标题： 各向异性海森堡自旋链中的边缘模式和边界杂质 显示英文标题

标题： Edge modes and boundary impurities in the anisotropic Heisenberg spin chain

摘要： 我们对自旋$\frac{1}{2}$各向异性海森堡链（XXZ-$\frac{1}{2}$）在带隙反铁磁相中的边界现象进行了全面分析，特别关注分数化自旋$\frac{1}{4} $边缘模式与边缘耦合的自旋$\frac{1}{2}$杂质之间的相互作用。 通过结合 Bethe 安萨茨、精确对角化和密度矩阵重归一化群（DMRG）方法，我们探索了当杂质以可积或非可积方式与链耦合时出现的复杂相图。 对于可积的反铁磁杂质耦合，我们识别出两个不同的相：库诺相，其中杂质被多粒子库诺效应屏蔽；以及反铁磁束缚模式相，其中指数局域化的束缚态在基态中屏蔽杂质。 当保持可积性而以铁磁方式耦合时，杂质表现为自由自旋$\frac{1}{2}$，导致形成铁磁束缚模式相，在基态中杂质保持自由但可能在较高能量激发时被屏蔽，或者未屏蔽（或局部矩）相，在这个相中杂质在每一个本征态中都保持未屏蔽；而对于非可积的铁磁耦合，杂质不是自由的。 在非可积反铁磁耦合的情况下，第三种相出现，其特征是在库诺相和反铁磁束缚模式相之上的带隙中存在两个低于质量间隙的简并态，进一步丰富了相图。 我们的研究突显了带隙反铁磁系统中边界杂质的微妙行为，为在分数化边缘模式和体反铁磁序存在的情况下理解库诺效应和杂质屏蔽提供了新的见解。 显示英文摘要

摘要： We present a comprehensive analysis of boundary phenomena in a spin-$\frac{1}{2}$ anisotropic Heisenberg chain (XXZ-$\frac{1}{2}$) in the gapped antiferromagnetic phase, with a particular focus on the interplay between fractionalized spin-$\frac{1}{4} $ edge modes and a coupled spin-$\frac{1}{2}$ impurity at the edge. Employing a combination of Bethe Ansatz, exact diagonalization, and density matrix renormalization group (DMRG) methods, we explore the intricate phase diagram that emerges when the impurity is coupled either integrably or non-integrably to the chain. For integrable antiferromagnetic impurity couplings, we identify two distinct phases: the Kondo phase, where the impurity is screened by a multiparticle Kondo effect, and the antiferromagnetic bound mode phase, where an exponentially localized bound state screens the impurity in the ground state. When coupled ferromagnetically while maintaining integrability, the impurity behaves as a free spin-$\frac{1}{2}$, leading to either a ferromagnetic bound mode phase, where the impurity remains free in the ground state but may be screened at higher energy excitations or an unscreened (or local moment) phase where impurity remains unscreened in every eigenstate whereas for non-integrable ferromagnetic coupling, the impurity is not free. In the case of non-integrable antiferromagnetic coupling, a third phase emerges, characterized by mid-gap excitations with two degenerate states below the mass gap on top of the Kondo and antiferromagnetic bound mode phases, further enriching the phase diagram. Our findings highlight the nuanced behavior of boundary impurities in gapped antiferromagnetic systems, offering new insights into Kondo effects and impurity screening in the presence of fractionalized edge modes and bulk antiferromagnetic order.