# Book Review
## Quantum Chaos and
Quantum Dots
Katsuhiro Nakamura and Takahisa
Harayama
Oxford University Press, New York, 2004
199 pp., $109.50 hb
ISBN 0-19-852589-3
Reviewed by Arjendu K. Pattanayak
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*Quantum Chaos and Quantum Dots*, written
by Katsuhiro Nakamura of Osaka City
University and his collaborator Takahisa
Harayama, is an interesting review of some
quantum-transport and related problems in
solid-state systems. The text should prove
useful to two categories of physicists: those
in solid-state physics looking for an entrée
to issues in quantum chaos, and those in
quantum chaos interested in learning about
this specific application.
Semiconductor quantum dots are stateof-
the-art structures fabricated at semiconductor
heterojunctions, and consist of a
mesoscopic scattering region connected to
external reservoirs. The length scales are
such that electrons travel essentially ballistically,
so these systems act like quantum versions
of classical billiards. They are excellent
playgrounds for experimentalists to
explore novel transport phenomena.
One research focus since the early 1990s
has been quantum chaos: given certain dot
shapes, classical trajectories bounce around
chaotically, resulting in the rapid separation
of initially close trajectories. In regular dots,
individual classical trajectories travel in
bands, such that small perturbations grow
slowly. These two different behaviors influence
the quantum dynamics of the electrons
in radically different ways, resulting in
different universality classes for transport
properties in such systems. For example,
several groups showed that at low temperatures,
the magnetoresistance of the dots
exhibits reproducible fluctuations. The
spectra of the fluctuations are different,
depending on whether the dot is chaotic or
regular.
Theoretical explanations use a
semiclassical expansion for the Green’s
function within the Landauer–Buttiker conductance
formula, which emphasizes the
role of the unstable periodic orbits of the
chaotic system. Another approach uses the
phenomenological technique of random
matrix theory, for example.
Nakamura has contributed significantly
to the theory, and this monograph is a
somewhat selective review of these issues,
focused closely on his own research in this
field.
The authors tell a compelling story,
especially in the earlier chapters. They
begin with the experimental evidence for
the difference between chaotic and regular
dots, followed by an interlude that establishes
the needed ideas of classical Lyapunov
exponents and escape rates (classical
ideas emerge on a need-to-know basis in
this book, a strategy that works well). After
discussing the original results on conductance
fluctuations, the authors give a short
pedagogical treatment of some quantum
chaos results, including the Gutzwiller trace
formula for the spectra of chaotic systems.
They follow this with a tour of various more
recent analyses, including treatments of
orbital diamagnetism and persistent currents,
quantum interference and weaklocalization
issues in single open billiards,
Coulomb blockade, and so on. They have
some interesting comparisons of results
using both the Landauer–Buttiker transmission
formula and a semiclassical Kubo linear
response theory.
The style will not surprise those who
have read Nakamura’s now out-of-print
book *Quantum Chaos: A New Paradigm of
Nonlinear Dynamics* (Cambridge University
Press, 1993), although there is little overlap
in content. The present book is far more
focused and could be used for a special topics
course on quantum chaos in billiard systems,
although it is idiosyncratic in content.
It should definitely be useful for
researchers, including those at the graduate
level. While not definitive in any way, it is a
good addition to the still-small collection of
books in the fascinating and wide-open
field of quantum chaos. Arjendu K. Pattanayak is
an assistant professor of physics at Carleton College in Northfield,
Minnesota. He has published several
papers over the past decade on quantum
chaos, particularly as related to decoherence
and the classical limit. |