发表机构
College of Physics and Materials Science, Tianjin Normal University; Interdisciplinary Center, Tianjin Normal University（天津师范大学物理与材料科学学院; 天津师范大学交叉科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出反向分级定向跳变的有限开放Krawtchouk网络，实现内部皮肤聚焦与定向镜像转移，可关联局域化几何、可公度谱与定向响应，是可精确求解的有限网络框架。
AI 中文摘要
空间分级非互易性可将皮肤权重从边界移开，但通常无法保持实的可公度谱或解析可控动力学。我们研究具有反向分级定向跳变的有限开放Krawtchouk网络。在宽的中间不对称范围内，其局域虚规范场在体内变号，其累积坐标生成正对角相似映射，该映射的归一化平方项形成偏二项式包络。同一映射将开放链转化为自旋旋转生成元2gJ_x，因此聚焦与等间距谱由同一种分级产生。在该聚焦 regime 中，精确的右、左及双正交本征向量表明，包络宽度和参与数均按√N缩放，确定了次广延性内部聚焦。在物理节点基中，自旋旋转产生具有方向选择性放大与衰减的完美镜像反转，其增益互为倒数。定向格林函数共享其极点，而其残差相差同一相似比。闭合链暴露出规范不变虚通量，任一精确单向极限产生N阶异常点。 onsite 无序保持定向预解式比，而独立跳变无序破坏干净解析Krawtchouk映射并降低可公度性与转移。因此该模型提供了一个可精确求解的有限网络框架，将局域化几何与本征向量非正交性关联到可公度谱及节点分辨的定向响应。
英文摘要
Spatially graded nonreciprocity can move skin weight away from a boundary, but it does not generally preserve a real commensurate spectrum or analytically controlled dynamics. We study a finite open Krawtchouk network with oppositely graded directed hoppings. For a broad intermediate range of asymmetries, their local imaginary gauge field changes sign in the bulk, and its accumulated coordinate generates a positive diagonal similarity map whose normalized squared entries form a biased-binomial envelope. The same map converts the open chain into the spin-rotation generator \(2gJ_x\), so the focus and equally spaced spectrum follow from one grading. In this focusing regime, exact right, left, and biorthogonal eigenvectors show that the envelope width and participation number both scale as \(\sqrt N\), identifying a subextensive interior focus. In the physical node basis, spin rotation produces perfect mirror inversion with direction-selective amplification and attenuation whose gains are mutually inverse. The directional Green functions share their poles, while their residues differ by the same similarity ratio. Closing the chain exposes a gauge-invariant imaginary flux, and either exact one-way limit yields an exceptional point of order \(N\). Onsite disorder preserves the directional resolvent ratio, whereas independent hopping disorder breaks the clean analytic Krawtchouk map and degrades commensurability and transfer. The model therefore provides an exactly solvable finite-network framework linking localization geometry and eigenvector nonorthogonality to commensurate spectra and node-resolved directional response.
Comments17 pages, 8 figures
