Hofstadter’s butterfly confirms a new tie between physics and mathematics

In the late 1960s, mathematician Robert Langlands tied together number theory and harmonic analysis. This novel connection developed into what is today known as the Langlands program, a mathematical conjecture that connects many different areas of mathematics. Originally, the Langlands program was isolated in mathematics. But eventually researchers were able to relate it to physics.

Thus far, the Langlands program’s connections to physics have been limited to particle or high energy physics. However, in the Journal of Mathematical Physics, author Kazuki Ikeda relates the Langlands program to condensed matter physics, or low energy physics, for the first time. “My work links physics and mathematics in a new way,” Ikeda said.

Specifically, Ikeda found novel connections between the Langlands program and Hofstadter’s butterfly. Hofstadter’s butterfly is a fractal pattern that describes the behavior of electrons in a magnetic field. The pattern is named such because of its uncanny resemblance to a butterfly.

Ikeda looked at Hofstadter’s butterfly on a two-dimensional square lattice system, housing Bloch electrons in a uniform magnetic field perpendicular to the system. He was able to relate the mathematical conjecture of Langlands program to this realistic example of condensed matter physics.

Previous work on the Langlands program was based only on assumptions from high energy physics based on supersymmetry, which has not yet been observed. But Hofstadter’s butterfly can be, and has been, observed in various experimental studies. And now, because of the results of this research, Hofstadter’s butterfly has extended the Langlands program into low energy physics in a way that is consistent with previous results in both high energy physics and the Langlands program.

Source: “Hofstadter’s butterfly and Langlands duality,” by Kazuki Ikeda, Journal of Mathematical Physics (2018). The article can be accessed at https://doi.org/10.1063/1.4998635 .