Supersymmetry approach to disordered quantum systems
Lead Research Organisation:
University of Nottingham
Department Name: Sch of Mathematical Sciences
Abstract
The rapid development of fabrication technology for small electronic structures allows us now tostudy systems with dimensions of a few nanometres (nanodevices). At these scales the wave nature of electronsbecomes important and so their behaviour is described by quantum mechanics. Another essential feature of many nanodevices is an inevitable presence of imperfections or disorder. Therefore the statistical description is most appropriate in this situation. Statistical properties of quantum systems containing different kind of disorder isthe subject of the theory of disordered quantum systems. Developed initially to study electronic properties of solids, it was applied later to other areas of physics, in which interplay between interference effects and disorder is essential. Among them, for instance, are light propagation in disordered media or physics of ultra cold atomic gases.One of the most powerful tools in the theory of disordered quantum systems is the field-theoretical approach.Originated in high-energy physics, the approach plays an increasingly important role in the modern condensed matter theory. In particular, its application to disordered quantum systems was extremely successful. In spite of this, a number of fundamental problems in the field resisted rigorous understanding.One of the central problems of that kind is the Anderson localization phenomenon. In 1958, P. W. Andersonconjectured that the diffusive propagation of an electron subject to a random potential can be completelysuppressed due to the destructive interference effects. Despite its long history and many important insights obtained, a rigorous theory of Anderson localization in dimensions D>1 is still lacking. The issue attracts a lot of interest from physicists and mathematicians. Indeed, it will be the subject of a forthcoming six-month programme ``Mathematics and Physics of Anderson localization: 50 Years After'' at the Isaac Newton Institutein Cambridge.The goal of this project is to develop novel field-theoretical techniques allowing to solve various problems in the field of quantum disordered systems. Application of those techniques to particular systems, such as, for example, the multidimensional Anderson model, should improve significantly our understanding of the properties of quantum disordered systems, both in the localized and the critical regimes.
Organisations
- University of Nottingham (Lead Research Organisation)
- FOM Institute AMOLF (Collaboration)
- UNIVERSITY OF NOTTINGHAM (Collaboration)
- Abdus Salam International Centre for Theoretical Physics (Collaboration)
- University of Murcia, Spain (Collaboration)
- Ludwig Maximilian University of Munich (LMU Munich) (Collaboration)
- University of Warwick (Collaboration)
People |
ORCID iD |
| Alexander Ossipov (Principal Investigator) |
Publications
Faez S
(2011)
Critical scaling of polarization waves on a heterogeneous chain of resonators
in Physical Review B
Fyodorov Y
(2009)
The Anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices
in Journal of Statistical Mechanics: Theory and Experiment
Kravtsov V
(2011)
Return probability and scaling exponents in the critical random matrix ensemble
in Journal of Physics A: Mathematical and Theoretical
Kravtsov V
(2010)
Dynamical scaling for critical states: Validity of Chalker's ansatz for strong fractality
in Physical Review B
Ossipov A
(2011)
Criticality without self-similarity: a 2D system with random long-range hopping.
in Journal of physics. Condensed matter : an Institute of Physics journal
Ossipov A
(2013)
Anderson localization on a simplex
in Journal of Physics A: Mathematical and Theoretical
Ossipov A
(2012)
Level-number variance and spectral compressibility in a critical two-dimensional random-matrix model.
in Physical review. E, Statistical, nonlinear, and soft matter physics
Ossipov A
(2012)
Virial expansion of the nonlinear sigma model in the strong coupling limit
in Journal of Physics A: Mathematical and Theoretical
Rushkin I
(2011)
Universal and non-universal features of the multifractality exponents of critical wavefunctions
in Journal of Statistical Mechanics: Theory and Experiment
| Description | One of the most fascinating phenomenon in the theory of disordered quantum systems is the Anderson localization phenomenon. It describes how complex dynamics of quantum particles in disordered environment such as electrons in a metal containing impurities is affected by their quantum mechanical wave nature. In particular, it predicts that by increasing the strength of disorder a sudden transition between diffusive-like dynamics expected classically and quantum localization may occur. In the context of solid state physics, the Anderson transition separates two phases: a metallic phase (characterised by high electrical conductivity) and an insulating phase (characterised by low electrical conductivity). The corresponding quantum wave functions are extended in the metallic phase and localized in the insulating phase. At the point of the transition a typical wave function is neither localized nor extended, it has a fractal nature characterised by self-similar fluctuations. The aim of this project was to develop new approaches and analytical techniques to understand better the Anderson localization and the Anderson transition. The research outcomes of the project can be roughly divided into three categories: (i) development of new models exhibiting Anderson localization and Anderson transition, (ii) analytical study of models, which were studied before using only numerical simulations, (iii) development and application of new techniques. (i) In [7] (see the list of references in Published Outcomes) we proposed a new random matrix model exhibiting Anderson transition. The model attracted