Academic Year 2009

In the doorway picture the strength of a particular mode is coupled via the doorway state to surrounding states, which are often, but not always, assumed to be chaotic. This picture is very useful to describe several physical systems and has been very much utilized in nuclear physics [1, 2]. We are interested in situations where the coupling is relatively weak, so that the doorway strength gets concentrated over a small number of complex states. Different physical situations where this picture applies include: superscars in billiards [3], decay out of superdeformed states [4], isobaric analogue states [5] , giant dipole resonance in light nuclei.

To describe the experimental detection of the strength it is important to understand the distribution of the doorway strength on surrounding states. On the average the doorway strength follows a Breit-Wigner function. But if the doorway coupling is weak, the strength will be concentrated on a small number of states. In this case more information on the strength distribution is often needed. Of particular interest is the state that carries the largest strength. How large is this strength in average, and what is its distribution function? The statistical information of this extreme value can be calculated in a model that couples the doorway strength to a complex surrounding, modeled by random matrix theory [3]. The important parameter of the model is the doorway coupling strength divided by the mean level spacing, i.e. V/d. By allowing for different statistics for the surrounding states, different physical situations can be considered. In pseudointegrable barrier billiards, where the eigenstates are semi-Poisson distributed, the spreading of the strength of a superscar on surrounding states can be measured [6], and has been compared to result of the model [3].

Another quantity of interest is how many states carry a certain threshold doorway strength, de¯ned by the experi- mental possibility to detect the strength of the doorway in a state. This provides information of how many discrete states can be observed. For a strong doorway coupling the strength is fragmented and a few or no discrete states can be measured. For a weak doorway coupling the strength is concentrated, and one or a few states obtain measurable strength. For a certain strength the largest number of states be observed. We study this average as well as its distribution function?

[1] A. Bohr and B.R. Mottelson, Nuclear Structure (Benjamin, New York, 1969), Vol.1.
[2] T. Guhr, A. MÄuller-Groeling and H.A. WeidenmÄuller, Phys. Rep. 299, 189 (1998).
[3] S. Åberg, T. Guhr, M. Miski-Oglu and A. Richter, Phys. Rev. Lett. 100 (2008) 204101.
[4] S. Åberg, Phys. Rev. Lett. 82 (1999) 299.
[5] S. Åberg, A. Heine, G.E. Mitchell and A. Richter, Phys. Lett. B598, 42 (2004).
[6] E. Bogomolny, B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, F. Sch"afer and C. Schmit, Phys. Rev. Lett. 97 (2006) 254102.

We address the question where eigenvalues are located in the complex plane, comparing four different symmetry classes of Ginibre and chiral type. The distributions of individual complex eigenvalues can be found from the so-called gap probability, the probability that a set is empty of eigenvalues. Choosing a circle of radius R we can provide an exact answer for maximal non-Hermiticity, that is a rotationally symmetric ensemble, and an approximate answer for intermediate non-Hermiticity. A relation between complex and symplectic models as well complex level spacing repulsion are discussed. As an example our analytic predictions are compared to complex Dirac operator eigenvalues from the theory of strong interactions with finite quark density.

A nano-size metallic grain with irregular boundaries in which the single-particle dynamics are chaotic is described by an effective Hamiltonian known as the universal Hamiltonian. The single-particle part of this Hamiltonian is described by random matrix theory, and its interacting part includes a superconducting pairing term and a ferromagnetic exchange term. These interaction terms compete with each other: pairing correlations favor minimal ground-state spin while the exchange interaction favors maximal spin polarization. We study the tunneling conductance through an almost-isolated grain in the fluctuation-dominated regime where the bulk pairing gap is comparable to the single-particle mean level spacing. We find signatures of the coexistence of superconductivity and ferromagnetism in the mesoscopic fluctuations of the conductance peak heights and peak spacings [1].
[1] S. Schmidt and Y. Alhassid, Phys. Rev. Lett. 101, 207003 (2008).

