A colored Hofstadter butterfly

A colored Hofstadter butterfly: This figure represents the phase diagram of Bloch electrons in a uniform magnetic field. The horizontal axis indicates the chemical potential and the vertical axis the flux through the system. Each color corresponds to a distinct topological phase with a particular quantized value of the Hall conductance.Credit: Yosi Avron

Figure 1: A colored Hofstadter butterfly: This figure represents the phase diagram of Bloch electrons in a uniform magnetic field. The horizontal axis indicates the chemical potential and the vertical axis the flux through the system. Each color corresponds to a distinct topological phase with a particular quantized value of the Hall conductance.

The quantum Hall effect (QHE) is the remarkable observation of quantized transport in two dimensional electron gases placed in a transverse magnetic field: the longitudinal resistance vanishes while the Hall resistance is quantized to a rational multiple of h/e2. The theory of the QHE is built largely around the special properties of single-particle free-electron states in a magnetic field—the celebrated Landau levels. This is particularly true of the F(ractional)QHE, where the construction of model wave functions with built-in analyticity forced by a restriction to the lowest Landau level has played an extremely important role in the theoretical development. Now, in three papers published back-to-back in Physical Review Letters [1, 2, 3], three research groups develop lattice models lacking this Landau level structure whose realizations could, in principle, exhibit a FQHE in the absence of an external magnetic field.

While the simplicity of the setting in which the QHE was first discovered was extremely helpful in unraveling its explanation, condensed matter physicists have periodically returned to the challenge of generalizing the reach of the phenomenon itself. There are two specific questions that have focused their attention. First, in the standard electron gases exhibiting the QHE, the surrounding solid has a fairly modest effect on their properties, which can be captured by a change in the (effective) mass of the electrons away from its value in free space. Does the QHE survive when the solid affects electronic motion more seriously and one needs to take the formation of energy bands into account? Second, a uniform magnetic field does two things: it breaks time reversal symmetry, but it also affects electron dynamics at long wavelengths in a specific fashion, as captured in the formation of Landau levels. Are both essential for the QHE?

The answer to both questions is known for the I(nteger)QHE, which is mostly a single-particle phenomenon. In a landmark paper in 1982, Thouless, Kohmoto, Nightingale, and den Nijs [4] analyzed the uniform-field Hall effect in a strong periodic potential that was known to lead to an intricate spectrum, the so-called Hofstadter butterfly (see Fig. 1); they showed that it gives rise to an integer QHE under certain conditions, i.e., whenever the chemical potential lies in a gap. Indeed, the Hall conductance was shown to map to a topological invariant associated with filled bands—the (first) Chern number. Six years later, in another striking development, Haldane [5] answered the second question, showing by an explicit construction of a tight-binding model on a honeycomb lattice that a quantized Hall conductance can arise from a fully filled band even in the absence of a net magnetic field. In his model, time-reversal symmetry is broken by a spatially inhomogeneous magnetic field with zero average, and the Hall conductance again equals the Chern number of the band.

The three recent papers [1, 2, 3] take up this development and address the next logical question: Can the FQHE, canonically a property of interacting electrons in a fractionally filled Landau level, also be separated from the weak lattice and uniform magnetic field limit? More precisely, they ask the following: If it is true for independent electrons that a filled Chern band is equivalent to a filled Landau level, then is it also true for interacting electrons that a fractionally filled Chern band is equivalent to a fractionally filled Landau level?

A Landau level involves a set of exactly degenerate single-particle states and thus, at a fractional filling, the kinetic energy alone does not select a ground state, but instead, it falls to the interactions to force the issue. By contrast, a Chern band typically will have a significant dispersion that will select a unique kinetic-energy-dominated ground state at reasonable interaction strengths, as it does in all metals. Recognizing this, all three papers devote considerable effort to constructing lattice models with nearly flat (degenerate) Chern bands. Neupert et al. [2] construct a flattened version of Haldane’s model on a square lattice. They note that while a fully flattened model requires the inclusion of electron hopping over arbitrarily large distances, the hopping amplitudes decrease exponentially, which allows a relatively flat band to be constructed by keeping a small set of hopping amplitudes. The relevant flatness parameter, which should be large for the effects of interactions to be important, is the ratio of the band gap (which sets a bound on the strength of the interactions one can safely include) to the bandwidth and they show how to get this number up to seven with just second-neighbor interactions. Similarly, Tang et al. [1], and Sun et al. [3] construct models on the kagome and checker-board lattices, which also exhibit large values of the flatness parameter.

With a flat Chern band in hand, Neupert et al. introduce interactions and study the system at a fractional filling of 1/3 through numerical computation on a modestly sized system. They find two of the classic signatures of the 1/3 FQHE state: a fractional quantum Hall conductance that was close to the filling fraction, and a nontrivial ground-state degeneracy with periodic boundary conditions. As a test, they vary the band structure continuously to a topologically trivial band and find that these features go away. In a related piece of unpublished work, another group finds similar results at fillings of 1/3 and 1/5 [6]. Altogether, this work offers strong evidence that fractionally filled Chern bands do indeed exhibit the FQHE.