Physics:Quantum topology: Difference between revisions
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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> | 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= | == 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> | 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" /> | 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= | == 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> | 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" /> | 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= | == 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 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" /> | 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= | == 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" /> | 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" /> | ||
Revision as of 22:11, 17 May 2026
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.[1]
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.[1]
This makes topology especially important in quantum mechanics, quantum field theory, and quantum materials. In such systems, physical effects may depend less on local details and more on the global structure of the quantum state.[2]
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.[1]
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 quantum information and fault-tolerant quantum computation.[3]
Wave functions and phase
Topology enters quantum mechanics through the phase structure of the 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.[4]
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.[4]
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.[5]
This idea is central to topological phases and to the study of topological insulators and semimetals. The boundary behavior is tied to the topology of the bulk quantum state.[5]
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.[6]
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.[6]
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.[3]
Because local disturbances cannot easily change a global topological invariant, topological protection is considered a possible route toward more stable quantum information processing.[7]
See also
Table of contents (217 articles)
Index
Full contents
References
- ↑ 1.0 1.1 1.2 "The Nobel Prize in Physics 2016". Nobel Prize Outreach. https://www.nobelprize.org/prizes/physics/2016/summary/.
- ↑ Schirber, Michael (7 October 2016). "Nobel Prize—Topological Phases of Matter". Physics 9: 116. doi:10.1103/Physics.9.116.
- ↑ 3.0 3.1 Nayak, Chetan; Simon, Steven H.; Stern, Ady; Freedman, Michael; Das Sarma, Sankar (2008). "Non-Abelian anyons and topological quantum computation". Reviews of Modern Physics 80 (3): 1083–1159. doi:10.1103/RevModPhys.80.1083.
- ↑ 4.0 4.1 Berry, M. V. (1984). "Quantal phase factors accompanying adiabatic changes". Proceedings of the Royal Society A 392 (1802): 45–57. doi:10.1098/rspa.1984.0023.
- ↑ 5.0 5.1 Hasan, M. Z.; Kane, C. L. (2010). "Colloquium: Topological insulators". Reviews of Modern Physics 82 (4): 3045–3067. doi:10.1103/RevModPhys.82.3045.
- ↑ 6.0 6.1 Witten, Edward (1989). "Quantum field theory and the Jones polynomial". Communications in Mathematical Physics 121 (3): 351–399. doi:10.1007/BF01217730.
- ↑ Field, Benjamin; Simula, Tapio (2018). "Introduction to topological quantum computation with non-Abelian anyons". arXiv:1802.06176 [quant-ph].
Source attribution: Quantum topology