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In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons.[1] In general, the operation of exchanging two identical particles, although it may cause a global phase shift, cannot affect observables. Anyons are generally classified as abelian or non-abelian. Abelian anyons (detected by two experiments in 2020)[1] play a major role in the fractional quantum Hall effect. Non-abelian anyons have not been definitively detected, although this is an active area of research.


The statistical mechanics of large many-body systems obeys laws described by Maxwell–Boltzmann statistics. Quantum statistics is more complicated because of the different behaviors of two different kinds of particles called fermions and bosons. Quoting a recent, simple description from Aalto University:[2]

In the three-dimensional world we live in, there are only two types of particles: "fermions," which repel each other, and "bosons," which like to stick together. A commonly known fermion is the electron, which transports electricity; and a commonly known boson is the photon, which carries light. In the two-dimensional world, however, there is another type of particle, the anyon, which doesn't behave like either a fermion or a boson.

In a two-dimensional world, two identical anyons change their wavefunction when they swap places in ways that can't happen in three-dimensional physics:[3] two dimensions, exchanging identical particles twice is not equivalent to leaving them alone. The particles' wavefunction after swapping places twice may differ from the original one; particles with such unusual exchange statistics are known as anyons. By contrast, in three dimensions, exchanging particles twice cannot change their wavefunction, leaving us with only two possibilities: bosons, whose wavefunction remains the same even after a single exchange, and fermions, whose exchange only changes the sign of their wavefunction.

This process of exchanging identical particles, or of circling one particle around another, is referred to by its mathematical name as "braiding." "Braiding" two anyons creates a historical record of the event, as their changed wave functions "count" the number of braids.[4]

Microsoft has invested in research concerning anyons as a potential basis for topological quantum computing. Anyons circling each other ("braiding") would encode information in a more robust way than other potential quantum computing technologies.[5] Most investment in quantum computing, however, is based on methods that do not use anyons.[5]

Abelian anyons[edit]

In quantum mechanics, and some classical stochastic systems, indistinguishable particles have the property that exchanging the states of particle i with particle j (symbolically ) does not lead to a measurably different many-body state.

In a quantum mechanical system, for example, a system with two indistinguishable particles, with particle 1 in state and particle 2 in state , has state in Dirac notation. Now suppose we exchange the states of the two particles, then the state of the system would be . These two states should not have a measurable difference, so they should be the same vector, up to a phase factor:

Here, is the phase factor. In space of three or more dimensions, the phase factor is or . Thus, elementary particles are either fermions, whose phase factor is , or bosons, whose phase factor is . These two types have different statistical behaviour. Fermions obey Fermi–Dirac statistics, while bosons obey Bose–Einstein statistics. In particular, the phase factor is why fermions obey the Pauli exclusion principle: If two fermions are in the same state, then we have

The state vector must be zero, which means it is not normalizable, thus it is unphysical.

In two-dimensional systems, however, quasiparticles can be observed that obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by Jon Magne Leinaas and Jan Myrheim of the University of Oslo in 1977.[6] In the case of two particles this can be expressed as

where can be other values than just or . It is important to note that there is a slight abuse of notation in this shorthand expression, as in reality this wave function can be and usually is multi-valued. This expression actually means that when particle 1 and particle 2 are interchanged in a process where each of them makes a counterclockwise half-revolution about the other, the two-particle system returns to its original quantum wave function except multiplied by the complex unit-norm phase factor e. Conversely, a clockwise half-revolution results in multiplying the wave function by e. Such a theory obviously only makes sense in two-dimensions, where clockwise and counterclockwise are clearly defined directions.

In the case θ = π we recover the Fermi–Dirac statistics (e = −1) and in the case θ = 0 (or θ = 2π) the Bose–Einstein statistics (e2πi = 1). In between we have something different. Frank Wilczek in 1982 explored the behavior of such quasiparticles and coined the term "anyon" to describe them, because they can have any phase when particles are interchanged.[7] Unlike bosons and fermions, anyons have the peculiar property that when they are interchanged twice in the same way (e.g. if anyon 1 and anyon 2 were revolved counterclockwise by half revolution about each other to switch places, and then they were revolved counterclockwise by half revolution about each other again to go back to their original places), the wave function is not necessarily the same but rather generally multiplied by some complex phase (by e2 in this example).

