Questions tagged [berry-pancharatnam-phase]
The phase difference acquired over the course of a closed loop which results from the geometrical properties of the parameter space of the Hamiltonian.
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Quantum geometry and Berry Curvature
Since Quantum geometric tensor has two parts real and imaginary. The imaginary part is the Berry Curvature. the real part of the tensor also has some significance. Can someone give me a clear idea of ...
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Does the total Zak phase always sum to zero?
In 2D, the sum of the Chern numbers over all bands is zero. However, this result relies on the ability to define a Berry curvature, which is only possible in $d \geq 2$ dimensions. In 1D it is ...
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Can you evaluate the Berry phase integral? [closed]
This is my first post. Can anyone simplify the integral in eq(8.16) in the picture. How the integral is evaluated ? How the sign function came to the scenarion? The pic taken from the book "...
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Deriving the non-abelian Berry connection
I'm slightly confused about a manipulation in Section 1.5.4 of Tong's notes on the Quantum Hall Effect.
This concerns the derivation of the non-abelian Berry phase.
Setup:
We have an $N$-dimensional ...
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Non-abelian Berry connection : clashing time-ordering conventions, and component-wise form
Let $\mathcal{M}$ be a $k$-dimensional parameter space associated to a quantum system with an $N$-dimensional ground state. As usual, we assume the system is subject to some adiabatic tuning of ...
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Gauge freedom issues when numerically calculating overlap of states
Calculations of quantities of the form:
$$\langle\psi(k)|\partial_k|\psi(k) \rangle \ ,$$
require a smooth choice of the wave function $|\psi(k)\rangle$ over the whole BZ. This is not a problem. ...
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Strange manipulation of Hamiltonian operator and gradient
I'm reading M. V. Berry's Quantal Phase Factors Accompanying Adiabatic Changes and came across an unfamiliar identity in eq. (8), namely $\langle m | \nabla _Rn \rangle = \frac{\langle m | \nabla_R \...
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How can I calculate this equation if we know that there is non zero Berry-phase between the valence and conduction band
The geometric phase can be interpreted as a Berry curvature in the momentum space. My guess is $(q^2+\text{Berry-phase}/\text{lattice constant}^2)/\text{direct gap}$.
$$\langle\psi_{n',\mathbf{k}+\...
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Calculating the Berry potential: Questions
[Reference: Modern Quantum Mechanics, J.J. Sakurai, Chapter 5]
The Berry potential is defined by,
$$ \mathbf{A}_{n} (\mathbf{R}) \ = \ i \langle n | \mathbf{\nabla}_{\mathbf{R}}|n\rangle$$
Here, $\...
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Physical meaning of Berry phase
The Berry phase is defined as the phase acquired by the wave function of an electron when passing along some closed contour along the Brillouin zone.
In the case of topological materials and, in ...
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Berry connection in SSH model
In the SSH model for the 1D case, we get the eigenvectors as $$|(\pm)k>= \begin{pmatrix} \pm e^{-i\phi} \\ 1 \end{pmatrix}$$ where $\phi = tan^{-1}(\frac{wsin(k)}{v+wcos(k)})$. We can calculate the ...
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Berry phase for nuclear magnetic resonance
I am sure that I am making some dumb mistake somewhere here so please bear with me. I have the Hamiltonian for $B>0$
\begin{equation}
\hat{H}(t) = B[\underbrace{\cos(\theta) \hat{\sigma}_z}_{\hat{H}...
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Berry curvature in cartesian coordinates
I am trying to find the Berry curvature in Cartesian coordinates for the spin-1/2 Hamiltonian given by
\begin{equation}
\hat{H} = \sum_{i}d_i \hat\sigma_i
\end{equation}
Where $\mathbf{d} = \mathbf{d(...
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Meaningful topological invariants and quantities for the description of 3D topological insulators
I'm currently trying to understand the classification of 3D topological insulators (like Bi2Se3). Most reviews dealing with this topic start with the introduction of the Quantum Hall effect since this ...
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What is natural about the Berry connection?
So I asked a similar question here and even though I still believe it's a valid question, the formulation may have been a bit too complicated to pique people's interest, so let me try to break it up ...
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Concise formulation of Berry phase as holonomy of "natural" connection
I've been trying to understand the Berry phase (abelian/non-abelian) as the holonomy of some "natural connection". I almost have all the pieces together, but there are a few parts that are a ...
