nLab 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

For free-fermion topological insulators

On the classification (now often referred to, somewhat rudimentarily, as the ten-fold way) of free fermion topological phases of matter (condensed matter with gapped Hamiltonians, specifically topological insulators) by twisted equivariant topological K-theory:

Precursor discussion phrased in terms of random matrix theory instead of K-theory:

The original proposal making topological K-theory explicit:

  • Alexei Kitaev, Periodic table for topological insulators and superconductors, talk at: L.D.Landau Memorial Conference “Advances in Theoretical Physics”, June 22-26, 2008, In:AIP Conference Proceedings 1134, 22 (2009) (arXiv:0901.2686, doi:10.1063/1.3149495)

Further details:

The technical part of the argument always essentially boils down (implicitly, never attributed this way before Freed & Moore 2013) to the argument for Karoubi K-theory from:

More on this Clifford algebra-argument explicit in view of topological insulators:

  • Michael Stone, Ching-Kai Chiu, Abhishek Roy, Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock, Journal of Physics A: Mathematical and Theoretical, 44 4 (2010) 045001 [[doi:10.1088/1751-8113/44/4/045001]]

  • Gilles Abramovici, Pavel Kalugin, Clifford modules and symmetries of topological insulators, International Journal of Geometric Methods in Modern PhysicsVol. 09 03 (2012) 1250023 [[arXiv:1101.1054, doi:10.1142/S0219887812500235]]

The proper equivariant K-theory formulation expected to apply also to topological crystalline insulators:

Further discussion:


Generalization to include interacting topological order:

Via cobordism cohomology:

Relation to the GSO projection:

With application of the external tensor product of vector bundles to describe coupled systems:

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 July 13, 2022 at 16:02:02. See the history of this page for a list of all contributions to it.