text:
\section{Introduction.}
\label{sec:intro}

In classical chaos, the Lyapunov exponent was used to characterize the exponential growth of two classical trajectories
    when there are just a tiny difference in the initial conditions. The classical concept of Lyapunov exponent
    can be extended to its quantum analog which can be used to characterize the exponential growth of two initially
    commuting operators in the early time (up to the Ehrenfest time) under the evolution of a quantum chaotic Hamiltonian \cite{larkin,chaosbutter,chaosbound}.
    There are recent flurry of research activities to extract
    the Lyapunov exponent $ \lambda_L $ of the Sachdev-Ye-Kitaev (SYK) model, its various variants and its possible
    dual asymptotic $ AdS_2 $ bulk quantum black holes through
    evaluating Out of time ordered correlation (OTOC) functions \cite{SY,SY3,Kit,subir3,Pol,Mald,CSYKnum,Gross,longtime,Ads2Mald,liu1,liu2}.
    The quantum analog of  $ \lambda_L $  need to be evaluated away from the thermodynamic limit by a $1/N $ expansion in the SYK models,
    also away from the conformal invariant limit due to a leading irrelevant operator  and
    at a finite temperature (ranging from low to high temperatures).

    From a completely different perspective, the quantum chaos can also be characterized by the system's energy level statistics (ELS) and
    level-level correlations encoded in the spectral form factor (SFF) through Random
    Matrix Theory (RMT)\cite{MBLSPT,randomM0,randomM}.
    The ELS and SFF are always evaluated at a finite but sufficiently large system.
    The ELS of SYK can be described by the Wigner-Dyson (WD) distributions in a $ N (mod 8) $ way \cite{MBLSPT,randomM0,randomM}:
    $ N=2,6 $ Gaussian unitary ensembles (GUE);
    $ N=0 $ Gaussian orthogonal ensemble (GOE), $ N=4 $ Gaussian  symplectic ensemble (GSE).
    The quantum chaos in the SYK models are due to the quenched disorders.
    However, it inspired a new class of clean models called (colored or un-colored) Tensor (Gurau-Witten) model \cite{tensor1,tensor2,tensor3}
    which share similar quantum chaotic properties as the SYK at least in the large $ N $ limit.

    In classical chaos, the classical  Kolmogorov-Arnold-Moser (KAM) theorem states that if an integrable Hamiltonian $ H_0 $ is disturbed by a small perturbation $ \Delta H $,
    which makes the total Hamiltonian $ H=H_0+ \Delta H $ , non-integrable. If the two conditions are satisfied:
    (a) $  \Delta H $ is sufficiently small (b) the frequencies $ \omega_i $ of $ H_0 $ are in-commensurate, then the system remains quasi-integrable. Despite some previous efforts,  the quantum analogue of the KAM theorem  remains elusive.
    Just like the quantum analog of Lyapunov exponent can be studied in the context of the SYK models \cite{SY,SY3,Kit,subir3,Pol,Mald,CSYKnum,Gross,longtime,Ads2Mald,liu1,liu2},
    it is important to explore the quantum analog of KAM theorem in various hybrid Sachdev-Ye-Kitaev models
    which contain both the chaotic $ q=4 $ SYK with interaction $ J $ and the integrable $ q=2 $ SYK  with interaction $ K $.
    Integrable or not maybe judged by the ELS and its evolution due to the competition of the two energy scales $ J/K $.
    Here we study two types of hybrid Sachdev-Ye-Kitaev models:  Type-I  preserving and Type II violating the P-H symmetry of the $ q=4 $ SYK
    respectively. The Type-I  include the hybrid Majorana fermion, complex fermion and hard bosonic SYK.
  The Type-II  include $ q=4 $ SYK with interaction $ J $ and $ (q=2)^2 $ SYK with the interaction $ \sqrt{K} $
  either Majorana or complex fermion.
  We study the quantum analog of the KAM theorem and the stability of quantum chaos in the context of the energy level statistics (ELS) in
  these hybrid SYK models.

  We first introduce a new universal ratio which is the ratio of next nearest neighbor (NNN) energy spacing
  to characterize the Random matrix classes and also establish its relations with the well
  known ratio of the nearest neighbor (NN) energy spacing.
  When a doubly degenerate level is split by a small perturbation, the NN ratio is nearly zero and also rapidly changing,
  the new NNN ratio can be  used to very effectively characterize the correct RM behaviours.
  We make exact symmetry analysis to classify the possible symmetry class in the 10-fold way,
  then perform exact diagonization on all these hybrid models. We evaluate the NN ratio and also NNN ratio when
  a doubly degenerate level is split by a small perturbation.
  In the Type I hybrid SYK, as $ K/J $ changes, there is always a chaotic to non-chaotic transition (CNCT) from GUE to Poisson
  in all the hybrid fermionic models, but not the hybrid bosonic model. Starting from the $ q=4 $ SYK,
  for the GOE case, there is  a GOE to GUE crossover first, then a CNCT from the GUE to the Poisson;
  for the GUE case, then a direct CNCT from the GUE to the Poisson;
  for the GSE case, any small $ K $ destroys the double generally of the GSE, then rises to GUE, then a CNCT from the GUE to the Poisson.
  Then in this case, the NNN can be most effectively applied to describe the stability regime of GSE and
  may also be used to describe the whole crossover until to the KAM side.
  In the Type II hybrid SYK, the Symmetry analysis alone may not be able to distinguish between
  the $ q=4 $ SYK  and   $ ( q=2 )^2 $ SYK.
  There are always CNCT from the corresponding WD distribution dictated by the symmetry of the  $ q=4 $ SYK to the
  Poissonan dictated by the  square of  $ q=2 $ SYK.
  When there is a double degeneracy at the $ ( q=2 )^2 $ side, the NNN can be most effectively applied to describe
  the KAM side and may also be used to describe the whole crossover until to the quantum chaos side.
  We compare quantitatively the stability of quantum chaos and KAM theorem near the integrability in all these hybrid SYK models.
  Poissble intrinsic connections between the onset of quantum chaos characterized by the RMT and
  the onset of quantum phase transitions characterized by Renormalization group (RG) are discussed.

\section{A new ratio to describe RMT:
ratio of next nearest neighbor (NNN) energy spacing}

In this section, we first review the known results on the statistics of
 the nearest neighbour (NN) energy level spacing $ ( r, \widetilde{r} ) $, then introduce
  a new ratio which is the NNN energy level spacing $ ( r', \tilde{r}' ) $,
  then establish an approximate, but quite accurate relation between the two.

\subsection{Review on the ELS of nearest neighbor energy level spacings}

Let $E_n$ be an ordered set of energy levels and $s_n=E_{n+1}-E_{n}$
the nearest-neighbor (nn) spacings.
By considering a $2\times2$ matrices system,
Wigner derived a simple approximate expression for $P(s)$,
\begin{align}
    P_{w,\beta}(s)=a_\beta s^\beta e^{-b_\beta s^2}
\label{PW}
\end{align}
where $\beta=1,2,4$ is the Dyson index for GOE, GUE and GSE respectively.

It is also known that independent random energy levels would yield a Poisson distribution
\begin{align}
    P_{p}(s)=e^{-s}
\label{PP}
\end{align}
However, in order to compare different results from different systems,
the energy levels will need an unfolding procedure,
which is not convenient when large enough statistics is not available.

To get rid of the dependence on the local density of states, it is convenient to look at the distribution
of the ratio of two adjacent energy level spacings \cite{ratiosta,MBLSPT}  $
r_{n}=\frac{s_{n}}{s_{n+1}} $ which distributes around $1$. This quantity has the advantage that it requires no unfolding
since ratios of consecutive level spacings are independent of the local density of states.

By considering  $3\times3$ matrices system, the authors in \cite{ratiosta} obtained the Wigner-like surmises of
the ratio of consecutive level spacings distribution
\begin{equation}
p_{p}\left( r\right) =  \frac{1}{\left( 1+r\right) ^{2}},~~~
p_{w}\left( r\right)  =  \frac{1}{Z_{\beta}} \frac{\left( r+r^2\right)^{\beta} }{\left( 1+r+r^{2}\right)
^{1+3 \beta/2}}
\end{equation}
where $\beta=1,2, 4 $ and $  Z_{\beta}=8/27, 4 \pi/81\sqrt{3},  4 \pi/729\sqrt{3}$ for GOE, GUE and GSE respectively.
The distribution $P_W(r) $ has the same level repulsion at small $r$ as $P_W(s)$,
namely, $P_W(r)\sim r^\beta$. However, the large $r$ asymptotic behavior $P_W(r)\sim r^{-(2+\beta)}$ is
dramatically different than the fast exponential decay of $P_W(s)$.

One may also compute the distribution of the logarithmic ratio \cite{ratiosta,MBLSPT}  $P(\ln r)=p\left( r\right) r$.
Because $p \left( \ln r\right)dr $ is symmetric under $ r \leftrightarrow 1/r $, one may confine $0<r<1$ and
double the possibility density $p\left( \widetilde{r}\right) =2p\left(
r\right) $. Therefore, the above two distributions have two different expected values of
$\widetilde{r} = min \{r, 1/r \} $:
\begin{eqnarray}
\left\langle \widetilde{r}\right\rangle _{p} & = & \int_{0}^{1}2 r p_{p}(r) dr=2\ln 2-1 \sim 0.38629 ,    \nonumber   \\
\left\langle \widetilde{r}\right\rangle _{w} & =  & \int_{0}^{1} 2 r p_{w}(\beta=1,2,4;r) dr
\end{eqnarray}%
 which is $ 4-2\sqrt{3} \sim0.53590, \frac{2\sqrt{3}}{\pi}-1/2 \sim 0.60266, \frac{32\sqrt{3}}{15\pi}-1/2 \sim 0.67617 $
 for GOE,GUE and GSE respectively.

\subsection{Introduction and calculation of the ELS of NNN energy level spacings }

When there are nearly double degenerated energy levels, we find that
it is convenient to introduce next nearest-neighbor (NNN) spacings
$s'_n=e_{n+2}-e_{n}$ and ratios $r'_n=s'_n/s'_{n-1} $.
Here we study the  distribution of the ratio of the two NNN spacings by exploring
the exact calculation for $5\times5$ matrices.

For a Poisson ensemble, it is more convenient to work with $P(s_1,s_2,s_3,s_4)$
and the ratio of consecutive NNN level spacings is
$r'=(s_3+s_4)/(s_1+s_2)$.
The NNN ratio distribution can be calculated from
\begin{align}
    P_P^{(2)}(r')=\int \prod_{i=1}^4 ds_i
	P(s_1,s_2,s_3,s_4)\delta\Big(r'-\frac{s_3+s_4}{s_1+s_2}\Big)\>.
\end{align}
Since energy levels are not correlated in Poisson ensemble,
one can rewrite $P(s_1,s_2,s_3,s_4)=\prod_{i=1}^4P_P(s_i)$
and evaluate the integral to obtain a simple result
\begin{align}
    P_P^{(2)}(r')=\frac{6r'}{(1+r')^4}\>.
\label{eq:PPr'}
\end{align}
It is easy to see that $P_P^{(2)}(r')\sim r'$ when $r'$ is small,
and $P^{(2)}(r')\sim {r'}^{-3}$ when $r'$ is large.
Interestingly, there is a NNN level repulsion in Poisson ensemble
which intuitively can be understood due to
the interruption of the intermediate level.

