nLab topological order

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Topological order

This entry is about topological orders of materials in condensed matter physics. For topological orders of directed acyclic graphs in graph theory, see linear extension of a partial order.

Context

Topological physics

Solid state physics

Topological order

Idea

In solid state physics, by topological order (Wen 89, Wen & Niu 90, Wen 91, 93, 95, Gu & Wen 09, p. 2, Chen, Gu & Wen 10) one refers to a phenomenon that may (but need not) be exhibited by quantum materials that are in a topological phase of matter. Hence there is an implication

topological order⇒topological phase, \text{topological order} \;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\; \text{topological phase} \,,

but not the other way around.

(“While ostensibly coined in 1989 by Xiao-Gang Wen, the term ‘topological order’ has been in use since 1972 to describe the behavior of the classical xy model. [arXiv:2308.01765])

Specifically, the ground state of a topologically ordered material is rich in structure (besides being topologically stable), which need not be the case for a general topological phase. In particular a topologically ordered ground state is “degenerate” (a standard but somewhat unfortunate jargon in this context: it really refers to the energy eigenvalues being degenerate) in that there is a ≥2\geq 2-dimensional Hilbert space of quantum states that all have the same lowest energy (within experimentally relevant approximation).

Moreover, for this to qualify as topologically ordered, one typically demands that in its degenerate ground state the system may exhibit “anyonic defects”. A popular more succinct way of making this somewhat more precise is to say that the dynamics of (the electrons in) a topologically ordered material, when restricted to the energy=0 ground states, is a topological field theory equal or akin to a Chern-Simons theory with Wilson line-insertions: these Wilson lines being the worldlines of the anyon-defects.

In short then, topological order is meant to be that aspect of topological phases of matter which is related to the existence of anyons in the material, in one way or other. Via this relation, topological order is closely related to considerations in topological quantum computation.

Via degenerate anyonic ground states

The definition is traditionally a little vague, but the hallmark of a topological order is meant to be the presence, in a topological phase, of some or preferably all of the following phenomena (Gu & Wen 09, p. 2, called the “modern day philosopher’s stone” in Tsvelik 2014b):

  1. degenerate ground state (Wen 89, p. 4, Wen 95, Sec. 1.1);

  2. Berry connection with non-abelian holonomy;

  3. anyon excitations/defects whose wavefunctions constitute braid group representations (“fractional statistics”).

The original articles (Wen 89, Wen & Niu 90, Wen 91, 93, 95) proposed to declare that topological phases with distinct ground state degeneracy exhibit distinct topological order. The demand that also the Berry connection should be non-abelian for there to be a topological order seems to appear first in Wen 91 review (then also Gu & Wen 09, p. 2). Or maybe the claim is rather that two distinct topological orders may have the same ground state degeneracy but be distinguished by their Berry connections.

Remark

(relation between these conditions)
By “degenerate ground state” one just means that the sub-Hilbert space of quantum states (for crystalline materials these are the quantum states of electrons in the crystal lattice of atomic nuclei) of those that have the lowest possible energy, has dimension ≥2\geq 2.

(It has been proposed that the presence of “short-range entanglement” in quantum materials implies that the ground state Hilbert space is 1-dimensional. In this sense, the condition that the ground state be degenerate implies that there is no “short-range entanglement” if there is “topological order”.)

Since (by Bloch-Floquet theory) the quantum states of these electrons form the space of sections of the Bloch bundle, and since (by assumption of a topological phase) the ground state involves sections only of the gapped sub-bundle which is called the valence bundle, this is closely related to the valence bundle having rank ≥2\geq 2.

(In making these statement there is some tacit switching between single-electron theory and its second quantization involved, sorting out of which would be necessary for being more precise about these matters.)

But since the Berry connection is a connection on the valence bundle, it can have non-abelian holonomy (namely in U(n) for n≥2n \geq 2) only if the bundle’s rank is ≥2\geq 2.

Similarly, the presence of anyons broadly means that there are families of adiabatic defomrations of the material which have the effect of transforming the ground state by linear operators forming a braid group representation. This is most interesting when the representation is non-abelian, which again requires that the Hilbert space of ground states that it is represented on has dimension ≥2\geq 2.

