image: Each filling factor ν = p/q is represented by a dot in a polar form with angle θ = 2πp/q. The radial axis represents the denominator q in an inverted manner: q = 1 is at the outermost edge, and larger denominators lie progressively closer to the center (up to q = 55). In general, the distance to the origin is chosen as r(q) = 55 − q. As a consequence, simpler fractions appear near the rim, whereas higher-denominator fractions cluster toward the center. For example, ν = 1/3 is located at angle 2π/3 and radius 52. Red dots are fractional quantum Hall (FQH) states found in their measurements, black dots are FQH states that have been reported in the literature , and blue dots are all possible fractions with odd denominators smaller than 52 but not yet observed in experiments. Black, green, blue, pink, and orange lines are used to connect fractions that correspond to integer quantum Hall (IQH) states of composite fermions (CFs) with 2, 4, 6, 8, and 10 attached fluxes, respectively. Blue shaded areas enclose the fractions that cannot be understood as IQH states of CFs. Green shaded area encloses ν > 2/3 fractions that may be particle-hole conjugates of certain states at ν < 1/3 or FQH states of CFs.
Credit: ©Science China Press
Technological advances of confining electrons in two dimensions paved the way for the observation of quantum Hall effect in high magnetic fields. In low temperature electrical transport measurements, the Hall resistance exhibits plateaus at certain quantized values whereas the longitudinal resistance is exponentially suppressed. If the Hall resistance of a state is an integer or fractional multiple of h/e^2, two important physics constants, it is called an integer or fractional quantum Hall (FQH) state. The integer ones can be understood as multiple Landau levels fully occupied by free electrons such that an energy gap appears in the bulk of a system, while many-body interactions must be incorporated to generate FQH states, thus making their experimental and theoretical studies much more challenging.
A new experimental study, published in National Science Review, unveils an appealing pattern into which about 100 FQH states can be organized and tries to establish theoretical connections between them. One major challenge in experimental investigations is that many FQH states are extremely fragile — they only emerge and remain stable at sufficiently low temperatures. The current state-of-the-art method for reaching the lowest environmental temperatures is nuclear adiabatic demagnetization. However, such fridges are difficult to build due to their technical complexity and usually consume large amounts of liquid helium, so they are available only in a limited number of laboratories. In recent years, with the widespread adoption of cryogen-free (dry) technology, nuclear adiabatic demagnetization refrigerators whose pre-cooling does not rely on liquid helium have begun to appear. One system recently developed at Peking University can reach the ultra-low temperature of 0.09 mK, which is the present world record for dry refrigerators.
Leveraging this powerful apparatus, the FQH states in ultra-high mobility GaAs quantum wells grown at Princeton University have been systematically investigated, and a pattern that organizes the observed states is proposed. In this pattern, each state is represented by one point in the polar coordinate system, with its angle determined by the filling factor and its distance to the origin associated with the denominator (smaller ones outside, larger ones inside). This leads to an interesting shape that mimics butterfly wings. The observed states are analyzed in the framework of composite fermion (CF) theory. It postulates that CFs emerge in a collection of strongly correlated electrons as bound states of bare electrons and an even number of quantized vortices and they behave as non-interacting particles in many cases. One can see clearly in the emerging pattern that most FQH states cluster along edges of the “wing” since they correspond to CFs forming integer quantum Hall states. In contrast, FQH states within the edges can only be explained if the interactions between CFs are taken into account, which leads to further fractionalization. The experimental data are also compared with the hierarchy theory, but it seems that a more intuitive and transparent picture is provided by the CF theory.
Going beyond GaAs, the same pattern can also be employed to understand the results in graphene, WSe2, and other 2D materials. In view of the drastically different properties of these platforms and the differences in data acquisition, a comprehensive summary of existing data from state-of-the-art samples is highly desirable. Obviously, if some features consistent with this pattern are spotted in upcoming devices, it would be quite convincing to claim the existence of new FQH states. In this regard, the states that have simple explanations within the CF theory could serve as a backbone.
As the history of low temperature physics has shown, many breakthroughs were completely unanticipated. It is quite possible that two-dimensional systems also host other exciting phenomena not within our imagination. Therefore, a violation of the pattern might imply novel physics and call for more in-depth studies.