Transport Peak in the Thermal Spectral Function of $\mathcal{N}=4$ Supersymmetric Yang-Mills Plasma at Intermediate Coupling

We study the structure of thermal spectral function of the stress-energy tensor in $\mathcal{N}=4$ supersymmetric Yang-Mills theory at intermediate ‘t Hooft coupling and infinite number of colors. In gauge-string duality, this analysis reduces to the study of classical bulk supergravity with higher-derivative corrections, which correspond to (inverse) coupling corrections on the gauge theory side. We extrapolate the analysis of perturbative leading-order corrections to intermediate coupling by nonperturbatively solving the equations of motion of metric fluctuations dual to the stress-energy tensor at zero spatial momentum. We observe the emergence of a separation of scales in the analytic structure of the thermal correlator associated with two types of characteristic relaxation modes. As a consequence of this separation, the associated spectral function exhibits a narrow structure in the small frequency region which controls the dynamics of transport in the theory and may be described as a transport peak typically found in perturbative, weakly interacting thermal field theories. We compare our results with generic expectations drawn from perturbation theory, where such a structure emerges as a consequence of the existence of quasiparticles.

Figure 1

Left: Poles of the $\mathcal{N}=4$ SYM stress-energy tensor correlator $G_{xy,xy}^R(\mathfrak{w},0)$ for $\gamma ={10}^{-3}$ (black solid circles), $\gamma ={10}^{-2}$ (blue crosses), and $\gamma =2\times {10}^{-2}$ (red empty circles) in the complex plane of normalized frequency $\mathfrak{w}=\omega /2\pi T$. Right: Imaginary parts of the poles closest to the origin normalized by the real part of the first complex mode [see Eq. (8)] as functions of the (inverse) coupling.

Figure 2

Dimensionless spectral function ${\overline{\rho }}_{xy,xy}(\omega ,q)\equiv ({\pi }^2/N_c^2{r}_0^4){\rho }_{xy,xy}(\omega ,q)$ as a function of $\mathfrak{w}=\omega /2\pi T$ for $\gamma ={10}^{-3}$ (black, solid), $\gamma ={10}^{-2}$ (blue, dot-dashed) and $\gamma =2\times {10}^{-2}$ (red, dashed). The left and right panels show different frequency ranges.