text:
\section{Introduction}
\label{sec:I}
The non-Fermi liquid (NFL) is a state of gapless fermionic matter that does not have long-lived quasiparticles due to its strongly interacting nature~\cite{Stewart2001,Abanov2003}. It is widely believed to be the microscopic origin of the ``strange metal" state observed in a broad range of materials, such as Cu-based ~\cite{Keimer2015} and Fe-based~\cite{ZhaoyuLiu2016,YanhongGu2017} high-temperature superconductors, heavy-fermion compounds~\cite{Custers2003,BinShen2019}, and recently in twisted 2D heterostructures~\cite{YCao2019,shen2019observation}. Additionally, the understanding of the unconventional superconducting phase in these systems naturally hinges on the understanding of the NFL ``normal state". Moreover, recently from the studies of the Sachdev-Ye-Kitaev (SYK) models~\cite{SY,K,SYK2,SYK3}, it has been realized that NFLs host a hidden connection {between strange metals~\cite{guo-gu-sachdev-2020} and} holographic quantum matters that saturate the upper bound for quantum chaos, opening an entirely new avenue in understanding the behavior of NFLs~\cite{YingfeiGu2017}.

The term ``non-Fermi liquid" captures the failure of conventional perturbative approach in treating interacting fermion systems with weak interactions, which poses a challenge in the theoretical understanding of such systems. In a general context, NFL behavior often occurs via electron interactions mediated by gapless bosonic modes~\cite{max,max2,max_last,raghu_15,steve_sam,lawler-barci-fernandez-2006,lawler-fradkin-2007,XiaoYanXu2017,ZiHongLiu2018,ZiHongLiu2019,XiaoYanXu2019,XiaoYanXuReview2019} that render the electrons incoherent. Such gapless bosons typically arise in the vicinity of
a quantum-critical point (QCP) or in quantum gauge theories. Despite the simplicity of the setup, the analytical solution to these models remains challenging due to the lack of a natural small control parameter. Advancement has been made via modifying the model to a large $N$ limit with $N$ the number of fermion flavors and a leap of faith that the same physics holds down to $N=O(1)$, while these large-$N$ approaches face important subtleties in two spatial dimensions~\cite{sslee}.

Along a separate path, there has been great progress in the numerical front in recent years, in particular in designer Hamiltonian of critical bosons Yukawa-coupled to Fermi surfaces~\cite{Berg2012,Berg2019,XiaoYanXuReview2019}. Recent results in minus-sign-problem-free quantum Monte Carlo (QMC) simulations~\cite{XiaoYanXu2017,ZiHongLiu2018,ZiHongLiu2019,XiaoYanXu2019,XiaoYanXuReview2019,Berg2019} have shown strong evidence of NFL states in a range of such boson-fermion models
{with gapless bosons from a nematic~\cite{Schattner2016} and ferromagnetic~\cite{XiaoYanXu2017,XiaoYanXu2020} quantum critical points and with gauge fields~\cite{XiaoYanXu2019,ChuangChen2019,ChuangChen2020}}
 (see Ref.~\cite{XiaoYanXuReview2019} for a recent review). It is now possible to obtain accurate and reliable information about the scaling behaviors in the close vicinity of these QCPs, testing and improving our theoretical knowledge about these challenging problems.

To reveal NFL physics in numerics, this class of models require tuning the mass of the boson to a critical value, while away from the QCP the system restores Fermi liquid behavior. However, the precise determination of the quantum critical point and region of NFL are subject to finite size effects, and the position of the QCP is not universal but system dependent. Moreover, recenlty it is found that to reveal the clear signature of NFL in fermion self-energy in these QCP systems, one would also need to control the strength of the effective coupling between fermions and bosons, as well as deduct the nonnegligible thermal contributions to the fermionic self-energy~\cite{XiaoYanXu2020}. These difficulties make it harder to directly reveal the scaling form of the NFL self-energies in these systems.

\begin{figure}[!htp]
	\centering
	\includegraphics[width=0.8\columnwidth]{figure_1.pdf}
	\caption{Yukawa-SYK model. There are $M$ quantum dots labeled by $\{i,j\}$, and each dot havs $N$ flavors labeled by $\{\alpha,\beta\}$. Bosons are given by antisymmetric fields $\phi_{\alpha\beta}$. Fermions are coupled to bosons through a  random Yukawa coupling $t_{i\alpha,j\beta }$.}
	\label{fig:1}
\end{figure}

