water molecules. If one were to
write down Newton's equations or Schrödinger's
equation for these molecules,
they would be far too complicated to solve. Even if one could solve them,
the solution would be useless without very precise knowledge of the initial
conditions -- all of the positions and velocities at some given time.
If we were
to make even a very small error in these initial conditions, we might
find that the solution describes a block of ice. A slightly larger error,
and we might find that the equations describe a physics professor.
(Only 2/3 water, but you get the idea.) Thus, the microscopic
equations which describe
an individual water molecule are of limited use when it comes to describing
the macroscopic properties of matter, as are more refined
descriptions at the atomic or subatomic scales.
In essence, the reductionist approach fails.
An entirely new set of concepts -- which
are not inherent in the microscopic equations -- must be introduced,
such as temperature,
entropy, and phase. The concept of temperature, for instance, makes
no sense for an individual water molecule, but one can hardly
understand a macroscopic
body of water without it. These properties only emerge at
macroscopic scales, which are the most
interesting and important scales for humankind.
In trying to understand the properties of matter, one must contend with the
fact that matter is made up of a large number of microscopic consituents.
As Einstein showed in his investigation of diffusion nearly 100 years
ago, therein lie the secrets to many of the mysterious properties of
matter. In considering such a perspective, one is immediately faced with
such questions as why is a solid solid? If you open the door to a sauna
and try to push against the steam which streams out,
the water molecules near your hand will move out of the way, and your hand
(and, eventually the rest of you) will be bathed in steam. This is precisely
what you would naively expect of a collection of molecules, each of
which acts fairly independently of the others in moving
around your hand. However, if you try the same thing with
a block of ice, the whole block will
move as one. How can this be? The answer is that the water molecules interact
strongly with each other in order to form one highly correlated whole in which
the oxygen and hydrogen atoms sit as the sites of a rather rigid lattice.
The same is true of all crystaline solids. Such highly
correlated states are said
to be condensed; it is their study which is the subject of condensed
matter physics.
A sheet of ice, such as that at an ice-skating rink, can support the weight of a person just as well as a metal platform. A cloud of water vapor, made up of the very same water molecules, cannot. Thus, when it comes to understanding the properties of matter in ordinary circumstances, it may not be particularly useful to understand the difference between a water molecule and an iron atom. The more crucial information is contained in the difference between the gaseous phase and the solid phase.
There are an enormous number of consequences which follow from
the fact that a solid, such as a block of ice or the graphite in a pencil,
consists of on the order of atoms which have condensed into
a highly-correlated state in which they act in unison.
One of them is the propagation of longitudinal and
transverse sound waves
through the solid. These, in turn, affect
the specific heat, thermal conductivity,
and optical properties of a solid. Indeed, it is impossible to understand
any of the properties of a solid in detail without first appreciating
the consequences of the condensation phenomenon which makes
it solid. There are many other, more exotic condensed states
with fascinating properties which follow from the particular
type of correlations which give birth to them.
Superconductivity and magnetism are two examples; in these
states, the electrons are correlated in such a way that they exhibit a rigidity
akin to that of a solid, but in their ability to expel a magnetic
field or maintain a static magnetization rather than their
ability to support a person's weight.
One of the main challenges facing physicists is understanding other
states of matter involving the correlated behavior of a large number of
constituents.
The examples of the previous paragraph, crystallization,
superconductivity, and magnetism, are examples of
spontaneously broken symmetry. The ground state of a
crystal does not exhibit translational symmetry even though
the underlying microscopic equations do. The ground state
of a magnet has a preferred direction -- the direction in
which the magnetic moment points -- even though the
underlying microscopic equations do not. Some examples
of broken-symmetry states into which liquid crystals order
and the electrons in high-temperature superconductors
might organize themselves are depicted on the right.
Professors Brown,
Chakravarty,
Clark,
Grüner,
Kivelson,
Nayak,
Rudnick,
and Williams are
extending the frontiers of our understanding of broken-symmetry phases.
Such states are rigid because it is energetically
favorable for the symmetry to be broken in the
same way in different parts of the system. Fractionalization
is another phenomenon which leads to robust condensed states.
Professors
Jiang
and
Kivelson
have made seminal contributions
to our understanding of these states.
In fractionalized states, there are particle-like excitations which
have quantum numbers which are fractions of the quantum numbers
of electrons. Such phenomena are robust because small perturbations
cannot change these fractions continuously; they can only jump by discrete
amounts and only as a result of relatively drastic changes.
The presence of a fractionalized
excitation in one part of the system implies the presence of
compensating fractions elsewhere in the system since the
quantum numbers of the total system are integral multiples
of those of an electron. Thus, there are global constraints
on the system which endow it with a certain rigidity.
For instance, the Hall conductance is fixed incredibly
precisely to a quantized value.
Part of Professor
Nayak's work is concerned with
a deeper understanding of fractionalized states
and their various incarnations, with an eye towards
exploiting their rigidity for quantum computation.
