2144-5 workshop on localization phenomena in...
TRANSCRIPT
![Page 1: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/1.jpg)
2144-5
Workshop on Localization Phenomena in Novel Phases of Condensed Matter
ALTLAND Alexander
17 - 23 May 2010
Universitaet zu Koeln, Institut fuer Theoretische Physik Zulpicher Strasse 77, D-50937 Cologne
GERMANY
Localization Phenomena in the Quantum Kicked Rotor
![Page 2: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/2.jpg)
Chirikov, 79
![Page 3: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/3.jpg)
QKR is:
1. a paradigmatic model system of quantum chaos
2. experimentally realizable
3. a system of rich phenomenology
![Page 4: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/4.jpg)
![Page 5: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/5.jpg)
QKR is:
1. a paradigmatic model system of quantum chaos
2. experimentally realizable
3. a system of rich phenomenology
![Page 6: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/6.jpg)
field theory approach to the quantum kicked rotor
Alexander Altland & Chushun Tian, Cologne University
kicked rotor
and its field theory
Trieste May 18th, 2010
arXiv:1001.4322, NJP 12 43043 (2010)
![Page 7: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/7.jpg)
kicked rotor
![Page 8: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/8.jpg)
standard map
classical system characterized by a single parameter .
quantum system sensitive to value of .
Chirikov, 79
here:
(semiclassical dynamics)
(chaotic dynamics)
![Page 9: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/9.jpg)
Fishman et al. 82
standard map: analogies to disordered systems
standard map
Rechester-White (1980) corrections ( corrections to diffusion constant)
accelerator modes, regular islands at special values of (Karney, 1983)
-classics, reentrant classical behaviour at (Fishman, 2004)
-resonances, absence of localization at (Casati et al., 1979)
chaotic scattering
correlations in angular momentum space
diffusion
localization
impurity scattering
velocity correlations
diffusion
localization
disordered quantum systems Fishman et al. 82standard map
Rechester-White (1980) corrections coorrections to diffusion constant) (
accelerator modes, regular islands at special values of (Karney, 1983)
-classics, reentrant classical behaviour at (Fishman, 2004)
-resonances (Casati et al., 1979), absence of localization at
chaotic scattering
correlations in angular momentum space
diffusion
localization
impurity scattering
velocity correlations
diffusion
localization
disordered quantum systems
-resonances, absence of localization at (Casati et al., 1979)-resonances, absence of localization at (Casati et al., 1979)
diffusion
localization
![Page 10: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/10.jpg)
energy diffusion
Describe system in terms of (angular momentum) correlation function
Floquet operator:
and (kinetic) energy increase
time in units of kick time
![Page 11: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/11.jpg)
localization
Expected behavior of energy increase
: diffusion constant
: localization length (Chirikov et al. 81)
Moore et al. 95
![Page 12: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/12.jpg)
QKR resonances
For different things happen (Izrailev & Shepelyansky, 81)
Floquet operator:
At resonant values: : translation operator
![Page 13: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/13.jpg)
reminder: Bloch picture of periodic quantum systems
![Page 14: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/14.jpg)
QKR resonances
For different things happen (Izrailev & Shepelyansky, 81)
Floquet operator:
At resonant values: : translation operator
Consequences (phenomenological):
energy expectation value:
short times, : diffusion,
large times, : single level parametric correlations,
![Page 15: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/15.jpg)
Large scale picture (Izrailev & Shepelyansky, 80)
Numerics:
QKR resonances (cont’d)
![Page 16: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/16.jpg)
field theory
![Page 17: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/17.jpg)
1st go
prototypical field theory of the rotor, Zirnbauer, aa 96
������������ ��������������
![Page 18: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/18.jpg)
Field integral representation
Zirnbauer, aa 96:
![Page 19: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/19.jpg)
Effective action
soft modes: fluctuations whose action can be continuously tuned to zero. For generic irrational values of
all other field configurations ( ) are massive (a linear decomposition )
![Page 20: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/20.jpg)
Effective low energy action
Integration over gapped modes produces
Bloch angle
nonlinear sigma model action on a ring subject to (Bloch gauge) flux
similar to effective action of disordered multichannel quantum wire
action encapsulates symmetries/system properties of the rotor
(Shepelyansky 87)
epsilon-classics symmetry (Fishman, 03)
resonances
![Page 21: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/21.jpg)
off resonance
Localization theory (Efetov & Larkin, 83) translated to the present context (Zirnbauer, a.a., 96):
![Page 22: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/22.jpg)
on resonance
Action
describes parametric correlations in mesoscopic systems (Simons & Altshuler, 93, Taniguchi & Altshuler, 93)
Result:
cf. Wojcik & Dorfman, 84
![Page 23: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/23.jpg)
discussion
the present theory:
analytically describes crossover ballistic motion/diffusion/localization
knows about QKR ‘fine structure’ (Shepelyansky/Rechester White corrections to diffusion constant,
epsilon classics, resonances)
provides analytical results for correlation functions on and off resonance
for specific parameter constellations (accelerator modes, certain arithmetic values of Planck’s
constant), the construction of the theory in its present form breaks down
![Page 24: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/24.jpg)
general ramifications
![Page 25: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/25.jpg)
������������ � � ��������������������
��
����
������
����
��
���
��
relation ?
![Page 26: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/26.jpg)
� ���������� �������������
� �� ������� ���
����������������� ��� �������� ���� �� ������ ����
�� ���� ��� ����������� ����������� � �
!��"��� �����#� ����$
�� ���� ��� ����� �������
���� �����
������������� ����� �������� ������������
semiclassics vs quasiclassics
![Page 27: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/27.jpg)
quasiclassical approach to chaos
field theory of standard map (Zirnbauer, aa, 96; Tian, aa, 10)
universal spectral correlations in quantum graphs (Gnutzmann, aa, 05)
strong Anderson localization in arrays of chaotic cavities (Brouwer, aa, 08)
spectral gap of the Andreev billiard (Micklitz, aa, 09)
universal spectral correlations from ballistic sigma model (Micklitz, Müller, aa, 06)
In cases of overlap, full agreement with semiclassics; non-perturbative applications
Applications:
Conceptual status:
soon after introduction (Altshuler et al., 95; Khemlnitskii & Muzykhantskii, 95): conceptual problems
(‘weak localization problem’, ‘mode locking problem’, ‘zero mode problem’, ‘repetition problem’, ...)
... these are basically solved (personal opinion)
![Page 28: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/28.jpg)
Effective action cont’d
quadratic integration over massive modes:
projector on massive modes
explicit representation
Shepelyansky (1987)
![Page 29: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/29.jpg)
Effective action (cont’d)
generalization beyond quadratic order
discussion:
localization at length scales
diffusion at length scales
at rational values : system effectively acquires ring topology. Circumference ; (nonabelian)
gauge flux.
theory sensitive to -classics, reentrant classical behaviour at
effective low energy theory of disordered multi-channel quantum wire (Efetov & Larkin, 80)
![Page 30: 2144-5 Workshop on Localization Phenomena in …indico.ictp.it/event/a09147/material/7/8.pdfquasiclassical approach to chaos field theory of standard map (Zirnbauer, aa, 96; Tian,](https://reader035.vdocuments.site/reader035/viewer/2022071107/5fe1b5154802e86c643c3f9c/html5/thumbnails/30.jpg)
summary
quasiclassical methods can be powerful alternative to semiclassics
often implemented in field theoretical framework
good in non-perturbative settings (localization, gap formation, non-perturbative correlations)
and not-so-good in others (repetitive dynamics, low order perturbation theory)
It is good to know both, semiclassical and quasiclassical concepts
time