For various Gaussian ensembles, it is good to start from
the joint probability distribution
$\rho(e_1,e_2,e_3,e_4,e_5)$ and the ratio of consecutive NNN level spacings is
$r'=(e_5-e_3)/(e_3-e_1)$.
The NNN ratio distribution can be calculated from
\begin{align}
    P_{W,\beta}^{(2)}(r')=\int \prod_{i=1}^5 de_i
	\rho(e_1,e_2,e_3,e_4,e_5)\delta\Big(r'-\frac{e_5-e_3}{e_3-e_1}\Big)\>,
\end{align}
where the joint probability distribution takes form
$\rho(e_1,e_2,e_3,e_4,e_5)
=C_{\beta} \prod_{1 \leq i\leq j\leq5}|e_i-e_j|^\beta \prod_{i=1}^5 e^{-e_i^2/2}$.
The integral can be simplified as
\begin{align}
    P_{W,\beta}^{(2)}(r')
	\sim  \int_0^\infty dx\int_0^{rx} dy\int_0^{x} dz
	&x[r'(1+r')x^3(r'x-y)(r'x+z)y(x+y)(x-z)z(y+z)]^\beta     \nonumber\\
	&\times  e^{-\frac{1}{5}[(2+2r^{\prime 2}+r)x^2+2z^2+2y^2+yz+(1-r')x(y-z)]}
\label{eq:PWr'}
\end{align}
where $\sim$ means the normalization constant is ignored.
The integrals can be evaluated analytically,
but its expression is lengthy.
Here, we just show analytically its asymptotic behaviour:
$P_{W,\beta}^{(2)}(r')\sim {r'}^{3\beta+1}$ when $r'$ is small,
and $P_{W,\beta}^{(2)}(r')\sim {r'}^{-3(\beta+1)}$ when $r'$ is large.
If comparing them with those of $P_W(r)$,
one can see that the asymptotic behaviours of the NNN level statistics with index $\beta$
are the same as those of the NN  with index $3\beta+1$
(See Eq.\ref{eq:PWW2} below for all the ranges).
Similar to the Poisson case discussed above,
the intermediate energy level induces much stronger level repulsions between NNN.
In Fig.\ref{nnn1}, we compare $P_{W,\beta}^{(2)}$ obtained from Eq.\eqref{eq:PWr'}
against the numerical simulations of the corresponding WD ensembles.
We find nearly perfect agreement in all ranges of of $r'$.

\begin{figure}[!htb]
\centering
    \includegraphics[width=0.55\linewidth]{NNN.pdf}
\caption{Distribution of the ratio of consecutive
	 NNN level spacings $P^{(2)}(r')$ for Poisson and RMT ensembles:
	 solid lines are from exact evaluation of integral Eq.\eqref{eq:PWr'}.
	 The points are numerical data.
	 The Poisson data are obtained from $10^6$ independent generated random numbers.
	 The random matrix data are obtained by diagonalizing the corresponding
	 GOE (real), GUE (complex) and GSE (quaternion) matrices of
	 size $ N=1000 $ with Gaussian distributed entries,
	 averaged over $10^5$ histograms.}
\label{nnn1}
\end{figure}

It is easy to check that $P^{(2)}(r')$ have the same symmetry as $P(r)$, namely,
    $P^{(2)}(r')=\frac{1}{r^{\prime 2}}P^{(2)}(\frac{1}{r'})$,
thus we can still define a $\tilde{r}'$.
From the NNN ratio distribution given in Eq.\eqref{eq:PPr'} for Poisson and Eq.\eqref{eq:PWr'} for WD,
one can calculate expectation value for $r'$ and $\tilde{r}'$ as
\begin{align}
    \langle r'\rangle=\int_0^\infty dr' r'P(r'),\quad
    \langle \tilde{r}'\rangle=\int_0^1 dr' 2r'P(r')
\label{eq:r2}
\end{align}
which are listed in Table 1.

\begin{table}[!htb]
\caption {
List of numerical values of averages
$\langle r\rangle$ and $\langle \tilde{r}\rangle$
as well as $\langle r'\rangle$ and $\langle \tilde{r'}\rangle$
for various ensembles.
The values of $\langle r\rangle$ and $\langle \tilde{r}\rangle$
are taken from Ref.\cite{ratiosta}. Those of $\langle r'\rangle$ and $\langle \tilde{r}'\rangle$
are obtained from evaluation of Eq.\eqref{eq:r2}.
We also list the values of $\langle r'\rangle_{3\beta+1}$
and $\langle \tilde{r}'\rangle_{3\beta+1}$
by using the $\beta\to3\beta+1$ rule in Eq.\eqref{eq:PWW2}.}
\label{tab:title}
\centering
\vspace{1.5mm}
\begin{tabular}{l*{4}{@{\hskip 0.25in}c}}
\hline\hline
Ensembles			& Poisson   & GOE & GUE & GSE  \\
\hline
$\langle r\rangle$		& $\infty$  & 1.75	& 1.3607    & 1.1747   \\
$\langle \tilde{r}\rangle$      & 0.38629   & 0.5359	& 0.6027    & 0.6762   \\ \hline
$\langle r'\rangle$		& 2	    & 1.1736	& 1.0972    & 1.0516   \\
$\langle\tilde{r}'\rangle$	& 0.5	    & 0.6769	& 0.7344    & 0.7910   \\  \hline
$\langle r'\rangle_{3\beta+1}$		& - & 1.1747	& 1.0980    & 1.0681   \\
$\langle\tilde{r}'\rangle_{3\beta+1}$	& - & 0.6762	& 0.7335    & 0.7672   \\
\hline\hline
\end{tabular}
\end{table}

\subsection{An approximate relation between the ELS of NNN and those of NN }

In fact, instead of lengthy results from exact evaluation of integral,
we find an approximate relation between $P^{(2)}(r')$ and $P(r)$:
\begin{align}
    P_{W,\beta}^{(2)}(r)\approx P_{W,3\beta+1}(r)
\label{eq:PWW2}
\end{align}

  The difference of the two was shown in Fig.\ref{nnn2}. The very small deviation
  shows that the approximation in Eq.\eqref{eq:PWW2} is quite accurate.
  In the last two lines in the TABLE I, we also list the numerical values of
  $ \langle r'\rangle_{3\beta+1} $ and  $\langle\tilde{r}'\rangle_{3\beta+1}$
  using $ P_{W,3\beta+1}(r) $. All these values are very close to those using $ P_{W,\beta}^{(2)}(r) $ in Eq.\ref{eq:PWr'}.
  For example, just take $ \langle\tilde{r}'\rangle $ for the GSE case, one can see the relative difference
  $\frac{0.7672\, -0.7910}{0.7910}=-3\%$ is very small.

  In fact, one can see Eq.\ref{eq:PWW2} can work well when putting $ \beta=0 $. Namely, when the NN satisfies Poisson, the NNN
  seems fit GOE  approximately. For example, from the TABLE I, $ \langle r'\rangle $ and $\langle\tilde{r}'\rangle$
  for Poisson $ 2, 0.5 $ are not too much away from  $ \langle r\rangle $ and  $\langle \tilde{r}\rangle$ for GOE
  $ 1.75, 0.53590 $ respectively. At least, one can use GOE for an quick eye guides to judge the NNN for the Poisson as we did in all the following figures.

\begin{figure}[!htb]
\centering
\includegraphics[width=0.55\linewidth]{NNNA.pdf}\quad
    \caption{ Difference of $\delta P^{(2)}=P_{W,\beta}^{(2)}(r)-P_{W,3\beta+1}(r)$.
    The solid line is from Eq.\ref{eq:PWr'}.
	The numerical data in $\delta P^{(2)}=P_\text{num}^{(2)}(r)-P_{W,3\beta+1}(r)$ are taken from Fig.\ref{nnn1}.
	Small $|\delta P^{(2)}|$ justifies validation of Eq.\eqref{eq:PWW2}}.
\label{nnn2}
\end{figure}

  In the following sections, we will apply both NN and NNN ELS to study the quantum analog of Kolmogorov-Arnold-Moser theorem
  in the two kinds of hybrid Sachdev-Ye-Kitaev models.

\section{Hybrid Sachdev-Ye-Kitaev models}

\subsection{The hybrid of $ q=2 $ and $ q=4 $  Majorana fermion SYK }

The Quantum analog of KAM theorem may be first investigated
in  the hybrid of $ q=2 $ and $ q=4 $ Majorana fermion SYK:
\begin{equation}
H_{M}=  \sum^{N}_{i<j<k<l} J_{ijkl} \chi_{i} \chi_{j} \chi_k \chi_l + i \sum^N_{i<j}
K_{ij} \chi_{i} \chi_{j}
\label{mix1}
\end{equation}
  where  $  J_{ijkl}, K_{ij} $ are real and  satisfy the Gaussian distributions with
  $ \langle J_{ijkl} \rangle=0,  \langle J^2_{ijkl} \rangle= 3! J^2/N^3 $ and
  $ \langle K_{ij} \rangle=0,  \langle K^2_{ij} \rangle=  K^2/N $ respectively.

   In the $ J=0 $ limit,  the $ q=2 $ SYK breaks the particle-hole symmetry. It
   is non-interacting, so must be integrable. Its energy level statistics (ELS)
   satisfies the Poisson distribution \cite{Mald,Gross,randomM}.
   Its single particle density of state (DOS) satisfies the semi-circle law \cite{Gross}.
   Its low energy excitation level spacing is $ \sim 1/N $,  the quasi-particle picture holds.
   Its $ T=0 $ entropy $ s_0$ vanishes.
   This is very similar to the $ U(1)/Z_2 $ Dicke model \cite{u1largen,gold,comment,strongED,KAM}
   in the $ U(1) $ limit ( inside the superradiant phase ) where the ELS  also satisfies the Poisson distribution, its
   single particle low energy excitation level spacing is also $ \sim D \sim 1/N $.

   In the $ K=0 $ limit, the $ q=4 $ SYK is non-integrable at any finite $ N $. In the following, we discuss when $ N $ is even or odd
   respectively.

   For even number of sites $ N=2N_c $, one can split the sites to even and odd, then introduce $ N_c $ complex fermions
   \cite{CSYKnum,MBLSPT,randomM} by  $ c_i= ( \chi_{2i}-i \chi_{2i-1} )/\sqrt{2},
   c^{\dagger}_i= ( \chi_{2i}+i \chi_{2i-1} )/\sqrt{2} $ and define the particle-hole symmetry operator
   to be $
   P= K \Pi^{N_c}_{i=1} ( c^{\dagger}_i + c_i ) $.
   It is easy to show $ P^2= (-1)^{[\frac{N_c}{2}]},  P c_i P= \eta c^{\dagger}_i, P c^{\dagger}_i P= \eta c_i,
   P \chi_i P= \eta \chi_i $
   where $ \eta=(-1)^{[\frac{N_c-1}{2}]} $. The total number of  fermions $ Q_c= \sum^{N_c}_{i=1} c^{\dagger}_i  c_i $.
   It is not a conserved quantity, but the parity $ (-1)^{Q_c} $ is in $ H_4 $.
   Then  $ P Q_c P^{-1}= N_c-Q_c $ which justifies $ P $ as an anti-unitary PH transformation. $ P $ also commutes with the Hamiltonian \cite{TR}
   $ [ P, H_4] =0 $. Depending on $ N \pmod 8 $, the ELS satisfies GUE
   when $ N mod(8)=2,6 $, GOE when $ N mod(8)=0 $, GSE when $ N mod(8)=4 $  \cite{CSYKnum,MBLSPT,randomM}.
   The ELS, the degeneracy at a given parity sector and the total parity sector are listed in the Table I.
   Its low energy excitation level spacing is $ e^{-s_0 N} $ which leads to extensive $ T=0 $ entropy $ s_0 $
   ( in the  $ N  \rightarrow \infty $ limit before $ T \rightarrow 0 $ limit )\cite{SY3,subir3,Mald}.
   The system's many body DOS ( different from the single particle DOS ) has been worked out  in \cite{Mald} to be different than the semi-circle DOS.
   The quasi-particle picture completely breaks down. This is similar to the $ U(1)/Z_2 $ Dicke model
   in the $ Z_2 $ limit  ( inside the superradiant phase ) where the ELS satisfies the GOE distribution \cite{strongED,KAM}, its
   two lowest energy splitting between two different parities is $ e^{-N} $ which is due to the instanton quantum tunneling (QT) process.
   It is not clear if the $ e^{-N} $ level spacing in the $ q=4 $ SYK is also due to some non-perturbative instanton QT process, which
   is inaccessible to the perturbation  $1/N $ expansion in \cite{Pol,Mald,CSYKnum,Gross,longtime} anyway.

