  The inhomogeneous transverse field Ising models mainly impurity based and the
joint chain are analysed analytically using Jordan-Wigner transformations. The
effects of inhomogeneities on the phase transition have been discussed in
detail. We constructed an ansatz to diagonalize the two models which are taken
into consideration. The inhomogeneity is quantified by a coupling parameter,
which can be tuned to control the occurrence of quantum phase transition in
these models. We have shown a systematic setup using which we can generalise
the solution to a system with an arbitrary number of impurity sites and
junctions, which are separated by at least two lattice sites. We have analysed
the quantum critical point by calculating the correlation functions, transverse
magnetization and the gap between the ground state and the first excited state.

Quantum Physics