50 Years of Anderson Localization

610 Pages·2010·7.77 MB·English

by  Elihu Abrahams

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Published by
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50 YEARS OF ANDERSON LOCALIZATION
Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd.
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ISBN-13 978-981-4299-06-0
ISBN-10 981-4299-06-5
ISBN-13 978-981-4299-07-7 (pbk)
ISBN-10 981-4299-07-3 (pbk)
Desk Editor: Tan Hwee Yun
Printed in Singapore.
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April13,2010 13:26 WorldScienti(cid:12)cReviewVolume-9.75inx6.5in 00a_introduction
INTRODUCTION
At the time of the publication of this volume, more than (cid:12)fty years have
passed since the appearance in The Physical Review of Philip W. Ander-
son’s landmark paper titled Absence of Di(cid:11)usion in Certain Random Lat-
1
tices. During the decades since, the phenomenon predicted and explained
inthat paperbecame knownas\Andersonlocalization" andhasbeenwidely
recognized as one of the fundamental concepts in the physics of condensed
matter and disordered systems. Anderson’s 1977 Nobel Prize, shared with
Nevill Mott and John Van Vleck, is based in part on that seminal paper.
Anderson was initially motivated to understand the in(cid:13)uence of disorder
on spin di(cid:11)usion and on electron transport. In the years since, the concepts
and results that he created have found their way across a wide range of
othertopics. Amongthemarenano{andmeso{scale technology, seismology,
acoustic waves, quantum optics, ultracold atomic gases, localization of light.
The chapters contributed by Phil Anderson, David Thouless and T. V.
Ramakrishnanexplain clearly some ofthe early history of the understanding
of the localization phenomenon. Earlier discussions of the background and
2
content of Anderson’s 1958 paper may be found in Thouless’ 1970 review
3
and in Anderson’s 1977 Nobel Lecture.
In the area of electron transport, not much was done on the localization
problem for more than a decade after the 1958 paper. What might be called
the modern era of localization began in the 1970s, with the introduction of
4 5 6
scaling ideas by Licciardello and Thouless, Wegner and Schuster. As a
matter of fact, it was the Schuster paperthat set PhilAnderson, Don Liccia-
rdello, T. V. Ramakrishnan, and I thinking about the statistical mechanics
analogy, one{parameter scaling and the beta function of scaling theory. The
7
consequence was our 1979 Physical Review Letter, often called the \gang of
four"paper(\G4"). Asiswellknown,weconcludedthatthemetal{insulator
transition is continuous, i.e. there is no minimum metallic conductivity and
that all states in two dimensions are localized. The history of these develop-
ments is beautifully reviewed by David Thouless in his contribution to this
volume.
v
April13,2010 13:26 WorldScienti(cid:12)cReviewVolume-9.75inx6.5in 00a_introduction
vi 50 Years of Anderson Localization
There are a number of papers that are not often quoted now, although
they made signi(cid:12)cant impact when they appeared. Here, I take this oppor-
tunity to mention some of them andto place them inhistorical context. The
functional integral formulation for correlation functions of a disordered elec-
tron system and disorder averaging by means of the n!0 replica trick was
8
developed by several people, notably Amnon Aharony and Yoseph Imry.
9
John Cardy, in 1978, reformulated the functional integral representation
and the n-replica method. He showed how to control the saddle point of the
equivalent Ginzburg{Landau action and obtained power laws for the energy
dependence of the density of states.
Following upon G4, Shinobu Hikami, Anatoly Larkin and Yosuke
10
Nagaoka incorporated scattering mechanisms with di(cid:11)erent symmetries
(spin{orbit scattering, magnetic impurities), inelastic scattering, and cru-
cially, magnetic (cid:12)eld into the treatment of the crossed graphs of Langer and
11
