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Questions tagged [topological-phase]

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Topological phase transitions in BKT-like system: relation to directed percolation

I am studying a non-equilibrium system that transitions between an unbound vortex state to a bound vortex state, that 'looks' like properties of BKT system. I am interested in the self-organising ...
Andy's user avatar
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How to numerically calculate Zak phase for SSH3 model?

The k-space hamiltonian of SSH3 model with nearest neighbour hopping is given by H(k)= \begin{bmatrix} 0 & u & w e^{-ika} \\ u & 0 & v \\ w e^{ika} & v & 0 \end{bmatrix} ...
SUMANTA SANTRA's user avatar
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Is the Berry phase at a degeneracy trivial or non-trivial?

This question is about the original paper on Berry phases by M. Berry (1984). There, the Berry phase $\phi_{B}$ is defined as $$ \phi_{B}= \oint_\mathcal{C}\underbrace{ \left\langle n(\mathbf{R}) \...
xabdax's user avatar
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Can the edge degeneracy in spin-$2$ AKLT go away on an arbitrarily small $SO(3)$-symmetric bulk perturbation?

I am learning about SPTs, or symmetry-protected-topological phases. There is a rich structure in antiferromagnetic spin chains. The Heisenberg point is gapless in half-integer-spin antiferromagnets ...
user196574's user avatar
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Is there possible isentropic phase transition?

Suppose we have a complex system changing in state but without order parameter jump. Is there possible that during this change particular dynamic of system chsnges enough to name this phase change, ...
kakaz's user avatar
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2 votes
1 answer
100 views

Motivation for the shape of the theta vacua

I understand that the reason why we construct the theta vacua is because instantons allow tunnelling between different vacuum states, $\left|n\right>$. This means that we have to consider a real ...
Gabriel Ybarra Marcaida's user avatar
2 votes
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34 views

Mathematical references for gauge theory in condensed matter physics

I am currently trying to go through some literature on the classification of symmetry protected topological phases. Primarily, I am interested in the classical of topological phases using mathematical ...
1 vote
1 answer
102 views

Resources: Tensor Categories and Topological Phases of Matter

For a mathematician with knowledge of tensor categories who is interested in the growing application of categorical techniques in topological phases of matter and topological order, along with their ...
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1 answer
125 views

Is there a Majorana representation for toric code

Kitaev's toric code is known to be the Z2 gauge field theory, which suggests that there might exists a Majorana representation for the toric code, e.g., Majorana + Z2 gauge field. Hence, I wonder if ...
Richard's user avatar
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4 votes
1 answer
128 views

Higher category's consistency relations

I have been reading on higher category and symTFTs. It appears to me that, for higher categories, people seldom mention the consistency relations (like the MacLane coherence theorem in the category ...
Waterfall's user avatar
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Continuum formulation of the Kitaev chain

In Kitaev's seminal paper (https://arxiv.org/abs/cond-mat/0010440), the Kitaev chain is described in a lattice formulation. On the other hand, many of the original papers on the the related ...
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Dealing with discontinuous phase issue in computing winding number numerically

Consider a 1D SSH model with winding number given by $$\nu = \frac{1}{2\pi i}\int_{-\pi}^\pi d\phi,$$ where $d\phi$ is the change in phase of the eigenvectors between nearby $k$ points. The phase is ...
Sean's user avatar
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1 answer
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Symmetry Protected Topology and Edge Modes

I have a spin 1/2 chain with open boundary conditions described by Hamiltonian $H=\sum_i \sigma_{2i}^z \sigma_{2i+1}^z$. From $H$ it's clear that boundary sites are decoupled from the rest of the ...
Barry's user avatar
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1 vote
0 answers
62 views

Quantum phase transition in condensed matter

I want to know that, for any spin-chains in condensed matter Physics like X-Y spin model, Kitaev model 1-D only in which degenerate point is critical point. Is it necessary that the critical points ...
Sadaf's user avatar
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2 answers
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If the curl of the gradient is always zero why isn't it in vorticity definition? Kosterlitz - Thouless - Berezinsky topological transition

Is a well estabilished property that the curl of a gradient is always zero (i.e. $\nabla\times\nabla\Phi=0$) and it's possible to prove it in many ways. e.g. If $(\nabla\times\nabla\Phi)_i = \...
Cuntista's user avatar
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How to exactly diagonalize a system with $Z_2\times Z_2$ symmetry?

I am studying the localization protected SPT phase, and try to compute the level spacing ratio of Hamiltonian, $$H=\sum_kJ_kZ_{k-1}X_kZ_{k+1}+\sum_kh_kX_{k}X_{k+1}.$$ We can check that this ...
Benjamin Jiang's user avatar
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1 answer
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Why does the energy gap of sublattice-symmetric systems never close?

