Dec. 21, 2012 — Forget solid, liquid, and
gas: there are in fact more than 500 phases of matter. In a major paper in a
recent issue of Science,
Perimeter Faculty member Xiao-Gang Wen reveals a modern reclassification of all
of them.
Condensed
matter physics -- the branch of physics responsible for discovering and
describing most of these phases -- has traditionally classified phases by the
way their fundamental building blocks -- usually atoms -- are arranged. The key
is something called symmetry.
To understand symmetry, imagine
flying through liquid water in an impossibly tiny ship: the atoms would swirl
randomly around you and every direction -- whether up, down, or sideways --
would be the same. The technical term for this is "symmetry" -- and
liquids are highly symmetric. Crystal ice, another phase of water, is less
symmetric. If you flew through ice in the same way, you would see the straight
rows of crystalline structures passing as regularly as the girders of an
unfinished skyscraper. Certain angles would give you different views. Certain
paths would be blocked, others wide open. Ice has many symmetries -- every
"floor" and every "room" would look the same, for instance
-- but physicists would say that the high symmetry of liquid water is broken.
Classifying the phases of matter
by describing their symmetries and where and how those symmetries break is
known as the Landau paradigm. More than simply a way of arranging the phases of
matter into a chart, Landau's theory is a powerful tool which both guides
scientists in discovering new phases of matter and helps them grapple with the
behaviours of the known phases. Physicists were so pleased with Landau's theory
that for a long time they believed that all phases of matter could be described
by symmetries. That's why it was such an eye-opening experience when they
discovered a handful of phases that Landau couldn't describe.
Beginning in the 1980s, condensed
matter researchers, including Xiao-Gang Wen -- now a faculty member at
Perimeter Institute -- investigated new quantum systems where numerous ground
states existed with the same symmetry.
Wen pointed out that those new states contain a new kind of order: topological
order. Topological order is a quantum mechanical phenomenon: it is not related
to the symmetry of the ground state, but instead to the global properties of
the ground state's wave function. Therefore, it transcends the Landau paradigm,
which is based on classical physics concepts.
Topological order is a more
general understanding of quantum phases and the transitions between them. In
the new framework, the phases of matter were described not by the patterns of
symmetry in the ground state, but by the patterns of a decidedly quantum
property -- entanglement. When two particles are entangled, certain
measurements performed on one of them immediately affect the other, no matter
how far apart the particles are. The patterns of such quantum effects, unlike
the patterns of the atomic positions, could not be described by their symmetries.
If you were to describe a city as a topologically ordered state from the
cockpit of your impossibly tiny ship, you'd no longer be describing the girders
and buildings of the crystals you passed, but rather invisible connections
between them -- rather like describing a city based on the information flow in
its telephone system.
This more general description of
matter developed by Wen and collaborators was powerful -- but there were still
a few phases that didn't fit. Specifically, there were a set of short-range
entangled phases that did not break the symmetry, the so-called
symmetry-protected topological phases. Examples of symmetry-protected phases
include some topological superconductors and topological insulators, which are
of widespread immediate interest because they show promise for use in the
coming first generation of quantum electronics.
In the paper featured in Science, Wen and
collaborators reveal a new system which can, at last, successfully classify
these symmetry-protected phases.
Using modern mathematics --
specifically group cohomology theory and group super-cohomology theory -- the
researchers have constructed and classified the symmetry-protected phases in
any number of dimensions and for any symmetries. Their new classification system
will provide insight about these quantum phases of matter, which may in turn
increase our ability to design states of matter for use in superconductors or
quantum computers.
This paper is a revealing look at
the intricate and fascinating world of quantum entanglement, and an important
step toward a modern reclassification of all phases of matter.
