Quantum Fisher information and nonlocal operators in one-dimensional quantum chains

This study investigates quantum Fisher information (QFI) with respect to a class of nonlocal operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}$ in one-dimensional quantum chains. Based upon the triangular matrix-product-operator theory, we establish a universal scaling behavior of QFI with respect to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}$ for general infinite matrix-product states (iMPSs): $\mathcal{F}\ensuremath{\sim}{\ensuremath{\varrho}}_{2}{N}^{2}+{\ensuremath{\varrho}}_{1}N+{\ensuremath{\varrho}}_{0}$, where the coefficients ${\ensuremath{\varrho}}_{i}$ can be efficiently identified by iMPS algorithms. Applied to a spin-1 Bilinear-Biquadratic model and an XXZ model, the scaling coefficients ${{\ensuremath{\varrho}}_{i}}$ distinguish various phases through distinct scaling behaviors: For instance, the topologically ordered Haldane phase exhibits characteristic quadratic scaling (${\ensuremath{\varrho}}_{2}\ensuremath{\ne}0$), while the trimer and dimer phases are featured with linear scaling (${\ensuremath{\varrho}}_{1}\ensuremath{\ne}0$). Notably, since the QFI scaling behavior undergoes fundamental changes between distinct phases, these coefficients provide signals at first-order, second-order, and Berezinskii-Kosterlitz-Thouless type quantum phase transitions in these models. Furthermore, finite bond-dimension ($D$) scaling analysis reveals a dichotomy. For gapped phases, the coefficients converge rapidly as $D$ increases. Conversely, for critical systems, the coefficients exhibit $D$-scaling effects that depend on the specific model. By bypassing finite-size fittings with limited $N$, our approach accesses the thermodynamic-limit physics for QFI with respect to nonlocal operators, offering new avenues for characterizing exotic quantum phases in one-dimensional quantum systems.