1. Ilya A. Gruzberg (University of Chicago)
Anderson transitions, critical wave functions, and conformal invariance
Ilya A. Gruzberg (University of Chicago)
Collaborators: A. Subramaniam (U of C), A. Ludwig (UCSB)
F. Evers, A. Mildenberger, A. Mirlin (Karlsruhe)
A. Furusaki , H. Obuse (RIKEN, Japan)
Papers: PRL 96, (2006); PRL 98, (2007);
PRB 75, (2007); Physica E 40, 1404 (2008);
PRL 101, (2008); PRB 78, (2008)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

2. Motivations and comments
Metal-insulator transitions (MIT) and critical behavior nearby
Anderson transitions are MIT driven by disorder: clearly posed problem
Physics and math are very far apart:
- Existence of metals: obvious to a physicist but an unsolved
problem in math
- What has been proven rigorously (existence of localized states)
was a shock to physicists, when predicted by Anderson
Critical region may be both more interesting and easier to treat rigorously
(through stochastic geometry methods: SLE and conformal restriction)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

3. Metal-insulator transitions
What do we want to find out?
- How MIT happens
- Phases on both sides
- Universal properties (exponents)
Key ingredients:
- quantum interference
- disorder
- interactions: neglect in this talk
Today: importance of boundaries, two dimensions
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

4. Metal-insulator transitions
Metals: conduct electricity
Insulators: do not conduct electricity
Sharp distinction at
a quantum phase transition
Scaling near MIT (similar to stat mech)
T. F. Rosenbaum et al. ‘80
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

5. Connection to what has been discussed: Eigenfunctions in chaotic billiards
Pictures courtesy of Arnd Bäcker
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

6. Statistics of eigenvalues and RMT
Bohigas-Gianonni-Schmidt conjecture
Same as in disordered metallic grains (nature of ensembles are different!)
All modes are extended (more or less…)
Pictures courtesy of Arnd Bäcker
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

7. Extended eigenmodes Even scarred eigenmodes are extended
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

8. Connection to what has been discussed: Eigenfunctions in fractal billiards
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

9. Eigenfunctions in fractal billiards
Some extended, some localized… Is there a transition?
Are there other types of eigenfunctions?
Pictures from B. Sapoval, Th. Gobron, A.Margolina, Phys. Rev. Lett. 67, 2974 (1991)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

10. Problem of localization
Single electron in a random potential (no interactions)
Possibility of a metal-insulator transition (MIT) driven by disorder
Ensemble of disorder realizations: statistical treatment
Classical analogues: waves in random media
More complicated variants (extra symmetries)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

11. Symmetry classification of disordered electronic systems
A. Altland, M. Zirnbauer ‘96
Integer quantum Hall
Spin-orbit metal-insulator
transition
Spin quantum Hall
Thermal quantum Hall
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

12. Weak localization Qualitative semiclassical picture
Superposition: add probability amplitudes, then square
Interference term vanishes for most pairs of paths
D. Khmelnitskii ‘82
G. Bergmann ‘84
R. P. Feynman ‘48
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

13. Weak localization Paths with self-intersections
- Probability amplitudes
- Return probability
- Enhanced backscattering
Reduction of conductivity
L. P. Gor’kov, A. I. Larkin, D. Khmelnitskii ‘79
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

14. Strong localization
P. W. Anderson ‘58
As quantum corrections may reduce conductivity to zero!
Depends on nature of states at Fermi energy:
- Extended, like plane waves
- Localized, with
- localization length
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

15. Strong localization
Nature of states depends on where they are in the spectrum
Mobility edge separates extended and localized states
Anderson transition at
- Localization length diverges
- Density of states smooth: no obvious order parameter
or broken symmetry
N. F. Mott ‘67
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

16. Anderson transitions Scaling theory of localization
E. Abrahams et al. “Gang of four” ‘79
Scaling theory of localization
all states are localized!
Anderson MIT
Sometimes happens in 2D:
- quantum Hall plateau transition
- spin-orbit interactions (symplectic class)
Modern approach: supersymmetric -model
describes metallic grains, gives RMT level statistics
K. B. Efetov ‘83
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

17. Anderson transitions
Metal-insulator transitions (MIT) induced by disorder
Quantum phase transitions with highly unconventional properties:
no spontaneous symmetry breaking!
Critical wave functions are neither localized nor truly extended,
but are complicated scale invariant multifractals
characterized by an infinite set of exponents
For most Anderson transitions no analytical results are available,
even in 2D where conformal invariance is expected to provide
powerful tools of conformal field theory (CFT)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

18. Wave functions across Anderson transition
Metal
Critical point
Insulator
Disorder or energy
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

19. Wave function across Anderson transition in 3D
R. A. Roemer
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

20. Wave function at quantum Hall transition
Insulator
Critical point
Insulator
Energy
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

21. Multifractal wave functions
Anderson transition:
Moments of the wave function intensity
- Divergent localization length
- No broken symmetry or order parameter
F. Wegner `80
C. Castellani, L. Peliti `86
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

22. Multifractals “A fractal is a set, a multifractal is a measure”
Clumpy distribution with density
Self-similarity, scaling
Characterizes a variety of complex systems: turbulence, strange
attractors, diffusion-limited aggregation, critical cluster boundaries…
Member of an ensemble
Two way to quantify: scaling of moments and singularity spectrum
B. Mandelbrot ‘74
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

