Questions tagged [topological-phase]
The topological-phase tag has no usage guidance, but it has a tag wiki.
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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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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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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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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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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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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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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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Berry curvature flux around a Weyl node
How can I formally show (or at least argue) that, given the crystal Hamiltonian expansion around a Weyl node in a three-dimensional Brillouin Zone located at $\vec{k}_{0}$,
$\hat{H}=f_{0}(\vec{k}_{0})\...
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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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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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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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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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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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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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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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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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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
$$S[A,B] = \...
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How can we judge the topological property of a material by looking at it's band structure?
I am a beginner of studying topological insulator. I want to ask some general question in this area to clarify my understanding. May be I am asking wrong, hope you can point me out.
If certain ...
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Topological order in Weyl Semimetal
Is the topological phase in a Weyl semimetal is intrinsic or symmetry protected? How can we realize that?
If symmetry protected, which symmetry protects the topological phase of non-centrosymmetric ...
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Must helical edge states be protected by time-reversal symmetry?
In a lattice system that exhibits quantum spin Hall effect (QSHE), like topological insulators in 2D or 3D, we call a pair of counter-propagating gapless edge states with opposite spin helical edge ...
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Topology of Helium 3A and 3B
The question concerns the topology and dimensions of Helium 3A and 3B
A. The Helium 3A phase shows the same low energy excitations as those of a 2 spatial dimensional chiral p-wave superconductor --- ...
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Why topologically non-trivial materials are robust againist any external perturbations or defects?
Topologically non-trivial materials are insensitive to perturbations or defects. How can I prove it mathematically?
I thought of making the first-order perturbation term zero.
$$\left< \psi \right|...
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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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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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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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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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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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Local explanation of the Aharonov-Bohm effect in terms of force fields
Here is an interesting paper for the Physics SE community:
On the role of potentials in the Aharonov-Bohm effect. Lev Vaidman. Phys. Rev. A 86 no. 4, 040101 (R) (2012). arXiv:1110.6169 [quant-ph].
...
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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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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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Are Weyl and Dirac points topological defects in nodal semimetals?
Recently, I heard the Weyl and Dirac points are topological defects in nodal semimetals. I do not really get it. And the definition of topological defects is confusing to me.
Are the topological ...
CommunityBot
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Why does an energy band crossing the Fermi energy mean the gap closes?
This online course on topology in condensed matter states the following:
We say that two gapped quantum systems are topologically equivalent if their Hamiltonians can be continuously deformed into ...
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Why is short-range entanglement defined in terms of its possible deformations?
After reading the question and answers in Definition of short range entanglement I wonder why the definition of a short-range entangled state is given in terms of its possible deformations -
A SRE ...
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Bosonic SPT phases with time reversal and a $Z_2$ symmetry
Consider a bosonic system with time reversal symmetry $\mathcal{T}$ and a unitary on-site $\mathbb{Z}_2$ symmetry. Suppose the symmetry is realized in a special way such that $$\mathcal{T}^2= (-1)^B$$ ...
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What is so topological about topological phase transitions?
I am studying the KT-transition, which is called a topological phase transition. The phase transition is driven by vortices in a 2-D superfluid, where it is shown that at a critical temperature $T_c$ ...
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What are gapless superconductors? [duplicate]
How are superconducting materials classified as gapped or gapless, also is this same as saying that a superconductor is conventional or unconventional? Could you explain how this is linked to topology ...
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The surface states and Fermi arcs in Weyl semimetals
I'm confused about surface states in Weyl semimetals. Assume that we have a single pair of Weyl points and the Fermi level turned to this points. In this https://arxiv.org/abs/1301.0330 paper the ...
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Exploring potential landscape with Monte Carlo
I am using a Monte Carlo approach for studying folding of a polymer chain. The polymer may fold in many configurations, corresponding to local potential minima, studying which is what interests me (i....
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Must the energy of a topological corner state in a 2D material vanish?
It seems that all the literature (just to name a few: Phys. Rev. Lett. 124, 166804, Phys. Rev. Research 2, 013330, Phys. Rev. Lett. 123, 073601, and Phys. Rev. Lett. 123, 256402) says yes to the ...
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Simple models with non-abelian anyons [closed]
It is well known, that in 1d and 2d there are particles with anyone statistics.
Which 1d and 2d models have such excitations? Which model with anyons is simplest?
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Entropy of $n$ topological excitations in the classical XY-model
I am dealing with Kosterlitz-Thouless phase transitions in the classical XY-model and trying to derive a formula for the entropy as a function of the number of vortices. In most textbooks, the entropy ...
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How to get algebraic PSG solutions once we got the constraints?
The question is more technical than conceptual. I've been trying to understand the classification of spin liquids as done by Prof.Wen. I have got the constraints on IGG(Invariant gauge group) elements ...
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Is this a topological $\mathbb Z_2$ (Majorana-)invariant in *any* dimension?
Consider a non-interacting superconducting Hamiltonian in an arbitrary dimension. This is most conveniently expressed in terms of Majorana modes, which are defined as $$\gamma_{2n-1} = c_n + c_n^\...
CommunityBot
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Mutual statistics between dyons (charge-monopole composite)
I am asking for some intuitive understanding between two dyons with $(e,m)$ in 3-dimensional space. Here the magnetic charge $m$ is normalized as
\begin{eqnarray}
m=\int_{S^2}\frac{B}{2\pi}\in\mathbb{...
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What are $U(n)$ or $\mathbb{Z}_2$ quantum spin liquids?
Quantum spin liquid is a state of matter in which spins are correlated and fluctuate even at zero temperature.
My question is about these terms in general. When we say that a state or a quasi-...
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