Recently a class of SYK-like models featuring random Yukawa interactions between bosons and fermions has been put forward to analyze the NFL pairing problem~\cite{wang-2019, esterlis-2019, schmalian-2019,wang-chubukov-2020}. Analytically solvable in a large-$N$ limit similar to the SYK models, the ``Yukawa-SYK model" takes a different approach from the perturbative one to address the interacting fermion system by eliminating kinetic energies from the outset. Physically, such a theoretical approach is of relevance to systems where the Fermi energy is small, e.g., systems with low electron density such as SrTiO$_3$, and Moir\'e flat band systems such as twisted bilayer graphene~\cite{Bistritzer2011,YDLiao2020}. The Yukawa-SYK models have been shown to be maximally chaotic~\cite{kim-cao-altman-2019} and thus likely to admit a holographic dual description. Compared with the SYK models that only involve interacting fermions, the inclusion of a dynamical bosonic degree of freedom in the Yukawa-SYK makes it ideal to model strongly interacting fermionic systems near a QCP.

Unlike finite dimensional models with quantum-critical points, within large-$N$ approximation these models have been shown to ``self-tune" to quantum criticality, i.e., the system becomes critical due to the strong mutual feedback between the bosonic and fermionic sectors, independent of the bosonic bare mass. In addition, the pairing behavior at large-$N$ has been analytically studied~\cite{wang-2019, esterlis-2019, schmalian-2019}. Depending on details of the Yukawa coupling, these models either show exotic pairing of incoherent fermions, or a NFL phase that is stable to pairing down to $T=0$. While the onset temperature of pairing may be finite, the feedback effects of pairing fluctuations to the fermion Green's function are small ($\sim\mathcal{O}(1/MN)$), without affecting the NFL behavior of the normal state.

As in the original SYK model, these analytical results of the NFL behavior are formally obtained using the replica trick and then taking the replica diagonal ansatz. This is equivalent with replacing the quenched disorder with annealed disorder, which is usually justified by the fact that replica-nondiagonal processes are suppressed by $1/N$~\cite{gu-qi-2016}.
However, the validity of this ansatz is far from obvious~\cite{baldwin-swingle-2019}, since it is not clear whether the summation of the subdominant processes, each small in $1/N$, is convergent. If the system breaks replica symmetry, the true ground state is then a spin glass. For example, replica symmetry breaking occurs in the bosonic SYK model~\cite{gps-2001,fu-sachdev-2016}, and in fact it has been shown recently that similar situations occur for all random interacting bosonic models~\cite{baldwin-swingle-2020}. On the other hand, for the fermionic SYK model, there is now strong numerical and analytical evidence that a glass phase is absent and the NFL state persists down to $T=0$~\cite{gps-2001,fu-sachdev-2016,gur-ari-2018}. For this reason the validity of the large-$N$ analytical result of the Yukawa-SYK model needs to be carefully investigated, especially since the model involves both fermions and bosons. To this end, unbiased numeric calculations, similar in the spirit to the aforementioned critical bosons Yukawa-coupled to Fermi surfaces systems~\cite{Berg2012,Berg2019,XiaoYanXuReview2019,XiaoYanXu2017,ZiHongLiu2018,ZiHongLiu2019,XiaoYanXu2019,Schattner2016,Gerlach2017,ChuangChen2019}, are highly desirable.

With this motivation in mind, here we address such a timely issue by showing that at finite $N$ the Yukawa-SYK model can be exactly solved by determinantal QMC simulations, thanks to the bosonic degree of freedom. A simple extension of the original model, introducing an antiunitary time-reversal symmetry, eliminates the minus-sign problem without altering the essential physics. {To enable a direct comparison with QMC, we numerically solve the self-energies of the Yukawa-SYK model within large-$N$ at finite temperatures with discrete imaginary time steps. At low temperatures, this indeed agrees with the analytical solution of the Schwinger-Dyson equations with an emergent time reparametrization symmetry. This emergent symmetry indicates that the effect of thermal fluctuations can be easily incorporated in the time domain (see Ref.~\cite{Klein2020,XiaoYanXu2020} for subtleties in the frequency domain), enabling a direct identification of the NFL behavior at finite temperatures.

