Dataset of Haldane Fractional Statistics for 1D Heisenberg Spin XXX Chain

Haldane's fractional exclusion statistics (FES) describes a generalized Pauli exclusion statistics, which can be regarded as an emergent quantum statistics induced by the intrinsic dynamical interaction. A non-mutual FES has been identified at the quantum criticality of the one-dimensional (1D) and 2D interacting Bose Gas [Nat. Sci. Rev. 9, nwac027 (2022)]. It is naturally asked if such a non-mutual FES can be induced by the spin-spin interaction in the antiferromagnetic spin-1/2 XXX chain? In this article, we first represent the Bethe ansatz equations of spin strings in terms of the FES equations of different species. Then we show that the 1D spin XXX chain remarkably possesses the non-mutual FES in the critical region. We observe that the equation of state in terms of the FES gives rises to full statistical properties of the model at quantum criticality, which are in good agreement with the results obtained from the thermodynamic Bethe ansatz (TBA) equations of the model. From the non-mutual FES, we also precisely determine the quantum scaling functions, which further agree well with the previous TBA results [Phys. Rev. B 96, 220401(R) (2017)]. Finally, we also build up an exact mapping between the scaling functions of the Lieb-Liniger model and the spin Heisenberg spin chain at quantum criticality. Our method provides deep insights into the critical phase of matter from quantum FES point of view.

霍尔丹分数排斥统计（Haldane's fractional exclusion statistics, FES）描述了一类广义泡利排斥统计，可被视为由内禀动力学相互作用诱导的涌现量子统计。此前，在一维（1D）与二维（2D）相互作用玻色气体的量子临界性中，已观测到非互易FES现象 [Nat. Sci. Rev. 9, nwac027 (2022)]。那么一个自然的问题是：反铁磁自旋1/2 XXX链中的自旋-自旋相互作用，是否也能诱导出这类非互易FES？本文首先基于不同种类的FES方程，重构了自旋弦的贝塞特拟设（Bethe ansatz）方程；随后证明，一维自旋XXX链的临界区域中确实存在显著的非互易FES。我们发现，基于FES框架的物态方程，可以完整给出该模型在量子临界态下的全部统计性质，这与通过模型的热力学贝塞特拟设（thermodynamic Bethe ansatz, TBA）方程得到的结果高度吻合。基于非互易FES，我们还精确确定了量子标度函数，其结果同样与此前基于TBA的研究结论一致 [Phys. Rev. B 96, 220401(R) (2017)]。最后，我们还建立了量子临界态下李布-林格（Lieb-Liniger）模型与自旋海森堡链的标度函数之间的精确映射关系。本研究从量子FES的视角，为理解物质的临界相提供了深刻的洞见。