\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{MajoranaSYK_Type1_Even.pdf}
\caption{ The evolution of the ELS for type-I hybrid Majorana SYK Model with even $N $.
    $\langle\tilde{r}\rangle$ (black curve), and $\langle\tilde{r}'\rangle$ (orange curve) for the GSE case in (d)
	evaluated for  $N_\chi=14,16,18,20$ and are averaged over  $100$, $80$, $60$, $40$ samplings  respectively.
Notably, in the GSE case in (d), the NN ratio $ \langle \tilde{r}  \rangle $ is rapidly changing near the $ q=4 $ side, so it
is quite difficult to determine the stability regime of the quantum chaos. Fortunately,
the NNN ratio $ \langle \tilde{r^{\prime}}  \rangle $ shows a nice plateau regime near the $ q=4 $ side, the quantum chaos stability regime can be
easily identified and listed in the Table 3-6. This dramatic advantage of the new  NNN ratio over the known NN ratio was
further demonstrated in all the following relevant figures. 	}
\label{MajoranaSYK_Type1_Even}
\end{figure}

\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{MajoranaSYK_Type1_Odd.pdf}
\caption{ The evolution of the ELS for type-I hybrid Majorana SYK Model with odd $N $.
$\langle\tilde{r}\rangle$ (black curve),
	and $\langle\tilde{r}'\rangle$ (orange curve) for the GSE case in (a) and (d)
	evaluated for $N_\chi=13,15,17,19$ and are averaged over $100$, $80$, $60$, $40$ samplings respectively.
    Note the advantages of using $ \langle \tilde{r^{\prime}}  \rangle $ over $ \langle \tilde{r}  \rangle $,
    especially in the quantum chaos side, in (a) and (d). }
\label{MajoranaSYK_Type1_Odd}
\end{figure}

   However, when N is odd, the above procedures for even $ N $ needs to be modified.
   In fact, one can still take the advantage of the above representation with $ N $ even case by
   adding $ \chi_{N+1} =\chi_{ \infty} $  to make the parity conservation  explicitly.
   Similar strategy was used before to study the symmetry protected topological phase of
   odd number of Majorana chain \cite{kitSPT} and the ELS of the SYK model with $ N $ odd \cite{MBLSPT}.
   Then one can still define $ P $ with $ N_c=\frac{N+1}{2} $ as before.
   All the commutation relations still apply. One can also define another anti-unitary operator $ Z= P \chi_{\infty} $.
   The combination of the two operators $ P $ and $ Z $ leads to the following tables:

\begin{table}[!htb]
\centering
\begin{tabular}{ |*{5}{c|}  }
\hline\hline
$N\pmod 8$  & 0		& 2	    & 4		& 6	    \\  \hline
ELS         & GOE	& GUE	    & GSE	& GUE	    \\  \hline
$ \beta $   & 1		& 2	    & 4		& 2	    \\  \hline
$ Q $       & $d=1$	& $d=1$	    & $d=2$	& $d=1$	    \\  \hline
$ Q_t $	    & $d_t=1$   & $d_t=1+1$ & $d_t=2$   & $d_t=1+1$ \\   \hline
\hline
$N\pmod 8$  & 1		& 3	    & 5		& 7	    \\  \hline
ELS         & GOE	& GSE	    & GSE	& GOE	    \\  \hline
$ \beta $   & 1		& 4	    & 4		& 1	    \\  \hline
$ Q $       & $d=1$	& $d=2$	    & $d=2$	& $d=1$	    \\  \hline
$ Q_t $	    & $d_t=1+1$ & $d_t=2+2$ & $d_t=2+2$ & $d_t=1+1$   \\
\hline\hline
\end{tabular}
\caption
The ELS and degeneracy of the Majorana fermion $ q=4 $ SYK model at N even or odd.
The degeneracy $ d $ is at a given parity sector $ Q_c $.
The total degeneracy $ d_t $ is at both parity sectors.
When $N$ is even, there is only one anti-unitary operator $ P $.
When $N \pmod 8=2,6$, $ P $ maps $ Q_c $ into $  Q_c + 1 $.
When $N \pmod 8=0,4$, $ Q_c $ and $ Q_c + 1$ are completely dependent, no operator connects between the two opposite parities.
When $N$ is odd, after adding one Majorana fermion at $ N+1=\infty $, one doubles the Hilbert space,
there are two anti-unitary operators $ P $ and $ Z $.
When $N \pmod 8=3, 7$,  $ P $ maps $ Q_c $ to itself, $ Z $ maps $ Q_c $ into $  Q_c + 1 $.
When $N \pmod 8=1, 5$, $ Z $ maps $ Q_c $ to itself, $ P $ maps $ Q_c $ into $  Q_c + 1 $.
So $ P $ and $ Z $ exchange their roles in the two cases. So the $ d_t $ in the odd N case
is the degeneracy in the enlarged Hilbert space which is the twice of the original one. }
\end{table}

   Now we apply the PH transformation to the hybrid SYK model Eq.\ref{mix1}.
   The parity  $ (-1)^{Q_c} $ remains to be conserved.
   However, $ P $ ( or $ Z $ when $ N mod 8=1, 5 $ ) is not conserved anymore due to $ \{ P, H_2 \} =0 $ ( or $ \{ Z, H_2 \} =0 $ ).
   So the hybrid SYK does not have the PH symmetry anymore.
   Just from symmetry point of view ( the 10 fold way classification scheme ),
   the hybrid Majorana SYK  Eq.\ref{mix1} belongs to the class A, so may satisfy GUE for any ratio of $ K/J $.
   Our Exact Diagonization (ED) studies at a given parity $ (-1)^{Q_c} $ sector in
   Fig.\ref{MajoranaSYK_Type1_Even} and \ref{MajoranaSYK_Type1_Odd}
   show that this is true only in the intermediate regimes of $ K/J $.
   The KAM theorem applies in the two end regimes $ K/J \ll 1 $ where the ELS remain Poisson
   and $ K/J \gg 1 $ where the ELS becomes the WD determined by the $ q=4 $ SYK.

   One may also understand the quantum analog of the KAM theorem from a dual point of view, namely,
   the stability of quantum chaos of
   the quantum chaotic $ q=4 $ SYK against the non-chaotic perturbation as one turns on $ K/J $.
   The dual form of the KAM theorem states that    as $ K/J $ increases, there should be a crossover
   from the WD to the GUE, then a CNCT from the GUE to the Poissonian. Our ED studies shown in
   Fig.\ref{MajoranaSYK_Type1_Even} and \ref{MajoranaSYK_Type1_Odd} confirms this  global picture.

 We applied the same ED to study the CNCT in the $ U(1)/Z_2 $ Dicke model to study
 the CNCT in the hybrid SYK model.
 The only extra work needed  here is that one need to draw the random couplings
 $ J_{ijkl}, K_{ij} $ from the corresponding Gaussian distributions $ P[J] $ and $ P[K] $,
 then get ELS at a given set of $ J_{ijkl}, K_{ij} $.
 Then one need to repeat the same calculations over 40 to 100 samples of such a random realizations of $ J_{ij,kl}, K_{ij} $,
 then  perform the average of ELS over these 40 to 100 samples \cite{noise}.
 The $ \widetilde{r} $  for GOE,GUE and GSE are drawn  as reference lines in
 Fig.\ref{MajoranaSYK_Type1_Even} and \ref{MajoranaSYK_Type1_Odd}.

     The even $ N $ cases are shown in Fig.\ref{MajoranaSYK_Type1_Even}. There are 3 cases:

    (a) For $ N \pmod 8=0 $, the $ q=4 $ Majorana fermion SYK at $ K=0 $ is in GOE,
    the  hybrid is in the GUE around $ K/J =e^{-3.5} \sim 0.03 $ to $1$,
    there is a crossover from GOE to GUE first,
    then a CNCT from GUE to the Poissonian as $ K/J $ increases.

    (b) For $ N \pmod 8=2,6 $, the $ q=4 $ Majorana fermion SYK at $ K=0 $ is in GUE,
    it stays in the GUE until $ K/J =1$,
    then there is a CNCT from GUE to the Poissonian as $ K/J $ increases \cite{studied}.

    (c) For $ N \pmod 8=4 $,  the $ q=4 $ Majorana fermion SYK at $ K=0 $ is in GSE at $ K=0 $.
    Because $ P^2=-1 $, any energy level is doubly degenerate at a given parity sector,
    so when doing ELS, we only pick up one of the  doubly degenerate levels to demonstrate the GSE.
    Any small $ K $ breaks the degeneracy, then we may consider both sets of energy levels,
    a small $ K $ makes $ \langle \tilde{r} \rangle $ small,
    so $ \langle \tilde{r} \rangle $ starts from zero and increases as $ K/J $ increases,
    then reaches the GUE in the range from $ e^{-4} $ to $ e^0.5 $.
    There is a CNCT from GUE to the Poissonian as $ K/J $ increases.
    So in this case, using the NN ELS is not enough.
    One may start to use the combination of NN and the NNN ELS presented in Sec.2.
    The $ \langle\tilde{r}'\rangle $ was also shown in
    Fig.\ref{MajoranaSYK_Type1_Even}d.

    It is complete to combine both $\langle \tilde{r}\rangle$ and $ \langle\tilde{r}'\rangle $ into the same Fig.3d.
    When $\langle \tilde{r}\rangle$ in (d) is close to be zero,
    the NNN $ \langle\tilde{r}'\rangle $ in (d) still shows GSE until $ K/J \sim e^{-5} $.
    When $\langle \tilde{r}\rangle$ in (d) reaches the plateau value
    $ \sim 0.60266 $ of GUE with $ \beta=1 $, according to Eq.\ref{eq:PWW2},
    then $ \langle\tilde{r}'\rangle $ in (d) reaches a corresponding plateau value
    listed in Table I as $ \sim 0.7344 $ with a RMT index $ 3 \beta + 1=4 $.
    When $\langle \tilde{r}\rangle$ in (d) reaches the plateau value
    $ \sim 0.38629 $ of Poisson, according to Eq.\ref{eq:PWW2},
    $ \langle\tilde{r}'\rangle $ in (d) reaches a corresponding plateau value
    listed in Table I as $ \sim 0.5 $.
    As pointed out in Sec.2, it is only slightly below the GOE value $ 0.53590 $
    with the RMT index $ 3 \beta + 1=1 $.

     The odd $ N $ cases are shown in  Fig.\ref{MajoranaSYK_Type1_Odd}.
     Due to the absence of the GUE, at the $ q=4 $ side,  there are are only
     the GSE case in (a) and (d), the GOE case in (b) and (c).
     They show similar evolution patterns as the corresponding GSE and GOE cases
     at the $ q=4 $ side for even $ N $ cases shown in Fig.\ref{MajoranaSYK_Type1_Even}.

   Obviously, at any given disorder realization of $ K_{ij} $, the eigen-energies
   of $ q=2 $ SYK is in-commensurate ( so the disorder in SYK may play a similar role
   as the Berry phase in the $ J-U(1)/Z_2 $ Dicke model ), the quantum KAM theorem should apply when $ J/K $ is sufficiently  small.
   So the ELS changes from the Poissonian to GUE  at the corresponding $ N_c mod (4) $ )
   around some critical $ (J/K)_c $ value.
   The mean value $ \langle \tilde{r} \rangle $ also changes from its Poissonain value to the corresponding GUE value.
   It is similar to the $ \beta_{U(1)} $ in the $ U(1)/Z_2 $ Dicke model where the
   ELS changes from  the Poisson to GOE when inside the superradiant phase $ g/g_c \geq 1 $ ( Fig.4a in \cite{KAM} ).
   Of course, in contrast to the Dicke model, due to the quenched disorders, there is no regular regime ( see Fig.4b in \cite{KAM} )
   at any values of $ J/K $.  It may be important to see how the low energy levels evolve
   from the $ 1/N $  spacing at the $ q=2 $ SYK to the
   $ e^{-s_0 N } $ spacing at the $ q=4 $ SYK and the roles of possible instanton effects during such a evolution.
   Inspired by the insights from  the Dicke model ( see Fig.3 in \cite{KAM} ),
   we expect the CNCT happens at $ (J/K)_c $ where the $ e^{ -N } $ spacings
   just emerge at the lowest energy levels. The quasi-particle picture starts to break down at the lowest energies, but still hold
   in slightly higher energies.  Then as $ J/K $ increases further,
   the regime where the quasi-particle picture holds shrink.
   At the $ q=4 $ SYK, all the low energy level spacing becomes $ e^{-s_0 N } $,
   the quasi-particle picture completely breaks down.