In this paper, we found that as one progressively increase $N$, the Green's functions from QMC simulations do approach the large-$N$ result and display self-tuned criticality and NFL behavior with power-law self-energies. Additionally we found that as $N$ increases, the QMC results with different realizations of the random interaction self-average, i.e., the variance of the Green's function decreases with increasing $N$ and we obtained a good match the large-$N$  results.  This is strong evidence that the system is free from glassy behavior at least within the temperature range accessible to QMC. By comparing with large-$N$ results, we analyze the behavior of finite-$N$ corrections and show that it is consistent with those from replica off-diagonal fluctuations and pairing fluctuations, which decreases with $N$. By contrast, we consider a model~\cite{wang-2019} in a similar form with a crucial difference that the random coupling is of a lower rank. In such a model with less randomness, replica off-diagonal processes are less suppressed. We numerically show that the bosonic Green's function exhibits glassy behavior.

\section{The model}
The model studied here describes $M$ quantum dots each hosting $N$ flavors of fermions interacting with $N^2$ flavors of matrix bosons via all-to-all random Yukawa interactions. The Hamiltonian of this Yukawa-SYK model is given by
\begin{align}
		H =& \sum_{i,j=1}^{M}\sum_{\alpha,\beta=1}^{N} \sum_{m,n}^{\uparrow,\downarrow}\(\frac{i}{\sqrt{MN}} t_{i\alpha,j\beta}\phi_{\alpha\beta}c^\dagger_{i \alpha;m}\sigma^z_{m,n}c_{j\beta ;n}
		\) \nonumber\\
		&+\sum_{\alpha , \beta =1}^N\(\frac{1}{2}\pi_{\alpha\beta}^2+\frac{m_0^2}{2}\phi_{\alpha\beta}^2\),
		\label{eq:eq1}
	\end{align}
where the random coupling between fermion and boson satisfies $\langle t_{i\alpha,j\beta}\rangle=0$, $\left\langle t_{i \alpha, j \beta} t_{k \gamma, l \delta}\right\rangle=\left(\delta_{\alpha \gamma} \delta_{i k} \delta_{\beta \delta} \delta_{j l}+\delta_{\alpha \delta} \delta_{i l} \delta_{\beta \gamma} \delta_{j k}\right) \omega_{0}^{3}$. This model is very similar to that studied in Ref.~\cite{wang-2019}, the only difference being that here the random coupling $t_{i\alpha,j\beta}$ has a higher rank than that in Ref.~\cite{wang-2019} thta does not depend on $\alpha$ and $\beta$.  As we will see in Sec.~III, the high rank randomness of the Yukawa coupling $t_{i\alpha,j\beta}$ is crucial for stabilizing the non-Fermi liquid behavior. $\pi_{\alpha \beta}$ is the canonical momentum of $\phi_{\alpha \beta}$. Hermiticity of the first term requires $\phi_{\alpha \beta}=-\phi_{\beta \alpha}$. As schematically depicted in Fig.~\ref{fig:1}, here $(\alpha, \beta)$ are flavor indices and $(i,j)$ are site indices. $\sigma^z$ is the Pauli matrix in the fermion spin space for each flavor. In the absence of a chemical potential term $\mu=0$, the model has an exact particle-hole symmetry under $c\to c^\dagger$. The general case with $\mu\neq 0$ has also been recently analytically solved at the $N,M\to \infty$ limit~\cite{wang-chubukov-2020} and will be left for our future numerical studies. Importantly, compared to the model studied in  Ref.~\cite{wang-2019}, this model has a time reversal symmetry $c^\dagger \to c^\dagger i\sigma^y \mathcal{K}$, where $\mathcal{K}$ is the complex conjugation operator, which guarantees the absence of minus-sign problem of the QMC simulation.  For the sake of simplicity we set $\omega_0=1$ as the energy unit throughout the paper. The only other energy scale in Eq.~\eqref{eq:eq1} is the bosonic bare mass $m_0$. We refer to  situations with small and large $\omega_0/m_0$ as ``weakly coupled" and ``strongly coupled".

