Operators, algebras and their invariants for aperiodic tilings. (English) Zbl 1458.52010
Akiyama, Shigeki (ed.) et al., Substitution and tiling dynamics: introduction to self-inducing structures. Lecture notes from the research school on tiling dynamical systems, CIRM Jean-Morlet Chair, Marseille, France, Fall 2017. Cham: Springer. Lect. Notes Math. 2273, 193-225 (2020).
Summary: We review the construction of operators and algebras from tilings of Euclidean space. This is mainly motivated by physical questions, in particular after topological properties of materials. We explain how the physical notion of locality of interaction is related to the mathematical notion of pattern equivariance for tilings and how this leads naturally to the definition of tiling algebras. We give a brief introduction to the \(K\)-theory of tiling algebras and explain how the algebraic topology of \(K\)-theory gives rise to a correspondence between the topological invariants of the bulk and its boundary of a material.
For the entire collection see [Zbl 1454.37001].
For the entire collection see [Zbl 1454.37001].
MSC:
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 52C99 | Discrete geometry |
| 55N15 | Topological \(K\)-theory |
Keywords:
tilings of Euclidean space; pattern equivariance for tilings; \(K\)-theory of tiling algebrasReferences:
| [1] | J.E. Anderson, I.F. Putnam, Topological invariants for substitution tilings and their associated C^∗-algebras. Ergod. Theory Dyn. Syst. 18(3), 509-537 (1998) · Zbl 1053.46520 · doi:10.1017/S0143385798100457 |
| [2] | F. Baboux, E. Levy, A. Lemaitre, C. Gómez, E. Galopin, L.L. Gratiet, I. Sagnes, A Amo, J. Bloch, E. Akkermans, Measuring topological invariants from generalized edge states in polaritonic quasicrystals. Phys. Rev. B 95, 161114(R) (2017) |
| [3] | A. Avila, S. Jitomirskaya, The ten Martini problem. Ann. Math. 170, 303-342 (2009) · Zbl 1166.47031 · doi:10.4007/annals.2009.170.303 |
| [4] | M. Baake, U. Grimm, Aperiodic Order (Vol. 1) (Cambridge University Press, Cambridge, 2013) · Zbl 1295.37001 |
| [5] | M. Baake, M. Schlottmann, P.D. Jarvis, Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability. J. Phys. A Math. General 24(19), 4637 (1991) · Zbl 0755.52006 |
| [6] | J. Bellissard, K-theory of C^∗-Algebras in solid state physics, in Statistical Mechanics and Field Theory: Mathematical Aspects, (Springer, Berlin, 1986), pp. 99-156 · Zbl 0612.46066 |
| [7] | J. Bellissard, Gap Labeling Theorems for Schrödinger Operators, ed. by M. Waldschmidt, P. Moussa, J.-M. Luck, C. Itzykson. From Number Theory to Physics (Springer, Berlin, 1995) |
| [8] | J. Bellissard, D.J.L. Herrmann, M. Zarrouati, Hull of aperiodic solids and gap labelling theorems. Directions Math. Quasicrystals 13, 207-258 (2000) · Zbl 0972.52014 · doi:10.1090/crmm/013/08 |
| [9] | B. Blackadar, K-theory for Operator Algebras. Mathematical Sciences Research Institute Publications, vol. 5, 2nd edn. (Cambridge University Press, Cambridge, 1998) · Zbl 0913.46054 |
| [10] | A. Clark, L. Sadun, When shape matters: deformations of tiling spaces. Ergod. Theory Dyn. Syst. 26(1), 69-86 (2006) · Zbl 1085.37011 · doi:10.1017/S0143385705000623 |
| [11] | A. Connes, Non-commutative Geometry (Academic, San Diego, 1994) · Zbl 0818.46076 |
| [12] | A. van Daele, K-theory for graded Banach algebras I. Quarterly J. Math. 39(2), 185-199 (1988) · Zbl 0657.46050 · doi:10.1093/qmath/39.2.185 |
| [13] | A. van Daele, K-theory for graded Banach algebras II. Pacific J. Math. 135(2), 377-392 (1988) · Zbl 0622.46049 · doi:10.2140/pjm.1988.134.377 |
| [14] | D. Damanik, M. Embree, A. Gorodetski, Spectral properties of Schrödinger operators arising in the study of quasicrystals, in Mathematics of Aperiodic Order (Birkhäuser, Basel, 2015), pp. 307-370 · Zbl 1378.81031 |
| [15] | K. Davidson, C*-Algebras by Example, vol. 6 (American Mathematical Society, Providence, 1996) · Zbl 0958.46029 |
| [16] | M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Pub. Math. IHES 49, 5-234 (1979) · Zbl 0448.58019 · doi:10.1007/BF02684798 |
| [17] | A. Forrest, J. Hunton, J. Kellendonk, Topological Invariants for Projection Method Patterns (No. 758) (American Mathematical Society, Providence, 2002) · Zbl 1011.52008 |
