Table of contents

We study the integrable XXZ model with general non-diagonal boundary terms at both ends. The Hamiltonian is considered in terms of a two-boundary extension of the Temperley–Lieb algebra.

We use a basis that diagonalizes a conserved charge in the one-boundary case. The action of the second boundary generator on this space is computed. For the L-site chain and generic values of the parameters we have an irreducible space of dimension 2^L. However, at certain critical points there exists a smaller irreducible subspace that is invariant under the action of all the bulk and boundary generators. These are precisely the points at which Bethe ansatz equations have been formulated. We compute the dimension of the invariant subspace at each critical point and show that it agrees with the splitting of eigenvalues, found numerically, between the two Bethe ansatz equations.

We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe–Peierls approximation appears naturally as the saddle-point approximation. The corrections to the saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetization and correlation function.

I construct a sandpile model for the evolution of the energy spectrum of water waves in finite basins. This model takes into account loss of resonant wave interactions in discrete Fourier space and restoration of these interactions at larger nonlinearity levels. For weak forcing, the wave action spectrum takes a critical ω⁻¹⁰ shape where the nonlinear resonance broadening overcomes the effect of the Fourier grid spacing. The energy cascade in this case takes the form of rare weak avalanches on the critical slope background. For larger forcing, this regime is replaced by a continuous cascade and a Zakharov–Filonenko ω⁻⁸ wave action spectrum. For intermediate forcing levels, both scalings will be relevant, ω⁻¹⁰ at small frequencies and ω⁻⁸ at large frequencies, with a transitional region in between, characterized by strong avalanches.

We consider a chain of coupled and strongly pinned anharmonic oscillators subject to a non-equilibrium random forcing. Assuming that the stationary state is approximately Gaussian, we first derive a stationary Boltzmann equation. By localizing the involved resonances and identifying the umklapp processes, we next invert the linearized collision operator and compute the heat conductivity. In particular, we show that the Gaussian approximation yields a finite conductivity κ ∼ 1/λ²T², for λ the anharmonic coupling strength.

We study a junction of three quantum wires enclosing a magnetic flux. The wires are modelled as single-channel spinless Tomonaga–Luttinger liquids. This is the simplest problem of a quantum junction between Tomonaga–Luttinger liquids in which Fermi statistics enter in a non-trivial way. We study the problem using a mapping onto the dissipative Hofstadter model, describing a single particle moving on a plane in a magnetic field and a periodic potential coupled to a harmonic oscillator bath. Alternatively we study the problem by identifying boundary conditions corresponding to the low energy fixed points. We obtain a rich phase diagram including a chiral fixed point in which the asymmetric current flow is highly sensitive to the sign of the flux and a phase in which electron pair tunnelling dominates. We also study the effects on the conductance tensor of the junction of contacting the three quantum wires to Fermi liquid reservoirs.

Within the framework of field-theoretical description of second-order phase transitions via the three-dimensional O(N) vector model, accurate predictions for critical exponents can be obtained from (resummation of) the perturbative series of renormalization-group functions, which are in turn derived—following Parisi's approach—from the expansions of appropriate field correlators evaluated at zero external momenta.

Such a technique was fully exploited 30 years ago in two seminal works of Baker, Nickel, Green and Meiron, which led to the knowledge of the β-function up to the six-loop level; they succeeded in obtaining a precise numerical evaluation of all needed Feynman amplitudes in momentum space by lowering the dimensionalities of each integration with a cleverly arranged set of computational simplifications. In fact, extending this computation is not straightforward, due both to the factorial proliferation of relevant diagrams and the increasing dimensionality of their associated integrals; in any case, this task can be reasonably carried on only in the framework of an automated environment.

On the road towards the creation of such an environment, we here show how a strategy closely inspired by that of Nickel and co-workers can be stated in algorithmic form, and successfully implemented on a computer. As an application, we plot the minimized distributions of residual integrations for the sets of diagrams needed to obtain RG functions to the full seven-loop level; they represent a good evaluation of the computational effort which will be required to improve the currently available estimates of critical exponents.

The social network formed by the collaboration between rappers is studied using standard statistical techniques for analysing complex networks. In addition, the community structure of the rap music community is analysed using a new method that uses weighted edges to determine which connections are most important and revealing among all the communities. The results of this method as well as possible reasons for the structure of the rap music community are discussed.

'Oscillations' occur in quite different kinds of many-particle systems when two groups of particles with different directions of motion meet or intersect at a certain spot. In this work a model of pedestrian motion is presented that is able to reproduce oscillations with different characteristics. The Wald–Wolfowitz test and Gillis' correlated random walk are shown to include observables that can be used to characterize different kinds of oscillations.

We report on the computer study of a lattice system that relaxes from a metastable state. Under appropriate nonequilibrium randomness, relaxation occurs by avalanches, i.e., the model evolution is discontinuous and displays many scales in a way that closely resembles the relaxation in a large number of complex systems in nature. Such apparent scale invariance simply results in the model from summing over many exponential relaxations, each with a scale which is determined by the curvature of the domain wall at which the avalanche originates. The claim that scale invariance in a nonequilibrium setting is to be associated with criticality is therefore not supported. Some hints that may help in checking this experimentally are discussed.

We use multifractal detrended fluctuation analysis (MF-DFA), to study sunspot number fluctuations. The result of the MF-DFA shows that there are three crossover timescales in the fluctuation function. We discuss how the existence of the crossover timescales is related to a sinusoidal trend. Using Fourier detrended fluctuation analysis, the sinusoidal trend is eliminated. The Hurst exponent of the time series without the sinusoidal trend is 0.12 ± 0.01. Also we find that these fluctuations have multifractal nature. Comparing the MF-DFA results for the remaining data set to those for shuffled and surrogate series, we conclude that its multifractal nature is almost entirely due to long range correlations.

We use the two-step density-matrix renormalization group method to study the effects of frustration in Heisenberg models for S = ½ to 4 in a two-dimensional anisotropic lattice. We find that as for S = ½ studied previously, the system is made up of nearly disconnected chains at the maximally frustrated point, J_d/J_⊥ = 0.5, i.e., the transverse spin–spin correlations decay exponentially. This leads to the following consequences: (i) all half-integer spins systems are gapless, behaving like a sliding Luttinger liquid as for S = ½; (ii) for integer spins, there is an intermediate disordered phase with a spin gap, with the width of the disordered state roughly proportional to the 1D Haldane gap.

We present experimental results for pattern formation in a thin horizontal fluid layer heated from below. The fluid was SF₆ at a pressure of 20.0 bar with a Prandtl number of 0.87. The cylindrical sample had an interior section of uniform spacing d = d₀ for radii r < r₀ and a ramp d(r) for r > r₀. For Rayleigh numbers R₀ > R_c in the interior, straight or slightly curved rolls with an average wavenumber (ε₀ ≡ R₀/R_c - 1) with were selected. The critical wavenumber depended sensitively on the cell spacing. For the largest the patterns were skewed-varicose unstable and dislocation pairs were generated repeatedly in the interior and for all ε. For slightly smaller time-independent rolls were stable for , but for larger ε the skewed-varicose instability was encountered near the sample centre and dislocation pairs were formed repeatedly for all samples. When stationary rolls were stable, their slight curvature and the width of their wavenumber distribution slowly increased with ε. This led to a complicated shape and overall broadening of the structure factor S(k). For the inverse width ξ₂ of S(k) was roughly constant and presumably limited by the finite sample size, but for larger ε we found .