Questions tagged [berry-pancharatnam-phase]

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Where does the Berry phase of $\pi$ come from in a topological insulator?

The Berry connection and the Berry phase should be related. Now for a topological insulator (TI) (or to be more precise, for a quantum spin hall state, but I think the Chern parities are calculated in ...
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89 views

Significance of geodesics of a Hamiltonian surface in condensed matter physics?

Many Hamiltonians in 2D quantum systems can be parameterized as a surface (such as the Bloch sphere) by their k-space coordinates. Another example is given by the (kx,ky) points of the Brillouin torus ...
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391 views

Calculation of a Berry phase in the Aharonov-Bohm effect

In his seminal paper, where he introduced the concept of geometric phase, Berry investigates, among other things, a quantum system in a box encircling the infinitely thin solenoid carrying flux $\Phi$....
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162 views

Where is wrong in the derivation about adiabatic theorem

There is a homework in Quantum Mechanics which is about adiabatic theorem. Let us argue where is wrong in following derivation. $$i \partial_t |\psi(t)\rangle = H(t) |\psi (t)\rangle \tag{1}$$ if $\...
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214 views

Monodromy, Holonomy and Braiding Phase

In quantum Hall effect, especially in the context of CFT description, these words come up often. I think I understand the braiding phase - as the phase gained by the wave function when a quasi ...
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884 views

How is the Geometric Phase measured in the experiment?

I had read some papers that have mentioned the geometric phase (Berry phase) can be used to detect the quantum phase transitions in a quantum many-body system. My question is: How is it measured in ...
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276 views

How does on-site energy $M$ influence Berry curvature and topological transitions in Haldane's model?

SOLUTION: The following papers almost fully-answer my question: https://arxiv.org/abs/0904.2117 https://arxiv.org/abs/1111.5020 Essentially, the Dirac points move and merge as M changes. I am ...
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176 views

In a spinless system with time reversal symmetry, is $E_n(k)=E_n(-k)$ always true?

I am studying TR-symmetry from: "Group Theory" by Dresselhaus, Dresselhaus and Jorio and there's a point that I cannot quite understand. The point is under eq. (16.17). In general, we know that the ...
3
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1answer
217 views

Positive and negative winding number related by time-reversal symmetry

I am now reading some articles about Dirac fermions in condensed matter physics and the most famous example is graphene. I am now trying to understand page 5 in this article : https://arxiv.org/abs/...
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291 views

Flux quantization and AB effect and Laughlin's argument of IQHE

I have a question essentially the same with this one "Aharonov-Bohm Effect and Flux Quantization in superconductors" which is why we can say the flux is quantized in superconducting disk but not in AB ...
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76 views

Examples of Chern number calculations where more than two $U(1)$ gauge of wavefunction has been used

While computing the Chern number of electronic wave functions \begin{align} \left|\psi\right\rangle = \begin{pmatrix}\cos\left(\frac{\theta}{2}\right) \\ \sin\left(\frac{\theta}{2}\right)e^{i \phi} \...
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Is Chern number still well defined with band touching?

Consider a 2 band system in 2d with band crossing on a ring. The coupling opens a gap. If the coupling is zero at some points of the ring, the band is still touching at these points. The berry ...
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81 views

How to interpret overlap in Hamiltonian if it is not a degeneracy?

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ h_x +...
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Getting $h_x, h_y, h_z$ Components of Hamiltonian after Gauge Transformation

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ ...
2
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1answer
186 views

Attempt at proving $-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$ from Kane and Fu's paper

I am trying to prove result (3.4) of the following paper: http://li.mit.edu/S/2d/Paper/Fu07Kane.pdf namely, that $$-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$$ ...
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262 views

Why the integral of Berry curvature over a closed surface is not zero?

I read [1,2] that for a spin-1/2 particle under magnetic field, the Berry curvature is a monopole, $$ \mathbf F_{\pm} = \mp\frac{\mathbf B}{2B^3}, $$ of which the integral over a closed surface is $2\...
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1answer
350 views

Is it valid to calculate Berry phase when there is energy level crossing?

