Questions tagged [topological-phase]

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What is a bulk state and bulk bands?

I am a bachelor student and I started studying topology and I came across two terms I have never seen before: Bulk band structure and bulk states. Can someone explain these two terms or provide me a ...
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Why does particle-hole symmetry in 1D lead to a $Z_2$ topological invariant?

From the well-known AZ Tenfold Classification Table, a 1D system with square-positive particle-hole symmetry belong to class D and hence is characterized by a $Z_2$ topological invariant. I suppose ...
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Uniqueness of AKLT Ground State vs. SU(2) symmetry and Lieb-Schultz-Mattis theorem

I have a question in my mind regarding the uniqueness of AKLT ground state. Currently I am watching a video clip of MPS and I am curious why the AKLT ground state model is unique gapped ground state. ...
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51 views

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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60 views

What is $\nu$ in this equation regarding phases of anyons under exchange?

From the Quantiki article on anyons: After exchanging two identical particles the quantum mechanics predicts that the wave function gain a phase factor: $$ \Psi\to e^{i\theta}\Psi $$ For bosons $θ = ...
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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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43 views

Homotopy group for spin-1 BEC

Homotopy group can be used to classify topological defects. The procedure is Find the Lie group $G$ that leaves the free-energy functional invariant when transforming $\psi$, where $\psi$ is the ...
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Phase freedom of the edge states in topological insulator

Suppose that we consider the BHZ-like Hamiltonian of the form $$ H_{bulk}=\left(M-B k^{2}\right) \tau_{z}-A k_{x} \tau_{y}+A k_{y} \sigma_{z} \otimes \tau_{x} $$ where $\tau_i $ acts on the orbital ...
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1answer
32 views

Classification of topolgical phases when eigenstates belong to complex Grassmannian

I want to understand the paper which belongs to Ludwig (I put it below). I do not understand why exactly he got the new space $U(m+n)/U(m) \times U(n)$. My understanding from Grassmannian Manifold is ...
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68 views

Topological phases of matter

So according to this, scientists have discovered more than 5 states of matter we usually had that is the solid, liquid, gases, and Bose-Einstein-Condensate, and plasma. So how many topological phases ...
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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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Related to Topological phases of matter

I have a very fundamental doubt, I am new to this field and I learned that the quantum hall effect was the first topological state of matter discovered, so is every step in the graph of the quantum ...
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21 views

Empirical definition of gapped quantum system

We can define a gapped quantum system theoretically by placing some conditions on the energy eigenvalues of (the elements of) a sequence of lattice hamiltonians in the thermodynamic limit, cf. this ...
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30 views

Homotopy classification in ten-fold way

I am trying to understand algebraic invariants in topological insulators and topological superconductors through homotopy. But I encounter kind of a conceptual question. Let's say we have a second ...
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45 views

Berry phase of generic two-dimensional gapless Dirac Hamiltonian

Reference: Topological Insulators and Superconductors, B. Andrei Bernevig, Taylor L. Hughes: Chapter 8, problem 1 The generic Bloch Hamiltonian $H(k)=k_i\mathcal{A}_{ij}\sigma_j$, with $i\in\{1,...
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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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Interpretation of adiabatic assumption in quantum mechanics

This thought just occurred to me. I recall from my quantum mechanics courses that adiabatic transformation is defined as a process in which a band-gap is kept open while the process is carried in a ...
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1answer
89 views

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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32 views

Is the Berry phase defined in terms of the periodic part of the Bloch wavefunction or the wavefunction itself?

In this paper, the berry phase is approximated to be $e^{-i\theta} = \prod_{i=1}^{N} \langle\psi_{n,k_i} | \psi_{n,k_{i+1}} \rangle$. The authors claim that "each Bloch wavefunction appears twice ...
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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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1answer
66 views

Missing factor 1/2 when using generalized Stokes theorem

I'm doing the following homework question: By invoking Stokes' theorem, according to which the integral of a vector field (which equals the field strength) over any two-dimension surface S that is ...
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Partition functions of descendent SPTs of the Haldane chain

The Haldane chain can be viewed as a $1+1$ D SPT protected by an $SO(3)$ symmetry. If this SPT is put on a triangulated closed manifold $X$, its partition function can be written as $$ e^{i\pi\...
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The meaning of phase operator in Majorana zero mode

In some article, such as Phys. Rev. B 94, 235446 (2016), they define the Majorana mode operator as follow $$ \gamma_j=\int \mathrm{d}r \ [\xi_j(r)e^{-i\theta/2}c^\dagger(r)+\xi_j^*(r)e^{i\theta/2}c(r)]...
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Order of topological phase transitions

I heard in a talk that topological phase transitions are generally higher order than two, and are described by non-local order parameters. Is there an argument why the order is greater than 2? Is ...
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68 views

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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42 views

Magnetization and Polarization in an electromagnetic field theory

I am currently reading through a paper by Hughes and Ramamurthy (ref: https://arxiv.org/abs/1508.01205), which describes the electromagnetic response of a line-node semimetal by the action $$S[A,B] = \...
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1answer
64 views

Argument for number of edge states as topological invariant for SSH model

I am currently reading the book "A short introduction to Topological insulators" by Asboth etal. In the first chapter on SSH model, they argue (see sec 1.5.3) that number of edge states is a ...
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44 views

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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39 views

How can we judge the topological property of a material by looking at it's band structure?

