nLab K-theory classification of topological phases of matter

Contents

Context

Solid state physics

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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.

Properties

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)

References

Topological phases of matter via K-theory

For free-fermion topological insulators

On the classification 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:

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)

Holographic

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 May 29, 2022 at 06:39:32. See the history of this page for a list of all contributions to it.