The important outstanding problems in physics
Date: Wednesday, January 04, 2006 @ 19:55:07 GMT Topic: Science
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 lowenergy 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 wavelength 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 quantummechanical 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 evendoubled state to the odddoubled 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 brokensymmetry 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 midgap 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.
Finitetemperature deviations from exact quantization are activated or obey
the Mott variablerange 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 longrange
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
solidstate 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 smallscience 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 solidstate 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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