Replica manifolds, pole skipping, and the butterfly effect

A bstract The black hole butterfly effect is a signal of quantum chaos in holographic theories that can be probed in different ways, including out-of-time-order correlators (OTOCs), pole skipping (PS), and entanglement wedge (EW) reconstruction. Each of these three phenomena can be used to define a butterfly velocity that measures the speed at which chaos spreads. In a general quantum system the three velocities $$ {v}_B^{\textrm{OTOC}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>v</mml:mi> <mml:mi>B</mml:mi> <mml:mtext>OTOC</mml:mtext> </mml:msubsup> </mml:math> , $$ {v}_B^{\textrm{PS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>v</mml:mi> <mml:mi>B</mml:mi> <mml:mi>PS</mml:mi> </mml:msubsup> </mml:math> , and $$ {v}_B^{\textrm{EW}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>v</mml:mi> <mml:mi>B</mml:mi> <mml:mi>EW</mml:mi> </mml:msubsup> </mml:math> can be different, but it is known from explicit calculations that they are all equal in certain holographic theories dual to Einstein gravity plus higher-curvature corrections. A conceptual explanation for this apparent coincidence is lacking. We show that it follows from a deeper relationship: the pole-skipping mode, added to the black hole background, can be reinterpreted as the gravitational replica manifold for the late-time entanglement wedge, and its imaginary part is the shockwave that computes the OTOC. Thus pole skipping is directly related to entanglement dynamics in holographic theories, and the origin of the pole-skipping mode is an extremal surface on the horizon. This explains the coincidence $$ {v}_B^{\textrm{OTOC}}={v}_B^{\textrm{PS}}={v}_B^{\textrm{EW}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>v</mml:mi> <mml:mi>B</mml:mi> <mml:mtext>OTOC</mml:mtext> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>v</mml:mi> <mml:mi>B</mml:mi> <mml:mi>PS</mml:mi> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>v</mml:mi> <mml:mi>B</mml:mi> <mml:mi>EW</mml:mi> </mml:msubsup> </mml:math> in known cases, and extends it to general theories of gravity with a pole-skipping mode having the usual behavior.