Questions tagged [topological-phase]

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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 ...
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49 views

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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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 ...
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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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36 views

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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66 views

Partial Transpose in Gapped Time-reversal Symmetric Spin Chains

Suppose you have a one-dimensional quantum spin system with on-site Hilbert spaces $\mathcal{S}$. Suppose there is an anti-unitary, anti-linear operator $C$ on $\mathcal{S}$ inducing an anti-linear, ...
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167 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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150 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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1k views

Definition of 'Majorana Number' in the Kitaev Chain

I have some questions about the Kitaev toy model for Majorana fermions (arXiv:cond-mat/0010440). First of all, his proof for the definition of the 'Majorana number' is not so clear to me. $$P(H(L_{1} ...
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60 views

Gapless modes at the boundary between topological insulator and normal insulator

I am currently learning about topology in condensed matter physics. I think I understand most of how topological indeces come about and differences between Z and Z2 indeces and the symmetries that ...
3
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69 views

Symmetry Protected Topological (SPT) phases of spin-1 chains

Let's consider this family of 1D spin-1 of hamiltonians: $$\sum_{i}[S^x_{i}S^{x}_{i+1}+S^y_{i}S^{y}_{i+1}+\lambda S^z_{i}S^{z}_{i+1} + D(S^{z}_{i})^2].$$ If I understand it right, these models have: ...
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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 ...
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430 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}[ \...
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2k views

Compute $Z_2$ Invariant of 2D Topological Insulators without Computing the Eigenstates

For 2D Time-Reversal Invariant systems ($T H(\vec{k}) T^{-1} = H(-\vec{k}) $), there is a formula by Fu-Kane-Mele in order to determine whether the system belongs to either one of distinct topological ...
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135 views

What is the physical mechanism of the topological phase transition driven by temperature?

The topological property of topological insulators (TIs) is characterized by the non-trivial topological invariants of their band structures, such as $Z_{2}$ topological invariants. While it's clearly ...
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177 views

What is the first excited state of the honeycomb Kitaev model in its gapped phase?

As we know, there are both gapless and gapped phases of the Kitaev model, and let's fix the couplings $J_x,J_y,J_z$ such that the model being in the gapped phase. My question is, what is the first ...
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48 views

The connection between symmetry and classifying spaces of a group

I recently read the following statement: "For any type of mathematical object, an object of that type with $G$ symmetry “is” a map from [its classifying space] $BG$ to the space of all objects ...
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51 views

Topological soliton objects in Minkowski v.s. Euclidean spacetime?

What makes the distinctions between the soliton objects in Minkowski or in Euclidean spacetime? It looks that usually, the Euclidean path integral is easier to be performed in many cases. In fact, ...
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42 views

Relation of SPT phases with different boundary conditions

Using the definition that two SPT phases are distinct if they can't be connected by a symmetric finite depth local unitary, how does one relate systems with different boundary conditions? For example,...
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117 views

Is there a precise definition for a “parity”, “time-reversal” or “chiral” symmetry in general quantum spin systems?

By "quantum spin system" I mean a physical system with qu-$d$-its (called "spins", for possibly different $d$) distributed somehow over space and a Hamiltonian that is a sum of arbitrary local ...
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46 views

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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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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172 views

R-matrix for $SU(N)_k$ anyon model

Does anyone know the $R$-move or $R$-matrix for $SU(N)_k$ anyon model? Thanks! For the definition of $R$-move or $R$-matrix, please see the definition in Eq.(2.30) of this paper: http://arxiv.org/abs/...
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100 views

Polarization to define trivial and non trivial topological phases?

Polarization is well defined for particle hole symmetry systems, so can we use polarization to identify topological phases? for example polarization can have possible value $$P=0 \quad or \quad1/2$$ ...
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32 views

Critical points of vector field with zeros in the magnitude

I am studying a vector field which has critical points (sources, sinks, saddle points and centers). The magnitude of the vector field goes to zero smoothly in these points, however. Contrast that to ...
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24 views

BKT transition: nature of topological transition

BKT-transition is one of the most well-known topological transition in $O(2)$ model.But I misunderstand the physical interpratation of this transition. I started from the low-temperature expansion of ...
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32 views

The surface states and Fermi arcs in Weyl semimetals

I'm confused about surface states in Weyl semimetals. Assume that we have a single pair of Weyl points and the Fermi level turned to this points. In this https://arxiv.org/abs/1301.0330 paper the ...
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0answers
39 views

Lattice hopping at boundary in graphene lattice with magnetic field

Let's say I have a tight binding model for graphene, where I have a two-atom basis and three nearest neighbor vectors. I've applied a homogenous magnetic field $B$ in the z-axis, and can take the ...
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0answers
75 views

On string-like excitations in (3+1)d discrete gauge theory

(3+1)d discrete $G$-gauge theory (untwisted Dijkgraaf-Witten theory) has both point-like and loop-like excitations; Point-like excitation is an electric charge labeled by an irreducible ...
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17 views

Linking phase of flux lines and excitation energy of monopole

I am reading this paper and on the left-hand side of pp.10 it states the following relation between linking phase and excitation energy of monopole: Now the $\theta = \pi$ term in the bulk implies ...
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132 views

Significance of topology in topologically ordered systems

The topology on which a lattice is placed plays an important role in topologically ordered systems, for example in toric code the degeneracy in the ground states is given by $4^{g}$ where $g$ is the ...
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90 views

PT-symmetry in Energy Band of Crystals

According to this source, it is proved that, in absence of spin-orbit coupling, spatial inversion symmetry (as a part of point-group symmetry which operates as $\hat{S}\psi(\vec{r})=\psi(-\vec{r})$) ...
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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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248 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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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 ...
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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 ...
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195 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 ...
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51 views

Do we have a good model for what happens to the manifold when a topological phase transition occurs?

To support this question I'm going to use a theoretical example. Suppose we have some Hamiltonian (H) which is a function of a continuous parameter, say $\alpha$. Also suppose that as a function of ...
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159 views

Interacting Chiral topological invariants using Green function

We can calculate the topological invariants for 1D interacting topological insulators as $n=\frac{\text{Tr}}{2\pi i}\oint_cG\partial_kG^{-1} $ where as for interacting chiral topological ...
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337 views

Graphene Chern number for Dirac nodes

Why do we add winding number at two Dirac nodes to determine topological phase?
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270 views

Finding parity eigenvalues from a character table

The all-electron code Wien2K will optionally calculate the character tables for a specified list of $k$-points. I'd like to know the parity eigenvalue for a given $k$-point and band index. Is there ...
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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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14 views

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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Does the polarized Kagome antiferromagnet contain Dirac or Weyl points?

I've been reading about frustrated quantum magnets lately and a prominent topic is the study of antiferromagnets on the Kagome lattice. A calculation of the spectrum for the sort of model I have in ...
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Are there any gapped systems that aren't invertible?

Assume the following definitions: A gapped phase of matter is a collection of (quantum-mechanical) systems with a unique ground state and an energy gap to all excitations in the limit of infinite ...
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48 views

Classical statistical model based on group multiplication

For a (finite) group $G$, consider the following classical statistical model on a 2 dimensional lattice with oriented edges: Each edge carries a classical degree of freedom that can take values in ...
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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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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 ...