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I need help with a result in the Berry Connection definition [closed]
I was reading B. Andrei Bernevig's "Topological Insulators and Topological Superconductors" and on the chapter simple model for Chern insulators there is a step where he states the Berry ...
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Significance of Berry's phase
I'm familiar with the adiabatic theorem and have basic knowledge of quantum mechanics. While doing some independent learning, I stumbled on the concept of Berry's phase.
I would like to understand the ...
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Berry phase and Wilson loop
According to the definition, the Wilson loop is
\begin{equation}
W[\mathcal{C}] =\operatorname{Tr}\left[\mathcal{P} \exp\left\{i\oint _{\mathcal{C}} A_{\mu } dx^{\mu }\right\}\right]
\end{equation}
...
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How does the fiber bundle perspective on geometric phase lead to a certain connection one-form?
I'm trying to understand why the relevant connection one-form when calculating geometric phase in quantum systems is
$$\mathcal{A}_\psi(X):=i \text{Im}\langle \psi | X\rangle.$$
Set-up: I'll set the ...
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Discrepancy of berry phase from two methods, in a tight binding basis and a Wannier basis
Let us consider a one-dimensional system with only one occupied band $|\psi_k\rangle$.
The berry phase is defined as
$$
\gamma = i \int_{BZ} dk \langle \psi_k | \partial_k |\psi_k \rangle.
$$
The ...
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Berry phase from Bloch wave functions in the basis of Wannier functions
The formulate to calculate berry phase for Bloch wave functions is
$$
\gamma = i \sum_{n\in occ}\int_{\mathcal{C}} dk \langle \psi_k^n|\partial_k|\psi_k^n\rangle,
$$
where $|\psi_k^n\rangle$ is a ...
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The integral for Berry curvature dipole
In this paper, the author proposed that the nonlinear Hall effect can exist in time-reversal-invariant systems, driven by Berry curvature dipole (BCD), in which a two-band model
$$H_{s\Lambda}=s\alpha ...
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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 ...
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What "really" happens to an evolving state which goes through degeneracy? (and relation to Berry phase)
Say I have a time dependent Hamiltonian $H(t)$ and I let an initial eigenstate $|\psi_n\rangle$ evolve according to the Schrodinger equation, so that at time $t$ the state is $|\psi_n(t)\rangle$ with ...
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What is the effect on Berry’s phase of a non-closed path in configuration space
I’m following Professor Zweibach’s MIT lecture series $8.06$ on and looking at the section on Berry’s phase for a system undergoing changes in configuration. He explains that in order for the process ...
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What is symmetry in this context?
I'm reading a paper that was written by a senior undergraduate at my college and I'm having a trouble trying to understand a part of his paper. His paper is on Measuring nonlinear Phase accumulation ...
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Berry's phase for an electron in a two-level system
I am trying to reconcile two seemingly contradictory statements about Berry's phase for a two-level system in a magnetic field. Consider the Hamiltonian $H(t) = -\mathbf{B}(t)\cdot\pmb{\sigma}$ where $...
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Berry Phase Computation
Equation $(6.7.16)$ in the book Lectures on Quantum Mechanics (second edition) by Weinberg reads
$$\gamma_{n}[C]=n \iint_{A[C]} d A \frac{\mathbf{B} \cdot \mathbf{e}[\mathbf{B}]}{|\mathbf{B}|^{3}}\tag{...
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Parallel transport geometric phase derivation
In Modern Foundations of Quantum Optics by Vlatko Vedral, he derives an expression for the change in phase under parallel transport after completing a small loop as
$$\begin{align} \delta \theta &=...
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Why not all Berry phase just vanished?
I just learned that for any real wavefuntions, berry phase equals zero. But in Griffiths' Problem 2.1(b), he proved that any complex wavefuntion can be written as linear combination of REAL ...
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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 ...
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Numerical calculation of berry curvature of Haldane model
I'm currently trying to simulate the Haldane model of graphene and am looking into the calculation of Berry curvature of the finite lattice. I'm using a tight-binding model of graphene with nearest-...
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Question about a step in Griffith's QM book on Berry's phase for an electron in time-varying magnetic field
Some Context
I am going through David Griffith's Introduction to Quantum Mechanics (Second Edition), Chapter 10, Section 10.2 on Berry's phase.
