Topological order is a new kind of order in quantum matter, which corresponds to pattern of long-range quantum entanglement. See http://en.wikipedia.org/wiki/Topological_order

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What is the difference between Bosonic and Fermionic symmetry protected topological phases (SPT)

I am reading the paper ``Braiding statistics approach to Symmetry Protected Topological Phases'' by Levin and Gu. In this paper two spin models considered describe spin-1/2 particles in (1+2) ...
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What does it mean for a topological phase to be “symmetry protected”?

I have seen some very nice and enlightening awnsers to questions related to topological order and insulators, such as here, or here. However, I'm still puzzled by the concept of "symmetry protection" ...
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Simplify $K$-matrix

2+1D Abelian topologically ordered states are believed to be described by multicomponent $U(1)$ Chern-Simons theories, with Lagrangian \begin{equation} ...
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On tests of topological superconductivity

What experimental procedures determine if a superconducting crystal (say 2D i.e. layered material) is a topological superconductor/not?
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TQFT's as effective theories of the groundstate subspace

I often hear: "The degenerate groundstate subspace of a QFT is often a TQFT". I'm trying to work out an example of this for, say, superconductors: In the context of condensed matter physics, the ...
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Triplet pairing hard to observe?

I am reading a review on topological superconductivity and Majorana Fermions by Flensberg and Leijnse and at some point they state Triplet pairing has been predicted for the ground state of the ...
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About the $Z_2$ topological invariant

In Kitaev 2001 it is shown that the topological invariant $Z_2$ in a topological superconductor (Class D or BDI, one dimensional) can be defined as $$ (-1)^\nu={\rm sign\, Pf} [ A ]={\rm sign\, Pf}[ ...
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About Weyl superconductors and fractionalized Weyl semimetals

Recently, the experimental observations of Weyl fermion semi-metal have been made. Weyl fermion becomes very hot in condensed matter physics. I am confused about the Weyl superconductors and ...
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Difference between $\nu=5/2$ quantum Hall state, chiral p-wave superconductor, He 3

I am interested in the relation between the following three phases of matter (in 2D): chiral $p$-wave superconductor (spineless $p_x + i p_y$ pairing) $\nu=5/2$ fractional quantum Hall state A-phase ...
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53 views

What does a zero topological S matrix element mean?

I realize that for nonabelian anyons, their S matrix elements could be zero (eg. the Ising anyons). I'm confused by the meaning of a zero S matrix element. Does it mean that the corresponding braiding ...
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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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String operators in the string-net model

In the string-net model http://arxiv.org/abs/cond-mat/0404617, quasiparticles are created by the string operators (defined in eq.(19)). An easier pictorial way to define string operators ...
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Why are there $F$-symbols in the splitting in anyon theory?

I am learning some basic knowledge of anyon theory by reading P. Bonderson's thesis: http://thesis.library.caltech.edu/2447/2/thesis.pdf. $F$-symbols and $R$-symbols are two basic operations on ...
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298 views

Chern insulator vs topological insulator

What is the basic distinction between a Chern Insulator and a Topological Insulator? Right now I know that a Chern Insulator has "topologically non-trivial band structure" and that a Topological ...
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86 views

How can a torus admit half a flux quantum, and why does a vortex induce an AB phase?

There is an issue that I have with the argument given in “Topological Degeneracy of non-Abelian States for Dummies” http://arxiv.org/abs/cond-mat/0607743 , regarding the ground state degeneracy of the ...
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85 views

Berry phase in the toric code model and 2D chiral $p$-wave superconductors

When we derive the exchange statistics by moving quasiparticles around a circle in the toric code model we do not mention any Berry phase contribution. Is the Berry phase contribution trivial or it ...
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88 views

why Hall conductance quantized

When I am studying quantum Hall effect, the quantum Hall conductance can be represented by Green function $\left(\text{up to}\ \frac{e^2}{h}\large \right)$: I cannot understand why it is an integer? ...
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136 views

1/m Laughlin state and $U(1)_M$ chiral CFT

I am a little confused that people claim that the edge theory of a 1/m Laughlin state corresponds to a $U(1)_m$ chiral CFT. This indicates there should be $m$ primary field operators in $U(1)_m$ ...
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127 views

Interpretation of Majorana's wave function

Given the BdG Hamiltonian of a 1D p-wave superconductor we can obtain the zero-energy excitation solution as Eq. 16.24 from Topological Insulators and Topological Superconductors (Bernevig & ...
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Dualities in 2+1D lattice gauge theories

A nice way to understand $\mathbb{Z}_2$ gauge theories is via duality transformations. For example, it is illustrated in http://arxiv.org/abs/1202.3120 that a $\mathbb{Z}_2$ gauge theory (with Ising ...
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225 views

Can a nondegenerate fermionic topological Mott insulator (TMI) state support an emergent bosonic topological order?

Based on my recent study and motivated by a recent paper, I have a naive question. Consider a 2d Hubbard model for electrons at half filling $H=\sum c_k^\dagger h_k c_k+U\sum n_{i\uparrow ...
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Basic questions about fusion of two anyons

Suppose we have two anyons $a$ and $b$ on a manifold, and we use $|a\otimes b\rangle$ to label the corresponding wavefunction. Based on the fusion rule: $a\otimes b=\oplus_c N_{ab}^c c$, we may ...
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R-matrix for $SU(N)_k$ anyon model

Does anyone know the $R$-move or $R$-matrix for $SU(N)_k$ anyon model? Thanks! For the definition of $R$-move or $R$-matrix, please see the definition in Eq.(2.30) of this paper: ...
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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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Total quantum dimension of excitations in the Toric code

In the Toric code, the excitations are e, m, fermion $\epsilon$ and vacuum. Thus, the total quantum dimension is $D= \sqrt{\sum{d_{a}^{2}}} = 2$. It seems one takes into account all sorts of possible ...
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112 views

Ground state of AKLT chain invariant under time-reversal?

