Questions tagged [topological-phase]

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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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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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56 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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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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60 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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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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56 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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242 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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167 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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How do we obtain Eq. 3.15 in Fu and Kane's PRB 74, 195312 Time Reversal polarization and a $Z_2$ adiabatic spin pump

This is a probably a very noob question. I attempted to derive Fu and Kane's Eq. 3.14 from the defnition of $A^I$ in (3.12) but ran into a bit of trouble with the minus signs with the $A^I(-k)$ ...
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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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28 views

quantum order in a (generalized) thermal steady state

It is known that starting from an initial product state, non-integrable systems will thermalize and eventually local observables can be described by a Gibbs ensemble. It has also been argued that a ...
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Does flat band imply localization?

Consider the Kitaev chain, whose Hamiltonian is as follows: $$ H = -\mu \sum_n c_n^\dagger c_n -t\sum_n (c_n^\dagger c_{n+1} + \mathrm{h.c.}) +\Delta \sum_n (c_n c_{n+1} + \mathrm{h.c.}) $$ I have ...
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About symmetry constraints in momentum space

When people study symmetry protected topological phases, certain symmetry constraints are enforced on the Hamiltonian. Specifically, for non-interacting fermionic systems, we could focus on the ...
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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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what is the relationship between topological charge and chern number in topological materials?

I would like to ask, what is the relationship between topological charge and chern number in topological materials? Why the topological charge of the Dirac cone is $0$?
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Meaning of complex pairing terms in Kitaev chain

I am studying some properties of the one dimensional Kitaev chain, which has the following form: $ H = -\mu \sum_n c_n^\dagger c_n - t \sum_n (c_{n+1}^\dagger c_n + h.c.) + \Delta \sum_n (c_n c_{n+1} ...
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Algebra of Time Reversal and Particle Hole Symmetry in 10-fold Classification of Topological Insulator/superconductor

In the ten fold classification of TI/TSC, when time reversal symmetry $\mathcal{T}$ and particle hole symmetry $\mathcal{P}$ are both present, i.e., in the symmetry classes BDI, DIII, CII, CI, for all ...
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Time-reversal (explicitly) broken surface of $(3+1)$-dimensional topological insulator

Let us consider the surface of $(3+1)$-dimensional topological insulator, which is protected by the charge conservation $U(1)_Q$ and a time-reversal symmetry $\mathbb{Z}_2^T$. Such a surface, if not ...
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Does a topological charge always need to be an integer

Does a topological charge always need to be an integer, I see many papers where people talk about non-integer topological charges due to boundary conditions. According to the formula for the ...
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Why are degenerate ground states interesting?

Studying the Su-Schrieffer-Heeger chain I have learned that the model has two different phases, one which is called topological and the other one trivial. In the notes it says that these phases are ...
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Topological phases and quantum information

I am concerned about the theorem saying that there is no topological order in 1d. According to the seminal paper https://arxiv.org/pdf/1008.3745.pdf, there are no non-trivial topological phases in 1d (...
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Calculating topological invariants under different conventions of tight-binding models

There are two widely used conventions to construct the Bloch-like basis in a tight-binding model [1]. Convention I: $$ \psi_\mathbf{k}=\frac{1}{\sqrt{N}}\sum_{\mathbf{R},j}c_j(\mathbf{k})e^{i\mathbf{...
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What is the critical temperature for a BKT transition in the 2D quantum XY model with $S=1$ (not $S=1/2$)?

What is the critical temperature for a BKT transition in the 2D quantum XY model with $S=1$ (not $S=1/2$)? For instance, the classical XY model has KTc/J = 0.898 and the quantum XY model with S=1/2 ...
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How is group cohomology in SPT's related to the 't Hooft anomaly on the boundary?

I understand that group cohomology description for symmetry protected topological phases (SPT) comes from discrete nonlinear sigma models. A tutorial on this can be found in the excellent lectures by ...
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What happens to topological insulators at finite temperature?

There is a similar question here, but I had a few things I wanted to ask. So basically pretty much all analysis/ theory of topological insulators is for pure wave-functions and conservative ...

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