Nobel Lecture: Topological quantum matter^*

Nobel Lecture, presented December 8, 2016, Aula Magna, Stockholm University. I will describe the history and background of three discoveries cited in this Nobel Prize: The “TKNN” topological formula for the integer quantum Hall effect found by David Thouless and collaborators, the Chern insulator or quantum anomalous Hall effect, and its role in the later discovery of time-reversal-invariant topological insulators, and the unexpected topological spin-liquid state of the spin-1 quantum antiferromagnetic chain, which provided an initial example of topological quantum matter. I will summarize how these early beginnings have led to the exciting, and currently extremely active, field of “topological matter.”

Figure 1

Simple energy-level picture for the integer quantum Hall effect, with an energy gap in the bulk stabilized by pinning of the Fermi level by gapless edge states. (The energy levels are shown as functions of the radius in a disk-shape sample.)

Figure 2

The “Hofstadter butterfly” spectrum of electrons on a periodic lattice plus a uniform magnetic field, showing energy levels as a function of magnetic flux through a unit cell. The structure in the lower left corner becomes that of a system of simple Landau levels. Colors in gaps between subbands represent the different integer quantizations of the Hall effect if the Fermi level is in that gap. Colored spectrum provided by D. Osadchy and J. Avron.

Figure 3

Berry phase for the adiabatic evolution of the state of a quantum spin aligned along a moving axis. The Berry phase ${\mathrm{\Phi }}_{\mathrm{\Gamma }}$ is the spin quantum number $S$ times the solid angle subtended by the closed path $\mathrm{\Gamma }$ of the alignment axis $\widehat{\mathbf{\Omega }}$. After the axis returns to its initial orientation, the final quantum state is the initial state times the factor $\exp i{\mathrm{\Phi }}_{\mathrm{\Gamma }}$ that depends geometrically on the path taken. Since the “solid angle subtended by the path” is ambiguous modulo $4\pi$, $2S$ is topologically required to be an integer (which it is).

Figure 4

The simple graphenelike tight-binding “toy model” ([15]) for the “broken-time-reversal topological insulator” or “Chern insulator” that exhibits a zero-field “quantum anomalous Hall effect.” Electrons “hop” along nearest-neighbor bonds (solid lines) with a real matrix element, and along second-neighbor bonds (dashed lines) with a complex matrix element, which has a positive phase for hopping in the direction of the arrow. Two conjugate copies (one for up-spin, one for down-spin electrons) were later combined by Kane and Mele to model a time-reversal-invariant topological insulator. The complex phases for hopping between second neighbors introduces broken-time-reversal symmetry, which could come from a ferromagnetically-ordered magnetic dipole at the center ($*$) of each hexagonal cell, pointing normal to the 2D plane. The dipoles give rise to different magnetic flux through regions $a$, $b$, and $c$ of the unit cell, but no net magnetic flux, leaving the standard Bloch structure intact.

Figure 5

“Zigzag” edge of graphene after perturbation by terms that break inversion or time-reversal symmetry. The unperturbed edge has an edge state joining the projections of the two Dirac points where the filled valence band (red) touches the empty conduction bands (green). When a gap is induced by the perturbation, there are four ways the edge state can be connected, two of which are topological, and connect conduction and valence bands.

Figure 6

The Jordan-Wigner transformation maps the $S=\frac{1}{2}$ Heisenberg chain with zero magnetization into a half-filled interacting band of spinless fermions, where $4k_F$ is a Bragg vector.

Figure 7

In $(1+1)\mathrm{D}$ space-time, the analog of the 2D vortex is an instanton tunneling process where the topological winding number of the easy-plane spin chain changes. This process is centered on a bond between consecutive sites on which the local Néel order breaks down for a short time interval.

Figure 8

The $S=1$ AKLT state treats each spin as a symmetric combination of two $S=\frac{1}{2}$ “half spins,” one of which forms a singlet valence bond with a half spin of the neighbor to the right, the other with the neighbor to the left. An unused spin $\frac{1}{2}$ is left at each open end of the chain, and the “entanglement spectrum” consists of a single doublet.