“Pour connaître la rose, quelqu’un emploie la géométrie et un autre emploie le papillon”, wrote Paul Claudel as he returned from a trip to Japan in 1927. We, mathematicians, must use both the rigor of geometry and the poetry of butterflies to explain not only the rose but also the physical world.

To understand the meaning of this introspection, we have to step back and start with a physical problem. Transport properties of electrons in crystalline solids (like metals and semiconductors) are extremely important in solid-state physics, as well as for future technological applications in the electronics industry. The theoretical understanding of these transport properties is a difficult task. In nanometric devices, the electron’s motion cannot be explained by the laws of Classical Mechanics, and one is forced to use the general laws of Quantum Mechanics, the theory governing the dynamics of elementary particles in molecules and atoms at the microphysical scale. The motion of the electrons is described by the celebrated Schrödinger equation, discovered by the Viennese physicist in the middle of the 1920s.

Among the most amazing phenomena that we study is the quantum Hall effect, which was discovered in 1980 and helped Klaus von Klitzing receive the Nobel Prize in 1985. Essentially, von Klitzing discovered that when a very thin plate of a crystalline solid (like gold or silicon) is immersed in a perpendicular magnetic field at a very low temperature, every measurement of a specific transport property of the electrons (the transverse conductance) provides outcomes that are integer multiples of a fundamental constant (known today as the von Klitzing constant, denoted here by the abbreviation “K”). In other words, in each measurement one obtains an outcome that is an integer number times the constant K (3K, 7K or 23K, for example), while non-integer multiples (like 3.2K or 7.52K) never occur when measurements are taken. The outcome is always an integer number times the von Klitzing constant K. This fact is independent from every experimental detail. The shape and the material of the plate, as well as the intensity of the magnetic field, do not matter.

The theoretical understanding of this phenomenon has been a challenge for the theoreticians of the last decades. The solution arrived, surprisingly, from a domain of mathematics supposedly far removed from the problem: differential geometry. Starting with a seminal paper by Thouless, Kohmoto, Nightingale and de Nijis (1982), the theoreticians understood that the integer numbers measured in the quantum Hall effect correspond to what mathematicians call topological invariants.

To provide clarification of this principle, we will use an example. Imagine a convex polyhedron in the three-dimensional space (a cube, a pyramid, a tetrahedron, etc.) and count the number of faces (F), the number of edges (E) and the number of vertices (V) of the geometric figure you have imagined. You will be surprised to notice that no matter what shape you have chosen, the number obtained by computing F – E + V is always equal to 2. The cube, for example, has 6 faces, 12 edges and 8 vertices, and, as expected, 6 – 12 + 8 = 2. This easy-to-compute integer number is the simplest example of a topological invariant. The great discovery of the last decades is that a similar topological invariant, the Chern number, rules the dynamics of the electrons in the quantum Hall effect. Indeed, just as the integer number F – E + V is the topological invariant of a simple geometric entity, the integer number that multiplies the von Klitzing constant (the 3 in 3K, the 7 in 7K or the 23 in 23K, to refer to the earlier example) is the topological invariant of a sophisticated geometric entity (called a vector bundle). This topological invariant deserves a special name: the Chern number. Just as the number F – E + V doesn’t depend on the shape you have imagined, the Chern number does not depend on the shape of the plate or other details of the experiment.

My personal interest for this topic originated as love at first sight; I saw a picture, known today as the coloured Hofstadter butterfly (see Figure 1). Every colour of the image corresponds to an integer number: warm colours correspond to positive numbers, cold colours to negative numbers, and white denotes zero. In the two images of Figure 1, the Chern numbers mentioned earlier are represented by different colours, which visually show how the Chern numbers vary according to the intensity of the magnetic field (horizontal axis) and the energy of the system (vertical axis).

Figure 1. The coloured Hofstadter butterfly.

Isn’t it amazing? The variation of the Chern numbers paints a mysterious and charming picture that resembles the wings of a butterfly; we never would have expected to encounter something like this in the microphysical world. Both the Chern number and the von Klitzing constant are needed to describe the activities of the electrons; understanding how electrons move helps us understand the existing solids and to imagine new artificial materials that could have amazing physical properties.

However, practical consequences are not the main motivation for many scientists. We are deeply curious about the mysteries of Nature, and we are fascinated when the intrinsic beauty of natural laws becomes clear. Paul Claudel was therefore correct—butterflies are also essential to understand and describe the world of science.