Nonlinear electron waves

The audience of my blog is mainly nonphysicists. The only physicists who are reading this blog (that I am aware of) are my father, and Doug the propreitor of this blog. (Added: and Ilya. Added: and Austen. Wow, I'm actually getting a couple of real physicists reading me!). At the risk of scaring everyone else off, I am going to actually write about physics here. Maybe my blog will start to attract other physicists too. Perhaps this attempt to discuss physics will be an abject failure, but why not try? (Added: Yes, I am also showing off that I can typeset pretty equations in my blog. Those interested in doing the same can find out how here).

Yesteday I heard an extremely interesting talk by Paul Weigmann about work with Sasha Abanov. (Full disclosure: I've spoken to them both about this work several times in the last year). The general idea is to study the (nonlinear) hydrodynamics of electronic motion along the edge of drops of electron fluids (in particular quantum Hall edges). They find, after some long and detailed arguments that I don't completely follow, that the motion of the electrons is very similar to the motion of certain waves in deep water (and in particular to the so called Benjamin-Ono equation). One of the results of this work is that when a large wave is introduced it will eventually break up into individual pulses of quantized charge. If this is true, it overturns about 20 years of dogma about "edge" physics.

The idea is quite pretty, but I still am not sure if I believe it. For the experts in the audience, here is the argument that gives me concern.

Let us start by considering the following special case

(1) Take the special inter-electron interaction V_1 for which the Laughlin wavefunction is exact

(2) Take the Harmonically confinement U=\alpha r^2

The reason I choose (1) and (2) is because for this combination, the Laughlin wavefunction remains the exact ground state.

Assuming (1) and (2) it turns out that the edge state spectrum is trivial. In the conventional language of bosonic edge modes, we have

H = \sum_{m \geq 1} \epsilon_m b^\dagger_m b_m + \mbox{higher order terms}

where in the usual Luttinger liquid story, one neglects the higher order terms. Keeping assumptions (1) and (2), it is an exact result that higher order terms all vanish and

\epsilon_m = \tilde \alpha \, m

where \tilde \alpha is the coefficient of the harmonic confinement \alpha times some constants. This structure differs markedly from that of Weigmann and Abanov. Lets call this "Issue 1"

Further let us relax condition (1) and consider more general inter-particle interactions. The following remains an exact statement only assuming the quadratic confinement. Given an eigenstate of the system \Psi_a with energy E_a then you can generate another descendant eigenstate

|\Psi_{a,j} \rangle = (b^\dagger_1)^j |\Psi_a \rangle

where

E_{a,j} = E_a + j \tilde \alpha

This is simply the statement thatb^\dagger_1 excites the center of mass degree of freedom of the disk.

As far as I can tell, the Weigmann Abanov theory does not have this structure built into it. Lets call this "Issue 2". However, I do not think this is necessarily cause to throw one's hands up in the air in disgust. It may, in fact, be easy to "mod-out" the center of mass degrees of freedom and then more or less avoid Issue 2. (Mind you, this step has not been done yet, but I can imagine it being done).

So suppose we remove issue 2, we are left with issue 1 to cause us to lose sleep. But in fact, this may not be so bad either. The special Hamiltonian we have chosen may be a particularly singular point. Even perturbing slightly away from this point will introduce nonlinearities in the spectrum. It is possible (again, not yet proven in my mind) that these nonlinearities, though small, will result in precisly the nonlinear wave equation that they have predicted --- with only a length scale that is nonuniversal. This length scale may become infinite precisely in the above limit, thus not causing any contradiction.

So to summarize, the story in my mind is quite interesting enough to be studied. Right now I see problems with it but if one is optimistic it is possible that these problems can be overcome. I hope to work with Weigmann and Abanov over the next few months to see if it is possible.