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]].

***

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### Context
#### Topological physics
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[[!include topological physics -- contents]]
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#### Solid state physics
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[[!include solid state physics -- contents]]
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=--

# Topological order
* table of contents
{: toc}

## Idea
 {#Idea}

In [[solid state physics]], by *topological order* ([Wen 89](#Wen89),  [Wen & Niu 90](#WenNiu90), [Wen 91](#Wen91), [93](#Wen93), [95](#Wen95), [Gu & Wen 09, p. 2](#GuWen09), [Chen, Gu & Wen 10](#ChenGuWen2010)) 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

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

but not the other way around.

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 $\geq 2$-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 "[[anyon|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
 {#ViaDegenerateAnyonicGroundStates}

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 matter|topological phase]], of some or preferably all of the following phenomena ([Gu & Wen 09, p. 2](#GuWen09), called the "modern day philosopher's stone" in [Tsvelik 2014b](parafermion#Tsvelik14b)):

1. degenerate [[ground state]] ([Wen 89, p. 4](#Wen89), [Wen 95, Sec. 1.1](#Wen95));

1. [[Berry connection]] with [[non-abelian group|non-abelian]] [[holonomy]];

1. [[anyon]] excitations/defects whose [[wavefunctions]] constitute [[braid group representations]] ("[[anyon statistics|fractional statistics]]").

The original articles ([Wen 89](#Wen89),  [Wen & Niu 90](#WenNiu90), [Wen 91](#Wen91), [93](#Wen93), [95](#Wen95)) proposed to declare that [[topological phase of matter|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](#Wen91Review) (then also [Gu & Wen 09, p. 2](#GuWen09)). 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.

\begin{remark}\label{RelationBetweenDefiningConditions}
**(relation between these conditions)**
\linebreak
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 of a vector space|dimension]] $\geq 2$.

{#RelationToSRE} (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 of matter|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 of a vector bundle|rank]] $\geq 2$.

(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 a vector bundle|connection]] on the [[valence bundle]], it can have non-abelian [[holonomy]] (namely in [[unitary group|U(n)]] for $n \geq 2$) only if the bundle's rank is $\geq 2$.

Similarly, the presence of [[anyons]] broadly means that there are families of [[quantum adiabatic theorem|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 $\geq 2$.

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.

\end{remark}

### Via topological entanglement entropy
 {#ViaTopologicalEntanglementEntropy}

It is now thought that some/all of the [above](#ViaDegenerateAnyonicGroundStates) characteristics may be captured [[quantum information theory|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](#ChenGuWen2010)) witnessed by a non-vanishing *[[topological entanglement entropy]]* ([Kitaev & Preskill 2006](entanglement+entropy#KitaevPreskill06), [Levin & Wen 2006](entanglement+entropy#LevinWen06)).

Often one sees topological order being related also to (1) *strong interaction* and/or (2) *strong quantum correlation* ([Wen 91 review](#Wen91Review)) 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]]).

{#BewareOfCorrelation} Beware that the use of the term "correlation" in the context of topological order (cf. [Wen 91 review ](#Wen91Review)) 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](#ZCZW19) and generally [Luo & Luo 03, p. 3](entanglement#LuoLuo03)). 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:

\begin{tikzcd}[
  column sep=12pt,
  row sep=-2pt,
]
  \mathrlap{
    \mbox{
      \hspace{-2cm}
      topological phase with...
    }
  }
  \\
    \fbox{
      \begin{tabular}{c}
      strong/long-range
      \\
      interaction
      \end{tabular}
    }
    \ar[r, Rightarrow]
    &
    \fbox{
      \begin{tabular}{c}
      strong/long-range
      \\
      quantum correlation
      \end{tabular}
    }
    \ar[r, Leftrightarrow]
    &
    \fbox{
      \begin{tabular}{c}
        long-range
        \\
        entanglement
      \end{tabular}
    }
    \ar[r, Rightarrow]
    &
    \fbox{
      \begin{tabular}{c}
        topological
        \\
        entanglement
        \\
        entropy
      \end{tabular}
    }
    \ar[r, Leftrightarrow]
    &
    \fbox{
      \begin{tabular}{c}
      topological
      \\
      order
      \end{tabular}
    }
    \ar[r, Leftrightarrow]
    &
    \fbox{
      \begin{tabular}{c}
      anyons
      \end{tabular}
    }
\end{tikzcd}

> (diagram adapted from [SS22](#SatiSchreiber22))

The first steps in this sequence is intuitively plausible and widely expected to hold (eg. [Lu & Viyah 2022, p. 1](entanglement+entropy#LuVijay22)) but not currently derivable from first principles ([Zaanen, Liu, Sun & Schalm 2015, p. 527](entanglement+entropy#ZaanenLiuSunSchalm15)):

If the [[electromagnetic field|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
 {#Examples}

> under construction

* [[quantum Hall effect]]

* [[non-abelian anyon|non-Abelian quantum Hall state]] (see [Wikipedia](http://en.wikipedia.org/wiki/Anyon#Non-abelian_anyons))

* [[chiral spin state|chiral]] [[spin liquid]] (see [Wikipedia](http://en.wikipedia.org/wiki/Topological_order#Chiral_spin_states))

* [[Z2 topological order|$\mathbb{Z}/2$]] [[spin liquid]] (see [Wikipedia](http://en.wikipedia.org/wiki/Quantum_spin_liquid#Realizations_of_.28stable.29_RVB_states))

## Related concepts

* [[topological state of matter]],

  * [[quantum Hall effect]]

* [[TQFT]], [[Chern-Simons theory]]

* [[string-net model]]

* [[quantum computing]]

* [[modular tensor category]]

* [[entanglement]]

## Literature

### Original articles
 {#OriginalArticles}

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

* {#Wen89} [[Xiao-Gang Wen]], *Vacuum degeneracy of chiral spin states in compactified space*, Phys. Rev. B **40** (1989) 7387(R) $[$[doi:10.1103/PhysRevB.40.7387](https://doi.org/10.1103/PhysRevB.40.7387)$]$

* {#WenNiu90} [[Xiao-Gang Wen]],  Q. Niu, *Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces*, Phys. Rev. B **41** (1990) 9377 $[$[doi:10.1103/PhysRevB.41.9377](https://doi.org/10.1103/PhysRevB.41.9377)$]$

* {#Wen91} [[Xiao-Gang Wen]], *Non-Abelian statistics in the fractional quantum Hall states*, Phys. Rev. Lett. **66** (1991) 802 [[doi:10.1103/PhysRevLett.66.802](https://doi.org/10.1103/PhysRevLett.66.802), [pdf](https://xgwen.mit.edu/sites/default/files/documents/nab.pdf)]

* {#Wen93} [[Xiao-Gang Wen]], *Topological order and edge structure of $\nu = 1/2$ quantum Hall state*, Phys. Rev. Lett. **70** (1993) 355 $[$[doi:10.1103/PhysRevLett.70.355](https://doi.org/10.1103/PhysRevLett.70.355)$]$

* {#Wen95} [[Xiao-Gang Wen]], *Topological orders and Edge excitations in FQH states*, Advances in Physics **44** (1995) 405 $[$[arXiv:cond-mat/9506066v2](https://arxiv.org/abs/cond-mat/9506066v2), [doi:10.1080/00018739500101566](https://doi.org/10.1080/00018739500101566)$]$

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

* {#GuWen09} [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], p. 2 of:  *Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order*, Phys. Rev. B **80** 155131 (2009) $[$[arXiv:0903.1069](https://arxiv.org/abs/0903.1069), [doi:10.1103/PhysRevB.80.155131](https://doi.org/10.1103/PhysRevB.80.155131)$]$

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

* {#ChenGuWen2010} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], Section V of: _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B **82** 155138 (2010) $[$[arXiv:1004.3835](http://arxiv.org/abs/1004.3835), [doi:10.1103/PhysRevB.82.155138](https://doi.org/10.1103/PhysRevB.82.155138)$]$

### Review

* {#Wen91Review} [[Xiao-Gang Wen]], *Topological orders and Chern-Simons theory in strongly correlated quantum liquid*,  International Journal of Modern Physics B **05** 10 (1991)  1641-1648 $[$[doi:10.1142/S0217979291001541](https://doi.org/10.1142/S0217979291001541)$]$

* [[Jason Alicea]], [[Matthew Fisher]], [[Marcel Franz]], [[Yong-Baek Kim]],  *Strongly Interacting Topological Phases*, report on Banff workshop [15w5051](http://www.birs.ca/events/2015/5-day-workshops/15w5051) (2015) [[pdf](https://www.birs.ca/workshops/2015/15w5051/report15w5051.pdf), [[AliceaEtAl-InteractingTopPhases.pdf:file]]]

Textbook accounts:

* {#ZCZW19} [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Part III of: *[[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](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

* [[Tudor D. Stanescu]], Section 6.2 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

Further review:

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman|M. Freedman]], [[Sankar Das Sarma]], _Non-Abelian anyons and topological quantum computation_, Rev Mod Phys __80__:3 (Aug 2008) 1083–1159 [MR2009g:81041](http://www.ams.org/mathscinet-getitem?mr=2443722) [doi](http://dx.doi.org/10.1103/RevModPhys.80.1083)

* [[Xiao-Gang Wen]], _An introduction of topological order_ 2009 ([pdf slides](http://dao.mit.edu/~wen/talks/09QHtop.pd), [article](http://dao.mit.edu/~wen/topartS3.pdf))

* Michel Fruchart, David Carpentier, _An Introduction to Topological Insulators_, Comptes Rendus Physique 14 (2013) 779-815 ([arXiv:1310.0255](http://arxiv.org/abs/1310.0255))

* [[Philip Ball]], *Making the world from topological order*, National Science Review **6** 2 (2019) 227–230 $[$[doi:10.1093/nsr/nwy116](https://doi.org/10.1093/nsr/nwy116)$]$

  > (an interview with [[Xiao-Gang Wen]])

See also:

* Wikipedia: [topological order](http://en.wikipedia.org/wiki/Topological_order)

### Early discovery articles

* [[Xiao-Gang Wen]], [_Vacuum Degeneracy of Chiral Spin State in Compactified Spaces_](http://dao.mit.edu/~wen/pub/dgn.pdf), Phys. Rev. B, 40, 7387 (1989).

