Questions tagged [anyons]
Anyons is the generic name for the particles which interchange among other according to the representation(s) of the braid group.
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Fusion 2-categories for string-like excitations: a more concrete description?
I'm familiar with how fusion categories describe the fusion of point-like excitations, and how braided fusion categories describe the fusion of anyons in 2+1D topological order. Concretely, a fusion ...
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Does gravity exist in two spatial dimensions? [duplicate]
I've been studying anyons and was wondering if gravity exists in two spatial dimensions and how it affects these particles?
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Non-abelian Berry connection : clashing time-ordering conventions, and component-wise form
Let $\mathcal{M}$ be a $k$-dimensional parameter space associated to a quantum system with an $N$-dimensional ground state. As usual, we assume the system is subject to some adiabatic tuning of ...
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How can Anyons be Possible? [duplicate]
I'm studying Identical Particles in Quantum Mechanics and somewhere I saw that some particles, called Anyons, can generalize the concept of Fermions and Bosons, showing a symmetry under permutation ...
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Can anyons exist on a torus without any additional conditions?
While learning recently some more "advanced" stuff about path integral formalism I was introduced to the topological conditions that specify the process of construction of the propagator, i....
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Understanding the possible values of orbital angular momentum of an electron orbiting a magnetic flux tube
I am reading through the paper Magnetic Flux, Angular Momentum, and Statistics by Frank Wilczek (https://doi.org/10.1103/PhysRevLett.48.1144) and had some questions about parts (B) and (C) as labeled ...
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Rigorously, what is the permutation operation in Quantum Mechanics?
I am an undergraduate wanting to understand anyons (in order to understand topological quantum computing). The foundational observation upon which the existence of anyons rests seems to be that the ...
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Why must a Hamiltonian be gapped to have "local" excitations?
On page 4 of Kitaev's "Anyons in an Exactly Solved Model and Beyond" he states
The notion of anyons assumes that the underlying state has an energy gap (at least for topologically ...
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Decomposition of Two-Particle Statistic in Chern-Simons Theory
In Fradkin's book "Field Theories of Condensed Matter Physics," the statistic of two particles under the Chern-Simons theory is examined. While I understand how the writhing numbers $R(\...
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Phase of wave-function in two-dimensional systems
I have a specific question about how different phases could appear on wave function (which interpreted as fractional statistics in 2-domensional system).
Here is what I know:
Laidlaw and DeWitt proved ...
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What does it mean by "condensation" of anyons?
My question is motivated from the paper Boundary degeneracy of topological order by Juven Wang and Xiao Gang Wen.
Consider a (2+1)D system with boundary, described by abelian Chern-Simons theory. Due ...
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Anyon and state spaces
I start learning about anyons, but I'm confused by a few Hilbert spaces.
First of all, it is said that anyons are "excitations" with anyonic statistics. By that I would imagine they are ...
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Does Fibonacci anyon come from a representation category of Hopf algebra?
I have heard that the UMTC(unitary modular tensor category) of Fibonacci anyon comes from a quantum group, but the representation category of Hopf algebra is equipped with a forgetful functor to $\...
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Union of two loops in the toric code is a loop?
In the toric code , configurations of the vertex operator $\sum_v A_v=\sum_v\sigma^z \otimes \sigma^z \otimes \sigma^z\otimes \sigma^z$ with eigenvalue $+1$ form closed loops of $|1\rangle$ states ...
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Role of fusion/splitting spaces in TQFT
In his book on topological quantum field theories Steven Simon writes that 2+1D TQFTs are objects that assign topologically invariant numbers to labeled links embedded in arbitrary 3-manifolds. They ...
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Fermions, Bosons, Anyons on a 2-manifold
My understanding of fermions, bosons, and anyons is that anyons are disallowed in 3+1 dimensions (or $n+1 | n\geq3$) because of the topology of spacetime. The paths of swapping two particles twice is ...
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What is a difference between solitons and anyons?
In the article Creation and annihilation of mobile fractional solitons in atomic chains the authors claim that they prepared 1D solitons which can be used in topological quantum computing. Based on ...