I am a beginner of studying topological insulator. I want to ask some general question in this area to clarify my understanding. May be I am asking wrong, hope you can point me out. If certain ...
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89 views

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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Some questions about Axion Electrodynamics

We know that the action of Maxwell's equation can add another term, that is $\theta$ term: S$\theta$=$\theta\int\vec{E}\cdot\vec{B}$. My questions are: 1.Why in TI(topological insulator), the $\theta$ ...
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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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29 views

Must helical edge states be protected by time-reversal symmetry?

In a lattice system that exhibits quantum spin Hall effect (QSHE), like topological insulators in 2D or 3D, we call a pair of counter-propagating gapless edge states with opposite spin helical edge ...
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Topology of Helium 3A and 3B

The question concerns the topology and dimensions of Helium 3A and 3B A. The Helium 3A phase shows the same low energy excitations as those of a 2 spatial dimensional chiral p-wave superconductor --- ...
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Single crystals for observing toplogical phase

The topological devices are mostly fabricated by single crystal growth technique. Is it necessary to have monocrystal for observing topological phase? If yes, why so? Else, can we observe the ...
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1answer
62 views

Why topologically non-trivial materials are robust againist any external perturbations or defects?

Topologically non-trivial materials are insensitive to perturbations or defects. How can I prove it mathematically? I thought of making the first-order perturbation term zero. $$\left< \psi \right|...
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351 views

What is the relation between non-local order parameters and topological phases?

I know of several definitions of phases of matter: The first is the "old" one, Landau theory and symmetry breaking. In this definition we pick a local order parameter $m$ (as far as I can ...
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Physical meaning of gapped path between Hamiltonians in the same phase

I'm reading this famous paper about the classification of quantum phases, and I'm wondering about the physical meaning of the definition of phases the authors use. They say that two Hamiltonians $H_0$ ...
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46 views

Staggered Zeeman field in topological magnetic insulators

I was reading the following paper. However, I do not understand a crucial part of their argumentation. They add a parity (P) and time (T) symmetry breaking term to the Hamiltonian in eq (2). Then they ...
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1answer
133 views

What is topological in Kitaev Chain

What is topological in Kitaev Chain? Realspace or the space of Pauli spins or the space of fermions? My Understanding I understand that majorana-zero modes are which are spatially separated, are ...
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1answer
55 views

One question about topological excitation in quantum many body system

I attended a lecture given by Professor Wen Xiaogang. In the lecture, Prof.Wen gave an example of topological excitation: For a state $$(\uparrow\downarrow)(\uparrow\downarrow)(\uparrow\downarrow)(\...
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Quantum Hall effects with an additional uniform unit flux on a compact manifold

I have two questions: Let us imagine that we have an integer quantum Hall system with electric Hall conductance as $\sigma_\text{H}$ on a two-dimensional (spatial) torus with size $L_1\times L_2$. If ...
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1answer
215 views

Difference between “ordinary” quantum Hall effect and quantum anomalous Hall effect

I am reading a review article on Weyl semimetal by Burkov where he writes, top of page 5: A 3D quantum anomalous Hall insulator may be obtained by making a stack of 2D quantum Hall insulators [Ref. ...
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Why don't certain decorated domain wall constructions for SPTs lead to spontaneous symmetry breaking?

There is a construction of symmetry protected topological (SPT) states which roughly goes as follows. We start with a $d$-dimensional system with symmetry $\mathbb{Z}_2 \times G$ in the phase where ...
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Construction of symmetry group algebra

In Kitaev's reasoning of constructing the algebra of symmetry group, he said, "considering the symmetry group G of a fermionic system and a map $\alpha$ $$ \alpha: G \rightarrow \mathbb{Z}_2 $$ ...
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Superfluids in areogel and porous media: why?

Aerogels are materials that are like ~90% or more air. As I understand, the topology of the material (i.e. of that part of the aerogel that is not air) is not such that air is contained into bubbles. ...
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1answer
182 views

Deriving the non-abelian Aharonov-Bohm effect as a Berry phase

I am trying to derive the non-abelian Aharonov-Bohm effect by generalising Michael Berry's derivation to the case of non-abelian gauge field $A$. My derivation so far We require a degenerate ...
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What is topological about topological (Dirac or Weyl) semimetals?

The following is my rough understanding of topological phases of matter (please let me know if it is incorrect.) Topologically ordered phases of matter are topological in the sense that they are ...
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65 views

Chern number for nonintracing hamiltonian while bands crossing

Is it possible to define and calculate chern number for two bands while they're crossing each other?

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