Models, Materials and Experiments of Topological
Insulators
Shou-Cheng Zhang, Stanford University
Most
quantum states of matter are categorized by the symmetries they break. For
example, the crystallization of water into ice breaks translational symmetry or
the magnetic ordering of spins breaks rotational symmetry. However, the
discovery in the early 1980s of the integer and fractional quantum Hall effects
has taught us that there is a new organizational principle of quantum matter.
In the quantum Hall state, an external magnetic field perpendicular to a
two-dimensional electron gas breaks time-reversal symmetry and causes the
electrons to circulate in quantized orbits. The “bulk” of the electron gas is
an insulator, but along its edge, electrons circulate in a direction that
depends on the orientation of the magnetic field. The circulating edge states
of the quantum Hall state are different from ordinary states of matter because
they persist even in the presence of impurities.
In the last few years, a
number of theorists realized that the same “robust” conducting edge states that
are found in the quantum Hall state could be found on the boundary of
two-dimensional band insulators with large spin-orbit effect, called
topological insulators, even in the presence of time-reversal symmetry. In
these insulators, spin-orbit effects take the role of an external magnetic
field, with spins of opposite sign counter-propagating along the edge. In 2006,
my colleagues and I predicted this effect (later confirmed) on the edge of HgTe
quantum wells — the first experimentally realized quantum spin Hall state. In
2007 Liang Fu and Charles Kane of the University of Pennsylvania predicted that
a three-dimensional form of the topological insulator with conducting surface
states could exist in Bi1-xSbx, an alloy in which
spin-orbit effects are large. Recent photoemission measurements of the surface
of Bi1-xSbx supported this picture, strongly suggesting
that Bi1-xSbx is the first realization of a topological
insulator in three dimensions and that its surface is a topological metal in
two dimensions. I will describe basic ideas behind these theoretical and
experimental works and more recent developments. I will also discuss a number
of other promising materials for the discovery of topological insulators.