Questions tagged [topological-phase]

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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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1answer
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Are there topological non-trivial states in zero dimension?

The periodic table of topological insulators and superconductors suggests that there can be topological non-trivial phases in zero dimension in non-interacting system with certain symmetries. A 0D ...
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1answer
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Has The Aharonov-Bohm Effect Been Experimentally Proven?

I have encountered two contradicting papers on the Aharonov-Bohm Effect: One supporting, The Aharonov-Bohm Effects: Variations on a Subtle Theme. H Batelaan and A Tonomura. Physics Today 62 pp. 38-...
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Why is the phase picked up during identical particle exchange a topological invariant?

I've been wondering about the standard argument that the only possible identical particles in three dimensions are bosons or fermions. The argument goes like this: Consider exchanging the positions ...
12
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1answer
796 views

Aharonov-Bohm effect as a geometric phase-Adiabatic transfer not needed?

In his 1984 paper, Michael Berry proved that the Aharonov-Bohm effect is the same as a geometric phase. He did this by transferring a box containing charged particles around a solenoid. However, he ...
12
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1answer
544 views

Has anyone experimentally shown the quantized thermal hall conductivity in Quantum Hall systems?

For background: In a $D=2+1$ state with edge modes described by a chiral $( c_L \neq c_R )$ CFT there is a predicted thermal Hall conductance associated with the gravitational anomaly at the edge. ...
11
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2answers
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Why do lattice models of fermions need a spin structure?

It is well-known that in order to define a relavistic quantum-field theory containing fermions on a general manifold $M$, the manifold $M$ needs to be equipped with a spin structure. The spin ...
10
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2answers
681 views

1+1d TSC as $Z_2^f $ symmetry breaking topological order?

I have been struggling recently with a comprehensive problem on the relationship between topological superconductor and topological order. My question originates from reading a work conducted by Prof. ...
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4answers
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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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Definition of short range entanglement

When studying Symmetry Protected Topological phases, one needs to define what a short range entangled (SRE) states means. But there appears to be different definitions that are not equivalent to each ...
9
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2answers
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Is band-inversion a 'necessary and sufficient' condition for Topological Insulators?

According to my naive understanding of topological insulators, an inverted band strucure in the bulk (inverted with respect to the vaccum/trivial insulator surrounding it) implies the existence of a ...
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3answers
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What does the Chern number physically represent?

In 2D the Chern number can be written as $$C_m=\frac 1{2\pi}\int_{BZ}\Omega_m(\mathbf k)\cdot d^2 \mathbf k$$ where we are integrating over the Brillouin zone. In 2D this is equivalent to finding ...
9
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1answer
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Does the Kosterlitz–Thouless transition connect phases with different topological quantum numbers?

The Kosterlitz-Thouless transition is often described as a "topological phase transition." I understand why it isn't a conventional Landau-symmetry-breaking phase transition: there is no local ...
8
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1answer
577 views

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 $T_c$ ...
8
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1answer
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Jordan Wigner Transformation in 1d Majorana chain

So, I was reading the paper by Fidkowski and Kitaev on 1d fermionic phase http://arxiv.org/abs/1008.4138. It explains the classification of 1d fermionic SPT phases with $\mathbb{Z}_2^T$ symmetry for ...
8
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1answer
204 views

Relation between a change in the topological invariant and the closure of the gap

I would like to understand the relation between a change of the topological invariant (e.g. when the Chern number changes from 1 to 2) and the closure of the gap of a condensed matter system. I know ...
8
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Physics uses of $SO(8)$ and Spin(8) triality [closed]

Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation ...
7
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1answer
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What does it mean for a topological phase to be “symmetry protected”?

I have seen some very nice and enlightening awnsers to questions related to topological order and insulators, such as here, or here. However, I'm still puzzled by the concept of "symmetry protection" ...
7
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1answer
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AKLT state and Nobel physics prize 2016

The AKLT Hamiltonian and the chain is described in Wikipedia, and also the page 17 of this year Nobel Prize advanced information I have questions concerning the info released by nobelprize.org, and ...
7
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1answer
291 views

Meissner effect and topology

Is there any analogy between Quantum Hall Effect and Meissner Effect? In other words, can we relate the existence of edge currents in superconductors (of type 1) with edge modes in QHE (that yield ...
6
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1answer
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Topological number of 1-D p-wave superconductor---Kitaev model/wire

After learning the Kitaev model, I tried to reformulate it and encounter some conceptual loopholes of my own. Here the setting: Given the 1-D chain Hamiltonian (differed from original form proposed ...
6
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1answer
487 views

Homotopy Theory for Topological Insulators

I'm trying to understand topological insulators in terms of homotopy invariants. I understand that in 2 spatial dimensions, we have Chern insulators since $$\pi_2(S^2) = \mathbb{Z}$$ One subtlety that ...
6
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1answer
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$Z_2$ topological insulator: odd vs. even number of edge state pairs

I am having trouble in understanding why in $Z_2$ topological insulators odd number of Kramers' pairs on one edge are protected by time reversal symmetry against elastic backscattering while even ...
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0answers
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What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
5
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3answers
283 views

Small confusion about the Aharonov-Bohm effect

I am mostly aware of the Aharonov-Bohm effect's (AB effect) physical interpretation, as well as the corresponding mathematical/differential geometric interpretation. What does confuse me slightly ...
5
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2answers
357 views

An exactly solvable model of 2D Majorana zero modes

The Kitaev's Majorana Model is an exactly solvable model of p-wave superconductor with localized Majorana zero modes in 1D quantum wire. For the 2D case, the general theory of Majorana zero modes near ...
5
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1answer
191 views

Is topological surface state always tangential to bulk bands?