\subsection{The hybrid of $ q=2 $ and $ q=4 $  complex fermion SYK }

    The Majorana fermion SYK  was extended to the complex fermion which has a $ U(1) $
    charge symmetry  \cite{CSYKnum}.
    One may also add a chemical potential $ \mu $ to fix the conserved
    fermion filling factor \cite{can} $ q_c= \sum_{i} ( c^{\dagger}_i c_i-1/2)  $.
    The above procedures used for the Majorana fermion can also be applied to study the $ q=2 $ and $ q=4 $ hybrid complex SYK:
\begin{equation}
H_{C}=  \sum^{N}_{i<j,k<l} J_{ij;kl} c^{\dagger}_{i} c^{\dagger}_{j} c_k c_l +  \sum^N_{i<j}
K_{ij} c^{\dagger}_{i} c_{j}- \mu \sum^{N}_{i=1 } c^{\dagger}_i c_i
\label{mix2}
\end{equation}
    where  $ \langle J_{ij;kl} \rangle=0,  \langle | J_{ij;kl} |^2 \rangle= 3! J^2/N^3 $ drawing from
  the Gaussian distribution $ P[J_{ij;kl}] \sim e^{ - A  |J_{ij,kl}|^{2}/2 J^2 } $ where $ A= N^3/3 ! $.
  In general, $ J_{ij;kl}=-J_{ji;kl}, J_{ij;kl}=-J_{ij;lk}, J^{*}_{ij;kl}= J_{kl;ij} $.
 {\sl  We also take the four site indices $ i,j; k,l$ are all different to keep the PH symmetry explicit at $ \mu=0 $.}
    $ K^{*}_{ij} =K_{ji} $ is  a Hermitian matrix satisfying
    $ \langle K_{ij} \rangle=0,  \langle |K_{ij}|^2 \rangle=  K^2/N $.

 In the $ K/J=0 $ limit, the $ q=4 $ complex SYK is non-integrable at any finite $ N $.
 Under the PH transformation, $ q_c \rightarrow -q_c $. So $ q_c=0 $ is the PH symmetric and only happens when $ N $ is even.
 At the half-filling $ q_c=0 $, the system also has the maximum zero temperature entropy $ S_0 $ \cite{CSYKnum}.
 Away from the half-filling $ \mu \neq 0 $, it breaks the $ P $ symmetry,
 $ -N/2 < q_c \neq 0 < N/2 $  corresponds to a non-vanishing electric field
 in the charged black hole in asymptotic $ AdS_2 $ bulk side \cite{subir3}.
    It was shown in \cite{MBLSPT} that when $ N \, mod (4)=0,2 $, the ELS is GOE and GSE respectively.
    But when $ N \, mod (4)=1,3 $ and $ q_c=\pm 1/2 $, it is GUE.
   ( In fact, as long as $ q_c \neq 0 $,  there is no P-H symmetry anymore,
    so it is in GUE regardless of the value of $ N \,mod (4) $ ).

   Now we apply the PH transformation to the hybrid SYK model Eq.\ref{mix2} when $ K/J \neq 0 $.
   The fermion number remains to be conserved.
   However, $ P $ is not conserved anymore  due to $ \{ P, H_2 \} =0 $.
   So the hybrid SYK does not have the PH symmetry anymore.
   Just from symmetry point of view, the hybrid complex SYK belongs to the class A in the 10 fold way classification,
   so may satisfy GUE for any ratio of $ K/J $.
   Our Exact Diagonization (ED) studies \cite{JW} at a given $q_c $ in Fig.\ref{ComplexSYK_Type1} shows
   that this is true only in the intermediate regimes of $ K/J $.
   The KAM theorem applies in the two end regimes $ K/J \ll 1 $ where the ELS remains Poissonian
   and $ K/J \gg 1 $ where the ELS becomes the WD determined by the $ q=4 $ SYK.

\begin{figure}[!htb]
\centering
\includegraphics[width=0.75\linewidth]{ComplexSYK_Type1_All.pdf}
\caption{ The evolution of the ELS for type-I hybrid complex SYK Model.
$\langle\tilde{r}\rangle$ (black curve),
	and $\langle\tilde{r}'\rangle$ (orange curve) for GSE case in (c)
	evaluated at $N_c=12,13,14$ and are averaged over $100$, $50$, $50$ samplings respectively.
Note the advantages of using $ \langle \tilde{r^{\prime}}  \rangle $
over $ \langle \tilde{r}  \rangle $ in (c), especially in the quantum chaos side.}
\label{ComplexSYK_Type1}
\end{figure}

 In our ED studies \cite{JW}, we look at $ q=4 $ SYK first to reproduce the results listed in the table II(a) in \cite{MBLSPT}.
 Then look at the $ q=2 $ SYK to show that it indeed satisfies the Poissonian distribution.
 Then we study the hybrid model Eq. \ref{mix2}.
  As shown in Fig.\ref{ComplexSYK_Type1}, there are also  3 cases:

 (a) For $ N_c \, mod (4)=0 $, at the half filling $ q_c=0 $, the hybrid complex fermion SYK is in the GUE in a wide range near $ K/J =1 $,
  there is a crossover from GOE to GUE, then a CNCT from GUE to the Poissonian as $ K/J $ increases.

   (b) Away from the half filling $ q_c \neq 0 $, regardless of $ N_c \, mod (4) $, the $ q=4 $ SYK is in GUE.
   The hybrid complex fermion SYK stays in the GUE across $ K/J =1 $ until to $ K/J \sim e^{1.5} $,
   then there is a CNCT from GUE to the Poissonian as $ K/J $ increases further.

    (c) For $ N_c \, mod (4)=2 $, at the half filling $ q_c=0 $,  the complex $ q=4 $ fermion SYK is in GSE at $ K=0 $.
        Because $ P^2=-1 $, any energy level is doubly degenerate, so when doing ELS,
        we only pick up one of the  doubly degenerate levels to demonstrate the GSE at $ K/J=0 $.
        Any small $ K $ breaks the degeneracy, then we may need to consider both sets of energy levels, then
        it is easy to see that any small $ K $ makes $ \langle \tilde{r} \rangle $ small, so $ \langle \tilde{r} \rangle $
        starts from zero and increases as $ K/J $ increases.
        The hybrid complex fermion SYK is in the GUE in a wide range near $ K/J =1 $.
        There is a CNCT from GUE to the Poissonian as $ K/J $ increases further.

        So in this case, using the NN ELS is not enough.
	One may start to use the combination of NN and the NNN ELS presented in Sec.2.
        The $ \langle\tilde{r}'\rangle $ was also shown in Fig.\ref{ComplexSYK_Type1}c.
	It is complete to combine both $ \langle\tilde{r}\rangle $ and $ \langle\tilde{r}'\rangle $ into the same figure.
        When $\langle \tilde{r}\rangle$ is close to be zero,
        the NNN $ \langle\tilde{r}'\rangle $ still shows GSE until $ K/J \sim e^{-5} $.
        When $\langle \tilde{r}\rangle$ reaches the plateau value $ \sim 0.60266 $ of GUE with $ \beta=1 $,
        then according to Eq.\ref{eq:PWW2},
        $ \langle\tilde{r}'\rangle $ reaches an  corresponding plateau value listed in Table I as
        $ \sim 0.7344 $ with a RMT index $ 3 \beta + 1=4 $.
        When $\langle \tilde{r}\rangle$ reaches the plateau value $ \sim 0.38629 $ of the Poisson,
        according to Eq.\ref{eq:PWW2}, $ \langle\tilde{r}'\rangle $ reaches an  corresponding plateau value listed in Table I as
        $ \sim 0.5 $. It is only slightly below the GOE value $ 0.53590 $ with the RMT index $ 3 \beta + 1=1 $.

\subsection{The particle-hole $P$ or $ Z $ conserving hybrid $ q=4 $  Majorana fermion SYK }

It may also be interesting to study the KAM in the following $ q=4 $ hybrid
  Majorana fermion SYK which keeps the $ P $ symmetry at any ratio of $ K/J $:
\begin{equation}
    H_{M2,\pm}
	=  \sum^{N}_{i<j<k<l} J_{ijkl} \chi_{i} \chi_{j} \chi_k \chi_l
	    \pm [ i \sum^N_{i<j}K_{ij} \chi_{i} \chi_{j} ]^2
\label{mix12}
\end{equation}
where  $J_{ijkl}$, $K_{ij}$ are real and  satisfy the Gaussian distributions with
$\langle J_{ijkl}\rangle=0$, $\langle J^2_{ijkl} \rangle= 3! J^2/N^3 $ and
$ \langle K_{ij} \rangle=0$, $\langle K^2_{ij} \rangle= K/N $ respectively.
 Note that here we use $\langle K^2_{ij} \rangle\sim K$  to make K/J dimensonless.

  Obviously, the second term can be written as $ H^2_{M2} $. In contrast to Eq.\ref{mix1},  it still keeps the $ P $ symmetry.
   So symmetry analysis alone can not distinguish between $ H_{M4} $ and $ H^2_{M2} $ despite the former is chaotic, the latter is integrable.
  It is easy to show that because $ H_{M2} $ is integrable, so is $ H^2_{M2} $.
  Furthermore, the eigenvalue of $ H^2_{M2} $ is always positive.
  This can be most conveniently seen from the NN ratio  $ r_n= s_n/s_{n+1} $ of the $ H_{M2} $.
  Then $ R_n $ of the $ H^2_{M2} $ can be written as:
\begin{equation}
  R_n= \frac{ E^2_n - E^2_{n-1} }{ E^2_{n+1} - E^2_{n} } \sim \frac{ E_n - E_{n-1} }{ E_{n+1} - E_{n} }= r_n
\label{Rrn}
\end{equation}
   where similar to density of states, the center of two NN energies just cancels in the ratio. Very similarly, one can show that
   the ratio of the NNN energy spacing $ R^{\prime}_n \sim r^{\prime}_n $.
  So the  ELS of $ H^2_{M2} $ remains Poissonian.

  This model with both $ \pm $ sign was studied before in \cite{MBLSPT} by $ 1/N $ expansion at $ N=\infty $ and
  by the Renormalization Group (RG) analysis. By performing the RG on the
  coefficient of the $ H^2_{M2} $ term around
  the $ q=4 $ SYK Conformally invariant fixed point, it was found that the $ + $ sign is marginally irrelevant ( Fig.\ref{sykrg}b2 ),
  so the $ q=4 $ non-integrable SYK NFL fixed point is stable in the IR against the $ + H^2_{M2} $ perturbation.
  However,  the $- $  is marginally relevant ( Fig.\ref{sykrg}b1 ),
  it flows to the integrable FL fixed point controlled by $ -H^2_{M2} $.
  However, our ED results show that there is very little differences between the two signs.
  This could be explained as follows:
\begin{eqnarray}
-H_{M2,-}[J^{\prime}, K] & = & \sum^{N}_{i<j<k<l} J^{\prime}_{ijkl} \chi_{i} \chi_{j} \chi_k \chi_l + [ i \sum^N_{i<j}
K_{ij} \chi_{i} \chi_{j} ]^2 = H_{M2,+}[J, K]
\label{pmeq}
\end{eqnarray}
   where $   J^{\prime}_{ijkl}=- J^{\prime}_{ijkl} $. So $ J^{\prime} $ and $ J $ satisfy the same distribution.

