Questions tagged [berry-pancharatnam-phase]

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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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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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33 views

Does $dH/dt$ mean anything in 2+1D condensed matter systems?

In condensed matter literature, I often see the derivative of the Hamiltonian with respect to momentum $k$, $dH/dk$. For example, it is often used in expressions for Berry curvature in 2D $k$-space. ...
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35 views

Polarization of a one-dimensional chain of singly charged alternating anions and cations

In the following paper https://arxiv.org/pdf/1202.1831.pdf, the author claculated the polarization of a 1D chain of singly charged alternating anions and cations. The value of polarization depends ...
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57 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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43 views

Berry phase in graphene

I was recently reading about the non-Abelian Berry phase and understood that it originates when you have an adaiabatic evolution across a degenerate manifold of states. Am I correct in thinking that a ...
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34 views

How to answer Berry phase example calculation

I'm reading David Vanderbilt's Berry phase book and can't reproduce a simple example. On pdf page 93 (book page 77), he gives a set of vectors in equation 3.2 (that I assume are in an orthonormal ...
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23 views

Berry phase mathematical manipulations

I’m learning about Berry phase and in the source they use the following without explaining the steps in between—any help interpreting how it’s done would be greatly appreciated \begin{aligned} & \...
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38 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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58 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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63 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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67 views

Berry curvature from spherical polar coordinates to Cartesian coordinates

Let us consider the typical example of calculating the Berry phase: a spin in a magnetic field. We start with the Hamiltonian $H=-\textbf{B} \cdot \boldsymbol{\sigma} + B$ where $\textbf{B}$ is the ...
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58 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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407 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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63 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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90 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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25 views

Are there conditions for vanishing of geometrical phases in QM?

Basically as the title says. Are there theorems for sufficient and necessary conditions for the vanishing of Berry and/or Wilzeck-Zee phases on a given quantum mechanical system?
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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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99 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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Derivation of the Berry Curvature and Bloch Magentic Moment in Graphene

I am attempting to derive equations 2 and 6 from Xiao et al. paper "Valley contrasting physics in graphene" (Link to paper). The Hamiltonian for graphene with a staggered sublattice potential (in ...
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49 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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115 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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58 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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133 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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55 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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210 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 ...
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48 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
121 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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296 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
211 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
102 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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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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157 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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187 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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190 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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229 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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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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169 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, ...
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64 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{...
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500 views

Inversion symmetry restrictions to the Berry curvature in 2D

It is said that if a lattice has inversion symmetry, then the Berry curvature, $\vec{\Omega}(\vec{k})$ is even in $\vec{k}$, i.e. $$\vec{\Omega}(\vec{k})=\vec{\Omega}(-\vec{k})$$ I have also derived ...
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132 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 ...
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135 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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1answer
968 views

Relation of Berry phase and winding number

I am reading the following article dealing with the properties of Dirac fermion in condensed matter physics : https://arxiv.org/abs/1410.4098 In the page 5 of this article, the formula for the ...
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1answer
177 views

Berry curvature in rotating traps

A quantum system in a rotating (harmonic) trap is equivalent to a stationary system in the presence of a vector potential $\mathbf{A}$. The proof can be found in chapter 5 here, but in short it goes ...
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381 views

When is the Berry phase only dependent on path topology?

Background Suppose we have a Hamiltonian $H(\mathbf{R})$ which depends on some parameters $\mathbf{R}$. For each value of $\mathbf{R}$, the Hamiltonian will have some set of eigenvectors $\{ | \phi_{...
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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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186 views

Visual representation of quantum state/phase

Is there a known good way to visualize a quantum state, composed of the sum of eigenstates, with a phase rotating on each state. I am looking for a way to keep up with the state and the phase. In a ...
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117 views

Aharonov-Bohm phase picked up when a magnetic dipole goes around a charge

When a particle with charge $q$ traverses a loop that encloses a magnetic flux $\Phi$, it picks up a phase $e^{iq\Phi}$ (I have set $c$ and $\hbar=1$). This is the usual Aharonov-Bohm phase. Now, let ...
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325 views

Questions about Berry Phase

I'm learning about the Berry Phase from the original paper, and from the TIFR Infosys Lectures The Quantum Hall Effect by David Tong (2016). I have some questions regarding the original derivation of ...
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512 views

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 ...