Questions tagged [topological-phase]

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4
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1answer
191 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
278 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
720 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 ...
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2answers
435 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
760 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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0answers
189 views

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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0answers
107 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
143 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
241 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
226 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 ...
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2answers
2k 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
70 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
216 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
876 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
119 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
418 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 ...
11
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1answer
952 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
134 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
414 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 ...
3
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1answer
364 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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0answers
53 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
208 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
98 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 ...
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1answer
371 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
317 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
474 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
106 views

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
197 views

Discontinuity of the geometric phase

Does the geometric phase accumulated along a closed trajectory (in some parameter space) has to be continuous?
12
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2answers
820 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
83 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
612 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
240 views

Quantum ising/heisenberg model, states representation

I am working with a hamiltonian which looks like this (Heisenberg model) $$ \hat{H} = -\frac{1}{2}\sum_{j=1}^N \left( J_x\sigma_j^x\sigma_{j+1}^x +J_y\sigma_j^y\sigma_{j+1}^y +J_z\sigma_j^z\sigma_{j+...
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0answers
704 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
625 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) ...
3
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1answer
428 views

Paradox in topological phase of SSH model

Consider the SSH model, i.e. the dimerized tight-binding model with Hamiltonian $$H = \sum_i (t+\delta t) c^\dagger_{Ai} c_{Bi} + (t-\delta t) c_{A(i+1)}^\dagger c_{Bi} + \text{h.c.}.$$ This describes ...
7
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1answer
2k views

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" ...
2
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1answer
2k views

Berry phase in 1D materials

The Berry phase $\phi_B$ is the phase that an eigenstate acquires after its momentum vector goes around a circle at constant energy around the Dirac point. It is defined as $\phi_B = -i \int \langle\...
9
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1answer
253 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 ...
13
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2answers
594 views

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 ...
4
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0answers
511 views

About the $Z_2$ topological invariant

In Kitaev 2001 it is shown that the topological invariant $Z_2$ in a topological superconductor (Class D or BDI, one dimensional) can be defined as $$ (-1)^\nu={\rm sign\, Pf} [ A ]={\rm sign\, Pf}[ \...
2
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0answers
209 views

About Weyl superconductors and fractionalized Weyl semimetals

Recently, the experimental observations of Weyl fermion semi-metal have been made. Weyl fermion becomes very hot in condensed matter physics. I am confused about the Weyl superconductors and ...
2
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1answer
251 views

What does a zero topological S matrix element mean?

I realize that for nonabelian anyons, their S matrix elements could be zero (eg. the Ising anyons). I'm confused by the meaning of a zero S matrix element. Does it mean that the corresponding braiding ...
3
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3answers
359 views

Different anyon condensations that share the same phase

In Kitaev's notes, he reviewed the toric code model. Consider on square lattice the Hamiltonian $H=-J_e \sum_s A_s-J_m \sum_p B_p,\ A_s=\prod_{j\in vertices} \sigma_j^x,\ B_p=\prod_{j\in plaquettes} \...
4
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2answers
614 views

String operators in the string-net model

In the string-net model http://arxiv.org/abs/cond-mat/0404617, quasiparticles are created by the string operators (defined in eq.(19)). An easier pictorial way to define string operators $W_{\alpha}(P)...
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1answer
286 views

Why are there $F$-symbols in the splitting in anyon theory?

I am learning some basic knowledge of anyon theory by reading P. Bonderson's thesis: http://thesis.library.caltech.edu/2447/2/thesis.pdf. $F$-symbols and $R$-symbols are two basic operations on ...
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4answers
9k views

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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1answer
614 views

can a gapless system be a topological state?

For a gapless system without boundary (i.e. in the bulk there is gapless excitation while no clear meaning of boundary excitations like QFT), can it be a topological state? What is the property of EE ...
1
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1answer
193 views

Berry phase in the toric code model and 2D chiral $p$-wave superconductors

When we derive the exchange statistics by moving quasiparticles around a circle in the toric code model we do not mention any Berry phase contribution. Is the Berry phase contribution trivial or it ...
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2answers
431 views

What are the conditions to observe gapless modes at the boundary for 1D case?

We observe gapless modes at the boundary for the case of SSH model or Polyacetylene The Hamiltonian for SSH model has particle hole and time reversal symmetries, it also has a Dirac like spectrum. It ...
2
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1answer
317 views

Can a nondegenerate fermionic topological Mott insulator (TMI) state support an emergent bosonic topological order?

Based on my recent study and motivated by a recent paper, I have a naive question. Consider a 2d Hubbard model for electrons at half filling $H=\sum c_k^\dagger h_k c_k+U\sum n_{i\uparrow }n_{i\...