The AKLT chain is an example of an SPT phase protected by time-reversal symmetry. The Hamiltonian of the system has time-reversal symmetry. The ground state wave function can be pictured as follows ...
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60 views

String operator in the string-net model

The string operator is a way to study the quasiparticle excitations in the string-net model http://arxiv.org/abs/cond-mat/0404617. It is claimed in the above reference (Eq.(19), p.9) that for string ...
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101 views

6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ ...
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Why is the plaquette operator in the string-net model a projection operator?

In the string-net model, the plaquette operator is defined as $B_P = \sum_{s}a_s B_{P}^{s}$, where $s$ runs over the string types $\{0,1,2,\dots,n\}$. It is claimed on page 19 of ...
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When do gauge theories have protected gapless excitations?

Goldstone's theorem states that a system in which a continuous symmetry is spontaneously broken necessarily has gapless excitations. (A hand-waving "proof" of Goldstone's theorem can be given by ...
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222 views

Unpaired Majoranas in the Kitaev chain

How can we see unpaired Majoranas for a Kitaev chain in topological non-trivial phase? By looking at the equation below (obtained by making ...
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$Z_2$ invariant and Wannier states switching partner

I have been reading about $Z_2$ topological invariant recently. However, after some literature survey, I still cannot understand $Z_2$ invariant in language of time reversal polarization. Basically, ...
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$Z_2$ invariants in quantum spin hall (QSH)

In a recent literature survey, I learned that $Z_2$ topological invariant is defined as zeros of Pfaffians in half a Brillouin Zone, where Pfaffians are defined as $P(k)=Pf[<u_i(k)|T|u_j(k)>]$. ...
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Topological insulators literature

I started learning things on topological insulators and I got lost in dozens of existing papers on this topic. Could anyone recommend me appropriate literature that explains deeply enough what ...
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Topological order and entanglement in quantum quench problem

I would like to ask about useful reviews, must-read papers on the study of topological order and entanglement in quantum quench problems that give a good introduction to the topic.
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Choice of framing in Gravitational Chern-Simons

I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360). There, it was mentioned, the ...
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Fermion version of Gauss-Milgram sum?

For Bosonic topological order, a very useful formula was proved to be true: $\sum_a d_a^2 \theta_a=\mathcal{D} \exp(\frac{c_-}{8}2\pi i) $ (for more detail: $d_a$ is the quantum dimension of anyon ...
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100 views

How useful is the study of entanglement entropy to quantum computing? [closed]

My question is somewhat conceptual: how, exactly, can we get closer to building a quantum computer by studying entanglement entropy? I've been reading all about the AdS/CFT correspondence and watching ...
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204 views

Definition of Topological Order in terms of categories

I have a question regarding the definition of topological order as defined in Wen's review article http://www.hindawi.com/journals/isrn/2013/198710/. Is the distinction between boundary-gapped ...
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161 views

Can a symmetry-preserving unitary transformation that goes from a trivial SPT to a non-trivial SPT be local?

This question concerns the very interesting paper: ''Symmetry protected topological (SPT) orders and the group cohomology of their symmetry group'' by Chen et al., http://arxiv.org/abs/1106.4772 In ...
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180 views

Topological Quantum Computing beyond Anyonic Braiding

In materials such as those that exhibit fractional quantum hall states, the ground-state topological degeneracy is known to be robust against external perturbations. This ultimately tells us that we ...
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173 views

What does it mean physically if pentagon identity or hexagon identity doesn't have any answers?

Imagine I write a fusion rule for some anyons on a paper. Then, I try to solve Pentagon identity and Hexagon identity, imagine finally I find out for example the Hexagonal equation doesn't have any ...
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Real World application of Topological Quantum Field Theory

What is a "killer-app" for the formalism of topological quantum field theory in "established real world physics"? To be more precise, I'm looking for an actual physical experiment, state of matter or ...
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Dilemma: Fusion space from a direct sum of anyons or NOT

In Preskill's note, 9.1.2 in page 44, concerning the fusion space, it states that: The fusion rules of the model specify the possible values of the total charge $c$ when the constituents have ...
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Topology of Fermi surface

In The universe in a Helium droplet, Grigory Volovik relates the stability of a fermi surface to topology of a Green function. There he gives the example of a Fermi gas and says that the Green ...
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128 views

Topological invariant for interacting systems using single particle green functions?

Why Single particle green's function is (preferred) used to find topological for interacting systems? $N_1 =\frac{\epsilon_{ijk}}{24 \Pi ^2} \int dw d^3k G \partial_i ...
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208 views

Topological theta term as a topological quantum field theory?

It is well known that the theta term $\int d^4x\frac{\theta}{4\pi}Tr[F\wedge F]=\int d^4x\frac{\theta}{4\pi}\epsilon_{\mu\nu\sigma\lambda}Tr[F^{\mu\nu}F^{\sigma\lambda}]$ is a topological term, ...
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effective field theory of the projective semion model

The "projective semion" model was considered in http://arxiv.org/abs/1403.6491 (page 2). It is a symmetry enriched topological (SET) phase. There is one non-trivial anyon, a semion $s$ which induces a ...
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A naive question about topologically ordered wavefunction?

Topological entanglement entropy (TEE, proposed by Levin, Wen, Kitaev, and Preskill) is a direct characterization of the topological order encoded in a wavefunction. Here I have some confusions, and ...
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What makes a superconductor topological?

I have read a fair bit about topological insulators and proximity induced Majorana bound states when placing a superconductor in proximity to a topological insulator. I've also read a bit about ...