Two indices Sachdev-Ye-Kitaev model

We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the $SU(M)$ complex fermion representation is that it need no local constraints, so a $1/M$ expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices $J_{ij}$ instead of four indices $J_{ijkl}$, so it may fit the bulk string theory better. By performing a $1/M$ expansion at $N=\infty$, we show that a quantum spin liquid (QSL) state remains stable at a finite $M$. The $1/M$ corrections are exactly marginal, so the system remains conformably invariant at any finite $M$. The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at $N=\infty$ at any finite $M$, nor at $M=\infty$ at any finite $N$. By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in $M$. We work out a criterion for the two large numbers $N$ and $M$ to satisfy so that the QSG instability may be avoided. We speculate that at any finite $N$, the quantum chaos appears at the order of $1/M^{0}$, which is the subleading order in the $1/M$ expansion. When the $1/N$ quantum fluctuations at any finite $M$ are considered, from a general reparametrization symmetry breaking point of view, we argue that the eThis work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.ffective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos.