Questions tagged [topological-phase]

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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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50 views

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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36 views

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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42 views

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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33 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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80 views

What are the applications of edge states in 1D topological systems?

In 2D, we get robust conducting edges. In 3D, we get robust conducting surfaces. These are interesting because we can possibly utilise this robustness for protected electron transport (or light ...
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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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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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40 views

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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176 views

Basics of topological order and its relation to entanglement

What is a topological order that drives a topological phase transition? How is it different from say magnetic ordering or the superfluid ordering? What is its relation with entanglement? Please ...
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49 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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1answer
102 views

How can I find edge states given a bulk Hamiltonian for a topologically ordered phase?

Suppose we have a momentum space tight binding Hamiltonian $H(\vec{k})$ that describes some topologically ordered system. It could be a Chern insulator in two dimensions, or a Weyl semimetal in three ...
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44 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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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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Spin-1 Heisenberg model, the AKLT model, and their ground states

I am reading literature on quantum spin chains and matrix product states, and I notice similar arguments regarding the spin-1 antiferromagnetic Heisenberg model, $H_{H} = \sum_i S_i \cdot S_{i+1}$, ...
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57 views

Integer quantum Hall conductance and time-reversal symmetry

If we have a (2+1)-dimensional electronic gapped system with a unique ground state and it has a nonzero integer quantum Hall conductance, then the system (or its ground state) must break the time-...
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1answer
60 views

Do topological transitions only occur at Dirac points?

Topological phase transitions happen when the band gap closes. It is not true that all band crossings are topological. There are Dirac (linear) band crossings, quadratic band crossings, Dirac-like ...
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1answer
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Aharonov-Casher effect vs Spin-Orbit coupling

The Aharonov-Casher phase is the electromagnetic dual of the Aharonov-Bohm phase. It arises when a neutral particle with a magnetic moment encircles, for example, a line charge, or moves on a closed ...
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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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647 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 $T_c$ ...
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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 ...
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70 views

Are fractional quantum hall effect system symetry enriched topological phases?

In the papers I review they first start to talk about topologically ordered phases of matter. Their standard example of it is FQHE. Than they give another set examples which are quantum spin liquids, ...
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1answer
84 views

Is the quantum Hall state a topological insulating state?

I am confused about the quantum Hall state and topological insulating states. Following are the points (according to my naive understanding of this field) which confuse me: Topological insulator has ...
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76 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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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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1answer
64 views

Experimental confirmation of Majorana modes in Kitaev chain

I'm confused about majorana modes at the edge of Kitaev chain, what do we seek in experiment? When we first define this one we write the creation and annihilation operators as: $$a^{+}=\frac{1}{2}(\...
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45 views

In a class of parametrized symmetric Hamiltonians, should its symmetry operator be parametrized the same way?

I would like to ask the following in the context of symmetry-protected topological phase. Consider a class of Hamiltonians parametrized by $\{a_1,a_2,...\}$ denoted by $H(a_1,a_2,...)$. Suppose there ...
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81 views

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 ...
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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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3answers
303 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 ...
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1answer
199 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 ...
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216 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^{\...
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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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117 views

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 ...
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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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116 views

Winding number of SSH model 3

SSH model can be written as $$H=-\sum_n\big[Jc_n^\dagger d_n + J'd_n^\dagger c_{n+1}\big]+h.c.$$ in Fourier space $$H(k)= \begin{bmatrix} c_k^\dagger && d_{k}^\dagger \end{bmatrix} \begin{...
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349 views

Why does spin-orbit coupling lead to a nonzero Berry curvature?

Many theories consider spin orbit coupling to be a prerequisite for a nonzero Berry curvature, and therefore, for the classical anomalous Hall effect. Here, the spin orbit coupling is defined as: $$ ...
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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 ...
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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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1answer
54 views

How do I understand different realizations of symmetry in the absence of fractionalization?

To use a simple example to ask my question, consider the two dimensional toric code with a $Z_2$ global symmetry acting in two ways: The most boring trivial way possible. By permuting the charge and ...
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1answer
111 views

Models for non-universal topological quantum computation

Anyon models do not lead in general to universal topological quantum computation (= existence of a universal set of quantum gates) when only the braiding is used for implementing gates. The Fibonacci ...
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134 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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101 views

Why is the flux quantized in 4D quantum Hall effect?

I am reading "Topological Field Theory of Time-Reversal Invariant Insulators" by Qi, Hughes, and Zhang (https://arxiv.org/abs/0802.3537). It argues that time reversal invariant (TRI) insulators in 2+1 ...
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2answers
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How do you design an object that looks different after you spin 360 degrees?

According to quantum mechanics, after a 360 rotation electrons have the opposite phase. If you rotated yourself 360 degrees, your electrons would have the opposite phase to the electrons in the ...
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174 views

Does a gap closing mean an occurrence of a quantum phase transition?

If we have observed a closing of the excitation gap in the energy spectrum of a certain model, can we safely conclude that a quantum phase transition occurs?