nLab topological phase of matter

Idea

General

In solid state physics, a phase of matter (of a quantum material) is called topological (sometimes: a topological state of matter), within a given energy bound $\Delta$ (the “gap”) if external deformations of the system that are gentle enough – namely “adiabatic” – not to excite modes (quanta) $\gt \Delta$ above the ground state leave the main properties of the system invariant. This is vaguely reminiscent of how topology is concerned with properties of spaces that are invariant under “gentle” – namely “continuous” – deformations. Indeed, it turns out or is expected that topological phases of matter may be characterized (classified) by certain homeomorphism classes or rather by homotopy classes of the kind studied in topology, or rather in homotopy theory and generalized cohomology theory (e.g. the K-theory classification of topological phases of matter).

Such topological phases are fundamentally different from classical phases of matter in that they are not controlled by the Landau theory of phase transitions.

Topological insulators

For instance, a topological insulator-phase of a crystalline material (a prime example being graphene, see also below) is one where the valence band of energies occupied by the crystal’s electrons is separated by such an energy gap from the conduction band, and where the valence bundle of the electron’s occupied quantum states, as a topological vector bundle over the material’s Brillouin torus, has (see also fiber bundles in physics):

1. a non-trivial equivariant homeomorphism class, in fact

2. a non-trivial equivariant homotopy class of its classifying map, in fact

3. a non-vanishing generalized cohomology class in twisted equivariant topological K-theory.

This is part of the statement of the K-theory classification of topological phases of matter (for which there is some experimental and theoretical support but which, one should admit, remains a conjecture).

The idea is that no gentle/adiabatic deformation of such a topological insulator-phase (eg. by changing ambient pressure, tension, electric fields, etc.) can change the “topological” (rather: homotopy-theoretic) class of the valence bundle, and all the material’s properties implied by this non-trivial class (notably the nature of its “edge modes”) remain unchanged under such deformation.

Here the equivariance is with respect to any or all of:

1. spatial crystallographic point symmetries,

2. non-spatial symmetries, including:

   1. CPT symmetries like time-reversal symmetry,

   2. “internal” or “on-site” symmetries

      (such as spin-reversal symmetries in systems negligible magnetic field and spin-orbit coupling)

which all act on:

1. the Brillouin torus and/or

2. the observables of the system

   (eg. on the Hamiltonian by complex conjugation, for the case of time-reversal symmetry).

If the class of a topological phase of matter crucially depends on its equivariance under such symmetries, hence if the phase could/would decay under deformations which (albeit remaining adiabatic) break some of the symmetry, then one speaks of a symmetry protected topological phase. The richness of topological phases all comes from this symmetry protection.

For instance, without any symmetry protection the valence bundles in a topological insulator-phase of a realistic crystalline material (necessarily of effective dimension $\leq 3$) are characterized entirely by their first Chern class; one speaks of a Chern insulator-phase.

If a more fine-grained topological phase is “protected” by crystallographic symmetry, then one speaks of a topological crystalline insulator-phase, etc.

Topological semi-metals

More generally, there are topological phases where small adiabatic deformations have no effect, as before, but where the topological space of possible deformations has itself a non-trivial topology, for instance a non-trivial fundamental group, in which case the end result of a loop of deformations may have the effect of having transformed the system’s ground state by a unitary operator which depends on the homotopy class of this loop.

For example, if there is an energy gap except over a codimension$\geq 2$ submanifold of “band crossings” or “nodal loci” inside the Brillouin torus, where the gap closes right at the chemical potential, then one speaks of a topological semi-metal-phase. In this case the valence bundle is well-defined (only) on the complement of these “nodal” points, as before, and as such again invariant under gentle deformations, but now these external deformations (which will gently shift the energy bands) may move the position of the nodal band-crossing points through the Brillouin torus. If a loop of such deformations has the effect of braiding these nodal points around each other then it may in total have the effect of having transformed the ground state by an operation in a braid group representation.

If this braiding is non-abelian, and/or if the holonomy of the Berry connection around the nodal points is non-abelian, it follows in particular that the ground state is “degenerate” (“has multiplicity”) hence that there is a Hilbert space of states, all of the ground state energy at the chemical potential, whose dimension is $\geq 2$. In this case one says that the topological phase in addition exhibits topological order or that its ground states exhibits “long-range entanglement” (at least “no short-range entanglement”).

