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Informations

We expect to provide this course now, in the 2022 Spring semester.

Lecturer: Imre VARGA, assoc. prof. of Department of Theoretical Physics.

Requirements: good background in quantum mechanics, statistical mechanics and

condensed matter phyiscs if possible.

Topics on: Random matrix theory and physical applications

This is a previous list. This year there might be some refreshing but similar topics may appear

Examination: working out a topic or a paper that uses the basic notions of RMT.

1. Introduction. Invariance properties of the Hamiltonian matrix. The eigenvalue problem of a 2x2 matrix, level-repulsion, the Wigner-surmise.

Probability theory for matrices: measure, metric, maxent (entropy maximization) principle, Jacobian. The joint probability distribution

of the eigenvalues. Universality classes, the 3-fold way of Wigner and Dyson.

2. Analysis of ‘real’ spectra: unfolding, the Wigner-semicircle law, spectral rigidity, the Delta3-function. The hydrodynamic model

– Coulomb fluid. The mean-field theory, the Vandermonde determinant, the method of orthogonal polynomials.

The R_{2 }and the Delta3-function in the thermodynamic limit, pair correlation function, form function, linear statistics

and their variance, ergodicity. Statistical properties of the eigenvector components.

3. The statistical theory of transmission coefficients: scattering matrix and transfer matrix, polar decomposition, universality classes.

UCF, circular ensembles, the hydrodynamic model of transmission eigenvalues.

4. The BGS conjecture, quantum chaos, counterexamples. Further possibilities of generalization (soft confinement and strong repulsion).

Random band matrix models.

5. Crossover between ensembles. Level dynamics (level velocity and curvature). Local and global perturbation.

The Brownian motion model. The Pechukas-Yukawa model. The Lenz-Haake construction (convolution), the preferential basis

(Pichard-Shapiro), and the maxent method of Hussein-Pato.

6. The metal insulator transition induced by disorder. The basic tight-binding model, energy, length and time scales. The applicability

of RMT for the case of disordered metals and insulators. The problem of the critical statistics, dependence on boundary conditions,

the semi-Poissson statistics. The PBRM (power-law band random matrix ensemble), spectral statistics and multifractal eigenstates.

7. Application for headway traffic of buses in Mexico and highway traffic in Germany. RMT analysis of time series of EEG signals,

internet traffic and economical processes (stock market). Covariance matrices, Wishart-matrices, the RMT test and deviations.

8. RMT in many body physics, the Fock space, the TBRE (two-body random interaction ensemble), deformed TBRE, the Poisson-RMT

crossover, the Aberg-limit, delocalization in Fock-space. New ensembles using chirality (parity) and charge conjugation. Application

to the Dirac problem and hybrid (superconductor-normal) systems. The 10-fold way of Altland and Zirnbauer.