# Disordered Systems and Quantum Chaos

1 July - 19 December 1997

Organisers:JP Keating (Bristol), DE Khmelnitskii (Cambridge), IV Lerner (Birmingham), P Sarnak (Princeton)

### Scientific Background and Objectives

The motion of a particle in a random potential in two or more dimensions is chaotic, and the trajectories in deterministically chaotic systems are effectively random. It is therefore no surprise that there are links between the quantum properties of disordered systems and those of simple chaotic systems. The question is, how deep do the connections go? And to what extent do the mathematical techniques designed to understand one problem lead to new insights into the other?

The quantum properties of disordered systems have been the focus of considerable attention in many branches of physics, most notably nuclear physics and condensed matter physics. In the last few years, advances in the field of microelectronics have shifted the focus in condensed matter physics to the study of mesoscopic systems, such as electronic devices that are large on the atomic scale but sufficiently small for quantum coherence effects to be important. In some circumstances the behaviour of these devices is governed by the fact that they have randomly distributed impurities; that is, they are disordered.

The canonical problem in the theory of disordered mesoscopic systems is that of a particle moving in a random array of scatterers. The aim is to calculate the statistical properties of, for example, the quantum energy levels, wavefunctions, and conductance fluctuations by averaging over different arrays; that is, by averaging over an ensemble of different realizations of the random potential. In some regimes, corresponding to energy scales that are large compared to the mean level spacing, this can be done using diagrammatic perturbation theory. In others, where the discreteness of the quantum spectrum becomes important, such an approach fails. A more powerful method, developed by Efetov, involves representing correlation functions in terms of a supersymmetric nonlinear $\sigma$ model. This applies over a wider range of energy scales, covering both the perturbative and non-perturbative regimes.

In quantum chaos, the goal is to understand the semiclassical asymptotics of the quantum properties of individual classically chaotic systems, examples of which include simple atoms (eg helium), molecules, and also mesoscopic quantum devices.

 Electron micrograph of a typical ballistic quantum dot taken from Marcus et al, Chaos, Soliton & Fractals, 8, 1261 (1997). Electrons are confined vertically to the ground state of a quantum well located at a GaAs/AlGaAs interface, and form a two-dimensional electron gas. A mean free path and coherence length of the order of 10 microns ensures that the carriers are coherent and ballistic. STM image of the surface electron density of a quantum stadium corral made from Iron atoms deposited on Copper (111), taken from Crommie et al, Nature 363, 524 (1993).

In this case the main tool is Gutzwiller's trace formula, which, in its most general form, links the quantum energy levels and eigenfunctions in a given system to the periodic orbits of the underlying classical dynamics. Considerable attention has been devoted to understanding the conditions of applicability of this formula, in particular with regard to its accuracy and convergence properties. It is now understood how the trace formula can be used to calculate approximations to the energy levels, wavefunctions and matrix elements of a particular system from its periodic orbits, and hence how to relate quantum fluctuation statistics to statistical properties (e.g. ergodicity, rate of mixing etc) of the corresponding classical motion. Despite the differences between the two approaches - one is based on ensemble averaging, the other on properties of the classical dynamics - during the last few years work on both the electronic structure of disordered systems and on quantum chaos has, to a considerable degree, been focused on the same fundamental problems, namely localization, fluctuation statistics, and correlations in energy spectra, with a particular emphasis on their role in mesoscopic physics. The general aims of the programme were therefore to bring together researchers in the two fields, to encourage the exchange of ideas on these problems, and to concentrate on understanding the connections between field-theoretic techniques and those based on the trace formula.

The concept of localization in disordered systems was introduced by Anderson in 1958. Localized quantum states do not contribute to the mobility in an infinite system. This leads to the metal--insulator transition when the disorder is so strong that all states in a sufficiently large system are localized. Mott showed in 1960 that, for a given disorder, all states may either be localized or extended as a function of energy, the boundary separating these two regimes being called the mobility edge. The metal--insulator transition takes place when the electrons' Fermi energy crosses the mobility edge. The modern theory of this transition in three-dimensional disordered systems is based on a one-parameter scaling conjecture (in one- and two-dimensional infinite systems all states are localized at any energy).

