Study of quantum nonlocality by CHSH function and its extension in disordered fermions

Quantum nonlocality is an important concept in quantum physics. In this work, we study the quantum nonlocality in a fermion many-body system under quasi-periodic disorders. The Clauser-Horne-Shimony-Holt (CHSH) inequality is systematically investigated, which quantifies quantum nonlocality between two sites. We find particular behaviors of the quantifiers of quantum nonlocality around the extended and critical phase transitions in the system and further that the CHSH inequality is not broken in the globally averaged picture of the maximum value of the quantum nonlocality, but the violation probability of the CHSH inequality for two site pairs in the system becomes sufficiently finite in the critical phase and on a critical phase boundary. Further, we investigate an extension of the CHSH inequality, Mermin-Klyshko-Svetlichny (MKS) polynomials, which can characterize multipartite quantum nonlocality. We also find a similar behavior to the case of CHSH inequality. In particular, in the critical regime and on a transition point, the adjacent three-qubit MKS polynomial in a portion of the system exhibits a quantum nonlocal violation regime with a finite probability in the critical phase.