We study a version of the 2-body Sachdev-Ye-Kitaev (SYK₂) model whose complex fermions exhibit twisted boundary conditions on the thermal circle. As we show, this is physically equivalent to coupling the fermions to a 1-dimensional external gauge field A(t). In the latter formulation, the gauge field itself can be thought of as arising from a radial symmetry reduction of a (2 + 1)-dimensional Chern-Simons gauge field A_μ(t, x). Using the diagnostic tools of the out-of-time-order correlator (OTOC) and spectral form factor (SFF), which probe the sensitivity to initial conditions and the spectral statistics respectively, we give a detailed and pedagogical study of the integrable/chaotic properties of the model. We find that the twisting has no effect on the OTOCs and, by extension, the early-time chaos properties of the model. It does, however, have two notable effects on the spectral form factor; an enhancement of the early-time slope and the emergence of an explicit twisting-dependent timescale needed for the manifestation of zero modes. These zero modes are responsible for the late-time exponential ramp in the quadratic SYK model.

我们研究了 2 体 Sachdev-Ye-Kitaev (SYK ₂ ) 模型的一个版本，其复杂的费米子在热循环上表现出扭曲的边界条件。正如我们所展示的，这在物理上相当于将费米子耦合到一维外部规范场A ( t )。在后一个公式中，规范场本身可以被认为是由 (2 + 1) 维 Chern-Simons 规范场A _μ ( t, x ) 的径向对称约化产生的。使用乱序相关器（OTOC）和谱形状因子（SFF）的诊断工具，分别探测对初始条件和谱统计的敏感性，我们对可积/混沌进行了详细的教学研究模型的属性。我们发现扭曲对 OTOC 没有影响，进而影响模型的早期混沌特性。然而，它确实对光谱形状因子有两个显着的影响：早期斜率的增强以及零模态表现所需的明确的扭曲相关时间尺度的出现。这些零模式导致二次 SYK 模型中的后期指数斜坡。