Thermal two-point functions in SYK and complex-time singularities

Technical Summary: Thermal Two-Point Functions in SYK and Complex-Time Singularities

Problem Statement
The paper investigates the analytic structure of the finite-temperature two-point function in the large-$N$ Sachdev-Ye-Kitaev (SYK) model, specifically focusing on the regime of intermediate couplings away from the infrared (IR) fixed point. While the holographic gauge/gravity correspondence has established that thermal correlators in strongly coupled theories are governed by the quasinormal modes of dual black hole geometries, the explicit time-domain singularities of these correlators in field theory remain less understood. In particular, the paper seeks to determine the location and behavior of complex-time singularities in the SYK model as a function of temperature, extending previous analyses that were limited to infinite temperature or the strict IR limit. The authors aim to identify whether these singularities persist down to zero temperature and to interpret their physical significance in terms of emergent geometry and operator complexity.

Methodology
To solve the Schwinger-Dyson (SD) equations for the thermal two-point function, the authors employ two complementary methods:

1. Double-Expansion Perturbative Solution:

   • The authors treat the correlator as a function of two dimensionless parameters: $\tau/\beta$ (Euclidean time scaled by inverse temperature) and $\beta J$ (coupling scaled by temperature).
   • They perform a perturbative expansion around the free ultraviolet (UV) point (small $\beta J$) and the conformal limit (large $\beta J$).
   • The expansion coefficients are determined iteratively. To extend the results beyond the finite radius of convergence of the power series (which is limited by fermionic thermal factors), they utilize Padé approximants.
   • Specifically, they first resum the series in $\beta^2$ to access finite and low temperatures, and then apply a second Padé approximation in the time variable $\tau$ to probe the analytic structure in the complex plane.

2. Numerical Solution:

   • The authors implement a damped fixed-point iteration procedure to solve the SD equations directly.
   • The algorithm discretizes both time and frequency axes, introducing a UV cutoff $\Lambda$.
   • The iteration cycle updates the self-energy $\Sigma$, the retarded Green's function $G_R$, the spectral density $\rho$, and the Wightman functions $G_{>}$ and $G_{<}$.
   • To extract singularities from the numerical data (which is defined on real time), the authors approximate the resulting functions using the AAA algorithm (a rational approximation method) to identify poles in the complex plane.

Key Contributions and Results

• Agreement of Methods: The authors demonstrate that the double-expansion method (with Padé resummation) and the direct numerical solution yield consistent results for the thermal two-point function, particularly at high temperatures ($\beta \lesssim 12$).
• Existence of Complex Singularities: Both methods confirm that the thermal two-point function possesses an infinite number of singularities in the complex time plane ($\tau$-plane) at any finite temperature.
• Leading Singularity (Effective Temperature):
  • The leading singularity is located on the imaginary axis at $i\tau^*$ (where $\tau^* > 0$).
  • This singularity controls the short-time dynamics of operator complexity and defines an "effective temperature" for the correlator.
  • Crucially, $\tau^*$ exhibits only mild temperature dependence. As $\beta \to \infty$ (zero temperature), $\tau^*$ appears to approach a constant value of approximately $2.8$, rather than diverging or vanishing. This implies the correlator remains analytic in a strip wider than that required by standard thermal KMS invariance.
• Next-to-Leading Singularities:
  • The next-to-leading singularities lie off the imaginary axis. Their real and imaginary parts also appear to converge to constant values as $\beta \to \infty$.
  • These singularities control the high-frequency behavior of the spectral density.
  • The authors note that the location of these subleading poles is reminiscent of time-domain singularities arising from "bouncing null geodesics" in emergent black hole geometries.
• Persistence to Zero Temperature: A central finding is that these singularities persist all the way down to zero temperature. This contrasts with the expectation that thermal singularities might disappear or merge in the zero-temperature limit, suggesting a robust structural feature of the SYK model's dynamics even away from the IR conformal fixed point.

Significance and Claims
The paper claims to provide a field-theoretic derivation of the analytic structure of thermal correlators in a solvable large-$N$ model, complementing previous gravitational descriptions. By tracking singularities from infinite to zero temperature, the authors argue that the SYK model retains signatures of emergent black hole geometry (specifically related to null geodesics) even in the intermediate coupling regime.

The identification of a temperature-independent "effective temperature" (via the leading singularity) suggests a universal mechanism governing operator growth and complexity in the model. The persistence of these singularities to zero temperature indicates that the emergent geometric picture is not merely an IR phenomenon but is deeply embedded in the model's UV and intermediate dynamics. The authors explicitly state that their work focuses on the complex-time domain, distinguishing it from related recent work that analyzes the frequency-space quasinormal spectrum.