Asymptotic Time Evolution of a Partitioned Infinite Two-sided Isotropic $XY$-chain

The system under consideration is that of a two-sided infinite isotropic $XY$-chain partitioned into two distinct regions. Each side is initially in thermal equilibrium. We investigate the situation when the partition is removed at time $t=0$. For $t\rightarrow\infty the system approaches thermal equilibrium if the two sides were at the same temperature. If initially the two sides were at different temperatures then the system approaches a steady state.