At the phase transitions which separate various
phases, remarkable phenomena occur. At a first-order
phase transition, a system will segregate itself
into different regions in which one or the other
phase occurs. At a second-order phase transition,
the system is in neither phase. Rather, it is
in a self-similar state which is poised at an instability
to either phase. In the figure at the right, we have
a depiction of the percolation transition.
There are both connected black regions and connected
grey regions which extend across the system. If grey
is favored even slightly more, then there will be localized
puddles of black in a grey landscape. If black is favored
even slightly more, then there will be localized islands of
grey in a sea of black. The phase transition point between
these two is the isolated special point at which black
and grey are on the same footing. The unbiased competition
between them leads to the fractal pattern in the
figure. Professors
Brown,
Chakravarty,
Clark,
Grüner,
Jiang
Kivelson,
Nayak,
Rudnick,
and
Williams
all study the properties of phase transitions, some of which are driven by
thermal fluctuations while others are due to quantum fluctuations.
The preceeding discussion has focussed on systems in or near thermal equilibrium. Much of the matter which matters to us is not in equilibrium. If it were heated up and cooled very slowly so that it could equilibrate, ordinary window glass would form crystalline quartz. When it is cooled quickly, the silicon and oxygen atoms get stuck in a random, disordered configuration instead. Though it is far from equilibrium, it appears to the casual eye to simply be a transparent brittle solid, not so very different from other covalent solids. On the other hand, some of its properties, such as its specific heat and thermal conductivity are quite different. Although it might not be appropriate to think of it as a condensed phase, it has macrosopic properties which are as robust as those of matter in equilibrium. Consider another example, shaving cream. It is made up predominantly of water and air bubbles. However, it can maintain its shape as a solid would, but is not quite solid since it does not resist shear stresses. It is really an entirely different state of matter. Professors Durian and Rudnick investigate the physics of these ill-condensed states by experimental and theoretical techniques, respectively. Related questions about the effects of impurities (which, by virtue of being stuck at random locations, are out of equilibrium) on the properties of solids are under study by professors Chakravarty, Grüner, Jiang, Kivelson, and Nayak. In the figure near the top of this page, there is an example of the type of strange behavior which can occur in the presence of impurities or some other degree of freedom which is randomly distributed and static and, thus, out of equilibrium; the figure shows the wavefunction of an electron moving in a randomly varying magnetic field.
Consider, again, our glass of water. Imagine stirring it vigorously or even shaking it. The water will be driven out of equilibrium and, eventually, it will become turbulent. By stirring or shaking the water, we will continually cause eddies to form; these eddies will break up and form smaller eddies. These features are essentially the same as those of turbulent air flows in the atmosphere -- the kind which can lead to a rather unpleasant airplane flight -- and a wide variety of other fluids. Again, we have a set of universal physical behaviors which cannot be understood in quite the same terms as an equilibrium phase -- temperature and entropy, for example, are not very useful concepts here -- but are no less commonplace or fundamental than other forms of matter. Professor Putterman's work deals with the cascade of energy from large scales to small scales in systems far-from-equilibrium, as does one facet of Professor Williams'.
Finally, there is life itself. A person is clearly not a crystalline solid, but a person certainly has many of the same rigidity properties (at least on time scales of $\sim 80$ years) as any other solid. How do the physical principles with which we are familiar from other contexts apply to something like a living being? Is it possible that there are new physical principles waiting to be discovered? These and related questions are of interest to professors Bruinsma, Grüner, Rudnick, and Zocchi. In the figure below, there is a representation of a possible configuration of a segment of supercoiled, closed DNA, as modelled by the Euler-Lagrange equation for a bent and twisted rod. There is a more detailed description of the biophysics program at UCLA at the Molecular Biphysics website.
There are important physical principles at play at all different
scales, from the subatomic to the cosmological. The great
advantage of studying matter at the scale of everyday experience
is that controlled experimental observations can be far more readily
performed. Consider the most surprising experimental discoveries
of the last 25 years: the fractional quantum Hall effect,
high-temperature superconductivity, superconductivity in
buckyballs, sonoluminescence, an apparent metal-insulator
transition in Silicon MOSFETS.
They were all in condensed matter physics, and they were all essentially
table-top experiments conducted by two or three people in collaboration.
The experimentalists in the UCLA CMP group are all engaged in precisely these
types of experiments. Some of these experiments may uncover
surprises of the type mentioned. Others may yield the
answers to long-standing puzzles or culminate
and consolidate years of study of interesting materials.