Topological field theory

Therefore, in this case of topological order, the adiabatic dynamics of the ground state of the topological material, say as the nodal points are braided around each other, is characterized by:

1. a finite-dimensional Hilbert space space of quantum states

   (the degenerate ground state),

2. a vanishing Hamiltonian

   (being the restriction of the full Hamiltonian of the material to its energy=0 eigenstates),

3. global dynamics depending (only) on the “topology” (really: the homotopy type) of the trajectory

   (such as on the braid formed by the worldlines of the nodal points).

These are exactly the characteristics of a topological field theory of the type of Chern-Simons theory, which here one may understand as the tiny but highly interesting subsector “below the gap” of the full non-topological quantum field theory that describes all the higher excitations of the given material (which is at least as rich as quantum electrodynamics in the background of the Coulomb potential of the atomic nuclei on the crystal sites, see also here).

Anyons

Generally, it is expected that codimension=2 defects in a topological phase of matter exhibit such a (symmetry protected) topological order in that the adiabatic braiding of them around each other inside the topological host material is described by a topological quantum field theory and in particular constitutes a braid group representation on the systems ground state.

In this case one refers to these defects also a anyons.

The CS-WZW correspondence between Chern-Simons theory and the Wess-Zumino-Witten model then predicts that the wavefunctions that constitute these anyonic ground states in topological phases are given by conformal blocks of a 2d CFT, and it seems that this is indeed the case.

The idea to use the resulting monodromy braid representations constituted by the conformal blocks as quantum gates in a quantum computer underlies the subject of topological quantum computation.

The technical problem with implementing this and related ideas is that, in general, the energy gap of a topological phase of matter is small, so that it is first of all hard to detect and second it will be hard to preserve (hence hard to remain in the topological phase) as one tries to put the “topological properties” of a quantum material to use.

Examples

Quantum Hall effect

• quantum Hall effect, quantum spin Hall effect

Graphene

The first and archetypical example of all these phenomena is seen in graphene, where a band gap almost closes over two nodal Dirac points (up to a tiny shift by the spin-orbit coupling).

The topological phase/order of graphene is “symmetry protected”:

That this must be the case follows already from the fact that any plain complex vector bundle (such as the valence bundle may naively seem to be) is necessarily trivial over the complement of two Dirac points inside the Brillouin 2-torus (a punctured torus, see here). Indeed, the dynamics of the electrons in graphene is “$T I$”-symmetric, which forces the valence bundle to be a real vector bundle with a class not in KU-theory but in KO-theory, which may be non-trivial over a punctured torus (see at Möbius strip). This non-triviality of the $T I$-symmetric valence bundles is exhibited by non-trivial Berry phases around the two nodal points.

Haldane model

• Haldane model

Related entries

• phase of matter, quantum material

• gapped Hamiltonian

• topological field theory

• tensor network state

• quantum Hall effect

• topological insulator

• topological order

• quantum spin liquid

• strange metal

• Weyl semimetal

• in metamaterials:

  • topological photonics

  • topological phononics

• topological quantum computing

• quantum spin Hall effect

• entanglement

References

Reviews

Textbook accounts:

• Sanju Gupta, Avadh Saxena, The Role of Topology in Materials, Springer Series in Solid-State Sciences 189, 2018 (doi:10.1007/978-3-319-76596-9)

• David Vanderbilt, Berry Phases in Electronic Structure Theory – Electric Polarization, Orbital Magnetization and Topological Insulators, Cambridge University Press (2018) (doi:10.1017/9781316662205)

• Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:

  Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) $[$arXiv:1508.02595, doi:10.1007/978-1-4939-9084-9$]$

• Tudor D. Stanescu, Part II of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press 2020 (ISBN:9780367574116)

Review and survey:

• Shou-cheng Zhang, Viewpoint: Topological states of quantum matter, APS Physics 1, 6 (2008) doi:10.1103/Physics.1.6

• Ching-Kai Chiu, Jeffrey C.Y. Teo, Andreas P. Schnyder Shinsei Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88 035005 (2016) $[$arXiv:1505.03535, doi:10.1103/RevModPhys.88.035005$]$

• Vishal Bhardwaj, Ratnamala Chatterjee, Topological Materials – New Quantum Phases of Matter, Resonance 25 (2020) 431–441 [doi:10.1007/s12045-020-0955-5, pdf]

