Ubiquity of conical points in topological insulators
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 507-532.

We show that generically, the degeneracies of a family of Hermitian matrices depending on three parameters have a conical structure. Our result applies to the study of topological phases of matter. It suggests that adiabatic deformations of two-dimensional topological insulators come generically with Dirac-like propagating currents, whose total conductivity equals the chiral number of conical points.

Nous montrons que les valeurs propres dégénérées de matrices dépendant de trois paramètres possèdent généralement une structure conique. Nous appliquons ce résultat à l’étude des phases topologiques de systèmes quantiques. Nous montrons que les déformations adiabatiques entre deux isolants topologiques distincts ont une conductivité globale égale au nombre chiral de points de Dirac.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.152
Classification: 47A13,  81Q10,  81Q05
Keywords: Dirac cones, topological insulators, Chern numbers
Alexis Drouot 1

1 Padelford Hall, University of Washington W Stevens Way NE, Seattle, WA 98105, USA
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Alexis Drouot. Ubiquity of conical points in topological insulators. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 507-532. doi : 10.5802/jep.152. https://jep.centre-mersenne.org/articles/10.5802/jep.152/

[ADH20] H. Ammari, B. Davies & E. O. Hiltunen - “Robust edge modes in dislocated systems of subwavelength resonators”, 2020 | 2001.10455

[ADHY19] H. Ammari, B. Davies, E. O. Hiltunen & S. Yu - “Topologically protected edge modes in one-dimensional chains of subwavelength resonators”, 2019 | 1906.10688

[AFH + 18] H. Ammari, B. Fitzpatrick, E. O. Hiltunen, H. Lee & S. Yu - “Honeycomb-lattice Minnaert bubbles”, 2018 | 1811.03905

[Arn95] V. I. Arnold - “Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect”, Selecta Math. (N.S.) 1 (1995) no. 1, p. 1-19 | Article | MR: 1327227 | Zbl: 0841.58008

[AS78] J. E. Avron & B. Simon - “Analytic properties of band functions”, Ann. Physics 110 (1978) no. 1, p. 85-101 | Article | MR: 475384

[ASBVB13] J. C. Avila, H. Schulz-Baldes & C. Villegas-Blas - “Topological invariants of edge states for periodic two-dimensional models”, Math. Phys. Anal. Geom. 16 (2013) no. 2, p. 137-170 | Article | MR: 3063955 | Zbl: 1271.81210

[Bal19a] G. Bal - “Continuous bulk and interface description of topological insulators”, J. Math. Phys. 60 (2019) no. 8, p. 081506, 20 | Article | MR: 3993755 | Zbl: 1435.82027

[Bal19b] G. Bal - “Topological protection of perturbed edge states”, Commun. Math. Sci. 17 (2019) no. 1, p. 193-225 | Article | MR: 3960646 | Zbl: 1421.35305

[Bal19c] G. Bal - “Topological invariants for interface modes”, 2019 | 1906.08345

[Ber84] M. V. Berry - “Quantal phase factors accompanying adiabatic changes”, Proc. Roy. Soc. London Ser. A 392 (1984) no. 1802, p. 45-57 | MR: 738926 | Zbl: 1113.81306

[BKR17] C. Bourne, J. Kellendonk & A. Rennie - “The K-theoretic bulk-edge correspondence for topological insulators”, Ann. Inst. H. Poincaré Phys. Théor. 18 (2017) no. 5, p. 1833-1866 | Article | MR: 3635969 | Zbl: 1372.82023

[Col91] Y. Colin de Verdière - “Sur les singularités de van Hove génériques”, in Analyse globale et physique mathématique (Lyon, 1989), Mém. Soc. Math. France (N.S.), vol. 46, Société Mathématique de France, Paris, 1991, p. 99-110 | Numdam | Zbl: 0773.47034

[DE99] L. Dieci & T. Eirola - “On smooth decompositions of matrices”, SIAM J. Matrix Anal. Appl. 20 (1999) no. 3, p. 800-819 | Article | MR: 1685053 | Zbl: 0930.15014

[Dom11] M. Domokos - “Discriminant of symmetric matrices as a sum of squares and the orthogonal group”, Comm. Pure Appl. Math. 64 (2011) no. 4, p. 443-465 | Article | MR: 2796513 | Zbl: 1219.15008

[DP12] L. Dieci & A. Pugliese - “Hermitian matrices depending on three parameters: coalescing eigenvalues”, Linear Algebra Appl. 436 (2012) no. 11, p. 4120-4142 | Article | MR: 2915273 | Zbl: 1244.15005

