IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009 Anderson transitions, critical wave Anderson transitions, critical wave functions, functions, and conformal invariance 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, 126802 (2006); PRL 98, 156802 (2007); PRB 75, 094204 (2007); Physica E 40, 1404 (2008); PRL 101, 116802 (2008); PRB 78, 245105 (2008)
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Anderson transitions, critical wave functions, and conformal invariance
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) - PowerPoint PPT Presentation
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IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
Anderson transitions, critical wave functions, Anderson transitions, critical wave functions, and conformal invarianceand 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, 126802 (2006); PRL 98, 156802 (2007);
PRB 75, 094204 (2007); Physica E 40, 1404 (2008);
PRL 101, 116802 (2008); PRB 78, 245105 (2008)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
Motivations and commentsMotivations 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
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
Anderson transitionsAnderson 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
Wave functions across Anderson transitionWave functions across Anderson transition
InsulatorCritical pointMetal
Disorder or energy
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
Wave function across Anderson transition in 3DWave function across Anderson transition in 3D
R. A. Roemer
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
Wave function at quantum Hall transitionWave function at quantum Hall transition
InsulatorCritical pointInsulator
Energy
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
• Critical wave functions in a finite system with corners
• Anomalous dimensions
Multifractality and conformal invarianceMultifractality and conformal invariance
H. Obuse et al., PRL 98, 156802 (2007)
L
surface
bulk
corner• Corner multifractality
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
• Assumption: corresponds to a primary operator
• Boundary CFT relates corner and surface dimensions:
• Singularity spectrum:
Multifractality and conformal invarianceMultifractality and conformal invariance
H. Obuse et al., PRL 98, 156802 (2007)
J. Cardy ‘84
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
• SU(2) model: 2D tight-binding model
• Different geometries
22
WWi
jijiij
iiii V
,
ccRccH 32
10
cossinsincos
sinsincoscos
ττ
ττR
ijijijij
ijijijijij
h
2D spin-orbit class2D spin-orbit class
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
• Multifractal spectrum depends on the region
• Boundaries dominate due to stronger fluctuations at the boundary
for Bulk, Surface, Corner , whole cylinder
Multifractal spectra for bulk, surface, cornerMultifractal spectra for bulk, surface, corner
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
• Direct evidence for conformal invariance!
Relation between corner and surface
Multifractality and conformal invarianceMultifractality and conformal invariance
for
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
Theory proposals and conjectured multifractality for the Theory proposals and conjectured multifractality for the integer quantum Hall transition 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
Actual multifractality at the IQH plateau transitionActual 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
Actual multifractality at the IQH plateau transitionActual multifractality at the IQH plateau transition
• Our numerics
• Fit
• are not parabolic, ruling out WZ-type theories!
• Surface exponents have stronger q-dependence
H. Obuse et al., PRL 101, 116802 (2008)
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009
Actual multifractality at the IQH plateau transitionActual 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
Wave functions across IQH plateau transitionWave functions across IQH plateau transition
• Recent experiments (Morgenstern et al. 2003-2008)
• Scanning tunneling spectroscopy of 2D electron gas
• Authors attempted multifractal analysis of earlier data
IPAM workshop “Laplacian Eigenvalues and Eigenfunctions”, February 13, 2009