* [[Xiao-Gang Wen]], [_Topological Orders in Rigid States_](http://dao.mit.edu/~wen/pub/topo.pdf), Int. J. Mod. Phys. B4, 239 (1990)

* [[Xiao-Gang Wen]] and Qian Niu,  [_Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces_](http://dao.mit.edu/~wen/pub/topWN.pdf), Phys. Rev. B41, 9377 (1990)

* E. Keski-Vakkuri and [[Xiao-Gang Wen]], [Ground state structure of hierarchical QH states on torus and modular transformation](http://dao.mit.edu/~wen/pub/kw.pdf)
Int. J. Mod. Phys. B7, 4227 (1993).

### Topologically ordered materials

On [[topological semimetals]] with degenerate ground states:

* Huaiming Guo, Yongfei Jia, *Interaction-driven phases in a Dirac Semimetal: Exact Diagonalization Results*, J. Phys.: Condens. Matter **26** 475601 (2014) [[arXiv:1402.4274](https://arxiv.org/abs/1402.4274), [doi:10.1088/0953-8984/26/47/475601](https://doi.org/10.1088/0953-8984/26/47/475601)]

On [[topological semimetals]] which gap to topologically ordered phases:

* Jinmin Yi, Xuzhe Ying, Lei Gioia, A. A. Burkov, *Topological order in interacting semimetals*, Phys. Rev. B **107** 115147 (2023) [[arXi:2301.03628](https://arxiv.org/abs/2301.03628), [doi:10.1103/PhysRevB.107.115147](https://doi.org/10.1103/PhysRevB.107.115147)]

### Other articles

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

* [[Zhuan Li]], [[Roger S. K. Mong]], *Detecting topological order from modular transformations of ground states on the torus*, Phys. Rev. B **106** (2022) 235115 [[doi:10.1103/PhysRevB.106.235115](https://doi.org/10.1103/PhysRevB.106.235115), [arXiv:2203.04329](https://arxiv.org/abs/2203.04329)]

See also:

* [[Davide Gaiotto]], [[Anton Kapustin]], _Spin TQFTs and fermionic phases of matter_, [arxiv/1505.05856](http://arxiv.org/abs/1505.05856)

* [[Nicholas Read]], [[Subir Sachdev]], _Large-$N$ expansion for frustrated quantum antiferromagnets_, Phys. Rev. Lett. 66 1773 (1991) (on $Z_2$ 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 $Z_2$ topological order)

* [[Xiao-Gang Wen]], [Non-Abelian Statistics in the FQH states](http://dao.mit.edu/~wen/pub/nab.pdf)
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_](http://dao.mit.edu/~wen/pub/nabdw.pdf)
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](http://arxiv.org/abs/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](http://arxiv.org/abs/1210.6672)

* {#WilczekZee1984} [[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](https://doi.org/10.1103/PhysRevLett.52.2111)$]$

* {#JamadagniWeimer20} Amit Jamadagni, Hendrik Weimer, _An Operational Definition of Topological Order_ ([arXiv:2005.06501](https://arxiv.org/abs/2005.06501))

* Nathanan Tantivasadakarn, Ashvin Vishwanath, Ruben Verresen, *A hierarchy of topological order from finite-depth unitaries, measurement and feedforward* [[arXiv:2209.06202](https://arxiv.org/abs/2209.06202)]

Discussion via [[twisted equivariant K-theory]]:

* {#SatiSchreiber22} [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Anyonic topological order in TED K-theory]]* [[arXiv:2206.13563](https://arxiv.org/abs/2206.13563)]

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](https://arxiv.org/abs/2310.17740)]

[[!include topological entanglement entropy -- references]]

[[!include anyonic topological order via braided fusion categories -- references]]

### Further resources

**Research groups:**

* Univ. of Maryland, [joint quantum institute](http://jqi.umd.edu/research/topological-matter)

* [Topology and correlation in condensed matter](http://tccm.pks.mpg.de) (Germany)

* [Center for topological matter](https://sites.google.com/site/ctmsrc) (Korea)

* [Microsoft Research Station Q](http://research.microsoft.com/en-us/labs/stationq) (KITP, Santa Barbara)

**Conference and seminar cycles:**

* seminar in Koeln [Topological states of matter](http://www.thp.uni-koeln.de/trebst/Lectures/2012-TopoSeminar.html)

* Topological Phases of Matter: Simons Center, June 10-14, 2013, videos [available](http://scgp.stonybrook.edu/archives/3464)

* CECAM 2013, Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations [description](http://www.cecam.org/workshop-867.html)

[[!redirects topological order]]
[[!redirects topological orders]]


topological phase of matter

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

### General
 {#GeneralIdea}

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 theorem|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 map|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 [[crystal|crystalline]] material (a prime example being [[graphene]], see also [below](#ExampleGraphene)) 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 bundle|trivial]] [[equivariant bundle|equivariant]] [[homeomorphism class]], in fact

1. a non-trivial [[equivariant homotopy class|equivariant]] [[homotopy class]] of its [[classifying map]], in fact

1. a non-vanishing [[Whitehead-generalized cohomology|generalized]] [[cohomology class]] in [[twisted equivariant K-theory|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 [[experiment|experimental]] and [[theoretical physics|theoretical]] support but which, one should admit, remains a [[conjecture]]).

The idea is that no gentle/[[quantum adiabatic theorem|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 group|crystallographic point]] symmetries,

1. *non-spatial* symmetries, including:

   1. [[CPT theorem|CPT symmetries]] like [[time-reversal symmetry]],

   1. "[[internal symmetry|internal]]" or "on-site" symmetries

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

which all [[group action|act]] on:

1. the [[Brillouin torus]] and/or

1. the [[quantum observables|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 [[quantum adiabatic theorem|adiabatic]]) [[symmetry breaking|break]] some of the symmetry, then one speaks of a *[[symmetry protected topological phase]]*. The richness of topological phases all comes from this [[symmetry protected topological phase|symmetry protection]].

For instance, without any symmetry protection the [[valence bundles]] in a [[topological insulator]]-[[phase of matter|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 "[[symmetry protected topological phase|protected]]" by [[crystallographic group|crystallographic symmetry]], then one speaks of a *[[topological crystalline insulator]]*-phase, etc.

### Topological semi-metals

More generally, there are topological phases where small [[quantum adiabatic theorem|adiabatic]] deformations have no effect, as before, but where the [[topological space]] *of possible deformations* has itself a non-trivial [[topological space|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 [[braid group|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 group|non-abelian]], and/or if the [[holonomy]] of the [[Berry connection]] around the nodal points is [[non-abelian group|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 of a vector space|dimension]] is $\geq 2$. In this case one says that the topological phase in addition exhibits *[[topological order]]* or that its ground states exhibits "[[short-range entanglement|long-range entanglement]]" (at least "no [[short-range entanglement]]").

### Topological field theory
 {#TopologicalFieldTheory}

Therefore, in this case of [[topological order]], the [[quantum adiabatic theorem|adiabatic]] [[dynamics]] of the [[ground state]] of the topological material, say as the nodal points are [[braid group|braided]] around each other, is characterized by:

1. a [[finite-dimensional vector space|finite-dimensional]] [[Hilbert space]] [[space of quantum states]]

   (the degenerate ground state),

1. a vanishing [[Hamiltonian]]

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

1. global dynamics depending (only) on the "[[topology]]" (really: the [[homotopy type]]) of the trajectory

   (such as on the [[braid group|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 field|background]] of the [[electromagnetic field|Coulomb potential]] of the [[atomic nuclei]] on the crystal sites, see also [here](Dirac+field#FreeDiracFieldInCoulombBackground)).

