Quantum mechanics in fractal geometry

String theory notwithstanding, we live in three-dimensional space. But physics in reduced dimensionality need not be a purely theoretical exercise. Atomically thin materials such as graphene are well described as 2D systems, and polymers and quantum wires have many 1D characteristics.

What if the number of dimensions could be tuned continuously between 1 and 2? A century ago mathematician Felix Hausdorff developed a new way of measuring dimensionality that could take fractional values. For most familiar shapes—points, lines, planes, and so on—the Hausdorff dimension is equal to the usual topological dimension. But fractals, due to their infinite complexity, have noninteger Hausdorff dimension.

Now Ingmar Swart , Cristiane Morais Smith , and colleagues at Utrecht University in the Netherlands have taken a step toward experimentally studying quantum physics in a fractional-dimensional system. On a (111) surface of copper they placed carbon monoxide molecules (black indentations in the figure) to corral the surface electrons into a simplified Sierpinski triangle .

The full Sierpinski triangle, a fractal made up of infinitely many nested smaller triangles, has a Hausdorff dimension of 1.58. If drawn from line segments, it has infinite length, and if carved from a solid triangle, it has zero area, so intuitively, it’s neither 1D nor 2D but something in between. The Cu(111) surface-electron density inside the triangle is an approximation of the fractal, just as a graphene sheet is an approximation of an infinitely thin plane. But like graphene, the surface-electron system inherits the dimensional properties of its mathematical idealization.

Probing the interesting questions of fractal quantum mechanics will require a more intricate experiment. In particular, ensembles of interacting fermions behave in qualitatively different ways in 1D and 2D (see Physics Today, September 1996, page 19 ). Their behavior in 1.58 dimensions is an open question—but the surface electrons of Cu(111) interact only weakly. Surfaces of other materials, however, might allow the researchers to study interactions in fractal geometry. (S. N. Kempkes et al., Nat. Phys., 2018, doi:10.1038/s41567-018-0328-0 .)