Quantum topology

Short description: Use of topological ideas in quantum physics

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]

Topological quantum systems are characterized by global properties that remain stable under smooth deformations.

## Contents

* 1 Overview
* 2 Topological phases
* 3 Wave functions and phase
* 4 Band topology
* 5 Quantum fields and knots
* 6 Quantum information
* 7 See also
* 8 Table of contents (185 articles)
* 9 Index
* 10 Full contents
* 11 References

# 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 (185 articles)

# Index

Core theory

1\. Foundations

2\. Conceptual and interpretations

3\. Mathematical structure and systems

4\. Atomic and spectroscopy

5\. Wavefunctions and modes

6\. Quantum dynamics and evolution

7\. Measurement and information

8\. Quantum information and computing

Applications and extensions

9\. Quantum optics and experiments

10\. Open quantum systems

11\. Quantum field theory

12\. Statistical mechanics and kinetic theory

13\. Condensed matter and solid-state physics

14\. Plasma and fusion physics

15\. Timeline

16\. Advanced and frontier topics

Book II

* Matter by scale

Book III

* Methods and tools

Book IV

* Data Analysis Techniques

# Full contents

1\. Foundations (11) ↑ Back to index

1\. Physics:Quantum basics

2\. Physics:Quantum Postulates

3\. Physics:Quantum Hilbert space

4\. Physics:Quantum Observables and operators

5\. Physics:Quantum mechanics

6\. Physics:Quantum mechanics measurements

7\. Physics:Quantum state

8\. Physics:Quantum system

9\. Physics:Quantum superposition

10\. Physics:Quantum probability

11\. Physics:Quantum Mathematical Foundations of Quantum Theory

2\. Conceptual and interpretations (14) ↑ Back to index

12\. Physics:Quantum Interpretations of quantum mechanics

13\. Physics:Quantum Wave–particle duality

14\. Physics:Quantum Complementarity principle

15\. Physics:Quantum Uncertainty principle

16\. Physics:Quantum Measurement problem

17\. Physics:Quantum Bell's theorem

18\. Physics:Quantum Hidden variable theory

19\. Physics:Quantum nonlocality

20\. Physics:Quantum contextuality

21\. Physics:Quantum Darwinism

22\. Physics:Quantum A Spooky Action at a Distance

23\. Physics:Quantum A Walk Through the Universe

24\. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

25\. Physics:Quantum measurement problem

3\. Mathematical structure and systems (13) ↑ Back to index

26\. Physics:Quantum Density matrix

27\. Physics:Quantum Exactly solvable quantum systems

28\. Physics:Quantum Formulas Collection

29\. Physics:Quantum A Matter Of Size

30\. Physics:Quantum Symmetry in quantum mechanics

31\. Physics:Quantum Angular momentum operator

32\. Physics:Quantum Runge–Lenz vector

33\. Physics:Quantum Approximation Methods

34\. Physics:Quantum Matter Elements and Particles

35\. Physics:Quantum Dirac equation

36\. Physics:Quantum Klein–Gordon equation

37\. Physics:Quantum pendulum

38\. Physics:Quantum configuration space

4\. Atomic and spectroscopy (14) ↑ Back to index

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

39\. Physics:Quantum Atomic structure and spectroscopy

40\. Physics:Quantum Hydrogen atom

41\. Physics:Quantum number

42\. Physics:Quantum Multi-electron atoms

43\. Physics:Quantum Fine structure

44\. Physics:Quantum Hyperfine structure

45\. Physics:Quantum Isotopic shift

46\. Physics:Quantum defect

47\. Physics:Quantum Zeeman effect

48\. Physics:Quantum Stark effect

49\. Physics:Quantum Spectral lines and series

50\. Physics:Quantum Selection rules

51\. Physics:Quantum Fermi's golden rule

52\. Physics:Quantum beats

5\. Wavefunctions and modes (9) ↑ Back to index

