Contents

cohomology

# 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.

## References

### Topological phases of matter via K-theory

#### General

Relation to the GSO projection:

• 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, $C^\ast$-algebras and K-theory, 2005 (pdf)

• Ian F. Putnam, Non-commutative methods for the K-theory of $C^\ast$-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, $C^\ast$-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:

• 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.