Questions tagged [topological-order]

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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Absence of topology in semi-dirac materials

Good morning everybody, I am facing a problem when calculating the topological invariant in a semi-dirac system, whose Hamiltonian is: $$ H=k_x^2\sigma_x+k_y\sigma_y $$ My question is that this ...
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"Universal" versus "nonuniversal" in the topological entanglement entropy

In their seminal paper, Kitaev and Preskill consider a 2D area law for entanglement of the form $$ S(\rho) = \alpha |\partial A| - \gamma + \cdots. $$ Here $\rho$ denotes the density matrix associated ...
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Duality between topological order and SPT in $K$-matrix formalism

It is a well-known fact that the low energy effective theory of intrinsic topological order is multi-components Chern-Simons theory $\frac{K_{I J}}{4 \pi} \int_{\mathcal{M}} d t d^{2} x \epsilon^{\mu \...
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Why are topological materials/phases "exotic"?

From what I understand, when a system has topological order, any local perturbation doesn't change the phases and thus its properties. This would suggest that it should be really easy to find ...
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How does one 'glue' together the different $k$-hilbert spaces to make them continuous in $k$?

In a typical introduction to toplogical phase transitions via e.g. the SSH chain or the transverse-field Ising model, the agument around using the Berry phase usually proceeds as follows: The ...
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Gauged topological BdG systems

Can anyone suggest literature showing in detail that in 2+1D, a gauged $p+ip$ or a DIII BdG superconductor is a $Z_2$ gauge theory? I know this work https://arxiv.org/abs/cond-mat/0404327 , but I am ...
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Can we write the effective field theory for the toric code model?

If not, then why not? If yes, then what is the effective field theory?
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Quadruple index and the existence of corner state

I have been following this paper, which seems to discuss the connection between corner state and a quadruple index calculated $q(k_z)$ by equation (3) of it. At two special point $k_z=0 /\pi$, the ...
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Order parameters in topologically ordered systems

Topologically ordered phases are characterized by a non-vanishing ground-state expectation value of a non-local operator. These operators are supported on sites whose number grows with the system size....
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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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What's string operators in a Chern-Simons theory?

In a $\mathbb{Z}_2$ lattice gauge theory or a toric-code model, we have e particles, m particles and their composition, and we have string operators which can create two anyons at the ends of a string,...
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Laughlin wavefunction

I have just started reading about the Fractional Quantum Hall Effect (FQHE) and had these doubts: In the review article [1, 2] by Prof. Wen, he writes that electron always dances anti-clockwise so ...
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Quantum Double Model and Chern-Simons with finite gauge group

Is there a relationship between Kitaev's quantum double model for a finite group $ G $ and a Chern Simons theory with finite gauge group $ G $. They are apparently both related to quantum groups and ...
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Code distance and other questions about Quantum double model as an error correcting code

Kitaev's quantum double model is an error correcting code, see: https://arxiv.org/abs/1908.02829 I am in a class on quantum error correction and the professor commented that a quantum double model for ...
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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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Why number of left-moving and right-moving edge states on a finite lattice system is equal?

I read an arguments about number of left-movers and right-mover in finite system in paper titled as Antichiral Edge States in a Modified Haldane Nanoribbon. In second paragraph of introduction, it ...
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Can we construct a Chern-Simons theory provided that the ground state is degenerate and gapped, like the Abelian fractional quantum Hall effect?

I am studying the Chern-Simons approach to fractional quantum Hall effect, which a special focus on the topological order in the context of Abelian fractional quantum Hall effect. To me, the logic to ...
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How to describe SSH chain with odd number of sites?

Usually when we discuss SSH(Su-Schrieffer–Heeger) chain, we discuss a chain with 2N atoms, with v the intra-cell coupling and w the inter-cell coupling. When N is infinite, the system becomes bulk, ...
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Is the superconducting gap the same as the many-body gap in superconductor?

In conventional superconductors, it is well known that there is a gap $\Delta_{\text{sc}}$ to electronic or quasiparticle excitations. This is the gap that shows up when one measures the temperature ...
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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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How to rigorously prove that the toric code ground state has non-trivial topological order?

Consider the unique ground state $|\psi\rangle$ of Kitaev's toric code model on a sphere. Has it been rigorously proved that $|\psi\rangle$ cannot be transformed into a trivial product state by ANY ...
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Could there exist gauge-symmetry-protected topological order?

More precisely, let $\hat{H}_1, \hat{H}_2$ be locally-interacting, translation-invariant quantum many-body Hamiltonians (defined on the same quantum system) that both has a gauge symmetry $G$, and ...
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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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Zero energy eigenstates of the Kitaev chain Hamiltonian

I am thinking about a two-site Kitaev Hamiltonian. Namely, $$H = -\mu c^\dagger_1 c_1 -\mu c^\dagger_2 c_2-t c^\dagger_{2} c_1-t c^\dagger_{1} c_2 +\Delta c_1 c_{2}+\Delta c_2^\dagger c_{1}^\dagger.$$ ...
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How to get the generators of $\mathfrak{so}(3)$ in the paper by Fidkowski and Kitaev?