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Filling factors and implementation for non-Abelian models
Currently reading through Pachos' Introduction to Topological Quantum Computation, and perusing other related articles and papers online. Have seen in many places that the 5/2 filling factor for ...
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How to detect anyonic statistics without calculating Berry phase diretly?
My question is: given a model, it is possible to know if it can support a specific kind of anyon (like Fibonacci or Ising) without having to explicitly calculate the Berry phase after a braiding?
I've ...
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Topological invariant for the Toric code
My understanding is that the Toric code is a model with topologically non-trivial ground state. The ground state is degenerate on a Torus and is robust to local perturbations. The model has anyonic ...
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Which topological orders described by TQFT and tensor category theories are not known to be microscopically realizable?
Topological order refers to long-range-entangled phases of matter that cannot be smoothly deformed into ordinary phases characterized by Landau’s symmetry breaking theory. A large number of ...
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Distinguishable ways of splitting/fusing anyons
I have a difficulty in understanding the possibility that two simple anyons can fuse into one simple anyon in distinguishable ways:
\begin{eqnarray}
a\times b= 2c.
\end{eqnarray}
Let us put it in the ...
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Underlying Hilbert space of Kitaev's exactly solvable models
In Kitaevs's paper (Anyons in an exactly solved model and beyond) section 2.1-2.2, he seems to be extending the Hilbert space of a multi-spin system using Majorana operators. More specifically, if ...
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Braiding anyons in one dimension
In the Rev. Mod. Phys. 80, 1083 (2008) Non-Abelian Anyons and Topological Quantum Computation, they make an aside in Section II.1.a that
as an aside, we mention that in 1 + 1D, quantum statistics is ...
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What is $\nu$ in this equation regarding phases of anyons under exchange?
From the Quantiki article on anyons:
After exchanging two identical particles the quantum mechanics predicts that the wave function gain a phase factor:
$$
\Psi\to e^{i\theta}\Psi
$$
For bosons $θ = ...
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How many of Kitaev's "Odds and Ends" in his 2006 anyon paper have been solved?
In Kitaev's 2006 paper Anyons in an exactly solved model and beyond, he lists nine open questions under the Section 10 "Odds and Ends". Briefly, these are
Find a condensed matter ...
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Are anyons non local?
Studying anyonic statistics in 2 dimensions, I naturally thought to ask the question of whether anyons are non local, since as we braid one around another, no matter the distance between the two, one ...
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The meaning of phase operator in Majorana zero mode
In some article, such as Phys. Rev. B 94, 235446 (2016), they define the Majorana mode operator as follow
$$
\gamma_j=\int \mathrm{d}r \ [\xi_j(r)e^{-i\theta/2}c^\dagger(r)+\xi_j^*(r)e^{i\theta/2}c(r)]...
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Excitations & Pentagon axiom in algebraic theory for anyons
I have been reading the anyon theory by Kitaev and Wang. I have two possibly related questions:
Why is the Pentagon equation/axiom sufficient for characterizing associative relations?
Are there anyon ...
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Topological spin in $Z_2$ toric code
On page 20 of this paper, Kitaev shows that the composite particle $\varepsilon = e \times m$ is a fermion. He also said that it is easy to show $e$ is a boson (i.e. carries a topological spin of 1). ...
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Rotation of a string operator in a string-net liquid
I am reading a review article on topological order. On page 6 of Ref. 1, the author introduces a 360-degree rotation of the string. And, it is said that a straight string state (i.e. an equivalence ...
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Anyon statistics in a lattice Moore-Read state
I'm trying to understand this paper1, in particular the remark after Eq. 26.
Let me rephrase the problem. According to the paper, one can write the Berry phase as
$$
\theta_B=i\oint_\Gamma\frac{1}{C}\...
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Deducing fusion rules of non-abelian fluxons
I have been reading about non-abelian fluxons in John Preskill's lectures notes on topological quantum computing and I do not understand how he deduced the fusion rules for fluxons in the example he ...
