Chiral Sachdev-Ye model: Integrability and chaos of anyons in 1+1d

Abstract

We construct and study a chiral Sachdev-Ye (SY) model consisting of $N$ chiral $\mathrm{SU}{(M)}_{1}$ Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the $0+1\mathrm{d}$ quantum chaotic SY spin model into $1+1\mathrm{d}$ chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of $2+1\mathrm{d}$ gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers $N$ and $M$, which is integrable and decomposes into a chiral $\mathrm{SU}{(M)}_{N}$ WZW model and its coset with a different speed of light. When $N=M=2$, the model maps to a free Majorana fermion model. The second limit is the large $N$ and $M$ limit with random interactions, which is solvable to the leading $\frac{1}{NM}$ order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound $2\ensuremath{\pi}/\ensuremath{\beta}$ at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$.