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

53\. Physics:Quantum Wavefunction

54\. Physics:Quantum Superposition principle

55\. Physics:Quantum Eigenstates and eigenvalues

56\. Physics:Quantum Boundary conditions and quantization

57\. Physics:Quantum Standing waves and modes

58\. Physics:Quantum Normal modes and field quantization

59\. Physics:Number of independent spatial modes in a spherical volume

60\. Physics:Quantum Density of states

61\. Physics:Quantum carpet

6\. Quantum dynamics and evolution (17) ↑ Back to index

62\. Physics:Quantum Time evolution

63\. Physics:Quantum Schrödinger equation

64\. Physics:Quantum Time-dependent Schrödinger equation

65\. Physics:Quantum Stationary states

66\. Physics:Quantum Perturbation theory

67\. Physics:Quantum Time-dependent perturbation theory

68\. Physics:Quantum Adiabatic theorem

69\. Physics:Quantum Scattering theory

70\. Physics:Quantum S-matrix

71\. Physics:Quantum tunnelling

72\. Physics:Quantum speed limit

73\. Physics:Quantum revival

74\. Physics:Quantum reflection

75\. Physics:Quantum oscillations

76\. Physics:Quantum jump

77\. Physics:Quantum boomerang effect

78\. Physics:Quantum chaos

7\. Measurement and information (9) ↑ Back to index

79\. Physics:Quantum Measurement theory

80\. Physics:Quantum Measurement operators

81\. Physics:Quantum Projective measurement

82\. Physics:Quantum POVM

83\. Physics:Quantum Weak measurement

84\. Physics:Quantum Measurement collapse

85\. Physics:Quantum entanglement

86\. Physics:Quantum Zeno effect

87\. Physics:Quantum limit

8\. Quantum information and computing (10) ↑ Back to index

88\. Physics:Quantum information theory

89\. Physics:Quantum Qubit

90\. Physics:Quantum Entanglement

91\. Physics:Quantum Gates and circuits

92\. Physics:Quantum Computing Algorithms in the NISQ Era

93\. Physics:Quantum Noisy Qubits

94\. Physics:Quantum random access code

95\. Physics:Quantum pseudo-telepathy

96\. Physics:Quantum network

97\. Physics:Quantum money

9\. Quantum optics and experiments (5) ↑ Back to index

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

98\. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

99\. Physics:Quantum optics beam splitter experiments

100\. Physics:Quantum Ultra fast lasers

101\. Physics:Quantum Experimental quantum physics

102\. Physics:Quantum optics

103\. Template:Quantum optics operators

10\. Open quantum systems (9) ↑ Back to index

104\. Physics:Quantum Open systems

105\. Physics:Quantum Master equation

106\. Physics:Quantum Lindblad equation

107\. Physics:Quantum Decoherence

108\. Physics:Quantum dissipation

109\. Physics:Quantum Markov semigroup

110\. Physics:Quantum Markovian dynamics

111\. Physics:Quantum Non-Markovian dynamics

112\. Physics:Quantum Trajectories

11\. Quantum field theory (20) ↑ Back to index

Structural dependency map of quantum field theory.

113\. Physics:Quantum field theory (QFT) basics

114\. Physics:Quantum field theory (QFT) core

115\. Physics:Quantum Fields and Particles

116\. Physics:Quantum Second quantization

117\. Physics:Quantum Harmonic Oscillator field modes

118\. Physics:Quantum Creation and annihilation operators

119\. Physics:Quantum vacuum fluctuations

120\. Physics:Quantum Propagators in quantum field theory

121\. Physics:Quantum Feynman diagrams

122\. Physics:Quantum Path integral formulation

123\. Physics:Quantum Renormalization in field theory

124\. Physics:Quantum Renormalization group

125\. Physics:Quantum Field Theory Gauge symmetry

126\. Physics:Quantum Non-Abelian gauge theory

127\. Physics:Quantum Electrodynamics (QED)

128\. Physics:Quantum chromodynamics (QCD)

