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Long-time tails in the SYK chain from the effective field theory with a large number of derivatives

We study the nonlinear energy diffusion through the SYK chain in the framework of Schwinger-Keldysh effective field theory. We analytically construct the interacting effective Lagrangian up to $40^{th}$ order in the derivative expansion. According to this effective Lagrangian, we calculate the first order loop correction of the energy density response function, the pole of which is the dispersion relation of energy diffusion. As expected, we see that the standard derivative expansion of that dispersion relation, $ω=-i D_{(1)} k^2- i D_{(2)} k^4+\mathcal{O}(k^6)$, breaks down due to the long-time tails. However, we find that the nonlinear contribution of order $n$ to the self-energy is proportional to $\left(k^{2}\right)^{n+1/2}$. This suggests to modify the dispersion relation by splitting it into two dispersion relations and double the number of transport coefficients at any order as $ω=-i k^2\big( D_{(1,1)}\pm i D_{(1,2)} \left(k^2\right)^{1/2}\big)-i k^4\big( D_{(2,1)}\pm i D_{(2,2)} \left(k^2\right)^{1/2}\big)+\mathcal{O}(k^6)$. We find that the modified series, which include the effect of long-time tails, are convergent. The radius of convergence is proportional to the ratio of thermal conductivity to diffusion constant.