Many of the phenomena of everyday experience, such as temperature and magnetization, can be quantitatively studied through thermodynamic measurements of quantities (Profs. Putterman, Williams). Others can be probed through transport measurements, in which one observes how a material conducts electricity or heat (Profs. Brown, Jiang, Braunstein). Still others involve the scattering of electromagnetic waves or neutrons by a material (Profs. Durian, Grüner, Patel, Zocchi). The ability of the material to absorb the incoming energy or create a diffraction pattern by scattering it bespeaks of its structure and dynamics (such measurements are a natural extension of the way in which we see things). In magnetic resonance measurements, one applies a static magnetic field to a material, thereby enabling one to see the nuclear magnetic moments or those of the electrons ( Clark, Brown, Holczer). Recently, microscopy techniques which measure the electromagnetic forces between the surface of a material and a probe have opened up a new way of ``seeing'' the spatial variations in the structure of matter (Prof. Holczer). Several faculty members (Profs. Brown, Clark) are engaged in projects which take advantage of the facilities available at national laboratories such as the Los Alamos National Laboratory and the National High Magnetic Field Laboratory.
These probes, and others, have guided our discovery of new phases of matter and the fundamental concepts which underlie them. It is sometimes said that, in the absence of observation, physicists might have guessed the existence of gaseous and crystalline solid phases, but would have never deduced the existence of a liquid phase. The holds for many of the important concepts in physics. Many of the concepts which grew out of this interaction between condensed matter theorists and experimentalists have had a profound influence on other fields since they, too, deal with problems involving a large number of particles, but in somewhat different regimes where experiments cannot always be done. In particular, there has been a great deal of cross-fertilization between condensed matter physics and high-energy physics. For instance, the concept of spontaneous symmetry breaking, which was developed in the study of magetism, superfluidity, and superconductivity, found its way into high-energy physics, where it underlies the the theory of electroweak interactions and of chiral symmetry. The renormalization group and the effective field theory approach, which grew out of the study of phase transitions, also had a profound influence on high-energy physics. More recently, the study of phase transitions in two-dimensional systems proved to be closely related to the study of the two-dimensional world-sheet description of string theory.
In some materials, such as lead, essentially the same physics is observed from high temperatures all the way down to a few degrees Kelvin. Equivalently, the physical properties of the material are qualitatively similar at all scales from a few angstroms all the way to thousands of angstroms, which is already a macroscopic scale. The best understood materials are those fall into this category: the interesting ordering phenomena occur at low temperatures and long length scales. However, this is not the only possibility. Sometimes, a funny thing happens on the way to the forum. This is clearly the case in the biological context. Life can range in scale from a micron to nearly 100 meters, but its appurtenances -- DNA, RNA, proteins -- are formed on the nanometer scale. One can imagine that structures could form spontaneously on such length scales in other forms of matter as well. Indeed it has been conjectured that distributions of charge (Prof. Kivelson) or current (Profs. Chakravarty, Nayak) which are inhomogeneous on such scales play a role in the physics of high-temperature superconductors. It is not clear what physical principles operate at this scale; clearly these phenomena are distinct from macroscopic order, but they are also too complex to be understood from a purely microscopic perspective. Grappling with with issue is one of the fundamental challenges facing biophysicists, such as Profs. Bruinsma, Rudnick, and Zocchi. It is now becoming possible to manufacture controlled physical systems -- such as quantum dots and carbon nanotubes -- on these scales. Progress in the development of experimental techniques has enabled one to probe them. At the California Nanoscience Institute (CNSI), UCLA is developing state-of-the-art facilities to investigate this regime, and is bringing together researchers from physics, chemistry, biology, and engineering.
There are special mathematical
challenges associated with understanding how
particles can behave in a highly-correlated manner. Ideas from
group theory, topology, differential geometry, functional analysis,
and other branches of mathematics have given physicists a precise
language in which to formulate their understanding of the properties
of matter. Similarly, the challenges of understanding the physical
world has provided the impetus for the development of these
areas of mathematics. At the
Institute for Pure and
Applied Mathematics
(IPAM) at UCLA, this interaction between the two fields -- and other
natural sciences -- is being furthered through workshops which
bring together leading mathematicians and scientists from around the world.
Human history has been delineated by our ability to manipulate different forms of matter: the stone age, the iron age, the bronze age, etc.. It is natural for us to be curious about why the materials which we see around us behave in the way that they do. There are clearly basic scientific issues here; they are complemented by practical motivations. Indeed, condensed matter physics had a very pragmatic orientation in its early days, as Carnot, Kelvin, and others came up with such fundamental concepts as entropy and free energy in their effort to understand steam engines. Today, condensed matter physics is still the closest branch of physics to modern technological applications. There are many examples of this, one of the most dramatic of which is the transistor, the basis of modern electronics, which was the culmination of thirty years of basic research on semiconductor physics. Conversely, basic research in condensed matter physics benefits from technological advances. The investigation of the quantum Hall effect has been assisted by the concurrent rise to prominence of GaAs as a key component of cellular telephones. This symbiotic relationship between basic science and technology is fostered here at UCLA, where several faculty members (Profs. Holczer, Grüner, Putterman, Nayak) have collaborations with groups in industry. More generally, the UCLA condensed matter physics group is developing today the science which could lead to tomorrow's technologies, from quantum computation to superconductivity to the transport of granular materials to biomedical applications.