Synthesis of enantiopure cyclopentanones by desymmetrization of allylic cyclobutanols

The interest in five-membered ring molecules derives from their important application in many different fields, such as pharmaceutical and agrochemical areas. A common strategy for their formation is four-membered ring expansion, which also allows to add molecular complexity and functional handles within one single operation starting from readily available starting materials. Organocatalysis can be exploited to promote the reaction and to obtain a good enantio- and diastereoselection. This technique involves the exclusive use of organic molecules as catalysts, without resorting to metals. The aim of this work is to obtain enantiopure cyclopentanones starting from achiral allylic cyclobutanols. The reaction consists in a ring expansion promoted by the addition of a halogen to the double bond of the substrate, with formation of a haliranium ion as intermediate, followed by a semipinacol rearrangement to afford the cyclopentanone. The reaction is catalysed by a chiral phosphoric acid that, besides accelerating the rate of the reaction, transmits a specific chirality thanks to its chiral structure, following the asymmetric catalysis principles. Starting from symmetric trans-allylic cyclobutanols, the whole reaction is a desymmetrization and leads to the formation of two new stereogenic centres: a mixture of diastereoisomers is obtained, each as couple of enantiomers; the ratio between the possible configurations is determined by the relative position that the chiral catalyst and the reagent occupy during the reaction. Since the reaction is already optimized, the original aim was to study the scope: first, the synthesis of a set of allylic cyclobutanols and their relative precursors, in order to have a wider range of substrates; then, the identification of the type of substrate that undergoes the expansion, with the study of enantio- and diastereoselectivity obtained in each case. Due to the Covid-19 emergency, most of the work was developed as a bibliographic study.

A two-body study of dynamical expansion in the XXZ spin chain and equivalent interacting fermion models

Lo scopo di questa tesi è studiare l'espansione dinamica di due fermioni interagenti in una catena unidimensionale cercando di definire il ruolo degli stati legati durante l'evoluzione temporale del sistema. Lo studio di questo modello viene effettuato a livello analitico tramite la tecnica del Bethe ansatz, che ci fornisce autovalori ed autovettori dell'hamiltoniana, e se ne valutano le proprietà statiche. Particolare attenzione è stata dedicata alle caratteristiche dello spettro al variare dell'interazione tra le due particelle e alle caratteristiche degli autostati. Dalla risoluzione dell'equazione di Bethe vengono ricercate le soluzioni che danno luogo a stati legati delle due particelle e se ne valuta lo spettro energetico in funzione del momento del centro di massa. Si è studiato inoltre l'andamento del numero delle soluzioni, in particolare delle soluzioni che danno luogo ad uno stato legato, al variare della lunghezza della catena e del parametro di interazione. La valutazione delle proprietà dinamiche del modello è stata effettuata tramite l'utilizzo dell'algoritmo t-DMRG (time dependent - Density Matrix Renormalization Group). Questo metodo numerico, che si basa sulla decimazione dello spazio di Hilbert, ci permette di avere accesso a quantità che caratterizzano la dinamica quali la densità e la velocità di espansione. Da queste sono stati estratti i proli dinamici della densità e della velocità di espansione al variare del valore del parametro di interazione.

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).