In recent years, progress has been made in understanding the status of random matrix theory in the physics of classically chaotic quantum systems by methods of semiclassical analysis. Methods based on the counting of periodic orbits (the Gutzwiller trace formula) have been successfully applied to obtain both perturbative and non-perturbative contributions to the spectral correlation functions of individual chaotic systems. I will review these developments and discuss the current status of the field.

The two-level bosonic $k$-body ensemble of random matrices is defined by distributing $N$ bosons on two single-particle states; the $N$ bosons interact through $k$-body forces which are taken as independent Gaussian distributed random variables with zero mean and fixed variance. For $k=1$ this ensemble is equivalent to a chain of randomly interacting harmonic oscillators while for $k=N$ its definition coincides with the canonical Gaussian ensembles (GOE or GUE). This ensemble is Liouville integrable in the classical limit ($N\to\infty$). In this talk I will present numerical results on the spectral statistics and the phase space structure as the rank of the interaction $k$ is varied, and discuss the applicability of a theorem by Schnirelman as $k$ approaches $N$.

The observed preponderance of ground states with angular momentum L=0 in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different energy eigenvalues. The geometric analysis of Chau et al is reinterpreted in purely statistical terms, allowing to relate this results with other unexpected statistical effects in correlated data. It is shown that the present method which, strictly speaking, is valid for diagonal interactions only, can be extended to non-diagonal interactions by means of perturbation theory.
Ref. Barea, Bijker and Frank, arXiv:0811.1240

We study the RPA equations in their general form by taking the matrix elements appearing in the RPA equations as random variables. The spectrum of the resulting random-matrix model is determined by solving a generalized Pastur equation. Two independent dimensionless parameters govern the behaviour of the system: the distance between the centers of positive and negative eigenvalues and the strength of their coupling. By increasing this coupling the two original semicircles are deformed and pulled toward each other, till they begin to overlap and as a consequence, the system becomes unstable and eigenvalues leave the real axis.

The chiral Lagrangian of QCD in a finite volume is intimately related to chiral Random Matrix Theory. The link is normally considered from the point of view of the so-called epsilon-regime of the theory. I discuss how chiral Random Matrix Theory has far wider applicability in the finite-volume theory, with a systematic expansion around it.

We measured the transmission and reflection spectra of a chaotic microwave billiard in the regime of isolated and overlapping resonances. By inserting a ferrite inside the cavity breaking of time-reversal (T) invariance was induced. The T-breaking effect of the ferrite varies with the excitation frequency. Only partial T-breaking is achievable. For the random-matrix theoretical description of fluctuation properties of the complex scattering matrix elements extracted from the measured spectra a scattering approach originally developed in the context of compound nucleus reactions is used. Based on this scattering formalism and the supersymmetry method we derived analytic expressions for the autocorrelation and the cross correlation function of pairs of complex conjugate scattering matrix elements. This model is applicable in the regime of isolated and of overlapping resonances and depends on a parameter, which defines the strength of the T-breaking. Fitting the scattering data we determined the strength of T-violation for the whole range of the excitation frequency. Furthermore, the elastic enhancement factor was determined as function of the excitation frequency. A comparison with the theoretical predictions derived from the fits to the auto- and cross correlation functions provides another confirmation of the model.
*Supported by the DFG within the SFB

We suggest a general scheme based on the supersymmetry technique that allows one to consider low lying excitations in electron systems. The same approach allows one to consider correlations with close eigenvalues in the theory of random matrices. The effective field theory obtained in this scheme is a generalization of the supermatrix \sigma-model and can be derived exactly. It is demonstrated how one can symplify the general model to known limiting cases.

Giant resonances and doorway status, first dealt with in nuclei, appear also in atoms, molecules and clusters. This spreading width phenomenon has been recently observed in microwave resonators, which are classical systems. In this work we present several instances of classical systems which also show this phenomenon. We demonstrate both theoretically and experimentally that the compressional oscillations of an elastic rod, formed by a small rod coupled to a long one, show a giant resonance. This phenomenon is also observed in the seismic response of sedimentary valleys. The spreading width phenomenon is, therefore, a unifying concept in Physics.

