Questions tagged [topological-phase]
The topological-phase tag has no usage guidance, but it has a tag wiki.
189
questions
2
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0answers
13 views
Mutual statistics between dyons (charge-monopole composite)
I am asking for some intuitive understanding between two dyons with $(e,m)$ in 3-dimensional space. Here the magnetic charge $m$ is normalized as
\begin{eqnarray}
m=\int_{S^2}\frac{B}{2\pi}\in\mathbb{...
2
votes
0answers
53 views
Phases and order in condensed matter beyond Landau
All materials in our world consist from electrons and nucleus. They non-trivially interact and can create different states of matter with very different properties. Some states of matter (or phases) ...
6
votes
0answers
69 views
Is there an AC version of the Quantum Hall Effect?
The quantum Hall effect has the well-known signature of plateaus in the Hall conductivity $\sigma_{xy}=n e^2/h$ for integer (or rational) n.
This quantization is extremely precise, and can go up to ...
3
votes
1answer
72 views
Relativistic Dispersion In One Space Dimension
I'm now reading Three Lectures On Topological Phases Of Matter by
Edward Witten and face some statements that are unclear to me.
According to lectures:
As I understand, electronic excitations in ...
0
votes
0answers
21 views
What are topological states of matter/topological symmetry breaking in biomechanical systems?
What is topological order? How topological states of matter applied to mechanical/classical systems? How this definition of topological order is applicable to classical systems? Is it something ...
4
votes
1answer
150 views
One dimensional phase transitions
Due to R. Peierls argument there is not phase transitions is one dimensional lattice systems.
Argument in $d=1$ goes like that: flipping of one spin in system of N spins will lead to change of free ...
1
vote
1answer
65 views
Alternatives for calculating topological invariants in topological materials
My questing is regarding the different alternatives for calculating topological invariants in topological materials protected by symmetry, specially time-reversal invariant topological insulators, ...
1
vote
1answer
88 views
Where is the connection between $U(1)$ gauge field and $\mathbb{Z}_2$ gauge theory?
I am a graduate student in condensed matter physics and today I was reading the Wikipedia article Topological Order.
There is the part:
Note that superconductivity can be described by the ...
5
votes
1answer
75 views
Gauging a symmetry-protected topological (SPT) phase
In this answer, it is said that gauging the symmetry which protects a symmetry-protected topological (trivial) phase gives something "morally very similar" to a phase with a topological order. What ...
0
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0answers
13 views
How do we obtain Eq. 3.15 in Fu and Kane's PRB 74, 195312 Time Reversal polarization and a $Z_2$ adiabatic spin pump
This is a probably a very noob question.
I attempted to derive Fu and Kane's Eq. 3.14 from the defnition of $A^I$ in (3.12) but ran into a bit of trouble with the minus signs with the $A^I(-k)$ ...
2
votes
0answers
32 views
Spin structure, Intersection form and 2D TQFT
I am reading this (page 20) and watching this Anton Kaputins' talk (33:24).
Here, he tried to explain how to define a spin structure on a lattice in a closed oriented (1+1)-D manifold $M$ (or at ...
1
vote
0answers
42 views
Gauged DIII superconductor and Z_2 topological order
I keep hearing that a gauged 2D topological superconductor with preserved time-reversal symmetry that belongs to the DIII class is equivalent to the Z_2 gauge theory (Z_2 topological order with ...
9
votes
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188 views
Research: Mott insulator and topological order
I'm an experimentalist who is mainly focusing on strongly correlated electron systems (SCES), in particular Metal-insulator (Mott) transitions in the classical example $V_2 O_3$. Recently I decided to ...
1
vote
0answers
50 views
Reason for Kitaev chain Pfaffian being calculated in momentum space
In Kitaev's paper where Kitaev defines his toy model for observation of Majorana fermions (MFs), he suggests a method of determining whether a system contain MFs at it ends. This method is to ...
1
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0answers
28 views
quantum order in a (generalized) thermal steady state
It is known that starting from an initial product state, non-integrable systems will thermalize and eventually local observables can be described by a Gibbs ensemble. It has also been argued that a ...
2
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0answers
61 views
Does flat band imply localization?
Consider the Kitaev chain, whose Hamiltonian is as follows:
$$ H = -\mu \sum_n c_n^\dagger c_n -t\sum_n (c_n^\dagger c_{n+1} + \mathrm{h.c.}) +\Delta \sum_n (c_n c_{n+1} + \mathrm{h.c.}) $$
I have ...
2
votes
0answers
84 views
About symmetry constraints in momentum space
When people study symmetry protected topological phases, certain symmetry constraints are enforced on the Hamiltonian. Specifically, for non-interacting fermionic systems, we could focus on the ...
3
votes
1answer
114 views
What are $U(n)$ or $\mathbb{Z}_2$ quantum spin liquids?
Quantum spin liquid is a state of matter in which spins are correlated and fluctuate even at zero temperature.
My question is about these terms in general. When we say that a state or a quasi-...
2
votes
0answers
37 views
what is the relationship between topological charge and chern number in topological materials?
I would like to ask, what is the relationship between topological charge and chern number in topological materials? Why the topological charge of the Dirac cone is $0$?
0
votes
1answer
66 views
Meaning of complex pairing terms in Kitaev chain
I am studying some properties of the one dimensional Kitaev chain, which has the following form:
$ H = -\mu \sum_n c_n^\dagger c_n - t \sum_n (c_{n+1}^\dagger c_n + h.c.) + \Delta \sum_n (c_n c_{n+1} ...
1
vote
0answers
29 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 ...
4
votes
1answer
92 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
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0answers
72 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
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0answers
43 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
71 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 (...
1
vote
0answers
82 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{...
1
vote
0answers
34 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 ...
5
votes
0answers
78 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
119 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
38 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
142 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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vote
0answers
44 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
76 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, ...
1
vote
1answer
334 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 ...
2
votes
0answers
126 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
156 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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0answers
59 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
1answer
182 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
76 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
90 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
113 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 ...
3
votes
0answers
64 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 ...
9
votes
1answer
1k 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
83 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
97 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
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1answer
117 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
77 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
95 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].$$
If I understand it right, these models have:
...
3
votes
1answer
73 views
Experimental confirmation of Majorana modes in Kitaev chain
I'm confused about majorana modes at the edge of Kitaev chain, what do we seek in experiment? When we first define this one we write the creation and annihilation operators as:
$$a^{+}=\frac{1}{2}(\...
1
vote
1answer
47 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 ...
