Spectral statistics of chaotic many-body systems

We derive a trace formula that expresses the level density of chaotic many-body systems as a smooth term plus a sum over contributions associated to solutions of the nonlinear Schr\"odinger (or Gross-Pitaevski) equation. Our formula applies to bosonic systems with discretised positions, such as the Bose-Hubbard model, in the semiclassical limit as well as in the limit where the number of particles is taken to infinity. We use the trace formula to investigate the spectral statistics of these systems, by studying interference between solutions of the nonlinear Schr\"odinger equation. We show that in the limits taken the statistics of fully chaotic many-particle systems becomes universal and agrees with predictions from the Wigner-Dyson ensembles of random matrix theory. The conditions for Wigner-Dyson statistics involve a gap in the spectrum of the Frobenius-Perron operator, leaving the possibility of different statistics for systems with weaker chaotic properties.Comment: 29 pages, 3 figure

Neveu-Schwarz and operators algebras III: Subfactors and Connes fusion

This paper is the third of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we introduce the local von Neumann algebra of the Neveu-Schwarz algebra, to obtain Jones-Wassermann subfactors for each representation of the discrete series. Then using primary fields we prove the irreducibility of these subfactors; to next compute the Connes fusion ring and obtain the explicit formula of the subfactors indices.Comment: 54 page

Classification of Module Categories for SO(3)2mSO(3)_{2m}

The main goal of this paper is to classify $\ast$-module categories for the $SO(3)_{2m}$ modular tensor category. This is done by classifying $SO(3)_{2m}$ nimrep graphs and cell systems, and in the process we also classify the $SO(3)$ modular invariants. There are module categories of type $\mathcal{A}$, $\mathcal{E}$ and their conjugates, but there are no orbifold (or type $\mathcal{D}$) module categories. We present a construction of a subfactor with principal graph given by the fusion rules of the fundamental generator of the $SO(3)_{2m}$ modular category. We also introduce a Frobenius algebra $A$ which is an $SO(3)$ generalisation of (higher) preprojective algebras, and derive a finite resolution of $A$ as a left $A$-module along with its Hilbert series.Comment: 56 pages, many figures; corrected error at the end of Section 4 about $\mathcal{E}_8$ nimrep, and corrected computational error in Theorem 5.10 about $\mathcal{E}_8^c$. The main theorem, Theorem 5.12, has been modified to reflect these correction

Complex Behaviour in Coupled Oscillators, Coupled Map Lattices and Random Dynamical Systems.

PhD ThesesThis thesis consists of three detailed studies on complex behaviour of deterministic and random dynamical systems. In the rst study, coupled nonlinear oscillator systems are proposed to model a future detection scheme of axionic dark matter particles in a Josephson junction environment. By studying initial value problems we observe rich phase space structures under variations of parameters such as the coupling constant, the strength of an external magnetic eld and the initial conditions. In the limit of small elongations, we obtain analytic solutions to the linearised equations of motion, with and without dissipation, and prove a time-shifted synchronisation between the two oscillators, with comparisons to the nonlinear case. The second study focuses on distinguished correlation properties of a family of shifted Chebyshev maps (TN;a). We present analytic results for two-point and higher-order correlation functions and show that TN;0 are most random-like among all smooth one-dimensional maps conjugated to an Nary shift, in the sense that they have least higher-order correlations. We discuss the eigenfunctions of the Perron-Frobenius operator for TN;a. The spectrum is degenerate for odd N. We then consider coupled map lattices of TN;a and numerically investigate zeros of the spatial and temporal nearestneighbour correlations. The third study concerns a random dynamical system that samples between a contracting and a chaotic map with a certain probability p in time. We rst derive an explicit expression for the invariant density. Due to long memory of history we consider a Markovian approximation for this problem and study two-point correlation functions when varying the parameter p, with emphasis on transitions between an exponential decay (at p = 1) and a power-law decay (when p ! 1=2). Finally we work towards determining the type of di usion generated by sums of iterates of this random map

Propagating wave correlations in complex systems

We describe a novel approach for computing wave correlation functions inside finite spatial domains driven by complex and statistical sources. By exploiting semiclassical approximations, we provide explicit algorithms to calculate the local mean of these correlation functions in terms of the underlying classical dynamics. By defining appropriate ensemble averages, we show that fluctuations about the mean can be characterised in terms of classical correlations. We give in particular an explicit expression relating fluctuations of diagonal contributions to those of the full wave correlation function. The methods have a wide range of applications both in quantum mechanics and for classical wave problems such as in vibro-acoustics and electromagnetism. We apply the methods here to simple quantum systems, so-called quantum maps, which model the behaviour of generic problems on Poincaré sections. Although low-dimensional, these models exhibit a chaotic classical limit and share common characteristics with wave propagation in complex structures