• Jérôme Cayssol, Jean-Noël Fuchs, Topological and geometrical aspects of band theory, J. Phys. Mater. 4 (2021) 034007 (arXiv:2012.11941, doi:10.1088/2515-7639/abf0b5)

• Delft Topology Course, Online course on topology in condensed matter (2015-) [topocondmat.org]

• Tanmoy Das, A pedagogic review on designing model topological insulators, Journal of the Indian Institute of Science 96 77-106 (2016) [arXiv:1604.07546, ISSN:0970-4140]

• Jing Wang, Shou-Cheng Zhang, Topological states of condensed matter, Nature Materials 16 (2017) 1062–1067 [doi:10.1038/nmat5012]

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

• Muhammad Ilyas, Quantum Field Theories, Topological Materials, and Topological Quantum Computing [arXiv:2208.09707]

• Alexander S. Sergeev: Topological insulators and geometry of vector bundles, SciPost Physics Lecture Notes 67 (2023) [arXiv:2011.05004, doi:10.21468/SciPostPhysLectNotes.67]

In view of (fabrication of) quantum materials:

• Kang L. Wang, Yingying Wu, Christopher Eckberg, Gen Yin, Quanjun Pan: Topological quantum materials, MRS Bulletin 45 (2020) 373–379 [doi:10.1557/mrs.2020.122]

• Nitesh Kumar, Satya N. Guin, Kaustuv, Manna, Chandra Shekhar, Claudia Felser: Topological Quantum Materials from the Viewpoint of Chemistry, Chemical Reviews 121 5 (2020) [doi:10.1021/acs.chemrev.0c00732]

• Junjie Wu, Ying Zhang, Bin Xiang: Synthesis, properties, and applications of topological quantum materials, JUSTC 53 10 (2023) 1002 [doi:10.52396/JUSTC-2023-0024]

Discussion with emphasis of the role of quantum anomalies:

• Jürg Fröhlich, Gauge Invariance and Anomalies in Condensed Matter Physics, Journal of Mathematical Physics [arXiv:2303.14741]

See also:

• Wikipedia, Topological insulator

• Wikipedia, Topological order

• Xiao-Gang Wen, Topological Orders and Edge Excitations in FQH States, Advances in Physics 44, 405 (1995). cond-mat/9506066. (for topological order)

• Chetan Nayak, Steven H. Simon, Ady Stern, M. Freedman, Sankar Das Sarma, Non-Abelian anyons and topological quantum computation, Rev Mod Phys 80:3 (Aug 2008) 1083–1159 MR2009g:81041 doi (for topological order)

• M. Z. Hasan, C. L. Kane, Topological insulators, Reviews of Modern Physics 82 (4): 3045 (2010) arXiv:1002.3895 doi (for topological insulator)

• C. L. Kane, An insulator with a twist, Nature physics 4, May 2008, pdf (for topological insulator)

• Class for Physics of the Royal Swedish Academy of Sciences, Topological phase transitions and topological phases of matter, Scientific Background on the Nobel Prize in Physics 2016, (2+)26 pages, pdf

In the context of topological quantum computation:

• Eric Rowell, Zhenghan Wang, Section 4 of: Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)

In the context of chemistry:

• Bradlyn et al. Topological quantum chemistry, Nature 547 (2017) 298–305 [doi:10.1038/nature23268]

Early discovery articles

• Xiao-Gang Wen, Qian Niu, Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B41, 9377 (1990) (for topological order described by TQFT)
• E. Keski-Vakkuri, Xiao-Gang Wen, Ground state structure of hierarchical QH states on torus and modular transformation Int. J. Mod. Phys. B7, 4227 (1993). (for topological order described by TQFT)
• Liang Fu, C. L. Kane, Topological insulators with inversion symmetry, Physical Review B 76 (4): 045302. arXiv:cond-mat/0611341 doi; Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100: 096407, arXiv:0707.1692 doi (for topological insulator, a special case of SPT states)
• B. Andrei Bernevig, Taylor L. Hughes, Shou-Cheng Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 15 December 2006: 314, n. 5806, pp. 1757-1761 doi (for topological insulator, a special case of SPT states)

Classification and symmetries

• Michael Levin, Xiao-Gang Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B, 71, 045110 (2005). (uses unitary fusion category to classify 2+1D topological order with gapped boundary)

• Alexei Kitaev, Periodic table for topological insulators and superconductors, Proc. L.D.Landau Memorial Conf. “Advances in Theor. Physics”, June 22-26, 2008, Chernogolovka, Russia, arxiv/0901.2686 (uses K-homology, Bott periodicity etc. to classify free fermion gapped phases with symmetry)

• Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, arXiv:1106.4772; A short version in Science 338, 1604-1606 (2012) pdf (uses group cohomology of the symmetry group to classify gapped interacting bosonic states with symmetry and trivial topological order)

• Daniel Freed, Gregory Moore, Twisted equivariant matter, arxiv/1208.5055 (uses equivariant K-theory to classify free fermion gapped phases with symmetry)

• Juven Wang, Zheng-Cheng Gu, Xiao-Gang Wen, Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, Phys. Rev. Lett. 114 (2015) 031601 [arxiv:1405.7689, doi:10.1103/PhysRevLett.114.031601]

• Daniel Freed, Short-range entanglement and invertible field theories (arXiv:1406.7278)

• Guo Chuan Thiang, On the K-theoretic classification of topological phases of matter [arXiv:1406.7366]

• Edward Witten, Fermion path integrals and topological phases, arxiv/1508.04715

• Ping Ye, (3+1)d anomalous twisted gauge theories with global symmetry, arxiv/1610.08645

Classification of topological phases of matter via tensor network states:

• C. Wille, O. Buerschaper, Jens Eisert, Fermionic topological quantum states as tensor networks, Phys. Rev. B 95, 245127 (2017) (arXiv:1609.02574)

• Andreas Bauer, Jens Eisert, Carolin Wille, Towards a mathematical formalism for classifying phases of matter (arXiv:1903.05413)

Relation of extended 3d TQFT to lattice models for topological phases of matter (like the 3d toric code):

• Daniel S. Freed, Michael J. Hopkins, Constantin Teleman: Discrete quantum systems from topological field theory [arXiv:2506.05131]

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:

Early detailed discussion (ignored by the revival of the idea following Kitaev 2009)

• Jean Bellissard: K-theory of $C^\ast$-Algebras in solid state physics, in: Statistical mechanics and field theory: Mathematical Aspects, Lecture Notes in Physics 2577, Springer (1986) 99–156 [doi:10.1007/3-540-16777-3_74, inSpire:222243, scan:pdf]

Later 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 (2009) 22-30 [doi:10.1063/1.3149495, arXiv:0901.2686]

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 Poincaré 17 4 (2016) 757-794 [doi:10.1007/s00023-015-0418-9, 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) as expressing the bulk-boundary correspondence:

• 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]

Further on the bulk-boundary correspondence:

• Danilo Polo Ojito, Emil Prodan, Tom Stoiber: $C^\ast$-framework for higher-order bulk-boundary correspondences [arXiv:2406.04226]

Discussion via cobordism cohomology:

• Anton Kapustin, Ryan Thorngren, Alex Turzillo, Zitao Wang: Electrons Fermionic Symmetry Protected Topological Phases and Cobordisms, JHEP 1512:052 (2015) [doi:10.1007/JHEP12(2015)052, 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)

Unstable classification of topological phases

Arguments that some effects in topological phases of matter are “unstable” or “fragile” in that the relevant deformation class of their valence bundles over the Brillouin torus is not their class in topological K-theory (as assumed by the K-theory classification of topological phases of matter) but an unstable homotopy class (what may be called a class in generalized nonabelian cohomology) such as of maps to a Grassmannian space (or more general flag variety) classifying (systems of) sub-bundles of a trivial vector bundle of fixed finite rank:

Influential precursor discussion:

• 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]

More explicit highlighting of the role of the unstable case (and coinage of the term “fragile” for “unstable”):

• Hoi Chun Po, Haruki Watanabe, Ashvin Vishwanath: Fragile Topology and Wannier Obstructions, Phys. Rev. Lett. 121 (2018) 126402 [doi:10.1103/PhysRevLett.121.126402]

• Hoi Chun Po, Liujun Zou, T. Senthil, Ashvin Vishwanath: Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Phys. Rev. B 99 (2019) 195455 [doi:10.1103/PhysRevB.99.195455, arXiv:1808.02482]

