# Berry phase without a magnetic field

• I
• spaghetti3451
In summary, the Berry phase is a phase gained by a wavefunction moving around a closed path and is often used as a diagnostic of nontrivial topology in the absence of any physical field. It can be calculated using the loop integral of the Berry curvature, which is similar to a gauge potential in momentum space. In a Z2 topological insulator, the Berry phase corresponding to the topological invariant is the integral of half the Brillouin zone.
spaghetti3451
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 adiabatic process, in this case, the magnetic field.

I was wondering what the Berry phase is for a system that has no external fields. Say, you have a particle in a box and the box rotates in three-dimensional box about some point. How do you compute the Berry phase for this system?

Taken at face value, your equation says that the Berry phase is zero if the magnetic field is zero. Do you have some reason to think that should not be the case?

The Berry phase is the phase gained by a wavefunction moving around a closed path. In a solid it would be from the variance of the wavefunction within the unit cell. You can formulate it as the loop integral of the Berry curvature, which is like a gauge potential in momentum space. This is just a general definition.

The Berry phase is often used as a diagnostic of nontrivial topology in the wavefunction without any physical field. For example, in a Z2 topological insulator, the Berry phase corresponding to the topological invariant is the integral of half the Brillouin zone. The spin orbit interaction acts as a magnetic field for spin up and spin down electrons but there is no external magnetic field.

## 1. What is the Berry phase without a magnetic field?

The Berry phase without a magnetic field, also known as the geometric phase, is a quantum mechanical phenomenon that describes the phase difference acquired by a quantum system as it evolves along a closed path in parameter space.

## 2. How is the Berry phase without a magnetic field different from the Berry phase with a magnetic field?

The main difference is that the Berry phase without a magnetic field is solely determined by the geometry of the parameter space, while the Berry phase with a magnetic field is also affected by the magnetic field strength and direction.

## 3. What are some real-life applications of the Berry phase without a magnetic field?

The Berry phase without a magnetic field has been observed in various physical systems, such as cold atoms, superconducting circuits, and quantum dots. It has also been utilized in topological insulators and quantum computation.

## 4. Can the Berry phase without a magnetic field be experimentally measured?

Yes, the Berry phase without a magnetic field can be measured through interferometric techniques, such as the Aharonov-Bohm effect, which can detect the phase difference between two paths of a quantum system.

## 5. How does the Berry phase without a magnetic field relate to other quantum mechanical phenomena?

The Berry phase without a magnetic field is closely related to other quantum mechanical phenomena, such as the Aharonov-Bohm effect, the Aharonov-Casher effect, and the quantum Hall effect. It also has connections to topological properties of materials and quantum entanglement.

• Quantum Physics
Replies
22
Views
4K
Replies
1
Views
1K
• Quantum Physics
Replies
2
Views
812
• Quantum Physics
Replies
1
Views
2K
• Quantum Physics
Replies
5
Views
920
• Quantum Physics
Replies
1
Views
932
• Atomic and Condensed Matter
Replies
6
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
344
• Quantum Physics
Replies
2
Views
571
• Introductory Physics Homework Help
Replies
10
Views
347