Correlations and Quantum Dynamics of 1D Fermionic Models: New Results for the Kitaev Chain with Long-Range Pairing

In the first part of the thesis, we propose an exactly-solvable one-dimensional model for fermions with long-range p-wave pairing decaying with distance as a power law. We studied the phase diagram by analyzing the critical lines, the decay of correlation functions and the scaling of the von Neumann entropy with the system size. We found two gapped regimes, where correlation functions decay (i) exponentially at short range and algebraically at long range, (ii) purely algebraically. In the latter the entanglement entropy is found to diverge logarithmically. Most interestingly, along the critical lines, long-range pairing breaks also the conformal symmetry. This can be detected via the dynamics of entanglement following a quench. In the second part of the thesis we studied the evolution in time of the entanglement entropy for the Ising model in a transverse field varying linearly in time with different velocities. We found different regimes: an adiabatic one (small velocities) when the system evolves according the instantaneous ground state; a sudden quench (large velocities) when the system is essentially frozen to its initial state; and an intermediate one, where the entropy starts growing linearly but then displays oscillations (also as a function of the velocity). Finally, we discussed the Kibble-Zurek mechanism for the transition between the paramagnetic and the ordered phase.

Development of an innovative strategy to enhance the efficacy of gene therapy for CDKL5 deficiency disorder

CDKL5 (cyclin-dependent kinase-like 5) deficiency disorder (CDD) is a severe X-linked neurodevelopmental disease caused by mutations in the CDKL5 gene, characterized by early-onset epileptic seizures, intellectual disability, motor and visual impairment and respiratory dysregulation. Although pharmacological treatments are used to control seizures, there is currently no cure to ameliorate symptoms for CDD. Albeit delivery of a wild-type copy of the mutated gene to cells represents the most curative approach for a monogenic disease, proof-of-concept studies highlight significant efficacy caveats for brain gene therapy. The major one regards the low efficiency of gene delivery to the CNS by viral vectors. We used a secretable Igk-TATk-CDKL5 protein to enhance the efficiency of a gene therapy for CDD. In view of the properties of the Igk-chain leader sequence, the TATk-CDKL5 protein produced by infected cells is secreted via constitutive secretory pathways. Importantly, due to the transduction property of the TATk peptide, the secreted CDKL5 protein is internalized by cells. We compared the effects of a CDKL5 gene therapy with an IgK-TATk-CDKL5 gene therapy in a Cdkl5 KO mouse model to validate whether the Igk-TATk-CDKL5 approach significantly improve the therapeutic efficacy. We found that, although AAVPHP.B_Igk-TATk-CDKL5 and AAVPHP.B_CDKL5 vectors had similar brain infection efficiency, the AAVPHP.B_Igk-TATk-CDKL5 vector led to a higher CDKL5 protein replacement and Cdkl5 KO mice treated with the AAVPHP.B_Igk-TATk-CDKL5 vector showed a behavioral and neuroanatomical improvement in comparison with Cdkl5 KO mice treated with the AAVPHP.B_CDKL5 vector.