Why is the nearest hopping kept real in Haldane model?

In summary, the Haldane model of graphene involves a complex hopping strength with a phase from the Aharonov-Bohm effect in the presence of a magnetic field. The model also considers the loop integral along a path of nearest bonds, which vanishes due to the net flux of a unit cell being zero. However, there may be confusion about the connection between the vanishing loop integral and the unchanged nearest hopping. This can be explained by considering the hopping strength between three sites a, b, and c, and the phase picked up by the electron as it hops around a closed contour. If there is a nonzero magnetic field within the triangle formed by these sites, the hopping phases will also be nonzero. But if there is no net
  • #1
lichen1983312
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I am leaning the Haldane model :
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.2015
Haldane imaged threading magnetic flux though a graphene sheet, and the net flux of a unit cell is zero.
He argued that since the loop integral ##\exp [ie/\hbar \oint {A \cdot dr} ]## along a path of nearest bonds vanishes, the nearest hopping is not changed.

However, I cannot see the connect between the vanishing loop integral and the unchanged nearest hopping, can anybody help?
 
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  • #2
In the Haldane model of graphene the hopping strength can be complex with a phase coming from the Aharonov-Bohm effect in the presence of a magnetic field. If one moves a particle around a closed contour, then the phase difference between the final and initial states is proportional to the magnetic flux enclosed by the contour ##\phi =\frac{e}{h} \iint \mathbf B \cdot \mathbf S = \frac{e}{h} \oint \mathbf A \cdot \mathbf {\mathcal l}##.
Consider three sites a, b and c. The hopping strength between these three sites is ##t_{ab}##, ##t_{bc}## and ##t_{ca}##. If a particle hops from a to b and then to c, then the hopping strength around the loop is: $$t_{ab}t_{bc}t_{ca}=\left | t_{ab}t_{bc}t_{ca} \right | e^{i(\phi_{ab} + \phi_{bc} + \phi_{ca})}$$
The phase picked up by the electron is: $$\phi_{ab} +\phi_{bc} + \phi_{ca} = \frac{e}{h} \iint \mathbf B \cdot \mathbf S$$
If B is nonzero inside the triangle formed by these three sites, the phase for these hoppings are nonzero. On the other hand; no net field means no phase change.
 

1. Why is the nearest hopping kept real in Haldane model?

The Haldane model is a theoretical model used to describe the behavior of electrons in a material. In this model, the nearest hopping refers to the movement of electrons between neighboring atoms. Keeping this hopping term real is necessary in order to accurately describe the properties of the material, such as its conductivity and magnetic properties.

2. What happens if the nearest hopping is not kept real in Haldane model?

If the nearest hopping term is not kept real, the Haldane model would not accurately describe the behavior of electrons in the material. This could lead to incorrect predictions of the material's properties and behavior.

3. How is the nearest hopping term kept real in Haldane model?

The nearest hopping term is kept real in Haldane model by ensuring that the Hamiltonian (a mathematical operator used to describe the dynamics of a system) is Hermitian. This means that the Hamiltonian is equal to its own conjugate transpose, which ensures that the nearest hopping term remains real.

4. What is the significance of keeping the nearest hopping term real in Haldane model?

Keeping the nearest hopping term real is crucial in order to accurately describe the properties and behavior of materials in the Haldane model. It allows for the model to accurately predict the material's conductivity, magnetic properties, and other important characteristics.

5. Are there any exceptions to keeping the nearest hopping term real in Haldane model?

In some cases, it may be necessary to introduce complex hopping terms in the Haldane model to better describe certain materials. This is often done in more complex models that take into account additional factors such as spin-orbit coupling or lattice distortions.

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