Holographic tensor network for double-scaled SYK

TL;DR

This work constructs a concrete holographic tensor network for the boundary theory of double-scaled SYK (DSSYK) by mapping the moments of the transfer matrix $T=a_+ + a_-$ to a matrix product state (MPS) of a spin chain and promoting a height direction to a holographic coordinate, yielding a finite-bond-dimension network built from a 4-index tensor with bond dimension $3$. The construction employs a Dyck-path tiling and a height-dependent operator $A_h$ with $A_h = \sqrt{\frac{1-q^h}{1-q}}(t_1+t_2) + t_3 + t_4 + t_5$, reproducing the $q$-deformed oscillator matrix elements and recovering the Fredkin-chain MPS in the $q\to 0$ limit. It connects to ASEP and the Fredkin spin chain, and naturally extends to matter two-point functions within the same tensor-network framework. The work provides a concrete, solvable holographic model with finite bond dimension that may shed light on bulk discretization, holographic error correction, and extensions to multi-particle chord sectors and ETH-type matrix models, opening several avenues for future exploration such as entanglement structure and continuum limits.

Abstract

We construct a holographic tensor network for the double-scaled SYK model (DSSYK). The moment of the transfer matrix of DSSYK can be mapped to the matrix product state (MPS) of a spin chain. By adding the height direction as a holographic direction, we recast the MPS for DSSYK into the holographic tensor network whose building block is a 4-index tensor with the bond dimension three.