Questions tagged [topological-phase]

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Qualitative understanding of Hamiltonian Terms for Quantum Phases

I have been reading up on topological order and quantum phases which are continually being discovered in condensed matter systems. (Here's a great article...https://www.quantamagazine.org/physicists-...
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Why is the seperation of a tight pair of vortices a Topological Phase Transition?

I have been doing some research on Topology in Physics and so I came across this picture Source is this link. Now the way I understood Topology so far is that you can classify specific ...
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1answer
25 views

Relative phase between neighbouring states in continous parameter space

The relative phase of two quantum states $|\psi_1\rangle,|\psi_2\rangle$ can be written, $$\gamma_{12}=-\text{arg}\langle\psi_1|\psi_2\rangle=-\text{arg}\big[|\langle\psi_1|\psi_2\rangle|e^{-i\gamma_{...
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 ...
4
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0answers
166 views

Different definitions of topological phases

When doing classification of topological phases, one need to formalize the problems mathematically. But, it seems that there are two not obviously equivalent ways to describe topological phases. In ...
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208 views

Machine Learning toric code ground states and phase transition under perturbation

I was wondering if the following is a viable method using machine learning and neural networks to get to the ground states of the toric code and also understand the phase transition in the presence of ...
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1answer
278 views

Heisenberg ferromagnet in continuum limit

I consider the case of the simple, say 2D, Heisenberg ferromagnet with exchange interaction between the nearest neighbors. The Hamiltonian is: $$H = -J \sum_{<ij>} \mathbf S_i \mathbf S_j,$$ ...
3
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2answers
386 views

Effects of Topological Terms: Hopf, $\Theta$, Chern-Simons, WZW, Berry phase term

What are the effects and the differences of Topological Terms? For example, I had known and heard several of them are called Topological, (1) Hopf term, (2) $\Theta$ term, (3) Chern-Simons term,...
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1answer
120 views

Part 2 - - Terminology — phase space embedding

I am facing difficulties in understanding the definition of Takens' delay embedding from a non-physicist point of view....too much technical jargons. Can somebody please provide a simpler way to ...
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439 views

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 ...
2
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0answers
124 views

Status of the discovery of non-abelian anyons and topological quantum computation?

This week Microsoft announced that it will make available the programming language for quantum computer available by the end of this (2017) year. https://news.microsoft.com/features/new-microsoft-...
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2answers
609 views

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 ...
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435 views

Chiral symmetry vs quantized Zak phase

I've been doing some condensed matter research about the topological phases in one dimension system and have some questions. I've heard that the chiral symmetry leads to the $\pi$-quantization of Zak ...
2
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1answer
116 views

Units related to chemical potential and orbital magnetization

I am studying this paper: Physical Review B 74, 024408 (2006) (arxiv) ABSTRACT We derive a multi-band formulation of the orbital magnetization in a normal periodic insulator (i.e., one in which ...
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247 views

About Kitaev p-wave superconductor model and Majorana Zero mode

The Kitaev $p$-wave spinless superconductor model has Hamiltonian as $$H = \sum_{j=1}^{N-1} tc_j^\dagger c_{j+1} + \Delta c_jc_{j+1} + h.c. + \sum_{j=1}^N \mu c_j^\dagger c_j $$ which has topological ...
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1answer
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How can an object ,moing on surface of Mobius ,be changed from left-handed to right handed without moving it in 3 dimension, as Mobius claims? [closed]

While Mobius claim that the object on it's surface moves in two dimension i.e. " on the surface" but if you could see, the object is actually moving in 3 dimension because the surface is curved which ...
4
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1answer
133 views

Is this a topological $\mathbb Z_2$ (Majorana-)invariant in *any* dimension?

Consider a non-interacting superconducting Hamiltonian in an arbitrary dimension. This is most conveniently expressed in terms of Majorana modes, which are defined as $$\gamma_{2n-1} = c_n + c_n^\...
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2answers
208 views

Trivial phase in Kitaev chain, trivial phase in what sense?

I am reading 1D p-wave SC proposed by Kiteav---Kitaev chain recently. This paramedic model somehow bothers me with the word "trivial." In this SE post, the answer pointed out what we mean by "...
6
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1answer
469 views

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 ...
0
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2answers
358 views

Chern number for the systems with open boundary conditions

For two-dimensional materials with periodic boundary conditions, we can solve the Bloch states and substitute them into the definition of Chern number, as shown in the picture: In the case of open ...
2
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1answer
572 views

Chern number in one-dimensional system

As the title, could we define Chern number for condensed matter systems with one spatial dimension? E.g. the 1D Su-Schrieffer–Heeger (SSH) model.
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Kosterlitz-Thouless in the XXZ chain: instanton condensation?