I am studying from this famous site some symmetries useful for topological quantum matter. At some point, talking about the particle-hole symmetry, it states: You can however notice that, unlike in ...
kBoltzmann's user avatar
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Quantum (higher-form) anomaly at finite temperature

At finite temperature, anomaly is generally known to be contaminated, and thus the 't Hooft anomaly matching does not work after thermal compactification. Meanwhile, I have read paper saying that ...
Richard's user avatar
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2 votes
0 answers
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Normalization in the Abelian Chern-Simons action

In all the places I looked (such as chapter 5 in the lecture notes of David tong (http://www.damtp.cam.ac.uk/user/tong/qhe.html) and E. Witten (https://arxiv.org/abs/1510.07698)) the action for the ...
Tuhin Subhra Mukherjee's user avatar
4 votes
1 answer
178 views

Is IQHE a degenerate case of FQHE? What is the role of topological orders?

(As suggested by Tobias, I shall indicate that I will write "IQHE" for "Integer quantum Hall effect" and "FQHE" for "Fractional quantum Hall effect" below.) I ...
Yuezhao Li's user avatar
1 vote
1 answer
106 views

Time reversal in a two-band system

Suppose I have a 3D system of spinless fermions described by the following two-band model Hamiltonian: $$ H(\vec{k})=\vec{d}(\vec{k}) \cdot \vec{\sigma} $$ where $\vec{d}=\left(-\sin k_{x},-\sin k_{y},...
dnrk's user avatar
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1 answer
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Plaquette operator in Kitaev honeycomb model

In his honeycomb model, Kitaev defines link operators \begin{equation} K_{jk} = \begin{cases} \sigma_j^x \sigma_k^x & \text{if }(j, k)\text{ is an }x\text{-link;}\newline \sigma_j^x \sigma_k^y &...
xzd209's user avatar
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Relation between Displacement Operator and Winding number

I am trying to implement a paper [https://arxiv.org/abs/2003.06086] using quantum computing techniques. In the supplementary material[SM] with the main paper, they introduce a displacement operator ...
CuriousMind's user avatar
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1 answer
67 views

Sublattice symmetry and the Fermi level

I am a math student who is learning topological phases from this website. Let's assume the fermi level is zero. For the graphene, the sublattice symmetry $\sigma_z H \sigma_z = -H$ makes the ...
Justin Lien's user avatar
1 vote
0 answers
60 views

Topological Insulators with different spin band

To obtain a topological band insulator, we usually start with two bands with either spin up or down. If these bands now get 'inverted', they will cross. When there is coupling of these two bands such ...
sined's user avatar
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1 vote
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Topological Insulator [closed]

What effect on the Brillouin zone (torus) after applying the magnetic field? As in real space, pressure deforms the torus and up to a certain pressure, this remains invariant topologically. Similar to ...
Satyendra Singh Nirvan's user avatar
3 votes
0 answers
93 views

Majorana Fermions in superconductors (Wikipedia page)

In Wikipedia, Majorana Fermions in supercondutors are described as Mathematically, the superconductor imposes electron hole "symmetry" on the quasiparticle excitations, relating the ...
peep's user avatar
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3 votes
0 answers
70 views

Infinite stacking of integer quantum Hall systems

Let us consider a (3+1)-dimensional system $\mathcal{H}$ constructed by stacking (2+1)-dimensional integer quantum Hall systems $\mathcal{H}_\text{Hall}$, e.g., $E_8$ bosonic systems (or $\sigma_H=1$ ...
Yuan Yao's user avatar
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3 votes
1 answer
189 views

Topological phase and Chern number

the relation between topological phase and Chern numbers is unclear to me. For Haldane model if the Chern numbers of its two bands go from (+1,-1) to (0,0), we say that it goes from topological phase ...
Eric N's user avatar
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Berry phase from Bloch wave functions in the basis of Wannier functions

The formulate to calculate berry phase for Bloch wave functions is $$ \gamma = i \sum_{n\in occ}\int_{\mathcal{C}} dk \langle \psi_k^n|\partial_k|\psi_k^n\rangle, $$ where $|\psi_k^n\rangle$ is a ...
lsdragon's user avatar
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0 answers
87 views

How we can implement a periodic gauge in numerical calculations?

For a 1D system, there is a way to calculate the Zak phase in the discrete form. Suppose C is some closed path in k-space (a 1D BZ). If we suppose the path is discretized into (not necessarily ...
Alireza Baradaran's user avatar
3 votes
2 answers
332 views

Is there a zero correlation length spin-$1$ chain in the Haldane phase?