Suppose we have a system with a time-dependent ...
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Geometric Phase vs Dynamic phase
On the Wikipedia page for the Adiabatic Theorem (https://en.wikipedia.org/wiki/Adiabatic_theorem), formulas for both the dynamic and geometric phases are given. The dynamic phase is expressed as:
$$\...
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Quaternion formulation for parallel transport along a curve on a 2-sphere
One image that is often used to illustrate curvature in general relativity is the triangle on a 2-sphere, made out of great circle arcs. At the end of a geodesic transport along this triangle, the ...
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Berry flux of magnetic monopole
I am sorry if it looks stupid question. I want to ask how sin is transferred back from spherical to Cartesian and how $F_{ij}$ tells us about magnetic monopole in magnetic field. $F_{ij}$ is berry ...
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Would there be plateaus in IQHE in haldane graphene model
It's my understanding that the plateaus observed in the integer quantum hall effect are due to scattering due to impurities in the material. The haldane model for graphene does not include a model for ...
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How can i calculate the Berry Curvature for the Dirac points in Haldane graphene?
I want to calculate the berry curvature at the Dirac points in graphene with complex next nearest hopping (haldane model) in order to show that it is non-zero at the dirac points and use it to compute ...
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Numerically calculating the berry curvature for graphene
i'm trying to reproduce this density plot for the Berry curvature in the Brillouin zone of graphene from this website.
In order to do this I am attempting to use this equation for the berry curvature
...
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How to derive Dirac's quantization of electric and magnetic charge from Berry's phase?
I have seen two ways of deriving Dirac's quantization of electric and magnetic charge using Berry's phase, one from Sakurai's book Modern Quantum Mechanics P354~355 and the other from David Tong's ...
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Deriving the Berry phase from the Schroedinger equation
Let $|n(\mathbf{R})\rangle$ be eigenstates of the snapshot Hamiltonian $H|\mathbf{R} \rangle$, of eigenvalues $E_n(\mathbf{R})$. The vector $\mathbf{R}$ contains the parameters upon which the system ...
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Why is the Chern number an integer?
I have relatively limited knowledge on the Chern number, and I know that there exists high-level math proofs that the Chern number is an integer, but let me try to focus on the case I have in mind. $\...
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What is the physical meaning of adiabatically varying the wavevector $k$ as a parameter to calculate the Chern number for topological effects?
Could it mean something like applying a weak electric field and perturbing the band structure? Or some other weak perturbation? Or is that the wrong idea?
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‘Proof’ that non-Abelian Berry phase vanishes identically
For a degenerate system with Hamiltonian $H =H(\mathbf{R})$ and eigenstates $\left|n(\mathbf{R})\right\rangle$ the non-Abelian Berry connection is
$$A^{(mn)}_i=\mathrm{i}\left\langle m|\partial_in\...
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Is the local Berry connection difference gauge invariant?
The Berry connection in terms of the cell-periodic Bloch states $u_{n\mathbf{k}}=e^{-i\mathbf{k \cdot r}}\psi_{n\mathbf{k}}$ with band index $n$ and crystal momentum $\mathbf{k}$ is defined as
$$
A_{n}...
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Noncontractable loops in the 2D Brilluoin zone and the Chern number
I'm getting quite twisted around trying to figure out what all is quantized exactly in IQH looking at it from the Chern number perspective.
Let's suppose quantum hall on a torus -- I can apply a large ...
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Quantization of the Berry Phase
Let's consider the Aharanov-Bohm effect. Following Girvin & Yang, an infinitely long, very thin flux tube running along the $\hat z$ axis is surrounded by a strong potential barrier preventing ...
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How to understand degeneracy and singularity of field
In the online lecure given by Professor Wu Yongshi https://www.koushare.com/video/videodetail/4619 1:30‘, he says:
Suppose we have a state $|m,R(t)\rangle$, where $R(t)$ is controlled parameters ...
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What definition of integral is implied when expressing nonzero Chern number as the integral of Berry curvature?
In defining a nonzero Chern number as the integral of Berry curvature over the parameter manifold: $$n=\frac{1}{2\pi}\int_{S}{\mathcal{F}}{dS}$$ does the integral exist in a general Riemann sense, or ...
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