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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Symmetry Protected Topology and Edge Modes
I have a spin 1/2 chain with open boundary conditions described by Hamiltonian $H=\sum_i \sigma_{2i}^z \sigma_{2i+1}^z$. From $H$ it's clear that boundary sites are decoupled from the rest of the ...
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Why does the energy gap of sublattice-symmetric systems never close?
I am studying from this famous site some symmetries useful for topological quantum matter.
At some point, talking about the particle-hole symmetry, it states:
You can however notice that, unlike in ...
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Quantum (higher-form) anomaly at finite temperature
At finite temperature, anomaly is generally known to be contaminated, and thus the 't Hooft anomaly matching does not work after thermal compactification. Meanwhile, I have read paper saying that ...
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Is IQHE a degenerate case of FQHE? What is the role of topological orders?
(As suggested by Tobias, I shall indicate that I will write "IQHE" for "Integer quantum Hall effect" and "FQHE" for "Fractional quantum Hall effect" below.)
I ...
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Fusion 2-categories for string-like excitations: a more concrete description?
I'm familiar with how fusion categories describe the fusion of point-like excitations, and how braided fusion categories describe the fusion of anyons in 2+1D topological order. Concretely, a fusion ...
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Hastings' "two" definitions of trivial mixed states
In https://arxiv.org/abs/1106.6026 (Definition 3), Hastings defines a density matrix $\rho$ in Hilbert space $\mathcal{H}$ to be trivial if one can tensor in additional degrees of freedom $\mathcal{K}$...
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Plaquette operator in Kitaev honeycomb model
In his honeycomb model, Kitaev defines link operators
\begin{equation}
K_{jk} = \begin{cases} \sigma_j^x \sigma_k^x & \text{if }(j, k)\text{ is an }x\text{-link;}\newline
\sigma_j^x \sigma_k^y &...
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1D Hermitian Hamiltonians are topologically trivial?
I am currently studying the following paper:
https://arxiv.org/abs/1706.07435
The authors state:
"The Z/2 classification we found for separable nonHermitian Hamiltonians in one dimension is in ...
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Emergent higher symmetry breaking without topological order?
In this paper prof. Wen states that (p.6)
a spontaneous higher symmetry broken state always corresponds to a topologically ordered state.
Are there examples of simple (or not) quantum spin models ...
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Exact ground state degeneracy for quantum spin system with non commuting terms and its quantum phase transition?
Let's say I have a 2D quantum spin model of N spin-1/2 particles, with two terms:
$$
H = -J \sum_N \prod_{i \in G} \sigma^x_i - h \sum_N \prod_{i \in G'} \sigma^z_i
$$
The first is a collection of ...
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Well-definedness of commutation relation in commuting local Hamiltonians
I'm reading the famous paper by Haah: Local stabilizer codes in three dimensions without string logical operators. In the last sentence of the introduction, he wrote:
A logical operator is a Pauli ...
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How does one define a gapless phase of matter?
Often we define a quantum phase of matter as a set of ground states of a gapped family of Hamiltonians, where the gap does not close anywhere in the family. Equivalently, we can define it as a set of ...
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Why must a Hamiltonian be gapped to have "local" excitations?
On page 4 of Kitaev's "Anyons in an Exactly Solved Model and Beyond" he states
The notion of anyons assumes that the underlying state has an energy gap (at least for topologically ...
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Understanding of Band Inversion in $\rm Bi_2Se_3$ Topological Insulator [closed]
I'm currently trying to understand Fig. 2 in this Paper (http://dx.doi.org/10.1103/PhysRevB.82.045122) which aims to explain the origin of band inversion in Bi2Se3. I really want to get an ...
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What is the direct sum of 1d domain walls in Toric code model?
I have read this paper:"An invitation to topological orders and category theory" (https://arxiv.org/abs/2205.05565v2). In page 93 and page 109, they show the result of fusion of simple 1d ...
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Duality transformation in Kitaev's quantum double models
In Kitaev's quantum double models (https://arxiv.org/pdf/quant-ph/9707021.pdf), it is well known that the vertex operators $A_g$ and the plaquette operators $B_h$ are dual to each other, so there ...
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Infinite stacking of integer quantum Hall systems
Let us consider a (3+1)-dimensional system $\mathcal{H}$ constructed by stacking (2+1)-dimensional integer quantum Hall systems $\mathcal{H}_\text{Hall}$, e.g., $E_8$ bosonic systems (or $\sigma_H=1$ ...
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Topological phase and Chern number
the relation between topological phase and Chern numbers is unclear to me.
For Haldane model if the Chern numbers of its two bands go from (+1,-1) to (0,0), we say that it goes from topological phase ...
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What does it mean by "condensation" of anyons?
My question is motivated from the paper Boundary degeneracy of topological order by Juven Wang and Xiao Gang Wen.
Consider a (2+1)D system with boundary, described by abelian Chern-Simons theory. Due ...
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Anyon and state spaces
I start learning about anyons, but I'm confused by a few Hilbert spaces.
First of all, it is said that anyons are "excitations" with anyonic statistics. By that I would imagine they are ...
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Does Fibonacci anyon come from a representation category of Hopf algebra?
I have heard that the UMTC(unitary modular tensor category) of Fibonacci anyon comes from a quantum group, but the representation category of Hopf algebra is equipped with a forgetful functor to $\...
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Role of fusion/splitting spaces in TQFT
In his book on topological quantum field theories Steven Simon writes that 2+1D TQFTs are objects that assign topologically invariant numbers to labeled links embedded in arbitrary 3-manifolds. They ...
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How to formulate the braiding of instanton?
I'm reading the paper https://arxiv.org/abs/2108.08835, after imposing the $\mathbb{Z}_2$ on-site symmetry, in the $J=0$ symmetric phase of the 1d Ising chain, the topological sectors of operators ...
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How can I actually get to the AKLT state from a product state in finite depth?
I'm currently learning about symmetry-protected topological phases in one dimension. The ground state of the AKLT model provides one such example. In particular, the AKLT state for any length $L$ ...
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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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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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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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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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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 ...