Think of a topologically nontrivial $D$-dimensional system. Its bulk bands form a $D+1$-dimensional manifold ($+1$ from energy). Its surface/edge bands form a $D$-dimensional one. Is the latter always ...
5
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1answer
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Topological materials and fractionalized excitations

I've been told several times that topological materials (such topological insulators) must have "fractionalized" excitations. Equivalently, if a material does not have fractionalized excitations, it ...
5
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1answer
203 views

Why does a monopole operator break the global symmetry with topological current?

I am currently reading the paper "A Duality Web in 2+ 1 Dimensions and Condensed Matter Physics" by Seiberg et al, and on page 22 they add to the Lagrangian a monopole operator of the form $\phi^{\...
5
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1answer
393 views

Goldstone mode as spin wave in 2D?

I'm trying to understand how Goldstone modes destroy long range order in 1D and 2D spin lattice. I started with a spin chain, using 1D XY-model, which has continuous symmetry. $H=- \sum_{<i j>} ...
5
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1answer
147 views

Topological Phase Transition v Quantum Phase Transition v Phase Transition

What are the main differences between this 3 type of phase transition? I understand the phase transitions of gas/liquid/solid as well as ferromagnet/paramagnet(Ising Model). All of which are between ...
5
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2answers
511 views

SPTs and systems with Topological Order

I am an undergrad interested in Condensed Matter Theory. Particularly topological phases and systems exhibiting topological order. A potential research advisor doing a lot of work in Symmetry ...
5
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0answers
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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 ...
5
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1answer
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Formula for the topological invariant for each of the symmetry classes

Is there a reference that systematically derives the topological invariant/winding number for all the ten symmetry classes in Altland and Zirnbauer's periodic table? For example, in this answer, there ...
5
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0answers
264 views

What is the different between topological order and Landau's order in a system

I have thought about topological order for a long time, but I am still confused it. Roughly speaking in my understanding, the topological state is the eigen-state of some special symmetry such time ...
5
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0answers
164 views

Topolgical insulators order parameter

For topological insulators Is there any way to define order parameter for topological phase transitions?
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2answers
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Topological insulator vs. topological superconductors in any dimension

My question today is simple. What is the difference between a topological insulator and a topological superconductor? How that difference depends on the dimensionality of space(time)? What is the ...
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1answer
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Chiral anomaly in Weyl semimetal

In the presence of electromagnetic fields $E$ and $B$, four current is not conserved in a Weyl semimetal i.e. $\partial_{\mu} j^{\mu}\propto E\cdot B \neq 0$. There are some proofs in the literature ...
4
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1answer
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What is the difference between Bosonic and Fermionic symmetry protected topological phases (SPT)

I am reading the paper ``Braiding statistics approach to Symmetry Protected Topological Phases'' by Levin and Gu. In this paper two spin models considered describe spin-1/2 particles in (1+2) ...
4
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1answer
596 views

A naive question about topologically ordered wavefunction?

Topological entanglement entropy (TEE, proposed by Levin, Wen, Kitaev, and Preskill) is a direct characterization of the topological order encoded in a wavefunction. Here I have some confusions, and ...
4
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1answer
350 views

How to distinguish between a topological state from from a non-topological one?

How to distinguish between a topological state from from a non-topological one? Is there any standard procedure for identifying the topological features of a given hamiltonian? In general what are the ...
4
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1answer
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Kane and Mele's argument on the existence of edge states in quantum spin Hall effect of graphene

Borrowing from Laughlin's argument on quantum Hall effect, Kane and Mele argued why there must be edge states in graphene with spin-orbit coupling in one paragraph, which is above the one with ...
4
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1answer
261 views

Relation between cobordism and unitary fusion category classification of TQFT/ SPT phase

In the introduction part of Gaiotto's paper (https://arxiv.org/pdf/1712.07950.pdf), he says "in the context of topological field theory, homotopy-theoretic ideas also lead to the classification of ...
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1answer
862 views

Berry curvature of Landau levels

If we consider an electron on a two dimensional surface with a magnetic field normal to the surface, we know the states the electron can occupy are Landau levels. If we additionally impose periodic ...
4
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1answer
591 views

Vortices and chemical potential in topological superconductors

I am trying to read up some review articles about Majorana physics in topological material, but I am not really familiar with the condensed matter terminology (with condensed matter in general I ...
4
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1answer
690 views

Aharonov-Casher effect for charged particles

All the explanations of the Aharonov-Casher effect seem to imply it only "works" for neutral particles with a magnetic moment. This seems to stem from the duality of the A-C effect with the more known ...
4
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1answer
644 views

Relation of Berry phase and winding number

I am reading the following article dealing with the properties of Dirac fermion in condensed matter physics : https://arxiv.org/abs/1410.4098 In the page 5 of this article, the formula for the ...
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1answer
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why entanglement entropy is important in topological phases?

When mentioning interacting topological phases people always talk about entanglement entropy. why it is important? what is its physical meaning?
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1answer
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Is the topological index of a self-adjoint operator always zero?

By the Atiyah-Singer index theorem, the index of a self-adjoint opeartor D (e.g., Hamiltonian) is given by Index(D) = dim Ker(D) − dim Ker(D*), where D* is the adjoint operator of D. Since D is self-...
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1answer
238 views

Rigorous definition of superselection sector/quasiparticle type in anyon systems

The systems I have in mind are for example Kitaev's toric code model (arXiv:quant-ph/9707021) and Kitaev's honeycomb model (arXiv:cond-mat/0506438). What I'm looking for is a mathematically rigorous ...