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 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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Fluxes on a finite group $G$

So I've been studying about topological quantum computation and I have a few questions I haven't been able to solve. The first one is why fluxes take values on a finite group $G$? Does it have to do ...
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Math of anyons: Quantum dimension of 1 implies abelian charge

This question originates from the following statement in Bonderson's thesis: Link to Thesis page 16 or pdf-page 23: The quantum dimension $d_a$ of an anyon of charge $a$ satisfies $d_a \geq 1$ with ...
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Literature Anyons [duplicate]

In a few months I have to give a talk about Identical Particles in Quantum Physics, which should answer the following questions or explain following concepts: Why are there only Fermions and Bosons ...
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Associativity of fusion of anyons: Why are anyons ordered?

Anyon theories are required to be associative, i.e. when fusing three anyons with labels $a,b,c$, we have $$(a\times b) \times c = a\times (b \times c)$$ This associativity is extended to the fusion ...
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Why does every anyon need to have an antiparticle?

It seems to be a basic requisite to every anyonic model that every type of anyon, say $a$ in the theory comes with an antiparticle $\bar{a}$ (which can be itself) where $a$ and $\bar{a}$ fuse together ...
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Non-abelian anyons, relation between representation of braid group and fusion rules

As far as I understand, anyons correspond to fields that live in the representation space of some (unitary?) representation of the braid group. One-dimensional representations commute and give rise to ...
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Do anyonic statistics only arise from spatial degrees of freedom?

Elementary texts on quantum mechanics justify the existence of fermions and bosons using the simple argument that if we have a state of two indistinguishable particles $|a,b \rangle$, where $a$ and $b$...
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Why is the composite fermion not included in the anyon contents of FQH topological orders?

For example, both the $\nu=1/3$ Laughlin state and the Moore-Read state has a simple interpretation in terms of composite fermions, which are bound states of an electron and two fluxes. Both the ...
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Theory for free, non-interacting anyons?

This link suggests that one cannot make a free theory out of anyons, because of its Lorentz representation. How exactly does the $SO(2,1)$ representation enforce the $\pm1$ eigenvalues? How can one ...
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How do I derive Pauli's exclusion principle with path integrals?

I am trying to prove Pauli's exclusion principle using path integrals. My starting point is the configuration space $\mathcal{C}$ for two indistinguishable particles in 3D: $$ \mathcal{C} = \{ \{x_1,...
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Configuration space of identical particles - fractional statistics

In Khare's book of fractional statistics and quantum theory, when discussing why we need fractional statistics he arrives at the configuration space for a system of two identical particles in $d$ ...
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Is 2D toric code dual to something?

I can understand 2D toric code as a quantum Z2 gauge theory defined on a lattice. Is this model dual to some simpler spin model? A bit of motivation to clarify my intention: I know 3D classical Ising ...
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How does Laughlin argument for hierarchical fractional quantum Hall effect work?

For 1 level and 1 layer $1/q$ FQHE let's say $q=5$ we have the following argument for Laughlin gauge principle. It says that if we adiabatically increase the flux from $0$ to $q\phi_0$ of a corbino ...
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Braiding matrix in Chern-Simons theories$.$

Consider a Chern-Simons system with gauge group $G$ and level $k$. Such a system can be used to model anyons, where the latter are identified with the integrable representations of $G$. One of the ...
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What makes spin 1/2 anyons special?

If the spectrum of a TQFT contains a fermion, the theory becomes a spin-TQFT, and it depends on the spin structure of the manifold (cf. 1505.05856). On the other hand, if no such anyon exists, the ...
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Anyon condensation – what's the precise definition?

Say I have an anyonic system modelled as a Chern-Simons system with group $G$. If the centre of $G$ is non-trivial, one may also study the system described by $G/\Gamma$, where $\Gamma$ is a discrete ...
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Non-Abelian anyons in the path integral formalism

Background Homotopy classes in the path integral Following the answer to this question about the role of homotopy classes in path integrals, it seems reasonable to me that, when calculating the ...
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What determines if a quasiparticle is a spinon or a vison?

Across topological literature, I've seen references to spinons and visons. In Kitaev's famous "Anyons in an exactly solved model" paper, he mentions that visons are "spinless bosons" whereas spinons ...
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When is the Berry phase only dependent on path topology?

Background Suppose we have a Hamiltonian $H(\mathbf{R})$ which depends on some parameters $\mathbf{R}$. For each value of $\mathbf{R}$, the Hamiltonian will have some set of eigenvectors $\{ | \phi_{...
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Symmetry for the exchange of particles

Let's say we have two particles like in this picture - Here it is argued that in 3D after taking the left particle around trajectory 1 or 2 the system should be in the same state since trajectory 1 ...
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Question about the existence of non-Abelian anyon

My question from reading this paper: Michael G. G. Laidlaw and Cécile Morette DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles. Phys. Rev. D 3, 1375 (971). ...
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Is every fusion category with unitary $S$ a UMTC?

A tensor category whose solutions to the pentagon/hexagon equations are unitary and whose braiding is nondegenerate is called a unitary modular tensor category (UMTC). When trying to find UMTCs, ...
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Status of the discovery of non-abelian anyons and topological quantum computation?

