I've noticed the following:

Observation: Central results in the condensed matter physics rely on Aharonov Bohm-type arguments involving quasiparticles with fractional charge.

However, I can't see how these arguments are rigorous. In order for them to work, we would need a partition function of the form

$$Z[x,A]=\int Dx\,\, e^{\frac{i}{\hbar}S[x,A]},~~~~~~S[x,A]=\int_C(qA+\text{terms without $A$})$$

There are two issues with demonstrating the above: for one, this is describing a quantum-mechanical system, whereas quasiparticles are naturally mentioned in a field-theoretic context. So we'd need to find a way to reduce a field-theoretic partition function to a particle-partition function:

$$Z[\Psi,A]=\int D\Psi\,\, e^{\frac{i}{\hbar}S[\Psi,A]},~~~\to~~~Z[x,A]=\int Dx\,\, e^{\frac{i}{\hbar}S[x,A]}$$

The second issue would lie in finding the field-theoretic Lagrangian for a quasiparticle, but it seems like that is already known (i.e., for the fractionally charged anyons of the FQHE, just write down the Chern-Simons lagrangian). However, the first step is missing, and in a glaring way.

Last Thing:

An even better alternative, if possible, would be to derive everything in the Hamiltonian picture, taking advantage of the adiabatic theorem. The Hamiltonian for the quasiparticles would then take the form $$H=\sum_q E_q \gamma^\dagger_q \gamma_q$$

Does anyone else find it awfully surprising that no one has tried to prove the Aharonov Bohm effect in this context?


I'm not sure what you were really asking about in the first question, so I'll just focus on the second.

The reason that people do not go for such a "second quantized" description is that these $\gamma_q$ operators must be necessarily highly non-local for fractionalized quasiparticles. And there is typically no controlled way to "derive" such a second-quantized Hamiltonian since the system is usually strongly interacting. However, it does not mean that the fractional braiding phase has not been derived using adiabatic theorem. In fact, it was first shown in this way. Take $\nu=1/m$ Laughlin state as an example. First, following Laughlin, we have the (first-quantized) wavefunction for quasiholes, as a function of the complex coordinate of the quasihole. This was proposed by Laughlin. Then one can simply go and calculate the braiding phase of the quasiholes, as an adiabatic Berry phase. Under certain assumption about the plasma screening which has to be checked numerically, one can establish the AB phase as well as the braiding statisitcs. The original derivation was done in http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.53.722.

Now maybe your question is, how do we know that quasihole wavefunction written down by Laughlin is a good approximation to the actual quasiparticle? This is justified in the following way: first, one should ask why Laughlin's wavefunction for the ground state is a good approximation of the actual ground state. People checked that numerically, of course, for a small number of electrons (~ 13 electrons for 1/3, actually not that small). However a better argument is that the Laughlin wavefunction is the exact ground state of a Hamiltonian in the lowest Landau level with certain very short-ranged interaction, instead of Coulomb, and the quasihole wavefunction also represents eigenstates of this ideal Hamiltonian. How do we know then the ideal Hamiltonian is in the same "phase" as the Coulomb one? At this point numerics is involved to show that indeed these two are in the same phase. It's not rigorous but that's the best one can do. So in a sense, Laughlin's ingenious "guess" of the wavefunction is essential since we do not know, and probably never will, how to solve the problem of many-electrons with Coulomb interactions in a Landau level analytically.

  • $\begingroup$ I guess what I'm confused by is how you're using the word "charge". The charge of a quasiparticle is defined by Aharanov Bohm phase - shouldn't the answer be based on the strength of its resulting electric field at a distance r? $\endgroup$ – David Roberts Dec 6 '15 at 2:57
  • $\begingroup$ What's wrong with that definition? Charge determines how particles couple to electromagnetic field, and AB phase measures the charge in a region globally. In fact, I would say this is how charge should be defined quantum mechanically, while electric field strength depends a lot on details (remember the quasiparticle is really a collective excitation in a system with a macroscopic number of electrons). I'm not saying it is impossible to measure the electric field strength, but AB phase is just conceptually clearer. $\endgroup$ – Meng Cheng Dec 6 '15 at 4:47

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