Singular geometry of the momentum space: from wire networks to quivers and monopoles. (English) Zbl 1388.81124
Summary: A new nano-material in the form of a double gyroid has motivated us to study (non)-commutative \(C^\ast\) geometry of periodic wire networks and the associated graph Hamiltonians.
Here we present a general more abstract framework, which is given by certain quiver representations, with special attention to the original case of the gyroid as well as related cases, such as graphene. The resulting effective \(C^\ast\)-geometry is that of the momentum space, which parameterizes the quasi-momenta.
This geometry is usually singular, where the singularities describe so-called band intersections in physics. We give geometric and algebraic methods to study these intersections; their origin being singularity theory and representation theory. A technique we newly apply to this situation is the use of topological invariants, which we formalize and explain in the paper.
This uses \(K\)-theory and Chern classes as well as “slicing methods” for their computation. In this method the invariants can be computed using Berry’s connection in the momentum space. This brings monopole charges and issues of topological stability into the picture.
Adding a constant magnetic field or allowing projective representations makes the \(C^\ast\) geometry non-commutative. In this case, we can also use \(K\)-theory, albeit in a different way, to make statements about the band structure using gap labeling.
Here we present a general more abstract framework, which is given by certain quiver representations, with special attention to the original case of the gyroid as well as related cases, such as graphene. The resulting effective \(C^\ast\)-geometry is that of the momentum space, which parameterizes the quasi-momenta.
This geometry is usually singular, where the singularities describe so-called band intersections in physics. We give geometric and algebraic methods to study these intersections; their origin being singularity theory and representation theory. A technique we newly apply to this situation is the use of topological invariants, which we formalize and explain in the paper.
This uses \(K\)-theory and Chern classes as well as “slicing methods” for their computation. In this method the invariants can be computed using Berry’s connection in the momentum space. This brings monopole charges and issues of topological stability into the picture.
Adding a constant magnetic field or allowing projective representations makes the \(C^\ast\) geometry non-commutative. In this case, we can also use \(K\)-theory, albeit in a different way, to make statements about the band structure using gap labeling.
MSC:
| 81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 57M15 | Relations of low-dimensional topology with graph theory |
| 58B32 | Geometry of quantum groups |
| 81R60 | Noncommutative geometry in quantum theory |
References:
| [1] | A. A. Agrachev, On the Space of Symmetric Operators with Multiple Ground States, Funct. Anal. and its App. 45, 241–251 (2011) · Zbl 1271.46056 |
| [2] | V. I. Arnold, V.V. Goryunov, O.V. Lyashko and V.A. Vasil’ev, Singularity Theory I, Springer, Berlin Heidelberg, 1998 |
| [3] | V. I. Arnol’d, Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect. (English summary) , Selecta Math. (N.S.) 1 (1995) 119 |
| [4] | J. Bellissard, Gap labelling theorems for Schr”odinger operators, in: From number theory to physics (1992) 538–630.DOI: 10.1007/978-3-662-02838-4 12 · Zbl 0833.47056 |