a lot of attention from other researches working in this field. In 2011 the paper was in the top 5 of the most cited papers published in J. Stat. Mech in the last 2 years. Further investigations of this model carried out in [4] allowed us to answer an important question concerning universality of fractal dimensions for the long-range Hamiltonians. In [5] we studied few modifications of a well-known random matrix model, which are relevant to experiments on propagation of electromagnetic waves in a chain of near-resonant weakly-coupled scatterers. Our findings provided interesting predictions for outcomes of possible experiments on propagation of electromagnetic waves in artificially made structures. (ii) In [1] and [2] we carefully studied two-dimensional critical random matrix models. We discovered that the Anderson transition in these models is different from a conventional picture, which was expected before based on numerical simulations. In particular, we identified an usual metallic phase having some traces of the critical behaviour. (iii) A new analytical approach to study fractal dimensions and other critical exponents in the regime of strong multifractality was developed and applied in [3] and [6] to the well-studied power-law random matrix model. We were able to provide an analytical verification of a long-standing conjecture concerning properties of the multifractal quantum eigenstates in critical systems. In [8] a novel perturbative approach to study the supersymmetric non-linear sigma model was developed. The method allowed us to calculate explicitly the moments of the eigenfunctions and the two-level correlation function for the non-linear sigma model characterized by a generic coupling matrix in the strong coupling limit. We also revealed a surprising equivalence between the sigma-model approach and its dual counterpart used for description of a certain class of random matrices. |
| Exploitation Route | The research conducted in the project is fundamental in nature and so the short term beneficiaries of the project are researches working in the fields of disordered quantum systems and random matrix theory. The publications appeared during the work on the project are already cited by many other researches including world leading experts in that fields. In the long term, the outcomes of the project might be relevant for understanding electronic properties of nanodevices. The research assistant worked on this project has benefited from developing his skills and knowledge in the area of disordered systems and random matrix theory. |
| Sectors | Electronics |
| Description | Collaboration with Dr. Alberto Rodriguez |
| Organisation | University of Warwick |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | New collaboration with Dr. Alberto Rodriguez (University of Warwick) has been established during the work on the project. |
| Start Year | 2009 |
| Description | Collaboration with Dr. Emilio Cuevas |
| Organisation | University of Murcia, Spain |
| Country | Spain |
| Sector | Academic/University |
| PI Contribution | New collaboration with Dr. Emilio Cuevas (University of Murcia, Spain) has been established during the work on the project. |
| Start Year | 2010 |
| Description | Collaboration with Dr. Oleg Yevtushenko |
| Organisation | Ludwig Maximilian University of Munich (LMU Munich) |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | The work on particular parts of the project became successful due to existing collaboration with Dr, Oleg Yevtushenko, University of Munich, Germany. |
| Start Year | 2005 |
| Description | Collaboration with Dr. Sanli Faez and Prof. Ad Lagendijk |
| Organisation | FOM Institute AMOLF |
| Country | Netherlands |
| Sector | Public |
| PI Contribution | New collaboration with Dr. Sanli Faez and Prof. Ad Lagendijk (AMOLF, Amsterdam, The Netherlands) has been established during the work on the project: |
| Start Year | 2009 |
| Description | Collaboration with Prof. Vladimir Kravtsov |
| Organisation | Abdus Salam International Centre for Theoretical Physics |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | The work on particular parts of the project became successful due to existing collaboration with Prof. Vladimir Kravtsov, ICTP, Trieste, Italy |
| Start Year | 2003 |
| Description | Collaboration with Prof. Yan Fyodorov |
| Organisation | University of Nottingham |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | The work on particular parts of the project became successful due to existing collaboration with Prof. Yan Fyodorov, University of Nottingham, UK |
| Start Year | 2002 |
| Description | Conferences, workshops, seminars. |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Other academic audiences (collaborators, peers etc.) |
| Results and Impact | The results of the work were presented at the international conferences, workshops and seminars in the UK and overseas. Talks: 2010 February Seminar, Birmingham University 2010 March Seminar, LPMMC, Grenoble, France 2010 May Seminar, LPTMS, Orsay, France 2010 December International Workshop on Random matrix Theory, Brunel University, London 2011 July Euler Symposium on Theoretical and Mathematical Physics, St.-Petersburg, Russia Posters: 2009 December International Workshop on Random Matrix Theory, Brunel University, London 2010 May Workshop on Localization Phenomena, ICTP, Trieste, Italy 2010 June Conference NanoPeter 2010, St.-Petersburg, Russia 2010 August Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence, ICTP, Trieste, Italy 2011 June Workshop and School on Topological Aspects of Condensed Matter Physics, ICTP, Trieste, Italy 2011 December International Workshop on Random Matrix Theory, Bielefeld University, Germany. These meetings and discussions stimulated development of new research ideas and identifying of new and import research directions. |
| Year(s) Of Engagement Activity | 2009,2010,2011 |