The Topological Origin of the Quantum Hall Effect — Epoche C2
What the plateaux are, and why Landau levels do not by themselves explain them In a silicon inversion layer held at 1.5 K in a field of about 18 T, Klaus von Klitzing, Gerhard Dorda and Michael Pepper found in 1980 that the Hall resistance of a two-dimensional electron gas does not vary smoothly with magnetic field but sits on flat plateaux at the values $h/(ne^2)$ for integer $n$, and that the plateau value is unchanged by the sample's geometry, its mobility, or the amount of disorder in it. The reproducibility was at the level of parts per million in that first experiment; modern comparisons between devices made of entirely different materials agree at the level of parts in $10^9$. Since the revision of the SI in May 2019 fixed $h$ and $e$ exactly, the von Klitzing constant $R_K = h/e^2 = 25\,812.807\,45\ \Omega$ is itself exact by definition, and the effect is used to realise the ohm rather than to measure it. The standard account of why the plateaux exist begins with Landau quantisation. A two-dimensional electron gas in a perpendicular field $B$ has its kinetic energy collapsed into levels with a degeneracy per unit area of $eB/h$ — at 18 T that is $4.4 \times 10^{11}$ cm$^{-2}$, comparable to the carrier density of a typical inversion layer — and a filling factor $\nu = n_e h/(eB)$ counting how many of these levels are occupied. When the Fermi level lies in a gap between levels, there are no states available for small-angle scattering, longitudinal resistance falls to zero, and transport is dissipationless. That much is correct and is not in dispute. What it does not explain is the thing that makes the effect remarkable. In a clean, disorder-free system the Fermi level lies in a gap only at isolated values of the field: as $B$ is swept, $\nu$ varies continuously and so does the Hall conductance, $\sigma_{xy} = \nu e^2/h$. There are no plateaux at all. Plateaux require disorder, which broadens each Landau level into a band of states of which only those near the centre are extended, the rest being localised by the random potential. Changing the field then moves the Fermi level through localised states that carry no current, and $\sigma_{xy}$ does not change — hence a plateau. Disorder is therefore not an imperfection that the theory must tolerate; it is a necessary ingredient. But this rescue creates a sharper problem than the one it solves. If only a small fraction of the states in a Landau level carry current, why should the current they carry come out to exactly $ne^2/h$, to nine decimal places, in samples whose disorder configurations have nothing in common? Laughlin's gauge argument: exactness without a band structure Robert Laughlin answered that question in 1981 with an argument that uses no band structure and no assumption about the disorder, and it is worth following because the integer it produces is the same integer that reappears later as a topological invariant. Roll the two-dimensional sample into a loop, so that it becomes an annulus with two edges, and thread a magnetic flux $\Phi$ through the hole. The electrons never enter the region containing the flux, so the only effect of $\Phi$ is on the phases of their wavefunctions. Now increase $\Phi$ adiabatically by one flux quantum $\Phi_0 = h/e$. A flux quantum can be removed by a gauge transformation, so the final Hamiltonian is unitarily equivalent to the initial one and the spectrum maps onto itself. A localised state, being confined to a region that does not encircle the flux, is returned to itself unchanged. An extended state, which does encircle the flux, need not be: the spectrum can flow, and states can be permuted so that the net effect of the cycle is to transport some number of electrons from the inner edge to the outer. That number must be an integer, because charge comes in units of $e$ and the system has returned to a configuration indistinguishable from the one it started in. Call it $n$. The threading induces an electromotive force $\mathcal{E} = d\Phi/dt$ around the loop, which is the Hall voltage $V$ in this geometry, and the transported charge per cycle is $ne$. The current is therefore $$I = \frac{dQ}{dt} = \frac{ne}{\Phi_0}\,\frac{d\Phi}{dt} = \frac{ne}{h/e}\,V = n\,\frac{e^2}{h}\,V,$$ so that $\sigma_{xy} = I/V = n e^2/h$ exactly. The argument makes the role of disorder transparent rather than mysterious: localised states are exactly the states that cannot contribute, and their presence is what allows the Fermi level to move without changing $n$. Bertrand Halperin (1982) supplied the complementary picture in the same year, showing that the current in the plateau regime is carried by states running along the edges of the sample, and that a non-zero Hall conductance requires extended states to exist somewhere within each Landau level — so the localisation cannot be complete. The Chern number, and a correction to the acronym Laughlin's argument establishes that $n$ is an integer but says nothing about which integer, and it gives no way of computing it from a Hamiltonian. That was supplied in 1982 by David Thouless, Mahito Kohmoto, Peter Nightingale and Marcel den Nijs — the four names behind the acronym TKNN. The earlier version of this essay gave the authors as Thouless, Kohmoto, Halperin and den Nijs; that substitution is a common one and it is wrong. Halperin's 1982 paper on edge states, cited above, is a separate and independently important contribution. Their setting is a clean, periodic two-dimensional crystal, so that Bloch's theorem applies and the states are labelled by a crystal momentum $\mathbf{k}$ in a Brillouin zone. For each band $n$ define the Berry connection from the periodic parts $u_{n\mathbf{k}}$ of the Bloch functions, $$A^{(n)}_j(\mathbf{k}) = i\,\langle u_{n\mathbf{k}} | \partial_{k_j} u_{n\mathbf{k}}\rangle,$$ and its curl, the Berry curvature $\Omega^{(n)}_{xy} = \partial_{k_x}A^{(n)}_y - \partial_{k_y}A^{(n)}_x$. Expanding that curl, the terms involving second derivatives can