Questions tagged [topological-phase]
The topological-phase tag has no usage guidance, but it has a tag wiki.
170
questions
0
votes
0answers
15 views
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 ...
3
votes
1answer
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 ...
1
vote
0answers
45 views
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 ...
1
vote
0answers
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 ...
0
votes
0answers
51 views
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 (...
0
votes
0answers
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{...
0
votes
0answers
19 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 ...
4
votes
0answers
45 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 ...
5
votes
0answers
73 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 ...
1
vote
0answers
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 ...
2
votes
1answer
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 ...
1
vote
0answers
27 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 ...
4
votes
0answers
71 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, ...
0
votes
0answers
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 ...
1
vote
1answer
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 ...
1
vote
0answers
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 ...
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
0answers
16 views
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 ...
0
votes
1answer
93 views
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}$,
...
1
vote
1answer
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-...
1
vote
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 ...
1
vote
1answer
74 views
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 ...
2
votes
0answers
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 ...
8
votes
1answer
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$ ...
3
votes
0answers
66 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 ...
1
vote
1answer
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, ...
1
vote
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 ...
1
vote
0answers
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 ...
3
votes
0answers
79 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].$$
If I understand it right, these models have:
...
3
votes
1answer
64 views
Experimental confirmation of Majorana modes in Kitaev chain
I'm confused about majorana modes at the edge of Kitaev chain, what do we seek in experiment? When we first define this one we write the creation and annihilation operators as:
$$a^{+}=\frac{1}{2}(\...
1
vote
1answer
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 ...
5
votes
1answer
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 ...
2
votes
0answers
53 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, ...
5
votes
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 ...
5
votes
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 ...
5
votes
1answer
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^{\...
0
votes
0answers
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 ...
5
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
1answer
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
$$H(k)=
\begin{bmatrix}
c_k^\dagger && d_{k}^\dagger
\end{bmatrix}
\begin{...
1
vote
2answers
349 views
Why does spin-orbit coupling lead to a nonzero Berry curvature?
Many theories consider spin orbit coupling to be a prerequisite for a nonzero Berry curvature, and therefore, for the classical anomalous Hall effect. Here, the spin orbit coupling is defined as:
$$
...
10
votes
3answers
4k views
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 ...
2
votes
0answers
43 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,...
1
vote
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 ...
2
votes
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 ...
1
vote
0answers
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 ...
3
votes
1answer
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 ...
2
votes
2answers
77 views
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 ...
0
votes
1answer
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?