### Anyons

Generally, it is expected that [[codimension]]=2 [[defects]] in a topological phase  of matter exhibit such a ([[symmetry protected topological order|symmetry protected]]) *[[topological order]]* in that the [[adiabatic theorem|adiabatic]] [[braid group|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 [[AdS3-CFT2 and CS-WZW correspondence|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
 {#Examples}

### Quantum Hall effect

* [[quantum Hall effect]], [[quantum spin Hall effect]]

### Graphene
 {#ExampleGraphene}

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 topological phase|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 bundle|trivial]] over the [[complement]] of two Dirac points inside the Brillouin [[2-torus]] (a punctured torus, see [[torus#AsAHomotopyType|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](https://link.springer.com/book/10.1007/978-3-319-76596-9))

* {#Vanderbilt18} [[David Vanderbilt]],  *Berry Phases in Electronic Structure Theory -- Electric Polarization, Orbital Magnetization and Topological Insulators*, Cambridge University Press (2018) ([doi:10.1017/9781316662205](https://doi.org/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](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/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](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

Review and survey:

* Shou-cheng Zhang, _Viewpoint: Topological states of quantum matter_,  APS Physics 1, 6 (2008) [doi:10.1103/Physics.1.6](http://dx.doi.org/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](https://arxiv.org/abs/1505.03535), [doi:10.1103/RevModPhys.88.035005](https://doi.org/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](https://doi.org/10.1007/s12045-020-0955-5), [pdf](https://www.ias.ac.in/article/fulltext/reso/025/03/0431-0441))

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

* Delft Topology Course, *Online course on topology in condensed matter* (2015-) $[$[topocondmat.org](https://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](https://arxiv.org/abs/1604.07546), [ISSN:0970-4140](http://journal.library.iisc.ernet.in/index.php/iisc/article/view/4606)$]$

* {#WangZhang17} Jing Wang, [[Shou-Cheng Zhang]], *Topological states of condensed matter*, Nature Materials **16** (2017) 1062–1067 [[doi:10.1038/nmat5012](https://doi.org/10.1038/nmat5012)]

* [[Charles Zhaoxi Xiong]], *Classification and Construction of Topological Phases of Quantum Matter* [[arXiv:1906.02892](https://arxiv.org/abs/1906.02892)]

* [[Muhammad Ilyas]], *Quantum Field Theories, Topological Materials, and Topological Quantum Computing* [[arXiv:2208.09707](https://arxiv.org/abs/2208.09707)]

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](https://arxiv.org/abs/2303.14741)]

See also:

* Wikipedia, *[Topological insulator](http://en.wikipedia.org/wiki/Topological_insulator)*

* Wikipedia, *[Topological order](http://en.wikipedia.org/wiki/Topological_order)*

* [[Xiao-Gang Wen]], _Topological Orders and Edge Excitations in FQH States_, Advances in Physics 44, 405 (1995). [cond-mat/9506066](http://arxiv.org/abs/cond-mat/9506066). (for topological order)

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman|M. Freedman]], [[Sankar Das Sarma]], _Non-Abelian anyons and topological quantum computation_, Rev Mod Phys __80__:3 (Aug 2008) 1083–1159 [MR2009g:81041](http://www.ams.org/mathscinet-getitem?mr=2443722) [doi](http://dx.doi.org/10.1103/RevModPhys.80.1083) (for topological order)

* [M. Z. Hasan](http://www.princeton.edu/physics/people/display_person.xml?netid=mzhasan), C. L. Kane, _Topological insulators_, Reviews of Modern Physics __82__ (4): 3045 (2010) [arXiv:1002.3895](http://arxiv.org/abs/1002.3895) [doi](http://dx.doi.org/10.1103%2FRevModPhys.82.3045) (for topological insulator)

* C. L. Kane, _An insulator with a twist_, Nature physics __4__, May 2008, [pdf](http://www.physics.upenn.edu/~kane/pubs/p59.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](http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced-physicsprize2016.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](https://arxiv.org/abs/1705.06206), [doi:10.1090/bull/1605](https://doi.org/10.1090/bull/1605))

In the context of [[chemistry]]:

* Bradlyn et al. *Topological quantum chemistry*, Nature **547** (2017) 298–305 [[doi:10.1038/nature23268](https://doi.org/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_](http://dao.mit.edu/~wen/pub/topWN.pdf), 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](http://dao.mit.edu/~wen/pub/kw.pdf)
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](http://arxiv.org/abs/cond-mat/0611341) [doi](http://dx.doi.org/10.1103%2FPhysRevB.76.045302); _Superconducting proximity effect and Majorana fermions at the surface of a topological insulator_, Phys. Rev. Lett. __100__: 096407, [arXiv:0707.1692](http://arxiv.org/abs/0707.1692) [doi](http://dx.doi.org/10.1103%2FPhysRevLett.100.096407) (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](http://dx.doi.org/10.1126/science.1133734) (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](http://arxiv.org/abs/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](http://arxiv.org/abs/1106.4772); A short version in Science __338__, 1604-1606 (2012) [pdf](http://dao.mit.edu/~wen/pub/dDSPTsht.pdf) (uses [[group cohomology]] of the symmetry group to classify gapped interacting bosonic states  with symmetry and trivial topological order)

* {#FreedMoore12} [[Daniel Freed]], [[Gregory Moore]], _Twisted equivariant matter_, [arxiv/1208.5055](http://arxiv.org/abs/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](https://arxiv.org/abs/1405.7689), [doi:10.1103/PhysRevLett.114.031601](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.031601)]

* {#Freed14} [[Daniel Freed]], _Short-range entanglement and invertible field theories_ ([arXiv:1406.7278](http://arxiv.org/abs/1406.7278))

* Guo Chuan Thiang, _On the K-theoretic classification of topological phases of matter_,
[arXiv:1406.7366](http://arxiv.org/abs/1406.7366)

* [[Edward Witten]], _Fermion path integrals and topological phases_, [arxiv/1508.04715](http://arxiv.org/abs/1508.04715)

* Ping Ye, _(3+1)d anomalous twisted gauge theories with global symmetry_, [arxiv/1610.08645](https://arxiv.org/abs/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](https://arxiv.org/abs/1609.02574))

* Andreas Bauer, [[Jens Eisert]], Carolin Wille, _Towards a mathematical formalism for classifying phases of matter_ ([arXiv:1903.05413](https://arxiv.org/abs/1903.05413))

[[!include topological phases of matter via K-theory -- references]]

### Interacting topological phases
 {#ReferencesInteractingTopologicalPhases}

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](topological+order#ReferencesLongRangeEntanglementRelationToStrongInteraction) 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](https://arxiv.org/abs/0904.2197), [doi:10.1103/PhysRevB.81.134509](https://doi.org/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](http://www.birs.ca/events/2015/5-day-workshops/15w5051) (2015) [[pdf](https://www.birs.ca/workshops/2015/15w5051/report15w5051.pdf), [[AliceaEtAl-InteractingTopPhases.pdf:file]]]

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

* [Sati & Schreiber 2023](#SatiSchreiber23).

### 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](http://arxiv.org/abs/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](http://arxiv.org/abs/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](http://arxiv.org/abs/1306.5570) [doi:10.1142/S0217732314500643](http://dx.doi.org/10.1142/S0217732314500643)]

* Yosuke Kubota, _Controlled topological phases and bulk-edge correspondence_, [arxiv/1511.05314](http://arxiv.org/abs/1511.05314)

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

* {#Wang21} [[Juven Wang]], *Ultra Unification*, Phys. Rev. D **103**  (2021) 105024 [[arXiv:2012.15860](https://arxiv.org/abs/2012.15860), [doi:10.1103/PhysRevD.103.105024](https://doi.org/10.1103/PhysRevD.103.105024)]

Via [[higher gauge theory|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](https://ora.ox.ac.uk/objects/uuid:e789bf29-179b-4b79-9168-605bf8c035ba), [pdf](https://ora.ox.ac.uk/objects/uuid:e789bf29-179b-4b79-9168-605bf8c035ba/files/ddj52w5202)]

* [[Joe Huxford]], [[Steven H. Simon]], *Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview* [[arXiv:2202.08294](https://arxiv.org/abs/2202.08294)]

### Research groups

* Univ. of Maryland, [joint quantum institute](http://jqi.umd.edu/research/topological-matter)
* [Topology and correlation in condensed matter](http://tccm.pks.mpg.de) (Germany)
* [Center for topological matter](https://sites.google.com/site/ctmsrc) (Korea)
* [Microsoft Research Station Q](http://research.microsoft.com/en-us/labs/stationq) (KITP, Santa Barbara)

### Conference and seminar cycles

* seminar in Koeln [Topological states of matter](http://www.thp.uni-koeln.de/trebst/Lectures/2012-TopoSeminar.html)
* Topological Phases of Matter: Simons Center, June 10-14, 2013, videos [available](http://scgp.stonybrook.edu/archives/3464)
* A. Kitaev, _On the classification of short-range entangled states_, [video](http://scgp.stonybrook.edu/archives/7874)

* CECAM 2013, Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations [description](http://www.cecam.org/workshop-867.html)

[[!redirects topological phases of matter]]

[[!redirects topological phase]]
[[!redirects topological phases]]

[[!redirects topological state of matter]]
[[!redirects topological states of matter]]

[[!redirects topological matter]]


topological phases of matter via K-theory -- 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 field|free]] [[fermion]]* [[topological phases of matter]] ([[condensed matter physics|condensed matter]] with [[gapped Hamiltonians]], specifically [[topological insulators]]) by [[twisted K-theory|twisted]] [[equivariant K-theory|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](https://iopscience.iop.org/article/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](https://doi.org/10.1103/PhysRevB.78.195125), [arXiv:0803.2786](https://arxiv.org/abs/0803.2786)$]$

The original proposal making [[topological K-theory]] explicit:

* {#Kitaev09} [[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](https://arxiv.org/abs/0901.2686), [doi:10.1063/1.3149495](https://doi.org/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](#FreedMoore13)) 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](http://www.numdam.org/item/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](https://doi.org/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](https://iopscience.iop.org/article/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](https://arxiv.org/abs/1101.1054), [doi:10.1142/S0219887812500235](https://doi.org/10.1142/S0219887812500235)$]$

The proper [[equivariant K-theory]] formulation expected to apply also to [[topological crystalline insulators]]:

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], *Twisted equivariant matter*, Ann. Henri Poincaré **14** (2013)  1927-2023 $[$[arXiv:1208.5055](https://arxiv.org/abs/1208.5055), [doi:10.1007/s00023-013-0236-x](https://doi.org/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](https://arxiv.org/abs/1406.7366))

* [[Guo Chuan Thiang]], _Topological phases: isomorphism, homotopy and K-theory_, Int. J. Geom. Methods Mod. Phys. 12 1550098 (2015) ([arXiv:1412.4191](https://arxiv.org/abs/1412.4191))

* [[Ralph M. Kaufmann]], Dan Li, Birgit Wehefritz-Kaufmann, _Topological insulators and K-theory_ ([arXiv:1510.08001](https://arxiv.org/abs/1510.08001), [spire:1401095/](http://inspirehep.net/record/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](https://arxiv.org/abs/1701.08725), [doi:10.1103/PhysRevB.95.235425](https://doi.org/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](https://arxiv.org/abs/1811.02592), [doi:10.4310/ATMP.2021.v25.n3.a3](https://doi.org/10.4310/ATMP.2021.v25.n3.a3)$]$

* Charles Zhaoxi Xiong, _Classification and Construction of Topological Phases of Quantum Matter_ ([arXiv:1906.02892](https://arxiv.org/abs/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](https://doi.org/10.1103/PhysRevResearch.3.013052), [arXiv:2009.04486](https://arxiv.org/abs/2009.04486)$]$

* {#BHMP23} [[Agnès Beaudry]], Michael Hermele, Juan Moreno, [[Markus Pflaum]], Marvin Qi, Daniel Spiegel, *Homotopical Foundations of Parametrized Quantum Spin Systems* $[$[arXiv:2303.07431](https://arxiv.org/abs/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](https://arxiv.org/abs/1802.06694), [doi:10.1103/PhysRevB.106.165103](https://doi.org/10.1103/PhysRevB.106.165103)$]$

  > (using the [[Atiyah-Hirzebruch spectral sequence]])

Review:

* {#vanNigtevecht19} [[Sven van Nigtevecht]], *Topological phases and K-theory*, Amsterdam (2019) $[$[web](https://svann.science/publication/bachelors), [pdf](https://webspace.science.uu.nl/~0554804/publications/bachelor.pdf)$]$

Generalization to include interacting [[topological order]]:

* {#SatiSchreiber23} [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Anyonic topological order in TED K-theory]]*, Rev. Math. Phys. **35** 03 (2023) 2350001 $[$[arXiv:2206.13563](https://arxiv.org/abs/2206.13563), [doi:10.1142/S0129055X23500010](https://doi.org/10.1142/S0129055X23500010)$]$

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](https://arxiv.org/abs/1406.7329))

* [[Daniel Freed]], [[Michael Hopkins]], _Reflection positivity and invertible topological phases_ ([arXiv:1604.06527](https://arxiv.org/abs/1604.06527))

* [[Daniel Freed]], _Lectures on field theory and topology_ ([cds:2699265](https://cds.cern.ch/record/2699265))

Relation to the [[GSO projection]]:

* Justin Kaidi, Julio Parra-Martinez, [[Yuji Tachikawa]], _Topological Superconductors on Superstring Worldsheets_ ([arXiv:1911.11780](https://arxiv.org/abs/1911.11780))

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

* {#Mera20} [[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](https://arxiv.org/abs/2009.02268), [doi:10.1103/PhysRevB.102.155150](https://doi.org/10.1103/PhysRevB.102.155150)$]$

For [[quasicrystals]] via [[KK-theory]] of the [[noncommutative topology of quasiperiodicity]]:

* {#Bellissard03} Jean Bellissard, *The Noncommutative Geometry of Aperiodic Solids*, in: Geometric and Topological Methods for Quantum Field Theory, pp. 86-156 (2003) ([pdf](http://people.math.gatech.edu/~jeanbel/Publi/ncg02.pdf), [doi:10.1142/9789812705068_0002](https://doi.org/10.1142/9789812705068_0002))

* {#Ypma05} Fonger Ypma, *Quasicrystals, $C^\ast$-algebras and K-theory*, 2005 ([pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.214.4414&rep=rep1&type=pdf))

* {#Putnam10} 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](https://www.math.uvic.ca/faculty/putnam/r/0802_CMP.pdf), [doi:10.1007/s00220-009-0968-0](https://doi.org/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](https://doi.org/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](D-brane#ReferencesKTheoryDescription) in [[string theory]], allowing a dual description of the topological phases even at [[non-perturbative effects|strong coupling]] via [[AdS/CFT duality]]:

* {#RyuTakayanagi10a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Topological Insulators and Superconductors from D-branes_, Phys. Lett. B693:175-179, 2010 ([arXiv:1001.0763](https://arxiv.org/abs/1001.0763))

* Carlos Hoyos-Badajoz, Kristan Jensen, [[Andreas Karch]], _A Holographic Fractional Topological Insulator_, Phys. Rev. D82:086001, 2010 ([arXiv:1007.3253](https://arxiv.org/abs/1007.3253))

* {#RyuTakayanagi10b} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Topological Insulators and Superconductors from String Theory_, Phys. Rev. D82:086014, 2010 ([arXiv:1007.4234](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://arxiv.org/abs/1509.04676))

* Charlotte Kristjansen, Gordon W. Semenoff, _The D3-probe-D7 brane holographic fractional topological insulator_, JHEP10 (2016) 079 ([arXiv:1604.08548](https://arxiv.org/abs/1604.08548))


topological photonics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The field of _topological photonics_ deals with [[electromagnetism|electromagnetic]] phenomena (specifically the propagation of [[light]] in suitable materials) which mimic the behaviour of [[electrons]] in [[topological insulator]] materials: [[photonic crystals]].

## Related concepts

* [[photonic crystal]]

* [[topological insulator]]

* [[topological phases of matter]]

* [[topological field theory]]

* [[optics]]

## References

(For more see the references at *[[photonic crystal]]*.)

Original proposal:

* [[Duncan Haldane]], S. Raghu, *Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry*, Phys. Rev. Lett. **100** (2008) 013904 [[doi:10.1103/PhysRevLett.100.013904](https://doi.org/10.1103/PhysRevLett.100.013904)]

* Z. Wang, Y. D. Chong, J. D. Joannopoulos, M. Soljačić, _Reflection-free one-way edge modes in a gyromagnetic photonic crystal_, Phys. Rev. Lett., vol. 100, no. 1, p. 013905, 2008 ([doi:10.1103/physrevlett.100.013905](https://doi.org/10.1103/physrevlett.100.013905))

First realizations:

* Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, _Observation of unidirectional backscattering-immune topological electromagnetic states_, Nature, vol. 461, no. 7265, pp. 772–775, 2009 ([doi:10.1038/nature08293](https://doi.org/10.1038/nature08293))

* M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, et al., _Photonic Floquet topological insulators_, Nature, vol. 496, no. 7444, pp. 196–200, 2013 ([doi:10.1038/nature12066](https://doi.org/10.1038/nature12066))

Review:

* Ling Lu, John D. Joannopoulos, Marin Soljačić, _Topological photonics_, Nature Photonics **8** (2014) 821–829 [[nphoton:2014.248](https://www.nature.com/articles/nphoton.2014.248)]

* Tomoki Ozawa, et al., _Topological photonics_, Rev. Mod. Phys. 91, 015006 2019 ([doi:10.1103/RevModPhys.91.015006](https://doi.org/10.1103/RevModPhys.91.015006))

* Mordechai Segev, Miguel A. Bandres, _Topological photonics: Where do we go from here?_,  De Gruyter 2020 ([doi:10.1515/nanoph-2020-0441](https://doi.org/10.1515/nanoph-2020-0441))

* Charles Choi, _Topological Photonics: What It Is and Why We Need It_, IEEE Spectrum, Mar 2020 ([web](https://spectrum.ieee.org/semiconductors/optoelectronics/topological-photonics-what-it-is-why-we-need-it))


topological physics -- contents

**Topological Physics** -- Phenomena in [[physics]] controlled by the [[topology]] (often: the [[homotopy theory]]) of the [[physical system]].

General theory:

* ([[extended topological field theory|extended]]) [[topological field theory]]

* [[Chern-Simons theory]]

In [[solid state physics]]:

* [[topological state of matter]]

  * [[quantum material]]

  * [[gapped Hamiltonian]]

  * [[topological order]], [[symmetry protected trivial order]]

* [[topological insulator]]

  * [[quantum Hall effect]], [[quantum spin Hall effect]]

    [[anyons]], [[braid group statistics]]

In [[metamaterials]]:

  * [[topological photonics]] ([[light waves]])

  * [[topological phononics]] ([[sound waves]])

For [[quantum computation]]:

* [[topological quantum computation]]

In [[quantum hadrodynamics]]:

* [[Skyrmion]]

In [[quantum chromodynamics]]:

* [[instanton]]


topological property

+-- {: .rightHandSide}
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### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _topological property_ is a [[property]]/[[predicate]] of [[topological spaces]] which is [[invariant]] under _[[isomorphism]]_ of topological spaces, hence under [[homeomorphism]]: _homeomorphism invariant_.