We may also use θ = 2π s with particle spin quantum number s, with s being integer for bosons, half-integer for fermions, so that


At an edge, fractional quantum Hall effect anyons are confined to move in one space dimension. Mathematical models of one-dimensional anyons provide a base of the commutation relations shown above.

In a three-dimensional position space, the fermion and boson statistics operators (−1 and +1 respectively) are just 1-dimensional representations of the permutation group (SN of N indistinguishable particles) acting on the space of wave functions. In the same way, in two-dimensional position space, the abelian anyonic statistics operators (e) are just 1-dimensional representations of the braid group (BN of N indistinguishable particles) acting on the space of wave functions. Non-abelian anyonic statistics are higher-dimensional representations of the braid group. Anyonic statistics must not be confused with parastatistics, which describes statistics of particles whose wavefunctions are higher-dimensional representations of the permutation group.[8]:22

Topological equivalence[edit]

The fact that the homotopy classes of paths (i.e. notion of equivalence on braids) are relevant hints at a more subtle insight. It arises from the Feynman path integral, in which all paths from an initial to final point in spacetime contribute with an appropriate phase factor. The Feynman path integral can be motivated from expanding the propagator using a method called time-slicing,[9] in which time is discretized.

In non-homotopic paths, one cannot get from any point at one time slice to any other point at the next time slice. This means that we can consider homotopic equivalence class of paths to have different weighting factors.[10]

So it can be seen that the topological notion of equivalence comes from a study of the Feynman path integral.[8]:28

For a more transparent way of seeing that the homotopic notion of equivalence is the "right" one to use, see Aharonov–Bohm effect.


A group of theoretical physicists working at the University of Oslo, led by Jon Leinaas and Jan Myrheim, calculated in 1977 that the traditional division between fermions and bosons would not apply to theoretical particles existing in two dimensions.[11] Such particles would be expected to exhibit a diverse range of previously unexpected properties. In 1982, Frank Wilczek published in two papers, exploring the fractional statistics of quasiparticles in two dimensions, giving them the name "anyons."[12]

Laughlin quasiparticle interferometer scanning electron micrograph of a semiconductor device. The four light-grey regions are Au/Ti gates of undepleted electrons; the blue curves are the edge channels from the equipotentials of these undepleted electrons. The dark-grey curves are etched trenches depleted of electrons, the blue dots are the tunneling junctions, the yellow dots are Ohmic contacts. The electrons in the device are confined to a 2d plane.[13]

Daniel Tsui and Horst Störmer discovered the fractional quantum Hall effect in 1982. The mathematics developed by Wilczek proved to be useful to Bertrand Halperin at Harvard University in explaining aspects of it.[14] Frank Wilczek, Dan Arovas, and Robert Schrieffer verified this statement in 1985 with an explicit calculation that predicted that particles existing in these systems are in fact anyons.[15][16]

In 2020, two teams of scientists (one in Paris, the other at Purdue) announced new experimental evidence for the existence of anyons. Both experiments were featured in Discover Magazine's 2020 annual "state of science" issue.[1]

In April, 2020, researchers from the École normale supérieure (Paris) and the Centre for Nanosciences and Nanotechnologies (C2N) reported results from a tiny "particle collider" for anyons. They detected properties that matched predictions by theory for anyons.[17][18][19]

In July, 2020, scientists at Purdue University detected anyons using a different setup. The team's interferometer routes the electrons through a specific maze-like etched nanostructure made of gallium arsenide and aluminum gallium arsenide. "In the case of our anyons the phase generated by braiding was 2π/3," he said. "That's different than what's been seen in nature before."[20][21]

Non-abelian anyons[edit]

Unsolved problem in physics:

Is topological order stable at non-zero temperature?

In 1988, Jürg Fröhlich showed that it was valid under the spin–statistics theorem for the particle exchange to be monoidal (non-abelian statistics).[22] In particular, this can be achieved when the system exhibits some degeneracy, so that multiple distinct states of the system have the same configuration of particles. Then an exchange of particles can contribute not just a phase change, but can send the system into a different state with the same particle configuration. Particle exchange then corresponds to a linear transformation on this subspace of degenerate states. When there is no degeneracy, this subspace is one-dimensional and so all such linear transformations commute (because they are just multiplications by a phase factor). When there is degeneracy and this subspace has higher dimension, then these linear transformations need not commute (just as matrix multiplication does not).

Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in the fractional quantum Hall effect (FQHE).[23][24] While at first non-abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when Alexei Kitaev showed that non-abelian anyons could be used to construct a topological quantum computer. As of 2012, no experiment has conclusively demonstrated the existence of non-abelian anyons although promising hints are emerging in the study of the ν = 5/2 FQHE state.[25][26] Experimental evidence of non-abelian anyons, although not yet conclusive and currently contested,[27] was presented in October, 2013.[28]

Fusion of anyons[edit]

In much the same way that two fermions (e.g. both of spin 1/2) can be looked at together as a composite boson (with total spin in a superposition of 0 and 1), two or more anyons together make up a composite anyon (possibly a boson or fermion). The composite anyon is said to be the result of the fusion of its components.

If identical abelian anyons each with individual statistics (that is, the system picks up a phase when two individual anyons undergo adiabatic counterclockwise exchange) all fuse together, they together have statistics . This can be seen by noting that upon counterclockwise rotation of two composite anyons about each other, there are pairs of individual anyons (one in the first composite anyon, one in the second composite anyon) that each contribute a phase . An analogous analysis applies to the fusion of non-identical abelian anyons. The statistics of the composite anyon is uniquely determined by the statistics of its components.

Non-abelian anyons have more complicated fusion relations. As a rule, in a system with non-abelian anyons, there is a composite particle whose statistics label is not uniquely determined by the statistics labels of its components, but rather exists as a quantum superposition (this is completely analogous to how two fermions known to have spin 1/2 are together in quantum superposition of total spin 1 and 0). If the overall statistics of the fusion of all of several anyons is known, there is still ambiguity in the fusion of some subsets of those anyons, and each possibility is a unique quantum state. These multiple states provide a Hilbert space on which quantum computation can be done.[29]

Topological basis[edit]

Anticlockwise rotation
Clockwise rotation
Exchange of two particles in 2 + 1 spacetime by rotation. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in 2d space).

In more than two dimensions, the spin–statistics theorem states that any multiparticle state of indistinguishable particles has to obey either Bose–Einstein or Fermi–Dirac statistics. For any d > 2, the Lie groups SO(d,1) (which generalizes the Lorentz group) and Poincaré(d,1) have Z2 as their first homotopy group. Because the cyclic group Z2 is composed of two elements, only two possibilities remain. (The details are more involved than that, but this is the crucial point.)

The situation changes in two dimensions. Here the first homotopy group of SO(2,1), and also Poincaré(2,1), is Z (infinite cyclic). This means that Spin(2,1) is not the universal cover: it is not simply connected. In detail, there are projective representations of the special orthogonal group SO(2,1) which do not arise from linear representations of SO(2,1), or of its double cover, the spin group Spin(2,1). Anyons are evenly complementary representations of spin polarization by a charged particle.

This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group SO(2) has an infinite first homotopy group.

This fact is also related to the braid groups well known in knot theory. The relation can be understood when one considers the fact that in two dimensions the group of permutations of two particles is no longer the symmetric group S2 (with two elements) but rather the braid group B2 (with an infinite number of elements). The essential point is that one braid can wind around the other one, an operation that can be performed infinitely often, and clockwise as well as counterclockwise.

A very different approach to the stability-decoherence problem in quantum computing is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.[30][31]

Higher dimensional generalization of anyons[edit]

Fractionalized excitations as point particles can be bosons, fermions or anyons in 2+1 spacetime dimensions. It is known that point particles can be only either bosons or fermions in 3+1 and higher spacetime dimensions. However, the loop (or string) or membrane like excitations are extended objects can have fractionalized statistics. Current research works show that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the key signatures for identifying 3+1 dimensional topological orders. [32] [33] [34] The multi-loop/string-braiding statistics of 3+1 dimensional topological orders can be captured by the link invariants of particular topological quantum field theories in 4 spacetime dimensions.[34] Explained in a colloquial manner, the extended objects (loop, string, or membrane, etc.) can be potentially anyonic in 3+1 and higher spacetime dimensions in the long-range entangled systems.

