Physics:Quantum Topological phases of matter - Revision history

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	'''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, topologically ordered phases, topological insulators, topological superconductors, and strongly correlated quantum spin liquid states.		'''Topological phases of matter''' 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, topologically ordered phases, topological insulators, topological superconductors, 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 invariants. In condensed matter physics, TQFTs often appear as low-energy effective theories of topologically ordered states, including fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquids. Overview of topological phases of matter: global invariants, protected edge states, and examples such as quantum Hall systems and topological superconductors. In ordinary phases of matter, such as crystals or magnets, phases are often classified by broken symmetries and local order parameters.

	A major theoretical language for describing such phases is ~~'''~~topological quantum field theory~~'''~~ (~~'''~~TQFT~~'''~~), a type of quantum field theory that computes topological invariants. In condensed matter physics, TQFTs often appear as low-energy 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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	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.		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.		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 surfacess, manifolds with boundary, and spaces with nontrivial topology.

	== Topological phases in condensed matter ==		== Topological phases in condensed matter ==
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	Important examples include:		Important examples include:

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

	[[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.		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.		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.		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.

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	'''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.		'''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.		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.
	<div style="float:right; width:420px; background:#fff8dc; border:1px solid #e0d890; padding:6px; margin:0 0 1em 1em;">		<div style="float:right; width:420px; background:#fff8dc; border:1px solid #e0d890; padding:6px; margin:0 0 1em 1em;">
	~~[[File:Quantum_topological_phases_of_matter_infographic_yellow.jpg\|400px]]~~
	<div style="font-size:90%; text-align:center;">		<div style="font-size:90%; text-align:center;">
	Overview of topological phases of matter: global invariants, protected edge states, and examples such as quantum Hall systems and topological superconductors.		Overview of topological phases of matter: global invariants, protected edge states, and examples such as quantum Hall systems and topological superconductors.
	</div>		</div>
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			</div>

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			[[File:Quantum_topological_phases_of_matter_infographic_yellow.jpg\|thumb\|280px\|Quantum Topological phases of matter.]]
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	== Overview ==		== Overview ==

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	'''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.		'''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, topologically ordered phases, topological insulators, topological superconductors, 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.		A major theoretical language for describing such phases is '''topological quantum field theory''' ('''TQFT'''), a type of quantum field theory that computes topological invariants. In condensed matter physics, TQFTs often appear as low-energy 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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	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.		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.		In a TQFT, correlation functions 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 ==		== 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]].		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.		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.
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	== Anyons and braiding ==		== 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.		In two-dimensional topological phases, quasiparticles can behave as anyons. 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.		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.		Non-Abelian anyons are especially important for topological quantum computation, where quantum information can be stored nonlocally and manipulated by braiding quasiparticles.

	== Boundary states ==		== Boundary states ==