Integrability of a family of clean SYK models from the critical Ising chain

TL;DR

This work demonstrates the complete integrability of a family of clean SYK models with uniform $p$-body interactions by grounding them in the Yang–Baxter framework of the critical Ising chain. The authors construct transfer matrices from the Ising $R$-matrix, show mutual commutativity of the SYK charges $H_{2p}$ and $H_{2p+1}$, and derive their exact spectra and eigenstates via a Majorana-fermion based diagonalization. A central result is that the forward/backward monodromy matrices decompose in terms of the SYK transfer matrices, and the critical Ising Hamiltonian emerges from the logarithmic derivative of the transfer matrix. The approach provides a unified, exact solvable picture linking quantum chaos indicators in SYK to integrable Ising-chain structures, with potential extensions to hybrid models combining long-range SYK and short-range Ising charges.

Abstract

We establish the integrability of a family of Sachdev-Ye-Kitaev (SYK) models with uniform $p$-body interactions. We derive the R-matrix and mutually commuting transfer matrices that generate the Hamiltonians of these models, and obtain their exact eigenspectra and eigenstates. Remarkably, the R-matrix is that of the critical transverse-field Ising chain. This work reveals an unexpected connection between the SYK model, central to many-body quantum chaos, and the critical Ising chain, a cornerstone of statistical mechanics.