New phenomena, known by the general name of weak localization', arise in finite low-dimensional systems that are smaller than the localization length. While many of the most important weak localization effects are well understood, the development of a theory for the transport and thermodynamic properties of disordered systems in the strong localization regime, in particular at the mobility edge, remains one of the central problems of condensed matter physics. Their complete description in terms of a one-parameter scaling theory is now thought unlikely, because the wavefunctions at the mobility edge are multifractal; that is, they are characterized by a set of lengths which scale with different, independent exponents.

The quantum eigenfunctions of classically chaotic systems can also exhibit localization. In fact, in certain cases such systems can be mapped onto the Anderson model. Considerable effort has been put into relating this behaviour to properties of the classical trajectories in the asymptotic limit as $\hbar\to0$. Another kind of localization observed in these eigenfunctions is 'scarring', namely the concentration of intensity around individual periodic orbits. This too is far from being well understood.

One of the specific aims of the programme was to focus on localization phenomena and scarring in both randomly disordered and deterministically chaotic systems.

Localization has also stimulated a great deal of interest amongst mathematicians. That eigenfunctions localize in more than one dimension is still unproved and is recognised as an outstanding problem. Likewise, there are no rigorous results concerning scarring. Over the last few years these questions have led to the development of a theory for the spectral properties of random operators, and to the study of the statistical properties of the eigenfunctions of the Laplacian on surfaces of constant negative curvature (on which the geodesic motion is strongly chaotic). Another of the specific aims of the programme was to attract researchers from the mathematical community to interact with physicists on these problems.

The importance of ensemble fluctuations in disordered systems was realized about a decade ago. Until that time, the conventional wisdom was that all disordered samples are self-averaging, so that ensemble fluctuations are no more important than, for example, thermal fluctuations in the classical Boltzmann gas. It turns out, however, that quantum coherence effects result in ensemble fluctuations which, for low-dimensional systems at zero temperature, do not vanish, even in the limit of infinite system size. At finite temperatures, quantum coherence sets a new mesoscopic scale. Since the mid-eighties, properties of electronic devices on this scale have been studied in thousands of experiments. The fluctuations measured in conductance and magnetic properties are related to correlations between the energy levels. In a certain regime, corresponding the limit in which the dimensionless conductance tends to infinity, these correlations are universal and are directly related to those that exist between the eigenvalues of random matrices. This provides a highly successful phenomenological model which enables fluctuation properties to be calculated.

The link between the spectral correlations of randomly disordered systems in the ergodic regime (when the dimensionless conductance tends to infinity) and the eigenvalue correlations of random matrix theory was established by ensemble averaging using the supersymmetric nonlinear $\sigma$ model. Building upon this, there has been considerable recent progress in developing non-perturbative techniques that encompass finite conductance corrections. One of the specific aims of the programme was to focus on these new developments.

Energy level correlations also play a central role in the theory of quantum chaos. One of the main conjectures in the subject is that generically, in the semiclassical limit, the spectral statistics of a given classically integrable system are Poissonian, whilst those of a given classically chaotic system are the same as for random matrices. This is supported by extensive numerical computations, and by theoretical arguments based on the trace formula. Recently there there has been considerable progress in this direction, including calculations of the asymptotic form of the approach to the limit.

In the last two years there has been an attempt to unify these methods by constructing a nonlinear $\sigma$ model for individual systems rather than for ensembles. In this case the average would be over a dynamical parameter (such as the energy) instead of different realizations of a random potential. The hope was that this would prove the energy level correlation conjectures. One of the main aims of the programme was to focus on understanding the connections between this approach and the one based on the trace formula.

A complication in this story is that some classically integrable systems are known not to have Poissonian level statistics, and the spectra of some classically chaotic systems are known not to be random-matrix-correlated. The key questions then are: what makes these examples exceptional (or non-generic); and how can this fact be accommodated into theories based either on the trace formula, or on a $\sigma$ model?

There are also far-reaching connections between these problems and recent developments in number theory. First, one can construct classically integrable systems whose quantum energy levels are given by simple formulae involving sets of integers. The goal then is to prove that these are uncorrelated (i.e. Poisson distributed) in the semiclassical limit. Recent results have shown that this is a hard problem. In particular, it has been proved that for certain large classes of systems the levels are correlated, whilst for others they are not. Hence the question of what is meant by genericity is a key issue.

