# 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 ...
0answers
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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 ...
1answer
186 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 $$-i\ \langle{u_n|\nabla_k u_n}\rangle=-\dfrac{i}{2}tr[v^\dagger(k)\nabla_k v(k) ]$$ ...
1answer
88 views

### 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 ...
0answers
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 ...
0answers
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 +...
0answers
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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 ...
1answer
46 views

### 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 . I believe this is the standard scheme used by many....
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 ...
1answer
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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3k views

### 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 ...
0answers
46 views

### Does the fiber bundle approach for Berry connection contradict adiabatic theorem?

In Ref , 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 ...
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 ...
1answer
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 $\gamma_{n}(t)$ ...
1answer
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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 ...
1answer
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. , 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 ...
1answer
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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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 ...
1answer
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 ...
1answer
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" (...
2answers
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 ...
1answer
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 ...
1answer
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 ...
2answers
356 views

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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 ...
1answer
3k views

### 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 ...
0answers
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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 ...
1answer
731 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 ...
1answer
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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 ...
4answers
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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 ...
1answer
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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 - $$H=\frac{p^2}{2m}+\frac{P^2}{2M} + V(r;R)$$ ...
1answer
172 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 ...
0answers
276 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 ...
1answer
39 views