I have just started reading about the Fractional Quantum Hall Effect (FQHE) and had these doubts:
- In the review article [1, 2] by Prof. Wen, he writes that electron always dances anti-clockwise so the Laughlin wavefunction depends only on $z = x +iy$. How are these two facts related? Also, how do we know they move anti-clockwise in the first place? Is it just from the classical picture? Also, we don't write anything about the time-evolution of these states is it because we assume electrons are at rest at zero temperature? If yes, then how are they dancing anti-clockwise?
- The Laughlin wavefunction shows topological order (I don't know how maybe that's why this question), is the wavefunction unique? I feel (not sure) we can create more combinations of states from 1st Landau level solutions that show the 1/m filling fraction properties. Can we? If yes, are those states related to each other by topological deformations of something, and how exactly? Also, is the manifold of these states a torus and is it because of the periodic boundary conditions in real space?
Best Answer
I will try to answer some points based on my understanding.
The statement that the electrons always dance anti-clockwise could be understood by noticing that expectation value of the relative angular-momentum operator of two particles in the Laughlin state is always positive and in fact, it's bounded below by the specific filling that the state describes. To see this for example, check this question. If the Laughlin wavefunction had some dependence on $\bar{z}$ (apart from the exponential factor), this is no longer guaranteed based on the form of the angular momentum operator in complex coordinates, $J = z\partial - \bar{z}\bar{\partial} $.
In principle, one could create whatever states from lowest Landau level wavefunctions. If a certain state happens to be in the same universality class as the Laughlin state, meaning that they both show the same topological order (same fractional charge, quasi-particle statistics, ... etc), it means that it's possible in principle to adiabatically connect the two states without closing the many-body gap by changing some parameters in the Hamiltonian. The Laughlin wavefunction itself is not the true ground state but it believed to be very accurate description. There are many ways to see this. One numerically motivated way to understand this is based on calculations of the overlap between the Laughlin wavefunction and the true ground state. For example, this is explained in chapter 4 of the book. This is very different from finding the degeneracy of the vacuum states that depends on the topology of the manifold as explained by Arian in the comments.