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{{Short description|Use of topological ideas in quantum physics}}

{{Quantum book backlink|Advanced and frontier topics}}

'''Quantum topology''' refers to the use of topological ideas in quantum physics, especially where the global structure of a system determines physical behavior that is stable under small local changes. In quantum theory, topology appears in wave functions, phase factors, energy bands, quantum fields, and many-body states.<ref name="nobel2016">{{cite web |title=The Nobel Prize in Physics 2016 |url=https://www.nobelprize.org/prizes/physics/2016/summary/ |website=NobelPrize.org |publisher=Nobel Prize Outreach |access-date=7 May 2026}}</ref>

<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">
[[File:Quantum_topology_yellow.png|400px]]
<div style="font-size:90%;">Topological quantum systems are characterized by global properties that remain stable under smooth deformations.</div>
</div>

=Overview=

In ordinary geometry, two shapes may differ by distances, angles, or curvature. In topology, the emphasis is instead on properties that remain unchanged under continuous deformation. Quantum systems can possess similar robust features, where a phase, winding number, knot structure, band invariant, or boundary state cannot be removed without changing the underlying quantum state in a discontinuous way.<ref name="nobel2016" />

This makes topology especially important in [[Physics:Quantum mechanics|quantum mechanics]], [[Physics:Quantum field theory|quantum field theory]], and [[Physics:Quantum materials|quantum materials]]. In such systems, physical effects may depend less on local details and more on the global structure of the quantum state.<ref name="schirber2016">{{cite journal |last=Schirber |first=Michael |title=Nobel Prize—Topological Phases of Matter |journal=Physics |volume=9 |pages=116 |date=7 October 2016 |doi=10.1103/Physics.9.116}}</ref>

=Topological phases=

A '''topological phase''' is a quantum phase of matter distinguished by global invariants rather than by ordinary local order parameters. Such phases may have protected boundary states, unusual quasiparticles, or quantized transport properties.<ref name="nobel2016" />

Topological phases occur in systems such as quantum Hall states, topological insulators, topological superconductors, and certain strongly correlated many-body systems. Their robustness makes them important in studies of [[Physics:Quantum information and computing|quantum information]] and fault-tolerant quantum computation.<ref name="nayak2008">{{cite journal |last1=Nayak |first1=Chetan |last2=Simon |first2=Steven H. |last3=Stern |first3=Ady |last4=Freedman |first4=Michael |last5=Das Sarma |first5=Sankar |title=Non-Abelian anyons and topological quantum computation |journal=Reviews of Modern Physics |volume=80 |issue=3 |pages=1083–1159 |year=2008 |doi=10.1103/RevModPhys.80.1083}}</ref>

=Wave functions and phase=

Topology enters quantum mechanics through the phase structure of the [[Physics:Quantum mechanics#Wave functions|wave function]]. A quantum state may acquire a phase after being transported around a closed path. In some systems, this phase depends on the topology of the path rather than on the detailed motion along it.<ref name="berry1984">{{cite journal |last=Berry |first=M. V. |title=Quantal phase factors accompanying adiabatic changes |journal=Proceedings of the Royal Society A |volume=392 |issue=1802 |pages=45–57 |year=1984 |doi=10.1098/rspa.1984.0023}}</ref>

Examples include geometric phases, Berry phases, and the Aharonov–Bohm effect. These effects show that quantum behavior can be sensitive to global structure even when local forces appear absent.<ref name="berry1984" />

=Band topology=

In condensed-matter physics, topology is used to classify electronic energy bands. A material may have energy bands with nontrivial topological invariants, leading to protected conducting states at its edges or surfaces.<ref name="hasan2010">{{cite journal |last1=Hasan |first1=M. Z. |last2=Kane |first2=C. L. |title=Colloquium: Topological insulators |journal=Reviews of Modern Physics |volume=82 |issue=4 |pages=3045–3067 |year=2010 |doi=10.1103/RevModPhys.82.3045}}</ref>

This idea is central to [[Physics:Quantum materials/topological phase|topological phases]] and to the study of topological insulators and semimetals. The boundary behavior is tied to the topology of the bulk quantum state.<ref name="hasan2010" />

=Quantum fields and knots=

Topological ideas also appear in quantum field theory, where fields may support solitons, instantons, vortices, monopoles, and other structures classified by topological invariants. In some approaches, knots and links are related to quantum amplitudes and field-theoretic observables.<ref name="witten1989">{{cite journal |last=Witten |first=Edward |title=Quantum field theory and the Jones polynomial |journal=Communications in Mathematical Physics |volume=121 |issue=3 |pages=351–399 |year=1989 |doi=10.1007/BF01217730}}</ref>

Topological quantum field theory studies systems whose observables depend mainly on global topological features rather than on a metric geometry. This connects quantum physics with knot theory, low-dimensional topology, and mathematical physics.<ref name="witten1989" />

=Quantum information=

Topology is important in proposals for robust quantum computation. In topological quantum computing, information is encoded in global degrees of freedom that are less sensitive to local noise. Anyons and braiding operations are often used as conceptual models for such systems.<ref name="nayak2008" />

Because local disturbances cannot easily change a global topological invariant, topological protection is considered a possible route toward more stable quantum information processing.<ref name="field2018">{{cite arXiv |last1=Field |first1=Benjamin |last2=Simula |first2=Tapio |title=Introduction to topological quantum computation with non-Abelian anyons |eprint=1802.06176 |class=quant-ph |year=2018}}</ref>

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

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum topology|1}}

Physics:Quantum topology - Revision history

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