# Workshop Programme

## for period 2 - 6 November 2008

### Delocalization Transition and Multifractality

2 - 6 November 2008

Timetable

Sunday 2 November | ||||

12:30-13:30 | Lunch | |||

15:20-16:10 | Mirlin, A (Karlsruhe) |
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Anderson transitions: recent developments | ||||

Recent developments in the physics of Anderson transitions between localized and metallic phases in disordered systems are reviewed [1]. The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states. A particular emphasis is put on multifractality of critical wave functions. Also discussed are criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of delocalization and criticality in low-dimensional systems, and random Dirac Hamiltonians. |
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16:10-16:40 | Tea and Posters | |||

16:40-17:30 | Gruzberg, I (Chicago) |
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Critical wave functions, conformal invariance, and theories for quantum Hall transitions | ||||

Wave functions at Anderson localization-delocalization (LD) transitions are known to exhibit complicated scale-invariant behavior best characterized by an infinite set of multifractal exponents. I will present our recent study of multifractal spectra of critical wave functions at various Anderson transitions, focusing on finite systems with boundaries. In two dimensions the boundary behavior of the critical wave functions provides a way to answer the question of conformal invariance at these transitions. For the integer quantum Hall transition the multifractal spectra were conjectured to be exactly parabolic in a number of proposals of critical field theories for the transition. Our numerical results for the Chalker-Coddington network model firmly rule out the exact parabolicity. In addition, we provide an exact result for surface multifractal exponents for a related LD critical point in the BDI (chiral orthogonal) class. |
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17:30-19:00 | Discussions | |||

19:00-20:00 | Dinner |

Monday 3 November | ||||

09:00-09:50 | Feigelman, MV (Russian Academy of Sciences) |
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Fractal superconductivity near localization threshold | ||||

We develop a theory of a pseudogap state appearing near the superconductor-insulator transition in strongly disordered metals with attractive interaction. We show that such an interaction combined with the fractal nature of the single particle wave functions near the mobility edge leads to an anomalously large single particle gap in the superconducting state near SI transition that persists and even increases in the insulating state long after the superconductivity is destroyed. We give analytic expressions for the value of the pseudogap in terms of the inverse participation ratio of the corresponding localization problem. |
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09:50-10:40 | Kravtsov, VE (Abdus Salam International Centre) |
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Multifractality of single-particle wave functions and superconductivity near the Anderson transition | ||||

10:40-11:10 | Coffee and Posters | |||

11:10-12:00 | Furusaki, A (RIKEN) |
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Boundary multifractality and conformal invariance at 2D metal-insulator transition | ||||

Critical wave functions near sample boundaries exhibit multifractal scaling behavior which is different from multifractality of wave functions in the bulk of the sample. In this talk I will give an overview of our recent studies on the consequence of conformal invariance in the boundary multifractality: (a) angle dependence of corner multifractal exponents and (b) relation between \alpha_0 and normalized localization length in quasi-1D geometry. These results are illustrated for the symplectic class and the integer quantum Hall plateau transition. |
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12:00-12:50 | Evers, F (Forschungszentrum Karlsruhe) |
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Multifractality in model systems for quantum Hall critical points | ||||

The talk gives an overview about recent developments in the classification of quantum Hall critical points in terms of multifractal spectra. We will begin by reviewing results for multifractality at the integer quantum Hall transition. Thereafter,we shall address quantum Hall physics in superconducting systems of the symmetry class D. Our focus will be on the fermionic representation of the random bond Ising model, which belongs to this symmetry class. We will also discuss the Cho-Fisher model, a close relative of this system, where an unusual scaling of the distribution function of the inverse participation ratio may indicate the presence of strong disorder physics. |
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12:50-14:30 | Lunch and Discussion | |||

14:30-15:20 | Mudry, CM (Paul Scherrer Institute) |
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Quantum transport of 2D Dirac fermions: The case for a topological metal | ||||

The problem of Anderson localization in graphene has generated a lot of renewed attention since graphene flakes have been accessible to transport and spectroscopic probes. The popularity of graphene derives from it realizing planar Dirac fermions. I will show under what conditions disorder for planar Dirac fermions does not result in localization but rather in a metallic state that might be called a topological metal. |
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15:20-16:10 | Obuse, H (Kyoto) |
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Multifractality at the metal-quantum spin Hall insulator transition in two dimensions | ||||

