标题： 边缘模和各向异性海森堡自旋链的边界杂质 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-$\frac{1}{2}$）在能隙反铁磁相中的边界现象进行了全面分析，特别关注分数量子化自旋-$\frac{1}{4} $边缘模态与边缘耦合的自旋-$\frac{1}{2}$杂质之间的相互作用。 采用贝特假设、精确对角化和密度矩阵重正交化群（DMRG）方法，我们研究了当杂质以可积或不可积方式耦合到链时出现的复杂相图。 对于可积的反铁磁杂质耦合，我们识别出两个不同的相：Kondo相，在该相中杂质通过多粒子Kondo效应被屏蔽；以及反铁磁束缚模相，在该相中一个指数局域化的束缚态在基态中屏蔽杂质。 当以保持可积性的方式进行铁磁耦合时，杂质表现为自由自旋-$\frac{1}{2}$，导致铁磁束缚模相，其中杂质在基态中保持自由，但在更高能量激发态中可能被屏蔽，或者无屏蔽（或局部磁矩）相，其中杂质在每个本征态中均未被屏蔽；而对于不可积的铁磁耦合，杂质并不自由。 在不可积的反铁磁耦合情况下，出现第三个相，其特征是在Kondo相和反铁磁束缚模相之上存在能隙内的激发态，具有两个简并态，进一步丰富了相图。 我们的发现突显了在能隙反铁磁系统中边界杂质的细微行为，为在存在分数量子化边缘模态和体反铁磁序的情况下Kondo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.