$\lambda$ represents the quantum Lyapunov exponent characterizing how fast the local information is scrambled over the system. A theorem [13] establishes an upper bound $\lambda \le 1$ due to unitarity, with $\lambda =1$ corresponding to maximal chaos—a saturation condition achieved by gravitational systems.

The notation in (6) and (7) are chosen such that the penalty factor $q$ for the SYK model is the same as our multi-color commuting SYK model.

This $h$ is the same as the transfer matrix $T$ in [34, 37] after a similarity transformation.

To show this, note that each term in the group need to overlap with another term with even numbers of Majorana fermions. Then one can always find a common 2-fermion basis like ${\mathcal{X}}_j^a$ for the largest possible group. Then this group is equivalent to one ${\tilde{H}}_a$.

The normalization $⟨{\mathbf{a}}_i|{\mathbf{a}}_i⟩$ in the denominator is due to the Hermitian construction of $H$. If we use the transfer matrix $T$ from [34, 37] in stead of $H$ in (93), we do not need to divide this factor.

There is a subtlety when we have imaginary eigenvalues. For the imaginary eigenvalue $n_{*}$ in ${\tilde{\mathcal{M}}}_{+}$, we have ${\sin }^2(n_{*}/4)<0$, which implies $n_{*}^2+(\gamma x\tan \frac{x}{2}{)}^2>0$. Therefore, when we do the integral over $\alpha$ in (169), it will hit a singularity of $1/F_{\pm }(\pm i\sqrt{\alpha })$ when $\pm i\sqrt{\alpha }=n_{*}$. To avoid this issue, note that the 1-loop determinant only cares about $|{\sin }^2(n/4)|$ and we can shift $n\to n+\epsilon$ in (168) for infinitesimal positive $\epsilon$ such that the average over $z=\pm n_{*}$ for $\log \frac{(z+\epsilon {)}^2}{(z+\epsilon {)}^2+(\gamma x\tan \frac{x}{2}{)}^2}$ gives the real value $\log |{\sin }^2(n_{*}/4)|$. Adding this $\epsilon$ leads to the absolute value in the log term of (eq:102). On the other hand, for the imaginary eigenvalue in ${\tilde{\mathcal{M}}}_{-}$, we have ${\cos }^2(n/4)>0$ and thus $n^2+(\gamma x\tan \frac{x}{2}{)}^2<0$. It follows that the $\alpha$ integral does not hit any singularity and we do not need the $\epsilon$-prescription.

For $d=9$ and $h=1/2$, the small $y$ expansion is different and the leading divergence is $\log y$.

Here we momentarily take $E_0=0$ for simplicity. Recovering $E_0$ simply replaces $E\to E-E_0$. Note that the $\beta$ contour in the inverse Laplace transformation is to the left of ${\beta }_{+}{\lambda }^{-1/2}$ and leads to the exponential decay for $E<0$.

Rigorously speaking, for $h=1/2+i\nu$ in Sec. 4d3 we should also consider the accumulation effect of large number of eigenvalues in $K_{\pm }$ because the ground energy (196) has an order $O(\sqrt{\lambda })$ gap from ${\tilde{E}}_0$ at large $d$ limit. However, this does not change the singularity of the partition function from the first null eigenvalue.

The exact gap between $T_{l.b.}$ and $T_{t.c.}$ is model dependent. For example, the finite $p$-spin SK model [29] has the Gaussian spectra and its critical temperature depends on $p$, which in our notation are in the same order as $T_{l.b}$. (213). In the $p\to \infty$ limit (noting it is within the regime of our double-scaled limit), the $p$-spin SK model becomes the random energy model [28, 56, 58] and the critical temperature becomes exactly (213). Since the tail (89) is essentially the same as the Gaussian tail, this assumption seems reasonable.

See also a recent interesting variant of the SYK model without RSB [74].

For $d=9$ and $h=1/2$, the approximate solution is $p/x=2n-1-2/(c_n+\log y)$ with $c_n$ a number depending on $n$ for $n\in {\mathbb{Z}}_{\mp }^{>0}$. The estimate (b3) and the numeric computation below still hold.

In principle, choosing larger $K_{\pm }$ and $K_{\pm }^{\prime }$ allows us to probe lower temperatures. However, Mathematica will loose accuracy for the associated Legendre functions for too large $n$ and makes the location of eigenvalues less reliable.