nLab topological phase of matter



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In solid state physics, a phase of matter (of a quantum material) is called topological, within a given energy bound Δ\Delta (the “gap”) if external deformations of the system that are gentle enough – namely “adiabatic” – not to excite modes >Δ\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.

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 3\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 codimension2\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 2\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).


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.


Quantum Hall effect


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 “TIT 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 TIT I-symmetric valence bundles is exhibited by non-trivial Berry phases around the two nodal points.

Haldane model



Textbook accounts:

Review and survey:

See also:

In the context of topological quantum computation:

Early discovery articles

Classification and symmetries

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)

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:

Further generalization to include interacting topological order:

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)


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)

Interacting topological phases

  • Strongly Interacting Topological Phases (2015) [[pdf]]
  • 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

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

Research groups

Conference and seminar cycles

  • seminar in Koeln Topological states of matter

  • Topological Phases of Matter: Simons Center, June 10-14, 2013, videos available

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

  • CECAM 2013, Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations description

Last revised on July 13, 2022 at 12:05:22. See the history of this page for a list of all contributions to it.