\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{MajoranaSYK_Type2_Even.pdf}
\caption{  The evolution of the ELS for type-II hybrid Majorana SYK Model when $ N_{\chi} $ is even.
$\langle\tilde{r}\rangle$ (black curve),
	and $\langle\tilde{r}'\rangle$ (orange curve) in (b)
	evaluated for $N_\chi=14,16,18,20$ and are averaged over  $100$, $80$, $60$, $40$  samplings respectively.
Notably, for $ N mod 8=0 $ case in (b), there is a double degeneracy at the $ (q=2)^2 $ side,
the NN ratio $ \langle \tilde{r}  \rangle $ is rapidly changing near the $ q=4 $ side, so it
is quite difficult to determine the KAM regime. Fortunately,
the NNN ratio $ \langle \tilde{r^{\prime}}  \rangle $ shows a nice plateau regime near the  $ (q=2)^2 $ side, the KAM regime can be
easily identified and listed in the Table 3-6. This dramatic advantage of the new NNN ratio over the known NN ratio were
further demonstrated in all the relevant figures.  For $ N mod 8=4 $ case in (d), all the energy levels are doubly
degenerate  at any $ J/K $.}
\label{MajoranaSYK_Type2_Even}
\end{figure}

\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{MajoranaSYK_Type2_Odd.pdf}
\caption{ The evolution of the ELS for type-II hybrid Majorana SYK Model when $ N_{\chi} $ is odd.
$\langle\tilde{r}\rangle$ (black curve),
	and $\langle\tilde{r}'\rangle$ (orange curve in (b) and (c) )
	evaluated for $N_\chi=13,15,17,19$ and are averaged over  $100$, $80$, $60$, $40$  samplings respectively.
For $ N mod 8=1, 7 $ GOE case in (b) and (c), one can see the advantages of using $ \langle \tilde{r^{\prime}}  \rangle $
over $ \langle \tilde{r}  \rangle $,  especially in the KAM side. For $ N mod 8=3, 5 $ GSE case in (a) and (d),  all the energy levels
are doubly degenerate at any $ J/K $.	}
\label{MajoranaSYK_Type2_Odd}
\end{figure}

   Let $E^{+}_n$ be an ordered set of energy levels of $ H_{M2,+}[J, K] $  and $s^{+}_n=E^{+}_{n+1}-E^{+}_{n} > 0 $
   the nearest-neighbor (NN) spacings. Then $ E^{-}_n=- E^{+}_n $ be an ordered set of energy levels
   of $ H_{M2,-}[J^{\prime}, K] $ and $ s^{-}_n=E^{-}_{n}-E^{-}_{n+1}=E^{+}_{n+1}-E^{+}_{n} = s^{+}_n $.
   Similarly, the NNN spacing $ s^{\prime -}_n=E^{-}_{n}-E^{-}_{n+2}=E^{+}_{n+2}-E^{+}_{n} = s^{\prime +}_n $.
   Because $ ( J^{\prime}, K ) $ and $ ( J, K ) $  satisfy the same distribution, so we conclude $ H_{M2, \pm} $
   satisfy the same ELS \cite{AFM}. This is confirmed by our ED. So we just show our $ + $ sign results in Fig.\ref{MajoranaSYK_Type2_Even} and \ref{MajoranaSYK_Type2_Odd}.
   However, there is an exchange between the ground state and the highest energy state in the $ \pm $ sign, so
   the $ H_{M2, \pm} $ will have completely different ground states as shown by the RG analysis in  \cite{MBLSPT}.
   This fact may show that the CNCT characterized by the RMT may not be directly related to QPT characterized by the RG.
   Further elucidations on the intricate relation between RG and RMT will be given in Sec.10.

  Because $ [P, H_{M2,\pm} ]=0 $ and $  [Z, H_{M2,\pm} ]=0 $,
  so the symmetry classification in Sec.3 still applies to this Type-II hybrid  Majorana fermion SYK at any $ J/K $.
  So the Table I still holds.  Our Exact Diagonization (ED) studies \cite{JW} in
  Fig.\ref{MajoranaSYK_Type2_Even} and \ref{MajoranaSYK_Type2_Odd} shows
  that this is true only when $ K/J $ is sufficiently small.
  The KAM theorem applies al small $ J/K \ll 1 $ where the ELS becomes Poissonian.
  There is a CNCT from the corresponding WD to the Possion as $ K/J $ increases.
  However, the degeneracy in Table I remains true at any $ J/K $.

  For $ N $ even shown in Fig.\ref{MajoranaSYK_Type2_Even},  there is always a CNCT from the corresponding WD to the Poissonian.
  Several salient features need to be stressed.
  For $ N \pmod 8=0 $ in (b),  one may also look at the ELS from the $ H^2_{M2} $ side when $ J/K \ll 1 $.
  Because $ \{ P, H_{M2} \} =0 $, then $ ( \psi, P \psi) $ have two different eigenvalues $ \pm \lambda $, so
  are orthogonal. They are also still in the same parity sector.
  but $ [ P, H^2_{M2} ] =0 $,  a pair of orthogonal eigenstates $ ( \psi, P \psi) $ have the same eigenvalue $ \lambda^2 $.
  so $ H^2_{M2} $ is doubly degenerate at $ J=0 $.
  However, the double degeneracy is broken by any $ J >0 $.
  Then using the NN ELS is not enough. One may start to use the combination of NN and the NNN ELS presented in Sec.2.
  In contrast to all the previous case with $ P $ or $ Z $ violating hybrid SYK models
  where the  doubly degeneracy comes from the $ q=4 $ non-integrable side,
  here the doubly degeneracy comes from the integrable side.
   It is best to combine $\langle \tilde{r}\rangle$ and $ \langle\tilde{r}'\rangle $
   in Fig.\ref{MajoranaSYK_Type2_Even}b and {\sl read them from the $ ( q=2 )^2 $ integrable side}.
        When $\langle \tilde{r}\rangle$ is close to be zero,
        the NNN $ \langle\tilde{r}'\rangle $ still shows Poisson until $ J/K \sim e^{-4} $.
        When $\langle \tilde{r}\rangle$ reaches the plateau value $ \sim 0.53590 $ of the GOE with $ \beta=1 $,
        then according to Eq.\ref{eq:PWW2},
        $ \langle\tilde{r}'\rangle $ reaches an  corresponding plateau value listed in Table I as
        $ \sim 0.6769 $ which is quite close to GSE with a RMT index $ 3 \beta + 1=4 $.

        For $ N \pmod 8=4 $ in (d),  there is a CNCT from GSE to Possion as $ K/J $ increases.
        As shown in Table I, there is always a double degeneracy at any $ J/K $.
        Although $\langle\tilde{r}'\rangle$ in (b) is quite similar
        to $\langle\tilde{r}\rangle$ in (d), both of which looks like to show a CNCT from GSE to Poission,
        they have very different physical meanings.
        As said above,  $ \langle\tilde{r}'\rangle $ with $ N \pmod 8=0 $ in (b) represents NNN ELS, so it stands  for a CNCT from GOE to Poission.  While  $\langle \tilde{r}\rangle$ with $ N \pmod 8=4 $ in (d) represents NN ELS, so it is  a true CNCT from GSE to Poission.

       For $ N \pmod 8=2,6 $ in (c) and (a),  there is no degeneracy at any $ J/K $. There is a CNCT from GUE to Possion as $ K/J $ increases.

       The odd $ N $ case is shown in Fig.\ref{MajoranaSYK_Type2_Odd}.  As shown in Table 1, one
       continue to use $ P $ symemtry when $ N Mod 8=3, 7 $, but  must use $ Z $ symmetry when $ N Mod 8=1,5 $.
       Due to the absence of GUE,  there are are only two cases: the CNCT from GSE to Possion in (a) and (d)
       and the CNCT from GOE to Possion in (b) and (c).
       They show similar evolution patterns as the corresponding CNCTs for even $ N $ case shown in Fig.\ref{MajoranaSYK_Type2_Even}.
       Again, although $\langle\tilde{r}'\rangle$ in (b) and (c) are quite similar
       to $\langle\tilde{r}\rangle$ in (a) and (d), both of which looks like to show a CNCT from GSE to Poission,
       they have very different physical meanings. The former is actually a CNCT from GOE to Possion.
       While the latter is a true a CNCT from GSE to Possion with a double degeneracy.

\subsection{The particle-hole  $P$  conserving  hybrid $ q=4 $  complex fermion SYK}

  It may also be interesting to study the KAM in the following $ q=4 $
  complex fermion SYK \cite{can} which also keeps the $ P $ symmetry at any ratio of $ K/J $:
\begin{equation}
H_{C2,\pm}=  \sum^{N}_{i<j,k<l} J_{ij;kl} c^{\dagger}_{i} c^{\dagger}_{j} c_k c_l \pm  [ \sum^N_{i<j}
K_{ij} c^{\dagger}_{i} c_{j} ]^2- \mu \sum^{N}_{i=1 } c^{\dagger}_i c_i
\label{mix22}
\end{equation}
  where $ K^{*}_{ij} =K_{ji} $ is  a Hermitian matrix satisfying
  $ \langle K_{ij} \rangle=0,  \langle |K_{ij}|^2 \rangle=  K/N $.
   Note that here we use $\langle K^2_{ij} \rangle\sim K$  to make K/J dimensonless.

\begin{figure}[!htb]
\centering
\includegraphics[width=0.75\linewidth]{ComplexSYK_Type2_All.pdf}
\caption{  The evolution of the ELS for type-II hybrid complex SYK Model.
$\langle\tilde{r}\rangle$ (black curve) and $\langle\tilde{r}'\rangle$ (orange curve) in (a)
	evaluated with $N_c=12,13,14$ and are averaged over $100$, $50$, $50$ samplings respectively.
For $ N_c mod 4=0, q_c=0 $ GOE case in (a), one can the advantages of using $ \langle \tilde{r^{\prime}}  \rangle $
over $ \langle \tilde{r}  \rangle $, especially in the KAM side. For $ N_c mod 4=2, q_c=0  $ GSE case in (c), all the energy levels
are doubly degenerate at any $ J/K $.}
\label{ComplexSYK_Type2}
\end{figure}

  Obviously, the second term can be written as $ H^2_{C2} $. So it still keeps the $ P $ and $ Z $ symmetry.
  So symmetry analysis along can not distinguish between $ H_{C4} $ and $ H^2_{C2} $.
  So the symmetry classification on complex SYK still applies to this $ P $ conserving hybrid $ q=4 $  complex fermion SYK.
  When $ N \pmod 4=0,2 $ and $ q_c=0 $, the ELS is GOE and GSE with the degeneracy $ d=1,2 $ respectively.
  But when $ N \pmod 4=1,3 $ and $ q_c=\pm 1/2 $, it is GUE with the degeneracy $ d=1 $.
  ( In fact, as long as $ q_c \neq 0 $,  there is no  $ P $ symmetry anymore,
  so it is in GUE regardless of the value of $ N \pmod 4$ ).
  Our Exact Diagonization (ED) studies \cite{JW} in
  Fig.\ref{ComplexSYK_Type2} shows that this is true only when $ K/J $ is sufficiently small.
  The KAM theorem applies al small $ J/K \ll 1 $ where the ELS becomes Poissonian.
  However, the degeneracy remains true at any $ J/K $. There is a CNCT from the corresponding WD to the Possion as $ K/J $ increases.

  The rest of discussions are quite similar to the $ P $ or $ Z $ conserving Type-II hybrid Majorana fermion SYK discussed in Sec.3.3.
  Similar to Eq.\ref{Rrn} in Sec.3.3,  it is easy to show that because $ H_{C2} $ is integrable, so is $ H^2_{C2} $,
  so its ELS remains Poission.
  Furthermore, the eigenvalue of $ H^2_{C2} $ is always positive.
  Both $ \pm $ sign in the second term need to be considered.
  In the RG sense, we expect the $ + $  (  $-$  ) sign in Eq.\ref{mix22} is irrelevant ( relevant ).
  However, Eq.\ref{pmeq} adopted to Eq.\ref{mix22} shows the ELS stay the same which is confirmed by our ED.
  So we just show our $ + $ sign results in Fig.\ref{ComplexSYK_Type2}.
  There is always a CNCT from the corresponding WD to the Poissonian as $ K/J $ increases.

  Notably, for $ N_c \pmod 4=0, q_c=0 $ in (a),  one may also look at the ELS from the $ H^2_{C2} $ side when $ J/K \ll 1 $.
  Because $ \{ P, H_{C2} \} =0 $, but $ [ P, H^2_{C2} ] =0 $, so $ H^2_{C2} $ is doubly degenerate at $ J=0 $.
  Then using the NN ELS is not enough. One may need to use the combination of NN and the NNN ELS presented in Sec.2.
  In contrast to the previous  $ P $ violating Type-I hybrid complex SYK model
  where the  doubly degeneracy comes from the $ q=4 $ chaotic side, here
  the doubly degeneracy comes from the integrable side. It is complete to combine
      both $\langle \tilde{r}\rangle$ and $ \langle\tilde{r}'\rangle $ in
      Fig.a and {\sl read them from the $ q=2 $ integrable side}.
        When $\langle \tilde{r}\rangle$ is close to be zero,
        the NNN $ \langle\tilde{r}'\rangle $ shows Poisson until $ J/K \sim e^{-6} $.
        When $ \langle \tilde{r}\rangle $ reaches the plateau value $ \sim 0.53590 $ for the GOE with $ \beta=1 $,
        then according to Eq.\ref{eq:PWW2},
        $ \langle\tilde{r}'\rangle $ reaches an  corresponding plateau value listed in Table I as
        $ \sim 0.6769 $ which is quite close to the GSE with a RMT index $ 3 \beta + 1=4 $.
        There is always a double degeneracy at any $ J/K $ in (c).
       Although $ \langle\tilde{r}'\rangle $ in (a) is quite similar to
       $ \langle\tilde{r}\rangle $ in (c), both of which looks like to show a CNCT from GSE to Poission, they have very different physical meanings.  As said above, $ \langle\tilde{r}'\rangle $ in (a) when $ N_c \pmod 4=0, q_c=0 $ represents NNN ELS, so it stands
       for a CNCT from GOE to  Poission.
       While $\langle \tilde{r}\rangle$ in (c) when $ N_c \pmod 4=2, q_c=0 $ represents NN ELS, so so it stands
       for a CNCT from GSE to  Poission and  each energy level is doubly degenerate at any $ K/J $.

