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Questions tagged [berry-pancharatnam-phase]

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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.}\...
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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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28 views

How to interpret Berry curvature in 2-band model?

While studying the 2-band Haldane model, I realized that I am missing an intuitive picture of how Berry curvature comes into play, especially when considering an adiabatic loop. A 2-band model has 2 ...
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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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45 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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31 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
132 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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27 views

Magnetic Flux in Unit Cell of Haldane Model

I am trying to understand how magnetic fluxes arise due to NN and NNN hopping in Haldane's model. In An Introduction to Topological Insulators by Fruchart et al., we see the following figure: ...
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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
47 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
96 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
72 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
55 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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55 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
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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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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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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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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/pdf/0904.2117.pdf https://arxiv.org/pdf/1111.5020.pdf Essentially, the Dirac points move and merge as M changes. I ...
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108 views

What does the sign of Berry curvature physically mean?

For Haldane’s model, I plotted the Berry curvature as follows: [UPDATE II: It appears as if the four peaks on the sides are due to Dirac points being in the process of moving as the on-site energy ...
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What is the closed-loop line integral of Berry curvature in a two-level model?

I am aware that integrating the Berry curvature over the entire Brillouin zone gives us the Chern number. However, I wonder what a closed loop line integral of the Berry curvature means. I think ...
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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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1answer
97 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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55 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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Berry phase in graphene

For a low-energy description of graphene the Hamiltonian is given by $$ H = v_F \left(\begin{array}{cc} 0 & p_x-ip_y\\p_x+ip_y & 0 \end{array} \right)$$ where $v_F$ is the Fermi-velocity and $...
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1answer
268 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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86 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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1answer
76 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
498 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
154 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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1answer
203 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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Sign change of Berry curvature with reversal of out-of-plane electric field

Why does the Berry curvature at K and K' valleys switch sign when we reverse the direction of applied perpendicular electric field? i.e., if for positive electric field Berry curvature at K is ...
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0answers
192 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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1answer
133 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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1answer
80 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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1answer
239 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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1answer
362 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 ...
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1answer
159 views

Why does calculating Berry phase with stokes theorem give you a different result from a direct integration?

For generic two level system $H = d(R) \cdot \mathbf{\sigma}$. $$C_{-} =\frac{1}{2\pi} \int_{S^2} d\theta d\varphi \Omega_{\pm,\theta\varphi}=+1$$ $$ \gamma _{n}=\int _{\mathcal {S}}d\mathbf {S} \...
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1answer
115 views

Geometric phase acquired by a photon on the Poincare sphere

I know that when a photon (spin 1) is parallely transported in the Poincare sphere in a closed loop, the geometric phase which it acquires is 'half' the solid angle subtended by the closed loop at the ...
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1answer
291 views

Why is Berry connection a connection?

The Berry connection, following the derivation of the Berry phase for a non degenerate system, is $\mathcal{A}_{k}(\lambda) = i \langle n|\frac{\partial}{\partial \lambda^{k}}|n\rangle$ This result ...
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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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1answer
359 views

Berry Curvature and Curvature Tensor

When the curvature tensor (from Einstein's theory) has a non-zero torsion, it is said to be an antisymmetric tensor in the last two indices composed of the connections of the field. Alternatively, the ...
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1answer
188 views

Berry phase in 2D Harmonic Oscillator

I am trying to figure out the Berry phase in case of 2D Harmonic Oscillator under adiabatic cyclic condition of Hamiltonian. I know that in 1 D since Hamiltonian has only 1 time dependent parameter (...
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1answer
237 views

Importance of the Pancharatnam–Berry Phase

As I understand it, the Pancharatnam–Berry phase first arises in the adiabatic approximation for the evolution of a quantum state. For the evolution of a quantum state parameterized by the set of ...
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2answers
342 views

Effects of Topological Terms: Hopf, $\Theta$, Chern-Simons, WZW, Berry phase term

What are the effects and the differences of Topological Terms? For example, I had known and heard several of them are called Topological, (1) Hopf term, (2) $\Theta$ term, (3) Chern-Simons term,...
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1answer
73 views

Topology of parameter space

How do we decide the topology of the parameter space? Does this topology depends on the Hamiltonian (the form of potential) we are using? I was studying Berry's phase and it involves rotation of the ...
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1answer
481 views

Is my attempt to prove that Berry's phase is quantized in inversion symmetric systems true? Do I violate gauge invariance?

The Berry's or Zak's phase is given as \begin{align*} \gamma & =\oint_\mathrm{BZ}d\mathbf{k}\mathcal{\mathcal{\mathcal{A}}}(\mathbf{k})\ \ \mbox{mod }2\pi\\ & =i\oint_\mathrm{BZ}d\mathbf{k}\...
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1answer
274 views

Flux quantization in a superconductor and Berry phase

In the derivation of flux quantization for a superconducting ring, we say that $$\oint_C \nabla\theta . dl = \theta_{2} - \theta_{1}$$ Then we equate this value to $2\pi n$. The reason cited in ...
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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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1answer
384 views

Berry curvature for metal and Anomalous Hall Effect

As much I understand, in a crystal Berry connection and Berry curvature is defined for a particular band. The Hall conductance is given by the total Chern number (integrated Berry curvature over the ...