nLab Bloch's theorem

Idea

(…)

Related concepts

• solid state physics

• electron band theory

• Floquet theory

References

For free electrons

The original article:

• Felix Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Zeitschrift für Physik 52 7-8 (1929) 555-600 [doi:10.1007/BF01339455]

Early developments:

• Walter Kohn: Theory of Bloch Electrons in a Magnetic Field: The Effective Hamiltonian, Phys. Rev. 115 (1959) 1460 [doi:10.1103/PhysRev.115.1460]

• Gregory H. Wannier: Wave Functions and Effective Hamiltonian for Bloch Electrons in an Electric Field, Phys. Rev. 117 (1960) 432 [doi:10.1103/PhysRev.117.432]

Textbook account in mathematical physics:

• Michael Reed, Barry Simon, Sec. XIII.16 Schrödinger operators with periodic potentials, of: Methods of Modern Mathematical Physics – IV: Analysis of Operators, Academic Press (1978) [ISBN:9780080570457, §XIII.16: pdf]

Textbook account in semiconductor-theory:

• Sheng San Li, Section 4.3 of: Semiconductor Physical Electronics, Springer (2006) [doi:10.1007/0-387-37766-2_4]

Review in the context of topological phases of matter:

• Jérôme Cayssol, Jean-Noël Fuchs, Section IV.B of: Topological and geometrical aspects of band theory, J. Phys. Mater. 4 (2021) 034007 [arXiv:2012.11941, doi:10.1088/2515-7639/abf0b5]

• Alexander S. Sergeev, Section 5.1.3 in: Topological insulators and geometry of vector bundles, SciPost Physics Lecture Notes 67 (2023) [arXiv:2011.05004, doi:10.21468/SciPostPhysLectNotes.67]

See also:

• Wikipedia: Bloch’s theorem

• Wikipedia: Crystal momentum

Further discussion:

• E. Dobardžić, M. Dimitrijević, M. V. Milovanović: Generalized Bloch theorem and topological characterization, Phys. Rev. B 91 (2015) 125424 [arXiv:1411.4920, doi:10.1103/PhysRevB.91.125424]

Variant for hyperbolic spaces:

• Joseph Maciejko, Steven Rayan, Hyperbolic band theory, Science Advances 7 36 (2021) [doi:10.1126/sciadv.abe9170]

• Adil Attar, Igor Boettcher, Selberg trace formula in hyperbolic band theory, Phys. Rev. E 106 034114 (2022) [arXiv:2201.06587, doi:10.1103/PhysRevE.106.034114]

For interacting electrons

• Yuejin Guo, Jean-Marc Langlois and William A. Goddard , Electronic Structure and Valence-Bond Band Structure of Cuprate Superconducting Materials, New Series, 239 4842 (1988) 896-899 $[$jstor:1700316$]$

• Jingsan Hu, Jianfei Gu, Weiyi Zhang, Bloch’s band structures of a pair of interacting electrons in simple one- and two-dimensional lattices, Physics Letters A 414 (2021) 127634 $[$doi:10.1016/j.physleta.2021.127634$]$

Wannier functions

The original article:

• Gregory H. Wannier: The Structure of Electronic Excitation Levels in Insulating Crystals, Phys. Rev. 52 (1937) 191 [doi:10.1103/PhysRev.52.191]

Further discussion:

• Nicola Marzari, David Vanderbilt: Maximally-localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56 (1997) 12847 [doi:10.1103/PhysRevB.56.12847, arXiv:cond-mat/9707145]

• Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, David Vanderbilt: Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84 (2012) 1419 [doi:10.1103/RevModPhys.84.1419, arXiv:1112.5411]

In the context of fractional Chern insulators exhibiting a fractional quantum anomalous Hall effect:

• Xiao-Liang Qi: Generic Wavefunction Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators, Phys. Rev. Lett. 107 (2011) 126803 [doi:10.1103/PhysRevLett.107.126803", arXiv:1105.4298]

• Maissam Barkeshli, Xiao-Liang Qi: Topological Nematic States and Non-Abelian Lattice Dislocations, Phys. Rev. X 2 (2012) 031013 [doi:10.1103/PhysRevX.2.031013, arXiv:1112.3311]

• Chao-Ming Jian, Xiao-Liang Qi: Crystal-symmetry preserving Wannier states for fractional chern insulators, Phys. Rev. B 88 (2013) 165134 [arXiv:1303.1787, doi:10.1103/PhysRevB.88.165134]