Neal, which are the basis of the scaling behavior derived in G4. Here, the
magnetoresistance was derived andthisbecame, andremains, the diagnostic
of choice for all subsequentexperiments. Inthis connection, see the chapters
of Bergmann, Dynes and Giordano in this book.
12
Around 1980, there were a number of discussions of the equivalence of
the localization problem and the matrix nonlinear (cid:27) model. An especially
13
transparent derivation was given by Shinobu Hikami in 1981. He showed
how the systematic perturbative treatment of the relevant di(cid:11)usion prop-
agators in the particle{hole (\di(cid:11)uson") and particle{particle (\cooperon")
channelsleadstoane(cid:11)ectiveHamiltonianofthenonlinear(cid:27)model. Hisanal-
ysis of the propagators and their interaction vertices became the standard
basis for subsequent perturbative treatments of various e(cid:11)ects, including
14
in particular early analyses of the e(cid:11)ect of electron{electron interactions.
Thedevelopmentofmatrixnonlinear(cid:27)modelmethodsisreviewedbyseveral
contributors to this volume: Efetov, Mirlin et al., Pruisken.
The physics of Anderson localization has had a pervasive in(cid:13)uence on a
broad variety of fundamental concepts and phenomena, including the quan-
tum Hall e(cid:11)ect, quantum criticality, symmetry and random matrix theory,
multifractality, electron{electroninteractionindisorderedmetals. Theseand
other issues are explored in many of the chapters of this book.
Someofthepioneersinthe(cid:12)eldofdisorderedsystems, boththeorists and
experimentalists, havecontributedtothisvolume. Itisamarkofthevitality
of Anderson localization physics (and indeed of the contributors) that while
a few chapters are essentially historical, the others report results of current
research. Unfortunately, space constraints have prevented a comprehensive
April13,2010 13:26 WorldScienti(cid:12)cReviewVolume-9.75inx6.5in 00a_introduction
Introduction vii
survey of all the past and current developments. In spite of this limitation
the hope is that the reader will acquire an appreciation of the history of the
physics of localization and its current manifestations.
References
1. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
2. D. J. Thouless, J. Phys. C: Solid State Phys. 4, 1559(1970).
3. P. W. Anderson, Rev. Mod. Phys. 50, 191 (1978).
4. D. C. Licciardello and D. J. Thouless, Phys. Rev. Lett. 35, 1475 (1974).
5. F. J. Wegner, Z. Physik B 25, 327 (1976).
6. H. G. Schuster, Z. Physik B 31, 99 (1978).
7. E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan,
Phys. Rev. Lett. 42, 673 (1979).
8. A. Aharony and Y. Imry, J. Phys. C: Solid State Phys. 10, L487 (1977).
9. J. L. Cardy, J. Phys. C: Solid State Phys. 11, L321 (1978).
10. S. Hikami, A. I. Larkin and Y. Nagaoka, Prog. Theor. Phys. 63, 707 (1980).
11. J. S. Langer and T. Neal, Phys. Rev. Lett. 16, 984 (1966).
12. F. J Wegner, Z. Phys. B 35, 207 (1979) and for example, K. B. Efetov, A. I.
Larkin and D. E. Khmelnitskii, JETP 52, 568 (1980); A. J. McKane and M.
Stone, Annals of Physics 131, 36 (1981); A. B. Harris and T. C. Lubensky,
Solid State Comm. 34, 343 (1980).
13. S. Hikami, Phys. Rev. B 24, 2671 (1981).
14. See the reviewof P. A. Lee andT. V. Ramakrishnan,Rev. Mod. Phys. 57,287
(1985).
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May26,2010 18:46 WorldScienti(cid:12)cReviewVolume-9.75inx6.5in contents
CONTENTS
Preface v
Chapter 1
Thoughts on Localization 1
P. W. Anderson
Chapter 2
Anderson Localization in the Seventies and Beyond 7
D. Thouless
Chapter 3
Intrinsic Electron Localization in Manganites 27
T. V. Ramakrishnan
Chapter 4
Self-Consistent Theory of Anderson Localization:
General Formalism and Applications 43
P. Wo(cid:127)l(cid:13)e and D. Vollhardt
Chapter 5
Anderson Localization and Supersymmetry 73
K. B. Efetov
Chapter 6
Anderson Transitions: Criticality, Symmetries and Topologies 107
A. D. Mirlin, F. Evers, I. V. Gornyi and P. M. Ostrovsky
Chapter 7
Scaling of von Neumann Entropy at the Anderson Transition 151
S. Chakravarty
ix