See also[edit]


  1. ^ a b c Ornes, Stephen (12 December 2020). "Physicists Prove Anyons Exist, a Third Type of Particle in the Universe - Physicists give us an early view of a third kingdom of quasiparticles that only arise in two dimensions". Discover. Retrieved 12 December 2020. This year brought two solid confirmations of the quasiparticles. The first arrived in April, in a paper featured on the cover of Science, from a group of researchers at the École Normale Supérieure in Paris...The second confirmation came in July, when a group at Purdue University in Indiana used an experimental setup on an etched chip that screened out interactions that might obscure the anyon behavior.
  2. ^ "Finally, anyons reveal their exotic quantum properties". Aalto University. 7 December 2018. Retrieved 24 September 2020. They were first proposed in the late 1970s, but direct experimental evidence of their quantum statistics hasn't been conclusively shown until now.
  3. ^ Shtengel, Kirilli (2007). "A home for anyon?". Nature Physics. 3 (11): 763. doi:10.1038/nphys767. Retrieved 30 November 2020. From a physicist's point of view, having two spatial dimensions is special: a pair of particles trading places behave very differently in two dimensions than they do in three. In three dimensions, any two sets of paths taken by two identical particles in the process of exchanging their positions can be continuously morphed into one another. But in two dimensions, particles can wind around each other in two distinct ways, clockwise or anticlockwise. A profound consequence of this observation for quantum mechanics is that in two dimensions, exchanging identical particles twice is not equivalent to leaving them alone.
  4. ^ Yirka, Bob (10 July 2020). "Best evidence yet for existence of anyons". News. Retrieved 30 November 2020. If a fermion or a boson were dragged around another of its kind, theory suggests, the action would not produce a record of what had occurred. But because anyons alter wave functions, they would create such a record.
  5. ^ a b Castelvecchi, Davide (3 July 2020). "Welcome anyons! Physicists find best evidence yet for long-sought 2D structures". Nature. 583 (7815): 176–177. Bibcode:2020Natur.583..176C. doi:10.1038/d41586-020-01988-0. PMID 32620884. S2CID 220336025. Retrieved 23 September 2020. Simon and others have developed elaborate theories that use anyons as the platform for quantum computers. Pairs of the quasiparticle could encode information in their memory of how they have circled around one another. And because the fractional statistics is 'topological' — it depends on the number of times one anyon went around another, and not on slight changes to its path – it is unaffected by tiny perturbations. This robustness could make topological quantum computers easier to scale up than are current quantum-computing technologies, which are error-prone.
  6. ^ Leinaas, Jon Magne; Myrheim, Jan (11 January 1977). "On the theory of identical particles" (PDF). Il Nuovo Cimento B. 37 (1): 1–23. Bibcode:1977NCimB..37....1L. doi:10.1007/BF02727953. S2CID 117277704.
  7. ^ Wilczek, Frank (4 October 1982). "Quantum Mechanics of Fractional-Spin Particles" (PDF). Physical Review Letters. 49 (14): 957–959. Bibcode:1982PhRvL..49..957W. doi:10.1103/PhysRevLett.49.957. If there is a generalized spin-statistics connection, we must expect that the flux-tube-particle composites have unusual statistics, interpolating between bosons and fermions. Since interchange of two of these particles can give any phase, I will call them generically anyons.
  8. ^ a b Khare, Avinash (2005). Fractional Statistics and Quantum Theory. World Scientific. ISBN 978-981-256-160-2.
  9. ^ Lancaster, Tom; Blundell, Stephen J. (17 June 2014). Quantum Field Theory for the Gifted Amateur. Oxford University Press. ISBN 978-0-19-969932-2.