| [5] | J. Bellissard, A. van Elst and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect. Topology and physics, J. Math. Phys. 35 (1994) 5373–5451.DOI: 10.1063/1.530758 · Zbl 0824.46086 |
| [6] | B. A. Bernevig, T. L. Hughes, S. Raghu and D. P. Arovas, Theory of the Three–Dimensional Quantum Hall Effect in Graphite, Phys. Rev. Lett. 99 (2007) 146804.DOI: 10.1103/PhysRevLett.99.146804 |
| [7] | M. V. Berry, Quantum phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392 (1984) 45–57.DOI: 10.1098/rspa.1984.0023 · Zbl 1113.81306 |
| [8] | A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009) 109.DOI: 10.1103/RevModPhys.81.109 |
| [9] | A. Connes, Noncommutative Geometry. Academic Press, Inc., San Diego, CA, 1994. |
| [10] | M. Demazure, Classification des germs ‘a point critique isol’e et ‘a nombres de modules 0 ou 1 (d’apr‘es Arnol’d). S’eminaire Bourbaki, 26e ann’ee, Vol. 1973/74 Exp. No 443, pp. 124-142, Lect. Notes Math. 431. Springer, Berlin 1975 |
| [11] | V. Ya. Demikhovskii and D. V. Khomitskiy, Quantum Hall effect in p–type heterojunction with a lateral surface superlattice, Phys. Rev. B 68 (2003) 165301.DOI: 10.1103/PhysRevB.68.165301 |
| [12] | C.F. Fefferman, M.I. Weinstein, Honeycomb Lattice Potentials and Dirac Points, preprint.arXiv:1202.3839 · Zbl 1316.35214 |
| [13] | D.Gieseker, H.Knoerrer, E.Trubowitz, An overview of the geometry of algebraic Fermi curves. in ”Algebraic geometry: Sundance 1988”, 19-46, Contemp. Math., 116, Amer. Math. Soc., Providence, RI, 1991. · Zbl 0751.14021 |
| [14] | N. Goldman and P. Gaspard, Quantum graphs and the integer quantum Hall effect, Phys. Rev B 77 (2008) 024302.DOI: 10.1103/PhysRevB.77.024302 |
| [15] | J. Goryo and M. Kohmoto, Berry Phase and Quantized Hall Effect in Three–Dimension, J. Soc. Jpn. 71 (2002) 1403.DOI: 10.1143/JPSJ.71.1403 · Zbl 1055.81082 |
| [16] | K. Grosse-Brauckmann and M. Wohlgemuth, The gyroid is embedded and has constant mean curvature companions, Calc. Var. Partial Differential Equations 4 (1996) 499.DOI: 10.1007/BF01261761 · Zbl 0930.53009 |
| [17] | K. Grove and G. K. Pedersen, Diagonalizing Matrices over C(X), J. Funct. Anal. 59 (1984) 65–89. DOI: 10.1016/0022-1236(84)90053-3 · Zbl 0554.46026 |
| [18] | D. A. Hajduk et. al., The gyroid - a new equilibrium morphology in weakly segregated diblock copolymers Macromolecules 27 (1994) 4063–4075.DOI: 10.1021/ma00093a006 |
| [19] | P.G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. London A68 (1955) 874–878.DOI: 10.1088/0370-1298/68/10/304 |
| [20] | D. Husem”oller, Fibre Bundles, Graduate Texts in Mathematics (Book 20), Springer, 1993 |
| [21] | R.M. Kaufmann,S.Khlebnikov,and B. Wehefritz–Kaufmann,The geometry of the double gyroid wire network:quantum and classical, Journal of Noncommutative Geometry 6 (2012) 623-664. DOI: 10.4171/JNCG/101 · Zbl 1262.81078 |
| [22] | R.M. Kaufmann,S.Khlebnikov,and B. Wehefritz–Kaufmann,The noncommutative geometry of wire networks from triply periodic surfaces,Journal of Physics:Conf. Ser. 343 (2012) 012054. DOI: 10.1088/1742-6596/343/1/012054 |
| [23] | R.M. Kaufmann, S. Khlebnikov, and B. Wehefritz–Kaufmann, Singularities, swallowtails and Dirac points. An analysis for families of Hamiltonians and applications to wire networks, especially the Gyroid. Annals of Physics 327 (2012) 2865.DOI: 10.1016/j.aop.2012.08.001 GEOMETRY OF THE MOMENTUM SPACE79 · Zbl 1254.81041 |
| [24] | R.M. Kaufmann, S. Khlebnikov, and B. Wehefritz–Kaufmann, Re-gauging groupoid, symmetries and degeneracies for Graph Hamiltonians and applications to the Gyroid wire network, accepted for publication in AHP (2015).arXiv:1208.3266 · Zbl 1351.82127 |
