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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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Is topological entanglement entropy independent on the state of the system? [closed]

I just read a paper by M. Levin and X.-G. Wen (https://arxiv.org/abs/cond-mat/0510613) where they present a general formula to evaluate the topological entanglement entropy $\gamma$. They also prove ...
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What is the relationship between long range order and long range entanglement?

I'm wondering whether there's a deep connection between long-range entanglement and long-range order. Here's my understanding of these two concepts. Entanglement. Short-range entanglement basically ...
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On the definition of topological order

I am currently reading the "Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems". I am trying to understand the proper definition ...
Truth and Beauty and Hatred's user avatar
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What does it mean that a state spontaneously break some symmetry?

I'm currently reading this paper and I found the following statements confusing: On page 4: To study symmetric SRE states, we further demand that the ensemble of states ${|\psi_{I}\rangle}$ does not ...
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An analytical solution to the zero-energy bound state in d-wave superconductor

In the review paper by Sato & Ando (page 11, equation 92), the author claims a solution for the zero mode at the boundary (with a flat dispersion). But when I plug it back to the equation, it is ...
Huu-Thong Le's user avatar
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Significance of the phase of the condensate as compared to that of the regular Fermi sea in the Anderson-Higgs mechanism

I do not fully understand how the phase of the charged Cooper pair condensate is different from the phase of e.g. the Fermi liquid in a regular metal. The state of the metal (any quantum state really) ...
Rooky's user avatar
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Is QHE the only topological order up to 3D?

At the section II.C in this review paper about topological superconductors, the author states that The symmetry of the Hamiltonian plays a crucial role in determining the topology of the occupied ...
Guilherme Queiroz Venâncio's user avatar
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Constructing a gapped family of Hamiltonians in the trivial paramagnet

Consider the trivial paramagnet, which has the Hamiltonian $$H = - \sum_i \sigma^x_i$$ Now let's say I have two different Hamiltonians $$H_0 = H + 2\sigma^x_{i_0} \qquad H_1 = H + 2\sigma^x_{i_1}$$ ...
pyroscepter's user avatar
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What is the meaning of the statistical gauge field in the fractional quantum Hall effect

I'm a grad student studying the fractional quantum Hall effect. To get started, I read chapter 9.5.1 of A. Altland and B. Simons' Condensed Matter Field Theory. They use the composite fermion (CF) ...
Steffen Bollmann's user avatar
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Resources: Tensor Categories and Topological Phases of Matter

For a mathematician with knowledge of tensor categories who is interested in the growing application of categorical techniques in topological phases of matter and topological order, along with their ...
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Is there a Majorana representation for toric code

Kitaev's toric code is known to be the Z2 gauge field theory, which suggests that there might exists a Majorana representation for the toric code, e.g., Majorana + Z2 gauge field. Hence, I wonder if ...
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Can Toric Code have a gapless boundary?

The toric code model is known to have two types of "gapped" boundaries, namely, the rough boundary and the smooth boundary. See, for example, Chap. 4.1 of this beautiful review https://arxiv....
Quasiphysics's user avatar
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Can we define topological order in the context of QFT?

Topological order is defined to be a phase that has ground state degeneracy (GSD) not described by the Landau SSB paradigm but exhibits some Long Range Entanglement property. Mathematically, it is ...
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Intuition for ground state degeneracy of Majorana checkerboard model

I'm now trying to learn about Fracton. In the very early paper studying Majorana checkerboard model, it is claimed that the ground state degeneracy ${D_0}$ on ${L \times L \times L}$ 3-torus is ${...
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Product of Majorana operators after a orthogonal transformation

The question I want to ask is the following: There are $N$ Majorana fermion modes: $\gamma_1, \gamma_2, \dots, \gamma_N$, and they satisfy the anti-commutation relation: $\{ \gamma_i, \gamma_j \} = 2\...
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Higher category's consistency relations

I have been reading on higher category and symTFTs. It appears to me that, for higher categories, people seldom mention the consistency relations (like the MacLane coherence theorem in the category ...
Waterfall's user avatar
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What is topological about topologically ordered states?

My naive understanding based off section 4.8 of Nakahara's Geometry Topology and Physics is that topologically ordered phases are ground states that are homotopically inequivalent to other possible ...
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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 ...
Barry's user avatar
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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 ...
kBoltzmann's user avatar
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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 ...
Yuezhao Li's user avatar
3 votes
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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 ...
TheQuantumMan's user avatar
1 vote
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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 ...
Hans Schmuber's user avatar
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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 ...
DeafIdiotGod's user avatar
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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 ...
Iris Cong's user avatar
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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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3 votes
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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 ...
Laplacian's user avatar
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2 votes
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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 $\...
edittide's user avatar
3 votes
1 answer
159 views

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 ...
Zarathustra's user avatar
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0 answers
30 views

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 ...
Daniel YUE's user avatar
2 votes
1 answer
387 views

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 ...
Feynnman pilows's user avatar
1 vote
1 answer
67 views

"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 ...
eepperly16's user avatar
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1 vote
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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 \...
GGBOND's user avatar
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4 votes
2 answers
265 views

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 ...
Another User's user avatar
3 votes
0 answers
56 views

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 ...
catalogue_number's user avatar
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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 ...
danport's user avatar
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1 answer
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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?
Aman Anand's user avatar
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32 views

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 ...
Tan Tixuan's user avatar
1 vote
1 answer
365 views

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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