\subsection{ The hybrid of $ q=2 $ and $ q=4 $  bosonic  SYK }

    The above procedures for fermions can also be applied to study the $ q=2 $ and $ q=4 $ hybrid bosonic SYK:
\begin{equation}
H_{B}=  \sum^{N}_{i<j,k<l} J_{ij;kl} b^{\dagger}_{i} b^{\dagger}_{j} b_k b_l +  \sum_{i<j}
K_{ij} b^{\dagger}_{i} b_{j}- \mu \sum^{N}_{i=1 } b^{\dagger}_i b_i
\label{mix3}
\end{equation}
  where, in general, $ J_{ij;kl}=J_{ji;kl}, J_{ij;kl}=J_{ij;lk}, J^{*}_{ij;kl}= J_{kl;ij} $ and
  $ \langle |J_{ij;kl}|^2 \rangle= 3! J^2/N^3 $.
  Following \cite{CSYKnum}, {\sl we take the four site indices $ i,j; k,l$ are all different to keep
  the PH symmetry explicit at $ \mu=0 $. }.
  $ K^{*}_{ij} =K_{ji} $ is  a Hermitian matrix satisfying  $ \langle K_{ij} \rangle=0,  \langle |K_{ij}|^2 \rangle=  K^2/N $.

  In the $ K/J=0 $ limit,  the $ q=4 $  bosonic  SYK was studied by the ED in \cite{CSYKnum},
  a QSG ground state was expected in the thermodynamic limit $ N=\infty $.
  One can define the particle-hole symmetry operator to be $ P= K \Pi^{N}_{i=1} ( b^{\dagger}_i + b_i ) $. The boson charge
    $ Q_b= \sum_{i} b^{\dagger}_i b_i, q_b= \sum_{i} ( b^{\dagger}_i b_i -1/2)  $.
   It is easy to show $ P^2=1,  P b_i P= b^{\dagger}_i, P b^{\dagger}_i P= b_i,
   P Q_b P= N- Q_b $. $ [P, H_4]=0 $.
   For $ N $ even, at half filling $ q_b=0 $, it is in GOE.
   However, as long as $ q_b \neq 0 $,  there is no P-H symmetry anymore,
   it is in GUE regardless of the value of $ N  $ is even or odd.

  Now we apply the PH transformation to the  bosonic hybrid SYK model Eq.\ref{mix3}.
  In contrast to the fermionic hybrid SYK models, $ [P,H_2]=0 $, so the PH symmetry is preserved in the  hybrid bosonic SYK model.
  In the $ J/K=0 $ limit, we expect that the ELS for $ q=2 $ bosonic SYK is the same as $ q=4 $ bosonic SYK:
  when $ q_b =0 $, it is in GOE, when  $ q_b \neq 0 $, it is in GUE.
  This is in sharp contrast to the $ q=2 $ fermionic SYK which is non-interacting, so integrable.
  While  the $ q=2 $ bosonic SYK is interacting ( the bosons on the same site behaves as fermions, but different sites
  as bosons ), so non-integrable and is already a quantum chaotic system.

\begin{figure}
\includegraphics[width=0.5\textwidth]{BSYKcrossover.pdf}
\caption{ (Color online)  ELS of the hybrid  bosonic SYK.
 (a) For $ N_b $ even, at the half filling $ q_b=0 $, it is always in GOE for any ratio of $ K/J $.
   (b) Away from the half filling $ q_b \neq 0 $, it is always in GUE for any ratio of $ K/J $, regardless of $ N_b $ is even or odd.
    In contrast to the fermion cases, there is no CNCT in the bosonic hybrid SYK. }
\label{bcross}
\end{figure}

   So we expect the ELS of the hybrid bosonic SYK stays the same from $ q=4 $, all the way down to $ q=2 $.
   This is indeed confirmed by our ED results \cite{JWb} shown in Fig.\ref{bcross}, there are only two cases here
   (a) For $ N_b $ even, at the half filling $ q_b=0 $, it is always in GOE for any ratio of $ K/J $.
   (b) Away from the half filling $ q_b \neq 0 $, it is always in GUE for any ratio of $ K/J $, regardless of $ N_b $ is even or odd.

  In short, in contrast to all the hybrid fermionic models discussed in this paper, the KAM theorem does not apply in
  the bosonic model,  there is no CNCT in the bosonic hybrid model.

\section{Contrast KAM theorem of integrability with stability of quantum chaos  }

Here from Fig.3-8 we summarize the stability of quantum chaos near $ q=4 $ SYK with the KAM theorem near the integrable $ q=2 $ SYK in Type I and  $ ( q=2 )^2 $ SYK in Type II in the following 4 Tables from which one can conclude the following trends:

(1) The validity regime of KAM theorem seem comparable in type I and type II.
    In all the case,  its validity regime  gets small as the system size gets large.
    We expect it gets to zero in the thermodynamic limit.
    However, how it approaches zero need to be investigated both theoretically and by ED in much large sizes.

(2) In Type  I,  the stability of GOE is comparable to that of GSE. They also also comparable
    to the KAM in the sizes we studied.
    But that of the GUE is much more robust than the two. It is also much more robust than the KAM.
    This is due to the fact that GUE is dictated by the symmetry classification anyway, so it
    exists as an intermediate regime in all the three cases, then the stability regime of the GUE at the
    $ q=4 $ side is greatly enhanced.
    We expect that in contrast to the KAM theorem, the stability of chaos remain a finite value in the thermodynamic limit.
    For example, although the Lyapunov exponent at low temperature may be quite sensitive to an integrable perturbation.
    its value at high or infinite temperature should be quite robust against the perturbation.
    The quantum chaos characterized by the RMT may correspond to the Lyapunov exponent at infinite temperature,
    because both involve all the energy levels.

(3) In Type  II,  The stability of  quantum chaos is even among GOE, GUE and GSE.
    This is because they are dictated by the symmetry classification at the corresponding $ N $ values anyway.
    They are also more robust than the KAM theorem.

\begin{table}[!htb]
\centering
\begin{tabular}{ |*{5}{c|} }
\hline\hline
$N_{\chi}$   &14	&16	&18	&20   \\
\hline
ELS at $K=0$ &GUE	&GOE	&GUE	&GSE\\
\hline
QCS: $K/J$   &$e^{0}=1$	&$e^{-5.5}\approx0.004$	    &$e^{0}$	&$e^{-5.5}$ \\
\hline
KAM: $J/K$   &$e^{-3}\approx0.05$ &$e^{-3.5}\approx0.04$ &$e^{-4}\approx0.02$ &$e^{-5}\approx0.01$\\\hline\hline
$N_{\chi}$   &13	&15	&17	&19   \\
\hline
ELS at $K=0$ &GSE	&GOE	&GOE	&GSE\\
\hline
QCS: $K/J$   &$e^{-4.5}\approx0.01$ &$e^{-5}\approx0.007$ &$e^{-5.5}\approx0.004$ &$e^{-5.5}$ \\
\hline
KAM: $J/K$   &$e^{-3}\approx0.05$ &$e^{-3.5}\approx0.04$ &$e^{-4}\approx0.02$ &$e^{-5}\approx0.01$\\
\hline\hline
\end{tabular}
\caption{
Contrast the stability of quantum chaos at $ q=4 $
with the KAM theorem of integrability at $ q=2 $ for
type-I hybrid Majorana fermion SYK. Notations: $ N_{\chi} $ is the number of Majorana fermions, ELS is Energy Level Statistics, QCS is Quantum Chaos stability. KAM is Kolmogorov-Arnold-Moser theorem. The same notations are used in the following tables.
The KAM shrinks as the size increases. The QCS of the GUE  is greatly enhanced. }
\end{table}

\begin{table}[!htb]
\centering
\begin{tabular}{ |*{4}{c|} }
\hline
$(N_{c},q_c)$	&$(12,0)$		&$(13,1/2)$		&$(14,0)$	\\  \hline
ELS at $K=0$	&GOE			&GUE			&GSE		\\  \hline
QCS: $K/J$	&$e^{-5.5}\approx0.004$	&$e^{1.5}\approx4.5$	&$e^{-6}\approx0.002$\\ \hline
KAM: $J/K$	&$e^{-5}\approx0.008$	&$e^{-5}$		&$e^{-5.5}$	\\ \hline
\end{tabular}
\caption{
Contrast the stability of quantum chaos at $ q=4 $
with the KAM theorem of Integrability at $ q=2 $ for
Type-I hybrid complex fermion SYK. The QCS of the GUE  is greatly enhanced. }
\end{table}

\begin{table}[!htb]
\centering
\begin{tabular}{ |*{5}{c|} }
\hline
$N_{\chi}$	&14	&16	&18	&20	\\  \hline
ELS at $K=0$	&GUE	&GOE	&GUE	&GSE	\\  \hline
QCS: $K/J$	&$e^{-2}\approx0.36$ &$e^{-2}$ &$e^{-2}$ &$e^{-2}$\\	\hline
KAM: $J/K$	&$e^{-3}=0.05$	&$e^{-4}=0.18$ &$e^{-5}=0.007$ &$e^{-6}=0.002$	\\
\hline\hline
$N_{\chi}$	&13	&15	&17	&19	\\  \hline
ELS at $K=0$	&GSE	&GOE	&GOE	&GSE	\\	\hline
QCS: $K/J$	&$e^{-2}\approx0.36$ &$e^{-2}$ &$e^{-2}$ &$e^{-2}$\\	\hline
KAM: $J/K$	&$e^{-3}=0.05$	&$e^{-4}=0.18$ &$e^{-5}=0.007$ &$e^{-6}=0.002$	\\
\hline\hline
\end{tabular}
\caption{
Contrast the stability of quantum chaos at $ q=4 $
with the KAM theorem of Integrability at $ q=2 $ for
Type-II hybrid Majorana fermion SYK. The QCS remains even among all WD and are much more robust than KAM. }
\end{table}

\begin{table}[!htb]
\centering
\begin{tabular}{ |*{4}{c|} }
\hline
$(N_{c},q_c)$	&$12,q_c=0$		&$13,q_c=1/2$		&$14,q_c=0$	\\  \hline
ELS at $K=0$	&GOE			&GUE			&GSE		\\  \hline
QCS: $K/J$	&$e^{-2}\approx0.14$	&$e^{-2}$		&$e^{-2}$	\\  \hline
KAM: $J/K$	&$e^{-6}\approx0.002$	&$e^{-7}\approx0.001$	&$e^{-7}$	\\  \hline
\end{tabular}
\caption{
Contrast the stability of quantum chaos at $ q=4 $
with the KAM theorem of Integrability at $ q=2 $ for
Type-II hybrid complex fermion SYK. The QCS remains even among all WD and are much more robust than KAM. }
\end{table}

\section{Random matrix theory versus  Renormalization group, CNCT versus Quantum phase transition. }

   It was well established that in terms of symmetries and the space dimension,
   the Renormalization Group (RG) ( including the DMRG, MPS and  tensor network )
   can be used to classify many body phases and quantum phase transitions at $ T=0 $ and classical phase transitions at
   finite temperatures \cite{aue,subirbook}.
   The RG focus on  infra-red ( IR ) behaviours of the system which are determined by the ground state and low energy excitations.
   The RG is also intimately connected to General Relativity (GR) through the holographic principle.

   While the 10 fold way RMT classification scheme to describe quantum chaos only depends on two anti-unitary symmetries,
   seems independent of space dimension.
   It covers all  the energy levels of the system,  so can  be used to characterize the CNCT.
   In a completely different context, the 10 fold way can also be used to classify topological equivalent class
   for non-interacting topological Insulators and  topological superconductors \cite{kane,zhang}.
   In this case, it also depends on the space dimension and  has the Bott periodicity $ d \rightarrow d +8 $.

