Polimeri diretti in ambiente aleatorio

In questo elaborato viene studiato il comportamento asintotico di un polimero diretto in ambiente aleatorio (o disordine) al variare della temperatura e della dimensione in cui lo si considera. Si dimostra che quando d=1,2 il polimero si comporta in modo super diffusivo, ovvero la catena polimerica dispone i suoi monomeri lontano dall'origine per raggiungere zone in cui l'ambiente risulta maggiormente favorevole; al contrario, quando d è > o = 3, si verifica una transizione tra una fase diffusiva(valori alti della temperatura), in cui le traiettorie della catena si comportano, su larga scala, come un moto browniano, ed una fase super diffusiva(basse temperature).

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.