129\. Physics:Quantum Electroweak theory

130\. Physics:Quantum Standard Model

131\. Physics:Quantum triviality

132\. Physics:Quantum confinement problem

12\. Statistical mechanics and kinetic theory (9) ↑ Back to index

133\. Physics:Quantum Statistical mechanics

134\. Physics:Quantum Partition function

135\. Physics:Quantum Distribution functions

136\. Physics:Quantum Liouville equation

137\. Physics:Quantum Kinetic theory

138\. Physics:Quantum Boltzmann equation

139\. Physics:Quantum BBGKY hierarchy

140\. Physics:Quantum Relaxation and thermalization

141\. Physics:Quantum Thermodynamics

13\. Condensed matter and solid-state physics (13) ↑ Back to index

142\. Physics:Quantum Band structure

143\. Physics:Quantum Fermi surfaces

144\. Physics:Quantum Semiconductor physics

145\. Physics:Quantum Phonons

146\. Physics:Quantum Electron-phonon interaction

147\. Physics:Quantum Superconductivity

148\. Physics:Quantum Topological phases of matter

149\. Physics:Quantum well

150\. Physics:Quantum spin liquid

151\. Physics:Quantum spin Hall effect

152\. Physics:Quantum phase transition

153\. Physics:Quantum critical point

154\. Physics:Quantum dot

14\. Plasma and fusion physics (8) ↑ Back to index

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

155\. Physics:Quantum Fusion reactions and Lawson criterion

156\. Physics:Quantum Plasma (fusion context)

157\. Physics:Quantum Magnetic confinement fusion

158\. Physics:Quantum Inertial confinement fusion

159\. Physics:Quantum Plasma instabilities and turbulence

160\. Physics:Quantum Tokamak core plasma

161\. Physics:Quantum Tokamak edge physics and recycling asymmetries

162\. Physics:Quantum Stellarator

15\. Timeline (8) ↑ Back to index

163\. Physics:Quantum mechanics/Timeline

164\. Physics:Quantum mechanics/Timeline/Pre-quantum era

165\. Physics:Quantum mechanics/Timeline/Old quantum theory

166\. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

167\. Physics:Quantum mechanics/Timeline/Quantum field theory era

168\. Physics:Quantum mechanics/Timeline/Quantum information era

169\. Physics:Quantum mechanics/Timeline/Quantum technology era

170\. Physics:Quantum mechanics/Timeline/Quiz

16\. Advanced and frontier topics (16) ↑ Back to index

171\. Physics:Quantum topology

172\. Physics:Quantum battery

173\. Physics:Quantum Supersymmetry

174\. Physics:Quantum Black hole thermodynamics

175\. Physics:Quantum Holographic principle

176\. Physics:Quantum gravity

177\. Physics:Quantum De Sitter invariant special relativity

178\. Physics:Quantum Doubly special relativity

179\. Physics:Quantum arithmetic geometry

180\. Physics:Quantum unsolved problems

181\. Physics:Quantum Yang-Mills mass gap

182\. Physics:Quantum gravity problem

183\. Physics:Quantum black hole information paradox

184\. Physics:Quantum dark matter problem

185\. Physics:Quantum neutrino mass problem

186\. Physics:Quantum matter-antimatter asymmetry problem

# References

1. ↑ 1.0 1.1 1.2 "The Nobel Prize in Physics 2016". Nobel Prize Outreach. https://www.nobelprize.org/prizes/physics/2016/summary/.
2. ↑ Schirber, Michael (7 October 2016). "Nobel Prize—Topological Phases of Matter". Physics 9: 116. doi:10.1103/Physics.9.116.
3. ↑ 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. ↑ 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. ↑ 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. ↑ 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.
7. ↑ Field, Benjamin; Simula, Tapio (2018). "Introduction to topological quantum computation with non-Abelian anyons". arXiv:1802.06176 [quant-ph].

Author: Harold Foppele

0.00

(0 votes)

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Quantum_topology&oldid=4328008"