nLab K-theory classification of topological phases of matter

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)

Related concepts

• K-theory classification of D-brane charge

References

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:

• Denis Bernard, André LeClair, A classification of 2D random Dirac fermions, J. Phys. A: Math. Gen. 35 (2002) 2555 $[$doi:10.1088/0305-4470/35/11/303$]$

• Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, Andreas W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 195125 (2008) $[$doi:10.1103/PhysRevB.78.195125, arXiv:0803.2786$]$

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:

• Michael Atiyah, Isadore Singer, Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’IHÉS, Tome 37 (1969) 5-26 $[$numdam:PMIHES_1969__37__5_0$]$

• Max Karoubi, Espaces Classifiants en K-Théorie, Transactions of the American Mathematical Society 147 1 (Jan., 1970) 75-115 $[$doi:10.2307/1995218$]$

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:

• Daniel Freed, Gregory Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013) 1927-2023 $[$arXiv:1208.5055, doi:10.1007/s00023-013-0236-x$]$

Further discussion:

• Guo Chuan Thiang, On the K-theoretic classification of topological phases of matter, Annales Henri Poincare 17(4) 757-794 (2016) (arXiv:1406.7366)

• Guo Chuan Thiang, Topological phases: isomorphism, homotopy and K-theory, Int. J. Geom. Methods Mod. Phys. 12 1550098 (2015) (arXiv:1412.4191)

• Ralph M. Kaufmann, Dan Li, Birgit Wehefritz-Kaufmann, Topological insulators and K-theory (arXiv:1510.08001, spire:1401095/)

• Ken Shiozaki, Masatoshi Sato, Kiyonori Gomi, Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups, Phys. Rev. B 95 (2017) 235425 $[$arXiv:1701.08725, doi:10.1103/PhysRevB.95.235425$]$

• Luuk Stehouwer, Jan de Boer, Jorrit Kruthoff, Hessel Posthuma, Classification of crystalline topological insulators through K-theory, Adv. Theor. Math. Phys, 25 3 (2021) 723–775 $[$arXiv:1811.02592, doi:10.4310/ATMP.2021.v25.n3.a3$]$

• Charles Zhaoxi Xiong, Classification and Construction of Topological Phases of Quantum Matter (arXiv:1906.02892)

• Eyal Cornfeld, Shachar Carmeli, Tenfold topology of crystals: Unified classification of crystalline topological insulators and superconductors, Phys. Rev. Research 3 (2021) 013052 $[$doi:10.1103/PhysRevResearch.3.013052, arXiv:2009.04486$]$

• Agnès Beaudry, Michael Hermele, Juan Moreno, Markus Pflaum, Marvin Qi, Daniel Spiegel, Homotopical Foundations of Parametrized Quantum Spin Systems $[$arXiv:2303.07431$]$

• Ken Shiozaki, Masatoshi Sato, Kiyonori Gomi, Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups, Phys. Rev. B 106 (2022) 165103 $[$arXiv:1802.06694, doi:10.1103/PhysRevB.106.165103$]$

  (using the Atiyah-Hirzebruch spectral sequence)

Review:

• Sven van Nigtevecht, Topological phases and K-theory, Amsterdam (2019) $[$web, pdf$]$

Generalization of the K-theory classification of free topological pgases to include interacting topological order:

• Hisham Sati, Urs Schreiber: Anyonic topological order in TED K-theory, Rev. Math. Phys. 35 03 (2023) 2350001 [doi:10.1142/S0129055X23500010, arXiv:2206.13563]

On T-duality in the K-theory classification of topological phases of matter, related to the Fourier transform between crystals and their Brillouin torus:

• Varghese Mathai, Guo Chuan Thiang, T-Duality of Topological Insulators, J. Phys. A: Math. Theor. 48 (2015) 42FT02 [doi:10.1088/1751-8113/48/42/42FT02, arXiv:1503.01206]

• Varghese Mathai, Guo Chuan Thiang: T-Duality Simplifies Bulk-Boundary Correspondence, Commun. Math. Phys. 345 (2016) 675–701 [doi:10.1007/s00220-016-2619-6, arXiv:1505.05250]

• Varghese Mathai, Guo Chuan Thiang, T-duality simplifies bulk-boundary correspondence: some higher dimensional cases, Annales Henri Poincaré 17 12 (2016) 3399-3424 [doi:10.1007/s00023-016-0505-6, arXiv:1506.04492]

• Keith C. Hannabuss, Varghese Mathai, Guo Chuan Thiang, T-duality trivializes bulk-boundary correspondence: the parametrised case, Adv. Theor. Math. Phys. 20 (2016) 1193-1226 [doi:10.4310/ATMP.2016.v20.n5.a8, arXiv:1510.04785]

• Keith C. Hannabuss, Varghese Mathai, Guo Chuan Thiang, T-duality simplifies bulk-boundary correspondence: the noncommutative case, Lett. Math. Phys. 108 5 (2018) 1163-1201 [doi:10.1007/s11005-017-1028-x, arXiv:1603.00116]

• Kiyonori Gomi, Guo Chuan Thiang, Crystallographic T-duality. J. Geom. Phys 139 (2019) 50-77 [doi:10.1016/j.geomphys.2019.01.002, arXiv:1806.11385]

Review:

• Guo Chuan Thiang, K-theory and T-duality of topological phases, Adelaide (2018) [ pdf]

• Keith C. Hannabuss, T-duality and the bulk-boundary correspondence, Journal of Geometry and Physics

  124 (2018) 421-435 [doi:10.1016/j.geomphys.2017.11.016, arXiv:1704.00278]

Discussion via cobordism cohomology:

• Anton Kapustin, Ryan Thorngren, Alex Turzillo, Zitao Wang, Electrons Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP 1512:052, 2015 (arXiv:1406.7329)

• Daniel Freed, Michael Hopkins, Reflection positivity and invertible topological phases (arXiv:1604.06527)

• Daniel Freed, Lectures on field theory and topology (cds:2699265)

Relation to the GSO projection:

• Justin Kaidi, Julio Parra-Martinez, Yuji Tachikawa, Topological Superconductors on Superstring Worldsheets (arXiv:1911.11780)

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

• Bruno Mera, The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator, Phys. Rev. B 102 155150 (2020) $[$arXiv:2009.02268, doi:10.1103/PhysRevB.102.155150$]$

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, $C^\ast$-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:

• Shinsei Ryu, Tadashi Takayanagi, Topological Insulators and Superconductors from D-branes, Phys. Lett. B693:175-179, 2010 (arXiv:1001.0763)

• Carlos Hoyos-Badajoz, Kristan Jensen, Andreas Karch, A Holographic Fractional Topological Insulator, Phys. Rev. D82:086001, 2010 (arXiv:1007.3253)

• Shinsei Ryu, Tadashi Takayanagi, Topological Insulators and Superconductors from String Theory, Phys. Rev. D82:086014, 2010 (arXiv:1007.4234)

• Andreas Karch, Joseph Maciejko, Tadashi Takayanagi, Holographic fractional topological insulators in 2+1 and 1+1 dimensions, Phys. Rev. D 82, 126003 (2010) (arXiv:1009.2991)

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)