  10. ^ Schulman, L. S. (February 1981). Techniques and Applications of Path Integration. Dover Publications. ISBN 0-471-76450-7.
  11. ^ Wilczek, Frank (January 2006). "From electronics to anyonics". Physics World. 19: 22–23. doi:10.1088/2058-7058/19/1/31. ISSN 0953-8585. In the early 1980s I named the hypothetical new particles 'anyons,' the idea being that anything goes – but I did not lose much sleep anticipating their discovery. Very soon afterwards, however, Bert Halperin at Harvard University found the concept of anyons useful in understanding certain aspects of the fractional quantum Hall effect, which describes the modifications that take place in electronics at low temperatures in strong magnetic fields.
  12. ^ "Anyons, anyone?". Symmetry Magazine. 31 August 2011. Retrieved 24 September 2020. In 1982 physicist Frank Wilczek gave these interstitial particles the name anyon...'Any anyon can be anything between a boson or a fermion,' Keilmann says. 'Wilczek is a funny guy.'
  13. ^ Camino, Fernando E.; Zhou, Wei; Goldman, Vladimir J. (17 August 2005). "Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics" (PDF). Physical Review B. 72 (7): 075342. arXiv:cond-mat/0502406. Bibcode:2005PhRvB..72g5342C. doi:10.1103/PhysRevB.72.075342. S2CID 52245802. Archived from the original (PDF) on 19 June 2015., see fig. 2.B
  14. ^ Halperin, B. I. (1984). "Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States". Phys. Rev. Lett. American Physical Society. 52 (18): 1583–1586. Bibcode:1984PhRvL..52.1583H. doi:10.1103/PhysRevLett.52.1583. The appearance of fractional statistics in the present context is strongly reminiscent of the fractional statistics introduced by Wilczek to describe charged particles tied to "magnetic flux tubes" in two dimensions.
  15. ^ Khurana, Anil (7 December 2018). "Bosons Condense and Fermions 'Exclude', But Anyons...?". Physics Today. doi:10.1063/1.2811205. Retrieved 26 November 2020. In 1984, two years after Wilczek discussed this seemingly arcane possibility, Bertrand Halperin (Harvard University) suggested that the excitations in the theory of fractional quantum Hall effect discussed by Robert Laughlin (Stanford University) behave like anyons. Later Wilczek, Daniel Arovas (University of California, San Diego) and Robert Schrieffer (University of California, Santa Barbara) confirmed the idea.
  16. ^ "Fractional Statistics and the Quantum Hall Effect" (D. Arovas and J. R. Schrieffer and F. Wilczek, 1984) Phys. Rev. Lett., 53, 722
  17. ^ H. Bartolomei, M. Kumar, R. Bisognin et al. Fractional statistics in anyon collisions // Science, 10 April 2020: Vol. 368, Issue 6487, pp. 173–177
  18. ^ Yirka, Bob (10 April 2020). "Anyon evidence observed using tiny anyon collider". Retrieved 12 December 2020. The work involved creating a very tiny 2-D anyon collider—so small they had to use an electron microscope to observe the action inside of it. The collider consisted of a 2-D plane set between another layered material. More specifically, the collider held a quantum Hall liquid that was kept inside of a strong magnetic field.
  19. ^ Najjar, Dana (12 May 2020). "'Milestone' Evidence for Anyons, a Third Kingdom of Particles". Quanta Magazine. Retrieved 12 December 2020. In 2016, three physicists described an experimental setup that resembles a tiny particle collider in two dimensions. Fève and his colleagues built something similar and used it to smash anyons together. By measuring the fluctuations of the currents in the collider, they were able to show that the behavior of the anyons corresponds exactly with theoretical predictions.
