With a brief introduction to one-dimensional channels and conductance quantization in mesoscopic systems, we discuss some recent experimental puzzles in these systems, which include reduction of quantized conductances and an interesting odd-even effect in the presence of an in-plane magnetic field. We then discuss a recent non-homogeneous Luttinger liquid model proposed by us, which addresses and gives an explanation for the reduced conductances and the odd-even effect. We end with a brief summary and discussion of future projects.

We investigate the solvability of a variety of well-known problems in lattice statistical mechanics. We provide a new numerical procedure which enables one to conjecture whether the solution falls into a class of functions called differentiably finite functions. Almost all solved problems fall into this class. The fact that one can conjecture whether a given problem is or is not 𝐷-finite then informs one as to whether the solution is likely to be tractable or not. We also show how, for certain problems, it is possible to prove that the solutions are not 𝐷-finite, based on the work of Rechnitzer $[1–3]$.

In this paper we review the quenching dynamics of a quantum $XY$ spin-1/2 chain in the presence of a transverse field, when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. We also extend the results to the cases in which the system starts with any arbitrary initial condition as opposed to the initial fully magnetically aligned state which has been extensively studied earlier. The evolution is non-adiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to non-adiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau–Zener problem. We show that in one dimension the density of defects in the final state scales as $1/\sqrt{\tau}$ irrespective of the initial condition, where 𝜏 is the quenching time-scale. However, the magnitude of density of defects is found to depend on the initial condition.