## Related concepts

* [[uniform property]]

* [[epsilontic analysis]]

* [[homotopy invariant]]

## References

* Wikipedia, _[Topological property](https://en.wikipedia.org/wiki/Topological_property)_

[[!redirects topological properties]]

[[!redirects topological invariant]]
[[!redirects topological invariants]]

[[!redirects homeomorphism invariant]]
[[!redirects homeomorphism invariants]]


topological quantum computation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In _topological quantum computation_ one aims to make use of [[quantum systems]] described by [[topological quantum field theory]] for [[quantum computation]], the idea being that the defining invariance of [[TQFTs]] under small deformations implements an intrinsic *fault tolerance* of the quantum computer against [[noise]] and [[decoherence]] (see also at *[[quantum error correction]]*).

### On Anyons in 2+1 d

The standard paradigm for potentially realizing topological quantum computation in practice ([Kitaev 03](#Kitaev03), [Freedman, Kitaev, Larsen & Wang03](#FreedmanKitaevLarsenWang03))
considers [[adiabatic quantum computation|adiabatic]] [[braiding]] of [defect anyons](braid+group+statistics#AsBraidingOfDefects) in effectively 2-dimensional [[quantum materials]], such as in the [[quantum Hall effect]] and [[effective field theory|effectively described]] by some kind of [[Chern-Simons theory]]/[[Reshetikhin-Turaev construction|Reshetikhin-Turaev theory]]:

\begin{imagefromfile}
    "file_name": "TQCOnAnyonsIdea.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": -10,
        "right": 0,
        "left": 10
    },
    "caption": "Adapted from [Rowell & Wang 18](#RowellWang18), [Rouabah 20](#Rouabah20)"
\end{imagefromfile}

\linebreak

Here topological [[quantum gates]] are encoded by [[braid group]]-elements
and are executed by [[actions]] through [[braid representations]] on the [[space of quantum states]]:

\begin{imagefromfile}
    "file_name": "QuantumGatesAsBraidGroupRepresentations.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": -10,
        "right": 0,
        "left": 10
    },
    "caption": "From Sati-Schreiber 21"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "QuantumBraidGateI.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -0,
        "bottom": -10,
        "right": 0,
        "left": 10
    },
    "caption": "From [Lahtinen & Pachos 17](#LahtinenPachos17)"
\end{imagefromfile}

### Extended TQC?

It is interesting to note that:

* ([[pure braid group|pure]]) [[braid group representations]] are equivalently degree-1 [[cocycles]] in the [[non-abelian cohomology]] of the [[configuration space of points]] (ordered) in the [[Euclidean plane]],;

* as such, [[braid representations]] are the first stage in a sequence that continues with [[weight systems]] on [[horizontal chord diagrams]], these being the [[complex cohomology]] in higher degree of the [[configuration space of points]] (ordered) in Euclideam 3-space (see at *[[weight systems are cohomology of loop space of configuration space]]*):

\begin{imagefromfile}
    "file_name": "BraidRepresentationsAndWeightSystemsAnalogy.jpg",
    "width": 700,
    "unit": "px",
    "margin": {
        "top": -10,
        "bottom": -10,
        "right": 0,
        "left": 10
    },
    "caption": "from Sati-Schreiber 2021"
\end{imagefromfile}

Here

* $Conf_N(\mathbb{R}^2) \simeq K( PBr(N), 1 ) \simeq B PBr(n)$ is an [[Eilenberg-MacLane space]]/[[classifying space]] for the [[pure braid group]]

* $Conf_N(\mathbb{R}^3)$ is a [[simply connected topological space|simply connected]] higher [[homotopy type]]:

This means that while every individual loop in $Conf_N(\mathbb{R}^3)$ is homotopically trivial (all "braid-gates" are equivalent) there is now non-trivial structure in higher-dimensional deformation families of braids (which is absent in $Conf_N(\mathbb{R}^2)$). Such structure would be reflected by [[extended TQFT]].

## Related concepts

* [[quantum Hall effect]]

* [[anyons]]

* [[topological order]]

* [[topological phase of matter]]

* [[quantum computation]]

  * [[adiabatic quantum computation]]

  * [[measurement-based quantum computation]]

## References
  {#References}

### Need for topological protection
  {#ReferencesNeedForTopologicalProtection}

Highlighting the need for topological stabilization mechanisms:

* [[Jay Sau]], *A Roadmap for a Scalable Topological Quantum Computer*, [Physics **10** 68 (2017)](https://physics.aps.org/articles/v10/68)

  > "small machines are unlikely to uncover truly macroscopic quantum phenomena, which have no classical analogs. This will likely require a scalable approach to quantum computation. [...] based on [...] topological quantum computation (TQC) as envisioned by [[Alexei Kitaev]] and [[Michael Freedman]] [...] The central idea of TQC is to encode [[qubits]] into states of [[topological phases of matter]]. Qubits encoded in such states are expected to be topologically protected, or robust, against the 'prying eyes' of the environment, which are believed to be the bane of conventional quantum computation."

* {#DasSarma22} [[Sankar Das Sarma]], *Quantum computing has a hype problem*, [MIT Technology Review (March 2022)](https://www.technologyreview.com/2022/03/28/1048355/quantum-computing-has-a-hype-problem/)

  > "The qubit systems we have today are a tremendous scientific achievement, but they take us no closer to having a quantum computer that can solve a problem that anybody cares about. $[\cdots]$ What is missing is the breakthrough $[\cdots]$ bypassing [[quantum error correction]] by using far-more-stable qubits, in an approach called topological quantum computing."

[[!include topological quantum computation with anyons -- references]]

[[!include anyons in the quantum Hall effect -- references]]

[[!include anyons in topological superconductors -- references]]

[[!redirects topological quantum computing]]

[[!redirects topological quantum computation]]

[[!redirects topological quantum computer]]
[[!redirects topological quantum computers]]


topological quantum computation with anyons -- references

### Topological quantum computation with anyons
 {#TopologicalQuantumComputationWithAnyons}

The idea of [[topological quantum computation]] via a [[Chern-Simons theory]] with [[anyon]] [[braiding]] [[defects]] is due to:

* {#Kitaev03} [[Alexei Kitaev]], _Fault-tolerant quantum computation by anyons_,  Annals Phys. 303 (2003) 2-30 $[$[arXiv:quant-ph/9707021](https://arxiv.org/abs/quant-ph/9707021), doi:10.1016/S0003-4916(02)00018-0$]$

* {#Freedman98} [[Michael Freedman]], *P/NP, and the quantum field computer*, Proc. Nat. Acad. Sci. **95** 1 (1998) 98-101 $[$[doi:10.1073/pnas.95.1.9](https://doi.org/10.1073/pnas.95.1.9)$]$

* {#FreedmanKitaevLarsenWang03} [[Michael Freedman]], [[Alexei Kitaev]], [[Michael Larsen]], [[Zhenghan Wang]], *Topological quantum computation*,  Bull. Amer. Math. Soc. __40__ (2003), 31-38 ([arXiv:quant-ph/0101025](https://arxiv.org/abs/quant-ph/0101025), [doi:10.1090/S0273-0979-02-00964-3](https://doi.org/10.1090/S0273-0979-02-00964-3), [pdf](http://www.ams.org/journals/bull/2003-40-01/S0273-0979-02-00964-3/S0273-0979-02-00964-3.pdf))

* {#FreedmanLarsenWang02} [[Michael Freedman]], [[Michael Larsen]], [[Zhenghan Wang]], _A modular functor which is universal for quantum computation_, Communications in Mathematical Physics. **227** 3 (2002) 605-622  $[$[arXiv:quant-ph/0001108](http://arxiv.org/abs/quant-ph/0001108), [doi:10.1007/s002200200645](https://doi.org/10.1007/s002200200645)$]$

  > (specifically via [[su(2)-anyons]])

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. **80** 1083 (2008) $[$[arXiv:0707.1888] (http://arxiv.org/abs/0707.1889), [doi:10.1103/RevModPhys.80.1083](https://doi.org/10.1103/RevModPhys.80.1083)$]$

* [[Dmitry Melnikov]], [[Andrei Mironov]], [[Sergey Mironov]], [[Alexei Morozov]], [[Andrey Morozov]], _Towards topological quantum computer_, Nucl. Phys. B926 (2018) 491-508 ([arXiv:1703.00431](https://arxiv.org/abs/1703.00431), [doi:10.1016/j.nuclphysb.2017.11.016](https://doi.org/10.1016/j.nuclphysb.2017.11.016))

and via a [[Dijkgraaf-Witten theory]] (like [[Chern-Simons theory]] but with [[discrete group|discrete]] [[gauge group]]):

* R. Walter Ogburn, [[John Preskill]], *Topological Quantum Computation*, in: *Quantum Computing and Quantum Communications*, Lecture Notes in Computer Science **1509**, Springer (1998) $[$[doi:10.1007/3-540-49208-9_31](https://doi.org/10.1007/3-540-49208-9_31)$]$