  20. ^ Tally, Steve (4 September 2020). "New evidence that the quantum world is even stranger than we thought". One characteristic difference between fermions and bosons is how the particles act when they are looped, or braided, around each other. Fermions respond in one straightforward way, and bosons in another expected and straightforward way. Anyons respond as if they have a fractional charge, and even more interestingly, create a nontrivial phase change as they braid around one another. This can give the anyons a type of "memory" of their interaction.
  21. ^ Nakamura, J.; Liang, S.; Gardner, G. C.; Manfra, M. J. (September 2020). "Direct observation of anyonic braiding statistics". Nature Physics. 16 (9): 931–936. Bibcode:2020NatPh..16..931N. doi:10.1038/s41567-020-1019-1. ISSN 1745-2481.
  22. ^ Fröhlich, Jürg (1988). "Statistics of Fields, the Yang–Baxter Equation, and the Theory of Knots and Links". Nonperturbative Quantum Field Theory. NATO ASI Series. 185. New York: Springer. pp. 71–100. doi:10.1007/978-1-4613-0729-7_4. ISBN 1-4612-8053-2.
  23. ^ Moore, Gregory; Read, Nicholas (19 August 1991). "Nonabelions in the fractional quantum hall effect" (PDF). Nuclear Physics B. 360 (2–3): 362–396. Bibcode:1991NuPhB.360..362M. doi:10.1016/0550-3213(91)90407-O.
  24. ^ Wen, Xiao-Gang (11 February 1991). "Non-Abelian statistics in the fractional quantum Hall states" (PDF). Phys. Rev. Lett. 66 (6): 802–5. Bibcode:1991PhRvL..66..802W. doi:10.1103/PhysRevLett.66.802. PMID 10043904. Archived from the original (PDF) on 26 March 2015.
  25. ^ Stern, Ady (2010). "Non-Abelian states of matter". Nature. 464 (7286): 187–93. Bibcode:2010Natur.464..187S. doi:10.1038/nature08915. PMID 20220836. S2CID 4362827.
  26. ^ An, Sanghun; Jiang, P.; Choi, H.; Kang, W.; Simon, S. H.; Pfeiffer, L. N.; West, K. W.; Baldwin, K. W. (15 December 2011). "Braiding of Abelian and Non-Abelian Anyons in the Fractional Quantum Hall Effect". arXiv:1112.3400 [cond-mat.mes-hall].
  27. ^ von Keyserling, Curt; Simon, S. H.; Bernd, Rosenow (2015). "Enhanced Bulk-Edge Coulomb Coupling in Fractional Fabry-Perot Interferometers". Physical Review Letters. 115 (12): 126807. arXiv:1411.4654. Bibcode:2015PhRvL.115l6807V. doi:10.1103/PhysRevLett.115.126807. PMID 26431008. S2CID 20103218.
  28. ^ R. L. Willett; C. Nayak; L. N. Pfeiffer; K. W. West (12 January 2013). "Magnetic field-tuned Aharonov–Bohm oscillations and evidence for non-Abelian anyons at ν = 5/2". Physical Review Letters. 111 (18): 186401. arXiv:1301.2639. Bibcode:2013PhRvL.111r6401W. doi:10.1103/PhysRevLett.111.186401. PMID 24237543. S2CID 22780228.
  29. ^ C. Nayak; S.H. Simon; A. Stern; M. Freedman; S. Das Sarma (28 March 2008). "Non-Abelian Anyons and Topological Quantum Computation". Reviews of Modern Physics. 80 (3): 1083–1159. arXiv:0707.1889. Bibcode:2008RvMP...80.1083N. doi:10.1103/RevModPhys.80.1083. S2CID 119628297.
  30. ^ Freedman, Michael; Alexei Kitaev; Michael Larsen; Zhenghan Wang (20 October 2002). "Topological Quantum Computation". Bulletin of the American Mathematical Society. 40 (1): 31–38. arXiv:quant-ph/0101025. doi:10.1090/S0273-0979-02-00964-3.
  31. ^ Monroe, Don (1 October 2008). "Anyons: The breakthrough quantum computing needs?". New Scientist (2676).
  32. ^ Wang, Chenjie; Levin, Michael (22 August 2014). "Braiding statistics of loop excitations in three dimensions". Physical Review Letters. 113 (8): 080403. arXiv:1403.7437. Bibcode:2014PhRvL.113h0403W. doi:10.1103/PhysRevLett.113.080403. ISSN 1079-7114. PMID 25192079. S2CID 23104804.
  33. ^ Wang, Juven; Wen, Xiao-Gang (15 January 2015). "Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory". Physical Review B. 91 (3): 035134. arXiv:1404.7854. doi:10.1103/PhysRevB.91.035134. ISSN 2469-9969. S2CID 13893760.
  34. ^ a b Putrov, Pavel; Wang, Juven; Yau, Shing-Tung (September 2017). "Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions". Annals of Physics. 384C: 254–287. arXiv:1612.09298. Bibcode:2017AnPhy.384..254P. doi:10.1016/j.aop.2017.06.019. S2CID 119578849.

Further reading[edit]