Anderson localisation transition: introductory training course

Workshop theme

When a single-particle quantum Hamiltonian system is subjected to a disorder potential, it is expected on physical grounds that a transition from localised to extended energy eigenstates takes place as a function of the disorder strength. Such a transition should be accompanied by a characteristic change in the energy spectrum: if the disorder is large enough for Anderson localisation to occur, the random Schrödinger operator is known to have dense point spectrum; on the other hand, if the disorder is weak and the space dimension larger than d = 2, then one expects the existence of absolutely continuous spectrum.

Giving a mathematical proof of this conjectured scenario, and clarifying the nature of the spectrum and the eigenfunctions at the transition point or in d = 2, remains an important and outstanding problem of mathematical physics. Many features of the scenario are believed to extend to a broader class of quantum systems including, most prominently, those exhibiting transitions of Quantum Hall type.

This training course is mainly directed at researchers in early stages of their careers. Its aim is to provide the participants with an introduction to the subject, by exposing them to ideas, terminology and analytical techniques of the rigorous as well as the heuristic kind. Methods used in the study of Anderson localisation by mathematicians and by theoretical physicists will be reviewed by experts from both communities. Reviewing the state of the art for both disciplines will hopefully help to bridge the existing language gap between the communities and create an environment conducive to fruitful collaboration between physicists and mathematicians during the rest of the program.

Tentative list of topics to be covered

phenomenology of Anderson localisation (T. Spencer)
introduction to the spectral theory of random Schrödinger operators (L. Pastur)
introduction to supermatrix techniques and the nonlinear σ-model (Y. Fyodorov)
rigorous techniques for 1D and quasi 1D systems (I. Goldsheid)
rigorous methods in the statistical mechanics of phase transitions (D. Brydges)
critical phenomena in two-dimensional disordered systems (A. Ludwig)

Applications for this workshop are now closed

The Institute kindly requests that any papers published as a result of this programme’s activities are credited as such. Please acknowledge the support of the Institute in your paper using the following text:

The author(s) would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Anderson localisation transition: introductory training course, where work on this paper was undertaken. This work was supported by EPSRC grant EP/F005431/1.

Timetable

Monday 14th July 2008
08:30 to 09:25	Registration	Room 1
09:25 to 09:30	David Wallace - welcome	Room 1
09:30 to 10:30	Introduction to spectral theory (1)	Room 1
10:30 to 11:00	Coffee	Room 1
11:00 to 12:30	Yan Fyodorov Queen Mary University of London Introduction to supermatrix techniques (1)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Introduction to spectral theory (2)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 16:45	Colin Benjamin University of Leeds Effect of disorder on positive shot noise cross-correlations in superconducting hybrids	Room 1
17:30 to 18:10	Welcome Wine Reception	No Room Required
18:15 to 19:00	Dinner at Churchill College	No Room Required

Tuesday 15th July 2008
09:00 to 10:30	Introduction to spectral theory (3)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	David Brydges University of British Columbia Statistical mechanics of phase transitions (1)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Introduction to spectral theory (4)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 16:45	Brian Winn Loughborough University Quasimodes of Seba billiards	Room 1
18:15 to 19:00	Dinner at Churchill College	No Room Required

Wednesday 16th July 2008
09:00 to 10:30	Introduction to spectral theory (5)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	Yan Fyodorov Queen Mary University of London Introduction to supermatrix techniques (2)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Rigorous techniques for 1D systems (1)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 17:30	Introduction to spectral theory (6)	Room 1
18:15 to 19:00	Dinner at Churchill College	No Room Required

Thursday 17th July 2008
09:00 to 10:30	David Brydges University of British Columbia Statistical mechanics of phase transitions (2)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	Yan Fyodorov Queen Mary University of London Introduction to supermatrix techniques (3)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Margherita Disertori Université de Rouen GUE density of states by SUSY	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 18:00	Poster session	Discussion Room
18:15 to 19:00	Dinner at Churchill College	No Room Required

