Quantum solitons in the XXZ model with staggered external magnetic field

The 1-D 1/2-spin XXZ model with staggered external magnetic field, when restricting to low field, can be mapped into the quantum sine-Gordon model through bosonization: this assures the presence of soliton, antisoliton and breather excitations in it. In particular, the action of the staggered field opens a gap so that these physical objects are stable against energetic fluctuations. In the present work, this model is studied both analytically and numerically. On the one hand, analytical calculations are made to solve exactly the model through Bethe ansatz: the solution for the XX + h staggered model is found first by means of Jordan-Wigner transformation and then through Bethe ansatz; after this stage, efforts are made to extend the latter approach to the XXZ + h staggered model (without finding its exact solution). On the other hand, the energies of the elementary soliton excitations are pinpointed through static DMRG (Density Matrix Renormalization Group) for different values of the parameters in the hamiltonian. Breathers are found to be in the antiferromagnetic region only, while solitons and antisolitons are present both in the ferromagnetic and antiferromagnetic region. Their single-site z-magnetization expectation values are also computed to see how they appear in real space, and time-dependent DMRG is employed to realize quenches on the hamiltonian parameters to monitor their time-evolution. The results obtained reveal the quantum nature of these objects and provide some information about their features. Further studies and a better understanding of their properties could bring to the realization of a two-level state through a soliton-antisoliton pair, in order to implement a qubit.

Passive based control on a kuka arm

L'interazione in maniera sicura e compliante è una caratteristica sempre più richiesta per i sistemi robotici. La modellazione di sistemi eseguita tramite l'uso di sistemi port-Hamiltoninani permette di comprendere cosa avviene a livello energetico durante l'interazione e aiuta nella progettazinoe di un controllore tale che il comportamento del sistema controllato sia passivo e sicuro durante essa. Ciò sfocia nel cosiddetto Controllore Intrinsicamente Passivo (IPC). Dal momento che questo un controllo impone la rigidezza desiderata al sistema controllato, è possibile, tra le altre cose, replicare il comportamento del dispositivo RCC (Centro Remoto di Complianza) e di migliorarlo in modo tale che durante l'azione di peg-in-hole il buco sia meno sollecitato dal robot.

Limiting theorems in statistical mechanics. The mean-field case.

The Curie-Weiss model is defined by ah Hamiltonian according to spins interact. For some particular values of the parameters, the sum of the spins normalized with square-root normalization converges or not toward Gaussian distribution. In the thesis we investigate some correlations between the behaviour of the sum and the central limit for interacting random variables.

Applications of elliptic functions to solve differential equations

In questa tesi si mostrano alcune applicazioni degli integrali ellittici nella meccanica Hamiltoniana, allo scopo di risolvere i sistemi integrabili. Vengono descritte le funzioni ellittiche, in particolare la funzione ellittica di Weierstrass, ed elenchiamo i tipi di integrali ellittici costruendoli dalle funzioni di Weierstrass. Dopo aver considerato le basi della meccanica Hamiltoniana ed il teorema di Arnold Liouville, studiamo un esempio preso dal libro di Moser-Integrable Hamiltonian Systems and Spectral Theory, dove si prendono in considerazione i sistemi integrabili lungo la geodetica di un'ellissoide, e il sistema di Von Neumann. In particolare vediamo che nel caso n=2 abbiamo un integrale ellittico.

Alternative derivative expansion in Functional RG and application

We give a brief review of the Functional Renormalization method in quantum field theory, which is intrinsically non perturbative, in terms of both the Polchinski equation for the Wilsonian action and the Wetterich equation for the generator of the proper verteces. For the latter case we show a simple application for a theory with one real scalar field within the LPA and LPA' approximations. For the first case, instead, we give a covariant "Hamiltonian" version of the Polchinski equation which consists in doing a Legendre transform of the flow for the corresponding effective Lagrangian replacing arbitrary high order derivative of fields with momenta fields. This approach is suitable for studying new truncations in the derivative expansion. We apply this formulation for a theory with one real scalar field and, as a novel result, derive the flow equations for a theory with N real scalar fields with the O(N) internal symmetry. Within this new approach we analyze numerically the scaling solutions for N=1 in d=3 (critical Ising model), at the leading order in the derivative expansion with an infinite number of couplings, encoded in two functions V(phi) and Z(phi), obtaining an estimate for the quantum anomalous dimension with a 10% accuracy (confronting with Monte Carlo results).