198. Fractional quantum Hall effect – a few words about theory

Finally, after going through the integer quantum Hall effect, we are a kind of ready to discuss one of the biggest puzzles in condensed matter theory: the fractional quantum Hall effect.

As you remember, it is also characterized by the quantization of the the Hall resistivity

, (1)

but the number in this formula acquires non-integer values like or with arbitrary integer and . This looks bizarre – since the denominator in the formula (1) also contains the charge of electron, it seems that elementary excitations in the quantum Hall effect carry fractional electric charge. This is why the work by Frank Wilczek on anyons made so many people in condensed matter theory excited. First of all, anyons exist only in 2-dimensional structures, and that’s exactly what we are dealing with studying quantum Hall effect. Second, they look like excitations carrying fractional charge (I explained it in details before).

Initially, Laughlin was the first who proposed the theory of fractionally charged quasi-particles (though, he did not associate them with anyons to my knowledge) and eventually he got the Nobel prize for that. The idea is basically the following. Since the explanation based on the model of weakly interacting electron gas fails for the fractional quantum Hall effect, we will say that electrons interact strongly (and this is actually the case) – more accurately, due to the strong Coulomb interaction between them electron gas becomes a non-compressible quantum liquid. Instead of fundamental electrons, we have to consider collective excitations in this liquid. Ground states of the liquid correspond to the values

of reduced concetration, but minima of the energy can be also achieved at

and other values of .

Excited states are separated by the gap from the ground states, and the gap allows to explain plateaus in the Hall resistivity as well as the minima in the diagonal resistivity similar to the way we did it in the previous post. If we increase and , the energy of the ground state grows and the value of the gap decreases, which makes the fractional quantum Hall effect less “visible” for higher peaks in the dependence .

Now, what Robert Laughlin did – he explicitly wrote the wavefunction of the ground state as well as excited states for the FQHE quantum liquid, but did not quite explain where does it come from I remember the attitude of people towards Laughlin in my Alma Mater very well – he was evil genius, who knew everything but did not want to explain to others how exactly he came to his amazing conclusion

Somewhat different understanding of the effect came a bit later with the works by Jain, Halperin, Lee and Read on composite fermions. What was said is that elementary excitations in the FQHE quantum liquid are elementary fermions with even number of magnetic field flux quanta attached to them (as you understand, this is very close to anyons already – kindly see how Frank Wilczek derives anyons in his recent review). One associates the integer quantum Hall effect with composite fermions and finds that effective is fractional. The interesting fact in support of composite fermions is that in the absence of magnetic field (if elementary excitations are composite fermions, and they are even in the absence of magnetic field – simply because the main effect influencing the spectrum of elementary excitations is strong Coulomb interaction). This is exactly what is seen on experiment.