In the $N\to \infty$, $M\to\infty$ limit, the ground state of the system has been analytically solved~\cite{wang-2019, esterlis-2019}, and the ground state is found to be a non-Fermi liquid. The large-$N$ result is based on the assumption that the replica symmetry of the random model is unbroken. In this work we verify the validity of the non-Fermi liquid solution by examining and extrapolating the system behavior at finite $N,M$. For such $N,M$ analytical calculations are uncontrolled.} Fortunately, due to the time-reversal symmetry in our designer Hamiltonian in Eq.~\eqref{eq:eq1} there is no minus-sign problem (the proof of this is given in Appendix.~\ref{sec:IIIC}).

\subsection{Normal-state results at $N, M\to \infty$}
\label{sec:IIA}
Before demonstrating our QMC results for the Yukawa-SYK model, we first briefly review the theoretical results at the $N, M\to \infty$ limit. In this limit one can show that the effective action has a saddle point given by the Schwinger-Dyson equations
\begin{align}
     \Pi (i\Omega_n) =& \frac{4M}{N}\omega_0^3T\sum_{m}
     G_f(i\omega_m-i\Omega_n/2) G_f(i\omega_m+i\Omega_n/2)\nonumber\\
    \Sigma(i\omega_m) =& -\omega_0^3
T\sum_{m}
     G_b(i\Omega_n) G_f(i\omega_m-i\Omega_n),
     \label{eq:eq2}
\end{align}
where $\Sigma, \Pi$ are fermionic and bosonic self-energies, and $G_f(i\omega_m) =\[
i\omega_m+\Sigma(i\omega_m)\]^{-1}$ and $G_b(i\Omega_n) =  \[\Omega_n^2+ \Pi(i\Omega_m)+m_0^2\]^{-1}$ are fermionic and bosonic Green's functions.

At $T=0$, it was found~\cite{wang-2019,esterlis-2019} that for $m_0\sim\omega_0$ and $\omega,\Omega\ll \omega_0$ the self-energies are given by
\begin{align}
	\Sigma(\omega) =&-G_f(\omega)^{-1}= ic \sgn(\omega) |\omega|^{x}\omega_0^{1-x},\nonumber\\
	\Pi(\Omega) =&G_b(\Omega)^{-1}= -m_0^2 +c^{-2} \alpha(x)|\Omega|^{1-2x}\omega_0^{1+2x},\nonumber\\
	\label{eq:eq3}
\end{align}
where $c$ is a non-universal $O(1)$ constant, and $0<x<1/2$ is determined by
\be
\frac{4M}{N}=\frac{1/x-2}{1+\sec(\pi x)}
\label{eq:eq4}
\ee
and
\be
\alpha(x) = -\frac{\Gamma^2(-x)}{4\pi \Gamma(-2x)}.
\ee
Compared to the results in Ref.~\cite{wang-2019}, Eq.~\eqref{eq:eq4} is different by a factor of 2 because the addition of the spin degree of freedom $m,n=\uparrow/\downarrow$ in the Hamiltonian in Eq.~\eqref{eq:eq1}. In particular, at $M=N$, one finds $x\approx 0.098$, and for $4M=N$, $x\approx 0.231$.

 \begin{figure}[! htp]
	\centering
	\includegraphics[width=\columnwidth]{figure_2.pdf}
	\caption{Theoretical result of $G_f$ and $G_b$ at $N=4M\to \infty$, $\omega_0=1, m_0=2$ and various temperatures. Here we show them in log-log plot. The  auxiliary dashed lines whose slopes are $1-x$ and $2x$ show that $G_f(\tau, 0) \propto \(\frac{\pi}{\beta\sin(\pi\tau/\beta)}\)^{1-x}$ and $G_b(\tau,0) \propto \(\frac{\pi}{\beta\sin(\pi\tau/\beta)}\)^{2x}$ at $\tau \rightarrow \frac{\beta}{2}$, when $\beta$ is large enough.}
	\label{fig:2}
\end{figure}

