Quantum chaos and pole skipping in two-dimensional conformal perturbation theory

Abstract

We analyze pole skipping of stress tensor two-point functions in two-dimensional quantum field theories perturbed away from conformality by a relevant deformation. The retarded two-point Green's function can be formally computed in conformal perturbation theory, though it results in singular expressions. We propose a natural interpretation of these expressions and compute the resulting Green's function to leading nontrivial order in the deformation. As a check of our results, we compare the Lyapunov exponents and butterfly velocities we find from our computed skipped poles to those obtained from both a leading-order conformal field theory analysis using Ward identities, as well as to a holographic gravitational dual perturbed by a massive scalar field; we find precise agreement. We comment on extensions to sub-leading order, where agreement with holographic expectations would no longer be expected.

Figures (6)

• Figure 1: Plots of $G_2^\text{E}(\Sigma)$ in the complex $\Sigma$ plane. From left to right, the plots correspond to $h = 0.4$, $h = h_* \approx 0.74$, and $h = 0.9$. $G_2^\text{E}(\Sigma)$ is even in $\Sigma$, single-valued, vanishes at $\Sigma = \pm 1$, and diverges at $\Sigma \to \infty$. $G_2^\text{E}(\Sigma)$ also exhibits two additional zeros besides $\Sigma = \pm 1$; for $h \in (0,h_*)$ these zeros lie in the interval $\Sigma \in (-1,1)$, while for $h > h_*$ they are on the imaginary axis.
• Figure 2: \ref{['subfig:tFouriercontour']}: the contour of integration in $t$ for evaluating the Fourier transform \ref{['eq:tFouriertransform']}. \ref{['subfig:tmodifiedcontour']}: the contour of integration can be deformed into a keyhole contour that runs along the branch cut $t > |x|$ and circles the branch point $t = |x|$. The branch point $t = x$ is singular, so when $x > 0$ we must be careful to keep track of the singular contributions from the circle of radius $\epsilon$ around the branch point and from the integral along the cut from $t = x + \epsilon$ to infinity. These contributions are individually singular as $\epsilon \to 0$, but the divergences must cancel out.
• Figure 3: The $O(\lambda^2)$-contribution to the deformed Lorentzian two-point function of the holomorphic component of the stress tensor, $G_2(t,x)$; c.f. equations \ref{['eq:DefLorGF']} and \ref{['eq:DeltaG2']}. Here we take $h= 1/2, \beta = 1$. It is zero everywhere outside the light cone, as expected; it is nonzero everywhere inside the future light cone, approaching zero at the left light cone and diverging at the right light cone.
• Figure 4: The perturbation to the butterfly velocity as a function of $h$. The blue curve shows the result \ref{['eq:vPSWard']} for the pole-skipping velocity obtained from the Ward identities, as well as the holographic result \ref{['eq:vBphi']} for the butterfly velocity obtained from the holographic calculation (the results coincide). The orange point shows the CFT result obtained in Section \ref{['sec:integrals']} for the special case $h = 1/2$. Near $h = 1$, the CFT results predict that $(v_\text{B}-1)$ should be quadratic in $(h-1)$, also consistent with the holographic results shown here.
• Figure 5: The contour of integration $\gamma$ for the integral \ref{['eq:Kcontourint']} obtained from applying Stokes' theorem. The large circle is a contour at infinity traversed counterclockwise, while the five infinitesimal circles go clockwise around the logarithmic singularity at $w' = 0$ and the four poles at $w' = (\Sigma \pm 1)/(w \pm 1)$.