K-theory classification of topological phases of matter



Solid state physics



Special and general types

Special notions


Extra structure





The phases of quantum materials with gapped Hamiltonian admit a “topological” (really: homotopy theoretical) classification by twisted equivariant topological K-theory, whence one speaks of topological phases of matter, topological states of matter, topological order, topological insulators, and the like.


Holographic relation to K-theory classification of D-brane charge

Under AdS/CFT duality in solid state physics the K-theory classification of topological phases of matter corresponds to the K-theory classification of D-brane charge (Ryu-Takayanagi 10a, Ryu-Takayanagi 10v)


Topological phases of matter via K-theory


Classification topological phases of matter (condensed matter with gapped Hamiltonians, topological insulators) by twisted equivariant topological K-theory:

Via cobordism cohomology:

Relation to the GSO projection:

For quasicrystals via KK-theory of the noncommutative topology of quasiperiodicity:

  • Jean Bellissard, The Noncommutative Geometry of Aperiodic Solids, in: Geometric and Topological Methods for Quantum Field Theory, pp. 86-156 (2003) (pdf, doi:10.1142/9789812705068_0002)

  • Fonger Ypma, Quasicrystals, C *C^\ast-algebras and K-theory, 2005 (pdf)

  • Ian F. Putnam, Non-commutative methods for the K-theory of C *C^\ast-algebras of aperiodic patterns from cut-and-project systems, Commun. Math. Phys. 294, 703–729 (2010) (pdf, doi:10.1007/s00220-009-0968-0)

  • Hervé Oyono-Oyonoa, Samuel Petite, C *C^\ast-algebras of Penrose hyperbolic tilings, Journal of Geometry and Physics Volume 61, Issue 2, February 2011, Pages 400-424 (doi:10.1016/j.geomphys.2010.09.019)


Under AdS/CFT duality in solid state physics the K-theory-classification of topological phases of matter translates to the K-Theory classification of D-brane charge in string theory, allowing a dual description of the topological phases even at strong coupling via AdS/CFT duality:

Relation to Yang-Mills monopoles as Dp/D(p+2)-brane intersections and Yang-Mills instantons as Dp/D(p+4)-brane intersections:

  • Koji Hashimoto, Taro Kimura, Band spectrum is D-brane, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 1 (arXiv:1509.04676)

  • Charlotte Kristjansen, Gordon W. Semenoff, The D3-probe-D7 brane holographic fractional topological insulator, JHEP10 (2016) 079 (arXiv:1604.08548)

Last revised on February 12, 2020 at 06:11:19. See the history of this page for a list of all contributions to it.