However, there are also non-trivial “abelian anyons”, namely braid group representations which are just complex 1-dimensional, and – even if not as interesting as their non-abelian cousins – these are counted as perfectly valid instances of the concept of anyons (in fact it was first a struggle and then a breakthrough to detect abelian anyons in experiment). Some authors, especially those using the language of fusion categories to speak about anyons, require of a “topological order” only that it contains anyons and possibly just abelian anyons, in which case it is maybe not so clear whether the ground state needs to be degenerate and/or the Berry connection be non-abelian.

Via topological entanglement entropy

It is now thought that some/all of the above characteristics may be captured quantum information theoretically in terms of the entanglement entropy of the ground state:

The non-degeneracy of the ground state is related to absence of “short-range entanglement”; and the existence of anyon excitations is related to the presence of “long-range entanglement” (Chen, Gu & Wen 10, Sec. 5) witnessed by a non-vanishing topological entanglement entropy (Kitaev & Preskill 2006, Levin & Wen 2006).

Often one sees topological order being related also to (1) strong interaction and/or (2) strong quantum correlation (Wen 91 review) between the electrons (while the classical electron band theory that gives rise to the widely-accepted K-theory classification of topological phases of matter assumes that it is sensible to neglect the interaction of electrons between each other and only retain their interaction to an effective Coulomb background field).

Beware that the use of the term “correlation” in the context of topological order (cf. Wen 91 review) is always meant as “quantum correlation” and specifically as “non-classical quantum correlation” and as such used as a synonym for quantum entanglement (cf. ZCZW 19, §1.5 and generally Luo & Luo 03, p. 3). In contrast, long-range classical correlation is indicative of non-topological Landau theory-phases and hence the opposite of what is relevant here.

Together with the entanglement-theoretic characterization just mentioned, the logic here seems to be the following sequence of schematic implications:

(diagram adapted from SS22)

The first steps in this sequence is intuitively plausible and widely expected to hold (eg. Lu & Viyah 2022, p. 1) but not currently derivable from first principles (Zaanen, Liu, Sun & Schalm 2015, p. 527):

If the Coulomb interaction between the electrons – which by itself is certainly strong and long-range – cannot be neglected (hence if the averaging- or screening-effects that make electron band theory work do not apply) then this strong interaction makes the electrons in the ground state be correlated with each other, one way or other, across non-negligible distances; and quantum mechanically this leads to the ground state’s entanglement entropy having long-range contributions, which, essentially by definition, means that it has a constant contribution by the topological entanglement entropy.

Examples

under construction

Literature

Original articles

The proposal that ground state degeneracy is a signature of “topological order”:

The additional requirement that the Berry connection be non-abelian (and/or the presence of anyons):

Suggestion that topological order goes along with long-range entanglement:

Review

Textbook accounts:

Further review:

See also:

Early discovery articles

Examples of topologically ordered materials

Definite examples of topologically ordered materials remain somewhat elusive, but a few candidates have been discussed.

[Manning-Coe & Bradlyn 2023:] outside of the fractional quantum Hall effect, the connection between the microscopic Hamiltonian for interacting electrons and the topological order of the ground state remains elusive.

Argument that superconductors are actually topologically ordered phases:

On topological semimetals with degenerate ground states:

On topological semimetals which gap to topologically ordered phases:

On detection of topological order by observing modular transformations on the ground state:

On thermodynamic stability of topological order:

See also:

  • Davide Gaiotto, Anton Kapustin, Spin TQFTs and fermionic phases of matter, arxiv/1505.05856

  • Nicholas Read, Subir Sachdev, Large-NN expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 1773 (1991) (on Z 2Z_2 topological order)

  • Xiao-Gang Wen, Mean Field Theory of Spin Liquid States with Finite Energy Gap and Topological orders, Phys. Rev. B 44 2664 (1991). (on Z 2Z_2 topological order)

  • Xiao-Gang Wen, Non-Abelian Statistics in the FQH states

    Phys. Rev. Lett. 66, 802 (1991).

