Pinning effect and QPT-like behavior for two particles confined by a core-shell potential

We study the ground state entanglement, energy and fidelities of a two-electron system bounded by a core-shell potential, where the core width is varied continuously until it eventually vanishes. This simple system displays a rich and complex behavior: as the core width is varied, this system is characterized by two peculiar transitions where, for different reasons, it displays characteristics similar to a few-particle quantum phase transition. The first occurrence corresponds to something akin to a second order quantum phase transition, while the second transition is marked by a discontinuity, with respect to the driving parameter, in the first derivatives of quantities like energy and entanglement. The study of this system allows to shed light on the sudden variation of entanglement and energy observed in S. Abdullah et al. [Phys. Rev. B 80, 235302 (2009)]. We also compare the core-shell system with a system where a core well is absent: this shows that, even when extremely narrow, the core well has a relevant `pinning' effect. Interestingly, depending on the potential symmetry, the pinning of the wavefunction may either halve or double the system entanglement (with respect to the no-core-well system) when the ground state is already bounded to the outer (shell) well. In the process we discuss the system fidelity and show the usefulness of considering the particle density fidelity as opposed to the more commonly used -- but much more difficult to access -- wavefunction fidelity. In particular we demonstrate that -- for ground-states with nodeless spatial wavefunctions -- the particle density fidelity is zero if and only if the wavefunction fidelity is zero.