3. The Fractional Quantum Hall E↵ect We’ve come to a pretty good understanding of the integer quantum Hall e↵ect and the reasons behind it’s robustness. Indeed, some of the topological arguments in the previous chapter are so compelling that you might think the Hall resistivity of an insulator has to be an integer. But each of these arguments has a subtle loophole and ultimately they hold only for non-interacting electrons. As we will now see, much more interesting things can happen when we include interactions. As with the integer quantum Hall e↵ect, these interesting things were first discovered by experimenters rather than theorists. Indeed, it came as a great surprise to the community when, in 1982, plateaux in the Hall resistivity were seen at non-integer 1 2 filling fractions. These plateaux were first seen at filling fraction ⌫ = 3 and 3 ,and 1 2 3 4 5 3 4 5 7 5 12 13 later at ⌫ = 5 , 5 , 7 , 9 , 9 , 5 ,... in the lowest Landau level and ⌫ = 3 , 3 , 5 , 2 , 5 , 5 ,... in higher Landau levels, as well as many others. There are now around 80 quantum Hall plateaux that have been observed. A number of these are shown below16: There’s two things that we can say immediately. First, the interactions between elec- trons must be playing some role. And second, the answer to why these plateaux form is likely to be very hard. Let’s see why. Suppose, for the sake of argument, that we have ⌫<1 so that the lowest Landau level is partially filled. Each Landau level can house = AB/Φ (spin polarised) N 0 16This data is from R. Willett, J. P. Eisenstein, H. L. Stormer, D. C. Tsui, A. C. Gossard and H. English “Observation of an Even-Denominator Quantum Number in the Fractional Quantum Hall E↵ect”, Phys. Rev. Lett. 59, 15 (1987). –74– E E Figure 25: Density of states in the lowest Figure 26: ...and with interactions (with Landau level without interactions only a single gap at ⌫ =1/3shown. electrons, where B is the magnetic field and A is the area of the sample. This is a macroscopic number of electrons. The number of ways to fill ⌫ of these states is ⌫ N (1 ⌫) 1 N 1 − N ⌫N which, using Stirling’s formula, is approximately ⌫ 1 ⌫ .Thisisa ridiculouslyN large number: an exponential of an exponential. The− ground state of any partially filled Landau level is wildly, macroscopically degenerate. Now consider the e↵ect of the Coulomb interaction between electrons, e2 V = (3.1) Coulomb 4⇡✏ r r 0| i − j| On general grounds, we would expect that such an interaction would lift the degen- eracy of ground states. But how to pick the right one? The approach we’re taught as undergraduates is to use perturbation theory. But, in this case, we’re stuck with extraordinarily degenerate perturbation theory where we need to diagonalise a macro- scopically large matrix. That’s very very hard. Even numerically, no one can do this for more than a dozen or so particles. We can, however, use the experiments to intuit what must be going on. As we mentioned above, we expect the electron interactions to lift the degeneracy of the Landau level, resulting in a spectrum of states of width E .Thedatawould ⇠ Coulomb be nicely explained if this spectrum had gaps at the filling fractions ⌫ where Hall states are seen. In the picture above, we’ve depicted just a single gap at ⌫ =1/3. Presumably though there are many gaps at di↵erent fractions: the more prominent the plateaux, the larger the gap. Then we can just re-run the story we saw before: we include some disorder, which introduces localised states within the gap, which then gives rise both to the plateaux –75– in ⇢xy and the observed ⇢xx =0.Thebiggerthegap,themoreprominenttheobserved plateaux. This whole story requires the hierarchy of energy scales, ~!B ECoulomb Vdisorder We will assume in what follows that this is the case. The question that we will focus on instead is: what is the physics of these fractional quantum Hall states? In what follows, we will take advantage of the difficulty of a direct theoretical attack on the problem to give us license to be more creative. As we’ll see, the level of rigour in the thinking about the fractional quantum Hall e↵ect is somewhat lower than that of the integer e↵ect. Instead, we will paint a compelling picture, using a number of di↵erent approaches, to describe what’s going on. 3.1 Laughlin States The first approach to the fractional quantum Hall e↵ect was due to Laughlin17,who described the physics at filling fractions 1 ⌫ = m with m an odd integer. As we’ve explained above, it’s too difficult to diagonalise the Hamiltonian exactly. Instead Laughlin did something very bold: he simply wrote down the answer. This was motivated by a combination of physical insight and guesswork. As we will see, his guess isn’t exactly right but, it’s very close to being right. More importantly, it captures all the relevant physics. 