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Recent progress on critical properties at Anderson transitions

Recent progress on critical properties at Anderson transitions. Ilya A. Gruzberg Ohio State University Joint work with S. Bera , F. Evers, D. Hernangómez -Pérez, A. Mirlin (also earlier work with A. Ludwig, N. Read, and M. Zirnbauer ). TexPoint fonts used in EMF.

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Recent progress on critical properties at Anderson transitions

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  1. Recent progress on critical properties at Anderson transitions Ilya A. Gruzberg Ohio State University Joint work with S. Bera, F. Evers, D. Hernangómez-Pérez, A. Mirlin (also earlier work with A. Ludwig, N. Read, and M. Zirnbauer) Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

  2. Anderson localization P. W. Anderson 1958 • Single electron in a random potential • Multiple scattering and quantum • interference may localize electrons • Nature of states depends on their • energy and spatial dimension • Anderson transition: a continuous quantum metal-insulator transition • Critical phenomena in transport and other observables • We focus on the critical wave functions Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  3. Wave functions across Anderson transitions • Spin-orbit metal-insulator transition in 2D Critical point Insulator Metal Disorder or energy H. Obuse Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  4. Wave function at the quantum Hall transition Critical point Insulator Insulator Energy F. Evers Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  5. AZ symmetry classification A. Altland, M. Zirnbauer ‘96 • Integer quantum Hall • Spin-orbit metal-insulator • transition • Spin quantum Hall • Thermal quantum Hall Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  6. Anderson localization: field theory • Field theory: a supersymmetric -model • Matrix field in a (super)coset space different for each AZ class: high degree of symmetry of the target space • Also possible topological or Wess-Zumino terms • Anderson transitions are typically strongly-coupled fixed points, • though perturbative results are available in • For ATs in 2D, the conformal symmetry is expected at critical points(assumption), leading to CFT with • We use field theory to focus on scaling of critical wave functions Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  7. Critical wave functions are multifractals F. Wegner `80 C. Castellaniand L. Peliti `86 • Critical wave functions are neither localized nor truly extended • characterized by an infinite set of multifractal (MF) exponents • Moments of the wave function • Scaling with the system size • Broad probability distribution of critical wave function intensity Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  8. Multifractal measures B. Mandelbrot ‘74 • Clumpy self-similar distribution exhibiting multiple scaling laws • Characterizes a variety of complex systems: turbulence, strange • attractors, diffusion-limited aggregation, critical cluster boundaries, … • Critical wave functions at AT: random measures from an ensemble Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  9. Multifractalmoments and spectrum • Probability measure with support in a cube of size • Divide the cube into boxes of size , • Measure of each box • (Complex) moments of the measure scale with • Multifractal spectrum Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  10. General properties of multifractal spectra • From the definition of it follows that for real • is non-decreasing: • is convex: • (dimension of the support) • (normalization of the measure) Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  11. Multifractal spectra: metal, insulator, and critical • Extreme cases • Extended wave function: uniform measure • “MF” spectrum is linear: • Wave function localized in volume • boxes filled with • is independent of • Critical wave function: anomalous scaling Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  12. Solvable case: Dirac fermion in a random gauge field • Dirac Hamiltonian in 2D • Hodge decomposition of gauge field • Exact zero energy wave functions • Disorder • is a free massless boson with • is a Liouville measure Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  13. Dirac fermion in a random gauge field: MF spectrum • Ludwig et al. `94 • Weak disorder : parabolic spectrum with termination point • Freezing transition at • Strong disorder : parabolic spectrum with freezing transition • Similar results for non-Abelian random gauge fields (WZW models) • How generic is the parabolic shape of the MF spectrum? • Chamon et al.`96; Castillo et al. `97 • Carpentier and LeDoussal `01 Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  14. Multifractality and field theory • Moments of critical wave functions • Moments of local DOS at the critical energy • are dimensions of operators in a field theory • Roughly speaking, • Precise correspondence exists within the sigma models • Multifractal spectrum Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  15. Multifractality and field theory • Dimensions are negative for sufficiently large • While natural for broad probability distributions of , • negative dimension are inconsistent with a unitary field theory • In 2D expect a non-unitary CFT with • Perturbative sigma-model results in are almost parabolic • Notice the symmetry • The symmetry turns out to a be general feature of all MF spectra Chamon et al.`96; Mudry et al. `96 Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  16. Exact results for MF spectra A. Mirlin,Y. Fyodorov, A. Mildenberger, F. Evers, `07 IAG, A. Ludwig, A. Mirlin, M. Zirnbauer `11 • There is such that • Existence of non-trivial with is inconsistent with a unitary field theory • Exact symmetry of MF spectra • This follows from the global conformal symmetry as well as group symmetries of the target spaces of sigma models • Symmetry points for all ten AZ classes Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  17. Exact results for MF spectra IAG, A. Ludwig, N. Read `99 • Class C (spin quantum Hall transition): mapping to classical percolation • Exact MF exponents • These four values lie on the parabola • What about other values of ? IAG, A. Ludwig, N. Read `99 A. Mirlin, F. Evers, A. Mildenberger `03 Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  18. Numerical MFspectra H. Obuse et al `08 IQH A. Mirlin et al `06 PRBM H. Obuse et al `07 A. Mirlin et al `03 SQH 2D SO • In all cases see deviations from parabolicity Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  19. Recent developments and conjectures • The relation suggests the Abelian fusion • In 2D, the assumptions • CFT with holomorphic factorization • are spinlessVirasoro primaries • Abelian OPE (single fusion channel) • lead to exactly parabolic MF spectra: • For IQH the Abelian fusion follows from supersymmetry in a lattice model R. Bondesan, D.Wieczorek, M. Zirnbauer `16 Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  20. Recent developments and conjectures M. Zirnbauer `18 • Proposal for a CFT of the IQH transition: a deformation of the • WZW model by a marginal perturbation • Includes a sector with a free boson, and predicts • Many problems: unconventional critical behavior, inconsistency with numerical simulations • Worth checking numerically for other symmetry classes H. Obuse et al `08 Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  21. Plan of action S. Bera, F. Evers, IAG, D. Hernangómez-Pérez, A. Mirlin `18 • MF analysis for class A (IQH) is difficult due to notorious finite-size • corrections to scaling from irrelevant operators • Consider a model of SQH transition in class C • Several advantages: • Corrections to scaling are small • Exact results are available and can be used as a control • Compute MF spectra • Check the symmetry relation as an additional control • Check the parabolicity Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  22. Results: symmetry relation • The ratio • approaches a constant as • in a wide interval around • Confirms the symmetry Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  23. Results: non-parabolicity • Substantial deviations from parabolicity! Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

  24. Conclusions and outlook • MF spectrum for the SQH transition in class C is not parabolic! • One or more assumptions (conformal invariance, Abelian fusion)fail • More likely, the Abelian fusion is not satisfied • Further numerical checks: • Boundary multifractality • Other symmetry classes, including IQH in Class A • Highly non-trivial CFTs for 2D Anderson transitions • Solving a network model for the IQH transition by mapping to a classical statistical model E. Bettelheim, IAG, A. W. W. Ludwig `12 E. Bettelheim, IAG, in progress Euler Symposium on Theoretical and Mathematical Physics, St. Petersburg, June 15, 2019

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