Why FQHE needs a lower energy state with gaped specrum

[knightq]

Hi, everyone. I have read some materials about fractional quantum hall effect,especially about Laughlin's wavefunction. However a lot of them claim Laughlin's wave function have lower energy and a gap in spectrum but seldom explain why that could explain the plateau. Also from my observation, it's not like phase transition, which was the first thing across my mind when I see "lower energy", since the energy at v=1/3 is lower than energy at different filling factors not, the energy of a "normal state" at the same v=1/3.
If anyone is a expert in this field , could you explain to me why a lower energy state at v=1/3 with gaped spectrum could explain the plateau? Thanks

 

[eljose79]

Hello, thank you for your question. The fractional quantum Hall effect (FQHE) is a phenomenon observed in two-dimensional electron systems at low temperatures and high magnetic fields. In this effect, the electrons in the system form a collective state known as a "quantum fluid" which exhibits quantized Hall conductance at specific fractions of the total number of electrons.

The Laughlin wavefunction, proposed by Robert Laughlin in 1983, is a theoretical description of this collective state. It describes the electrons as forming a complex, correlated pattern in the system, which is responsible for the quantized Hall conductance. This wavefunction has a lower energy compared to other states at the same filling factor, which is why it is considered the ground state of the system.

Now, to understand why this lower energy state with a gapped spectrum can explain the plateau in the FQHE, we need to look at the energy levels of the system. In a normal state, without the presence of a strong magnetic field, the energy levels of electrons in a two-dimensional system are continuous. However, in the presence of a strong magnetic field, these energy levels form discrete Landau levels, with a gap between them.

When the system is at a specific filling factor, such as 1/3, the electrons in the system can only occupy a certain number of Landau levels. The Laughlin wavefunction describes the electrons in a specific arrangement that minimizes the energy of the system, resulting in a lower energy state compared to other states at the same filling factor. This lower energy state is responsible for the plateau observed in the FQHE.

In other words, the lower energy state of the Laughlin wavefunction at a specific filling factor creates a gap in the energy spectrum of the system. This gap prevents the electrons from scattering and disrupting the collective state, leading to the quantized Hall conductance observed in the FQHE.

I hope this explanation helps to clarify why a lower energy state with a gapped spectrum can explain the plateau in the FQHE. Please let me know if you have any further questions.