Krylov complexity in the IP matrix model
Norihiro Iizuka, Mitsuhiro Nishida
Abstract
A bstract The IP matrix model is a simple large N quantum mechanical model made up of an adjoint harmonic oscillator plus a fundamental harmonic oscillator. It is a model introduced previously as a toy model of the gauge theory dual of an AdS black hole. In the large N limit, one can solve the Schwinger-Dyson equation for the fundamental correlator, and at sufficiently high temperature, this model shows key signatures of thermalization and information loss; the correlator decay exponentially in time, and the spectral density becomes continuous and gapless. We study the Lanczos coefficients b n in this model and at sufficiently high temperature, it grows linearly in n with logarithmic corrections, which is one of the fastest growth under certain conditions. As a result, the Krylov complexity grows exponentially in time as $$ \sim \exp \left(\mathcal{O}\left(\sqrt{t}\right)\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∼</mml:mo> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mrow> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msqrt> <mml:mi>t</mml:mi> </mml:msqrt> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> . These results indicate that the IP model at sufficiently high temperature is chaotic.