Fractal structure, as reflected in a time series by a 1/f power spectrum, may be a common behavior associated to both classical and quantum chaotical behavior. The dripping faucet is a paradigmatic example of transitions to chaos. We discuss this example, as well as chaos in some biological systems. We then show how these ideas can be also applied to quantum spectra in nuclei and other systems. We discuss the possible consequences of this analysis.

The quantum Loschmidt echo (also termed fidelity) has been introduced by Peres 1984 to study the stability of quantum dynamics. Only one year later, Verbaarschot, Weidenmüller and Zirnbauer published their supersymmetry formula (the VWZ-formula) for the statistical model of compound-nucleus scattering. In this contribution we will discuss recent developments and applications of the quantum Loschmidt echo from the perspective of random matrix theory (RMT). Our starting point is a universal prediction for the behaviour of the fidelity in a regime where wave phenomena are important (from the Fermi-golden-rule to the perturbative regime). Soon afterwards, exact formulas for the fidelity and related quantities became also available. In all cases, these exact results are closely related to the VWZ-formula from 1985. On the one hand, fidelity was then extended to scattering systems, which allowed to design experiments, where fidelity could be measured. On the RMT side, this extention was based on the statistical model mentioned above. Surprisingly this "scattering fidelity" is closely related to "coda wave interferometry", a theory used in earth sciences for the analysis of observational data. On the other hand, fidelity can be related to decoherence in information processing quantum devices. Here, recent work is devoted to a unifying description of fidelity loss and decoherence in many-qubit systems.

The quantum Loschmidt echo (also termed fidelity) has been introduced by Peres 1984 to study the stability of quantum dynamics. Only one year later, Verbaarschot, Weidenmüller and Zirnbauer published their supersymmetry formula (the VWZ-formula) for the statistical model of compound-nucleus scattering. In this contribution we will discuss recent developments and applications of the quantum Loschmidt echo from the perspective of random matrix theory (RMT). Our starting point is a universal prediction for the behaviour of the fidelity in a regime where wave phenomena are important (from the Fermi-golden-rule to the perturbative regime). Soon afterwards, exact formulas for the fidelity and related quantities became also available. In all cases, these exact results are closely related to the VWZ-formula from 1985. On the one hand, fidelity was then extended to scattering systems, which allowed to design experiments, where fidelity could be measured. On the RMT side, this extention was based on the statistical model mentioned above. Surprisingly this "scattering fidelity" is closely related to "coda wave interferometry", a theory used in earth sciences for the analysis of observational data. On the other hand, fidelity can be related to decoherence in information processing quantum devices. Here, recent work is devoted to a unifying description of fidelity loss and decoherence in many-qubit systems.

Complex scattering matrix elements have been measured in experiments with chaotic microwave billiards at Darmstadt. They exhibit fluctuations as a function of the incident frequency. The resonances of the cavity are weakly overlapping. This regime is crucial for a test of the correlation function derived by Verbaarschot, Weidenmueller and Zirnbauer. Within experimental accuracy the Fourier coefficients of the scattering matrix are statistically independent Gaussian random variables. Several ten thousand of coefficients have been obtained. They are the basis for the test of the VWZ-formula and thus of random matrix theory.

We discuss an approach for studying the properties of mesoscopic systems for which discrete and continuum parts of the spectrum are equally important. The approach can be applied to heavy nuclei and complex atoms near the continuum threshold, as well as to open mesoscopic devices with interacting electrons. We developed a new consistent version of the continuum shell model that simultaneously takes into account strong interaction between fermions and coupling to the continuum. Main attention is paid to the formation of compound resonances, their statistical properties, and correlations between the cross sections. Specific interest is in the properties of the Ericson fluctuations, in comparison with the conductance fluctuations.