Assume that we have electronic band structure and want to calculate Berry phase following the red line: If we have energy crossing in the path of integration, is it okay just to use the Berry phase ...
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134 views

Berry phase for instanton events in 2d quantum antiferromagnet

I am reading the paper "http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.1029" by Haldane where he calculates the berry phase associated with instanton or hedgehog events in the O(3) ...
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154 views

The Berry phase and anomalous commutator

The problem Recently I've read an article "Adiabatic theorem and anomalous commutator", written by Iida and Kuratsuji. In this article the authors relate the Berry phase with anomalous commutator of ...
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219 views

From Berry's phase to artificial Gauge potential

How a nonzero geometric phase in a loop is used to generate artificial gauge potentials? If possible, can you also tell how to generate the non-abelian artificial gauge potentials.
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Berry curvature and parallel transport gauge

As I understand it, the Berry connection in the parallel transport gauge is null. The Berry curvature however is gauge-invariant and we can compute it in any gauge we wish, including the parallel ...
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12 views

Why does the Berry Phase of π cause anti localisation in Dirac fermions?

I am learning about the theory of topological insulators and one point that puzzles me is the following: The Berry Phase aqcuired by forming a closed loop on a Dirac cones is π. The argument that I do ...
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61 views

Degenerate berry curvature, or not(?)

Recently I was trying to replicate some calculations concerning a particular hamiltonian, and I ran into some confusion concerning the berry curvature. Starting from, $$H= \frac{1}{2} \sum_k \psi_k^\...
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65 views

Topological invariants, what's that?

What's the difference between the Berry phase, the Euler number,the winding number and the Chern number? As far as I know they can all be computed by the same integral, but there seems to be some ...
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68 views

Berry connection in a solid

I am having troubles to understand an equation-sign for the Berry connection in a solid. The general formula reads \begin{equation} \vec{A}(\vec{R}) = \mathrm{i} \langle \Psi(\vec{R}) \, | \nabla_{\...
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118 views

Berry Curvature in a hexagonal lattice

I am having troubles to understand the concept of the Berry curvature in a hexagonal lattice. What I know is: The Berry curvature $\Omega_n (\vec{k})$ for the $n$-th band reads \begin{equation} \...
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184 views

Can we define Spin-Chern number for original QAHE Haldane model?

In Haldane's original paper [5], he discusses the quantum anomalous Hall effect as being characterized by the so-called Chern number that is the surface integral of Berry curvature over the entire ...
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95 views

Symmetry arguments on the Berry connection and the polarization charge

Consider the Berry connection $$ A_n(\mathbf{k})=i \langle n(\mathbf{k})|\nabla_{\mathbf{k}}|n(\mathbf{k})\rangle $$ and the polarization charge $$ \mathbf{P}=-\frac{1}{4\pi^2} \int_\mathrm{B.Z.}\...
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187 views

Berry Phase of Topological Insulators

I read all threads about that topic I could find, but didn't really find a sufficient answer for me, so I decided to ask my own: I read in Bernevig's book "Topological Insulators and Topological ...
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90 views

How to Explicitly Calculate z-Component of Berry Curvature?

While numerically playing with the 2-level Haldane model recently, I wondered how I could analytically calculate the z-component of the Berry curvature $F$. I framed my problem as needing an ...
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232 views

Why gauge-invariant Berry curvature commutator looks like torsion?

The Berry Curvature is defined as (for invariant gauge transformations) $$F_{ij} = [\partial_i, A_j] - [\partial_j,A_i] + [A_i,A_j]$$ The gauge covariance satisfies the transformation $$A_i \...
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72 views

From second quantisation Hamiltonian to general Hamiltonian form of Rice-Mele Model

I am concerning the following second quantization Hamiltonian: \begin{equation} H=\sum_j(\frac{t}{2}+(-1)^j\frac{\delta}{2})(c^\dagger_jc_{j+1}+H.c.)+\delta(-1)^jc_j^\dagger c_j \end{equation} ...
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504 views

Help to understand Non abelian berry phase

I'm trying to understand non abelian berry phase, but I'm having trouble with it. I'm reading the article "Appearance of Gauge Structure in Simple Dynamical Systems" of Wilczek and Zee (https://...
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73 views

three level system and holonomies

I'm doing an excercise from the book "Introduction to Topological Quantum Computation" of J. Pachos. This is excercise 2.1 related to the berry phase. The problem asks to consider the following ...
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286 views

