Two-Point Stabilizer Rényi Entropy: a Computable Magic Proxy of Interacting Fermions

TL;DR

This work tackles the challenge of quantifying non-stabilizerness, or quantum magic, in interacting fermions by introducing the two-point stabilizer Rényi entropy (SRE) and its mutual variant as computable probes. A determinant quantum Monte Carlo (DQMC) framework is developed to evaluate SRE for interacting fermions via Majorana monomials, with a tomography protocol to reconstruct two-site states and express the SRE in terms local observables. The two-point SRE is shown to detect phase transitions and critical behavior: in 1D, it sharply marks the Luttinger liquid to CDW transition with a BKT-like finite-size scaling; in 2D honeycomb lattices, the mutual SRE yields the Gross–Neveu–Ising critical exponent $\eta\approx 0.423$. Extending to fractional quantum Hall states, the approach reveals a spatial texture of magic in the Laughlin state, including a short-range plateau tied to exclusion constraints. Overall, the two-point SRE provides a versatile, local diagnostic linking quantum resource theory to fermionic criticality and topological order in strongly correlated matter.

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 Rényi 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.

Figures (5)

• Figure 1: Rank-2 SRE as a function of chain length $L$ for the one-dimensional half-filled $t$-$V$ model at various interaction strengths $V$ ($t=1$). Open and solid circles denote DQMC and DMRG results, respectively, while solid lines represent linear fits to the DMRG data. Both approaches confirm volume law scaling, $M_{2}\sim L$. The DQMC data for $L\ge 26$ exhibit large error bars due to insufficient sampling, as the high computational cost of the two-level Monte Carlo scheme limits access to large system sizes.
• Figure 2: Scaling behavior of the mutual two-point SRE in the one-dimensional half-filled spinless $t$-$V$ model. (a) Distance dependence of the mutual SRE in the LL phase (blue circles) and CDW phase (red squares) for $L=400$. Interaction strengths are chosen deep within each phase to minimize finite size effects, clearly exhibiting algebraic and exponential decay, respectively. (b) Mutual two-point SRE as a function of interaction strength $V$ for $L=900$ at various distances. A pronounced peak identifies the critical transition. (c) Finite size scaling of the critical points $V_c(L)$, it obeys the scaling relation predicted by BKT transition. The non-universal microscopic length scale $L_0$ is fitted to be $L_0\approx 1.7$ with data $L>100$ are considered in the fitting.
• Figure 3: Critical scaling on the honeycomb lattice. (a) Mutual two-point SRE $\tilde{\mathcal{M}}^{(2)}_{i,i+\vec{r}_{\max}}$ as a function of linear system size $L$ at various interaction strengths $V$ near the critical threshold ($V_c \approx 1.34$wangfohong2026). The dashed line indicates a power-law fit characteristic of the critical point. (b) Finite-size scaling collapse of the rescaled SRE, $\tilde{\mathcal{M}}^{(2)}L^{2+2\eta}$, plotted against the correlation ratio $R$. The optimal data collapse yields an anomalous dimension $\eta \approx 0.423$. Note that data for $V>1.36$ are excluded from the fit to ensure a robust estimation of $\eta$.
• Figure 4: Orbital texture of non-stabilizerness in the $\nu=1/3$ Laughlin liquid ($N=9$). The mutual two-point SRE (red circles) is plotted against the orbital distance $j$, alongside the two-body density correlation $P_{11}=\langle n_0 n_j \rangle$ (blue squares). The green dot-dashed line marks the magic critical threshold $x_c = \nu_{\text{eff}} - 1/4$. Note the pronounced plateau at short range ($j=1,2$) arising from the vanishing component of relative angular momentum $L_{\text{rel}}<m$, and the subsequent synchronization between magic revival and density modulation at intermediate distances.
• Figure S1: Analogous to Fig. \ref{['fig:Laughlin_WF']} for the $\nu=1/5$ Laughlin state with $N=6$ particles. The mutual two-point SRE displays an extended plateau at short ranges ($j<5$). At intermediate distances, the magic signal synchronously tracks the oscillations of the joint occupation probability $P_{11}$.