In the paper by Fidkowski and Kitaev, they aim to study the interaction of 8 parallel Majorana wires, and they work on $\mathfrak{so(8)}$ Lie Algebra. They first start with just 4 parallel Majorana ...
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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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Why stacking two $p+ip$ superconductors or superconductors with Chern number 1 ($C=1$) is a quantum hall state?

My question is based on the lecture by Bernevig in PiTP 2015 on "Category Theory and the Kitaev 16 Fold Way"41:00. Why by stacking two superconductors with Chern number $C^{(1)}=1$ we have ...
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Heat capacity from topolocal excitation vs local excitation

In an online lecture given by Professor Xiaogang Wen, he mentioned: Given the Hamiltonian $\hat{H}=-J\sum\limits_{i}^{L}\sigma^{x}_{i}\sigma^{x}_{i+1}-h\sum\limits_{i}^{L}\sigma^{z}_{i}.$ When $\frac{...
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How we can get the "Fermion Parity" and "Ground states" for Majorana fermions in Bernevig's talk PiTP 2015?

I have two questions regarding the talk, Topological Superconductors, Majorana...and Interactions, by Bernevig in PiTP 2015. How he gets the "Fermion Parity" for the ground states in the ...
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Properties of Topologically Ordered States

From what I've read so far, all topologically ordered states seem to possess the following properties: Gapped excitations with fractionalized statistics (anyons) Gauge theory structure (which may be &...
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Quasi-hole excitation of Laughlin sates?

My knowledge in condensed matter physics is limited and I am trying to understand fractional quantum Hall effect in a more detail. The Laughlin wave function at filling fraction $\nu=\frac{1}{m}$ is \...
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Eigenvalues of Majorana fermions

I have a somewhat naive question about Majorana fermions. Typically, two Majorana fermion modes $\gamma_{i,1}$ and $\gamma_{i,2}$ are defined by writing a single ordinary fermion $c_i$ in terms of &...
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What does "continuous transformation" mean with regard to the Hamiltonian of a system?

When dealing with topological phases of matter (topological insulators, quantum hall effect, etc...) one says that the system remains in the same phase as long as any continuous transformation of the ...
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Excitations of the string-net Hamiltonian

A quite general 2D topological order can be constructed through the string-net theory. Here, if the input data is some braided fusion category $C$ (i.e. the $F$-symbols), the elementary excitations ...
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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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Why is magnetic ’t Hooft loop operator independent of its path in$Z_2$ gauge theory?

One important concept for $Z_2$ gauge theory is magnetic ’t Hooft loop operator $\tilde{W}_{\tilde{\Gamma}}$ along a non-contractible loop $\tilde{\Gamma}$ on the dual lattice of the torus is: $$\...
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Particle-Hole Symmetry (Charge-conjugation Symmetry) in CMT

Charge-conjugation symmetry was defined in the bellow paper as follows: \begin{align} \hat{\mathcal{C}} \hat{\psi}_A \hat{\mathcal{C}}^{-1} &= \sum_B (U^{*\dagger}_C)_{A,B} \hat{\psi}^{\dagger}_B \...
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A pedagogical semi-rigorous review of topological phases, topological order, and related subjects

I'm looking for a pedagogical review or book about topological phases, topological order, TQFTs, and related subjects. The ideal thing would be a mix of rigorous definitions and physical examples, ...
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Why is Kitaev's toric code a $Z_2$ gauge theory?

I am reading Kitaev's 2003 paper. In the literature, it is often said that the model proposed in this paper is a $Z_2$ gauge theory. I don't quite see why it is the case. Where is the $Z_2$ gauge ...
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Why do we call fracton by its name?

I am reading on fractons. In the literature, it is said that factons are fractionalized excitations. My understanding about fractons is that it is energetically costly to move fractons, and in this ...
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Topological order and volume-law entanglement

Topological order is a property traditionally most associated with ground states of gapped Hamiltonians. However, using the notion that topological order is fundamentally about a form of "long-...
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$n$-fold degeneracy, ground state, toric code [closed]

I'm having a hard time understanding the degeneracy of the Kitaev toric code. Can anyone explain to me with details the following: How do we find the ground states of the toric code and its gape? ...
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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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Rotor representation of creation and annihilation operators

I read the paper "Edge–Entanglement correspondence for gapped topological phases with symmetry" Where a lattice $U(1)$ model is introduced with bosons both on the sites and links of the ...
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About Chern insulator

I know when we talk about Insulator, U(1)charge symmetry naturally exists. But in this article:Quantum phase transitions of topological insulators without gap closing, the author claims that: "...
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Topological order in Weyl Semimetal

Is the topological phase in a Weyl semimetal is intrinsic or symmetry protected? How can we realize that? If symmetry protected, which symmetry protects the topological phase of non-centrosymmetric ...
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