The way to obtain the Berry phase into the hamiltonian

I'm interested in how to capture the chiral anomaly effects in quasi-classical approach. Precisely, I want to derive Boltzmann equation for partition function for massless fermions with definite ...
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320 views

Massive Dirac model Chern number 1/2

Why is massive Dirac model have a Chern number as half? I know this is something related to anomalies. And for fermions we have half, for bosons we have integer. But I failed to find any good ...
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194 views

Classical - quantum analogy of Berry phase

I am reading this webpage about Berry phase: http://materia.fisica.unimi.it/manini/berryphase.html I am happily convinced about the first "elementary geometry" example they provide. Now I am trying ...
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74 views

Is it possible to change one quantum state to another state by a cyclic adiabatic process?

An example is applying magnetic flux through the axis of a cylinder (2D system with periodic boundary condition). When changing flux from 0 to 1 flux quanta adiabatically, it seems that we can ...
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599 views

Bloch's theorem and Bloch's state

The question is not so much about the theorem, but more about what it means in this context: see this link. So yes, because of Bloch's theorem the Hamiltonian eigenstates in a crystalline system can ...
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389 views

Nonablianity of Wilczek Zee Potential

Consider a hamiltonian with degeneracies of energy in parameter space $(R)$.Now the Geometric phase(Wilczek Zee Potential) will acquire a non abelian nature. To prove the non abelian nature of Wilczek ...
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How the wave vector $k$ change slowly and travel a loop in Brillouin zone when we calculate the Berry phase?

According to the definition of the Berry phase, there must have a slowly changing parameter that travel a loop. when we discuss topology in energy band, the slowly changing parameter seems the $k$. my ...
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Is the Berry phase defined in terms of the periodic part of the Bloch wavefunction or the wavefunction itself?

In this paper, the berry phase is approximated to be $e^{-i\theta} = \prod_{i=1}^{N} \langle\psi_{n,k_i} | \psi_{n,k_{i+1}} \rangle$. The authors claim that "each Bloch wavefunction appears twice ...
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Does the fiber bundle approach for Berry connection contradict adiabatic theorem?

In Ref [1], the authors show how the Berry connection is a geometric quantity using the fiber bundle approach. My question is about the idea of taking a local section of a fiber bundle (corresponding ...
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1answer
42 views

Aharonov-Bohm Effect and the Berry Phase: gradient of a line integral of a vector field

I need some advice on how to perform the gradient of a line integral of a vector field. My problem refers to the Aharonov-Bohm Effect as it is discussed in the QM book from David Griffiths, as it ...
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26 views

When/why does an evolving wavefunction loop intersect with itself?

Let's say I have a 2-state system described by a $2\times 2$ non-degenerate Hamiltonian in some 2D parameter space. This is in the context of condensed matter, but should be more fundamental quantum ...
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31 views

Berry curvature vanishes in TRS system

In spin 1/2 system with TR symmetry , the Berry curvature must vanish. Because Berry curvature is odd. How to prove it? \begin{equation} \langle\partial_{-k_x}u^{I}(-k)|\partial_{-k_y}u^{I}(-k)\rangle-...
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Understanding why Berry phase, as you parallel transport along the geodesic is not zero

Parallel transporting a state along a geodesic doesn't introduce any anholonomy angle, that's what I learned in general relativity. In quantum mechanics, this anholonomy for states are related to the ...
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36 views

What are the most general symmetries that a Hamiltonian of the form $H=\vec{k}\cdot\vec{\sigma}$ can have?

Hamiltonians of the form $H=\vec{k}\cdot\vec{\sigma}$ with $\vec{k}$ being the crystal momentum and $\sigma_i$ being the $i$-th Pauli matrix (an $su(2)$ generator), are pretty common in the study of ...
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21 views

Can Berry phase been carried by bulk electrons in TIs?

I'm studying 3D topological insulators and more in particular, weak antilocalization (WAL) effects, so I know that they are characterized by a $\pi$ Berry phase that gives rise to destructive ...
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77 views

Conserved quantity in Graphene

The computation of the band structure of Graphene basically leads to the diagonalization of the following Hamiltonian: $$ H = -t \left( \begin{array}{cc} 0 & \epsilon(\vec{k}) \\ \epsilon^*(\vec{...