Entanglement entropy in type II$_1$ von Neumann algebra: examples in Double-Scaled SYK

An intriguing feature of type II$_1$ von Neumann algebra is that the entropy of the mixed states is negative. Although the type classification of von Neumann algebra and its consequence in holography have been extensively explored recently, there has not been an explicit calculation of entropy in some physically interesting models with type II$_1$ algebra. In this paper, we study the entanglement entropy $S_n$ of the fixed length state $\{|n\rangle\}$ in Double-Scaled Sachdev-Ye-Kitaev model, which has been recently shown to exhibit type II$_1$ von Neumann algebra. These states furnish an orthogonal basis for 0-particle chord Hilbert space. We systematically study $S_n$ and its R\'enyi generalizations $S_n^{(m)}$ in various limit of DSSYK model, ranging $q\in[0,1]$. We obtain exotic analytical expressions for the scaling behavior of $S_n^{(m)}$ at large $n$ for random matrix theory limit ($q=0$) and SYK$_2$ limit ($q=1$), for the former we observe highly non-flat entanglement spectrum. We then dive into triple scaling limits where the fixed chord number states become the geodesic wormholes with definite length connecting left/right AdS$_2$ boundary in Jackiw-Teitelboim gravity. In semi-classical regime, we match the boundary calculation of entanglement entropy with the dilaton value at the center of geodesic, as a nontrivial check of the Ryu-Takayanagi formula.