Questions tagged [topological-phase]
The topological-phase tag has no usage guidance, but it has a tag wiki.
287
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Interface between two different Su-Schrieffer-Heeger (SSH) chains
When I was exploring the interface (edge) states between two different SSH chains, I noticed some strange solutions.
I was considering the following system. Two different SSH chains are connected as a ...
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Meaningful topological invariants and quantities for the description of 3D topological insulators
I'm currently trying to understand the classification of 3D topological insulators (like Bi2Se3). Most reviews dealing with this topic start with the introduction of the Quantum Hall effect since this ...
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Topological Insulators with different spin band
To obtain a topological band insulator, we usually start with two bands with either spin up or down. If these bands now get 'inverted', they will cross. When there is coupling of these two bands such ...
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Topological Insulator [closed]
What effect on the Brillouin zone (torus) after applying the magnetic field? As in real space, pressure deforms the torus and up to a certain pressure, this remains invariant topologically. Similar to ...
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Majorana Fermions in superconductors (Wikipedia page)
In Wikipedia, Majorana Fermions in supercondutors are described as
Mathematically, the superconductor imposes electron hole "symmetry" on the quasiparticle excitations, relating the ...
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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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Berry phase from Bloch wave functions in the basis of Wannier functions
The formulate to calculate berry phase for Bloch wave functions is
$$
\gamma = i \sum_{n\in occ}\int_{\mathcal{C}} dk \langle \psi_k^n|\partial_k|\psi_k^n\rangle,
$$
where $|\psi_k^n\rangle$ is a ...
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How we can implement a periodic gauge in numerical calculations?
For a 1D system, there is a way to calculate the Zak phase in the discrete form. Suppose C is some closed path in k-space (a 1D BZ). If we suppose the path is discretized into (not necessarily ...
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Is there a zero correlation length spin-$1$ chain in the Haldane phase?
The ground state of the spin-$1$ AKLT model gives an example of a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological (SPT) phase, the Haldane phase. This state is a nice example of the ...
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Calculating degeneracy in Landau Gauge with fixed boundary conditions
In another question I asked the differences between aplying periodic boundary conditions and fixed boundary conditions. The answer is that both of them for large enough samples present the same amount ...
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Landau Levels degeneracy in a finite sample
According to different sources: Tong lectures on IQHE (Tong), MIT Open courses (MIT) etc, when calculating the number of states in each Landau Level all of them impose (in the Landau gauge) periodic ...
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Edge state protection in Chern insulator
I have a confusion about the nature of topologically protected boundary states in the Chern insulator. Since the Chern insulator does not require any symmetries to be present in the system, what is ...
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Topological properties of twisted TMD homobilayers
I'm reading this article about twisted TMD homobilayers (https://arxiv.org/abs/1807.03311) and there are certain topological properties that I don't understand:
On page 3, in the paragraph next to Fig ...
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66
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Symmetry and corresponding operator
For a quantum symmetry, is the operator of symmetry necessary to be unitary?
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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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Plausible finite group on-site-SPTs in realistic materials
I am looking for some understanding of which on-site symmetries in realistic crystalline materials (i.e. not just in random lattice models) can plausibly be expected to be realized and to induce ...
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Berry curvature concentration around nodal points
It is well-known that in TI-symmetric semi-metals the Berry curvature on the Brillouin torus vanishes away from the nodal points (eg. [XCN10, III.B] [Van18, p. 105]).
But even for non-TI-symmetric ...
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How to stack two Haldane chains?
This questions is a follow up to a pervious question of mine:
Inverse of Haldane phase?
Now that I know that Haldane phase is it's own inverse, I am having trouble is visualizing how could we stack ...
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What's the difference of flux tube and vortex in FQHE (especially in Jain wavefuntion)
In the book Composite Fermion by Jainendra K.Jain, he mentioned the motivation of Jain wavefunction: attach flux tube of 2p flux quantum to fermions to form composite fermions. Naively, this is done ...