The quantum spin-Hall (QSH) effect occurs in a new kind of a topological insulator characterized by the $Z_2$ topological number. Since QSH system possesses time-reversal symmetry and broken spin-rotation symmetry, this system is naively expected to belong to the symplectic class if taking account only of symmetries but ignoring the topological nature. We investigate various critical properties at the Anderson transition of the QSH system in two-dimensions (2D) based on the network model. In Ref. [1], we show that the critical exponent characterizing the divergence of the localization length at criticality is identified with that of the ordinary symplectic class in 2D. We also investigate bulk and boundary multifractality in the QSH system [2]. When reflecting boundaries are imposed on the QSH network model, there exist two kinds of critical points depending on whether a boundary induces edge states in the adjacent insulating phase. We found that bulk multifractality in the QSH system are same as that of the ordinary symplectic class in 2D. It is also clarified that boundary multifractality at the critical point of the metal - ordinary insulator (absence of edge states) transition is same as that of the ordinary symplectic class, while boundary multifractality at the critical point of the metal-QSH insulator (presence of edge states) transition is completely different from that of the ordinary symplectic class. Therefore, boundary multifractality observed at the latter critical point is considered as a new boundary critical phenomenon in the symplectic class, reflecting the presence of topologically non-trivial edge states in the adjacent insulating phase. This is the first example for the presence of different boundary multifractality in the same universality class. [1] HO, A. Furusaki, S. Ryu, and C. Mudry, PRB 76, 075301 (2007) [2] HO, A. Furusaki, S. Ryu, and C. Mudry, PRB 78, 115301 (2008) |
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16:10-16:40 | Tea and Posters | |||

16:40-17:30 | Moerters, P (Bath) |
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Minimal supporting subtrees for the free energy of polymers on disordered trees | ||||

We consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this simple model undergoes a phase transition with respect to its localization behaviour. We show that, for high temperatures, the free energy is supported by a random tree of positive exponential growth rate, which is strictly smaller than that of the full tree. The growth rate of the minimal supporting subtree is decreasing to zero as the temperature decreases to the critical value. At the critical value and all lower temperatures, a single polymer suffices to support the free energy. Our proofs rely on elegant martingale methods from the theory of branching random walks. Related Links http://people.bath.ac.uk/maspm/ - First author's homepage http://people.bath.ac.uk/ma2mo/ - Second author's homepage |
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17:30-19:00 | Discussions | |||

19:00-20:00 | Dinner |

Tuesday 4 November | ||||

09:00-09:50 | Chalker, J (Oxford) |
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Class C network models, and quantum and classical delocalisation transitions. | ||||

We study the disorder-induced localisation transition in network models that belong to symmetry class C. The model represents quasiparticle dynamics in a gapless spin-singlet superconductor without time-reversal invariance. It is a special feature of network models with this symmetry that the conductance and density of states can be expressed as averages in a classical system of dense, interacting random walks. For a two-dimensional system the random walks are hulls of percolation clusters and their properties are known exactly. For multilayer and three-dimensional systems there are no exact results but the mapping provides a very efficient starting point for simulations. In particular, we present a more precise numerical study of critical behaviour at an Anderson transition than has been possible previously in any context. |
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09:50-10:40 | Kagalovsky, V (Sami Shamoon College of Engineering) |
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Chalker-Coddington network model and its applications to various quantum Hall systems | ||||

We start by detailed description of the original Cahlker Coddington network model and briefly discuss its various generalizations. We then study a physical system consisting of noninteracting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation invariance. This system belongs to class D within the recent classification scheme of random matrix ensembles, and its phase diagram contains three different phases: metallic and two distinct localized phases with different quantized thermal Hall conductances. We find that critical exponents describing different transitions (insulator-to-insulator and insulator-to-metal) are identical within the error of numerical calculations. Finally, we discuss localization-delocalization transition in quantum Hall systems with a random field of nuclear spins acting on two-dimensional electron spins via hyperfine contact Fermi interaction. The inhomogeneous nuclear polarization acts on the electrons as an additional confining potential and, therefore, introduces additional parameter p - the probability to find a polarized nucleus in the vicinity of a saddle point of random potential responsible for the change from quantum to classical behavior. In this manner we obtain two critical exponents corresponding to quantum and classical percolations. |
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10:40-11:10 | Coffee and Posters | |||

11:10-12:00 | Avishai, Y (Ben-Gurion) |
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Tight binding spectrum of electrons on the sphere, subject to the field of: I. Magnetic charge, II. Electric charge. A surprising relation between I. and II | ||||