* [[Carlos Mochon]], *Anyons from non-solvable finite groups are sufficient for universal quantum computation*, Phys. Rev. A **67** 022315 (2003) $[$[arXiv:quant-ph/0206128](https://arxiv.org/abs/quant-ph/0206128), [doi:10.1103/PhysRevA.67.022315](https://doi.org/10.1103/PhysRevA.67.022315)$]$

* [[Carlos Mochon]], *Anyon computers with smaller groups*, Phys. Rev. A **69** 032306 (2004) $[$[arXiv:quant-ph/0306063](https://arxiv.org/abs/quant-ph/0306063), [doi:10.1103/PhysRevA.69.032306](https://doi.org/10.1103/PhysRevA.69.032306)$]$

Textbook accounts:

* [[Zhenghan Wang]], *Topological Quantum Computation*, CBMS Regional Conference Series in Mathematics **112**, AMS 2010 ([ISBN-13: 978-0-8218-4930-9](http://www.ams.org/publications/authors/books/postpub/cbms-112), [pdf](http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf))

* [[Jiannis K. Pachos]], *Introduction to Topological Quantum Computation*, Cambridge University Press (2012) $[$[doi:10.1017/CBO9780511792908](https://doi.org/10.1017/CBO9780511792908)$]$

* [[Tudor D. Stanescu]], Part IV of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

* {#Simon21} [[Steven H. Simon]], *Topological Quantum*, 2021 $[$[pdf](http://www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2021/TopoBook-Sep1-2021.pdf), [webpage](http://www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2021/topocourse2021.html)$]$

Review:

* [[Louis Kauffman]], *Quantum Topology and Quantum Computing*, in: [[Samuel J. Lomonaco]] (ed.), *Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium*, Proceedings of Symposia in Applied Mathematics **58**, AMS (2002) [[pdf](https://redirect.cs.umbc.edu/~lomonaco/ams/lecturenotes/Kauffman.pdf), [doi:10.1090/psapm/058](https://doi.org/10.1090/psapm/058)]

  > (in relation to [[quantum topology]])

* [[Gavin K. Brennen]], [[Jiannis K. Pachos]], *Why should anyone care about computing with anyons?*, Proc. R. Soc. A **464** (2008) 1-24 $[$[arXiv:0704.2241](https://arxiv.org/abs/0704.2241), [doi:10.1098/rspa.2007.0026](https://doi.org/10.1098/rspa.2007.0026)$]$

* [[Ady Stern]], Netanel H. Lindner, *Topological Quantum Computation -- From Basic Concepts to First Experiments*, Science **339** 6124 (2013) 1179-1184 ([doi:10.1126/science.1231473](https://science.sciencemag.org/content/339/6124/1179.full.pdf+html))

* [[Eric C. Rowell]], *An Invitation to the Mathematics of Topological Quantum Computation*, J. Phys.: Conf. Ser. 698 (2016) 012012 ([doi:10.1088/1742-6596/698/1/012012](https://iopscience.iop.org/article/10.1088/1742-6596/698/1/012012))

* [[Ananda Roy]], [[David P. DiVincenzo]], *Topological Quantum Computing*, Lecture notes of the [48th IFF Spring School](https://www.fz-juelich.de/en/pgi/expertise/schools-and-courses/iff-spring-school/the-iff-spring-school-topics-and-history/springschool2017) (2017) $[$[arXiv:1701.05052](https://arxiv.org/abs/1701.05052)$]$

* {#LahtinenPachos17} [[Ville Lahtinen]], [[Jiannis K. Pachos]], _A Short Introduction to Topological Quantum Computation_,  SciPost Phys. **3** 021 (2017) $[$[arXiv:1705.04103](https://arxiv.org/abs/1705.04103), [doi:10.21468/SciPostPhys.3.3.021](https://scipost.org/SciPostPhys.3.3.021)$]$

* {#RowellWang18} [[Eric C. Rowell]], [[Zhenghan Wang]], _Mathematics of Topological Quantum Computing_, Bull. Amer. Math. Soc. 55 (2018), 183-238 ([arXiv:1705.06206](https://arxiv.org/abs/1705.06206), [doi:10.1090/bull/1605](https://doi.org/10.1090/bull/1605))

* Bernard Field, [[Tapio Simula]], _Introduction to topological quantum computation with non-Abelian anyons_, Quantum Science and Technology **3** (2018) 045004 $[$[arXiv:1802.06176](https://arxiv.org/abs/1802.06176), [doi:10.1088/2058-9565/aacad2](https://iopscience.iop.org/article/10.1088/2058-9565/aacad2)$]$

* [[Muhammad Ilyas]], *Quantum Field Theories, Topological Materials, and Topological Quantum Computing* $[$[arXiv:2208.09707](https://arxiv.org/abs/2208.09707)$]$

* [[Eric C. Rowell]], *Braids, Motions and Topological Quantum Computing* $[$[arXiv:2208.11762](https://arxiv.org/abs/2208.11762)$]$

Focus on *[[abelian anyons]]*:

* [[Jiannis K. Pachos]], *Quantum computation with abelian anyons on the honeycomb lattice*, International Journal of Quantum Information **4** 6 (2006) 947-954 ([arXiv:quant-ph/0511273](https://arxiv.org/abs/quant-ph/0511273))

* [[James Robin Wootton]], *Dissecting Topological Quantum Computation*, 2010 ([pdf](https://etheses.whiterose.ac.uk/1163/1/main.pdf), [[WoottonDissectingTQC.pdf:file]])

  > "non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models."

* Wade Bloomquist, [[Zhenghan Wang]], *On Topological Quantum Computing With Mapping Class Group Representations*, J. Phys. A: Math. Theor. **52** (2019) 015301 ([arXiv:1805.04622](https://arxiv.org/abs/1805.04622), [doi:10.1088/1751-8121/aaeea1](https://iopscience.iop.org/article/10.1088/1751-8121/aaeea1))

* [[Seth Lloyd]], *Quantum computation with abelian anyons*, Quantum Information Processing **1** 1/2 (2002) ([arXiv:quant-ph/0004010](https://arxiv.org/abs/quant-ph/0004010), [doi:10.1023/A:1019649101654](https://doi.org/10.1023/A:1019649101654))

Realization in [[experiment]]:

* Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt,

  *Experimental Quantum Computations on a Topologically Encoded Qubit*, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 ([arXiv:1403.5426](https://arxiv.org/abs/1403.5426), [doi:10.1126/science.1253742](https://science.sciencemag.org/content/345/6194/302))

  > (for [[quantum error correction]])

Simulation of [[Ising anyons]] in a lattice of ordinary [[superconductor|superconducting]] [[qbits]]:

* T. Andersen et al. *Observation of non-Abelian exchange statistics on a superconducting processor* $[$[arXiv:2210.10255](https://arxiv.org/abs/2210.10255)$]$

[[!include braid group representations -- references]]

See also:

* Yuanjie Ren, [[Peter Shor]], *Topological quantum computation assisted by phase transitions* \[arXiv:2311.00103\]

\linebreak

### Compilation to braid gate circuits
  {#CompilationToBraidGates}

On approximating (cf. the [[Solovay-Kitaev theorem]]) given [[quantum gates]] by (i.e. compiling them to) [[quantum circuit|cicuits]] of [[anyon]] [[braid representation|braid]] [[quantum gates|gates]] (generally considered for [[su(2)-anyons]] and here mostly for universal [[Fibonacci anyons]], to some extent also for non-universal [[Majorana anyons]]):

* [[Nicholas E. Bonesteel]], [[Layla Hormozi]], [[Georgios Zikos]], [[Steven H. Simon]], *Braid Topologies for Quantum Computation*, Phys. Rev. Lett. **95** 140503 (2005) $[$[arXiv:quant-ph/0505065](https://arxiv.org/abs/quant-ph/0505065), [doi:10.1103/PhysRevLett.95.140503](https://doi.org/10.1103/PhysRevLett.95.140503)$]$

* [[Layla Hormozi]], [[Georgios Zikos]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Compiling*, Phys. Rev. B **75** 165310 (2007) $[$[arXiv:quant-ph/0610111](https://arxiv.org/abs/quant-ph/0610111), [doi:10.1103/PhysRevB.75.165310](https://doi.org/10.1103/PhysRevB.75.165310)$]$

* [[Layla Hormozi]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Computing with Read-Rezayi States*, Phys. Rev. Lett. **103** 160501 (2009) $[$[doi:10.1103/PhysRevLett.103.160501](https://doi.org/10.1103/PhysRevLett.103.160501), [arXiv:0903.2239](https://arxiv.org/abs/0903.2239)$]$

* M. Baraban, [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Resources required for topological quantum factoring*, Phys. Rev. A **81** 062317 (2010) $[$[doi:10.1103/PhysRevA.81.062317](https://doi.org/10.1103/PhysRevA.81.062317), [arXiv:1002.0537](https://arxiv.org/abs/1002.0537)$]$

  > (focus on compiling [[Shor's algorithm]])

* Vadym Kliuchnikov, Alex Bocharov, Krysta M. Svore, *Asymptotically Optimal Topological Quantum Compiling*, Phys. Rev. Lett. **112** 140504 (2014) $[$[arXiv:1310.4150](https://arxiv.org/abs/1310.4150), journal: [doi:10.1103/PhysRevLett.112.140504](https://doi.org/10.1103/PhysRevLett.112.140504), talk recording: [doi:10.48660/13100129](https://doi.org/10.48660/13100129)$]$

