Two-Point Stabilizer R\'enyi Entropy: a Computable Magic Proxy of Interacting Fermions

Abstract

Quantifying non-stabilizerness (``magic'') in interacting fermionic systems remains a formidable challenge, particularly for extracting high order correlations from quantum Monte Carlo simulations. In this Letter, we establish the two-point stabilizer R\'enyi entropy (SRE) and its mutual counterpart as robust, computationally accessible probes for detecting magic in diverse fermionic phases. By deriving local estimators suitable for advanced numerical methods, we demonstrate that these metrics effectively characterize quantum phase transitions: in the one-dimensional spinless $t$-$V$ model, they sharply identify the Luttinger liquid to charge density wave transition, while in the two-dimensional honeycomb lattice via determinant quantum Monte Carlo, they faithfully capture the critical exponents of the Gross-Neveu-Ising universality class. Furthermore, extending our analysis to the fractional quantum Hall regime, we unveil a non-trivial spatial texture of magic in the Laughlin state, revealing signatures of short-range exclusion correlations. Our results validate the two-point SRE as a versatile and sensitive diagnostic, forging a novel link between quantum resource theory, critical phenomena, and topological order in strongly correlated matter.