Magnetic-field-induced oscillation of multipartite nonlocality in spin ladders

Spin-$\frac{1}{2}$ two-leg ladder models under a magnetic field have a well-known phase diagram. In this paper, we use multipartite nonlocality (a measure of multipartite quantum correlations) to characterize the quantum correlations in the ladder models at zero temperature. Both finite-size and infinite-size ladders are considered. We investigate the global nonlocality ${\mathcal{S}}_{g}=\mathcal{S}(|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle})$, which describes quantum correlations of the ground states $|{\mathrm{\ensuremath{\Psi}}}_{g}\ensuremath{\rangle}$ of the entire lattices, and the partial nonlocality ${\mathcal{S}}_{p}=\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n})$, which describes quantum correlations of the reduced states ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}$ of some sublattices in the ladders. We find that as the magnetic field $\ensuremath{\lambda}$ increases, the global nonlocality ${\mathcal{S}}_{g}$ presents a single-peak curve. Moreover, the logarithmic measure $ln{\mathcal{S}}_{g}$ changes dramatically at the two critical fields ${\ensuremath{\lambda}}_{{c}_{1}}$ and ${\ensuremath{\lambda}}_{{c}_{2}}$ of the models and thus signals the quantum phase transitions in the models. For the partial nonlocality ${\mathcal{S}}_{p}$, in the regions ${\ensuremath{\lambda}}_{{c}_{1}}l\ensuremath{\lambda}l{\ensuremath{\lambda}}_{{c}_{2}}$, we observe that the ${\mathcal{S}}_{p}(\ensuremath{\lambda})$ curve shows an oscillation. Numerical results reveal that the underling mechanism is the ``major component transitions'' in the reduced states ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{n}$ of the sublattices. More importantly, the oscillation of the partial nonlocality ${\mathcal{S}}_{p}$ is modulated by the single-peak curve of the global nonlocality ${\mathcal{S}}_{g}$. The result provides valuable clues about the relation between partial nonlocality and global nonlocality in low-dimensional quantum models.