Questions tagged [berry-pancharatnam-phase]

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Numerically calculating non-Abelian Berry curvature: Definition of multiplet in explicit $4\times 4$ system with 2-fold degeneracy?

I am trying to use eq 16 of the following paper to calculate the Chern number of a 4x4 degenerate system: https://arxiv.org/abs/cond-mat/0503172 [1]. I believe this is the standard scheme used by many....
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43 views

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
40 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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1answer
52 views

Curl of Berry connection

If $|n\rangle=|n( \textbf{R}(t) ) \rangle $ satisfies the equation $$H(\textbf{R}(t))|n(\textbf{R}(t)) \rangle = E_{n}(\textbf{R}(t))|n(\textbf{R}(t))\rangle$$ then the berry phase $\gamma_{n}(t)$ ...
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1answer
23 views

Aharonov-Bohm effect of doubly localized wavepacket

I want to imagine an exotic situation regarding Aharonov-Bohm effect. The wavefunction $\psi$ of the electron is even ($\psi(\mathbf r) = \psi(-\mathbf r)$) and localized in two spatially separated ...
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1answer
73 views

Berry's phase for non-normalized wave functions

Let $\hat{H}(t)$ be the Hamiltonian of a quantum system depending on time $t$ through $k$ parameters $R(t) = (R_1(t), R_2(t), \dots, R_k(t))$: $$ \hat{H}(t) = \hat{H}(R_1(t),R_2(t),\dots,R_k(t)). $$ ...
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1answer
79 views

In fiber bundle picture of Berry connection, what is the vertical basis if the horizontal basis is the underlying parameter space?

In Ref. [1], the authors show how The geometric (Berry) phase is shown to have its origin in the nontrivial geometry of the fiber bundle: Hilbert space --—> space of states. The nontrivial ...
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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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1answer
60 views

Berry Connection Calculation for a 2-Level System [closed]

Suppose we start with a state on the Bloch sphere given by: $$|\psi\rangle = \begin{pmatrix}\cos\left(\frac{\theta}{2}\right)\\e^{i\varphi} \sin\left(\frac{\theta}{2}\right)\end{pmatrix}$$ The Berry ...
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1answer
127 views

Numerically calculating Berry curvature in >2-band 2D systems?

The standard method for numerically calculating the Berry curvature of a 2D condensed matter system is given by Fukui-Hatsugai-Suzuki in this paper. They discretize $k$-space into a grid with tiny ...
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1answer
145 views

Deriving the non-abelian Aharonov-Bohm effect as a Berry phase

I am trying to derive the non-abelian Aharonov-Bohm effect by generalising Michael Berry's derivation to the case of non-abelian gauge field $A$. My derivation so far We require a degenerate ...
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29 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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28 views

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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66 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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57 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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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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350 views

Numerical Berry curvature for bosons

I am trying to numerically compute the Berry Curvature for a generic quadratic Bosonic Hamiltonian of the form $$H = \sum_{ij} A_{ij} b_{i}^\dagger b_j + \frac{1}{2} \sum_{ij}\left( B_{ij} b_i b_j + \...
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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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63 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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1answer
40 views

Can you give an experimental example showing the difference between global and relative phase in QM?

Let's say, that we are in possesion of a very simple quantum system, whose state can be written as $$ |\psi\rangle = c_0 |\psi_0 \rangle + c_1 |\psi_1\rangle.$$ Now, we can change this state in two ...
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1answer
49 views

Berry phase and an emergent gauge field

In Nakahara 10.6.2 the case of system with fast $r$ and slow $R$ degrees (might be more than one of each) of freedom is discussed. The Hamiltonian is - $$H=\frac{p^2}{2m}+\frac{P^2}{2M} + V(r;R)$$ ...
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1answer
159 views

How does magnetic monopole arise from Berry curvature?