From Eq.~\eqref{eq:eq3} we have
\be
m_0^2-\Pi(\Omega=0)=0,
\ee
indicating that the boson is critical. This was argued in Refs.~\cite{wang-2019, esterlis-2019} to be true for an \emph{arbitrary} $m_0^2$. No matter what the bosonic bare mass is, the system renormalizes it to zero via interaction effects. For this reason we dub this phenomenon ``self-tuned quantum criticality". This feature is certainly not present in any finite dimensional models such as those of critical bosons coupled to Fermi surface systems~\cite{max,max2,max_last,raghu_15,steve_sam,lawler-barci-fernandez-2006,lawler-fradkin-2007,XiaoYanXu2017,ZiHongLiu2018,ZiHongLiu2019,XiaoYanXu2019,XiaoYanXuReview2019,Gerlach2017,XiaoyuWang2017,Berg2019,Schattner2016,ChuangChen2019,XiaoYanXu2020} discussed in Sec.~\ref{sec:I}.

In the time domain, by a Fourier transform we obtain \footnote{The Fourier transform from $\Pi(\Omega)$ is tricky, since the positive power-law $|\Omega|^{1-2x}$ does not have a Fourier transform in the common sense as the Fourier integral is UV divergent for $0<x<1/2$. This divergence is canceled by the Fourier transform of the $m_0^2$ term in $\Pi(\Omega)$ with the UV information.}
\begin{align}
	\Pi(\tau,\tilde\tau) \propto& |\tau-\tilde\tau|^{-(2-2x)},\nonumber\\
	G_b(\tau,\tilde\tau) \propto& |\tau-\tilde\tau|^{-2x},\nonumber\\
	\Sigma (\tau,\tilde\tau) \propto& |\tau-\tilde\tau|^{-(1+x)}\sgn(\tau-\tilde\tau),\nonumber\\
	G_f(\tau,\tilde\tau) \propto& |\tau-\tilde\tau|^{x-1}\sgn(\tau-\tilde\tau).
\end{align}

At a finite temperature $T=1/\beta$, one can accordingly obtain the fermionic and bosonic Green's functions through a reparametrization symmetry transformation$\tau\to f(\tau)=\tan(\pi\tau/\beta)$~\cite{SY,K,SYK2,SYK3} , and we have at low-temperatures and long-time limit,
\begin{align}
	G_f(\tau, 0) \propto & \(\frac{\pi}{\beta\sin(\pi\tau/\beta)}\)^{1-x}\nonumber\\
	G_b(\tau,0) \propto& \(\frac{\pi}{\beta\sin(\pi\tau/\beta)}\)^{2x}.
	\label{eq:eq9}
\end{align}

To enable a direct comparison with the QMC data, we developed an iterative algorithm to solve the nonlinear equation in \eqref{eq:eq2} numerically at an arbitrary temperature. To ensure the convergence of the interations, the temperature dependence of $\Pi(0)$ was fixed using analytical results obtained in Ref.~\onlinecite{esterlis-2019}. As we shall see, the QMC simulations for the bosonic sector are performed on a time lattice with lattice constant $\Delta\tau$, the Matsubara frequencies are compact and defined in a frequency Brillouin zone $\omega_m,\Omega_n  \in (-\pi/\Delta\tau, \pi/\Delta\tau)$. We have incorporated the compactness of the frequency domain within our numerical solution of Eq.~\eqref{eq:eq2} as well, which ensures better match with QMC results especially at large frequencies.

In Fig.~\ref{fig:2} we plot the behavior of $G_f$ and $G_b$ at $N=4M$, $\omega_0=1, m_0=2$  and different temperature from iterative theoretical calculation, in particular, we see that at $\beta=256 \ (T=\frac{1}{256})$, the results matches well in the long-time limit with the approximate result obtained using time-reparametrization symmetry, exhibiting self-tuned criticality and NFL behaviors. This result will later be compared with numerical ones in Sec.~\ref{sec:num}.

