Questions tagged [berry-pancharatnam-phase]

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24
votes
4answers
834 views

Crystal momentum and the vector potential

I noticed that the Aharonov–Bohm effect describes a phase factor given by $e^{\frac{i}{\hbar}\int_{\partial\gamma}q A_\mu dx^\mu}$. I also recognize that electrons in a periodic potential gain a phase ...
19
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0answers
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 ...
16
votes
2answers
1k views

How is Berry phase connected with chiral anomaly?

Recently I've read in one article about very strange way to describe chiral anomaly on quasiclassical level (i.e., on the level of Boltzmann equation and distribution function). Starting from Weyl ...
15
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1answer
902 views

Fermi statistics and Berry phase

When the positions of two fermions are exchanged adiabatically in three-dimensional space, we know that the wave function gains a factor of $-1$. Is this related to Berry's phase?
12
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2answers
3k views

Adiabatic theorem and Berry phase

As far as I can check, the adiabatic theorem in quantum mechanics can be proven exactly when there is no crossing between (pseudo-)time-evolved energy levels. To be a little bit more explicit, one ...
12
votes
1answer
907 views

Aharonov-Bohm effect as a geometric phase-Adiabatic transfer not needed?

In his 1984 paper, Michael Berry proved that the Aharonov-Bohm effect is the same as a geometric phase. He did this by transferring a box containing charged particles around a solenoid. However, he ...
11
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1answer
3k views

Calculating the Berry curvature in case of degenerate levels (Non abelian Berry curvature): issue

The Berry phase accumulated on a path can be described by a matrix when we look at adiabatic time evolution with a Hamiltonian with degenerate energy levels. The Berry phase matrix is given by $$ \...
10
votes
1answer
2k views

Physical Interpretation of Relationship Between Hall Conductivity and Berry Curvature?

Why is the Hall conductivity in a 2D material $$\tag{1} \sigma_{xy}=\frac{e^2}{2\pi h} \int dk_x dk_y F_{xy}(k)$$ where the integral is taken over the Brillouin Zone and $F_{xy}(k)$ is the Berry ...
10
votes
1answer
582 views

Why is the Berry curvature odd under time reversal?

The question is: Why is the Berry curvature, defined as $$\mathcal{F}=-\mathrm i\, \epsilon_{ij}\, \left\langle\partial_{ki}u_{n}(k)\mid \partial_{kj}u_{n}(k) \right\rangle ,$$ odd if I apply time ...
9
votes
1answer
409 views

Why isn't Dirac credited with the discovery of the Aharonov-Bohm effect?

Above equation (8) of Dirac's famous 1931 paper in which he proposes his quantization condition for magnetic monopoles, he says "the change in [an electron's] phase around [a] closed curve [is] $2 \pi ...
9
votes
2answers
357 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 + \...
8
votes
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 ...
8
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1answer
989 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 ...
7
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1answer
2k views

What is curved in Berry Curvature?

Can anyone explain to me what is actually "curved" when we speak of a Berry Curvature?
7
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1answer
1k views

Topological insulators - Surface states have a phase?

When I look at the circle of the Dirac cone around the Dirac point of, let's say, $Bi_2Se_3$, then the electron winds around and it is true that it goes from momentum $-k$ and spin-up to $+k$ and spin-...
7
votes
1answer
840 views

Can the Berry Connection be derived from a metric?

The Berry Connection is $$A_\mu(R)=-i \langle \Psi(R) |\partial_\mu \Psi(R) \rangle$$ which allows us to parallel transport a state indexed by $R$. We can integrate the Berry Connection to get the ...
7
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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., $S^2, T^2$, ...). ...
6
votes
1answer
991 views

Aharonov-Bohm Effect and Integer Quantum Hall Effect

What is the relationship between Aharonov-Bohm effect and Integer Quantum Hall effect?
6
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1answer
1k views

Derivation of expression for Berry curvature

Many texts quote the expression for the Berry curvature for a two-level system, with Hamiltonian $\mathbf{h}(\mathbf{k})=(h_x,h_y,h_z)$ in terms of $\mathbf{k}=(k_x,k_y)$, as something like \begin{...
6
votes
1answer
732 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 ...
6
votes
1answer
1k views

Book on Berry phase and its relation to topology

I am searching for a book covering the Berry phase. Griffith has a good outline, but I would like a bit more detail, especially on the relation to topology. According to this post Ballentine also has ...
6
votes
2answers
2k views

How to include Berry connection in Hamiltonian?

When we calculate Berry connection, $A(R)=i<\psi(x,y)|\frac{d}{dR}|\psi(x,y)>\hat{R}$ corresponding to the Berry phase of any system, the gauge potential is related to the $R$ of the parameter ...
6
votes
1answer
553 views

Which charge to use in the Dirac quantization condition?

