• composite quantum many-body
• decoherence
• finite-size scaling
• Kitaev model
• quantum phase transitions

We analyze the equilibrium properties at zero temperature of a quantum many-body subsystem interacting with a many-body environment, assuming that the global system is in the ground state. We consider, as a paradigmatic model, two stacked one-dimensional quadratic fermionic chains homogeneously coupled through a hopping term.
A crucial role is played by quantum phase transitions. At zero temperature, they generally separate different phases and are associated with the arising of scaling behaviours that can be described, in finite-size systems, within a Finite-Size Scaling (FSS) framework.
We study analytically the global phase diagram, determining the transition lines of the model and the dispersion relation of the low-energy excitations at the gapless points.
Choosing one chain as the subsystem under observation, we quantify the loss of coherence, due to a small coupling with the environment being turned on, through the purity of the related reduced density matrix. We study how the scaling behaviour of the latter quantity depends on whether the environment is critical or not, and the crossover phenomena that can arise when the two chains develop different diverging time scales.
As a complementary tool for the analysis of the critical behaviour, we exactly compute numerically entanglement entropies of different bipartitions. While also following proper FSS behaviours, entanglement properties can give information about the nature of the critical behaviour in specific situations.