* [[Joren W. Brunekreef]],  *Topological Quantum Computation and Quantum Compilation*, Utrecht (2014) $[$[studenttheses.uu.nl:20.500.12932/17738](https://studenttheses.uu.nl/handle/20.500.12932/17738)$]$

* Yuan-Hang Zhang, Pei-Lin Zheng, Yi Zhang, Dong-Ling Deng, *Topological Quantum Compiling with Reinforcement Learning*, Phys. Rev. Lett. **125** 170501 (2020) $[$[doi:2004.04743](https://arxiv.org/abs/2004.04743), [doi:10.1103/PhysRevLett.125.170501](https://doi.org/10.1103/PhysRevLett.125.170501)$]$

*  [[Emil Génetay-Johansen]], [[Tapio Simula]], Section IV of: *Fibonacci anyons versus Majorana fermions -- A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models*, PRX Quantum **2** 010334 (2021) $[$[arXiv:2008.10790](https://arxiv.org/abs/2008.10790), [doi:10.1103/PRXQuantum.2.010334](https://doi.org/10.1103/PRXQuantum.2.010334)$]$

* Cheng-Qian Xu, D. L. Zhou, *Quantum teleportation using Ising anyons*, Phys. Rev. A **106** 012413 (2022) $[$[arXiv:2201.11923](https://arxiv.org/abs/2201.11923), [doi:10.1103/PhysRevA.106.012413](https://doi.org/10.1103/PhysRevA.106.012413)$]$

  > (focus on implemening the [[quantum teleportation]]-protocol with [[Ising anyons]])

Approximating all topological quantum gates by just the **weaves** among all [[braids]]:

* [[Steven H. Simon]], [[Nick E. Bonesteel]], [[Michael H. Freedman]], N. Petrovic, [[Layla Hormozi]], *Topological Quantum Computing with Only One Mobile Quasiparticle*, Phys. Rev. Lett. 96 (2006) 070503 ([arXiv:quant-ph/0509175](https://arxiv.org/abs/quant-ph/0509175), [doi:10.1103/PhysRevLett.96.070503](https://doi.org/10.1103/PhysRevLett.96.070503))

* [[Layla Hormozi]], [[Georgios Zikos]], [[Nick E. Bonesteel]], [[Steven H. Simon]], *Topological quantum compiling*, Phys. Rev. B 75, 165310 ([doi:10.1103/PhysRevB.75.165310](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.75.165310), [arXiv:quant-ph/0610111](https://arxiv.org/abs/quant-ph/0610111))

* {#Rouabah20} Mohamed Taha Rouabah, *Compiling single-qubit braiding gate for Fibonacci anyons topological quantum computation* ([arXiv:2008.03542](https://arxiv.org/abs/2008.03542))

* [Simon 2021, Sec. 11.4.1](#Simon21)


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This entry is about the article

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]]:

  \linebreak

  **Topological Quantum Field Theories from Compact Lie Groups**

  \linebreak

  in P. R. Kotiuga (ed.):

  *A celebration of the mathematical legacy of Raoul Bott*

  AMS (2010)

  [arXiv:0905.0731](http://arxiv.org/abs/0905.0731)

  [ISBN:978-0-8218-4777-0](https://bookstore.ams.org/view?ProductCode=CRMP/50)

on

1. in sections 3 and 8; a central topic in [[higher category theory and physics]]: the [[higher category theory|abstract higher categoretic]] conception of [[path integral]] [[quantization]] of classical [[action functionals]] to [[FQFT|extended quantum field theories]], realized here for finite [[higher gauge theories]] [[Dijkgraaf-Witten theory]]-type theories (see also at [[prequantum field theory]])

1. the [[extended TQFT]]-[[quantization]] of $G$-[[Chern-Simons theory]] for abelian [[Lie groups]] $G$.

More on the story of sections 3 and 8 is in

* [[Jacob Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_, talk at _Notre Dame Graduate Summer School on Topology and Field Theories_ and _Harvard lecture_ 2012 ([video part 1](http://www.youtube.com/watch?v=eQayYLDw1VA), [part 2](http://www.youtube.com/watch?v=OEShrQyvmS4), [part 3](http://www.youtube.com/watch?v=nOIcdn1iUR4) [part 4](http://www.youtube.com/watch?v=ZwnClYedaYM), [pdf lecture notes](http://www.math.northwestern.edu/~celliott/notre_dame_notes/Lurie_notes.pdf) by Chris Elliott)

#Contents#
* table of contents
{:toc}

## Preliminaries

The non-toy example application that gives the paper its title is to [[Chern-Simons theory]].

The notion of quantization discussed builds on the notion of $(\infty,n)$-categories of _families of $\infty$-groupoids_ that appears in some of the later sections of

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_ .

Together with a notion of $n$-vector spaces (the [[vertical categorification]] of [[vector space]] and [[2-vector space]]) the article sketches a general abstract formalsim making precise the notion of [[path integral]] [[quantization]] for "finite" theories such as [[Dijkgraaf-Witten theory]].

The development, sketching a rather grand picture, remains somewhat sketchy, though, possibly due to the fact that this is a conference proceedings. Also some of the ideas claimed to now be fully generalized have appeared elsewhere before. Notably the notion of the quantization map $Fam_n(C) \to C$ (see below) is effectively what [[John Baez]], [[Jim Dolan]] call in their program on [[groupoidification]] call _degroupoidification_ . The general idea underlying this, that spaces of states are computed as colimits of sections, has been made clear previously by [[Simon Willerton]] _The twisted Drinfeld double of a finite group via gerbes and finite groupoids_ ([arXiv:math/0503266](http://arxiv.org/abs/math.QA/0503266))

## The general abstract notion of quantization for discrete theories

> Here is a summary of the general quantization aspect of the article, together with some additional remarks on how to think of all this by [[Urs Schreiber|an nLab author]].

The following is the formalization of the notion of [[quantization]] for discrete theories (such as [[Dijkgraaf-Witten theory]]) as presented in the article.

Fix some $n \in \mathbb{N}$, the dimension of the [[quantum field theory]] to be described.

In

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_ .

are described the following two [[(∞,n)-category|(∞,n)-categories]]

* the [[(∞,n)-category of cobordisms]] $Bord_n$: its [[k-morphism]]s are roughly $k$-dimensional [[manifold]]s with boundaries and corners;

* for $C$ any [[(∞,n)-category]] the [[(∞,n)-category of spans]] over $C$, denoted $Fam_n(C)$,

  whose

  * [[object]]s are [[∞-groupoid]]s $P$ equipped with functors $F_P :P \to C$

  * [[morphism]]s are [[span]]s of [[∞-groupoid]]s with [[natural transformation]]s between the corresponding functors ([[bi-brane]]s)

    $$
      \array{
         && P
         \\
         & \swarrow && \searrow
         \\
         P_{in} &&\swArrow&& P_{out}
         \\
         & {}_{\mathllap{\exp(S(-))_{in}}}\searrow
         && \swarrow_{\mathrlap{\exp(S(-))_{out}}}
         \\
         && C
      }
    $$

  * "and so on".

For the application to [[quantization]] of [[sigma-model]] theories we want to be thinking of the data encoded by these $(\infty,n)$-categories as follows:

* An [[k-morphism]] $\Sigma$ in $Bord_n$ is a piece of $k$-dimensional "worldvolume" of some extended object, whose quantum dynamics we want to describe; we may roughly think of this as a [[cospan]]

  $$
    \array{
      && \Sigma
      \\
      & \nearrow && \nwarrow
      \\
      \Sigma_{in} &&&& \Sigma_{out}
     }
  $$

  where $\Sigma_{in}$ and $\Sigma_{out}$ are pieces of the boundary of $\Sigma$. We think of $\Sigma_{in}$ as the "incoming" piece of the object that we want to describe, which then experiences a self-interaction as described by the topology of $\Sigma$ and comes out in the shape of $\Sigma_{out}$ (for instace $\Sigma$ might be the three-holed sphere, $\Sigma_{in}$ the disjoint unions of two of its bounding circles and $\Sigma_{out}$ the remaining one, modelling the interaction where two strings merge to a single one).