Second, some of the most widely investigated classically chaotic systems correspond to geodesic motion on compact surfaces of constant negative curvature. The corresponding quantum energy levels are the eigenvalues of the Laplace-Beltrami operator and are related to the closed geodesics by the Selberg trace formula. These have been found numerically to exhibit the same correlations as those in random matrix theory for the surfaces generated by non-arithmetic groups. For arithmetic groups, it has been proved that the correlations are almost absent; that is, the eigenvalues are almost Poisson distributed. Thus these systems provide a rich class of examples showing both generic and non-generic behaviour.

Finally, extensive numerical evidence supports the conjecture that the zeros of the Riemann zeta function and other L-functions are also correlated like the eigenvalues of random matrices, and this in turn hints at a spectral interpretation for them. In this case the explicit formula that links the zeros to the primes plays the role of a trace formula.

Another key aim of the programme was to encourage further interactions between mathematicians working on these problems and physicists interested the analogous properties of disordered systems and quantum chaos.

### Organisation

The overall organisation was undertaken by Jon Keating, David Khmelnitskii and Igor Lerner in consultation with Peter Sarnak. The pre-programme arrangements were overseen by Jon Keating, and the day-to-day administration of the programme was carried out by Jon Keating from July to September and Igor Lerner from October to December.

Many of the participants also played a substantial role in the organization of workshops, principally Hans Weidenmüller for the NATO ASI, Jens Marklof for the Spitalfields Day, and Uzy Smilansky for the final conference.

Throughout the programme, we ran a regular seminar series on Tuesdays. There were usually three such talks each week.

The organization was greatly assisted by the dedicated, helpful and cooperative work of all of the staff at the Newton Institute. Without this the scientific programme could not have succeeded.

### Participation

The programme attracted 66 long-term participants (whose average stay was for about 7 weeks) and 71 short-term participants (on average, staying for 11 days). In addition, about 200 researchers visited the conferences and one-day meetings. Other than when conferences were being held, the number of participants at any one time was on average about 24.

The list of participants confirms the strong interest aroused by the programme: practically all of the leading theorists in mesoscopic physics and quantum chaology and many prominent mathematicians working on related problems attended. Geographically, there were participants from most European countries (including the former Soviet Union and Eastern Europe), North and South America, the Middle and Far East. In particular, support from the Leverhulme Trust enabled us to invite many scientists whose home institutions are in the former Eastern Bloc or in Latin America.

Younger researchers were encouraged to attend by making use of the Junior Membership scheme. This was particularly successful in helping to attract post-doctoral researchers and PhD students.

UK researchers were also specifically targeted. In total 13 of our long-term participants and 32 of our short-term visitors were from UK institutions. Many others came to the conferences and workshops.

### Meetings and Workshops

We had three long workshops and two one-day meetings. These were as follows.

EC Summer School on Disordered Systems and Quantum Chaos
Organisers: Jon Keating, David Khmelnitskii, Igor Lerner.

This School was held at the beginning of the programme, during the period 28 July - 1 August. The idea was that it would provide an impetus to the whole programme and would help set the agenda. The topics covered included the quantum field theory of disordered systems, quantum localization in classically chaotic systems, rigorous mathematical results concerning localization, symbolic dynamics, random matrix theory as a model for the statistical properties of quantum spectra, the Riemann zeta function, wavefunctions in disordered systems, rigorous results on spectral statistics, disordered superconductors, and quantum graphs. Lectures were specifically coordinated so as to allow participants to compare and contrast the various different approaches.

The Lecturers were: A Altland (Cambridge), MV Berry (Bristol), EB Bogomolny (Orsay), V Falko (Lancaster), S Fishman (Haifa), I Goldsheid (London), DE Khmelnitskii (Cambridge), IV~Lerner (Birmingham), YG Sinai (Princeton), U Smilansky (Weizmann Institute), BZ Spivak (Seattle), M Wilkinson (Strathclyde), S Zelditch (Baltimore), and MR Zirnbauer (Cologne).

The School attracted in total 57 advanced graduate students, postdoctoral fellows and young researchers, mainly from EC countries. All were invited to present posters at an afternoon-long session. Many did.