\subsection{Pairing at $N, M\to \infty$: Mean field theory}
\label{sec:IIB}

\begin{figure}[!htp]
	\centering
	\includegraphics[width=\columnwidth]{figure_3.pdf}
	\caption{Inverse transition temperature $\beta_c$  from NFL to superconductivity as a function of the ratio $\omega_0/m_0$ for $N=4M$ and $N=M$, obtained from solving Eq.~\eqref{eq:eliash} at large-$N$.}
	\label{fig:3}
\end{figure}

  \begin{figure*}[! htp]
	\centering
	\includegraphics[width=.83\textwidth]{figure_4.pdf}
	\caption{QMC results at $N=4M$, $m_0=2,\omega_0=1$ and $\beta=16$ for $M=2,3,4$. (a) Green's function of fermions $G_{f}(\tau,0)$ versus $\tau$ in the range of $\tau\in[0,\beta]$. Blue, red and yellow dots are DQMC data and the black dashed line is the large-$N$ result. (b) Green's function of bosons $G_{b}(\tau,0)$ versus $\tau$ in the range of $\tau\in[0,\beta]$. (c) and (d) The same as above, but in a special log-log scale as in Fig.~\ref{fig:2}.
	The convergence towards the large-$N$ results as $(M,N)$ increase is obvious. In all panels, the error bar denotes the variation of the Green's function for different disorder realizations. The progressively small error bars as $M$ increases indicate that the randomness of the coupling self-averages.}
	\label{fig:4}
\end{figure*}

It is straightforward to see that the leading pairing instability mediated by the critical boson mode is toward a spin-singlet, intra-dot, and intra-flavor channel:
\be
\Delta \sim  \sum_{i,\alpha}\langle c^\dagger_{i\alpha\uparrow}c^\dagger_{i\alpha\downarrow}\rangle.
\ee
Within mean-field theory, the pairing behavior is described by the Eliashberg equation
\be
\Delta (i\omega_n) = \omega_0^3  T\sum_{n} G_b(i\Omega_n)|G_f(i\omega_m+i\Omega_n)|^2 \Delta(i\omega_n+i\Omega_m),
\label{eq:eliash}
\ee
where the $1/MN$ factor given by the two Yukawa interaction vertices has been canceled by the summation of the site and flavor indices of the internal fermions.

 \begin{figure*}[!htp]
	\centering
	\includegraphics[width=0.83\textwidth]{figure_5.pdf}
	\caption{(a) and (c) show $G_b$ and $G_f$ of 20 different disorder realizations with $M=N=9$, $\beta=24$, $m_0=2$ and $\omega_0=1$. The Green's functions are very close to each other. (b) and (d) present the difference between theoretical (large-$N$) results $G^{Theory}(\beta/2,0)$ and QMC numerical simulation data $G(\beta/2,0)$. It is clear that as $M(N)$ increases, the distance between QMC and analytics gradually reduces. And the relative standard deviations in the QMC data are also decreasing. The  parameters are set at $M=N$, $\beta =24$, $m_0=2$ and $\omega_0=1$.}
	\label{fig:5}
\end{figure*}

At $T=0$, plugging in the analytical results in Eq.~\eqref{eq:eq3}, we have
\be
{\Delta(\omega) = \frac{2}{\alpha(x)}\int^{\omega_0}_{\Delta} \frac{d\omega'}{2\pi} \frac{ \Delta(\omega')}{|\omega-\omega'|^{1-2x}|\omega'|^{2x}}.}
\label{eq:eq13}
\ee
where $\Delta$ is the order of magnitude of the frequency-dependent gap $\Delta(\omega)$ that serves as an infrared cutoff of the Green's functions, and $\omega_0\sim m_0$ is an ultraviolet cutoff scale for the low-energy quantum-critical NFL behavior.

At finite temperatures, we can solve for the critical temperatures $T_c$ using the normal state results numerically obtained. To that end, we numerically solve Eq.~\eqref{eq:eliash} as an eigenvalue problem. As temperature lowers, the eigenvalues of the kernel increases, and $T
_c$ corresponds to the temperature at which the largest eigenvalue approaches one. For reference, we plot $\beta_c$ ($T_c =\frac{1}{\beta_c}$) as a function of the ratio $\omega_0/m_0$ for $N=4M$ and $N=M$ in Fig.~\ref{fig:3}. We see that as  the dimensionless coupling constant $\omega_0/m_0$ increases, $\beta_c$ decreases ($T_c$ increases) in both cases.

