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In the theory of Fractional Quantum Hall states, one often sees quasi-holes and quasi-electrons (or quasi-particles) being called "excitations" from the ground state initially given by Laughlin (Jastrow-Laughlin style wavefunctions). Usually when one speaks of excitations, it is to a higher energy level. However from the wavefunctions of the quasi-holes and quasi-particles, it seems to me that we are still in the Lowest Landau Level. My question is: in what sense are these "excitations"?

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  • $\begingroup$ They are excitations from the vacuum. The vacuum is the state where there are no quasiparticles. $\endgroup$ – Crimson Jan 31 '18 at 14:57
  • $\begingroup$ Could you be a bit more elaborate? i.e. How are the opposite kinds of excitations which can annihilate each other and be produced in pairs from the vacuum? $\endgroup$ – Sreekar Voleti Feb 1 '18 at 6:06
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I will first repeat some semiconductor physics and then move to the fractional quantum hall effect.

In an undoped semiconductor there are many real electrons and real protons that interact. For $T=0$ they condense into a stable neutral state. Let´s call this state the vacuum. We can create create quasi-electron and quasi-holes as fermionic excitations from this vacuum state. These quasi-electrons and quasi-holes are the non-interacting electrons and holes that provide charge transport in a semiconductor.

For the fractional quantum hall effect, there is not just one vacuum state but there are many. Each Laughlin-wavefunction-like state could be considered as the vacuum. So you can switch from one vacuum to another by changing the magnetic field. And in each of these vacua you can create excitations in the form of quasi-particles and quasi-holes. The properties of these quasiparticles will be different for each vacuum. These quasiparticles are typically anyons (somewhere between a fermion and a boson) and have fractional charge.

There are many good introductory texts about the fractional quantum hall effect. For instance this one by Girvin.

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The full microscopic Hamiltonian of the Hall effect

$$ H = \sum_j \frac{1}{2m} \left [\frac{\hbar}{i} \nabla_j + \frac{e}{c} A(\mathbf{r}_j) \right ]^2 + \sum_{i<j} \frac{e^2}{|\mathbf{r}_i - \mathbf{r}_j|}$$

In the quantum Hall effect, the dynamics is restricted to the lowest Landau level due to the application of very high magnetic fields.

Restricted to the Lowest Landau level the first part of the Hamiltonian is a multiple of the zero point energy, thus an unimportant constant. Thus the Hamiltonian of the restricted dynamics becomes:

$$H_{LL} = P_{LL} \sum_{i<j} \frac{e^2}{|\mathbf{r}_i - \mathbf{r}_j|} P_{LL}$$

Where $P_{LL}$ is the lowest Landau projector. It is of course very hard to write a closed analytic expression of the projected Hamiltonian.

The quasiparticle states are very good approximations to the excited eigenstates of this interaction Hamiltonian.

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