Matter exhibits weird properties at very cold temperatures. Take
superfluids, for example: discovered in 1937, they can flow without resistance
forever, spookily climbing the walls of a container and dripping onto the
In the past 100 years, 11 Nobel Prizes have been awarded to nearly two dozen
people for the discovery or theoretical explanation of such cold materials—superconductors
and Bose–Einstein condensates, to name two—yet a unifying theory of these
extreme behaviors has eluded theorists.
University of California, Berkeley, physicist Hitoshi
Murayama and graduate student Haruki Watanabe have now discovered a commonality
among these materials that can be used to predict or even design new materials
that will exhibit such unusual behavior. The theory, published online by Physical
Review Letters, applies equally to magnets, crystals, neutron stars, and
“This is a particularly exciting result because it concerns pretty much all
areas of physics; not only condensed matter physics, but also astrophysics,
atomic, particle and nuclear physics, and cosmology,” said Murayama, the
MacAdams Professor of Physics at UC Berkeley, a faculty senior scientist at
Lawrence Berkeley National Laboratory and director of the Kavli Institute for
the Physics and Mathematics of the Universe at the University of Tokyo. “We are putting together all of them into a single theoretical framework.”
The theorem Watanabe and Murayama proved is based on the concept of
spontaneous symmetry breaking, a phenomenon that occurs at low temperatures and
leads to odd behavior. This produces superconductors, which allow electric
currents to flow without resistance; or Bose-Einstein condensates, which have
such low energy that every atom is in the same quantum state.
By describing the symmetry breaking in terms of collective behavior in the
material—represented by so-called Nambu-Goldstone bosons—Murayama and Watanabe
found a simple way to classify materials’ weirdness. Boson is the name given to
particles with zero or integer spin, as opposed to fermions, which have
“Once people tell me what symmetry the system starts with and what symmetry
it ends up with, and whether the broken symmetries can be interchanged, I can
work out exactly how many bosons there are and if that leads to weird behavior
or not,” Murayama said. “We’ve tried it on more than 10 systems, and it works
out every single time.”
Anthony Leggett of the University of Illinois at Urbana Champaign, who won
the 2003 Nobel Prize in Physics for his pioneering work on superfluids, pointed
out that “it has long been appreciated that an important consequence of the
phenomenon of spontaneously broken symmetry, whether occurring in particle
physics or in the physics of condensed matter, is the existence of the
long-wavelength collective excitations known as Nambu-Goldstone bosons.
“In their paper, Watanabe and Maruyama have now derived a beautiful general
relation … (involving) Nambu Goldstone bosons … (that) reproduces the relevant
results for all known cases and gives a simple framework for discussing any
currently unknown form of ordering which may be discovered in the future.”
“Surprisingly, the implications of spontaneous symmetry breaking on the low
energy spectrum had not been worked out, in general, until the paper by
Watanabe and Murayama,” wrote Hirosi Ooguri, a professor of physics and
mathematics at Caltech. “I expect that there will be a wide range of
applications of this result, from condensed matter physics to cosmology. It is
a wonderful piece of work in mathematical physics.”
Symmetry has been a powerful concept in physics for nearly 100 years, allowing
scientists to find unifying principles and build theories that describe how
elementary particles and forces interact now and in the early universe. The
simplest symmetry is rotational symmetry in three dimensions: a sphere, for
example, looks the same when you rotate it arbitrarily in any direction. A
cylinder, however, has a single rotational symmetry around its axis.
Some interactions are symmetric with respect to time, that is, they look the
same whether they proceed forward or backward in time. Others are symmetric if
a particle is replaced by its antiparticle.
When symmetry is broken spontaneously, new phenomena occur. Following the
Big Bang, the universe cooled until its symmetry was spontaneously broken,
leading to a predicted Higgs boson that is now being sought at the Large Hadron
Collider in Geneva, Switzerland.
With solids, liquids, or gases, symmetry relates to the behavior of the
spins of the atoms and electrons. In a ferromagnetic material, such as iron or
nickel, the randomness of the electron spins at high temperatures makes the
material symmetric in all directions. As the metal cools, however, the electron
spins get locked in and force their neighbors to lock into the same direction,
so that the magnet has a bulk magnetic field pointing in one direction.
Nambu-Goldstone bosons are coherent collective behavior in a material. Sound
waves or phonons, for example, are the collective vibration of atoms in a
crystal. Waves of excitation of the electron spin in a crystal are called
magnons. During the cooling process of a ferromagnet, two symmetries were
spontaneously broken, leaving only one Nambu-Goldstone boson in the material.
In Bose-Einstein condensates, for example, “you start with a thin gas of
atoms, cool it to incredibly low temperature—nanokelvins—and once you get to
this temperature, atoms tend to stick with each other in strange ways,” Murayama
said. “They have this funny vibrational mode that gives you one Nambu-Goldstone
boson, and this gas of atoms starts to become superfluid again so it can flow
without viscosity forever.”
On the other hand, solid crystals, regardless of their compositions or
structures, have three Nambu-Goldstone bosons, equivalent to the three
vibrational modes (phonons).
“What this Nambu-Goldstone boson is, how many of them there are and how they
behave decide if something becomes a superfluid or not, and how things depend
on the temperature,” Murayama added. “All these properties come from how we
understand the Nambu-Goldstone boson.”
Yoichiro Nambu shared the 2008 Nobel Prize in Physics, in part, for
explaining that in some systems, the number of broken symmetries equals the
number of Nambu-Goldstone bosons.
The new theorem expands on Nambu’s ideas to the more general case, Watanabe
said, proving that in weird materials, the number of Nambu-Goldstone bosons is
actually less than the number of broken symmetries.
“What Nambu showed was true, but only for specialized cases applicable to
particle physics,” he said. “Now we have a general explanation for all of
physics; no exceptions.”
One characteristic of states with a low Nambu-Goldstone boson number is that
very little energy is required to perturb the system. Fluids flow freely in
superfluids, and atoms vibrate forever in Bose-Einstein condensates with just a
As a student at the University
of Tokyo, Watanabe had
proposed a theorem to explain materials’ properties through Nambu-Goldstone
bosons, but was unable to prove it until he came to UC Berkeley last year and
talked with Murayama. Together, they came up with a proof in two weeks of what
they call a unified theory of Nambu-Goldstone bosons.
“Those two weeks were very exciting,” Watanabe said.