NATO ASI Supersymmetry and Trace Formulae: Chaos and Disorder
Director: Igor Lerner. Organizing Committee: Jon Keating, David Khmelnitskii, Peter Sarnak and Hans Weidenmüller

This Advanced Study Institute was one of the main focal points of the programme. The aim of the lectures was both to present a coherent and comprehensive overview of the modern quantum theory of disordered and chaotic systems, and to discuss recent results in the field. The courses were complemented by selected research seminars, and by posters presented at an afternoon-long session.

The subjects covered in the courses included a quantum interpretation of some properties of the Riemann zeta function, the semiclassical theory of spectral statistics, the pair correlation of the Riemann zeros, classical trace formulae, quantum field theory and the nonlinear $\sigma$ model, non-exponential relaxation in disordered systems, semiclassical trace formulae, scarring in quantum wavefunctions, a field-theoretic approach for chaotic systems, parametric statistics, rigorous results for level correlations and the Riemann zeros, the scattering matrix approach and random matrix theory.

The lectures were given by: MV Berry (Bristol), EB Bogomolny (Orsay), P Cvitanovic (NBI), KB Efetov (Bochum), MC Gutzwiller (IBM), F Haake (Essen), E Heller (Harvard), JP~Keating (Bristol), DE Khmelnitskii (Cambridge), VE Kravtsov (Trieste), S Fishman (Haifa), P Sarnak (Princeton), BD Simons (Cambridge), U Smilansky (Weizmann Institute), M Wilkinson (Strathclyde), HA Weidenmüller (Heidelberg), and MR Zirnbauer (Cologne). Most will be published in a NATO ASI volume by Plenum.

The workshop was extremely successful and we received many favourable comments from the participants.

One-Day Conference on Physics in Mesoscopic Conductors
Organiser: David Khmelnitskii

This one-day meeting was sponsored by the Mathematical Physics Group of the Institute of Physics. The idea was to highlight new directions in experimental mesoscopic physics for the participants of the programme, mainly theoretical physicists and mathematicians. The Conference attracted about 50 participants from different UK institutions.

The talks were given by L Eaves (Nottingham), R Nicholas (Oxford), M Pepper (Cambridge), and V Petrashov (Royal Holloway).

Spitalfields Day: Zeta Functions and Spectra
Organisers: Jon Keating and Jens Marklof

This meeting was supported by the London Mathematical Society. The aim was to focus on the links between the theory of spectra and the properties of the Riemann zeta function, and to stimulate discussion of recent developments in the spectral interpretation of the Riemann zeros.

The lecturers were: A Connes (IHES), DR Heath-Brown (Oxford), D Hejhal (Minnesota), M Huxley (Cardiff), and Z Rudnick (Tel Aviv).

The meeting attracted over 100 participants.

Final Conference: Quantum Chaos and Mesoscopic Systems.
Organisers: Jon Keating, David Khmelnitskii, Igor Lerner and Uzy Smilansky

The final conference was an opportunity for participants to present work completed during their stay at the Institute. It also provided a forum to summarize the conclusions reached during the programme.

Lectures were given by: N Argaman (UCSB), H Baranger (Bell Labs), C Beenakker (Leiden), EB Bogomolny (Orsay), G Casati (Como), I Dana (Bar-Ilan), V Falko (Lancaster), M Fromhold (Nottingham), Y Gefen (Weizmann Institute), F~Izrailev (Universidad de Puebla, Mexico), L Kaplan (Harvard), S Kravchenko (City College, NY), A MacKinnon (Imperial College), C Marcus (Stanford), J Marklof (Bristol), G Montambaux (Orsay), T Monteiro (University College London), T Nieuwenhuizen (Amsterdam), F von Oppen (Heidelberg), J-L Pichard (Saclay), R Prange (Maryland), JM Robbins (Bristol), AD Stone (Yale), and A Voros (Saclay).

Topics included recent experiments on mesoscopic devices, intermediate statistics, scarring of quantum wavefunctions, spectral statistics in integrable systems, localization, and the Pauli exclusion principle.

### Outcome and Achievements

The main achievement of the meeting was that it brought together researchers from an uncommonly wide range of backgrounds, stretching from experimental condensed matter physics to number theory, to focus on fundamental issues in the theories of disordered systems and quantum chaos, and on their links with various problems in mathematics. This worked surprisingly well: almost all participants made serious attempts to interact with those from the other areas, and most who did reported positive results.

