[论文解读] Impurity screening by defects in (1+1)$d$ quantum critical systems

Ying-Hai Wu, Zhang, Yueshui+2 more|arXiv (Cornell University)|Jul 18, 2023

Physics of Superconductivity and MagnetismPhysics and Astronomy被引用 3

一句话总结

本文提出了一种在由共形场论（CFT）描述的(1+1)维量子临界系统中杂质屏蔽的新机制，其中杂质不仅可通过手征初级场屏蔽，还可通过CFT的拓扑缺陷线（TDL）屏蔽。当杂质的量子数与TDL匹配时发生屏蔽，导致对称性丰富CFT中的奇异边界条件，该机制在SU(3)₁和Spin(5)₁ CFT中通过自旋链的解析与数值研究得到证实，系统中边缘耦合了自旋-1/2和自旋-2杂质。

ABSTRACT

We propose a novel mechanism of impurity screening in (1+1)$d$ quantum critical states described by conformal field theories (CFTs). An impurity can be screened if it has the same quantum numbers as some gapless degrees of freedom of the CFT. The common source of these degrees of freedom is the chiral primary fields of the CFT, but we uncover that topological defect lines of the CFT may also take this role. Theoretical analysis relies on the insight that the impurities can be interpreted as edge modes of certain symmetry-protected topological (SPT) states. By stacking a SPT state with a CFT, one or two interfaces on which the SPT edge modes reside are created. If screening occurs due to topological defect lines, a symmetry-enriched CFT with exotic boundary states are obtained. The boundary conditions that appear in these cases are difficult to achieve using previously known methods. As a concrete example, we consider a spin-1 chain whose bulk is described by the SU(3)$_{1}$ CFT and edges are coupled to spin-1/2 impurities. We demonstrate that both the low-energy eigenstates and the extracted Affleck-Ludwig entropy are in excellent agreement with our theoretical predictions.

研究动机与目标

• 理解由共形场论（CFT）描述的无能隙(1+1)维量子临界系统中杂质屏蔽的机制。
• 识别CFT的拓扑缺陷线（TDL）在何种条件下可超越传统手征初级场，实现杂质屏蔽。
• 建立对称保护拓扑（SPT）态的边缘模式与CFT中边界条件之间的联系。
• 证明当TDL介导屏蔽时，可通过标准方法难以实现的奇异边界态可被实现。
• 通过SU(3)₁和Spin(5)₁ CFT中的自旋链数值模拟验证理论框架。

提出的方法

• 理论分析采用边界共形场论（BCFT），将杂质解释为SPT态的边缘模式，屏蔽对应于存在与杂质的投影表示匹配的G-对称边界态。
• 利用群上同调（H²(G, U(1)))对SPT相进行分类，以识别杂质的投影类ωπ，该类必须与边界态的类ωB匹配，屏蔽才能发生。
• 研究采用具有对角划分函数的有理CFT中边界态的语言，聚焦于保持全局对称群G的简单边界条件。
• 在具有开放边界条件的自旋链上进行数值模拟，实现体相中的SU(3)₁和Spin(5)₁ CFT，并将自旋-1/2或自旋-2杂质耦合至边缘。
• 哈密顿量包含二次和双二次相互作用，临界点位于ULS和Reshetikhin角，对应于SU(n)₁和Spin(n)₁ CFT。
• 通过低能能级的有限尺寸标度及能隙比（例如Lc=60时约为2.168–2.236）检验屏蔽与未屏蔽相的理论预测一致性。

Figure 1: Schematics of our model. (a) The middle part is a spin chain whose low-energy physics is described by a CFT. Its two edges are attached with spin-1/2 impurities. (b) The spin-1/2 impurities in panel (a) can be viewed as the edge states of the AKLT chain.

实验结果

研究问题

• RQ1 CFT中的拓扑缺陷线（TDL）是否可介导边缘杂质的屏蔽，超越手征初级场？
• RQ2 在对称性丰富CFT中，何种条件下边界态对应于具有给定投影表示的屏蔽杂质？
• RQ3 TDL如何导致通过传统边界态构造无法实现的奇异边界条件？
• RQ4 自旋链中SU(3)₁和Spin(5)₁ CFT的数值模拟在多大程度上证实了预测的屏蔽机制？
• RQ5 杂质的投影类ωπ在决定屏蔽是否通过TDL或手征场发生中起何种作用？

主要发现

• 在SU(3)₁ CFT中，自旋-1/2杂质的低能谱在Lc=60、J₁=10、J₂=0时显示能隙比约为2.236，与屏蔽情况的理论预测一致。
• 在相同模型中，当J₂=−3时，能隙比下降至2.168，表明有限尺寸效应被抑制，与理论预测的吻合度提高。
• 在Spin(5)₁ CFT中，两个自旋-2杂质的最低三重态具有量子数S=0,1,2，随着J₁减小，能隙比趋近理论预期值。
• 第四激发态在Lc=12时为S=7/2，在Lc=16时变为S=1/2（J₁=2），表明随着系统尺寸增大，其收敛至预测的屏蔽态。
• 在Reshetikhin点（Spin(n)₁ CFT）处，SO(n)旋量杂质始终被旋量Cardy边界态屏蔽，证实了n=5情况下的TDL介导屏蔽。
• 理论分析确认，当n为奇数时，唯一TDL Ds（携带旋量表示）满足Ds×Ds=∑aDa，可通过Verlinde线实现屏蔽，该机制与数值结果一致。

Figure 2: Numerical results on the ULS-AKLT chain. The ground state energy is shifted to zero. The total spin of each level is indicated. The gaps defined in the main text are illustrated in panels (a) and (c). Finite-size scaling results for the gaps are plotted in panels (b) and (d). The ratio $R=