Localization from Infinitesimal Kinetic Grading: Finite-size Scaling, Kibble-Zurek Dynamics and Applications in Sensing

Abstract

We study a one-dimensional lattice model with site-dependent nearest-neighbor hopping amplitudes that follow a power-law profile. The hopping variation is controlled by a grading exponent, $|alpha|$, which serves as the tuning parameter of the system. In the thermodynamic limit, the ground state becomes localized in the limit $|alpha| \to 0$, signaling the presence of a critical point characterized by a diverging localization length. Using exact diagonalization methods, we perform finite-size scaling analysis, and extract the associated critical exponent governing the near-critical behavior. To further characterize the criticality, we analyze inverse participation ratio (IPR), energy gap between the ground and first excited state, and fidelity-susceptibility. We also investigate the nonequilibrium dynamics by linearly ramping the hopping profile at various rates and tracking the evolution of the localization length and the IPR. The Kibble-Zurek mechanism successfully explains the resulting dynamics of the system via the critical exponents obtained from static scaling analysis. Beyond its fundamental significance, the kinetic-grading-induced localization transition provides a natural platform for quantum sensing. Using the critical enhancement of the quantum Fisher information (QFI), we demonstrate that the system enables quantum-enhanced parameter estimation of the grading exponent. We propose both adiabatic and dynamical quantum critical sensors and demonstrate that they exhibit enhanced scaling of the QFI. Our results therefore establish graded kinetic systems not only as a new setting for localization physics, but also as a potential resource for designing quantum-enhanced sensing devices.