Clustered quantum materials provide a new platform for the experimental study of many- body entanglement. Here we address a simple model featuring N interacting spins in a transverse field. The field can induce an entanglement transition (ET). We calculate the magnetisation, low-energy gap and neutron-scattering cross-section and find that the ET has distinct signatures, detectable at temperatures as high as 10% of the interaction strength. Unlike a quantum critical point, the signatures of the ET are stronger for smaller clusters.

The spin-ice materials Ho2Ti2O7 (HTO) and Dy2Ti2O7 (DTO) are part of a large family of compounds called magnetic pyrochlore oxides. Typically, the magnetism of these systems arise from the rare-earth ions RE3+ which sits at the vertices of a lattice of corner-sharing tetrahedra and couple with the degrees of freedom of the crystal leading to a wide spectrum of exotic phenomena. In spin-ices the magnetic moments of the individual RE3+ ions are large enough to let their mutual dipolar interactions be the leading factors for the thermodynamics. Moreover, the strength and the symmetries of their local crystalline environment are such that each ion behaves like a magnetic dipole with only two allowed configurations: it points either parallel or opposite to the ⟨111⟩ axis joining the vertex where it sits to the centre of the tetrahedron. As a result, the ground state of the system is macroscopically degenerate because the ferromagnetic interactions between the 4 Ising-like spins in each tetrahedron cannot be satisfied simultaneously, and in turn the manifold of possible configurations minimising the energy (2 spins inward - 2 spins outward) increases with the size of the system. This exotic ground state is such that the Ising configurations of the dipoles map to the disordered vector-displacements of the protons in the water molecules of conventional ice; hence the name spin-ice. Violations of the (2in-2out) ice rules take the systems out of the ground state into more energetic configurations. More precisely, the flipping of a spin between two adjacent ground-state tetrahedra creates a local excitation (1in-3out in a tetrahedron, 3in-1out in the neighbouring one) that is made of two fractionalised opposite magnetic fluxes. Once the two fluxes are created they can separate and freely hop across the lattice as their motion does not involve ahy higher order violation of the ice-rules. The low temperature properties of spin-ice is found to depend heavily, if not exclusively, on the density and mobility of such flux-defects which effectively behave as magnetic monopoles mutually interacting via a Coulomb potential. Beside the success of many experiments which exploit the physics of the monopoles in their exclusively classical formulation, there has been an increasing curiosity about the microscopic mechanisms which dictate their propagation across the lattice. At present, the dynamics of the monopoles are still puzzling showing different responses under different probes and non-identical behaviours between the two compounds HTO and DTO. As the monopoles themselves consist of packed magnetic fluxes originating from the magnetic moments of the RE3+ ions, investigating the microscopical mechanisms underlying their motion requires revisiting the foundations of the classicality which emerges form the quantum substrate of the interactions of the magnetic ions. This is the subject of the work presented in this thesis. With particular focus on the interplay between the local crystal-symmetries and the mutual interactions between the RE3+ ions, the present study gives an accurate de- scription of the microscopic mechanisms which occur in the pyrochlore substrate in the presence of a monopole. The results suggest that the motion of the monopoles is achieved thanks to the spin-tunnelling of the RE3+ ions which account for the flipping of the Ising spins, necessary for the propagation of a monopole. A major improvement we add to the standard theory of spin ice, is the role of the exchange interactions that are overcome by the dipolar ones in the ice-state but posses a dynamical resilience which manifests in the presence of a monopole. Furthermore, the present study brings to light the articulated statistical structure of the kinematic spin-constraints which are expected to dictate the diffusion of the free monopoles and their response under dif- ferent probes also in conditions out of equilibrium. From a more general perspective, the mathematical and physical models developed during this work promise to be of in- terest also in other magnetic systems. Primarily, in the other pyrochlore oxides whose microscopical structure is akin the spin-ice one, secondarily, in other RE3+ compounds where the interplay between quantum and classical physics leads to the manifestation of unusual dynamical effects.