A Study of the SYK$_{2}$ Model with Twisted Boundary Conditions

We study a version of the 2-body Sachdev-Ye-Kitaev (SYK$_{2}$) 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 $A_{\mu}(t,\mathbf{x})$. 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 disorder scale needed for the manifestation of zero modes. These zero modes are responsible for the late-time exponential ramp in the quadratic SYK model.