The important outstanding problems in physics
Posted on Wednesday, January 04, 2006 @ 19:55:07 UTC by vlad
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Dr. Jack Sarfatti writes: Excerpts from Nobel Lecture: Fractional quantization*
R. B. Laughlin
Department of Physics, Stanford University, Stanford,California
94305
Superfluidity, like the fractional quantum Hall effect, is an emergent phenomenon —a low-energy collective effect of huge numbers of
particles that cannot be deduced from the microscopic equations of motion
in a rigorous way and that disappears completely when the system is taken
apart (Anderson, 1972).
There are prototypes for
superfluids, of course, and students who memorize
them have taken
the first step down the long road to under standing the
phenomenon,
but these are all approximate and in the end not deductive at
all,
but fits to experi ment. The students feel betrayed and hurt by this
expe rience because they have been trained to think in reduc tionist
terms and thus to believe that everything not amenable to such
thinking
is unimportant. But nature is much more heartless than I am,
and those
students who stay in physics long enough to seriously
confront the
experimental record eventually come to understand that
the reductionist
idea is wrong a great deal of the time, and perhaps
always. One common
response in the early stages of learning is that
superconductivity and the
quantum Hall effect are not fundamental and
therefore not worth taking
seriously. When this happens I just open
up the AIP Handbook and show the
disbeliever that the accepted values
of e and h are defined by these
effects, and that ends that. The
world is full of things for which one’s
understanding, i.e., one’s
ability to predict what will happen in an
experiment, is degraded by
tak ing the system apart, including most
delightfully the standard
model of elementary particles itself. I myself
have come to suspect
most of the important outstanding problems in physics
are emergent in
nature, including particularly quantum gravity.
One
of the things an emergent phenomenon can do is create new
particles. When a
large number of atoms condense into a crystal, the
phonon, the elementary
quantum of sound, becomes a perfectly
legitimate par ticle at low energy
scales. It propagates freely, does
not decay, carries momentum and energy
related to wave-length and
frequency in the usual way, interacts by simple
rules that may be
verified experimentally, medi ates the attractive
interaction
responsible for conven tional superconductivity, and so forth,
and
none of these things depends in detail on the underlying equations of
motion. They are generic properties of the crystalline state. The
phonon ceases to have meaning when the crystal is taken apart,
however,
because sound makes no sense in an isolated atom. A somewhat
more esoteric
example, although a more apt one, is the Landau quasi
particle of a metal
(Pines and Nozie`res, 1966). This is an excited
quantum state that behaves
like an extra elec tron added to a cold
Fermi sea, but which is actually a
complex motion of all the
electrons in the metal. It is not possible to
deduce the existence of
quasiparticles from first principles. They exist
instead as a generic
feature of the metallic state and cease to exist if
the state does.
This problem is not limited to solids. Even the humble
electron, the
most elementary particle imaginable, car ries a polarization
of the
Dirac sea with it as it travels from place to place and is thus
itself a complex motion of all the electrons in the sea. In quantum
physics there is no logical way to distinguish a real particle from
an
excited state of the system that behaves like one. We therefore
use the
same word for both.
…
SOLITONS
The idea that particles carrying parts
of an elementary quantum
number might occur as an emergent phenom enon is
not new. …The
topological soliton or kink particle of Jackiw and Rebbi is
conceptually similar to the ’t Hooft–Polyakov monopole (’t
Hooft, 1974;
Polyakov, 1974) and the skyr mion (Skyrme, 1961), both
of which were
proposed as simplified models of real elementary
particles. …
This
breakup of the charge and spin quantum numbers of the electron
is the impor
tant new effect, for an excitation carrying charge but
no spin, or vice
versa, cannot be adiabatically deformed into a free
electron as a matter of
principle.
have charge e and spin 1/2. The alleged properties of solitons
were
therefore quite unprecedented and ex traordinary.
The property
of polyacetylene that causes solitons to exist is its
discrete broken
symmetry. If one wraps a mol ecule with an even
number of segments into a
ring, one discovers that it has two
equivalent quantum-mechanical ground
states, and that these transform
into each other under clockwise rotation
by one segment, a symmetry
of the underlying equations of motion. A
conventional in sulator
would have only one ground state and would
transform under this
operation into itself. These two states acquire
classical integrity
in the thermodynamic limit. When the ring is small,
local
perturbations, such as a force applied to one atom only, can mix the
two ground states in arbitrary ways, and in particular can tunnel the
system from the even-doubled state to the odd-doubled one. But this
tunneling becomes exponen tially suppressed as the size of the ring
grows, and even tually becomes insignificant. Classical integrity and
two fold degeneracy together make the broken-symmetry state
fundamentally different from the conventional in sulating state; one
cannot be deformed into the other in the thermodynamic limit without
encountering a quan tum phase transition.
