Questions tagged [topological-phase]

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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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21 views

360 degree 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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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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47 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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23 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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Calculating Winding Number Numerically in MATLAB

How can I calculate the Winding number numerically in Matlab? I mean, I want to calculate the $$W=\frac{1}{2\pi i}\int_{-\pi}^\pi dk \frac{d}{dk}Ln h(k)$$ by using a loop that changes k from $-\pi$ to ...
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55 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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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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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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72 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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51 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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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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52 views

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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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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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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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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66 views

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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Definition for long range entanglement (LRE) by generalized local unitary (gLU) and generalized stochastic local (gSL) transformations

I am studying this book: Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems (https://arxiv.org/abs/1508.02595). In chapter 7, it introduces ...
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What are gapless superconductors?

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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50 views

What does “parity eigenvalue” mean in Fu-Kane formula?

I'm studying the online course "Topology in Condensed Matter", in the QSHE section (<https://topocondmat.org/w5_qshe/fermion_parity_pump.html>), I've studied the Fu-Kane formula $$ Q=\...
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133 views

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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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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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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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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20 views

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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60 views

Topological invariants, what's that?

What's the difference between the Berry phase, the Euler number,the winding number and the Chern number? As far as I know they can all be computed by the same integral, but there seems to be some ...
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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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Phases and order in condensed matter beyond Landau

All materials in our world consist from electrons and nucleus. They non-trivially interact and can create different states of matter with very different properties. Some states of matter (or phases) ...
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Is there an AC version of the Quantum Hall Effect?

The quantum Hall effect has the well-known signature of plateaus in the Hall conductivity $\sigma_{xy}=n e^2/h$ for integer (or rational) n. This quantization is extremely precise, and can go up to ...
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Relativistic Dispersion In One Space Dimension

I'm now reading Three Lectures On Topological Phases Of Matter by Edward Witten and face some statements that are unclear to me. According to lectures: As I understand, electronic excitations in ...
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What are topological states of matter and what is topological symmetry breaking?

What is topological order? How topological states of matter applied to mechanical/classical systems? How this definition of topological order is applicable to classical systems? Is it something ...
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393 views

One dimensional phase transitions

Due to R. Peierls argument there is not phase transitions is one dimensional lattice systems. Argument in $d=1$ goes like that: flipping of one spin in system of N spins will lead to change of free ...
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Alternatives for calculating topological invariants in topological materials

My questing is regarding the different alternatives for calculating topological invariants in topological materials protected by symmetry, specially time-reversal invariant topological insulators, ...
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234 views

Where is the connection between $U(1)$ gauge field and $\mathbb{Z}_2$ gauge theory?

I am a graduate student in condensed matter physics and today I was reading the Wikipedia article Topological Order. There is the part: Note that superconductivity can be described by the ...
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Gauging a symmetry-protected topological (SPT) phase

In this answer, it is said that gauging the symmetry which protects a symmetry-protected topological (trivial) phase gives something "morally very similar" to a phase with a topological order. What ...
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Spin structure, Intersection form and 2D TQFT

I am reading this (page 20) and watching this Anton Kaputins' talk (33:24). Here, he tried to explain how to define a spin structure on a lattice in a closed oriented (1+1)-D manifold $M$ (or at ...
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Gauged DIII superconductor and Z_2 topological order

I keep hearing that a gauged 2D topological superconductor with preserved time-reversal symmetry that belongs to the DIII class is equivalent to the Z_2 gauge theory (Z_2 topological order with ...
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Research: Mott insulator and topological order

I'm an experimentalist who is mainly focusing on strongly correlated electron systems (SCES), in particular Metal-insulator (Mott) transitions in the classical example $V_2 O_3$. Recently I decided to ...
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Reason for Kitaev chain Pfaffian being calculated in momentum space

In Kitaev's paper where Kitaev defines his toy model for observation of Majorana fermions (MFs), he suggests a method of determining whether a system contain MFs at it ends. This method is to ...

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