arXiv 2404.010022024-04-01hep-latcond-mat.mes-hallhep-th

格点上弯曲畴壁费米子的研究

Study of Curved Domain-wall Fermions on a Lattice

Shoto Aoki

备注：: 83 pages, 28 figures, PhD thesis, Supervisor: H. Fukaya

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AI总结：

本文研究格点上弯曲畴壁费米子，发现畴壁上的手征边缘态通过自旋联络感受引力，并分析了U(1)规范场引起的T反常及其与拓扑项、强通量局域模式的抵消机制，最后将结果应用于拓扑绝缘体中涡旋或磁单极子获得分数电荷的微观描述。

摘要：

在本论文中，我们考虑在具有弯曲畴壁质量项的方形格点空间上的费米子系统。与平坦情形类似，我们发现无质量且手征的态局域在畴壁上。在嵌入方形格点的 $S^1$ 和 $S^2$ 畴壁情形中，我们发现这些边缘态通过诱导的自旋联络感受到引力。在更高维格点的常规连续极限下，我们发现与连续理论中的解析结果有良好的一致性。我们还确认旋转对称性会自动恢复。我们还讨论了 $U(1)$ 规范联络对具有 $S^1$ 畴壁质量项的二维格点费米子的影响。我们发现规范场通过 Aharanov-Bohm 效应改变了边界系统的本征值谱，并产生了时间反演（$T$）对称性的反常。我们的数值评估与 Atiyah-Patodi-Singer 指标一致，该指标描述了体系统上的拓扑项对 $T$ 反常的抵消。当我们在保持总通量不变的情况下将通量压缩到一个 plaquette 内时，反常流入发生了剧烈变化。强通量在通量周围产生了额外的畴壁。我们观察到在通量处出现了一种新的局域模式，它抵消了畴壁上的 $T$ 反常，而不是体中的拓扑项。我们将该研究应用于凝聚态物理中的一个问题。已知在拓扑绝缘体内部，涡旋或磁单极子会获得分数电荷并转变为双子（dyon）。通过将拓扑绝缘体描述为 Dirac 费米子的负质量区域，我们从动力学畴壁产生的角度为这一现象提供了微观描述。

英文摘要：

In this thesis, we consider fermion systems on square lattice spaces with a curved domain-wall mass term. In a similar way to the flat case, we find massless and chiral states localized at the wall. In the case of $S^1$ and $S^2$ domain-wall embedded into a square lattice, we find that these edge states feel gravity through the induced spin connection. In the conventional continuum limit of the higher dimensional lattice, we find a good consistency with the analytic results in the continuum theory. We also confirm that the rotational symmetry is recovered automatically. We also discuss the effect of a $U(1)$ gauge connection on a two-dimensional lattice fermion with the $S^1$ domain-wall mass term. We find that the gauge field changes the eigenvalue spectrum of the boundary system by the Aharanov-Bohm effect and generates an anomaly of the time-reversal ($T$) symmetry. Our numerical evaluation is consistent with the Atiyah-Patodi-Singer index, which describes the cancellation of the $T$ anomaly by the topological term on the bulk system. When we squeeze the flux inside one plaquette while keeping the total flux unchanged, the anomaly inflow undergoes a drastic change. The intense flux gives rise to an additional domain wall around the flux. We observe a novel localized mode at the flux, canceling the $T$ anomaly on the wall instead of the topological term in the bulk. We apply the study to a problem in condensed matter physics. It is known that inside topological insulators, a vortex or monopole acquires a fractional electric charge and turns into a dyon. Describing the topological insulator as a negative mass region of a Dirac fermion, we provide a microscopic description of this phenomenon in terms of the dynamical domain-wall creation.

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