I have a follow-up question to Dirac magnetic monopoles and quark fractional electric charge quantization, regarding whether the "unit of electric charge" in the Dirac quantization condition should be ...
6
votes
1answer
588 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_{...
6
votes
1answer
359 views

Dirac string and Nielsen–Ninomiya theorem

Nielsen–Ninomiya theorem states that in a lattice system one can not have just one chiral fermion. Fermions necessarily come in pairs of opposite chirality. I am wondering if one can "explain" this ...
6
votes
1answer
1k views

Detail of deriving Berry Curvature From Berry Connection

The Berry curvature of the $n^{\mathrm{th}}$ eigenstate of Hamiltonian $H$ for the vector of external parameters $\vec{R}$ can be derived in part by writing the following two lines: $$ B^n(\vec{R}) \...
6
votes
1answer
482 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 ...
6
votes
1answer
307 views

Equation 2.27 from Pachos's introduction to topological quantum computing

http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the ...
6
votes
1answer
298 views

How should I explain Aharonov-Bohm effect in QED?

Let me clarify my problem, we all know what Aharonov-Bohm effect is, where we treat the vector potential $A_\mu$ as a classical field. Calculate this effect under classical quantum mechanics using ...
5
votes
1answer
488 views

Why the Hamiltonian near degeneracies should be proportional to Pauli matrices?

When a quantum system have a double degenerescence at one point, the Hamiltonian should be proportional to Pauli matrices near this point (also known as diabolic point) [Ref.]. But, why the ...
5
votes
1answer
288 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 ...
5
votes
1answer
1k 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 ...
5
votes
1answer
449 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 ...
5
votes
2answers
292 views

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)...
5
votes
1answer
518 views

How to calculate the ground states' Berry phases with doubly degeneracy, such as that due to the particle-hole symmetry or time reversal symmetry?

Suppose the ground states of a system are doubly degenerate due to an anti-unitary symmetry $K$, which are $|\psi>$ and $|K\psi>$. If the system is an one-dimensional Fermion system and anti-...
4
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1answer
718 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 ...
4
votes
1answer
300 views

Sources to learn about Berry phases and Adiabatic Theorem

I recently went through Griffiths' Quantum Mechanics text and there is a chapter called the Adiabatic Theorem that includes Berry phase and the Aharonov-Bohm effect. As I found them very ...
4
votes
2answers
1k views

How to derive the Aharonov-Bohm effect result?

In the derivations of the Aharonov-Bohm phase, it is directly mentioned that due to the introduction of the vector potential $A$, an extra phase is introduced into the wavefunction for case $A\neq0$ i....
4
votes
2answers
619 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,...
4
votes
1answer
765 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 ...
4
votes
1answer
213 views

Discontinuity of the geometric phase

Does the geometric phase accumulated along a closed trajectory (in some parameter space) has to be continuous?
4
votes
1answer
655 views

Berry's phase: in which space does the degeneracy appear?

This question follows a previous one of mine: Adiabatic theorem and Berry phase. In his original paper [ M. V. Berry, Proc. R. Soc. Lond. A. Math. Phys. Sci. 392, 45 (1984) ], Berry discussed the ...
4
votes
1answer
583 views

Berry phase with density matrix approach

Berry phase, coming from Schrodinger equation, has well known form in terms of closed integral $$\gamma = \int_C A(\xi) d\xi $$ with Berry connection $$A(\xi) = i < \psi(\xi) | \partial_{\xi} | \...
4
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0answers
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 ...
4
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0answers
392 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 $\Phi$....
4
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0answers
162 views

Where is wrong in the derivation about adiabatic theorem

There is a homework in Quantum Mechanics which is about adiabatic theorem. Let us argue where is wrong in following derivation. $$i \partial_t |\psi(t)\rangle = H(t) |\psi (t)\rangle \tag{1}$$ if $\...
4
votes
0answers
214 views

Monodromy, Holonomy and Braiding Phase

In quantum Hall effect, especially in the context of CFT description, these words come up often. I think I understand the braiding phase - as the phase gained by the wave function when a quasi ...
4
votes
0answers
886 views

How is the Geometric Phase measured in the experiment?

I had read some papers that have mentioned the geometric phase (Berry phase) can be used to detect the quantum phase transitions in a quantum many-body system. My question is: How is it measured in ...
3
votes
2answers
632 views

Is the artificial gauge field a gauge field?

The so-called artificial gauge fields are actually the Berry connection. They could be $U(1)$ or $SU(N)$ which depends on the level degeneracy. For simplicity, let's focus on $U(1)$ artificial gauge ...
3
votes
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 ...