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{{Quantum book backlink|Condensed matter and solid-state physics}}

'''Topological phases of matter''' are quantum phases whose essential properties are not described by local symmetry breaking alone, but by global or topological features of the quantum state. They include systems such as the [[quantum Hall effect]], [[fractional quantum Hall effect]], [[topological order|topologically ordered]] phases, [[topological insulator]]s, [[topological superconductor]]s, and strongly correlated [[quantum spin liquid]] states.

A major theoretical language for describing such phases is '''topological quantum field theory''' ('''TQFT'''), a type of [[quantum field theory]] that computes [[topological invariant]]s. In condensed matter physics, TQFTs often appear as low-energy [[effective field theory|effective theories]] of topologically ordered states, including fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquids.
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Overview of topological phases of matter: global invariants, protected edge states, and examples such as quantum Hall systems and topological superconductors.
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== Overview ==

In ordinary phases of matter, such as crystals or magnets, phases are often classified by broken symmetries and local order parameters. Topological phases are different: their defining properties are encoded in global structures, such as ground-state degeneracy depending on spatial topology, protected boundary modes, quantized response coefficients, or exotic quasiparticle statistics.

Topological quantum field theories capture these properties because their physical predictions do not depend on local geometric details such as distances or angles. Instead, they depend on the topology of the underlying space or spacetime.

In a TQFT, [[correlation function]]s are metric-independent, meaning they remain unchanged under smooth deformations of spacetime. This makes them topological invariants rather than ordinary local dynamical observables.

== Topological quantum field theory ==

A '''topological quantum field theory''' is a quantum field theory whose observables compute topological invariants. Although invented by physicists, TQFTs have become important in mathematics, especially in [[knot theory]], [[four-manifold]] theory, [[algebraic topology]], [[moduli space]] theory, and [[algebraic geometry]].

Several major mathematical developments connected to TQFT have been recognized by Fields Medals, including work related to Donaldson theory, Jones polynomials, Witten’s quantum field theoretic interpretation of topology, and Kontsevich’s work on moduli spaces.

TQFTs are usually most interesting on curved or topologically nontrivial manifolds. Flat [[Minkowski spacetime]] is contractible, so many purely topological invariants become trivial there. For this reason, TQFTs are often applied to curved spacetimes, [[Riemann surface]]s, manifolds with boundary, and spaces with nontrivial topology.

== Topological phases in condensed matter ==

In condensed matter physics, TQFTs often arise as effective descriptions of gapped quantum phases. The microscopic material may contain electrons, spins, phonons, impurities, and lattice degrees of freedom, but at low energies the universal properties may be described by topological fields.

Important examples include:

* [[Fractional quantum Hall effect|Fractional quantum Hall states]]
* [[Topological order|Topologically ordered phases]]
* [[Quantum spin liquid]]s
* [[String-net condensation|string-net condensed states]]
* [[Topological insulator]]s
* [[Topological superconductor]]s
* [[Topological quantum computer|topological quantum computing]] systems

In these phases, the most important observables are often not local densities, but global quantities such as braiding phases, edge states, quantized conductance, and degeneracy depending on the topology of the sample.

== Metric independence ==

The central idea of a topological field theory is that its observables are independent of the metric. This means that stretching, bending, or smoothly deforming the spacetime does not change the value of the observable, provided the topology is unchanged.

This property is very different from ordinary quantum field theories, where distances, angles, masses, and local propagation strongly affect correlation functions. In a TQFT, the theory is sensitive instead to topological features such as holes, boundaries, knots, linking numbers, and cobordisms.

This makes TQFTs useful for describing robust quantum phenomena. In condensed matter systems, robustness against local perturbations is one of the key reasons topological phases are physically important.

== Schwarz-type TQFTs ==

Known topological field theories are often divided into two broad classes: '''Schwarz-type''' TQFTs and '''Witten-type''' TQFTs.

In Schwarz-type TQFTs, the action functional is explicitly independent of the metric. A simple example is the BF model, whose action in two dimensions can be written as

:<math>S=\int_M B F</math>

where ''B'' is an auxiliary field and ''F'' is a curvature two-form.

