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IEEE Transactions on Magnetics

Probing Dynamical Correlations, Structure Factor, and Entanglement in a Spin-1/2 Heisenberg Tetrahedron

Elham Shahhosseini Shahrabadi, Hana Vargová, Jozef Strečka

Published Mar 25, 2026• 0 citations
Publisher

Abstract

This study presents a rigorous analysis of the temperature and the magnetic field dependence of the dynamical autocorrelations and correlation functions in the spin-1/2 Heisenberg tetrahedron. Exact results for these quantities are employed to evaluate the corresponding static and dynamic structure factors, which can be directly probed through inelastic neutron scattering experiments. It is demonstrated that the combination of inelastic neutron scattering data with basic magnetometry measurements enables the extraction of the concurrence serving as a quantitative measure of bipartite entanglement. Notably, pronounced temperature-induced variations in both structure factors and concurrence emerge near two level-crossing fields associated with field-driven transitions between singlet–triplet and triplet–quintet ground states.

DOI
10.1109/tmag.2026.3677536

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This study presents a rigorous analysis of the temperature and the magnetic field dependence of the dynamical autocorrelations and correlation functions in the spin-1/2 Heisenberg tetrahedron. Exact results for these quantities are employed to evaluate the corresponding static and dynamic structure factors, which can be directly probed through inelastic neutron scattering experiments. It is demonstrated that the combination of inelastic neutron scattering data with basic magnetometry measurements enables the extraction of the concurrence serving as a quantitative measure of bipartite entanglement. Notably, pronounced temperature-induced variations in both structure factors and concurrence emerge near two level-crossing fields associated with field-driven transitions between singlet–triplet and triplet–quintet ground states.
The energy spectrum, ground-state phase diagram, pair correlation functions, and concurrence are exactly calculated for a spin-1/2 quantum Heisenberg distorted tetrahedron in a magnetic field.The ground-state phase diagram in the interaction ratio versus magnetic field plane totally involves five different ground states, which differ from one another by the total spin, local pair correlation functions, and/or concurrence.In addition to the classical fully polarized ferromagnetic ground state there also appear four outstanding quantum ground states, two of which belong to a singlet sector and another two to a triplet sector.A degree of bipartite entanglement quantified through the concurrence is examined as a function of magnetic field, temperature, and the interaction ratio.
Quantum correlations are a fundamental property of quantum many-body states. Yet they remain experimentally elusive, hindering certification of genuine quantum behavior, especially in quantum materials. Here we show that the momentum-dependent dynamical susceptibility measured via inelastic neutron scattering enables the systematic reconstruction of a general family of quantum correlation functions, which express the degree of quantum coherence in the fluctuations of two spins at arbitrary mutual distance. Using neutron scattering data on the compound KCuF3_33​ $\unicode{x2014}$ a system of weakly coupled S=1/2S=1/2S=1/2 Heisenberg chains $\unicode{x2014}$ and of numerically exact quantum Monte Carlo data, we show that quantum correlations possess a radically different spatial structure with respect to conventional correlations. Indeed, they exhibit a new emergent length scale $\unicode{x2014}$ the quantum coherence length $\unicode{x2014}$ which is finite at any finite temperature (including when long-range magnetic order develops). Moreover, we show theoretically that coupled Heisenberg spin chains exhibit a form of quantum monogamy, with a trade-off between quantum correlations along and transverse to the spin chains. These results highlight real-space quantum correlators as an informative, model-independent means of probing the underlying quantum state of real quantum materials.
Quantum correlations are a fundamental property of quantum many-body states. Yet they remain experimentally elusive, hindering certification of genuine quantum behavior, especially in quantum materials. Here we show that the momentum-dependent dynamical susceptibility measured via inelastic neutron scattering enables the systematic reconstruction of a general family of quantum correlation functions, which express the degree of quantum coherence in the fluctuations of two spins at an arbitrary mutual distance. Using neutron scattering data for the compound KCuF 3 —a system of weakly coupled S = 1 / 2 Heisenberg chains—and numerically exact quantum Monte Carlo data, we show that quantum correlations possess a radically different spatial structure with respect to conventional correlations. Indeed, they exhibit a different emergent length scale—the quantum coherence length—which is finite at any finite temperature (including when long-range magnetic order develops). Moreover, we show theoretically that coupled Heisenberg spin chains exhibit a form of quantum monogamy, with a trade-off between quantum correlations along and transverse to the spin chains. These results highlight real-space quantum correlators as an informative, model-independent means of probing the underlying quantum state of real quantum materials. Published by the American Physical Society 2024
Heisenberg spin-1/2 chains are the archetype of quantum integrable one-dimensional models describing magnetic properties of a wide range of compounds (like the KCuF 3 crystal), which can be probed experimentally through neutron scattering experiments, while being at the same time at the root of the invention of Bethe ansatz and Yang-Baxter structures that led in turn to quantum groups discovery. The aim of this contribution is to describe these algebraic ingredients and to show how to obtain from them (using combined analytical and numerical analysis) dynamical correlation functions of integrable Heisenberg spin-1/2 chains, the Fourier transform of which, the so-called dynamical structure factors, being directly measured in inelastic neutron scattering experiments. Our method is based on the algebraic Bethe ansatz and the resolution of the quantum inverse scattering problem. It leads to recent progress in the computation of integrable Heisenberg spin-1/2 chains correlation functions that we review here. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Inelastic neutron scattering is employed to study transverse spin correlations of a Heisenberg S=1/2S=1/2S=1/2 chain compound in a magnetic field of 7.5 T. The target compound is the antiferromagnetic Heisenberg S=1/2S=1/2S=1/2 chain material 2(1,4-dioxane)$\ifmmode\cdot\else\textperiodcentered\fi{}2({\mathrm{H}}_{2}\mathrm{O})\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{CuCl}}_{2}$, or CuDCl for short. The validity and the limitations of the scaling relation for the transverse dynamic structure factor are tested, discussed, and compared to the Tomonaga-Luttinger spin liquid theory and to Bethe-ansatz results for the Heisenberg model.
Three-dimensional Heisenberg model in the form of a tetrahedron lattice is investigated. The concurrence and multipartite entanglement are calculated through 2-concurrence C and 4-concurrence C 4 . The concurrence C and multipartite entanglement C 4 depend on different coupling strengths J i and are decreased when the temperature T is increased. For a symmetric tetrahedron lattice, the concurrence C is symmetric about J 1 when J 2 is negative while the multipartite entanglement C 4 is symmetric about J 1 when J 2 <2. For a regular tetrahedron lattice, the concurrence C of ground state is 1/3 for ferromagnetic case while C = 0 for antiferromagnetic case. However, there is no multipartite entanglement since C 4 = 0 in a regular tetrahedron lattice. The external magnetic field B can increase the maximum value of the concurrence C B and induce two or three peaks in C B . There is a peak in the multipartite entanglement C 4 B when C 4 B is varied as a function of the temperature T . This peak is mainly induced by the magnetic field B .