The peculiar quantum
numbers of the soliton are caused by the
formation of a mid-gap state in
the elec tron spectrum.
…
FRACTIONAL QUANTUM HALL STATE
The
fractional quantum Hall state is not adiabatically deformable to
any
noninteracting electron state. I am always astonished at how
upset people
get over this statement, for with a proper definition of
a state of mat ter
and a full understanding of the integral quantum
Hall effect there is no
other possible conclusion. The Hall
conductance would necessarily be
quantized to an integer because it
is conserved by the adiabatic map and is
an integer in the
noninteracting limit by virtue of gauge invariance and
the
discreteness of the electron charge. So the fractional quantum Hall
state is some thing unprecedented—a new state of matter.
Its
phenomenology, however, is the same as that of the integral
quantum Hall
state in almost every detail (Tsui et al., 1982). There
is a plateau. The
Hall conduc tance in the plateau is accurately a
pure number times e2/h.
The parallel resistance and conductance are
both zero in the plateau.
Finite-temperature deviations from exact
quantization are activated or obey
the Mott variable-range hopping
law, depending on the tempera ture. The
only qualitative difference
between the two effects is the quantum of Hall
conductance.
Given these facts the simplest and most obvious ex
planation, indeed
the only conceivable one, is that the new state is
adiabatically
deformable into something physically similar to a filled
Landau level
except with fractionally charged excitations. Adiabatic
winding of a
flux quantum—which returns the Hamiltonian back to itself
exactly—
must transfer an integral number of these objects across the
sample.
Localization of the objects must account for the existence of the
plateau. All the arguments about deformability and exactness of the
quantization must go through as before. As is commonly the case with
new emergent phenomena, it is the experi ments that tell us these
things must be true, not theories. Theories can help us better
understand the experiments, in particular by providing a tangible
prototype vacuum, but the deeper reason to accept these conclusions
is
that the experiments give us no alternative.
… FRACTIONAL
STATISTICS
Fractional quantum Hall quasiparticles exert a long-range
velocity-
dependent force on each other—a gauge force—which is unique in
the
physics literature in having neither a progenitor in the underlying
equations of mo tion nor an associated continuous broken symmetry. It
arises spontaneously along with the charge fractionaliza tion and is
an
essential part of the effect, in that the quantum states of the
quasiparticles would not count up properly if it were absent. This
force, which is called fractional statistics (Leinaas and Myrheim,
1977; Wilczek, 1982), has a measurable consequence …
REMARKS
The
fractional quantum Hall effect is fascinating for a long list of
reasons,
but it is important, in my view, pri marily for one: It
establishes
experimentally that both particles carrying an exact
fraction of the
electron charge e and powerful gauge forces between
these par ticles, two
central postulates of the standard model of
elementary particles, can arise
spontaneously as emer gent phenomena.
Other important aspects of the stan
dard model, such as free
fermions, relativity, renormal izability,
spontaneous symmetry
breaking, and the Higgs mechanism, already have apt
solid-state
analogues and in some cases were even modeled after them
(Peskin,
1995), but fractional quantum numbers and gauge fields were
thought
to be fundamental, meaning that one had to postulate them. This is
evidently not true. I have no idea whether the properties of the
universe as we know it are fundamental or emergent, but I believe the
mere possibility of the latter must give string theorists pause for
it
would imply that more than one set of microscopic equations is
consistent
with experiment—so that we are blind to the microscopic
equations until
better experi ments are designed—and also that the
true nature of these
equations is irrelevant to our world. So the
chal lenge to conventional
thinking about the universe posed by these
small-science discoveries is
actually troubling and very deep.
Fractional quantum Hall quasiparticles
are the el ementary
excitations of a distinct state of matter that cannot
be deformed
into noninteracting electrons with out crossing a phase
boundary.
That means they are dif ferent from electrons in the only
sensible
way we have of defining different, and in particular are not
adiabatic images of electrons the way quasiparticle excitations of
metals and band insulators are. ... I emphasize these things be cause
there is a regrettable tendency in solid-state phys ics to equate an
understanding of nature with an ability to model, an attitude that
sometimes leads to overlook ing or misinterpreting the higher
organizing principle ac tually responsible for an effect. In the case
of the inte gral or fractional quantum Hall effects the essential
thing
is the accuracy of quantization. No amount of modeling done on
any computer
existing or contem plated will ever explain this
accuracy by itself. Only a
thermodynamic principle can do this. The
idea that the quasiparticle is
only a screened electron is
unfortunately incompatible with the key
principle at work in these ex
periments. If carefully analyzed it leads to
the false con clusion
that the Hall conductance is integrally
quantized.
…. I similarly acknowledge Bert Halperin’s many outstand ing
contributions, including particularly his discovery that
quasiparticles
obey fractional statistics (Halperin, 1984).
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