We analyse an exactly solvable spin-1/2 chain which is a generalised version of Kitaev's honeycomb model. We show that every state of the system has a 2 N/4 fold degeneracy, where N is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localised, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (anti-kink) Z2 flux configurations. The unpaired Majorana modes can therefore be created and manipiulated if the Z2 flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wavefunctions can be tuned if the couplings of the model can be tuned.

We analyse a spin-1/2 chain with two-spin interactions which shown to exactly solvable by Lieb, Schultz and Mattis 1 . We show that the model can be viewed as a generalised Kitaev model that is analytically solvable for all defect sectors. We present an alternate proof that the defect free sector is the ground state, which is valid for a larger parameter range. We show that the defect sectors have degenerate ground states corresponding to unpaired Majorana fermion modes and that the degeneracy is topologically protected against disorder in the spin-spin couplings. The unpaired Majorana fermions can be manipulated by tuning the model parameters and can hence be used for topological quantum computation.

We study orbital Mott transition in two dimensional pyrochlore lattice, using a two orbital Hubbard model with only inter-orbital electronic hopping. We use a real space Monte Carlo based approach to study the model at nite temperature, and establish temperature-interaction phase diagram that highlights the Mott transition, orbital ordering, and spectral trends, and possible window of pseudogap. Due to only inter-orbital hopping, the Mott insulator 'generates' ferro exchange resulting in ferro-orbital ordering, with T corr /t peaked at ≈0.2 around U/t ≈ 6. The optical conductivity shows unusual two peak feature due to two dimensional pyrochlore lattice.