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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Why does the topological entropy scale with $\log(L)$ in 1D?

Why, in one dimension, does the topological entropy scale with the size of system as $S \sim \ln L$, while in a 2D system it scales with $S \sim L$? Why does dimensionality play such an important role ...
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Numerical Tools to find Braiding Statistics of Quasiparticles

While certain classes of systems that exhibit topological order can be solved exactly (such as the Toric Code, Abelian FQH Edges, etc.) there also exist systems (think of perturbed versions of the ...
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435 views

Quasi 1D insulators with strong spin-orbital interaction

We know that the spin-1 chain realizes the Haldane phase which is an example of symmetry protected topological (SPT) phases (ie short-range entangled phases with symmetry). The Haldane phase is ...
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1answer
310 views

How to understand topological order at finite temperature?

I have heard that in 2+1D, there are no topological order in finite temperature. Topological entanglement entropy $\gamma$ is zero except in zero temperature. However, we still observe some features ...
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1answer
260 views

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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72 views

The topological degeneracy and quasiparticles

I know this conclusion in topological order for a while: "the topological degeneracy on torus is equal to the number of quasiparticles types." But can anyone give a physical argument that supports ...
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1answer
174 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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1answer
119 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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1answer
262 views

Exact diagonalization to resolve ground state degeneracies

I am studying a perturbed Toric Code model that is not analytically solvable. On a torus the ground state degeneracy of the unperturbed model is 4. Once we turn on the perturbation there is a change ...
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1answer
86 views

$Z_2$ 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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1answer
73 views

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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Is the “particle number” of “electrons” well defined in Wen's string-net theory of elementary particles?

According to professor Wen's string-net theory, electrons can be viewed as the elementary excitations of string-net objects. Just like the phonons and magnons are the elementary excitations of ...
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220 views

Topological entanglement entropy only defined for a system in the ground state?

What happens to the topological entanglement entropy of a system, when it is driven out of its groundstate by increasing the temperature?
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146 views

Third-order topological quantum phase transition in p+ip superfluid

A two-dimensional spinless non-relativistic p+ip superfluid undergoes a quantum phase transition between the BCS (weakly-coupled) and BEC (strongly-coupled) regimes. This transition is driven by ...
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70 views

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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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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105 views

Topolgical insulators order parameter

For topological insulators Is there any way to define order parameter for topological phase transitions?
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100 views

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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110 views

Axiom approach for majorana fermions

This is the usual way of introducing majorana operators. First we have $N$ fermionic modes. The corresponding operators satisfy the commutation relations $$ \{c_i, c_j \}= \{c_i^\dagger, c_j^\dagger ...
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What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
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What is the definition of topological when talking about topological phases of matter?

What is the definition of topological when talking about topological phases of matter? Why do people think that the fractional quantum hall effect is topological? I think it means that the ground ...
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395 views

Jordan Wigner Transformation in 1d Majorana chain

So, I was reading the paper by Fidkowski and Kitaev on 1d fermionic phase http://arxiv.org/abs/1008.4138. It explains the classification of 1d fermionic SPT phases with $\mathbb{Z}_2^T$ symmetry for ...
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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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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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69 views

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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Weaker Formulations of Bulk-boundary Correspondence for Interacting Systems

From this post, it seems that bulk-boundary correspondence does not hold in general for interacting systems. What is meant by bulk-boundary correspondence there appears to be the existence of robust ...
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Explicit degeneracy in SPT phases

In the wikipedia article on symmetry protected topological phases the author states: If the boundary is a gapped degenerate state, the degeneracy may be caused by spontaneous symmetry breaking ...
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119 views

Construction of a spin chain Hamiltonian invariant under a finite subgroup of SO(3)

I would like to construct a 2-local Hamiltonian that acts on a 1D spin chain where each spin transforms as the 3D irrep of $A_4$ which is a subgroup of $SO(3)$. I know that an $SO(3)$ invariant ...
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150 views

Questions on the elementary excitations in the resonating-valence-bond(RVB) states?

It is known that the RVB states can support spin-charge separations and its elementary excitations are spinons and holons. But it seems that there are some different possibilities for the nature of ...
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19 views

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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100 views

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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97 views

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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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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189 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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57 views

What is the different between topological order and Landau's order in a system

I have thought about topological order for a long time, but I am still confused it. Roughly speaking in my understanding, the topological state is the eigen-state of some special symmetry such time ...
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72 views

Effective Theory of FQH Edge State

When I was learning Xiao-Gang Wen's paper about the edge theory of Fractional Quantum Hall(FQH) state, I had one question. The paper's link is as below:\ http://dao.mit.edu/~wen/pub/edgere.pdf As ...
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51 views

Momentum conservation in the Fractional Quantum Hall Effect

Generically an Abelian Fractional Quantum Hall Systems is described by chiral scalar fields $\hat{\Phi}^{\ }_{i}(t,x)$ with $i=1,\ldots,N$ and a Hamiltonian of the form $ \hat{H}^{\ }_{0}:= ...
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91 views

What is the first excited state of the honeycomb Kitaev model in its gapped phase?

As we know, there are both gapless and gapped phases of the Kitaev model, and let's fix the couplings $J_x,J_y,J_z$ such that the model being in the gapped phase. My question is, what is the first ...
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123 views

Equivalent Chern Simons Theories

This is a follow-up question to FQH Edge Theory as decoupled chiral bosons . The document that I will be refering to is http://dao.mit.edu/~wen/pub/toprev.pdf . On page 14 in Eq.(2.33) the author ...
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170 views

Central charges c and topological ground state degeneracy GSD

A 2+1D topological field theory (topologically ordered states), implies that the topological ground state degeneracy (GSD) on $T^2$ torus (2D manifold without boundary). For example a level k U(1) ...
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86 views

Conventions for Klein factors in bosonization of Quantum Hall edge states

I am not having much experience in the field of bosonization, hence the following question: In some papers (such as http://arxiv.org/pdf/cond-mat/9501007.pdf Eq. (6) ) a Quantum Hall edge is ...
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25 views

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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76 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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114 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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77 views

Can a conformal field theory with chiral central charge be gapped out?

Consider a 2-dimensional conformal field theory with nonzero chiral central charge (that is, the central charges of the holomorphic and antiholomorphic sectors are different.) I think that ...
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345 views

Topological disorder in condensed matter?

What is meant by topological disorder in condensed matter (both crystalline and amorphous)? For example, please see the following two papers from arxiv.org http://arxiv.org/pdf/0906.3848.pdf ...