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Inverse of Haldane phase?
Based on what I have learned so far, Haldane phases are a nontrivial SPT for 1D spin-1 chains. The trivial phase acts as an "identity" under the group of SPT phases ( with stacking as the ...
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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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Smooth deformation in topological systems
In various topological systems, it is common to encounter the concept of smooth deformation, which introduces changes in spectra of topological systems without allowing topological phase transitions. ...
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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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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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What kind of phase it is when a photon gain a quantum phase, a dynamical one or a geometrical one?
It's known$^1$ that the phase factor in quantum mechanics can be divided into geometric phase and dynamical phase. Since in quantum optics, light is treated as a quantum object, i.e., the photon. So ...
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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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What do the '4' and 'b' signify in a layer of a crystal called a '4Hb' crystal or material?
From Phys.org:
Study gathers evidence of topological superconductivity in the transition metal 4Hb-TaS2
Which, in turn, references:
Abhay Kumar Nayak et al, Evidence of topological boundary modes with ...
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Detection of topological phases
In the book A Short Course on Topological Insulators (https://arxiv.org/abs/1509.02295) the authors start with a simple toy model, the SSH-Chain, which is a bipartite one-dimensional lattice with ...
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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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The $\rm SO(8)$ invariant interaction piece in Fidkowski and Kitaev's model
In this paper (arXiv link), the authors demonstrate the existence of a quartic interaction $W$ involving the 8 majorana operators $c_1 \ldots c_8$ (eq. 8) which is invariant under an $\rm SO(7)$ ...
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What is the physical meaning of adiabatically varying the wavevector $k$ as a parameter to calculate the Chern number for topological effects?
Could it mean something like applying a weak electric field and perturbing the band structure? Or some other weak perturbation? Or is that the wrong idea?
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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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How to see that the trivial insulator is trivial?
I have been trying to better understand gapped phases of matter — which may be "topological" or "trivial" — and I have run into the problem that I don't really understand the ...
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Is there any sense in which the floor function is a "topological number"
As the title suggests, is there any sense in which the floor function can be thought of as some sort of topological number? Here by topological number I mean a topologically invariant robust quantity, ...
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What is the topological space in “topological materials/phases of matter”?
I’m embarrassed to admit that after sitting in on several “topological physics” seminars, I still don’t understand the basic ideas of the area. In particular, when physicists talk about the “topology” ...
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Spin accumulation in Rashba-Edelstein effect
I have been trying to figure out how to see the spin accumulation/spin current due to the Rashba-Edelstein effect from the k-space diagram. All that everybody says is that application of E-field ...
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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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Inversion Symmetry in Periodic Lattices
I am studying Short Course On Topological Insulator by J. K. Asboth, et.al.
In the context of inversion symmetry in section 3.2, the effect of inversion symmetry, $\Pi$, on the external degree of ...
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What definition of integral is implied when expressing nonzero Chern number as the integral of Berry curvature?
In defining a nonzero Chern number as the integral of Berry curvature over the parameter manifold: $$n=\frac{1}{2\pi}\int_{S}{\mathcal{F}}{dS}$$ does the integral exist in a general Riemann sense, or ...
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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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Phase in Quantum Theory
Can somebody please give the most general answer to:
"what is a Phase of a wavefunction in quantum theory?"
I have understandings and doubts so far as follows:
The non-relativistic quantum ...
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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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Which property lead to the conclusion that BZ is a torus?
In calculating the Chern number in 2D BZ, we assume that BZ is a torus. However, under periodic gauge, we have $\psi_k(r) = \psi_{k+G}(r)$, which means that $u_{k}(r) = u_{k+G}(r) e^{iG\cdot r}$. ...
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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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Calculation of Bulk and edge states in SSH model
I am reading “A Short Course on Topological Insulators” by János K. Asbóth. et.all., and want to calculate the Bulk and edge state of the SSH model (Chapter 1) to drive the energy spectrum in Fig. 1....
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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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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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