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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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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Confusion in change of basis for anyon fusion
I'm doing some introductory reading on anyons, and I'm a bit confused by the way the basis are changed.
Suppose we have the Ising anyons, which obey $1\times1=1$, $1\times\Psi = \Psi$, $1\times\sigma =...
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How to derive anyons fusion rules by given spin liquid Hamiltonian and PSG?
In many textbooks, the gapped Z2 spin liquid (equivalent to toric code) with parton method is well illustrated. It has 1, e, m, f four basic choson anyons and their fusion rules are easy to obtain and ...
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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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Topological Degeneracy in FQH Liquids
It is well known that fractional quantum Hall states have topological degeneracy: ground state degeneracy that depends on the topology of the closed manifold on which the state exists. As explained ...
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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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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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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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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 ...