At finite $N,M$, a true phase transition to a superconductor does not occur. Yet, pairing fluctuations, which become stronger upon lowering temperatures, do contribute to the fermion Green's function, making the fermions more incoherent. In the Yukawa-SYK model such effects are suppressed by $\mathcal{O}(1/MN)$, but can be detected at small $M,N$. In this sense the fermion Green's function receives two types finite-$N$ corrections -- both from replica-off-diagonal fluctuations and from pairing fluctuations. A true finite temperature phase transition to superconductivity,
on the other hand, can be obtained by a finite-size extrapolation of the pairing susceptibility  in the QMC simulations. However, the calculation of such observables is beyond the scope of this work and we leave it for future investigations. In this work, we focus on the NFL normal state, although we will discuss signatures of pairing fluctuations in $G_f$ obtained by QMC.

 \section{Numerical results}
 \label{sec:num}
 \subsection{NFL Green's functions}
 In this section, we report the key numerical findings in this paper, the NFL Green's function and self-tuned quantum criticality at finite values of $(M,N)$. We choose $\omega_0=1, m_0=2$, such that the dimensionless coupling $\omega_0/m_0$ is reasonably weak, and the pairing fluctuations discussed in Sec.~\ref{sec:IIB} do not significantly modify the normal state NFL behavior.

Fig.~\ref{fig:4} demonstrates the fermion and boson Green's functions obtained in QMC simulation. We focus on $G_b$ and $G_f$ obtained with $N=4M$, $\omega_0=1$, $m_0=2$ at $\beta=16$ for $M=2$, $M=3$ and $M=4$ respectively . Each data point is obtained by averaging over 20 disorder realizations in $\{t_{i\alpha,j\beta}\}$).  Fig.~\ref{fig:4} (a) and (b) are plotted in linear scale, and one can see the QMC curves are progressively close to the large-$N$ curve as $N$ increases. In Fig.~\ref{fig:4} (c) and (d), we present $G_b$ and $G_f$ versus ${\pi}/[{\beta\sin(\pi\tau/\beta)}]$ in a log-log scale, as suggested in Eq.~\eqref{eq:eq9}. It is clear that QMC results match very well with the large-$N$ result, and approach the latter as $N$ increases. The (rather small) error bars in Fig.~\ref{fig:4} denote the variance of the QMC results with different realizations of random couplings.~\footnote{Notice that this is not to be confused with the inherent error of the QMC simulations, whose magnitude is much smaller than the variance from disorder.} We see that such variance decreases upon increasing $N$. This is consistent with the self-averaging behavior of disordered systems and indicates our values of $M=3, N=12$ and $M=4, N=16$ can be reasonably regarded as close to ``large-$N$".

To further quantify the extrapolation to large-$N$, we simulated the model for $N=M$; such a parameter choice allows us to go deeper into the $M$ sequence, and the smaller system size also allows us to go to lower temperatures. In Fig.~\ref{fig:5} (a) and (c), we plot the QMC data of $G_b(\tau,0)$ and $G_f(\tau,0)$ with $M=N=9$, $\beta=24$, $m_0=2$ and $\omega_0=1$, averaged over 20 disorder realizations. As before we see the self-averaging behavior of disorder realizations. In Fig.~\ref{fig:5} (b) and (d), we plot the distance between the QMC disorder averaged Green's functions $G_{b}(\frac{\beta}{2},0)$ and $G_f(\frac{\beta}{2},0)$ and those from the large-$N$ analytical calculation $G^{Theory}_{b}(\frac{\beta}{2},0)$ and $G^{Theory}_{f}(\frac{\beta}{2},0)$. As $1/N\to 0$, indeed $G_b(\frac{\beta}{2},0)$ approaches its large-$N$ value. This indicates the replica-off-diagonal fluctuations are small and suppressed by $1/N$. As a result, glass behavior is absent in this model at least down to $\beta=24$.
 In the meantime, $G_f(\frac{\beta}{2},0)$ is quite close to its large-$N$ value, but remains slightly smaller up to $N=9$. Contrasting the behaviors of $G_b$ and $G_f$, it is tempting to attribute the deviation of $G_f$ to pairing fluctuations. This is consistent with the
fact that pairing fluctuations makes the fermions more incoherent, and that $T=1/24$ is quite close to the critical temperature with $T_c=1/36$, as shown in Fig.~\ref{fig:3} for the case of $N=M$.  We expect that as $N$ further increases, the effect of pairing fluctuations will be suppressed and the pairing transition at $N=\infty$ is mean-field like. We postpone a detailed study of the pairing transition to future work.

