Finite $N$ Bulk Hilbert Space in ETH Matrix Model for double-scaled SYK

TL;DR

This paper extends the concept of chord number to finite N in the ETH matrix model related to the SYK model, constructing operators that approximate bulk geometry and relate to complexity and black hole formation.

Contribution

It introduces a finite N chord number operator in the ETH matrix model, connecting it to bulk geometry and Krylov complexity, and explores its algebraic structure.

Findings

Chord number states form an over-complete basis at finite N.

Constructed chord number operator approximates q-deformed oscillator algebra.

Chord number zero state matches Krylov state complexity, showing black hole features.

Abstract

We extend the notion of chord number in the strict large $N$ double-scaled Sachdev-Ye-Kitaev (DSSYK) model to the corresponding finite $N$ ETH matrix model. The chord number in the strict large $N$ DSSYK model is known to correspond to the discrete length of the Einstein-Rosen bridge in the gravity dual, which reduces to the renormalized geodesic length in JT gravity at weak coupling. At finite $N$, these chord number states form an over-complete basis of the non-perturbative Hilbert space, as the structure of the inner product gets significantly modified due to the Cayley-Hamilton theorem: There are infinitely many null states. In this paper, by considering ``EFT for gravitational observables'' or a version of ``non-isometric code'', we construct a family of chord number operators at finite $N$. While the constructed chord number operator depends on the reference chord number state, it realizes approximate $q$-deformed oscillator algebra and reproduces semiclassical bulk geometry around the reference state. As a special case, we will show that when the reference is chosen to be the chord number zero state, the chord number operator precisely matches with the Krylov state complexity, leading to the ``ramp-slope-plateau'' behavior at late times, implying the formation of ``grey hole''.