Quantum metrology enhancement via higher-order spin interactions in an infinite-range model

Motivated by recent experimental demonstrations showing that metrological performance can be enhanced beyond the standard quantum limit in the infinite-range spin-1/2 Lipkin–Meshkov–Glick (LMG) model with second-order spin interactions, we investigate the impact of adding a quartic spin interaction. This extension, both theoretically relevant and experimentally accessible, yields quantum sensitivity surpassing that of the standard LMG model. To gain physical insight, we analyze the system in its classical limit, where higher-order interactions reshape the energy landscape and modify the stability of fixed points. In the conventional LMG model, metrological gain originates from the instability near a single saddle point in phase space, which leads to the exponentially fast growth of out-of-time-ordered correlators. In contrast, in the extended LMG model, we identify parameter regimes where two saddle points emerge, resulting in enhanced instability and even greater metrological gain. Our findings reveal that higher-order spin couplings provide a powerful means to achieve improved quantum metrological performance.