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

TL;DR

This paper investigates a twisted boundary condition version of the SYK$_{2}$ model, revealing that twisting does not affect early-time chaos but influences spectral properties and zero modes, with implications for understanding integrability and chaos.

Contribution

It provides a detailed analysis of how twisted boundary conditions affect the spectral and chaotic properties of the SYK$_{2}$ model, highlighting the role of zero modes.

Findings

Twisting does not alter early-time OTOC behavior.

Spectral form factor shows enhanced early-time slope due to twisting.

Zero modes emerge, affecting late-time spectral ramp.

Abstract

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.