* An [[morphism]] in $Fam_n(C)$ is to be thought of as

  * two **configuration spaces of fields** $P_{in}, P_{out}$
    of some field theory;

  * together with an [[action functional]] on it in the form of an
    higher vector bundle ("[[gerbe]]") $\exp(S_P()) : P \to C$;
    being the component of the [[natural transformation]]
    that assigns to each _path_ $P$ between field
    configuration a _phase_ ;

In these terms the **kinematics of a classical field theory** is a choice of $(\infty,n)$-functor

$$
  kin : Bord_n \to Fam_n(*)
$$

whereas the **dynamics of a classical field theory** -- the specificaton of an [[action functional]] on the given configuration spaces,  is a lift of that to $Fam_n(C)$

$$
  \array{
    && Fam_n(C)
    \\
    & {}^{\exp(S(-))}\nearrow & \downarrow
    \\
    Bord_n &\stackrel{kin}{\to}&
    Fam_n(*)
  }
$$

To illustrate this: specifically, if we consider a [[sigma-model]] [[quantum field theory]] that is induced from a **target space** geometry $X$, such that a field configuration on $\Sigma$ is a morphism $\phi : \Sigma \to X$, and with a [[gauge field|background field]] $\nabla : X \to C$, then we think of the corresponding functor

$$
  conf_X : Bord_n \to Fam_n(C)
$$

as given by homming a [[cobordism]] [[cospan]] of the form

$$
  \array{
    && \Sigma
    \\
    & \nearrow && \nwarrow
    \\
    \Sigma_{in} &&&& \Sigma_{out}
   }
$$

into $X$ to produce a [[span]] of path and configuration spaces

$$
  \array{
    && [\Sigma,X]
    \\
    & \swarrow && \searrow
    \\
    [\Sigma_{in},X] &&&& [\Sigma_{out},X]
   }
$$

equipped with the **transgressed background field** as the corresponding action functional

$$
  \array{
    && [\Sigma, X]
    \\
    & \swarrow && \searrow
    \\
    [\Sigma_{in},X] &&&& [\Sigma_{out},X]
    \\
    & \searrow && \swarrow
    \\
    && C
   }
   \,.
$$

With that in hand, the **[[quantization]]** of the given classical field theory $\exp(S(-)) : Bord_n \to Fam_n(C)$ is its "pushforward to the point", given by postcomposition with a functor

$$
  \int : Fam_n(C) \to C
$$

that over objects $\exp(S(-)) : P \to C$ is given by taking $n$-categorical [[colimit]]

$$
  (\exp(S(-)) : P \to C) \mapsto (\lim_\to \exp(S(-)))
$$

which in terms of [[coend]]-notation is indeed nicely suggestively written as

$$
  (\exp(S(-)) : P \to C) \mapsto (\int \exp(S(-)))
  \,.
$$

Taking such a colimit may be thought of as forming the space of [[section]]s of the action functional $n$-vector bundle $\exp(S(-)) : P \to C$. That this is the right general idea was maybe first amplified in

* [[Dan Freed]], _Higher Algebraic Structures and Quantization_ ([arXiv:hep-th/9212115](http://arxiv.org/abs/hep-th/9212115).

A first more categorical formulation of this is in

* [[Simon Willerton]] _The twisted Drinfeld double of a finite group via gerbes and finite groupoids_ ([arXiv:math/0503266](http://arxiv.org/abs/math.QA/0503266))

What exactly the functor $\int : Fam_n(C) \to C$ does to [[k-morphism]]s is apparently left as an exercise for the inclined reader. it requires that in $C$ limits and colimits coincide. This is the case notably for $C = Vect$.

The authors indicate in section 8 a general recursive procedure for defining higher categories of higher vector spaces, by iterating the bimodule-style definition of [[2-vector space]]s, as described there. This yields a notion $C = n Vect$, which should be the right codomain for $n$-dimensional QFTs. So we end up with a diagram

$$
  \array{
    && Fam_n(C) &\stackrel{\int}{\to}& C
    \\
    & {}^{\exp(S(-))}\nearrow & \downarrow
    \\
    Bord_n &\stackrel{kin}{\to}&
    Fam_n(*)
  }
$$

whose left bit is the kinematical and dynamical input given by a classical field theory, and whose composition to to the right is supposed to give the corresponding quantum field theory, which by the logic motivating the [[cobordism hypothesis]] is a functor $Z : Bord_n \to n Vact$:

$$
  Z_S  = \int \exp(S(-)) : Bord_n \stackrel{\exp(S(-))}{\to}
   Fam_n(n Vect) \stackrel{\int}{\to} n Vect
   \,.
$$

## 3d Chern-Simons as a fully extended TQFT
 {#3dCSFullyExtended}

A summary of the FHLT-argument about realizing 3d [[Chern-Simons theory]] as a fully [[extended TQFT]] is given in

* [[Konrad Waldorf]], _Chern-Simons theory and the categorified group ring_ ([[WaldorfTalbot2010.pdf:file]])

The following are some notes from a talk by [[Constantin Teleman]] on joint work with [[Dan Freed]], given at [ESI Program on K-Theory and Quantum Fields (2012)](http://maths-old.anu.edu.au/esi/2012/).

### CS as a fully extended TQFT

Goals

* i) describe [[Chern-Simons theory]] for compact Lie groups as an [[extended TQFT]], generated by some structure assigned to the point

* ii) relate to chiral [[WZW model]], also down to the point, have a formal framework for this

approach related to:

* [[Kevin Walker]]: 3d Chern-Simons theory is best understood as a boundary condition for a 4d theory.

* [[Chris Douglas]], [[Andre Henriques]]: _[[conformal nets]]_ and their relation to CS theory

construction is special case of [[Reshetikhin-Turaev construction]] which to a [[modular tensor category]] $\mathcal{B}$ assigns a 1-2-3 [[extended TQFT]] that assigns $\mathcal{B}$ to the [[circle]] $S^1$.

So the goal here is to extend this down to the point, to a 0-1-2-3 [[extended TQFT]], hence to an $(\infty,3)$-functor

$$
  Bord^{String}_{0,1,2,3} \to some\;symm\;mon.\;3\; category
$$

the standard choice on the right is a 3-category of (multi) [[fusion categories]], (see the reference by Douglas, Schommer-Pries and Snyder there) whose

* [[objects]] are [[fusion categories]];

* [[morphisms]] are bimodule categories;

* [[2-morphisms]]: bimodule homomorphism functors;

* [[n-morphisms|3-morphisms]]: [[natural transformations]] of these.

If this can be done, then

* $CS(*) = T$;

* $CS(S^1) = $ [[Drinfeld center]] of $T$ (again an MTC)

[[Witt group]] of [[modular tensor category]]: many abelian examples of CS give nontrivial classes

**Theorem** Given a [[modular tensor category]] $A$, there exists a [[symmetric monoidal (infinity,n)-category|symmetric monoidal 3-category]] $\mathcal{C}_A$ containing the [[fusion categories]] and a [[fully dualizable object]] in $X \in \mathcal{C}_A$ which [[cobordism hypothesis|generates]] a 0-1-2-3 [[extended TQFT]] whose 1-2-3 part agrees with the [[Reshetikhin-Turaev construction]] applied to $A$.

Here $\mathcal{C}_A$ and $X$ are formally constructed form $A$ by means of

* $X \otimes X^\vee \simeq A$

* $\mathcal{C}_A = FusionCat[X, X^\vee]$

**Remarks**

* i) This is a theory for "[[string structure]]" manifolds in the sense that the [[first Stiefel-Whitney class]] $w_1$ and the [[first Pontryagin class]] $p_1$ are trivialized, but not necessarily $w_2$.

* ii) This is the universal extension: every other one factors through it.

  guess: there is some kind of an "algebraic extension" of fusion categories in which the equation $X \otimes X^\vee = A$ can be solved for any MTC $A$.

* iii) a choice of cube root wil be needed to construct the theory for a [[framed manifold|framed]] 3-manifold

  change of framing: $n \mapsto \times \exp(\frac{2 \pi i c n }{24})$ for $c$ a "[[central charge]]"

because string [[bordism ring|bordism group]] in $dim = 3$ is $\mathbb{Z}_3$, in case of [[spin structure]]

$$
  \array{
    3 & \mathbb{Z}_{24}
    \\
    2 & \mathbb{Z}_2
    \\
    1 & \mathbb{Z}_2
  }
$$

for spin theories one would need categorical representations of this 3-groupoid, but at the moment not known.

**Theorem** ([[Kevin Walker]], in [[Jacob Lurie]]'s language)

* i) $A$ generates an invertible 4d [[extended TQFT]] for [[orientation|oriented]] manifolds;

* ii) $A$ is a valid boundary condition over itself;

The 3d boundary theory in dim 1-2-3 is equivalent to $ReshTur(A)$ after a choice of "bulking manifold"

**Comments** there is a symmetric monoidal 4-category  whose

* objects are bimodule tensor categories

* morphisms are bialgebra categories = tensor categories $T$ with braided monoidal functors

  $(B') \otimes B^{bop} \to DZ(T)$

* 2-morphisms are bimodule categories

* 3-morphisms are functors respecting the structure;

* 4-morphisms are natural transformations between these.

Invertibility of a TQFT $Z$ $\Leftrightarrow$ invertibility of $Z(*)$.

so then $Z(S^1) \simeq 1$

**Lemma** [[modular tensor categories]] are invertible in this sense

(4d anomaly theory) + (bulking manifold) = (3d standalone theory)

**Result** Have a 3d TQFT defined on the full subcategory of $Bord^String_{0-1-2-3-4}$ of manifolds which bound.

So the final step in the construction of the full 0-1-2-3-4 theory is to extend from that to the full category of bordisms.

### Boundary conditions: WZW theory

general metaphor:

* TQFT $\leftrightarrow$ algebra

* boundary condition $\leftrightarrow$ bimodule

other metaphor

* TQFT $Z$ determined by $Z(*)$

* boundary condition $\alpha : 1 \to Z(*)$ (left) or $\beta : Z(*) \to 1$ (right)

For the 3d RT theory this yields for boundary conditions $DZ(T)$-algebra categories.

**Theorem** ([[Graeme Segal]])

Chiral WZW model is a conformal boundary for CS theory

**Problem** make this work down to the point

## Related references

* [[Dan Freed]], _[[4-3-2 8-7-6]]_

category: reference