3.1.1 The Laughlin Wavefunction Laughlin’s wavefunction didn’t come out of nowhere. To motivate it, let’s start by considering an illuminating toy model. Two Particles Consider two particles interacting in the lowest Landau level. We take an arbitrary central potential between them, V = V ( r r ) | 1 − 2| Recall that in our first courses on classical mechanics we solve problems like this by using the conservation of angular momentum. In quantum physics, this means that we 17The original paper is “Anomalous quantum Hall e↵ect: An Incompressible quantum fluid with fractionally charged excitations”,Phys. Rev. Lett. 50 (1983) 1395. –76– work with eigenstates of angular momentum. As we saw in Section 1.4,ifwewantto talk about angular momentum in Landau levels, we should work in symmetric gauge. The single particle wavefunctions in the lowest Landau level take the form (1.30) m z 2/4l2 z e−| | B m ⇠ with z = x iy.Thesestatesarelocalisedonaringofradiusr = p2ml .The − B exponent m of these wavefunctions labels the angular momentum. This can be seen by acting with the angular momentum operator (1.31), @ @ ¯ J = i~ x y = ~(z@ z¯@) J m = ~m m @y − @x − ) ✓ ◆ Rather remarkably, this information is enough to solve our two-particle problem for any potential V ! As long as we neglect mixing between Landau levels (which is valid if ~!B V ) then the two-particle eigenstates for any potential must take the form 2 2 2 M m ( z1 + z2 )/4l (z + z ) (z z ) e− | | | | B ⇠ 1 2 1 − 2 where M,m are non-negative integers, with M determining the angular momentum of the centre of mass, and m the relative angular momentum. Note that here, and below, we’ve made no attempt to normalise the wavefunctions. It’s surprising that we can just write down eigenfunctions for a complicated potential V (r)withouthavingtosolvetheSchr¨odingerequation.It’sevenmoresurprisingthat all potentials V (r)havethesameenergyeigenstates.Itisourinsistencethatweliein the lowest Landau level that allows us to do this. Many-Particles Unfortunately, it’s not possible to generalise arguments similar to those above to uniquely determine the eigenstates for N>2 particles. Nonetheless, on general grounds, any lowest Landau level wavefunction must take the form, N 2 2 zi /4l (z1,...,zn)=f(z1,...,zN )e− i=1 | | B (3.2) P for some analytic function f(z). Moreover, this function must be anti-symmetric under exchange of any two particle z z ,reflectingthefactthattheunderlyingelectrons i $ j are fermions. –77– Laughlin’s proposal for the ground state wavefunction at filling fraction ⌫ =1/m is: m n z 2/4l2 (z )= (z z ) e− i=1 | i| B (3.3) i i − j i<j P Y Clearly this is anti-symmetric when m is an odd integer. For m an even integer, it can be thought of as a quantum Hall state for bosons. The pre-factor vanishes with azerooforderm whenever two electrons come together. Meanwhile, the exponential factor decreases quickly whenever the electrons get too far away from the origin. The wavefunction is peaked on configurations that balance these two e↵ects. Let’s first show that the wavefunction has the desired filling fraction. To do this, focus on what the wavefunction is telling us about a single particle, say z1.Theterms that depend on z1 in the pre-factor of the Laughlin wavefunction are N (z z )m (z z )m i − j ⇠ 1 − i i<j i=2 Y Y which tells us that there are m(N 1) powers of z . This, in turn, tells us that the − 1 maximum angular momentum of the first particle is m(N 1) and so its maximum − radius is R p2mNl .Correspondingly,theareaofthedropletisA 2⇡mNl2 ⇡ B ⇡ B (where we’ve replaced N 1withN). Recall that the number of states in the full − Landau level is AB/Φ = A/2⇡l2 mN.Thisargumentgivesusthefillingfraction 0 B ⇡ 1 ⌫ = (3.4) m as promised. It can be shown numerically that, at least for small numbers of particles, this wave- function has greater than 99% overlap with the true ground state arising arising from both the Coulomb repulsion (3.1)aswellasanumberofotherrepulsivepotentialsV . Heuristically this occurs because the wavefunction has a zero of order m whenever two electrons coincide. Of course, a single zero is guaranteed by Pauli exclusion, but the Laughlin wavefunction does more. It’s as if each electron carves out a space around it which helps it minimise the energy for repulsive potentials. The high numerical overlap with the true ground state is often put forward as strong evidence for the veracity of the Laughlin wavefunction.