[Dro19a] A. Drouot - “The bulk-edge correspondence for continuous honeycomb lattices”, Comm. Partial Differential Equations 44 (2019) no. 12, p. 1406-1430 | Article | MR: 4000843 | Zbl: 1428.82009

[Dro19b] A. Drouot - “Characterization of edge states in perturbed honeycomb structures”, Pure Appl. Anal. 1 (2019) no. 3, p. 385-445 | Article | MR: 3985090 | Zbl: 1423.35275

[Dro19c] A. Drouot - “Microlocal analysis of the bulk-edge correspondence” (2019), arXiv:1909.10474

[DW20] A. Drouot & M. I. Weinstein - “Edge states and the valley Hall effect”, Adv. Math. 368 (2020), p. 107142, 51 | Article | MR: 4085141 | Zbl: 1442.82003

[EGS05] A. Elgart, G. M. Graf & J. H. Schenker - “Equality of the bulk and edge Hall conductances in a mobility gap”, Comm. Math. Phys. 259 (2005) no. 1, p. 185-221 | Article | MR: 2169973 | Zbl: 1086.81081

[FC13] M. Fruchart & D. Carpentier - “An introduction to topological insulators”, Comptes Rendus Physique 14 (2013) no. 9, p. 779-815 | Article

[FK04] C. Fermanian Kammerer - “Semiclassical analysis of generic codimension 3 crossings”, Internat. Math. Res. Notices (2004) no. 45, p. 2391-2435 | Article | MR: 2076099 | Zbl: 1098.81038

[FKG03] C. Fermanian Kammerer & P. Gérard - “A Landau-Zener formula for non-degenerated involutive codimension 3 crossings”, Ann. Inst. H. Poincaré Phys. Théor. 4 (2003) no. 3, p. 513-552 | Article | MR: 2007256 | Zbl: 1049.81029

[FLTW16] C. L. Fefferman, J. P. Lee-Thorp & M. I. Weinstein - “Edge states in honeycomb structures”, Ann. PDE 2 (2016) no. 2, article ID 12, 80 pages | Article | MR: 3595458 | Zbl: 1404.35128

[FLTW18] C. L. Fefferman, J. P. Lee-Thorp & M. I. Weinstein - “Honeycomb Schrödinger operators in the strong binding regime”, Comm. Pure Appl. Math. 71 (2018) no. 6, p. 1178-1270 | Article | Zbl: 1414.35061

[FT16] S. Freund & S. Teufel - “Peierls substitution for magnetic Bloch bands”, Ann. PDE 9 (2016) no. 4, p. 773-811 | Article | MR: 3530192 | Zbl: 1343.81088

[FW12] C. L. Fefferman & M. I. Weinstein - “Honeycomb lattice potentials and Dirac points”, J. Amer. Math. Soc. 25 (2012) no. 4, p. 1169-1220 | Article | MR: 2947949 | Zbl: 1316.35214

[GP74] V. Guillemin & A. Pollack - Differential topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974 | Zbl: 0361.57001

[GP13] G. M. Graf & M. Porta - “Bulk-edge correspondence for two-dimensional topological insulators”, Comm. Math. Phys. 324 (2013) no. 3, p. 851-895 | Article | MR: 3123539 | Zbl: 1291.82120

[Hat93] Y. Hatsugai - “Chern number and edge states in the integer quantum Hall effect”, Phys. Rev. Lett. 71 (1993) no. 22, p. 3697-3700 | Article | MR: 1246070 | Zbl: 0972.81712

[Ily92] N. V. Ilyushechkin - “The discriminant of the characteristic polynomial of a normal matrix”, Mat. Zametki 51 (1992) no. 3, p. 16-23, 143 | Article | MR: 1172221 | Zbl: 0796.15009

[KP07] P. Kuchment & O. Post - “On the spectra of carbon nano-structures”, Comm. Math. Phys. 275 (2007) no. 3, p. 805-826 | Article | MR: 2336365 | Zbl: 1145.81032

[KRSB02] J. Kellendonk, T. Richter & H. Schulz-Baldes - “Edge current channels and Chern numbers in the integer quantum Hall effect”, Rev. Math. Phys. 14 (2002) no. 1, p. 87-119 | Article | MR: 1877916 | Zbl: 1037.81106

[Kuc16] P. Kuchment - “An overview of periodic elliptic operators”, Bull. Amer. Math. Soc. (N.S.) 53 (2016) no. 3, p. 343-414 | Article | MR: 3501794 | Zbl: 1346.35170