The Berry connection is defined as $$A_n(R)=i\left<\psi_n(R)\right|\nabla_R\left|\psi_n(R)\right>$$ and it is mathematically analogous to the vector potential. We can then naively define the ...
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1answer
39 views

Berry phase for effective gauge potential

On page 290 of Wens QFT he says that for the adiabatic motion of a single quasiparticle, for small t, $$ \left\langle\Psi^{h}\left(\xi(t+\Delta t), \xi^{*}(t+\Delta t)\right) | \Psi^{h}\left(\xi(t), \...
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1answer
129 views

Berry phase (Condensed matter)

I am reading Marder (Condensed Matter Physics). He talks about the geometric phases and then we derives an expression in which the Berry connection and phase comes up. The goal is to see how the ...
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0answers
93 views

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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1answer
71 views

What is the logic connection between these two statements?

What is the connection between these two statements: the berry curvature change sign under time-reversal operation If the system has the time-reversal symmetry, then berry curvature is odd in k. ...
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1answer
972 views

Numerical Calculation of Berry Curvature

I am trying to calculate some berry curvature (BC) in a 2D lattice and I have some things I am getting lost with. In the 2D lattice, we set up the eigenvalue problem $H|u_1\rangle = \epsilon_i|u_i\...
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0answers
67 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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111 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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1answer
108 views

Berry phase: Spin in a magnetic field parameter space manifold

Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce ...
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174 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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1answer
417 views

Derivation of the Berry Curvature and Bloch Magentic Moment in Graphene

(I found a workable solution, skip to the "Solution" part to see it) I am attempting to derive equations 2 and 6 from Xiao et al. paper "Valley contrasting physics in graphene" (...
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1answer
91 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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2answers
132 views

Magnitude of the cross product of two bra-kets?

From the mathematical perspective, one can take the magnitude of a cross product: $$ |a\times b|=|a| |b| \sin{\theta}\cdot n, $$ where $\theta$ is the angle between a and b in the plane containing ...
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72 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 \\ ...
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182 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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68 views

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 \\ ...
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1answer
285 views

How does the Berry curvature relate to the hopping strengths in the Haldane model?

Take Haldane's Hamiltonian, as quoted from Fruchart et al.'s An Introduction to Topological Insulators: 3.5.3. Haldane's Hamiltonian The first quantized Hamiltonian of Haldane's model can be written ...
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89 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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1answer
183 views

Is it possible understand Berry curvature as Gaussian curvature in some limit?

I would like to understand the Berry curvature and the Chern number from mathematical geometry-topology. I understand that in electronic QHE, there is a map from $k^2$ to a vector space where the ...
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1answer
467 views

Berry connection and time reversal symmetry

I am seeing how the Berry connection $\mathcal{A}(k)$ transforms under time reversal symmetry. I seem to have a hiccup over something simple. I may have overcomplicated things but I think it points to ...
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1answer
384 views

Why does gauge invariance HAVE to correspond to an observable?(Or is it the other way round)

Under the line integral of the geometrical Berry phase, a close-loop integral is gauge invariant as if we were to perform a gauge transformation of the initial state, with the end point of the path in ...
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1answer
172 views

Berry phase covariant derivative

I have been studying some simple examples of the covariant derivative for 2D surfaces and the way that it is constructed is by taking the usual derivative in the 3D Euclidean space at a point $p$ on ...
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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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1answer
180 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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253 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
271 views

Second Chern class in 2D Haldane model from Atiyah-Singer Index Theorem?

I was reading through a physics-centered exposition of the Atiyah-Singer index theorem and I wondered what it would mean to talk about Haldane's model for the case of a manifold with a boundary. It is ...
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271 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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4answers
2k views

Book recommendations - Topological Insulators for dummies

Is there a pedagogical explanation of what is a topological insulator for those that do not even know what the Berry phase is but have a basic understanding of quantum mechanics and solid state ...
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1answer
245 views

Berry Phase for Bloch electrons

I am new to the topic of Berry phase. The definition says that Berry phase depends only on the path in the parameter space of $R$, where the Hamiltonian is $H(R)$, but whatever problems I have seen, ...