This week Microsoft announced that it will make available the programming language for quantum computer available by the end of this (2017) year. https://news.microsoft.com/features/new-microsoft-...
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Ising anyon topological order and its edge $c=1/2$ CFT

We know that conformal field theories are closely related to two-dimensional topological orders via edge-boundary correspondence. An Ising topological order can be obtained by gauging the fermion ...
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Resource recommendation for fractional statistics

A gentle yet comprehensive introduction to the concept of abelian and non-abelian statistics will be much appreciated.
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Gapless anyons: are massless fermions not well-defined?

It is usually said that anyon statistics are not well-defined if the anyons are massless. If I understand it correctly, the intuition is that any braiding has a finite duration, which means that any ...
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A question about the anyon

Let's say we have two indistinguishable particles at positions (two spatial dimensional) $x^i_1$ and $x^i_2$ at some initial time and end up at positions $x^f_1$ and $x^f_2$ a time $T$ later. In the ...
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K-matrix for ppq Halperin states

In the literature, the K-matrix which describes a $(ppq)$ Halperin state is given as $$\begin{pmatrix} p&q\\ q&p \end{pmatrix}$$ with charge vector $\vec{q}=(1,1)$. Are there any equivalent ...
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Ising anyon (or Ising topological order) from spin lattice model

It is well-know that Kitaev’s honeycomb lattice model can give rise to non-abelian Ising anyon. I am wondering, is there another spin-lattice model in two spatial dimension that also can host Ising ...
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Anyon Lagrangian

The discussion of 2D quantum mechanics of two identical particles may be started (as written in the following article) with Lagrangian $$ L_s = \frac{\theta}{\pi} \dot{\phi} $$ where $\phi$ is ...
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How is a half twist of a diagram equivalent to exchanging anyons?

Consider the fusion of $a$ and $b$ to form $c$, with the exchange of anyons $a$ and $b$, as per Pachos' book Introduction to Topological Quantum Computation. I'm confused how the half twist of $c$ is ...
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Why is the phase picked up during identical particle exchange a topological invariant?

I've been wondering about the standard argument that the only possible identical particles in three dimensions are bosons or fermions. The argument goes like this: Consider exchanging the positions ...
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Different anyon condensations that share the same phase

In Kitaev's notes, he reviewed the toric code model. Consider on square lattice the Hamiltonian $H=-J_e \sum_s A_s-J_m \sum_p B_p,\ A_s=\prod_{j\in vertices} \sigma_j^x,\ B_p=\prod_{j\in plaquettes} \...
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What are the instances of usage of four color theorem in the theory of fractional statistics?

How important is four-color theorem (Hypothesis) in theory of Fractional Statistics?
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Which quasiparticles follow which statistics [closed]

Let me say beforehand that I know this is an ill-defined question, but I believe it is useful anyway. For these common quasiparticles: Phonons Holes Plasmons Excitons Plasmon-polaritons What ...
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Wen plaquette model with one kind of plaquette operator

Let us say that we modify the familiar Wen plaquette model so that only one kind of plaquette operator is in the Hamiltonian which is a product of $\sigma_z$s, what kind of topological order or state ...
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Mean field approach to toric code

Toric code is one of the few exactly solvable model in condensed matter, however, like the paper (http://arxiv.org/abs/1104.5485) that uses SU(2) slave fermion to "solve" Kitaev's honeycomb model, is ...
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What determines the anyon fusion outcome (if several are possible)?

Given the fusion rule (for anyons) "A x B = 1 + C" it is possible for the anyons A and B to fuse to the vacuum "1" or to fuse to the anyon C. What determines what will happen if they fuse? Is the ...
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Relationship between modular transformations and anyon braiding

In the context of anyon braiding, we have $S$ and $T$ matrices which describe the mutual and self statistics of anyons. In the context of conformal field theory on a torus, we have modular ...
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How can I derive fusion rules for anyons?

I am reading Pachos "Introduction to Topological Quantum Computation". Pachos writes that a model for anyons consists of a list of all anyons and a fusion rule for them. Given a model with anyonic ...
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properties of p-wave superconductors in a self consistent calculation

2D p+ip superconductors have zero energy mid-gap modes localized at the boundaries as well as in the vortex cores as pointed out in several places such as Read, Green and Ivanov. The modes result in ...
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Diagonal part of the configuration space of two indistinguishable quantum particles

Why is the configuration space of two indistinguishable particles given by $\frac{M^n-\Delta}{S_n}$? My question is about the $\Delta$. (Notation: $M$ is the configuration space of 1 particle. $M^n$ ...
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How to read the indicies of the Fusion Matrix F for Anyons

I am reading "Introduction to Topological Quantum Computation" from J. K. Pachos (first script from his Website: http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf). I can't seem to understand ...
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Limitations of Quantum Simulations on a Classical Computer

My question is on the simulation of a quantum computer on a classical machine. I understand that a classical computer to simulate any quantum algorithm--the problem is that the quantum computer does ...