Another basic example has action

:<math>S=\int_M A\wedge dA.</math>

The best-known Schwarz-type example is [[Chern–Simons theory]], which plays a central role in knot invariants, topological phases, and the effective field theory of quantum Hall states.

== Chern–Simons theory ==

[[Chern–Simons theory]] is one of the most important topological quantum field theories. Its action depends on a gauge connection but not on a spacetime metric. This makes it naturally topological.

In mathematics, Chern–Simons theory is deeply connected to knot invariants, including the [[Jones polynomial]]. In condensed matter physics, Chern–Simons effective theories describe the universal low-energy behavior of fractional quantum Hall states.

For quantum Hall systems, the topological field theory captures quantized Hall conductance, quasiparticle braiding, and fractional statistics. These features are insensitive to microscopic details and depend instead on the topological structure of the state.

== Witten-type TQFTs ==

'''Witten-type''' TQFTs, also known as cohomological field theories, need not be explicitly metric-independent at the level of the action. Instead, metric dependence becomes physically irrelevant after applying a topological twist or using a BRST-like symmetry.

A typical Witten-type theory has a symmetry operator <math>\delta</math> satisfying

:<math>\delta^2 = 0.</math>

The observables <math>O_i</math> satisfy

:<math>\delta O_i = 0.</math>

If the stress-energy tensor is itself <math>\delta</math>-exact,

:<math>T^{\alpha\beta} = \delta G^{\alpha\beta},</math>

then expectation values of physical observables are independent of changes in the metric. This is the mechanism by which the theory becomes topological.

Witten’s 1988 topological Yang–Mills theory was a major example and connected quantum field theory to Donaldson invariants of four-manifolds.

== Topological order ==

'''Topological order''' is a kind of order beyond the conventional symmetry-breaking classification of phases. A topologically ordered state may have no local order parameter, but it can still possess long-range quantum entanglement and robust global properties.

Typical signatures of topological order include:

* Ground-state degeneracy depending on spatial topology
* Fractionalized quasiparticles
* Anyonic statistics
* Long-range entanglement
* Protected edge modes
* Robustness against local perturbations

Fractional quantum Hall states are among the most important examples. Their quasiparticles can carry fractional charge and obey anyonic exchange statistics.

== Anyons and braiding ==

In two-dimensional topological phases, quasiparticles can behave as [[anyon]]s. Unlike bosons and fermions, anyons may acquire more general phases—or even transform among multiple internal states—when exchanged or braided.

This braiding behavior is naturally described by topological quantum field theory. Since braiding depends only on the topology of particle paths, it is robust against small local disturbances.

Non-Abelian anyons are especially important for [[topological quantum computer|topological quantum computation]], where quantum information can be stored nonlocally and manipulated by braiding quasiparticles.

== Boundary states ==

Many topological phases have protected boundary or edge states. These boundary modes occur because the topology of the bulk phase differs from that of the surrounding vacuum or material.

For example, quantum Hall systems have chiral edge states, while topological insulators have conducting boundary states protected by symmetry. These boundary modes are often robust against disorder and local perturbations, as long as the relevant topological protection is not destroyed.

This relationship between a topological bulk and protected boundary excitations is known as the bulk–boundary correspondence.

== Mathematical formulation ==

Mathematically, a TQFT can be formulated as a functor from a category of cobordisms to a category of vector spaces. This idea was developed in the Atiyah–Segal axiomatic approach.

In this framework, a spatial manifold is assigned a vector space, interpreted physically as a Hilbert space. A spacetime cobordism between spatial manifolds is assigned a linear map between the corresponding vector spaces.

The gluing of manifolds corresponds to composition of linear maps. Thus, the topology of spacetime determines algebraic operations on quantum state spaces.

This formulation provides a bridge between quantum physics and topology.

== Cobordism and quantum amplitudes ==

In the cobordism picture, a spacetime manifold can be interpreted as an evolution from one spatial boundary to another. In ordinary quantum theory, time evolution is governed by a Hamiltonian. In a topological theory, the Hamiltonian may effectively vanish, so the result depends not on local time evolution but on the topology of the interpolating manifold.