It was expected that those with interests in disordered systems and quantum chaos would find much in common, and this proved to be the case. More remarkable was the degree of common interest both groups found with the mathematicians who participated.

The programme focused on the following problems: the statistics of quenched (mesoscopic) fluctuations and spectral correlations, and, in particular, generic features in their deviations from universality; the existence of a wider set of universality classes within the nonlinear $\sigma$ model related to the Cartan classification of symmetric spaces; the limits of validity of the Bohigas--Giannoni--Schmit conjecture, which concerns the link between spectral statistics and random matrix theory in classically chaotic systems, the Berry-Tabor conjecture, which states that the corresponding statistics in classically integrable systems are Poissonian, and Berry's conjecture that the quantum eigenfunctions of chaotic systems exhibit Gaussian random fluctuations in the semiclassical limit; the scarring of eigenfunctions and the existence of nontrivial wavefunction statistics in quantum disordered systems; multifractality in eigenvalue and eigenfunction distributions; thermodynamic and transport properties of nanoscale quantum dots; non-exponential relaxation processes in ballistic and disordered quantum dots; eigenvalue correlations in systems described by non-hermitian Hamiltonians; and interaction effects in the description of spectral correlations in quantum dots.

The main outcome of the programme was the emergence of a more coherent view of the links between these problems. It is difficult to predict at this stage what long-term consequences this will have, but some of the main short-term achievements and conclusions are outlined below.

As anticipated, the main focus of the programme was on spectral statistics. First, the application of a field-theoretic approach within the framework of the nonlinear $\sigma$ model to the description of the energy level statistics of individual chaotic systems is a very exciting prospect, but has turned out to present a number of fundamental mathematical and physical problems. Some of these were overcome during the programme (mainly in work by A Altland, BD Simons and MR Zirnbauer), but many remain to be solved. The results of Zirnbauer's extension of these ideas to quantum maps suggests that the way forward might be to consider averages over a very small amount of disorder. However, at this stage the question of whether one can avoid an ensemble average altogether is, for typical systems, still open. This is likely to remain one of the key directions of research in the future.

There were also significant steps in the extension of semiclassical methods to the description of level statistics in disordered systems (IV Lerner and RA Smith in collaboration with BL Altshuler). These appear to be particularly useful in describing behaviour in the non-ergodic regime, where the spectral correlations do not exhibit random-matrix universality. In the ergodic regime the use of diagrammatic techniques allows one to identify the trajectories responsible for the deviations from the diagonal approximation at short times.

D Cohen, U Smilansky and G Tanner worked on the related problem of periodic orbit correlations in chaotic systems. Many participants contributed to discussions on this subject, which remains one of the most mysterious in the field. It is hoped that the numerical investigations started at the Institute, coupled with insights obtained by comparison with the diagrammatic method, will shed some new light on the problems involved.

U Smilansky investigated eigenvalue correlations for quantum graphs. This led to considerable interest in these models amongst both the mathematicians and those working on disordered systems, in particular because in this case the trace formula is exact rather than a semiclassical approximation.

S Fishman and F Haake both developed techniques to calculate the eigenvalues of the Perron-Frobenius operator as part of a project involving many of the participants to investigate the role these play in the description of deviations from random matrix theory in spectral statistics. Discussions with I Goldsheid were particularly helpful in characterising the mathematical basis of their results. These questions are also central to the approach based on the ballistic nonlinear $\sigma$ model (O Agam, BD Simons).

JP Keating worked with MV Berry and SD Prado to develop a semiclassical theory for the influence of orbit bifurcations on spectral statistics in mixed systems, and together with S Fishman used a construction involving the Aharonov-Bohm effect to calculate the range of validity of the diagonal approximation in theories of spectral statistics based on the trace formula.

There was also a great deal of activity on the autocorrelations of eigenstates in disordered and chaotic systems, and in particular on their relation to eigenvalue correlations in the non-ergodic regime. This is of central importance to the Anderson localization problem, because it has become clear that the new universal regimes for level correlations which emerge at the mobility edge (VE Kravtsov, IV Lerner; AD Mirlin) are crucially affected by the multifractal character of the eigenstates (JT Chalker, VE Kravtsov, IV Lerner, RA Smith).

