The three site interacting spin chain in staggered field: Fidelity vs Loschmidt echo

We study the the ground state fidelity and the ground state Loschmidt echo of a three site interacting XX chain in presence of a staggered field which exhibits special types of quantum phase transitions due to change in the topology of the Fermi surface, apart from quantum phase transitions from gapped to gapless phases. We find that on one hand, the fidelity is able to detect only the boundaries separating the gapped from the gapless phase; it is completely insensitive to the phase transition from two Fermi points region to four Fermi points region lying within this gapless phase. On the other hand, Loschmidt echo shows a dip only at a special point in the entire phase diagram and hence fails to detect any quantum phase transition associated with the present model. We provide appropriate arguments in support of this anomalous behavior.