Scaling and front dynamics in Ising quantum chains

We study the relaxation behaviour of the quantum Ising chain, focusing our attention onto the non-equilibrium dynamics of the transverse magnetization. The initial states, from which the magnetization relaxes, are product states such as those generated by setting in contact several systems, each initially equilibrated at a given temperature. Due to the free fermionic structure of the chain, the dynamics of the transverse magnetization is easily expressed in a compact form. In the completely factorized initial state, corresponding to a situation where all the spins are thermalized independently, we obtain in the scaling limit the Green function associated to the transverse magnetization. The relaxation behaviour is also considered in the system-bath case, where part of the chain called the system is thermalized at a temperature T_s and the remaining part is at a temperature T_b. The magnetization profiles show a scaling behaviour. Moreover, in the extreme case T_b=∞ and T_s=0, it is shown that the magnetization relaxes in quantized steps in the strong transverse field region.