• Adrien Bouhon, Tomáš Bzdušek, Robert-Jan Slager: Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102 (2020) 115135 [doi:10.1103/PhysRevB.102.115135, arXiv:2005.02044]

Applications:

• Adrien Bouhon, Robert-Jan Slager, around equation (3) in: Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs [arXiv:2203.16741]

• Zory Davoyan, Wojciech J. Jankowski, Adrien Bouhon, Robert-Jan Slager, section II.A in: Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index [doi:10.1103/PhysRevB.109.165125 arXiv:2308.15555]

Expositions with an eye towards non-abelian braiding of band nodes in momentum space:

• Gunnar F. Lange: Multi-gap topology & non-abelian braiding in $\mathbf{k}$-space, talk at University of Oslo (Feb 2023) $[$pdf]

• Adrien Bouhon: Non-Abelian and Euler multi-gap topologies in crystalline materials, talk at: Quantum Information and Quantum Matter, CQTS @ NYU Abu Dhabi (May 2023) $[$pdf]

See also:

• Barry Bradlyn, Zhijun Wang, Jennifer Cano, B. Andrei Bernevig: Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice, Phys. Rev. B 99 (2019) 045140 [doi:10.1103/PhysRevB.99.045140, arXiv:1807.09729]

Interacting topological phases

There is comparatively little discussion of the classification of topological phases of matter where the elementary excitations cannot be approximated as being non-interacting with each other.

Strong interaction is thought to be necessary for topological order, see there for references on this point.

Discussion for the example of the Kitaev spin chain:

• Lukasz Fidkowski, Alexei Kitaev, The effects of interactions on the topological classification of free fermion systems, Phys. Rev. B 81 (2010) 134509 [arXiv:0904.2197, doi:10.1103/PhysRevB.81.134509]

An overview:

• Jason Alicea, Matthew Fisher, Marcel Franz, Yong-Baek Kim, Strongly Interacting Topological Phases, report on Banff workshop 15w5051 (2015) [pdf, pdf]

A proposal for the generalization of the K-theory classification of topological phases to interacting systems

• Sati & Schreiber 2023.

Other articles

• N. Read, Subir Sachdev, Large-N expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 1773 (1991)

• Xiao-Gang Wen, Mean Field Theory of Spin Liquid States with Finite Energy Gap and Topological orders, Phys. Rev. B 44 2664 (1991).

• Alexei Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303:1, January 2003; Anyons in an exactly solved model and beyond, Annals of Physics 321:1, January 2006

• A. Kitaev, C. Laumann, Topological phases and quantum computation, arXiv/0904.2771

• Alexei Kitaev, John Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006)

• M. Levin, X-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Letts.,96(11), 110405, (2006)

• Jan Carl Budich, Björn Trauzettel, From the adiabatic theorem of quantum mechanics to topological states of matter, physica status solidi (RRL) 7, 109 (2013) arXiv:1210.6672

• Kumar S. Gupta, Amilcar Queiroz, Anomalies and renormalization of impure states in quantum theories, Modern Physics Letters A29 13 (2014) [arxiv/1306.5570 doi:10.1142/S0217732314500643]

• Yosuke Kubota, Controlled topological phases and bulk-edge correspondence, arxiv/1511.05314

Speculations on topological phases relevant in the standard model of particle physics:

• Juven Wang, Ultra Unification, Phys. Rev. D 103 (2021) 105024 [arXiv:2012.15860, doi:10.1103/PhysRevD.103.105024]

Via higher lattice gauge theory:

• Joe Huxford, The higher lattice gauge theory model for topological phases of matter, PhD thesis, Oxford (2022) [uuid:e789bf29-179b-4b79-9168-605bf8c035ba, pdf]

• Joe Huxford, Steven H. Simon, Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview [arXiv:2202.08294]

Research groups

• Univ. of Maryland, joint quantum institute
• Topology and correlation in condensed matter (Germany)
• Center for topological matter (Korea)
• Microsoft Research Station Q (KITP, Santa Barbara)

Conference and seminar cycles

• Markus Garst, Achim Rosch, Simon Trebst: Topological states of matter

• Simons Center: Topological Phases of Matter, June 10-14, 2013 [scgp:3464]

• Alexei Kitaev, On the classification of short-range entangled states, video

• CECAM 2013: Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations [webpage]