We study weak scattering of Gaussian random ocean waves by Gaussian random currents. Scattering creates persistent, branch-like, local energy density variations, which survive dispersion over wavelength and incoming wave propagation direction. These variations lead to a long, non-Gaussian tail in the wave height distribution. The resulting distribution depends primarily on the "freak index", which measures the strength of refraction relative to the angular spread of the incoming sea. For parameter values typically occurring in nature, the tail of the probability distribution can be easily enhanced by a factor of 10 to 100 as compared with the random wave model. Interesting analogies exist with 2DEG electron transport and recent experiments in disordered microwave cavities.

I will review some applications and implications of semiclassical resummation theory for spectral statistics.

Supersymmetry can be used even for non-Gaussian probability densities. There are two approaches which will be presented: the generalized Hubbard-Stratonovitch transformation and the superbosonization formula. I discuss connections and differences. The orthogonal, unitary and unitary-symplectic symmetry classes are treated in a unifying way. In the course of doing so, various new identities for functions in superspaces are found.

Transport models were first introduced to heavy ion physics at collision energies below the Coulomb energy. Using a random matrix model for the matrix elements of the coupling between the collective and the non collective degrees of freedom, Weidenmueller and his collaborators derived a transport equation from the coupled-channel reaction theory for understanding the so-called heavy ion deeply inelastic collisions. Transport models based on hadronic degrees of freedom were then used to study heavy ion collisions at high energies, resulting in significantly advanced knowledge on the nuclear equation of state at high densities and the in-medium properties of hadrons. Including also the partonic degrees of freedom in the initial stage of ultrarelativistic heavy ion collisions has further added to our understanding of the produced partonic matter and their effects on experimental observables. In this talk, some of the achieved results in heavy ion collisions based on transport models will be reviewed and possible improvements of the transport models will be discussed.

The doorway mechanism explains spectral properties in a rich variety of open mesoscopic quantum systems, ranging from atoms to nuclei. A distinct state and a background of other states couple to each other which sensitively affects the strength function. We provide an RMT description for the doorway mechanism and calculate exactly the distribution of the maximal coupling strength to the doorway state using a supersymmetric two matrix model.

We suggest and study Gaussian ensembles of random matrices which mimic all essential physics, both in the insulator and in the metal phase of disordered quantum systems close to the Anderson transition.
We compare the local DoS correlation function computed from such random matrix ensembles with that for the 3D Anderson model, studied numerically. Furthermore, we apply the random matrix ensembles to describe the electron paring close to Anderson transition.

Suspensions of small anisotropic particles, termed rheoscopic fluids, are used for flow visualisation. It is commonly assumed that the principal axis of the particles align with the straining direction of the flow (it is assumed that the particles follow the flow). But here we show that the direction field of the principal axes of small rod-like particles suspended in a thin layer of randomly moving fluid exhibits singularities not present in the underlying flow. We explain their occurence and characterise their forms employing a theory based on random monodromy matrices of the flow. We discuss how the singularities can be visualised in a light-scattering experiment. This is joint work performed with M. Wilkinson and V. Bezuglyy. It is based on the preprint: Fingerprints of Random Flows? M. Wilkinson, V. Bezuglyy, and B. Mehlig, submitted to Phys. Fluids (2008).

The statistical theory of complex wave interference phenomena is of considerable interest in many fields of physics: the statistical fluctuations of transmission and reflection of waves by disordered media is an important example. In this talk we shall be interested in those situations where the complexity derives from the quenched randomness of scattering potentials, as in the case of disordered conductors, or, more in general, disordered waveguides.

In studies performed in such systems one has found remarkable statistical regularities: the probability distribution for various macroscopic quantities involves a rather small number of relevant physical parameters, while the rest of the microscopic details serves as mere ``scaffolding". We shall review past work in which this feature was captured following a maximum-entropy approach within a random-matrix-theory framework, as well as later studies in which the existence of a limiting distribution, in the sense of a generalized central-limit theorem, was actually demonstrated. We shall then describe a potential model that was developed recently, which gives rise to a further generalization of the central-limit theorem --and thus to a generalized limiting macroscopic statistics-- and allows the description of certain statistical features which were not described properly by previous work.

