Higher-order Spatially Separated Correlations and Quantum Magic in the Long-range Transverse Field Ising Model

Abstract

Long-range interacting quantum systems exhibit interesting phenomena beyond the nearest-neighbours regime. Such interactions can reach further separated particles and change the behaviour of many-body correlations. Here, we analyse their structure in the antiferromagnetic long-range interaction transverse field Ising model with power-law interactions decaying as $1/r^α$. We examine the spatial behaviour of basis-independent total multipartite correlations, called sector lengths $S_k$, of its ground state in the vicinity of critical points for different interaction exponents $α$ and fixed system size. These correlation functions were shown to indicate entanglement and the existence of local realistic models, allowing us to make claims about the Bell nonlocality and entanglement in the system. We further investigate them in the context of non-stabiliserness, often referred to as quantum magic, which holds information on their classical simulability and usefulness for quantum computation. We evaluate R{é}nyi stabiliser entropy near and outside the criticality and show that cumulative magic, up to the critical point, increases with $α$ for all of the examined system sizes. Our results provide new insights into the interplay between the long-range interactions, observable non-classical phenomena and their classical simulability.

The paper studies how interaction range affects spatially separated multipartite correlations and quantum magic in the ground state of the one-dimensional antiferromagnetic transverse-field Ising model with couplings $\lambda/r^\alpha$. It uses sector lengths—sums of squared Pauli correlations invariant under local basis changes—and the second stabiliser Rényi entropy $M_2$. Ground states are obtained with DMRG, complemented by exact diagonalisation. Entanglement-entropy peaks for $L=50,100,200$ locate critical couplings ranging from approximately $1.533$ at $\alpha=5/3$ to $1.189$ at $\alpha=3$. For $L=100$, equally spaced configurations of $k=2,\ldots,7$ spins near criticality have sector lengths fitted by $S_k(r)=(a_k+b_kr)^{-\gamma_k}$, with fitted exponents between $0.248$ and $0.689$. Their dependence on $\alpha$ is nonmonotonic. Neighbouring five-spin configurations, and seven-spin configurations for $\alpha>5/3$, satisfy the entanglement witness $S_k>1$. The paper reports no violations of the examined two-setting, full-correlation Bell criteria. Magic is evaluated using tensor cross interpolation for systems up to $L=25$. Its maximum occurs near criticality, and the cumulative quantity $\int_0^{\lambda^*}M_2\,d\lambda$ increases with $\alpha$ over the reported coupling range. An analytic large-coupling calculation for real superpositions of the two Néel states gives $0\leq M_2\leq\log(4/3)$. The correlation fits describe fixed-size behaviour, not universal critical exponents; the nonmonotonic ordering may change at larger sizes. Magic results are restricted to small systems, with deviations at strong coupling for $L=25$ attributed to numerical convergence difficulties.