The autocorrelation properties of eigenfunctions were also investigated in the ergodic and weak-localization regimes (AD Mirlin; YV Fyodorov; M Srednicky; IV Yurkevich, VE Kravtsov). In the weak localization regime B Muzykantskii and DE Khmelnitskii have developed a new instanton approach, based on the existence of a non-trivial inhomogeneous saddle-point in the $\sigma$ model, that has led to results concerning multifractality and long-time relaxation processes due to untypical realizations of the disorder (B Muzykantskii and DE Khmelnitskii; V Falko and KB Efetov; KM Frahm; AD Mirlin). It turns out that this has a very natural counterpart in a time-dependent field-theoretic approach to the study of different models of turbulence at intermediate scales. Further research in this direction seems to be very promising.

On the question of localization in quantum chaotic systems, G Casati and R Prange both worked on compact billiards that exhibit diffusion in angular momentum (specifically, a Bunimovch stadium that is almost circular). Such systems have quantum wavefunctions that are localized in the angular momentum basis. J P Keating and G Tanner constructed a semiclassical periodic orbit description of the spectral statistics of these systems.

E J Heller and L Kaplan developed a very promising nonlinear theory of scars. This stimulated a lively debate on the links with an alternative approach based on the trace formula. The possibility of a unified description is under current investigation.

Several participants worked on applications of the methods described above. For example, M Wilkinson collaborated with B Mehlig and K Richter on applying semiclassical methods to calculate polarization effects in small metallic particles. H A Weidenmüller worked on extending the statistical description based on the $\sigma$ model to classical wave scattering.

The statistical distribution of the eigenvalues in non-hermitian systems proved to be another direction of research in which very different approaches appear to be converging. While non-hermitian systems have been studied in the context of turbulence for some time, their importance in condensed matter physics has been recognised only recently, mainly in relation to certain problems in the quantum Hall effect. KB Efetov calculated the distribution of the eigenvalues in the complex plane for a model associated with an imaginary vector potential using the supersymmetric $\sigma$ model, and JT Chalker applied diagrammatic perturbation theory to calculate the corresponding distribution for advective diffusion. Remarkably, many of the same models have also recently been considered in a purely mathematical context (I Goldsheid and B Khoruzhenko) and this stimulated considerable activity during the programme.

One of the most remarkable features in theory of disordered systems is that many experimental data can be well understood within the model of noninteracting electrons. However, new experimental results on the energy level correlations in nanoscale quantum dots (reported by C Marcus), and on the existence of a metal-insulator transition in two-dimensional silicon structures in contradiction to the standard scaling theory of the transition (S Kravchenko and collaborators), have highlighted the necessity to include interactions. Thus the treatment of strong correlations', which is central to many areas of condensed matter theory, is now beginning to become important on the mesoscopic scale. Considerable progress has been made in understanding the effects of interactions on level statistics and transport properties in quantum dots (Y Gefen and A Kamenev; B Spivak; Y Imry; S Levit), but the characterization of their influence on the metal-insulator transition remains an outstanding challenge.

As has already been mentioned, some of the most exciting developments centred around the interactions between physicists and mathematicians. In particular, A Connes presented his results concerning a spectral interpretation of the Riemann zeros, and a possible approach to proving the Riemann Hypothesis. Z Rudnick played a key role in explaining these ideas, and this in turn stimulated work by MV Berry and JP Keating in the same direction.

S Zelditch initiated work with M Zirnbauer to construct a rigorous mathematical framework for the $\sigma$ model, and with J Marklof and Z Rudnick to compute the level statistics for a class of integrable systems using estimates for trigonometric series.

J Marklof discovered a deep relationship between the pair correlation of the energy levels of rectangular billiards and the mixing properties of a dynamical system defined on a product space of hyperbolic surfaces. This allowed him to prove several strong results about the Poisson conjecture for the level statistics.

Finally, MV Berry and JM Robbins investigated the consequences for isospin of their work on the Pauli exclusion principle after discussions with D J Thouless.

It will be clear from the above that the range of collaborations started during the programme was extremely broad. As many of the participants commented, to a large extent this was due to the uniquely stimulating environment provided by the Newton Institute and maintained by its staff.