I will show recent results for the distribution of the S-matrix in the presence of both absorption and direct processes. Results for the average of the reflection coefficient will be given as well as the distribution for N-channels that we recently obtained.

Recently it has been shown that cavities with the property of unidirectionality may display spectral statistics close to the Gaussian unitary ensemble rather than to the expected orthogonal one. The later predicted for systems with time reversal invariance and classically chaotic dynamics. As a consequence, the scattering properties of such cavities are expected to have also the signatures of unitary ensemble statistics. Here we show, for a class of unidirectional cavities, that while a set of quantities (S-matrix elements, conductance distribution) show signatures of unitary ensemble statistics some others (variance of conductance, shot noise power) are well predicted by the orthogonal ensemble.

There are multiple connections between the nucleus and Random Matrix Theory (RMT). The nucleus can serve as a test laboratory to evaluate the predictions of RMT. One can also assume RMT and use the theory as a tool to determine nuclear properties. The former is illustrated by measuring the effect of symmetry breaking on the distributions of nuclear widths and spacings. We consider the approximate symmetry isospin. The latter approach is illustrated by the use of RMT to determine characteristics of parity violation in neutron resonances.

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Chaos occurs in quantum systems if the statistical properties of the eigenvalue spectrum coincide with predictions of random-matrix theory. Chaos is a typical feature of atomic nuclei and other self-bound Fermi systems. How can the existence of chaos be reconciled with the known dynamical features of spherical nuclei? Such nuclei are described by the shell model (a mean-field theory) plus a residual interaction. The question is answered using a statistical approach (the two-body random ensemble): The matrix elements of the residual interaction are taken to be random variables. Chaos is shown to be a generic feature of the ensemble and some of its properties are displayed, emphasizing those which differ from standard random-matrix theory. In particular, the existence of correlations among spectra carrying different quantum numbers is demonstrated. These are subject to experimental verification.

We present a random matrix model that uses the classical ensembles and apply it paradigmatically to the problem of decoherence and internal entanglement decay of non--interacting qubits. The average density matrix and purity are available in linear response approximation. Single qubit decoherence (as measured by purity) is obtained in linear response. Two qubit decoherence is solved using the spectator configuration. Entanglement decay, for tow qubits, is considered both numerically and analytically using tools of quantum information. Ensemble averages can be taken at different stages; resulting corrections are small, easing some calculation and relaxing the requirements for an eventual experimental setup. Various results are exemplified using a kicked spin chain as a toy model, shading light on how to apply our results to interacting systems. The linear response solutions are sufficient for any quantum information application, yet we shall outline the possibilities to find exacts answers at least for one qubit.

We study the properties of a two-body random matrix ensemble for distinguishable spins. We require the ensemble to be invariant under the group of local transformations and analyze a parametrization in terms of the group parameters and the remaining parameters associated with the "entangling" part of the interaction. We then specialize to a spin chain with nearest neighbour interactions and numerically find a new type of quantum phase transition related to the strength of a random external field i.e. the time reversal breaking one body interaction term.

We compare predictions from Random Matrix Theory (RMT) with detailed numerical calculations for scattering in mesoscopic devices, such as quantum dots [1] and superconducting hybrid structures ("Andreev systems") [2]. In particular, we investigate the question if the unavoidable presence of disorder in the experiment makes realistic mesoscopic systems amenable to a random matrix description. For this purpose we study the distribution of transmission eigenvalues in transport through quantum dots and the density of states in Andreev systems. Our results clearly show in which limits RMT works fine for describing disordered scattering devices and in which other limits alternative descriptions must be chosen.

