The Kubo Formula of Hall Conductivity

In summary, the conversation discusses a formula used in a two-dimensional system to calculate the Hall conductivity, which involves velocity operators in x and y directions. The formula can be derived from the Kubo identity, but there is confusion about the process. The formula involves a time correlation function of the observable and the applied field, with the current measured in the direction orthogonal to the field. It is suggested to perform time evolution in the basis of eigenstates to obtain the final result.
  • #1
cometzir
18
0
Several papers (eg. Di Xiao, et. al, Berry phase effects on electronic properties, RevModPhys, 82,2010)mentioned a formula to calculate the Hall conductivity(See the picture).This formula is used in an two dimensional system, v1 and v2 are velocity operators in x and y direction, Phi0 and PhiN are ground and excited state vector.
The papers claim that this formula can be derived from the Kubo identity, but I am not sure how this can be done, since the form of Kubo formla is quite different from this expression.
Could anyone help me with the derivation?
 

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  • #2
What are v1 and v2?
 
  • #3
DrDu said:
What are v1 and v2?

Sorry for unclearly description.
This formula is used in a two dimensional system. v1 and v2 are velocity operators in x and y direction
 
  • #4
Which part is unfamiliar to you? The general linear response formula involves the time correlation function of the observable and the applied field. In a Hall measurement, the current is measured in the direction orthogonal to the applied field, which is why vx and vy show up. Writing the field in terms of the current density then gives the time integral of a current-current (or velocity-velocity) correlation function and performing the time evolution in the basis of eigenstates (Lehmann representation) should give the final result.
 

1. What is the Kubo Formula of Hall Conductivity?

The Kubo Formula of Hall Conductivity is a mathematical equation that describes the relationship between the electric field, magnetic field, and current density in a material. It is used to calculate the Hall conductivity, which is a measure of how well a material conducts electricity in the presence of a magnetic field.

2. How is the Kubo Formula derived?

The Kubo Formula was first derived by Japanese physicist Ryogo Kubo in 1957. It is based on the quantum mechanical theory of electron transport, which takes into account the effects of both the electric and magnetic fields on the movement of electrons in a material.

3. What are the key assumptions of the Kubo Formula?

The Kubo Formula makes several key assumptions, including that the material is isotropic (has the same properties in all directions), that the electrons are non-interacting, and that there are no impurities or defects in the material. These assumptions allow for a simplified calculation of the Hall conductivity.

4. How is the Kubo Formula used in practical applications?

The Kubo Formula is used in many areas of condensed matter physics and materials science, including the study of semiconductors, metals, and superconductors. It is also used in the design and development of electronic devices, such as transistors and computer chips.

5. Are there any limitations to the Kubo Formula?

While the Kubo Formula is a useful tool for understanding the behavior of electrons in materials, it does have some limitations. For example, it does not take into account the effects of electron-electron interactions, which can play a significant role in certain materials. Additionally, the formula may not be applicable in extreme conditions, such as at very low temperatures or in the presence of strong magnetic fields.

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