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Why do we remove the origin from the topological space when deriving anyons?
When we derive the existence of anyons in 'r-space' where r = r1 - r2 and r1 and r2 are the parameters describing two particles called '1' and '2', we remove the point r=0 so that 'the two particles ...
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Simple models with non-abelian anyons [closed]
It is well known, that in 1d and 2d there are particles with anyone statistics.
Which 1d and 2d models have such excitations? Which model with anyons is simplest?
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Connection between Wilson loops and fusion rules in $Z_2$ topological order
I'm looking for references (reviews, original articles, lecture notes, etc.) that discuss the connection between the expectation value of Wilson loops (the "disorder parameter" of the system) and ...
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Why are particles either bosons or fermions in spatial dimensions $d>2$? (in Wigner classification)
This questions might have been asked several times, but I haven't seen a mathematical point of view, so here it is.
Based on Wigner classfication: A particle is a representation, because any theory ...
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Experimental progress: Topological phases of matter
It's been:
43 years since Leinaas & Myrheim's seminal paper
38 years since Wilczek coined the term anyon
29 years since Moore & Read's paper on non-Abelions in the Fractional Quantum Hall ...
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Anyons under weaker assumptions?
I. Duality assumption
In Anyons in an exactly solved model and beyond p.74, Kitaev says, "We will see that for theories with particle-antiparticle duality, condition 3 can be dispensed with.&...
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Why is the normal argument for fermions/bosons wrong?
If we have a state $\psi(x_1,x_2)$ of two identical particles and an exchange operator $O$ which swaps the particles. Obviously the physics must be the same and hence $O$ can only introduce an ...
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How many Anyons can be allowed in a state
For fermions, a state allows only one fermion to exist . For bosons, there can be infinite number of bosons in one state . But for anyons, how many can a state allow?How do we come to this conclusion?
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Non-physicality of 'strings' in Kitaev's anyon model
I was reading Kitaev's paper on arXiv (arXiv:quant-ph/9707021, 'Fault-tolerant quantum computation by anyons') and was wondering if someone could clear something up for me about the non-physicality of ...
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Can supersymmetry be extended to include anyons for 3+1D superstrings?
Are there no-go theorems and the like proving that there is no generalization of supersymmetry to fractional spins either in general or for superstrings in four dimensions? If so, are there ...
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Rotating three (or $n$) times to come back to itself
Hello I'm a quantum mechanics newbie.
I learned about spinors, and how they are different from vectors because unlike vectors, rotating them once does not give the original spinor, but the negative ...
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Fusion of anyons
I have been studying anyons and I have found the algebraic approach rather abstract and I am struggling to understand it as it seems quite different to the usual procedure of quantum mechanics.
I do ...
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Generalized commutator/anticommutator via phase factor
We know that the commutator between two operators $A$ and $B$ reads $[A,B]_{-}=AB - BA$, while the anticommutator reads $[A,B]_+=AB + BA$.
I am wondering if someone has ever used a generalized ...
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Exchange Non-abelian Anyons results in rotation in Vacuum degenerate space?
A basic notion when studying non-abelian anyons is that the system's groundstate is degenerate. Not only that, but exchanging two anyons' position rotates the state in this degenerate subspace. I'm ...
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Do anyons obey the exclusion principle?
In general, if we have two indistinguishable particles in states $\psi_1$ and $\psi_2$, then starting in the combined state $|\psi_1\psi_2\rangle$ and then exchanging them will produce the state $e^{i\...
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How to obtain this formula for 2+1 dimensional boosts
In this paper on anyons, a formula for boost transformations in 2+1 dimensional spacetime is given (equations 2.7--2.10).
The boost transformation here is defined as:
$$\displaystyle B(p) \hat{p} = p$$...
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Understanding Anyonic Exchange
In the book of "Introduction To Topological Quantum Computation" by Jiannis K. Pachos, in chapter 5, it tries to explain anyonic exchange.
In the following, the $m$ and $e$ quasi-particles are ...
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