# What is Berry phase: Definition and 28 Discussions

In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by T. Kato (1950), S. Pancharatnam (1956), and by H. C. Longuet-Higgins (1958) and later generalized by Sir Michael Berry (1984).
It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase.
It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385-386 of the textbook by Bunker and Jensen.In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be simply connected, or there will be nonzero holonomy.
Waves are characterized by amplitude and phase, and may vary as a function of those parameters. The geometric phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could involve rotations but also translations of particles, which are apparently undone at the end. One might expect that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursions correspond to a loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the geometric phase, and its occurrence typically indicates that the system's parameter dependence is singular (its state is undefined) for some combination of parameters.
To measure the geometric phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the geometric phase. This mechanics analogue of the geometric phase is known as the Hannay angle.

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1. ### A Is Chern number exactly an integer of half-integer?

I'm trying to calculate the Chern number for a specific system. When I calculate the Chern number, it's not an integer, for example, I have 0.9 for it. Is it the wrong result or can I consider it as 1?
2. ### Calculation of the Berry connection for a 2x2 Hamiltonian

This isn't technically a homework problem, but I'm trying to check my understanding of the geometric phase by explicitly calculating the Berry connection for a simple 2x2 Hamiltonian that is not a textbook example of a spin-1/2 particle in a three dimensional magnetic field solved via a Bloch...
3. ### I How to Find the Berry Phase in Slowly Varying Systems

I was trying to follow this page: https://phyx.readthedocs.io/en/latest/TI/Lecture notes/2.html With ##|n(\mathbf{R})\rangle## being the eigenstates of ##H|\mathbf{R} \rangle## of eigenvalue ##E_n(\mathbf{R})##, the task is to solve the Schroedinger equation ##i | \dot{\psi}(t) \rangle =...
4. ### Time dependent perturbation theory (Berry phase)

If I plug the solution into the Schrodinger equation I get $$(i \hbar \partial_t - H)\ket{\psi} = 0$$ Since I know that the zeroth-order expansion is lambda is already a solution I think this is equal to $$(i \hbar \partial_t - H)e^{i\phi} e^{-i\gamma}\ket{\delta n} = 0$$ If now I carry on with...
5. ### I How to recalculate Berry phase dipole moments (jumps over time)?

Is there anyone out there who can help a newbie understand how to deal with Berry phase dipole moment data? I ran a simulation to calculate dipole moments over time using the Berry phase method. The problem is that there are jumps in my data. There is an example at the end of this post (part...
6. ### A Berry phase in condensed matter

I am trying to understand how is topology used to characterize materials. So I understand that to calculate the Berry phase you will parameterize your Hamiltonian and change this parameter in some way and return to the initial value. What I do not understand is what does this changing of...
7. ### What is the effect on the Berry phase?

Homework Statement : Consider a Hamiltonian H[s ] that depends on a number of slowly varying parameters collectively called s(t). What is the effect on the Berry phase γn[C] for a given closed curve C, if H[s ] is replaced with f[s ] H[s ], where f[s ] is an arbitrary real numerical function of...
8. ### A Berry phase and parallel transport

Hello. In the following(p.2): https://michaelberryphysics.files.wordpress.com/2013/07/berry187.pdf Berry uses parallel transport on a sphere to showcase the (an)holonomy angle of a vector when it is parallel transported over a closed loop on the sphere. A clearer illustration of this can be...
9. ### A Is the Berry connection compatible with the metric?

Hello, Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)? Also, does it have torsion? It must either have torsion or not be...
10. ### A Is the Berry connection a Levi-Civita connection?

Hello! I have learned Riemannian Geometry, so the only connection I have ever worked with is the Levi-Civita connection(covariant derivative of metric tensor vanishes and the Chrystoffel symbols are symmetric). When performing a parallel transport with the L-C connection, angles and lengths are...
11. ### I Berry phase, Bra-Ket and gradient

Could somebody show me how to derive this equation? How can I get right side from left. Step by step, thanks...
12. ### A Integration along a loop in the base space of U(1) bundles

Let ##P## be a ##U(1)## principal bundle over base space ##M##. In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase ##\gamma = \oint_C A ## where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of...
13. ### Berry phase of 1/2 spin in slowly rotating magnetic field