Tight binding quantum mechanical spectrum of an electron hopping on spherical graph and subject to radial magnetic or electric fields is elucidated ( think for example of an electron hopping on the atoms of the Fullerene). In the first part, calculation of the spectrum in the presence of central magnetic charge g is studied. It should take into account the fact that g is quantized as g=(hc/2e)n where n is the monopole number. This restriction requires a meticulous determination of the phase factors appearing in the hopping matrix elements. Having solved this mathematical problem, the spectrum of the symmetric polytopes (tetrahedron, cube, octahedron, dodecahedron and icosahedron) is calculated analytically and shown to display a beautiful pattern, which is entirely distinct from that of the Hofstadter butterfly. In the second part, the spectrum in the central field of an electric charge Q is calculated in the second part. The radial electric field induces Rashba type spin-orbit interaction on the hopping electron. These are constructed leading to the tight-binding form of the familiar atomic L.S interaction. The spectra of the five symmetric polytopes are calculated analytically as function of Q and display rich and beautiful patterns with some unexpected symmetries. Finally, we expose a remarkable relation between the two seemingly distinct physical problems: The spectrum of the second system (electron in the field of central electric charge inducing spin-orbit interaction) is found to be identical with the spectrum of the first system (electron in the field of magnetic monopole) at n = 1. This means that it is principally possible to test the spectrum of an experimentally inaccessible system (magnetic monopole) in terms of an experimentally accessible one (electron subject to spin-orbit force induced by central electric charge ). |
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12:00-12:50 | Slevin, K (Osaka) |
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Numerical estimates of critical exponents of the Anderson transition | ||||

We will report numerical estimates of the critical exponents and other universal quantities for the Anderson transition in the Anderson model with various symmetries and in the Chalker-Coddington model. We will briefly explain the method used to take account of corrections to scaling. This method works well for the Anderson type models where the corrections to scaling decrease rapidly. However, successful analysis of the Chalker-Coddington model seems to require the inclusion of corrections to scaling that converge very slowly. |
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12:50-14:30 | Lunch and Discussion | |||

14:30-15:20 | Ohtsuki, T (Sophia) |
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Critical conductance distributions at the localization-delocalization transitions | ||||

Due to mesoscopic fluctuations, not the conductance but the distribution of the conductance becomes importantto study the quantum transport phenomena. We review and report the distributions of conductance at various types of localization-delocalization transitions, such as the Anderson transitions in three dimensions, the quantum Hall transition, and the quantum spin Hall transition. Reflecting the multifractal nature of the wave functions at the localization-delocalization transitions, the distributions take peculiar form, which is scale and model independent. We also examine the conductance distributions in the presence of perfectly conducting channels. Authors: Tomi Ohtsuki, Keith Slevin and Koji Kobayashi |
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15:20-16:10 | tba | |||

16:10-16:40 | Tea and Posters | |||

16:40-17:30 | Aoki, H (Tokyo) |
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Quantum hall effect and real-space wavefunctions --- multifractality and edge states in 2DEG and lattice systems | ||||

17:30-19:00 | Discussions | |||

19:00-20:00 | Dinner |

Wednesday 5 November | ||||

09:00-09:50 | Le Doussal, P (ENS) |
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Some applications of freezing transitions | ||||

09:50-10:40 | Fyodorov, Y (Nottingham) |
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Random potentials with logarithmic correlations: multifractality of Boltzmann-Gibbs weights and extreme value statistics | ||||

10:40-11:10 | Coffee and Posters | |||

11:10-12:00 | Vargas, V (CNRS) |
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Stochastic scale invariance and KPZ equation | ||||

In this talk, we will prove the KPZ equation (initially introduced in the framework of quantum gravity in 2 dimensions) for a class of multifractal log infinitely divisible measures (defined in all dimensions). More specifically, for a given set K, we will relate it's Hausdorff dimension under the Euclidian metric to it's Hausdorff dimension under the random metric induced by the multifractal measure. We will see how the notion of stochastic scale invariance is crucial in the proof of the aforementioned relation. |
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12:00-12:50 | Yevtushenko, O (Ludwig-Maximilians-Universität München) |
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Delocalization transition in unconventional random matrices | ||||

Critical Power-Law Banded Random Matrices (CRMT) are a powerful toy model which allows one to study universal features of the Anderson transition. Wave functions of CRMT are multifractal and the fractal dimensions are controlled by the matrix band- width. The \sigma-model can be successfully applied to explore CRMT with a large band-width (a weak multifractality regime). Recently, an alternative method of the virial expansion (VE) has been suggested to describe the opposite case of RMT with the small band-width. VE is an expansion in the number of localized states weakly interacting in the energy space due to the matrix off-diagonal elements. VE answers the question whether such weak interaction can lead to the criticality (in the strong multifractality regime) and to the delocalization. We briefly review VE, and use it to study spectral properties and statistics of wave-functions of CRMT. Analytical results will be compared with numerical ones. |
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12:50-14:30 | Lunch and Discussion | |||

14:30-15:20 | Mucciolo, E (Central Florida) |
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Universal vs. nonuniversal scaling of the conductivity in disordered graphene | ||||

In this talk I will review some key questions related to the conductivity of disordered graphene sheets near the neutrality point, with emphasis on possible critical behavior. Results of numerical simulations performed by our group and others will be presented and discussed. |
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15:20-16:10 | Markos, P (Slovak Academy of Sciences) |
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Universality of Anderson transition in disordered systems | ||||