| [25] | R.M. Kaufmann, S. Khlebnikov, and B. Wehefritz–Kaufmann, Topologically stable Dirac points in a threedimensional supercrystal. In preparation. · Zbl 1254.81041 |
| [26] | S. Khlebnikov and H. W. Hillhouse, Electronic Structure of Double-Gyroid Nanostructured Semiconductors:Perspectives for carrier multiplication solar cells,Phys. Rev. B 80 (2009) 115316. DOI: 10.1103/PhysRevB.80.115316 |
| [27] | M. Koshino, H. Aoki, K. Kuroki, S. Kagoshima and T. Osada, Hofstadter Butterfly and Integer Quantum Hall Effect in Three Dimensions, Phys. Rev. Lett. 86 (2001) 1062.DOI: 10.1103/PhysRevLett.86.1062 |
| [28] | C. A. Lambert, L. H. Radzilowski, E. L. Thomas, Triply Periodic Level Surfaces as Models for Cubic Tricontinous Block Copolymer Morphologies, Philos. Transactions: Mathematical, Physical and Engineering Sciences, Vol. 354, No 1715, Curved Surfaces in Chemical Structure (1996) 2009–2023 · Zbl 0881.92035 |
| [29] | G. De Nittis and G. Landi, Topological aspects of generalized Harper operators, The Eight International Conference on Progress in Theoretical Physics, Mentouri University, Constantine, Algeria, October 2011; Conference proceedings of the AIP, edited by N. Mebarki and J. Mimouni (2012) 58-65.DOI: 10.1063/1.4715400 |
| [30] | M. Marcolli and V. Mathai, Towards the fractional quantum Hall effect: a noncommutative geometry perspective, in: Noncommutative geometry and number theory: where arithmetic meets geometry, Caterina Consani, Matilde Marcolli (Eds.), Vieweg, Wiesbaden (2006) 235-263.DOI: 10.1007/978-3-8348-0352-8 12 · Zbl 1105.81068 |
| [31] | J. Mather, Notes on Topological Stability, photocopied notes, Harvard Univ., Cambridge, MA, 1970; Stratifications and mappings, in Dynamical Systems, 195–232, Proc. Sympos., Univ. Bahia (Salvador, Brazil, 1971) (M. M. Peixoto, ed.), Academic Press, New York, 1973. |
| [32] | G. Panati, H. Spohn and S. Teufel, Effective dynamics for Bloch electrons: Peierls substitution and beyond, Comm. Math. Phys. 242 (2003) 547-578.DOI: 10.1007/s00220-003-0950-1 · Zbl 1058.81020 |
| [33] | A.H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note No. TN D -5541 (1970) |
| [34] | B. Simon, Holonomy, The Quantum Adiabatic Theorem, and Berry’s Phase, Phys. Rev. Lett. 51, 2167–2170 (1983).DOI: 10.1103/PhysRevLett.51.2167 |
| [35] | R. Thom. Sous-vari’et’es et classes d’homologie des vari’et’es diff’erentiables. II. R’esultats et applications. C. R. Acad. Sci. Paris 236, (1953). 573–575 |
| [36] | R.Thom,Ensemblesetmorphismesstratifi’es,Bull.Amer.Math.Soc.75(1969),240–284. DOI: 10.1090/S0002-9904-1969-12138-5 |
| [37] | D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. den Nijs, Quantized Hall Conductance in a Two– Dimensional Periodic Potential, Phys. Rev. Lett. 49 (1982) 405–408.DOI: 10.1103/PhysRevLett.49.405 |
| [38] | V.N. Urade, T.C. Wei, M.P. Tate and H.W. Hillhouse. Nanofabrication of double-Gyroid thin films. Chem. Mat. 19, 4 (2007) 768-777.DOI: 10.1021/cm062136n |
| [39] | J. von Neumann and E. Wigner, ”Uber das Verhalten von Eigenwerten bei adiabatischen Prozessen, Z. Phys. 30, 467 (1929) · JFM 55.0520.05 |
| [40] | G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Chern Semimetal and the Quantized Anomalous Hall Effect in HgCr2Se4, Phys. Rev. Lett. 107 (2011) 186806.DOI: 10.1103/PhysRevLett.107.186806 Ralph M. Kaufmann, Department of Mathematics, Purdue University, West Lafayette, IN 47907 E-mail address: rkaufman@math.purdue.edu |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