For a closed manifold, the TQFT assigns a number, often interpreted as a partition function or vacuum expectation value. For a manifold with boundary, it assigns a state or linear transformation.

This viewpoint explains why TQFT is useful in both mathematical topology and condensed matter physics: it turns global geometric structures into quantum amplitudes.

== Relation to quantum computation ==

Topological phases of matter are important candidates for robust quantum information processing. In a topological quantum computer, quantum information is stored nonlocally in a topologically ordered state. Operations are performed by braiding quasiparticles, and the result depends only on the topology of the braid.

This gives a potential form of protection against local noise, because local perturbations cannot easily distinguish or corrupt globally encoded quantum information.

Systems supporting non-Abelian anyons, including certain fractional quantum Hall states and topological superconductors, are therefore of major interest.

== Importance ==

Topological phases of matter are important because they extend the classification of phases beyond symmetry breaking. They show that quantum matter can be organized by entanglement, topology, and global response rather than only by local order.

They also connect several major areas of physics and mathematics:

* Condensed matter physics
* Quantum field theory
* Knot theory
* Gauge theory
* Quantum information
* Algebraic topology
* Geometry of manifolds

This makes topological phases one of the central subjects in modern quantum physics.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

= References =
{{reflist|3}}

==Further reading==
* {{cite book | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | chapter=New invariants of three and four dimensional manifolds | title=The Mathematical Heritage of Hermann Weyl | year=1988a | volume=48 | publisher=American Mathematical Society | pages=285–299 | doi=10.1090/pspum/048/974342 | series=Proceedings of Symposia in Pure Mathematics | isbn=9780821814826 }}
* {{cite journal | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | title=Topological quantum field theories | url=http://www.math.ru.nl/~mueger/TQFT/At.pdf | year=1988b | journal=Publications Mathématiques de l'IHÉS | issue=68 | pages=175–186 | doi=10.1007/BF02698547 | volume=68 | s2cid=121647908 }}
* {{cite arXiv |last1=Gukov |first1=Sergei |last2=Kapustin |first2=Anton | title=Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories |year=2013 |eprint=1307.4793 |class=hep-th }}
* {{cite journal | last1=Linker | first1=Patrick | title=Topological Dipole Field Theory | year=2015 | journal=The Winnower | page=e144311.19292 | volume=2 | doi=10.15200/winn.144311.19292 | doi-access=free | url=https://tuprints.ulb.tu-darmstadt.de/4981/1/2658-topological-dipole-field-theory%20%281%29.pdf }}
* {{cite arXiv | last=Lurie | first=Jacob |author-link=Jacob Lurie | title=On the Classification of Topological Field Theories | year=2009 | eprint=0905.0465 | class=math.CT }}
* {{cite arXiv | last1=Schwarz | first1=Albert | author1-link=Albert Schwarz | title=Topological quantum field theories | year=2000 | eprint=hep-th/0011260 }}
* {{cite journal | last1=Segal | first1=Graeme | author1-link=Graeme Segal | title=Topological structures in string theory | year=2001 | journal=Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences | pages=1389–1398 | volume=359 | issue=1784 | doi=10.1098/rsta.2001.0841 | bibcode=2001RSPTA.359.1389S | s2cid=120834154 }}
* {{cite journal | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Super-symmetry and Morse Theory | year=1982 | journal=Journal of Differential Geometry | pages=661–692 | volume=17 | issue=4 | doi=10.4310/jdg/1214437492 | doi-access=free }}
* {{cite journal | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Topological quantum field theory | url=http://projecteuclid.org/euclid.cmp/1104161738 | year=1988a | journal=Communications in Mathematical Physics | volume=117 | issue=3 | pages=353–386 | doi=10.1007/BF01223371 | bibcode=1988CMaPh.117..353W | s2cid=43230714 }}
* {{cite journal | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Topological sigma models | year=1988b | journal=Communications in Mathematical Physics | volume=118 | issue=3 | pages=411–449 | doi=10.1007/bf01466725 | bibcode=1988CMaPh.118..411W | s2cid=34042140 }}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Topological phases of matter|1}}

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