Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras
Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operat...
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MDPI AG
2023-08-01
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| author | Elias Zafiris Albrecht von Müller |
| author_facet | Elias Zafiris Albrecht von Müller |
| author_sort | Elias Zafiris |
| collection | DOAJ |
| description | Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a continuous topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the quasi-momentum in a direction, when the edge of the zone is reached, we obtain a closed path. Then, if we lift this loop from the base space to the sections of the sheaf-theoretic fibration induced by the localization of the energy eigenfunctions, we obtain a global topological phase factor which encodes the topological structure of the Brillouin zone. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. In this communication, we show that it is the unitary representation theory of the discrete Heisenberg group in terms of commutative modular symplectic variables, giving rise to a joint commutative representation space endowed with an integral and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-invariant symplectic form that articulates the specific form of the topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework. |
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| format | Article |
| id | doaj.art-4d727896d79848d4b42725cae1ebc4d0 |
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| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-10T21:53:30Z |
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| spelling | doaj.art-4d727896d79848d4b42725cae1ebc4d02023-11-19T13:17:21ZengMDPI AGUniverse2218-19972023-08-019939210.3390/universe9090392Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier AlgebrasElias Zafiris0Albrecht von Müller1Parmenides Foundation, Center for the Conceptual Foundations of Science, Kirchplatz 1, Pullach, 82049 Munich, GermanyParmenides Foundation, Center for the Conceptual Foundations of Science, Kirchplatz 1, Pullach, 82049 Munich, GermanyTopological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a continuous topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the quasi-momentum in a direction, when the edge of the zone is reached, we obtain a closed path. Then, if we lift this loop from the base space to the sections of the sheaf-theoretic fibration induced by the localization of the energy eigenfunctions, we obtain a global topological phase factor which encodes the topological structure of the Brillouin zone. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. In this communication, we show that it is the unitary representation theory of the discrete Heisenberg group in terms of commutative modular symplectic variables, giving rise to a joint commutative representation space endowed with an integral and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-invariant symplectic form that articulates the specific form of the topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework.https://www.mdpi.com/2218-1997/9/9/392Heisenberg groupsymplectic geometrysheaf theoryAbstract Differential Geometryline bundlestheta functions |
| spellingShingle | Elias Zafiris Albrecht von Müller Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras Universe Heisenberg group symplectic geometry sheaf theory Abstract Differential Geometry line bundles theta functions |
| title | Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras |
| title_full | Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras |
| title_fullStr | Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras |
| title_full_unstemmed | Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras |
| title_short | Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras |
| title_sort | electron beams on the brillouin zone a cohomological approach via sheaves of fourier algebras |
| topic | Heisenberg group symplectic geometry sheaf theory Abstract Differential Geometry line bundles theta functions |
| url | https://www.mdpi.com/2218-1997/9/9/392 |
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