  • Moore, Gregory; Read, Nicholas. Nonabelions in the fractional quantum hall effect Nuclear Physics B 360 (2–3): 362 (1991).

  • Xiao-Gang Wen and Yong-Shi Wu, Chiral operator product algebra hidden in certain FQH states

    Nucl. Phys. B419, 455 (1994).

  • Alexei 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

  • Michael Levin, Xiao-Gang Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B, 71, 045110 (2005).

  • 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)

  • Levin M. and Wen X-G., 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

  • Frank Wilczek, Anthony Zee, Appearance of gauge structure in simple dynamical systems, Physical Review Letters 52 24 (1984) 2111 [[doi:10.1103/PhysRevLett.52.2111]]

  • Amit Jamadagni, Hendrik Weimer, An Operational Definition of Topological Order (arXiv:2005.06501)

  • Nathanan Tantivasadakarn, Ashvin Vishwanath, Ruben Verresen, A hierarchy of topological order from finite-depth unitaries, measurement and feedforward [arXiv:2209.06202]

Discussion in “reciprocal momentum space” via twisted equivariant K-theory:

Discussion of quantum measurement of topologically ordered states:

  • Yabo Li, Mikhail Litvinov, Tzu-Chieh Wei, Measuring Topological Field Theories: Lattice Models and Field-Theoretic Description [arXiv:2310.17740]

Topological entanglement entropy

On entanglement entropy in arithmetic Chern-Simons theory:

  • Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo, Entanglement entropies in the abelian arithmetic Chern-Simons theory [arXiv:2312.17138]

General

Identification of a contribution to entanglement entropy at absolute zero which is independent of the subsystem‘s size (“topological entanglement entropy”, “long-range entanglement”), reflecting topological order and proportional to the total quantum dimension of anyon excitations:

Review:

In terms of Renyi entropy (it’s independent of the Renyi entropy parameter):

  • Ulrich Schollwöck, (Almost) 25 Years of DMRG - What Is It About? (pdf)

and in the example of Chern-Simons theory:

Discussion in the dimer model:

  • Shunsuke Furukawa, Gregoire Misguich, Topological Entanglement Entropy in the Quantum Dimer Model on the Triangular Lattice, Phys. Rev. B 75, 214407 (2007) (arXiv:cond-mat/0612227)

Discussion via holographic entanglement entropy:

  • Ari Pakman, Andrei Parnachev, Topological Entanglement Entropy and Holography, JHEP 0807: 097 (2008) (arXiv:0805.1891)

  • Andrei Parnachev, Napat Poovuttikul, Topological Entanglement Entropy, Ground State Degeneracy and Holography, Journal of High Energy Physics volume 2015, Article number: 92 (2015) (arXiv:1504.08244)

See also:

  • Tatsuma Nishioka, Tadashi Takayanagi, Yusuke Taki, Topological pseudo entropy (arXiv:2107.01797)

Relation to strong interaction

Relation of long-range entanglement to strong interaction:

  • Jan Zaanen, Yan Liu, Ya-Wen Sun, Koenraad Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press 2015 [[doi:10.1017/CBO9781139942492]]

    In a way it appears obvious that the strongly interacting bosonic quantum critical state is subject to long-range entanglement. Nonetheless, the status of this claim is conjectural.

    It is at present impossible to arrive at more solid conclusions that are based on rigorous mathematical procedures. It does illustrate emphatically the central challenge in the pursuit of field-theoretical quantum information: there are as yet not general measures available to precisely enumerate the meaning of long-range entanglement in such seriously quantum field-theoretical systems. [[p. 527]]

  • Tsung-Cheng Lu, Sagar Vijay, Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface [[arXiv:2201.07792]]

    strongly interacting quantum phases of matter at zero temperature can exhibit universal patterns of long-range entanglement

Characterizing topological order

On characterizing anyon braiding / modular transformations on topologically ordered ground states by analysis of (topological) entanglement entropy of subregions:

Relation to irreducible correlation:

Simulation and experiment

Experimental observation:

Detection of long-range entanglement entropy in quantum simulations on quantum computers:

exposition in:

Anyonic topological order in terms of braided fusion categories

Claim and status

In condensed matter theory it is folklore that species of anyonic topological order correspond to braided unitary fusion categories/modular tensor categories.