23. Multifractal moments
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

24. Multifractal moments Extreme cases In general
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

25. Singularity spectrum Local view: look at the set of points where
The set is a fractal with dimension
Wave function distribution
In an ensemble may become negative
“Multifractal formalism”: Legendre transform
T. Halsey et al. ‘86
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

26. Multifractality and field theory
Anomalous exponents: In
- scaling dimensions of operators in a field theory
Infinity of operators with negative dimensions
corresponds to non-critical DOS
Connection trough Green’s functions and local DOS
F. Wegner ‘80
B. Duplantier, A. Ludwig ‘91
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

27. Boundary properties at Anderson transitions
Surface critical behavior has been extensively studied at ordinary
critical points, but not for Anderson transitions
Why are boundaries interesting?
- practical interest: leads attach to the surface
- multifractals with boundaries have not been studied before:
we find that the boundaries affect scaling properties even in
thermodynamic limit!
- in 2D boundaries provide access to elusive CFT description:
we use them to provide direct numerical evidence for presence
of conformal invariance
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

28. Boundary multifractality
A. Subramaniam et al., PRL 96, (2006)
By analogy with critical phenomena expect different scaling behavior
near boundaries
“Surface” (boundary) contribution to wave function moments
surface multifractal exponents, distinct from the bulk ones
Can calculate in and other cases
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

29. Multifractality in a finite system
A. Subramaniam et al., PRL 96, (2006)
Consider IPR for a whole sample, including bulk and surface
Does the boundary matter?
Naive answer: no, the weight of the boundary is down by
so expect
This is not correct even in the thermodynamic limit!
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

30. Boundary multifractality in (2+e)D
A. Subramaniam et al., PRL 96, (2006)
Perturbation theory gives ( )
Singularity spectra
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

31. Boundary multifractality in (2+e)D
Bulk dominates
Surface dominates
Legendre transform:
convex hull
of and
Domination of the boundary contribution due to stronger fluctuations
at the boundary
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

32. Boundary multifractality
Domination of the boundary should be a property of multifractals in general
Known case: charge distribution (harmonic measure) on a polymer
(and other SLE’s)
What about realistic Anderson transitions?
Investigate numerically for 2D MIT in the spin-orbit symmetry class
In 2D Anderson transitions should possess conformal invariance
Can test directly using boundary multifractal scaling of wave functions
E. Bettelheim et al., PRL 95, (2005)
I. Rushkin et al., J. Phys A 40, 2165 (2007)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

33. Spin-orbit metal-insulator transition
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

34. Multifractality and conformal invariance
H. Obuse et al., PRL 98, (2007)
Critical wave functions in a finite system with corners
Anomalous dimensions
surface
bulk q
Corner multifractality
corner L
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

35. Multifractality and conformal invariance
H. Obuse et al., PRL 98, (2007)
Assumption: corresponds to a primary operator
Boundary CFT relates corner and surface dimensions:
Singularity spectrum:
J. Cardy ‘84
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

36. 2D spin-orbit class h SU(2) model: 2D tight-binding model
Different geometries h
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

37. Multifractal spectra for bulk, surface, corner
for Bulk, Surface, Corner 　, whole cylinder
Multifractal spectrum depends on the region
Boundaries dominate due to stronger fluctuations at the boundary
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

38. Multifractality and conformal invariance
Relation between corner
and surface
Direct evidence for conformal invariance!
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

39. Theory proposals and conjectured multifractality for the integer quantum Hall transition
Variants of Wess-Zumino (WZ) models on a supergroup
(Zirnbauer, Tsvelik, LeClair)
Different target spaces (same Lie superalgebra )
Different proposals for the value of the “level”
A series of conjectures leads to a prediction of exactly parabolic
multifractal spectra
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

40. Actual multifractality at the IQH plateau transition
Only accessible by numerical analysis
Previous numerical study (Evers et al., 2001)
Both and
are parabolic within 1%
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

41. Actual multifractality at the IQH plateau transition
H. Obuse et al., PRL 101, (2008)
Our numerics
Fit
are not parabolic,
ruling out WZ-type theories!
Surface exponents have stronger q-dependence
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

42. Actual multifractality at the IQH plateau transition
Ratio of bulk and surface exponents
The ratio (the case for free field theories)
Our results strongly constrain any candidate theory for IQH transition
Amenable to experimental verification!
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

43. Wave functions across IQH plateau transition
Recent experiments (Morgenstern et al )
Scanning tunneling spectroscopy of 2D electron gas
Authors attempted multifractal analysis of earlier data
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

44. Experimental multifractality
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009

45. Conclusions
Scaling of critical wave functions contains a lot of useful information
Multifractal spectrum of a whole system is crucially modified by
presence of boundaries (should be property of multifractals in general)
Boundary and corner multifractal spectra provide direct numerical
evidence of conformal invariance at 2D Anderson transitions
Multifractal spectrum at the IQH plateau transition is not parabolic,
ruling out existing theoretical proposals
Stochastic geometry (conformal restriction): a new approach to
quantum Hall transitions (and other Anderson transitions)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009