Thermal and dynamical response of solids within and beyond perturbation theory

This thesis explores advanced aspects of lattice thermal transport and dynamical response in electronic systems, tackling unresolved questions in theoretical models of solid state physics. We address the general theory of thermal transport, introducing a response formalism in the presence of a temperature gradient. Using a many-body Green's function approach, we compute the conductivity in ultralow thermal conductivity materials, accounting for the intraband transport of phonons. We obtain a new Kubo formula valid for strongly anharmonic systems, where conventional perturbative methods fail and phonon interactions are computed by exploiting self-consistent approaches. We derive an original expression of thermal conductivity obtained from first principles valid in self-consistent schemes, thereby advancing our understanding of heat transport near phase transitions. We introduce a variational formulation for dynamical response functions in electronic systems including exchange interactions beyond the standard density functional theory picture. We demonstrate that this formulation yields a quadratic error dependence on the approximated electronic density matrix, enabling more accurate simulations of spectroscopic properties.