Questions tagged [topological-phase]

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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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8 votes
2 answers
380 views

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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Understanding polarization and Zak phase

I was trying to understand the arbitrariness of polarization and Zak phase. Consider the following example, for this 1D lattice with wavefunction mostly localized at the atomic sites, the polarization ...
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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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Topological hinge states at only one pair of diagonal edges?

High-order 3D topological systems can have hinge states at the edges like the figure below from this paper. But can we have only one pair of counter-propagating diagonal edges? I mean remove the red ...
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Is the derivative of Berry curvature with respect to band energy possible?

The Berry curvature is one of the most important quantities for a topological material like Dirac semimetal(DSM), Weyl semimetal(WSM) etc. Berry curvature is always momentum ($\mathbf{k}$) dependent ...
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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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Proof of correlation revival in Kitaev chain

I am searching for detailed notes on the Kitaev chain. In particular, I am looking for a proof of the correlation revival in the single-particle correlator from one edge to the other. Do you know how ...
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Reference for Kitaev chain correlation functions

I am searching for detailed notes on the Kitaev chain to get familiar with the computations that one should perform on it. In particular, I am looking for a proof of the correlation revival in the ...
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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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3 votes
0 answers
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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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Topological Degeneracy in FQH Liquids

It is well known that fractional quantum Hall states have topological degeneracy: ground state degeneracy that depends on the topology of the closed manifold on which the state exists. As explained ...
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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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Why stacking two $p+ip$ superconductors or superconductors with Chern number 1 ($C=1$) is a quantum hall state?

My question is based on the lecture by Bernevig in PiTP 2015 on "Category Theory and the Kitaev 16 Fold Way"41:00. Why by stacking two superconductors with Chern number $C^{(1)}=1$ we have ...
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2 votes
1 answer
217 views

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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1 answer
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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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3 votes
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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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2 votes
1 answer
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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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1 vote
1 answer
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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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2 answers
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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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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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7 votes
1 answer
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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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1 answer
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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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2 votes
1 answer
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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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1 vote
2 answers
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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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1 vote
0 answers
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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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1 vote
0 answers
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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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2 votes
1 answer
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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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