Homework Statement Homework Equations This is the way to solve when magnetic field B is arbitrary direction one. The Attempt at a Solution I got a eigenvalue of this Hamiltonian and eigenstates. but i have no idea how to set a coordinate to value the gradient
14. H

### A Why electrical polarization can be expressed as Berry phase?

Could we calculate (linear) electric polarizability from Berry phase?
15. ### I Defining exchange statistics of anyons in terms of Berry phase

In 2D, if we define exchange statistics in terms of the phase change of the wavefunction of two identical particles when there are exchanged via adiabatic transport (https://arxiv.org/abs/1610.09260), we would discover that this phase can be arbitrary due to the topology of relative...
16. ### A Question about Berry phase in 1D polyacetylene

Hi. I'm taking a look at some lectures by Charles Kane, and he uses this simple model of polyacetylene (1D chain of atoms with alternating bonds which give alternating hopping amplitudes) [view attached image]. There are two types of polyacetylene topologically inequivalent. They both give the...
17. ### I Berry phase without a magnetic field

It is common to calculate Berry phases for quantum systems in, for example, a magnetic field. In this case we compute the Berry phase ##\gamma## using $$\gamma[C] = i\oint_C \! \langle n,t| \left( \vec{\nabla}_R |n,t\rangle \right)\,\cdot{d\vec{R}} \,$$ where ##R## parametrizes the cyclic...
18. ### A Berry phase and gauge dependence

Hi, I have a question that has been bugging me recently. It's about the berry phase and something that I find contradictory. One can see that it is possible to get rid of 2π x (integer) part of the Berry phase by means of a gauge transformation. This in general applies to phases (gauge...
19. ### Adiabatic expansion of infinite square well

Homework Statement Suppose that an infinite square well has width L , 0<x<L. Nowthe right wall expands slowly to 2L. Calculate the geometric phase and the dynamic phase for the wave function at the end of this adiabatic expansion of the well. Note: the expansion of the well does not occur at...
20. ### Matrix elements of non-normalizable states

Although strictly quantum mechanics is defined in ##L_2## (square integrable function space)， non normalizable states exists in literature. In this case, textbooks adopt an alternative normalization condition. for example, for ##\psi_p(x)=\frac{1}{2\pi\hbar}e^{ipx/\hbar}## ##...

Hello, 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 interesting, I would appreciate if anyone could provide me with some good sources(books, internet...
22. ### Why there is mod 2pi in Berry phase?

Some books say that there is a gauge transform that we can put an extra phase e^{i \phi ( R(t))} to the wave function. Since R(t=0) = R(t=T), difference in \phi = 2 pi n, where n is any integers. As gauge transform would lead to 2 pi n difference, berry phase is determined up to 2pi n. However...
23. ### Help me check a fact about Berry phase

I have vague memories of having read somewhere that if you have a parameter manifold that a) Have a non-trivial Berry phase (meaning the line integral of the berry connection is non zero for some curve) and b) the Berry curvature is non-singular anywhere in that manifold then you can...
24. ### Where does the Berry phase of $\pi$ come from in a topological insulat

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 the same fashion for a 3D TI). I can follow the argument up to defining the Chern parity $\nu$...
25. ### Understanding the Vector Spaces in Berry's Geometrical Phase Paper

I am examining Berry's original 1984 paper of geometrical phases. I am having trouble understanding a portion of it, however (which is attached as a photo). In particular, equation (6) appears to have a bra-ket expression which includes the bra- complex conjugate of a state, <n(R)|, along with a...
26. ### How to Measure Berry Phase in a Magnetic Field?

i have no idea of how to measure a berry phase take a 1/2 spin in an external magnetic field it is well known that this system can exhibit berry phase if the external magnetic field B takes a loop on the sphere. but how to measure the berry phase? how to separate the berry phase from...
27. ### Connection, horizontal lift, and berry phase

i am now reading the PRL paper by simon on berry phase under adiabatic approximation, the wave function evolves as \langle \psi | d/dt | \psi \rangle=0. how to relate this equation to the connection on a fiber bundle? how to understand that the wave function is parallel transported?
28. ### Berry phase in degenerare case

Is there anyone familiar with berry phase? Wilczek and Zee have a classic paper PRL 52_2111 I can not understand their equation (6) I can not see why the first equality should hold \eta_a and \eta_b should be orthogonal to each other, but why should \eta_b be orthogonal to the time...