Quantum phase transition in the one-dimensional period-two and uniform compass model

Quantum phase transitions in the one-dimensional period-two and uniform quantum compass model are studied by using the pseudospin transformation method and the trace map method. The exact solutions are presented, the fidelity, the nearest-neighbor pseudospin entanglement, spin and pseudospin-correlation functions are then calculated. At the critical point, the fidelity and its susceptibility change substantially, the gap of pseudospin concurrence is observed, which scales as $1/N$ (N is the system size). The spin-correlation functions show smooth behavior around the critical point. In the period-two chain, the pseudospin-correlation functions exhibit an oscillating behavior, which is absent in the uniform chain. The divergent correlation length at the critical point is demonstrated in the general trend for both cases.