   The differences and connections between RG and the RMT are explicitly demonstrated in the two types of SYK models in this work.
   For example, in the thermodynamic limit $ N = \infty $, the fermions in the Type I hybrid SYK models have scaling dimension $ 1/2 $
   and $ 1/4 $ at the $ q=2 $ and $ q=4 $ SYK respectively.
   So the $ q=2 $ ( $ q=4 $ ) SYK is a stable ( unstable ) conformably invariant fixed point ( Fig.\ref{sykrg}a.
   Any $ J $ ( $ K $ ) is irrelevant ( relevant ) to the $ q=2 $ ( $ q=4 $ ) SYK.
   At any $ K/J \neq 0 $, the ground state is controlled by the $ q=2 $ SYK fix point, so it is always non-chaotic Fermi liquid
   with well defined low energy quasi-particle excitations.
   There is no quantum phase transition (QPT) in
   the Type  I hybrid SYK models. There is no finite temperature transitions either.
   However, as shown in Fig.3-8,
   there is always a chaotic to non-chaotic transition (CNCT) from the GUE to Poisson.
   Of course, the name of CNCT  is a little misleading, because it is really a crossover instead of a sharp phase transition \cite{studied}.
   As shown in these figures, there are also crossovers between different WD ensembles as $ K/J $ increases.
   In the Type  II hybrid SYK models, the coefficient of $ (q=2)^2 $ is marginally irrlevant/relevant when the sign is $ +/- $
   ( Fig.\ref{sykrg}b.
   So the ground state with $ + $ sign is the non-Fermi liquid without quasi-particle excitations controlled by $ q=4 $ SYK fixed point,
   while that with $ + $ sign is the Fermi liquid with well defined quasi-particle excitations controlled by $ ( q=2 )^2 $ SYK fixed point.
   So there is no quantum phase transition (QPT) and no finite temperature transitions either in the the Type II hybrid SYK models.
   However, independent of the $ +/- $ sign, there is always a CNCT from the corresponding WD of $ q=4 $ SYK to
   the Poissonian as shown in Fig.\ref{MajoranaSYK_Type2_Even},\ref{MajoranaSYK_Type2_Odd} and \ref{ComplexSYK_Type2}.
   In fact, as explained in Sec.3.3,3.4, despite the $ +/- $ sign leads to dramatically different ground states, the ELS is identical \cite{AFM}.
   So here we provides two interesting examples where the ELS are the same,
   but the ground states are dramatically different. This fact demonstrates explicitly the dramatic differences
   between the two classification schemes.
   For another example,
   it was known the ground states of the hybrid bosonic SYK is a quantum spin glass (QSG) which breaks ergodicity.
   There is a finite temperature phase transition from the QSG to a paramagnet where the ergodicity is restored.
   However, as shown in Fig.\ref{bcross}, the ELS stays as GOE or GUE. There is no CNCT.
   So it is difficult to see if there is a connection between CNCT and a finite temperature phase transition which is driven by
   the free energy \cite{studied}.

   QPT only involves the changing of the ground states and the low energy excitations,
   while the high energy states are irrelevant. There is a divergent length (or time ) scale
   and associated scalings near a QPT.
   While the ELS involves all the energy levels at a finite but large enough $ N $.
   So despite the absence of QPT in the two types of the hybrid SYK models,
   there could still be a CNCT which is not a true QPT \cite{studied}.
   There is no divergent length (or time ) scales, no associated scalings, therefore no finite size scalings near a CNCT \cite{studied}.

   Quantum chaos also arise in a quantum optics model which is completely unrelated to SYK models: the $ U(1)/Z_2 $ Dicke model\cite{u1largen,gold,comment,strongED}.
   It describes $ N $ atoms ( qubits ) interact with a single photon mode with both
   rotating wave term $ g $ and the counter-rotating wave term $ 0< g ^{\prime}/g= \beta < 1 $.
   In the thermodynamic $ N \rightarrow \infty $ limit, there is a quantum phase transition  at
   $ g =  g_c=\frac{ \sqrt{\omega_a \omega_b}} {1 + \beta } $ from the normal phase to the super-radiant phase.
   At any finite and large enough $ N $,  there is also quantum chaos and the associated
   CNCT from the GOE to the Poisson in the $ U(1)/Z_2 $ Dicke model \cite{KAM}.
   Of course, the quantum chaos in the SYK models are due to the interplay between interactions and disorders.
   While those in the Dicke models are solely due to the atom-photon interactions.
   Another difference between the two models is that the SYK models are fermionic.
   While the Dicke models are pure bosonic ones. Therefore,  the fermionic SYK has a finite Hilbert space
   $ 2^{N/2} $ or $ 2^{N} $ for Majorana or complex fermion ( including hard core bosons ) respectively,
   so ED can be performed directly in the finite Hilbert space
   at any finite $ N $. While due to the infinite number of occupations of photons, the Hilbert space of the $ U(1)/Z_2 $ Dicke model
   is infinite, so a energy cut-off $ \Lambda= n_c \omega_a $ with $ n_c=200 \sim 2000 $ must be introduced to perform an ED \cite{KAM}.
   The ELS can still be trusted when when the energy levels are well below the the enrgy cutoff.
   Despite these differences in the two microscopic Hamiltonians, they share similar relations
   on  Random matrix theory versus  Renormalization group, CNCT at a finite $ N $  versus Quantum or classical phase transitions at $ N=\infty $,
   Quatum chaos stability versus KAM near the integrability, etc.

   There maybe still possible deep connections between the onset of quantum chaos characterized by the RMT and
   the onset of quantum phase transition characterized by the RG which need to be explored further.
   The hybrid SYK models ( or the corresponding hybrid  GW tensor model ) and the $ U(1)/Z_2 $ Dicke models
   may supply new platforms to investigate possible relations between the two dramatically different classification  schemes.

\begin{figure}
\centering
    \includegraphics[width=0.5\linewidth]{sykrg.pdf}
\caption{RG flow of Type I and Type II hybrid SYK models.
	(a) The RG flow of the Type I hybrid SYK models.
	The $ K/J^{\prime}, J^{\prime}= \sqrt{ J^2+K^2} $ is relevant to the $ q=4 $ SYK.
	So the ground state is always Fermi liquid with well
	defined low energy quasi-particle excitations.
	(b) Upper, The RG flow of the Type II hybrid SYK models with $+$ sign.
	The $ K/J^{\prime} $ is marginally relevant to the $ q=4 $ SYK.
	So the ground state is always a Fermi liquid with well
	defined low energy  quasi-particle excitations.
	Lower, The RG flow of the Type II hybrid SYK models with $+$ sign.
	The $ K/J^{\prime} $ is marginally relevant to the $ q=4 $ SYK.
	So the ground state is always a Non-Fermi liquid without quasi-particle excitations. }
\label{sykrg}
\end{figure}

\section{Conclusions. }

   In this work, we only focused on the ELS of bulk spectrum, it may also be interesting to focus on the low energy spectrum, especially
   the edge spectrum in a future study. For example, it was known the ground state of both the $ q=2 $ and $ q=4 $ bosonic SYK
   is a quantum spin glass (QSG) state which is a ergodicity broken state  instead of a chaotic state.
   As shown in Fig.\ref{bcross}, its bulk ELS still shows the quantum chaotic behaviours. However, we expect its low energy (or edge) spectrum
   may show different behaviours indicating its broken ergodicity. This is also related to
   the issue of RG versus RMT, ground state versus all energy states  discussed in the last section.
   It may also be interesting to study the ELS of the two indices SYK model \cite{me}
   with two large numbers: $ N $ ( the number of sites ) and the $ M $ ( the group of $ O(M) $ or $ SU(M)$ ).
   Depending on the relations between $ N $ and $ M $, it may show chaotic QSL or non-chaotic QSG ground state.

  As alerted in Sec.4, it remains interesting to explore (1) What is the finite size dependence   of the  KAM theorem:
  will $ (J/K)_c \rightarrow 0 $  in the thermodynamic limit $ N \rightarrow \infty $.
  Namely, will the KAM theorem completely shrinks to zero as $ N \rightarrow \infty $ ?
  If so, what is its asymptotical behaviours.
  (2) What is the finite size dependence of the stability of quantum chaos:  $  K/J $.
  These questions need to be addressed analytically and ED in much bigger system sizes.

   As said in the introduction, in addition to the ELS in the RMT classifications,
   the CNCT may also be dynamically diagnosed from  the spectral form factor (SFF) in the RMT.
   The  SFF at $ \beta_T = 0 $  may also be used to measure the dynamic ( time-dependent ) chaotic behaviours of the
   two types of hybrid SYK models.
   A slope-dip-ramp-plateau structure in the time evolution was considered to be evidence for the chaotic behaviours.
   This feature may disappear at $ (K/J)_c $ and beyond.
   While the Lyapunov exponent  is a completely different way to characterize the dynamic nature of the quantum chaos.
   It seems quite in-sensitive to the 10-fold way global discrete symmetry classification, therefore
   independent of $ N $ mod (8)  or $ N_c $ mod (4). So it is important to investigate the KAM from the OTOC perspective.
   These results will be presented in a separate publication \cite{LZ}.

   As shown in \cite{pure},  the stability of quantum chaos of black holes against a constant $ q=2 $ SYK terms may be used to explore the interior behind the black hole horizon.
   Implications of the results achieved here on the bulk gravity side or on quantum error corrections
   need to be explored \cite{QEC}.

\acknowledgments
Y.Y thank Y.-Z. You for sharing the ED code used in Ref.\cite{MBLSPT}.
J. Ye thank C. Xu for helpful discussions.
F.S and J.Y acknowledge AFOSR FA9550-16-1-0412 for supports.
This research at KITP was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.
WL was supported by the NKBRSFC under grants Nos. 2011CB921502, 2012CB821305, NSFC under grants Nos. 61227902, 61378017, 11311120053.

redpajama_set_name: RedPajamaArXiv


Firstly, I extend a warm welcome to you all - from across the pond here in Belize. I hope you enjoy looking through the site, with its hundreds of catalogued photographs covering over 50 years of Regimental life. There are many different boards to view, from message to contact details, to fallen friends to joke boards to name just a few of the boards available for you to browse.
Go in and enjoy exploring them all. The aim of my site is to provide a useful meeting place that is both informative and amusing for people with an interest in The 1st Bn. Royal Sussex Regiment/The 3rd Bn. The Queens Regiment. We are not competing with any Regimental Associations or other organizations. Users can expect to find both reasoned argument and discussions peppered with military type banter and humour. We especially welcome new members, all I ask is, when you make your 'maiden post,' please explain a little about yourself for example, your name, the unit and dates you served and so on. Should you have any photographs relating to the Regiment then please e-mail them to Bob Jones . I will size and text them before I then upload them into the correct library.
Although we have no issues with challenging the "official" viewpoint we do not wish to bring the wrath of the authorities down upon the site. We wish to avoid attracting those with extremist views or personal agendas to the site - for they are not welcome on this site. Any visitor or member who feels that a posted message is objectionable or another user is behaving inappropriately is encouraged to contact us immediately by email at Web master We have the ability to remove or edit objectionable messages and we will make every effort to do so, within a reasonable time frame, if we determine that removal is necessary. This is a manual process, however, so please realize that we may not be able to remove or edit particular messages immediately. Any minor spats that can occasionally occur in any family unit, it's better to reconcile these by personal E-mail and/or Skype, rather than continue to prosecute your case, verbatim on the notice board.
There are very few rules that are set in stone, just common sense and a respect for all visitors and members alike is all that's needed.
1. You agree, through your use of the site, that you will not use it to post any material which is knowingly false and/or defamatory, inaccurate, vulgar, hateful, harassing, obscene, profane, threatening, invasive of a person's privacy, or otherwise vocative of any law. If you cannot type something nice about a topic, chances are, it probably isn't worth posting it in the first place.
The Queen's Regiment name and Badge are Copyright of the MoD and may not be used or reproduced without permission.
We act and have acted, as communicator for ex-members in cases to do with : Benefits, Claims, Pensions, Benevolence, Finding lost "brothers in arms", Reuniting families and friends, Finding war graves, Answering historical questions, Answering queries on badges, crest or colours, we run Regimental quizzes, write articles for various outlets.
liaise closely with The Regimental Museum, Regimental Archives and the Regimental Association.
We see ourselves as an allied and communicative extension of the Royal Sussex Regimental Association.


