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# Multipartite entanglement in the 1-D spin-$\frac{1}{2}$ Heisenberg Antiferromagnet

Dec 2022

Multipartite entanglement refers to the simultaneous entanglement betweenmultiple subsystems of a many-body quantum system. While multipartiteentanglement can be difficult to quantify analytically, it is known that it canbe witnessed through the Quantum Fisher information (QFI), a quantity that canalso be related to dynamical Kubo response functions. In this work, we firstshow that the finite temperature QFI can generally be expressed in terms of astatic structure factor of the system, plus a correction that vanishes as$T\rightarrow 0$. We argue that this implies that the static structure factorwitnesses multipartite entanglement near QCPs at temperatures below acharacteristic energy scale that is determined by universal properties, up to anon-universal amplitude. Therefore, in systems with a known static structurefactor, we can deduce finite temperature scaling of multipartite entanglementand low temperature entanglement depth without knowledge of the full dynamicalresponse function of the system. This is particularly useful to study 1Dquantum critical systems in which sub-power-law divergences can dominateentanglement growth, where the conventional scaling theory of the QFI breaksdown. The 1D spin-1/2 antiferromagnetic Heisenberg model is an importantexample of such a system, and we show that multipartite entanglement in theHeisenberg chain diverges non-trivially as $\sim \log(1/T)^{3/2}$. We verifythese predictions with calculations of the QFI using conformal field theory andmatrix product state simulations. Finally we discuss the implications of ourresults for experiments to probe entanglement in quantum materials, comparingto neutron scattering data in KCuF$_3$, a material well-described by theHeisenberg chain.

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