Questions tagged [berry-pancharatnam-phase]

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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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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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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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Does a Berry phase operator exist?

The closed-path Berry phase can have measurable effects and, if I am understanding correctly, is a measurable quantity in and of itself. If that is so, is there a Hermitian operator with Berry phases ...
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28 views

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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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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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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1answer
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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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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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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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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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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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53 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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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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86 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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93 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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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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61 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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445 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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877 views

What happens to Berry curvature under time reversal symmetries, in band structures?

First let we have a system with a periodic potential such that we can apply Bloch theorem to it. $|\psi_{n,k}>=e^{ikr}|u_{n,k}>,$ is our eigen function n and k are band indice and cyrstal ...
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155 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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151 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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356 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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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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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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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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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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What is the quantum / Berry-Pancharatnam phase for a spin-j state with z-component m?

Quantum phase arises when a spin-j state is sent through a sequence of transitions that return it to its original position. For example with spin$-1/2$, a state picks up a complex phase of $\pi/4$ ...
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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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A question on the Chern number and the winding number?

Let $\mid \psi(x,y) \rangle$ be a normalized wavefunction living in a $d$-dimensional Hilbert space and depend on two real parameters $(x,y)$ that belong to a closed surface (e.g., $S^2, T^2$, ...). ...
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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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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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Is the non-trivial topology on the torus reflected on the Bloch sphere?

Almost every text on topological insulators have the Bloch sphere example of a two level system showing the non triviality of the bundle of an eigenvector over the sphere: we can't define an ...
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44 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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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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58 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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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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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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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
143 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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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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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
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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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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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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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Calculation of Berry's phase due to monopole tunneling event of $O(3)$ NLSM on square lattice

I am currently reading the seminal paper by Duncan Haldane: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.1029 In this paper, he asserts that for a unit-vector field $\hat{\Omega}(x,y,t)...
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1answer
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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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183 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
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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.}\...