Pole skipping away from maximal chaos

TL;DR

The paper extends the pole-skipping phenomenon to non-maximally chaotic systems by tying the special point in the energy-density correlator to the stress-tensor contribution to chaos via a velocity-dependent Lyapunov framework. It introduces precise definitions of the stress-tensor induced Lyapunov exponent and butterfly speed, and proposes a universal bound u_B ≤ u_B^{(T)} with pole skipping located at (ω,p)ps = i λ^{(T)}(1,1/u_B^{(T)}). The authors test these ideas in the exactly solvable large-q SYK chain, deriving a closed-form energy-density two-point function across coupling, extracting the diffusion constant, pole motion, and all-order hydrodynamics, and thus linking chaos diagnostics to a tractable two-point observable. The results demonstrate that pole skipping encodes stress-tensor chaos data but does not generally reveal the full Lyapunov exponent, and they illuminate the analytic structure of thermal correlators and hydrodynamic convergence over the full coupling range, with implications for bounds on diffusion and chaos propagation.

Abstract

Pole skipping is a recently discovered subtle effect in the thermal energy density retarded two point function at a special point in the complex $(ω,p)$ planes. We propose that pole skipping is determined by the stress tensor contribution to many-body chaos, and the special point is at $(ω,p)_\text{p.s.}= i λ^{(T)}(1,1/u_B^{(T)})$, where $λ^{(T)}=2π/β$ and $u_B^{(T)}$ are the stress tensor contributions to the Lyapunov exponent and the butterfly velocity respectively. While this proposal is consistent with previous studies conducted for maximally chaotic theories, where the stress tensor dominates chaos, it clarifies that one cannot use pole skipping to extract the Lyapunov exponent of a theory, which obeys $λ\leq λ^{(T)}$. On the other hand, in a large class of strongly coupled but non-maximally chaotic theories $u_B^{(T)}$ is the true butterfly velocity and we conjecture that $u_B\leq u_B^{(T)}$ is a universal bound. While it remains a challenge to explain pole skipping in a general framework, we provide a stringent test of our proposal in the large-$q$ limit of the SYK chain, where we determine $λ,\, u_B,$ and the energy density two point function in closed form for all values of the coupling, interpolating between the free and maximally chaotic limits. Since such an explicit expression for a thermal correlator is one of a kind, we take the opportunity to analyze many of its properties: the coupling dependence of the diffusion constant, the dispersion relations of poles, and the convergence properties of all order hydrodynamics.

Figures (13)

• Figure 1: The two dimensional coupling space of the model. There is a critical line $v_*({{\gamma}})$ that separates the blue region, where above a critical velocity $u_*$ the VDLE is dominated by the pole and is maximal, from the white region, where the saddle always dominates, and the VDLE is nowhere maximal.
• Figure 2: We piece together the solution to \ref{['eq:greenseq']} using the homogeneous solution $\Psi_n^-$ for regions with red boundaries and $\Psi_n^+$ for regions with blue boundaries. Along green dashed lines, there are jumps in the first derivatives giving rise to the delta functions in the RHS of \ref{['eq:greenseq']}.
• Figure 3: Density plot of the retarded energy-energy two point function for imaginary momentum and frequency for $v=0.6,\, {{\gamma}}=1$. Hot lines (white) are pole lines, cold lines (blue) are zero lines. Black dots are the pole skipping points corresponding to chaos, whose location agrees with the proposal \ref{['PoleSkip2']}. Left: We plot the complete correlator. Right: We drop the momentum dependent contact term and plot only $\partial_\theta \log \psi_n(\theta_v)$, as we do in the rest of the density plots in this paper. The pole lines and the pole skipping points are unaffected by this, while the shape of the zero lines change. These shapes are therefore not physical.
• Figure 4: Density plot of the numerator and denominator of \ref{['eq:retTT']} overlaid, with pole skipping points marked with black dot. There are pole/zero lines starting from negative $n$ that make it to positive $n$ that are not visible on the overall density plot of Fig. \ref{['fig:poleskip']}. Each of the additional pole zero line pairs cross an odd number of times. The diffusion pair crosses once, the next one three times, the next one five times and so on. The pole skipping points that are not on the diffusion pole line do not contribute to chaos.
• Figure 5: Two point function for imaginary frequency and real momentum. The pole lines (hot) are confined to negative $n$ as they should be. There are various pole skipping points in this case too, marked by black dots. The left plot is for $v=0.6$, the right one is for $v=0.8$ with ${{\gamma}}=1$.