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Our 6 ball multibuy saver is available in all three sizes and two colourways.
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Size 3, 4, and 5 options.


text:
'The Nightingale' Movie Review | A Rant of Epic Proportions
I told myself I wouldn't review this film. I told myself to just put it out of my mind and go on with my life and yet, here we are … ready to rant.
I watched The Nightingale at the beginning of August and as soon as I left the theater, all I wanted was the memory of it immediately erased from my mind. The film follows Clare, a young Irish convict (played by Aisling Franciosi) who chases a British officer (played by Sam Clamflin) through the rugged Tasmanian wilderness and is bent on revenge for a terrible act of violence the man committed against her family. On the way, she enlists the services of Aboriginal tracker Billy, a black man, (played by Baykali Ganambarr) who is marked by trauma from his own violence-filled past.
Lets get the good out of the way: the acting performances were phenomenal. Every actor did the best with what they were given. Aisling beautifully allowed every single emotion to shine through her, Sam Clamflin was every bit of repulsive that he needed to be and Baykali was an absolute revelation. Cinematically the film was truly beautiful, combined with the interestingly-angled shots and careful editing: I found myself in awe (when I wasn't fighting the urge to walk out of the movie theater).
Yet, even with all of that, I have to say: I AM SICK AND TIRED OF WHITE SAVIOR FILMS!!!!
This movie is trauma porn; it uses shocking rape and death scenes to cause a visceral reaction from the audience. Though it's clear that the revenge storyline is meant to take precedent throughout this film, it somehow keeps finding its way back to the white savior narrative. It's clear that director Jennifer Kent was attempting to make a large statement about Feminism and the #MeToo movement but this movie was not the vessel for that message. It seems absolutely ridiculous to think that the story of the colonization of Australia and the terrible acts of violence that black people had to endure at the hands of the colonizers, was a story meant to be told by a WHITE WOMAN!
There were a few gratuitous violent scenes that did nothing for the actual storyline. There is NO NEED for us to get two different scenes of black men hanging from trees, we don't need a black woman being raped to understand the racist colonization of Australia. WE GET IT, THE MEN IN THIS FILM ARE TERRIBLE RACIST MEN!!
If I can convince you to do anything, ever, then I hope that thing is to never ever watch this film. Okay, that's a bit extreme and you're entitled to do with your life what you want. The thing is, all white savior films do is remind us that we haven't gotten as far in terms of inclusion as we'd like to think we have.

redpajama_set_name: RedPajamaCommonCrawl


text:
THIS ISLAND
SHARING & CARING
NEW TRIP ENQUIRY
If there is one thing you must do on a holiday in Sri Lanka that is to taste a cup of Ceylon tea. From a coffee rush to a tea tale, the start of the tea industry in Sri Lanka is as refreshing as the brew itself. The tale of the brew that put this teardrop-shaped island on the map unfolds amidst the rolling hills of the island.
In 1867, when the island was thriving with coffee plantations, James Taylor, a Scottish planter was entrusted to assess the commercial feasibility of some Assam tea shrubs, which marked the birth of Ceylon tea. Taylor, who was the superintendent of Loolecondera Estate near Kandy, planted eight hectares of tea, and perfected the art of growing the new crop. The historic Loolecondera tea estate can be visited even today where you can witness the fire place of his rustic log cabin that has stood the test of time and the fields in which the original tea was grown as well as a little museum.
With "Devastating Emily", a fungal disease which wiped out the coffee production of the country in 1869, Ceylon tea claimed the place and became a world-favourite brew. In Taylor's days, a pound of tea leaves was sold in Kandy for Rs 1.50. For the colonisers who suffered due to the coffee catastrophe, tea became their lucky charm that once again saved their financials that were almost in ruins.
The balmy breeze of the looming hills, rich red soil and nourishing weather of the hill country, among other natural features of the island, create the unique taste of Ceylon tea that comes in a myriad of flavours. From a light floral brew to a strong dark cup of tea, the various Tea Leaf Grades make for teas to reinvigorate everyone's palette. The one million Ceylon tea packets sold at the Chicago World's Fair in 1893, marked a milestone in Sri Lanka's history of tea. Known as "The Black Gold of Sri Lanka", Ceylon tea has become an essential part of the island's identity.
Today, decades after the departure of the British planters, aspects of the industry have changed much, but the distinctive taste of Ceylon tea has remained the same, preserving its hallmark quality. Whether you are indulging in a lavish afternoon tea spread at one of London's most luxurious hotels or enjoying a chit-chat with a warm croissant at the Blue Box Café at Tiffany & Co. in New York, chances are that you will be offered a cup of Ceylon tea.
Yet, the best way to experience the art of making Ceylon tea, the how, why and what of this trade, is to visit a working tea plantation. You can indulge in a spot of artisanal tea tasting or make your journey to misty hills of the island to glimpse the exciting journey from the bush to cup.
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text:
Department Of Revenue Of Oregon v. Acf Industries, Inc.
PETITIONER: Department Of Revenue Of Oregon
RESPONDENT: Acf Industries, Inc., et al.
LOCATION: Charleston Naval Base
DOCKET NO.: 92-74
DECIDED BY: Rehnquist Court (1993-1994)
LOWER COURT: United States Court of Appeals for the Ninth Circuit
CITATION: 510 US 332 (1994)
ARGUED: Nov 08, 1993
DECIDED: Jan 24, 1994
ADVOCATES:
Carter G. Phillips - on behalf of the Respondents
Kent L. Jones - as amicus curiae in support of the Petitioner
Richard A. Malm - for the Railway Progress Institute et al. as amici curiae urging affirmance
Thomas W. Andrews - for the Railway Progress Institute et al. as amici curiae urging affirmance
Virginia L. Linder - on behalf of the Petitioner
Facts of the case
Media for Department Of Revenue Of Oregon v. Acf Industries, Inc.
Opinion Announcement - January 24, 1994
Oral Argument - November 08, 1993
Audio Transcription for Oral Argument - November 08, 1993 in Department Of Revenue Of Oregon v. Acf Industries, Inc.
Audio Transcription for Opinion Announcement - January 24, 1994 in Department Of Revenue Of Oregon v. Acf Industries, Inc.
William H. Rehnquist:
The opinion of the Court in No. 92-74 Department of Revenue of Oregon versus ACF Industries will be announced by Justice Kennedy.
Anthony M. Kennedy:
Oregon imposes an ad valorem tax upon all real and personal property within its jurisdiction except property granted an express exception and there are various classes of business personal property that are exempt.
The respondents in these case are called the Carlines and there are eight companies at least railroad cars to railroad and shippers.
Now, the railroad cars are not exempt in taxation in Oregon and the Carlines brought claim that this differential exemption scheme voilates the federal statute and that statute prohibits certain taxes against railroad property.
It is called the Railroad Revitalization and Regulatory Reform Act of 1976, otherwise known as the 4-R Act.
The question presented is whether Congress violated the 4-R Act by imposing an ad valorem tax upon railroad property while exempting various other but not all classes of commercial and industrial property.
In an opinion for the Court of some 14 pages now we attempt in detial to examine the structure and the wording to the statute to resolve the issue presented.
Now, that analysis would be somewhat insidious to explain on the bench even in a brief statement.
Suffice it to say that our holding rejects the challenge to the taxation scheme.
The exemptions Oregon granted did not discriminate its railroad property within the meaning relevant statutory sections.
We are confident that had congress to limit.
The power of the State to exempt non-railroad property from a general property tax scheme it would have spoken with precision and clarity in order to do so.
We do not find that intent to the stature.
We reverse the judgment of the Court of Appeals for the Ninth Circuit and Justine Steven has filed a dissenting opinion.
Cases - 1993
Select Year 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987 1986 1985 1984 1983 1982 1981 1980 1979 1978 1977 1976 1975 1974 1973 1972 1971 1970 1969 1968 1967 1966 1965 1964 1963 1962 1961 1960 1959 1958 1957 1956 1955 1940-1955 1900-1940 1850-1900 1789-1850
← Davis v. United States
Turner Broadcasting System, Inc. v. Federal Communications Commission →


text:
La odintsovita és un mineral de la classe dels silicats. Rep el nom en honor del professor M. M. Odintsov (1911-1979), fundador de l'Institut de l'escorça terrestre d'Irkutsk.

Característiques
La odintsovita és un silicat de fórmula química K₂Na₄Ca₃Ti₂Be₄Si₁₂O38. Cristal·litza en el sistema ortoròmbic. La seva duresa a l'escala de Mohs es troba entre 5 i 5,5.

Formació i jaciments
Va ser descoberta al massís Malyi Murun, a la província d'Irkutsk, dins el Districte Federal de l'Extrem Orient (Rússia). També ha estat descrita al mont Koaixva, al massís de Jibiny, també a Rússia. Fora d'aquest país només ha estat trobada a Nakkaalaaq, un muntanya situada al Complex intrusiu d'Ilímaussaq, a la costa sud-oest de Groenlàndia.

Referències

Ciclosilicats
Minerals de potassi
Minerals de sodi
Minerals de calci
Minerals de titani
Minerals de beril·li
Minerals descoberts a la Província d'Irkutsk

redpajama_set_name: RedPajamaWikipedia


Designer: Stefan Jolie not Stephen Yearick -BRAND NEW NEVER BEEN USED!
Beautiful embroidery work on the top part of the dress with pearls and rhinestones.
Style: Ball gown dress – heart shaped top.
Dress is from a high end bridal store called Bijou Bridal.
Brand New! Never been worn!
Dress is still in bag with hanger. Never been worn.


text:
Google Pixel 5 design just leaked — and there's bad news
By Adam Ismail 21 August 2020
These Pixel 5 renders look like more of the same, which could result in an affordable flagship
(Image credit: @OnLeaks and Pricebaba)
If you were pining for a radical shift in design for the Google Pixel 5, it seems like you're going to be disappointed. Google itself teased the Pixel 4a 5G and Pixel 5 in an image from a side view. Now, leaker Steve Hemmerstoffer (@OnLeaks) and Indian tech blog PriceBaba have shared Pixel 5 renders of their own, teasing a very familiar design.
Based on these mockups, the Pixel 5 is more or less a Pixel 4a with a more premium-looking finish across its entire body, as well as a dual-lens rear camera. The lack of separation between the frame and back panel indicates the material used here could be plastic, rather than a combination of metal and glass.
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That may disappoint some, though I really like the luster Google's chosen; it reminds me of the glittery, eye-catching motif of one of my favorite old phones from back in the day, the Nexus 4. Just like the render Google itself shared earlier, the Pixel 5's power button here is clad in a dark chrome for a little contrast.
The most interesting thing about this design is how much it shares with the Pixel 4a. Of course, there's no 3D Face Unlock or Soli radar system in the top bezel, and the Pixel 4a's fingerprint sensor can be found again here, on the back. Given that all rumors to date have indicated Google is very much looking to cut costs with the Pixel 5, this is hardly surprising. It's already been confirmed that the Pixel 4a 5G will start at $499, so $599 could be Google's target here.
I've said before how great I thought the Pixel 4's Face Unlock system was. It's just so much faster and more cleverly-designed than Apple Face ID, which really hasn't been improved upon in any meaningful way since it launched on the iPhone X back in 2017.
I would have liked to see Google find a way to incorporate Face Unlock while still keeping the price relatively low, as Apple has proven able to with the $699 iPhone 11 — though that doesn't seem to be in the cards. Instead, we'll get an old-school rear fingerprint sensor, which may seem dated to some — though in my experience, they're still so much more dependable and less frustrating to use than modern optical and ultrasonic in-display sensors.
We've heard the Pixel 5 will incorporate Qualcomm's Snapdragon 765G chipset, which offers a cheap road to 5G connectivity, in addition to 8GB of RAM. Given that the Pixel 4a only comes in one 128GB storage configuration, 128GB seems like the starting point here as well. As Jon Prosser leaked yesterday, it could be offered in two shades: black, seen here, as well as green.
Again, the Pixel 5 might look a bit too familiar, but it seems Google is very much sticking to a strategy of keeping costs low to deliver a more affordable semi-premium phone. Considering how successful the Pixel 3a was last year, and how much of a failure the Pixel 4 line proved to be despite all its cutting-edge features, can you blame them?
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