APS Medal for Exceptional Achievement in Research: Topology and other tools in condensed matter physics^*

This article is an extended version of a talk given at the March 2019 meeting of the American Physical Society, in connection with the award to the author of the 2019 APS Medal for Exceptional Achievement in Research. The article will summarize some of the author’s work in areas linked to topological aspects of classical and quantum physics. Topics include quantum Hall effects, as well as the theory of melting in to dimensions and effects of vortex motion in a thin superconductor or a superfluid film.

Figure 1

Dislocation in a two-dimensional crystal. A square lattice is illustrated for ease of visualization. The Burgers vector $\mathbf{b}$, which characterizes the dislocation, is the missing lattice vector when one follows a path around the defect consisting of, say, $n$ steps in the $x$ direction, $n$ steps in the $y$ direction, $n$ steps in the $-x$ direction, and $n$ steps in the $-y$ direction.

Figure 2

Disclination in a square lattice. Following a closed path counterclockwise around the defect, the crystal axes are rotated by an angle of $\pi /2$. The disclination may also be characterized as having an extra quadrant of material. A disclination of the opposite sign (not shown), would have a missing quadrant of material and a rotation angle of $-\pi /2$.

Figure 3

Split annular geometry filled with a quantized Hall state. The left half of the annulus is occupied by ideal material in a quantized Hall state with filling factor $\nu$. The right half contains a disordered sample. If there are no extended states at the Fermi level in the bulk of the dirty material or at the interface between the two halves, then, at $T=0$ one can reach a steady state in which there are different voltages, $V_1$ and $V_2$, on the inner and outer edges of the annulus, but the voltage on a given edge is constant around the annulus. By conservation of charge, the current $I$ must also be constant around the annulus, so it must be given by $I=\nu e^2(V_1-V_2)/\hslash$ in both regions.