The ground state of the spin-$1$ AKLT model gives an example of a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological (SPT) phase, the Haldane phase. This state is a nice example of the ...
user196574's user avatar
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2 votes
1 answer
328 views

Landau Levels degeneracy in a finite sample

According to different sources: Tong lectures on IQHE (Tong), MIT Open courses (MIT) etc, when calculating the number of states in each Landau Level all of them impose (in the Landau gauge) periodic ...
Nacho Figueruelo's user avatar
1 vote
2 answers
243 views

Edge state protection in Chern insulator

I have a confusion about the nature of topologically protected boundary states in the Chern insulator. Since the Chern insulator does not require any symmetries to be present in the system, what is ...
JustAGuy's user avatar
1 vote
0 answers
73 views

Topological properties of twisted TMD homobilayers

I'm reading this article about twisted TMD homobilayers (https://arxiv.org/abs/1807.03311) and there are certain topological properties that I don't understand: On page 3, in the paragraph next to Fig ...
Eric N's user avatar
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Symmetry and corresponding operator

For a quantum symmetry, is the operator of symmetry necessary to be unitary?
Milad Jangjan's user avatar
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0 answers
66 views

Absence of topology in semi-dirac materials

Good morning everybody, I am facing a problem when calculating the topological invariant in a semi-dirac system, whose Hamiltonian is: $$ H=k_x^2\sigma_x+k_y\sigma_y $$ My question is that this ...
Feynnman pilows's user avatar
2 votes
0 answers
75 views

Plausible finite group on-site-SPTs in realistic materials

I am looking for some understanding of which on-site symmetries in realistic crystalline materials (i.e. not just in random lattice models) can plausibly be expected to be realized and to induce ...
Urs Schreiber's user avatar
10 votes
2 answers
1k views

Berry curvature concentration around nodal points

It is well-known that in TI-symmetric semi-metals the Berry curvature on the Brillouin torus vanishes away from the nodal points (eg. [XCN10, III.B] [Van18, p. 105]). But even for non-TI-symmetric ...
Urs Schreiber's user avatar
2 votes
0 answers
84 views

How to stack two Haldane chains?

This questions is a follow up to a pervious question of mine: Inverse of Haldane phase? Now that I know that Haldane phase is it's own inverse, I am having trouble is visualizing how could we stack ...
baba26's user avatar
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1 vote
1 answer
125 views

What's the difference of flux tube and vortex in FQHE (especially in Jain wavefuntion)

In the book Composite Fermion by Jainendra K.Jain, he mentioned the motivation of Jain wavefunction: attach flux tube of 2p flux quantum to fermions to form composite fermions. Naively, this is done ...
Black Monolith's user avatar
2 votes
1 answer
96 views

Inverse of Haldane phase?

Based on what I have learned so far, Haldane phases are a nontrivial SPT for 1D spin-1 chains. The trivial phase acts as an "identity" under the group of SPT phases ( with stacking as the ...
baba26's user avatar
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4 votes
2 answers
260 views

Why are topological materials/phases "exotic"?

From what I understand, when a system has topological order, any local perturbation doesn't change the phases and thus its properties. This would suggest that it should be really easy to find ...
Another User's user avatar
2 votes
1 answer
191 views

Smooth deformation in topological systems

In various topological systems, it is common to encounter the concept of smooth deformation, which introduces changes in spectra of topological systems without allowing topological phase transitions. ...
Shasa's user avatar
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0 answers
54 views

How to detect anyonic statistics without calculating Berry phase diretly?

My question is: given a model, it is possible to know if it can support a specific kind of anyon (like Fibonacci or Ising) without having to explicitly calculate the Berry phase after a braiding? I've ...
Cristobal Mendez's user avatar
1 vote
1 answer
173 views

Code distance and other questions about Quantum double model as an error correcting code

Kitaev's quantum double model is an error correcting code, see: https://arxiv.org/abs/1908.02829 I am in a class on quantum error correction and the professor commented that a quantum double model for ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
67 views

What kind of phase it is when a photon gain a quantum phase, a dynamical one or a geometrical one?

It's known$^1$ that the phase factor in quantum mechanics can be divided into geometric phase and dynamical phase. Since in quantum optics, light is treated as a quantum object, i.e., the photon. So ...
Sherlock's user avatar
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3 votes
0 answers
194 views

Topological invariant for the Toric code

My understanding is that the Toric code is a model with topologically non-trivial ground state. The ground state is degenerate on a Torus and is robust to local perturbations. The model has anyonic ...
Jeet Shah's user avatar
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1 answer
57 views

What do the '4' and 'b' signify in a layer of a crystal called a '4Hb' crystal or material?

From Phys.org: Study gathers evidence of topological superconductivity in the transition metal 4Hb-TaS2 Which, in turn, references: Abhay Kumar Nayak et al, Evidence of topological boundary modes with ...
Kurt Hikes's user avatar
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4 votes
1 answer
262 views

Detection of topological phases

In the book A Short Course on Topological Insulators (https://arxiv.org/abs/1509.02295) the authors start with a simple toy model, the SSH-Chain, which is a bipartite one-dimensional lattice with ...
nico's user avatar
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