We review our recent numerical results for the critical regime of Anderson transition for various disordered models. The universality of the metal-insulator transition in a three-dimensional Anderson model is confirmed by the numerical analysis of the scaling properties of the electronic wave functions. We prove that the critical exponent and the multifractal dimensions are independent on the microscopic definition of the disorder and universal along the critical line which separates the metallic and the insulating regime [1]. In the integer quantum Hall regime, we calculated the sample averaged longitudinal two-terminal conductance and the respective Kubo-conductivity. In the limit of large system size, both transport quantities are found to be the same within numerical uncertainty in the lowest Landau band, 0.60 +/- 0.02 and 0.58 +/- 0.03, respectively (in units of e^2/h). In the 2nd lowest Landau band, a critical conductance 0.61 +/- 0.03 is obtained which indeed supports the notion of universality. We argue that these values are consistent with the multifractal structure of critical wave functions [2]. For the symplectic two dimensional Ando model we calculate the critical two-terminal conductance and the spatial fluctuations of critical eigenstates. For square samples, we verify numerically the relation between critical conductivity and the fractal information dimension of the electron wave function. Through a detailed numerical scaling analysis of the two-terminal conductance we also estimate the critical exponent = 2.80 +/- 0.04 [3]. We study the localization properties and the two-terminal conductance of two-dimensional lattice systems with static random magnetic flux per plaquette and zero mean (systems with chiral symmetry). The influence of boundary conditions and of the oddness of the number of sites in the transverse direction are also studied. Our data are in perfect agreement with previous theoretical predictions. We also find a diverging localization length in the middle of the energy band and determine its critical exponent = 0.35 +/- 0.03. [4] [1] J. Brndiar and P. Markos, Phys. Rev. B 74, 153103 (2006); 77, 115131 (2008) [2] L. Schweitzer and P. Markos, Phys. Rev. Lett. 95, 256805 (2005) [3] P. Markos and L. Schweitzer, J. Phys. A 39, 3221 (2006) [4] P. Markos and L. Schweitzer, Phys. Rev. B 76, 115318 (2007) |
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16:10-16:40 | Tea and Posters | |||

16:40-17:30 | Ossipov, A (Nottingham) |
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Multifractality in delay times statistics | ||||

We reveal an exact relation between the statistics of delay times and statistics of the eigenfunction fluctuations. This relation allows us to derive explicit expressions for delay times distributions in various regimes. In particular, we show, how the eigenfunction multifractality reflects itself in the distribution of delay times. |
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17:30-19:00 | Discussions | |||

19:00-20:00 | Dinner |

Thursday 6 November | ||||

09:00-09:50 | Yudson, V (Russian Academy of Sciences) |
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One-dimensional Anderson model: devil's staircase of statistical anomalies | ||||

We consider one-dimensional Anderson model of localization on a lattice focusing attention at the probability distribution of the normalized random eigenfunctions. We derive the general formula for the joint probability distribution of the eigenfunction amplitude and phase in the bulk of a long chain in terms of the generation function which obeys the Fokker-Plank like (transfer-matrix) equation. This equation is shown to have anomalous terms at any energy that corresponds to the rational filling factor f (fraction of the states below this energy). At weak disorder the principle anomaly corresponds to f=1/2. The transfer matrix equation in this case is derived and exactly integrated. arXiv:0806.2118v1 |
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09:50-10:40 | Roemer, R (Warwick) |
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Multifractal analysis of the metal-insulator transition in the 3D Anderson model | ||||

We study the multifractal analysis (MFA) of electronic wavefunctions at the localisation-delocalisation transition in the 3D Anderson model for very large system sizes up to 240^3. The singularity spectrum f(alpha) is numerically obtained using the ensemble and the typical average of the scaling law for the generalized inverse participation ratios $P_q$, employing box-size and system-size scaling. The validity of a recently reported symmetry law [Phys. Rev. Lett. 97, 046803 (2006)] for the multifractal spectrum is carefully analysed at the metal-insulator transition (MIT). The results are compared to those obtained using different approaches, in particular the box- and system-size scaling approaches. System-size scaling with ensemble average appears as the most adequate method to carry out the numerical MFA. |
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10:40-11:10 | Coffee and Posters | |||

11:10-12:00 | Varga, I (Budapest Univ of Tech and Economics) |
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Power-law banded random matrices: a testing ground for the Anderson transition | ||||

In this talk we present numerical simulations using the Power-law band random matrix (PRBM) ensemble which became a useful testing tool for investigation of the bahavior of random systems at the Anderson transition. The effect of multifractality of the eigenstates in several physical situations are investigated, e.g. interaction at the Hartree-Fock level, entanglement, dynamics of wavepackets, magnetic impurities, etc. |
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12:30-13:30 | Lunch |