Friday 18th July 2008
09:00 to 10:30	Anderson localisation: phenomenology and mathematics (1)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	David Brydges University of British Columbia Statistical mechanics of phase transitions (3)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Rigorous techniques for 1D systems (2)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 16:45	Hatem Najar I.P.E.I.Monastir Tunisia Localization for random divergence operators	Room 1
16:45 to 17:30	Cavity approach to the spectral density of sparse symmetric random matrices	Room 1
18:15 to 19:00	Dinner at Churchill College	No Room Required

Monday 21st July 2008
09:00 to 10:30	David Brydges University of British Columbia Statistical mechanics of phase transitions (4)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	Yan Fyodorov Queen Mary University of London Introduction to supermatrix techniques (4)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Rigorous techniques for 1D systems (3)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 16:45	Yevgeny Krivolapov Technion - Israel Institute of Technology Anderson localisation and sub-diffusion for the nonlinear Schrodinger equtation: results and puzzles	Room 1
17:00 to 18:00	Philip Anderson Princeton University The historical origins of localisation	Room 1
18:15 to 19:00	Dinner at Churchill College	No Room Required

Tuesday 22nd July 2008
09:00 to 10:30	David Brydges University of British Columbia Statistical mechanics of phase transitions (5)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	Anderson localisation: phenomenology and mathematics (2)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Andreas Ludwig University of California, Santa Barbara Critical phenomena in 2D disordered systems (1)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 16:45	David Brydges University of British Columbia Statistical mechanics of phase transitions (6)	Room 1
16:45 to 17:30	Florina Halasan University of British Columbia Absolutely continuous spectrum for the Anderson model on more general trees	Room 1
18:15 to 19:00	Dinner at Churchill College	No Room Required

Wednesday 23rd July 2008
09:00 to 10:30	Andreas Ludwig University of California, Santa Barbara Critical phenomena in 2D disordered systems (2)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	Yan Fyodorov Queen Mary University of London Introduction to supermatrix techniques (5)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Rigorous techniques for 1D systems (4)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 16:45	Christian Sadel Friedrich-Alexander-Universität Erlangen-Nürnberg Lyapunov exponents of randomly coupled wires: a perturbative calculation	Room 1
19:30 to 23:00	Conference dinner - Emmanuel College (Old Library)	No Room Required

Thursday 24th July 2008
09:00 to 10:30	Andreas Ludwig University of California, Santa Barbara Critical phenomena in 2D disordered systems (3)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	Yan Fyodorov Queen Mary University of London Introduction to supermatrix techniques (6)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Anderson localisation: phenomenology and mathematics (3)	Room 1
15:30 to 16:00	Tea	No Room Required
16:00 to 16:45	Nadia Sidorova University College London A two cities theorem for the parabolic Anderson model	Room 1
18:15 to 19:00	Dinner at Churchill College	No Room Required

Friday 25th July 2008
09:00 to 10:30	Anderson localisation: phenomenology and mathematics (4)	Room 1
10:30 to 11:00	Coffee	No Room Required
11:00 to 12:30	Rigorous techniques for 1D systems (5)	Room 1
12:30 to 13:30	Lunch at Churchill College	No Room Required
14:00 to 15:30	Andreas Ludwig University of California, Santa Barbara Critical phenomena in 2D disordered systems (4)	Room 1
15:30 to 16:00	Tea	No Room Required
18:15 to 19:00	Dinner at Churchill College	No Room Required

Anderson localisation transition: introductory training course

Speakers

Philip Anderson Princeton University
Colin Benjamin University of Leeds
David Brydges University of British Columbia
Margherita Disertori Université de Rouen
Yan Fyodorov Queen Mary University of London
Florina Halasan University of British Columbia
Yevgeny Krivolapov Technion - Israel Institute of Technology
Andreas Ludwig University of California, Santa Barbara
Hatem Najar I.P.E.I.Monastir Tunisia
Christian Sadel Friedrich-Alexander-Universität Erlangen-Nürnberg
Nadia Sidorova University College London
Brian Winn Loughborough University

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