.
D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs, J. A. Barth, Heidelberg, 1995.
P. Delplace and G. Montambaux, WKB analysis of edge states in graphene in a strong magnetic field, Phys. Rev. B 82 (2010), 205412.
M. Embree and J. Fillman, Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials, J. Spectr. Theory, 9 (2019), 1063–1087.
C. Fefferman and M. Weinstein, Honeycomb lattice potentials and Dirac points, J. Amer. Math. Soc. 25 (2012), 1169–1220.
C. Fefferman and M. Weinstein, Edge States of continuum Schroedinger operators for sharply terminated honeycomb structures, arXiv:1810.03497.
C. Fefferman, J. P. Lee-Thorp and M. Weinstein, Honeycomb Schroedinger operators in the strong binding regime, Commun. Pure Appl. Math. 71 (2018), 1178–1270.
J. Fillman and R. Han, preprint in preparation.
D. Gieseker, H. Knörrer and E. Trubowitz, The Geometry of Algebraic Fermi Curves, Academic Press, Boston, MA, 1993.
R. Han, Absence of point spectrum for the self-dual extended Harper’s model, Int. Math. Res. Not. IMRN 9 (2018), 2801–2809.
R. Han, Dry Ten Martini problem for the non-self-dual extended Harper’s model, Trans. Amer. Math. Soc. 370 (2018), 197–217.
R. Han, S. Jitomirskaya, Full measure reducibility and localization for quasiperiodic Jacobi operators: A topological criterion, Adv. Math. 319 (2017), 224–250.
R. Han and S. Jitomirskaya, Discrete Bethe-Sommerfeld Conjecture, Commun. Math. Phys. 361 (2018), 205–216.
J. H. Han, D. J. Thouless, H. Hiramoto and M. Kohmoto, Critical and bicritical properties of Harper’s equation with next-nearest-neighbor coupling, Phys. Rev. B 50 (1994), 11365.
B. Helffer, P. Kerdelhué and J. Royo-Letelier, Chambers’s formula for the graphene and the Hou model with kagome periodicity and applications, Ann. Henri Poincaré 17 (2016), 795–818.
B. Helffer and A. Mohamed, Asymptotics of the density of states for the Schrödinger operator with periodic electric potential, Duke Math. J. 92 (1998), 1–60.
S. Jitomirskaya and C. A. Marx, Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model, Comm. Math. Phys. 316 (2012), 237–267.
Y. E. Karpeshina, Perturbation Theory for the Schrödinger Operator with a Periodic Potential, Springer, Berlin, 1997.
E. Korotyaev and N. Saburova, Schrödinger operators on periodic discrete graphs, J. Math. Anal. Appl. 420 (2014), 576–611.
H. Krüger, Periodic and limit-periodic discrete Schrödinger operators, preprint, arXiv:1108.1584
P. Kuchment, O. Post, On the spectra of carbon nano-structures, Comm. Math. Phys. 275 (2007), 805–882.
P. Van Mouche, The coexistence problem for the discrete Mathieu operator, Comm. Math. Phys. 122 (1989), 23–33.
K. Novoselov, Nobel lecture: Graphene: Materials in the flatland, Rev. Modern Phys. 83 (2011), 837–849.
L. Parnovski, Bethe-Sommerfeld conjecture, Ann. Henri Poincaré 9 (2008), 457–508.
L. Parnovski and A. V. Sobolev, On the Bethe-Sommerfeld conjecture for the polyharmonic operator, Duke Math. J. 107 (2001), 209–238.
L. Parnovski and A. V. Sobolev, Perturbation theory and the Bethe-Sommerfeld conjecture, Ann. Henri Poincaré 2 (2001), 573–581.
V. N. Popov and M. Skriganov, A remark on the spectral structure of the two dimensional Schrödinger operator with a periodic potential, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 109 (1981), 131–133.
O. Post, Spectral Analysis on Graph-Like Spaces, Springer, Heidelberg, 2012.
M. Skriganov, Proof of the Bethe-Sommerfeld conjecture in dimension two, Soviet Math. Dokl. 20 (1979), 89–90.
M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Proc. Steklov Math. Inst. 171 (1984), 3–122.
M. Skriganov, The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential, Inv. Math. 80 (1985), 107–121.
D. J. Thouless, Bandwidth for a quasiperiodic tight binding model, Phys. Rev. B 28 (1983), 4272–4276.
O. A. Veliev, Spectrum of multidimensional periodic operators, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen 49 (1988), 17–34.
Acknowledgements
We would like to thank Svetlana Jitomirskaya for comments on an earlier version of the manuscript, and Tom Spencer for useful discussions. R. H. would like to thank IAS, Princeton, for its hospitality during the 2017–18 academic year, and Virginia Tech for its hospitality during which part of the work was done. R. H. is supported in part by the National Science Foundation under Grant No. DMS-1638352 and DMS-1800689. J. F. was supported in part by an AMS Simons Travel Grant 2016–2018 and Simons Collaboration Grant #711663.
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Fillman, J., Han, R. Discrete Bethe-Sommerfeld conjecture for triangular, square, and hexagonal lattices. JAMA 142, 271–321 (2020). https://doi.org/10.1007/s11854-020-0138-z
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DOI: https://doi.org/10.1007/s11854-020-0138-z