[Lax98] P. D. Lax - “On the discriminant of real symmetric matrices”, Comm. Pure Appl. Math. 51 (1998) no. 11-12, p. 1387-1396 | Article | MR: 1639147 | Zbl: 0933.15013

[Lee16] M. Lee - “Dirac cones for point scatterers on a honeycomb lattice”, SIAM J. Math. Anal. 48 (2016) no. 2, p. 1459-1488 | Article | MR: 3490495 | Zbl: 1342.35283

[LTWZ19] J. P. Lee-Thorp, M. I. Weinstein & Y. Zhu - “Elliptic operators with honeycomb symmetry: Dirac points, edge states and applications to photonic graphene”, Arch. Rational Mech. Anal. 232 (2019) no. 1, p. 1-63 | Article | MR: 3916971 | Zbl: 1410.78024

[Mac15] I. G. Macdonald - Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015 | Zbl: 1332.05002

[Mon17] D. Monaco - “Chern and Fu-Kane-Mele invariants as topological obstructions”, in Advances in quantum mechanics, Springer INdAM Ser., vol. 18, Springer, Cham, 2017, p. 201-222 | Article | MR: 3588051 | Zbl: 1374.81101

[Moo01] J. D. Moore - Lectures on Seiberg-Witten invariants, Lect. Notes in Math., vol. 1629, Springer-Verlag, Berlin, 2001 | MR: 1830497 | Zbl: 1036.57014

[MP14] D. Monaco & G. Panati - “Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene”, J. Statist. Phys. 155 (2014) no. 6, p. 1027-1071 | Article | MR: 3207728 | Zbl: 1401.82064

[Pan07] G. Panati - “Triviality of Bloch and Bloch-Dirac bundles”, Ann. Inst. H. Poincaré Phys. Théor. 8 (2007) no. 5, p. 995-1011 | Article | MR: 2342883 | Zbl: 1375.81102

[Par02] B. N. Parlett - “The (matrix) discriminant as a determinant”, Linear Algebra Appl. 355 (2002), p. 85-101 | Article | MR: 1930139 | Zbl: 1018.15006

[Pet16] P. Petersen - Riemannian geometry, Graduate Texts in Math., vol. 171, Springer, Cham, 2016 | Article | MR: 3469435 | Zbl: 1417.53001

[PSB16] E. Prodan & H. Schulz-Baldes - Bulk and boundary invariants for complex topological insulators, Mathematical Physics Studies, Springer, Cham, 2016 | Article | Zbl: 1342.82002

[PST03] G. Panati, H. Spohn & S. Teufel - “Effective dynamics for Bloch electrons: Peierls substitution and beyond”, Comm. Math. Phys. 242 (2003) no. 3, p. 547-578 | Article | MR: 2020280 | Zbl: 1058.81020

[RH08] S. Raghu & F. D. M. Haldane - “Analogs of quantum-Hall-effect edge states in photonic crystals”, Phys. Rev. A 78 (2008), article ID 033834, 21 pages | Article

[See64] R. T. Seeley - “Extension of C functions defined in a half space”, Proc. Amer. Math. Soc. 15 (1964), p. 625-626 | Article | MR: 165392 | Zbl: 0127.28403

[Ser10] D. Serre - Matrices. Theory and applications, Graduate Texts in Math., vol. 216, Springer, New York, 2010 | Article | Zbl: 1202.15003

[Sim83] B. Simon - “Holonomy, the quantum adiabatic theorem, and Berry’s phase”, Phys. Rev. Lett. 51 (1983) no. 24, p. 2167-2170 | Article | MR: 726866

[Sin05] S. F. Singer - Linearity, symmetry, and prediction in the hydrogen atom, Undergraduate Texts in Math., Springer, New York, 2005 | MR: 2156403 | Zbl: 1088.81006

[Tey99] M. Teytel - “How rare are multiple eigenvalues?”, Comm. Pure Appl. Math. 52 (1999) no. 8, p. 917-934 | Article | MR: 1686977 | Zbl: 0942.47012

[vNW29] J. von Neumann & E. Wigner - “Über das Verhalten von Eigenwerten bei adiabatischen Prozessen”, Phys. Z. 30 (1929), p. 467-470 | Zbl: 55.0520.05

[Wal47] P. R. Wallace - “The band theory of graphite”, Phys. Rev., II. Ser. 71 (1947), p. 622-634 | Article | Zbl: 0033.14304

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