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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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DMRG for anyons

I want to do some DMRG calculations for anyons. For example, consider the golden chain model for fibonacci anyons. https://arxiv.org/pdf/cond-mat/0612341 I have two anyon types: $1, \tau$. However, ...
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Question about Path Integrals and Exchange Statistics in Steve Simon's "Topological Quantum"

In the introduction to the path integral approach leading to exchange statistics for many particles, Steve Simon breaks up the sum of paths into two types: paths where particles do not exchange (type ...
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Why the unit of quantum information for anyons systems should be the qubit?

I'm starting to learn more about anyons systems. I took a read on this article which is an introduction to topological quantum computing, and also took a look in other places like forums and some ...
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Chern-Simons (K matrix) theory and ${\rm Spin}^{\mathbb C}$ connections

If I understand correctly (e.g. from this paper), an Abelian bosonic Chern-Simons theory defined on $T^2\times \mathbb R$ is specified by a $K$ matrix via e.g. $S \sim \int_M K_{IJ}A^I \wedge dA^J$. ...
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Frobenius-Schur indicator and total orbital angular momentum

It is a well-known formula in the theory of anyons that $R_c^{ba}R_c^{ab}=\frac{\theta _c}{\theta _a\theta _b}$, where $R$ represents the braiding matrix and the $\theta$ are twists, i.e. the phase ...
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$F$-matrix rigidity condition in anyons

In Wang's "Topological Quantum Computation", the following condition of "rigidity" is imposed on the $F$-matrices characterizing basis changes of the topological Hilbert space for ...
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Anyons and Elementary particles in 2D [closed]

I'm doing my master's degree and I'm starting to learn more about Anyons. I want to understand more deeply why they can exist and how. I've done some research on the internet and found this question ...
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Qubits vs Fermions, Bosons and Anyons [closed]

I found out recently that qubits are different from fermions, bosons and anyons. And, which is why we use Jordan-Wigner Transformation to map them to their fermioinc counterpart. I think I am trying ...
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What physical properties give rise to abelian anyons as opposed to non-abelian anyons?

As far as I understand, abelian anyons are those which have a particle exchange operator which is a one-dimensional representation of the braid group. Non-abelian anyons, then, have a higher-...
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Since anyons cannot exist in our 3+1D world, what does it mean to have discovered them? Why should we study them?

There have been previous questions on this, for example see this and this question, but my question is different. I get that in 2+1D, mathematically speaking, exchanging two identical particles twice ...
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Can Toric Code have a gapless boundary?

The toric code model is known to have two types of "gapped" boundaries, namely, the rough boundary and the smooth boundary. See, for example, Chap. 4.1 of this beautiful review https://arxiv....
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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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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 ...
Martin Vesely's user avatar
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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 ...
Cristobal Mendez's user avatar
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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 ...
Andrew Hardy's user avatar
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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 ...
pyroscepter's user avatar
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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 ...
8 votes
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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.&...
Sachin Valera's user avatar
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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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