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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)

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Anderson transitions, critical wave functions, and conformal invariance

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  1. 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, 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

  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 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 Critical point Insulator Metal 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 Critical point Insulator 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, 126802 (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, 126802 (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, 126802 (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, 170602 (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 • 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, 156802 (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, 156802 (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 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. for Relation between corner and surface Multifractality and conformal invariance • 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, 116802 (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. 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

  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

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