[1] S. Rotter, F. Aigner, and J. Burgdoerfer, Phys. Rev. B 75, 125312 (2007).
[2] F. Libisch, J. Moeller, S. Rotter, M. Vavilov, and J. Burgdoerfer, Europhys. Lett. 82, 47006 (2008).

Statistical properties of the conductance and the shot-noise power are considered for a chaotic cavity with arbitrary numbers of open channels. In the framework of the random matrix approach, we establish the relevance of the Selberg's integral theory to problems of chaotic quantum transport, and apply it to calculate all the moments of transmission eigenvalues analytically up to the fourth order. As a result, we derive exact explicit expressions for the skewness and kurtosis of the conductance as well as for the variance of the shot-noise power in chaotic cavities. Relevant results for higher cumulants and distributions of these quantities are also discussed.

The development of a modern and more realistic nuclear energy density functional (EDF) for accurate predictions of properties of nuclei is the subject of enhanced activity, since it is very important for the study of properties of rare nuclei with unusual neutron-to-proton ratios that are difficult to produce experimentally and likely to exhibit interesting new phenomena associated with isospin, clusterization and the continuum.

Adopting the standard parametrization of the Skyrme type interactions, we have recently determined within the Hartree-Fock (HF) approximation a new and more realistic Skyrme interaction by carrying out, using the simulating annealing method (SAM), a fit to an extensive set of experimental data on; (i) binding energies, (ii) "bare single-particle energies, (iii) charge root-mean-square (rms) radii, (iv) rms radii of valence neutron orbits, and (v) the energies of isoscalar giant monopole resonances (ISGMR). We have also imposed additional constraints; (i) Landau stability constraints on nuclear matter (NM), and (ii) non-negativity of the slope of the symmetry energy density at high density of NM, which is of importance in the study of properties of neutron star.

A modern EDF with improved predictive power for properties of nuclei is being developed by addressing the issues of: (i) The isospin dependence of the spin-orbit (SO) interaction. (ii) The effects of long-range correlations on properties of nuclei. We point out that for consistency, one must require that the comparison with experimental data be made after the inclusion of the effects of correlations. (iii) The equation of state (EOS) of NM, i.e., the density dependence of energy. The EOS of NM is an important ingredient in the study of various properties of nuclei, heavy ion collisions, supernovae and neutron stars. Accurate values of the NM incompressibility, K, and the symmetry energy, J, coefficients are needed in order to extend our knowledge of the EOS of NM.

We developed and applied several mathematical methods and computerized tools to solve various problems in archeological research. I shall talk in particular about the analysis and classification of ancient pottery and pre-historical lithic tools. The power of our methods will be demonstrated by discussing a few studies where we were able to solve long standing archaeological debates, or to gain completely new insights.

To study the properties of a system such as its spectrum it has to be opened leading to a modification of the system properties. Thus every measurement unavoidably gives an unwanted combination of the properties of the system and the apparatus.

Scattering theory is the method of choice to cope with this situation. Originally developed in nuclear physics [1], it meanwhile has found numerous applications also in mesoscopic systems [2]. An example is the study of the transport properties of open quantum dots. These measurements, however, are difficult: (i) it is non-trivial determine the confining potential from the geometry of the gate electrodes, (ii) impurities are unavoidable and difficult to control, (iii) the systems typically are of sub-micron size, and (iv) temperatures of the order of mK or even lower are needed.

Here microwave resonators pose an alternative. For flat resonators there is a complete equivalence to the corresponding quantum dot system, as long as electron-electron interactions are negligible [3]. System sizes are of the order of centimeter, the measurements can be performed at room temperature, and the geometry is perfectly controllable. Furthermore, a detailed look into the system is possible, whereas standard quantum dot techniques only allow the study of global transport properties.

I shall illustrate these features of the microwave technique by three examples. First, results on the emission patterns of deformed micro-resonators will be presented. Shape optimization to enhance emission in given directions is a topical research subject in optoelectronics. Using microwaves the size of the resonators is promoted from the micron to the centimeter regime because of the much larger wavelengths of the microwaves as compared to those in the optical regime.