The origin of the claim is:

Early accounts re-stating this claim (without attribution):

Further discussion (mostly review and mostly without attribution):

Emphasis that the expected description of anyons by braided fusion categories had remained folklore, together with a list of minimal assumptions that would need to be shown:

An argument that the statement at least for SU(2)-anyons does follow from an enhancement of the K-theory classification of topological phases of matter to interacting topological order:

In string/M-theory

Arguments realizing such anyonic topological order in the worldvolume-field theory on M5-branes:

Via KK-compactification on closed 3-manifolds (Seifert manifolds) analogous to the 3d-3d correspondence (which instead uses hyperbolic 3-manifolds):

Via 3-brane defects:

Further discussion

Relation to ZX-calculus:

On detection of topological order by observing modular transformations on the ground state:

See also:

Anyonic topological order on tori

The theory of anyonic topologically ordered quantum materials is often discussed assuming periodic boundary conditions, making the space of positions of a given anyon a torus. (While this is a dubious assumption for position-space anyons in actual experiment, the intended ground state-degeracy crucially depends on this assumption.)

In fact, the notion of topological order was introduced already assuming torus-shaped materials:

Further discussion along these lines:

Explicit discussion of anyons on tori:

But anyonic states may alternatively be localized in more abstract spaces. Anyons localized not in position space but in “reciprocal momentum space”, namely on the Brillouin torus of quasi-momenta of electrons in a crystal, are considered in

Also proposals to classify (free or interacting) topological phases of matter by topological quantum field theory mean to consider them on all base space topologies, including tori. This idea may originate around:

  • Anton Kapustin: Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology [spire:1283873, arXiv:1403.1467]

    “Our basic assumption is that a gapped state of matter with short-range interactions can be put on a curved space-time of arbitrary topology […] At short distances a system is usually defined on a regular lattice, with short-range interactions. However, if we allow for disorder, then dislocations in the lattice are possible, and more general triangulations also become possible”

  • Anton Kapustin: Bosonic Topological Insulators and Paramagnets: a view from cobordisms [spire:1292830, arXiv:1404.6659]

    “SPT phases are usually defined on a spatial lattice, while time may or may not be discretized. In the effective action approach we want to allow space-time to have an arbitrary topology, thus we discretize both space and time and regard the system as being defined on a general triangulation KK of a dd-dimensional manifold XX.”

and is implicit also in the proposal of classification via invertible field theory of:

The broad idea that TQFT is the right language to speak about anyonic topological order is now often stated as if self-evident, e.g. in:

Critical commentary on the assumption of non-trivial topology in position space appears in the following (whose authors then suggest that using extended TQFT may ameliorate the problem, p. 2):

  • Davide Gaiotto, Theo Johnson-Freyd: Condensations in higher categories [spire:1736539, arXiv:1905.09566]

    “This relationship between gapped condensed matter systems and TQFTs is perplexing, particularly so if one takes a “global” approach to TQFTs, defining them à la Atiyah 1988 in terms of partition functions attached to non-trivial Euclidean space-time manifolds and spaces of states attached to non-trivial space manifolds. From that perspective, matching a given lattice system to a TQFT would require identifying a lot of extra structure to be added to the definition of the lattice system in order to define it on discretizations of non-trivial space manifolds and to define adiabatic evolutions analogous to non-trivial space-time manifolds.”

    “If one takes the information-theoretic perspective that a phase of matter is fully characterized by the local entanglement properties of the ground-state wavefunction of the system, with no reference to a time evolution, then even more work may be needed.”

    “These concerns are not just abstract. Given some phase of matter, in the lab or in a computer, it is hard to extract the data which would pin down the corresponding TQFT, or even know if the TQFT exists. For example, we can hardly place a three-dimensional material on a non-trivial space-manifold. We can only try to simulate that by employing judicious collections of defects in flat space.”

Further resources

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

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

Last revised on May 14, 2025 at 09:06:26. See the history of this page for a list of all contributions to it.

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