Technological advances in confining electrons to two dimensions opened the door to observing the quantum Hall effect under high magnetic fields. In low-temperature electrical transport measurements, the Hall resistance forms plateaus at certain quantized values while longitudinal resistance is exponentially suppressed. When the Hall resistance of a state equals an integer or fractional multiple of h/e^2 - two fundamental physics constants - the state is classified as either an integer or a fractional quantum Hall (FQH) state.

Integer quantum Hall states are explained by multiple Landau levels fully occupied by free electrons, producing an energy gap in the bulk of the system. FQH states, however, require many-body interactions to arise, making them considerably more challenging to study both experimentally and theoretically.

A new experimental study published in National Science Review now reveals a striking pattern in which approximately 100 FQH states can be organized, and attempts to establish theoretical connections linking them. Each state is represented as a point in a polar coordinate system, with its angle set by the filling factor and its distance from the origin tied to the denominator of that factor - smaller denominators placed toward the outside, larger ones clustering toward the center. The resulting arrangement resembles butterfly wings.

One central obstacle in this line of research is that many FQH states are extremely fragile, only emerging and remaining stable at sufficiently low temperatures. The leading method for reaching such temperatures is nuclear adiabatic demagnetization. These refrigerators are technically demanding to construct and typically require large quantities of liquid helium, limiting their availability to a small number of laboratories worldwide.

In recent years the adoption of cryogen-free, or dry, technology has enabled nuclear adiabatic demagnetization refrigerators that do not depend on liquid helium for pre-cooling. One such system, developed at Peking University, reaches an ultra-low temperature of 0.09 millikelvin - the current world record for dry refrigerators. Using this apparatus, the team systematically investigated FQH states in ultra-high mobility GaAs quantum wells grown at Princeton University and proposed the butterfly-wing organizational pattern for the observed states.

The observed states are analyzed within the framework of composite fermion (CF) theory, which posits that composite fermions emerge in collections of strongly correlated electrons as bound states of bare electrons and an even number of quantized vortices. In many cases these composite fermions behave as non-interacting particles. The butterfly pattern makes clear that most FQH states cluster along the outer edges of the wings, corresponding to composite fermions forming integer quantum Hall states. States appearing within those edges can only be explained when interactions between composite fermions are taken into account, leading to further fractionalization.

The experimental data were also compared with hierarchy theory, but the researchers concluded that composite fermion theory provides a more intuitive and transparent picture. The same butterfly-wing pattern can be extended beyond GaAs to interpret results in graphene, WSe2, and other two-dimensional materials. Given the markedly different properties of these platforms and the varying conditions under which data are acquired, a comprehensive summary of existing results from state-of-the-art samples is considered highly desirable.

The researchers note that if features consistent with the butterfly pattern appear in forthcoming devices, that consistency would constitute strong evidence for the existence of new FQH states. States with straightforward explanations within CF theory are expected to serve as a structural backbone for such identifications.

As the history of low-temperature physics has demonstrated, major breakthroughs are often unanticipated. The researchers note that a violation of the butterfly pattern in any future two-dimensional system could signal entirely novel physics and would warrant deeper investigation.

Research Report:Cascade of fractional quantum Hall states in two-dimensional system