From scanning probe microscopy studies it is known that the electron flow through quantum contacts does not show the diffraction pattern which might have been expected from elementary quantum mechanics but exhibiters an intricate branching structure instead [4]. It had been conjectured by Kaplan [5] that these features are a manifestation of caustics generated by the background potential due to impurities and charged donors. In a microwave realisation of the flow through a potential landscape we had been able to verify the caustics conjecture, where the potential had been mimicked by a variation of the resonator height. The result may be applied as well to the wave patterns produced in the ocean giving new insight into the formation of rogue waves.

In the last part I shall present some recent results on the analysis of the poles of the scattering matrix in the regime of strong overlap. There are numerous results on average quantities in open systems such as distribution and correlation functions of reflection and transmission coefficients. The direct measurement of the poles, however, had not been accessible up to now. Here a recent progress had been achieved by using the method of Harmonic Inversion [6]. In this technique poles and residua of the scattering matrix are obtained from the eigenvalues of a matrix which can be generated from the experimental spectra. No foreknowledge on the number of contributing eigenvalues is needed for this purpose. First results on the distributions of linewidths in the regime of strong overlap are presented.

[1] T. Guhr, A. M\"uller-Gr\"oling, H. Weidenm\"uller, Phys. Rep. 299, 189 (1998)
[2] C. Beenakker, Rev. Mod. Phys. 69, 731 (1997)
[3] H.-J. St\"ockmann, Quantum Chaos - An Introduction, Cambridge University Press 1999.
[4] M. Topinka et al., Science 289, 183 (2000)
[5] L. Kaplan, Phys. Rev. Lett. 89, 184103 (2002)
[6] J. Main, Phys. Rep. 316, 233 (1999)

Random states have the statistical properties of the Gaussian and Circular Ensemble eigenstates. Even though the components are correlated by the normalization constraint, for the unitary ensemble it is nevertheless possible to derive compact analytic formulae for the extreme intensity values' statistical properties for all dimensionalities. The maximum intensity result slowly approaches the Gumbel distribution even though the variables are bounded, whereas the minimum intensity result rapidly approaches the Weibull distribution. Since random matrix theory is conjectured to be applicable to chaotic quantum systems, we calculate the extreme eigenfunction statistics for the standard map with parameters at which its classical map is fully chaotic. The statistical behaviors are consistent with the finite-$N$ formulae. An approximation is also introduced for the orthogonal ensembles as well.

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.

Random Matrix Theory has become an established paradigm in quantum field theories. In particular, chiral random matrix theory is now an essential ingredient for understanding the i spectrum of the QCD Dirac operator. We explain the foundations of these ideas and review the main developments of this field during the past 15 years. We conclude with a short discussion of recent results for QCD at nonzero chemical potential.

The QCD Dirac operator becomes non-hermitian at nonzero quark density. Its spectral correlations can be computed in lattice QCD and described by non-hermitian random matrix theory. We have generalized the overlap lattice Dirac operator to nonzero density. This operator has exact zero modes at finite lattice spacing and therefore allows us to verify the RMT predictions for nontrivial topological charge. We have also generalized the domain-wall operator to nonzero density and shown that it converges to the overlap operator if the extent of the fifth dimension is taken to infinity.

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We sketch a proof of universality of local level correlation functions for non-Gaussian invariant random matrix ensembles, by using a method based on the so-called superbosonization formula in combination with elements of free probability theory. Superbosonization, a variant of the method of commuting and anticommuting variables, eclipses the traditional Hubbard- Stratonovich transformation in that it is not restricted to Gaussian probability distributions. In this talk we consider random matrices H distributed according to a probability measure of the form exp(-N Tr V(H)) dH with V being a polynomial. To apply the superbosonization formula, one needs to have control of the Fourier transform of the measure in the limit of infinite matrix size N. We show this Fourier transform to be determined by a key notion in free probability theory: the Voiculescu R-transform of the asymptotic level density.

