Evaluating many-body stabilizer R\'enyi entropy by sampling reduced Pauli strings: singularities, volume law, and nonlocal magic

We present a novel quantum Monte Carlo method for evaluating the $\alpha$-stabilizer R\'enyi entropy (SRE) for any integer $\alpha\ge 2$. By interpreting $\alpha$-SRE as partition function ratios, we eliminate the sign problem in the imaginary-time path integral by sampling \emph{reduced Pauli strings} within a \emph{reduced configuration space}, which enables efficient classical computations of $\alpha$-SRE and its derivatives to explore magic in previously inaccessible 2D/higher-dimensional systems. We first isolate the free energy part in $2$-SRE, which is a trivial term. Notably, at quantum critical points in 1D/2D transverse field Ising (TFI) models, we reveal nontrivial singularities associated with the \emph{characteristic function} contribution, directly tied to magic. Their interplay leads to complicated behaviors of $2$-SRE, avoiding extrema at critical points generally. In contrast, analyzing the volume-law correction to SRE reveals a discontinuity tied to criticalities, suggesting that it is more informative than the full-state magic. For conformal critical points, we claim it could reflect nonlocal magic residing in correlations. Finally, we verify that $2$-SRE fails to characterize magic in mixed states (e.g. Gibbs states), yielding nonphysical results. This work provides a powerful tool for exploring the roles of magic in large-scale many-body systems, and reveals intrinsic relation between magic and many-body physics.