Towards the kinetic resolution of mechanically planar chiral rotaxanes

In this work, we reported the synthesis and characterization of two [2]rotaxanes endowed with a central ammonium group and two triazolium recognition stations on either side, acting as complexation sites for a dibenzo-24-crown-8 ether macrocycle. These mechanically interlocked architectures were obtained through the interlocking of a functionalized achiral macrocycle with Cs symmetry (where the symmetry element is a mirror plane corresponding to plane of the ring) and a C∞v symmetric axle (where a mirror plane and a C∞ principal axis are aligned along the axle length). We took advantage of the reversible acid/base triggered molecular shuttling of the ring between two lateral triazolium units to switch the rotaxanes between prochiral and mechanically planar chiral forms, which exists as two rapidly-interconverting co-conformers. We exploited the reactivity of the central amino group to attach an optically pure chiral substituent, with the goal of demonstrating the enantiomeric nature of the co-conformers and to obtain a non-zero diastereomeric excess in the resulting diastereomeric products through a dynamic kinetic resolution. To this end, two enantiopure reagents were chosen that could perform clean and fast reaction with amines: a sulfonyl chloride and an acyl chloride. Only the acyl chloride successfully produced an amide in high yield with the deprotonated rotaxane. The group added to the central amine station acted as a stopper against the shuttling of the macrocycle along the axis, thus preventing the fast interconversion of the two mechanically planar enantiomers. We analysed the results through static and dynamic NMR spectroscopic techniques by varying temperature and solvent used. Indeed, the presence of diastereomers was recorded alongside the configurational isomers resulting from the slow rotation of the CN-CO bond of the amide moiety, thus paving the way for a dynamic kinetic resolution.

Hidden Schrödinger symmetry in FLRW and Bianchi I minisuperspace models of cosmology and black holes Schwarzschild dynamics

A broad sector of literature focuses on the relationship between fluid dynamics and gravitational systems. This thesis presents results that suggest the existence of a new kind of fluid/gravity duality not based on the holographic principle. The goal is to provide tools that allow us to systematically unearth hidden symmetries for reduced models of cosmology. The focus is on the field space of these models, i.e. the superspace. In fact, conformal isometries of the supermetric leave geodesics in the field space unaltered; this leads to symmetries of the models. An innovative aspect is the use of the Eisenhart-Duval’s lift. Using this method, systems constrained by a potential can be treated as free ones. Moreover, charges explicitly dependent on time, i.e. dynamical, can be found. A detailed analysis is carried out on three basic models of homogenous cosmology: i) flat Friedmann-Lemaître-Robertson-Walker’s isotropic universe filled with a massless scalar field; ii) Schwarzschild’s black hole mechanics and its extension to vacuum (A)dS gravity; iii) Bianchi’s anisotropic type I universe with a massless scalar field. The results show the presence of a hidden Schrödinger’s symmetry which, being intrinsic to both Navier-Stokes’ and Schrödinger’s equations, indicates a correspondence between cosmology and hydrodynamics. Furthermore, the central extension of this algebra explicitly relates two concepts. The first is the number of particles coming from the fluid picture; while the second is the ratio between the IR and UV cutoffs that weighs how much a theory has of “classical” over “quantum”. This suggests a spacetime that emerges from an underlying world which is described by quantum building blocks. These quanta statistically conspire to appear as gravitational phenomena from a macroscopic point of view.

Diagnosing criticality in symmetric and chiral clock models

Quantum clock models are statistical mechanical spin models which may be regarded as a sort of bridge between the one-dimensional quantum Ising model and the one-dimensional quantum XY model. This thesis aims to provide an exhaustive review of these models using both analytical and numerical techniques. We present some important duality transformations which allow us to recast clock models into different forms, involving for example parafermions and lattice gauge theories. Thus, the notion of topological order enters into the game opening new scenarios for possible applications, like topological quantum computing. The second part of this thesis is devoted to the numerical analysis of clock models. We explore their phase diagram under different setups, with and without chirality, starting with a transverse field and then adding a longitudinal field as well. The most important observables we take into account for diagnosing criticality are the energy gap, the magnetisation, the entanglement entropy and the correlation functions.