The anisotropic spin-$\frac{1}{2}$ Heisenberg chain $$H = \sum_n S^x_n S^x_{n+1} + S^y_n S^y_{n+1} + \Delta S^z_n S^z_{n+1}$$ is known to have the same physics as the two-dimensional classical XY ...
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94 views

Equivalence between different definition of winding numbers

At the moment I am reading the paper by A. Schynder and S. Ryu arXiv: 1011.1438. The general setup is a superconductor with time-reversal symmetry. I can write my Bogoliubov - de Gennes in the ...
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0answers
118 views

Can 1D SPT phase host non-Abelian topological defects?

People usually consider Kitaev chain as a topologically ordered phase in 1D, and the edge modes of Kitaev chain are Majorana zero modes (MZM), which are topological defects with non-Abelian mutual ...
2
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1answer
201 views

Are Weyl and Dirac points topogical defects in nodal semimetals?

Recently, I heard the Weyl and Dirac points are topogical defects in nodal semimetals. I do not really get it. And the definition of topological defects is confusing to me. Are the topoligical ...
2
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1answer
179 views

Transfer matrix approach to the topological phases

The transfer matrix contains all the information. i.e., information about the edges and bulk. What new insight does the transfer matrix approach provide in the study of the topological phases of ...
2
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2answers
1k views

What is the difference between symmetry protected topological (SPT) order and topological order?

As far as I know, the SPT orders(or SPT phases) are all gapped and protected by symmetry. However they are short range entangled, and the topological order phases are all long range entangled. So the ...
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0answers
58 views

Explain the characteristic features of long range entanglement

The long range entanglement (LRE) can exist in fermi liquids or lattice. The characteristic features of long range order entanglement could be the degeneracy, fractional excitation or the entanglement ...
4
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1answer
182 views

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
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 ...
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1answer
114 views

What are goldstone variables?

Pretty straightforward question (I hope). I am an experimentalist. I used to work in Nuclear Experiment (now I work in membrane biophysics). While reading about topological transitions in liquid ...
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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 ...
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1answer
776 views

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 ...
4
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1answer
119 views

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-...
3
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1answer
343 views

Have edge modes of the SSH model (or Kitaev Chain) been observed?

I am putting together a presentation on topological phase transitions in 1D tight binding models for a course in Solid State Physics, and while I have found many sources for theoretical descriptions I ...
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1answer
295 views

Why do people always talk about continuous topological phase transition?

I have a question which puzzles me for a long time. Usually when people talk about topological phase transition, they usually have a gap-closing picture in their mind. Namely, the phase transition ...
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48 views

For symmetry of wave function global phase is not allowed?

The key equation leading to the group cohomology classification of SPT phases is equation 12 on pp.8 of this paper which I reproduce below $$ \bigotimes_i{U^i}|\Psi_{pSRE}\rangle = |\Psi_{pSRE}\...
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1answer
182 views

Low dim physics: Examples of confinement-deconfinement phases of U(1) gauge theory in 2 dimensions

Please provide some examples of confinement-deconfinement phases of U(1) gauge theory in 2 spacetime dimensions (Low dimwnsional physics). U(1) gauge theory can be: pure U(1) gauge theory, or U(1) ...
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0answers
90 views

Is the 'Chern number' of a topological Kondo insulator an integer?

If you calculate the anomalous Hall conductance $\sigma_{xy}/\sigma_0$ for a simple complex hopping model at a whole band filling, this will equal an integer Chern number (given e=h=1). I would like ...
7
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1answer
321 views

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 ...
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>} ...
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2answers
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A problem related to Wick's theorem from RG analysis of KT transition

Recently, I was reading a review paper by John B. Kogut An introduction to lattice gauge theory and spin systems, when he was doing the RG analysis for the X-Y model, on page 702, to go from (7.61a) ...
2
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1answer
181 views

Discontinuity of the geometric phase

Does the geometric phase accumulated along a closed trajectory (in some parameter space) has to be continuous?
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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. ...
0
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1answer
78 views

Effects of interactions on the topological classification of free fermion systems

Can someone who read this paper (ArXiv, APS) by Fidkowski and Kitaev explain to me the two highlighted statements in the following extract? This is because for four Majorana chains the only ...
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 ...
0
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1answer
148 views

Quantum ising/heisenberg model, states representation

I am working with a hamiltonian which looks like this (Heisenberg model) I have made a program which computes this hamiltonian using Pauli matrices (spin 1/2). My working space is then the tensor ...
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0answers
569 views

How to calculate the string order parameter (for Haldane phase) in density matrix renormalization group?

The ground state of the spin-1 chain is the Haldane phase, which is known to be a symmetry protected topological phase and cannot be detected by conventional order parameter (beyond the Landau-...
4
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1answer
476 views

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) ...