Recently, the mass spectra of hadrons have been studied in the context of quantum chaos, leading to a contradiction between results based on the experimental masses (chaos) [1] and the results based on theoretical masses from quark models (regularity or integrability) [2], which has been explained as due to an inherent incapacity of quark models to reproduce the hadron mass-spectrum. In this contribution, we show the sensitivity of the statistical results on the process of unfolding that is used to separate the global from the local density variation of the hadron masses with excitation energy. We find that the apparent contradiction between experiment and theory is more probable to be an artifact of the unfolding process than an unsuitability of quark models to describe the lower part of the hadron mass-spectrum.
[1] V. Pascalutsa, Eur. Phys. J. A16 (2003) 149.
[2] C. Fernández-Ramírez and A. Relaño, Phys. Rev. Lett. 98 (2007) 062001.

A Detrended Fluctuation Analysis (DFA) method is applied to investigate the scaling properties of the energy fluctuations in the spectrum of 48-Ca obtained with a large realistic shell model calculation (ANTOINE code) and with a random shell model (TBRE) calculation. We compare the scale invariant properties of the 48-Ca nuclear spectrum with similar analyses applied to the RMT ensembles GOE and GDE. A comparison with the corresponding power spectra analysis is made.

We investigate the distribution of the lowest-lying energy states in a disordered Andreev billiard by solving the Bogoliubov-de Gennes equation numerically. Contrary to conventional predictions we find a decrease rather than an increase of the excitation gap relative to its clean ballistic limit. We relate this finding to the eigenvalue spectrum of the Wigner-Smith time delay matrix between successive Andreev reflections. We show that the longest rather than the mean time delay determines the size of the excitation gap. With increasing disorder strength the values of the longest delay times increase, thereby, in turn, reducing the excitation gap.

We are used to think in terms of real functions and orthogonal ensembles, whne dealing with time-reversal invariant problems in RMT. We also believe that averaging over the GOE or COE already implies a state average, and this is perfectly true, when dealing with the expectation values of Hermitean operators, i.e. with usual concept of measurement. Yet if we deal with nonlinear properties such as purity, fidelity, concurrence, or even properties as simple as the inverse participation ratio this is no longer true, and we shall briefly explain why, and what consequences we expect.

Important properties of complex scattering systems are revealed through a combined description of internal structure and reaction process. This allows one to probe the interplay between the degree of internal chaos and the irreversible decay into continuum. Main attention is paid to the important role of correlations between different cross sections; in particular, universal properties of conductance fluctuations are nicely described within the framework of our model. The approach can be applied to various many-body systems such as stable heavy nuclei, nuclei far from the region of stability and complex mesoscopic devices with interacting electrons. We also discuss a one-body analogue that can be applied to open microwave cavities.

We study the statistical properties of wave scattering in a disordered waveguide starting from a potential model. The scattering units consist of delta slices perpendicular to the longitudinal direction, the variation of the potential in the transverse direction being arbitrary. The parameters defining the various slices are statistically independent and identically distributed. In the dense-weak-scattering limit, in which the potential slices are very weak and their linear density is very large, so that the resulting mean free paths are fixed, the statistical properties of the waveguide depend only on the mean free paths and on no other property of the slice distribution. This universality demonstrates the existence of a generalized central-limit theorem. Our final result is a diffusion equation in the space of transfer matrices of the system: it describes the evolution of the transport properties with the length L of the waveguide. In contrast with earlier publications, in the present analysis the energy is fully taken into account. For one propagating mode we have solved the diffusion equation in some particular cases, the solution being in excellent agreement with microscopic calculations. In general we have not succeeded in solving the equation. We have thus developed a numerical simulation, called ”random walk in the transfer matrix space”, in which the universal statistical properties of a ”building block” of thicknesss δL, which are derived analytically, are first implemented numerically, and then the various building blocks are combined to construct the full waveguide. The reported results are in very good agree- ment with microscopic calculations, for both bulk and surface disorder.