| [18] | F. Gähler, Computer code, private communication |
| [19] | F. Gähler, J. Hunton, J. Kellendonk, Integral cohomology of rational projection method patterns. Algebr. Geometri. Topol. 13(3), 1661-1708 (2013) · Zbl 1270.52025 · doi:10.2140/agt.2013.13.1661 |
| [20] | B.I. Halperin, Quantized Hall conductance, current carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25(4), 2185 (1982) |
| [21] | Y. Hatsugai, Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71(22), 3697 (1993) · Zbl 0972.81712 |
| [22] | N. Higson, J. Roe, Analytic K-homology (OUP Oxford, Oxford, 2000) · Zbl 0968.46058 |
| [23] | J. Hunton, Spaces of projection method patterns and their cohomology, in Mathematics of Aperiodic Order (Birkhäuser, Basel, 2015), pp. 105-135 · Zbl 1382.52020 |
| [24] | S. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159-1175 (1999) · Zbl 0946.47018 · doi:10.2307/121066 |
| [25] | P. Kalugin, Cohomology of quasiperiodic patterns and matching rules. J. Phys. A Math. General 38(14), 3115 (2005) · Zbl 1065.37015 |
| [26] | J. Kellendonk, Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7(07), 1133-1180 (1995) · Zbl 0847.52022 · doi:10.1142/S0129055X95000426 |
| [27] | J. Kellendonk, Pattern-equivariant functions and cohomology. J. Phys. A Math. General 36(21), 5765 (2003) · Zbl 1055.53029 |
| [28] | J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces. Ergod. Theory Dynam. Syst. 28(4), 1153-1176 (2008) · Zbl 1149.52017 · doi:10.1017/S014338570700065X |
| [29] | J. Kellendonk, On the C^∗-algebraic approach to topological phases for insulators. Ann. Henri Poincaré 18(7), 2251-2300 (2017) · Zbl 1382.82045 · doi:10.1007/s00023-017-0583-0 |
| [30] | J. Kellendonk, E. Prodan, Bulk-boundary correspondence for Sturmian Kohmoto like models. Ann. Henri Poincaré 20(6), 2039-2070 (2019) · Zbl 1492.82024 · doi:10.1007/s00023-019-00792-5 |
| [31] | J. Kellendonk, I.F. Putnam, Tilings, C^∗-algebras and K-theory, in Directions in Mathematical Quasicrystals, ed. by M. Baake, R.V. Moody. CRM Monograph Series, vol. 13 (2000) (American Mathematical Society, Providence, 1999), pp. 177-206 · Zbl 0972.52015 |
| [32] | J. Kellendonk, S. Richard, Topological boundary maps in physics, in Perspectives in Operator Algebras and Mathematical Physics. Theta Series in Advanced Mathematics, vol. 8 (Theta, Bucharest, 2008), pp. 105-121 · Zbl 1199.81029 |
| [33] | J. Kellendonk, T. Richter, H. Schulz-Baldes, Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14, 87-119 (2002) · Zbl 1037.81106 · doi:10.1142/S0129055X02001107 |
| [34] | Y. Last, Spectral theory of Sturm-Liouville operators on infinite intervals: a review of recent developments, in Sturm-Liouville Theory (Birkhäuser Basel, Basel, 2005), pp. 99-120 · Zbl 1098.39011 |
| [35] | M. Pimsner, D. Voiculescu. Exact sequences for K-groups of certain cross products of C^∗-algebra s. J. Op. Theory 4, 93-118 (1980) · Zbl 0474.46059 |
| [36] | E. Prodan, H. Schulz-Baldes, Non-commutative odd Chern numbers and topological phases of disordered chiral systems. J. Funct. Analy. 271(5), 1150-1176 (2016) · Zbl 1344.82055 · doi:10.1016/j.jfa.2016.06.001 |
| [37] | E. Prodan, H. Schulz-Baldes, Bulk and Boundary Invariants for Complex Topological Insulators (Springer, Berlin, 2016) · Zbl 1342.82002 · doi:10.1007/978-3-319-29351-6 |
| [38] | J. Renault, A groupoid Approach to C*-algebras (Vol. 793). (Springer, Berlin, 2006) · Zbl 0433.46049 |
| [39] | M. Rieffel, C^∗-algebras associated with irrational rotations. Pacific J. Math. 93(2), 415-429 (1981) · Zbl 0499.46039 · doi:10.2140/pjm.1981.93.415 |
| [40] | M. Rordam, F. Larsen, N.J. Laustsen, An Introduction to K-theory ofC^∗-Algebras (Cambridge University Press, Cambridge, 2000) · Zbl 0967.19001 · doi:10.1017/CBO9780511623806 |
| [41] | L.A. Sadun, Topology of Tiling Spaces (Vol. 46) (American Mathematical Society, Providence, 2008) · Zbl 1166.52001 |
| [42] | L. Sadun, Cohomology of Hierarchical Tilings, in Mathematics of Aperiodic Order (Birkhäuser, Basel, 2015), pp. 73-104 · Zbl 1372.37040 |
| [43] | L. · Zbl 1038.37014 · doi:10.1017/S0143385702000949 |
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