\begin{figure}[!htp]
 	\centering
 	\includegraphics[width=\columnwidth]{figure_6.pdf}
 	\caption{QMC results of $G_b$ and $G_f$ are at $M=4,N=16$, $m_0=\omega_0=1$, $\beta=16,20,24$. In log-log plot, we see that as the $\beta$ increases, large-$N$ results are basically unchanged, while curves of QMC progressively deviate from the large-$N$ value due to increasing finite-$N$ corrections.}
 	\label{fig:6}
 \end{figure}
\begin{figure}[!htp]
	\centering
	\includegraphics[width=\columnwidth]{figure_7.pdf}
	\caption{Self-tuned quantum criticality with different boson masses in log-log plot. (a) $G_{b}(\tau,0)$ from a free boson model with $m_0=1$ and $\beta=16$. The exponential decay in imaginary time is evident with $\ln(G_b(\tau=\beta/2)) \sim -7$. (b) and (c) show the $G_{b}(\tau,0)$ from the Yukawa-SYK model in Eq.\eqref{eq:eq1} with different mass $m_0=\omega_0$ [(b)] and $m_0=2\omega_0$ [(c)] with $\omega_0=1$ at $M=4,N=16$ and $\beta=16$. Blue dots are DQMC data and the red dashed lines are large-$N$ result. It is clear to see power law decay of $G_b$ at low-temperatures and long-time limit in log-log plot. These results reveal the self-tuned quantum criticality in our system.}
	\label{fig:7}
\end{figure}

We emphasize that the randomness of the Yukawa coupling is crucial in stabilizing the NFL behavior. To demonstrate this in Appendix~\ref{sec:appC} we consider a very similar model, in which the random coupling $t_{ij}$ is of lower rank and does not depend on $\alpha, \beta$. This model was analyzed by one of us~\cite{wang-2019} using the Schwinger-Dyson equation at large-$N$, and the analytical results is practically identical to those here. However, our QMC studies have found that its low-temperature phase is actually a spin glass, as the bosonic Green's function has a large static component. Somewhat counterintuitively, the glass phase absent in our present model is realized in such a ``less random" model. Indeed, one can show that in this model, certain replica-off-diagonal diagrams that are not suppressed by $1/N$ survives the disorder averaging, thanks to the lower-rank randomness in the Yukawa coupling, and are expected to drive the glass transition~\cite{wang-unpublished}. Therefore the model studied in Ref.~\cite{wang-2019} needs to be modified.

It is also interesting to investigate the evolution of finite $N$ corrections as a function of temperature. In the original SYK model, it is well known that the strength replica-off-diagonal fluctuations increases with lowering temperature as $\sim 1/N \log^2(T)$, which have led to initial speculations of a glass transition at exponentially low temperatures. On the other hand,  the strength of pairing fluctuations also increases with lowering temperature. To enable a clear analysis of the fluctuation effects, we numerically simulated the intermediate coupling regime of our model with $\omega_0=m_0=1$. Shown in Fig.~\ref{fig:6}, indeed we see that in this case indeed the deviation between numerical and large-$N$ results are more pronounced, and increases upon lowering the temperature. Furthermore, the finite-$N$ corrections modify $G_b$ upward and $G_f$ downward. The upward deviation in $G_b$ is consistent with replica-off-diagonal fluctuations, since they make $G_b$ more static, just as in a glass transition. On the other hand, the downward deviation in $G_f$ is likely to predominantly come from pairing fluctuations.

\subsection{Self-tuned quantum criticality}

As discussed in the Sec.~\ref{sec:IIA}, the self-tuned quantum criticality occurs independently of the bare boson mass $m_0$, at least at the large-$N$ limit. We numerically tested this expectation in QMC simulation with $M=4, N=16$. The results are shown in Fig.~\ref{fig:7}. As a comparison with our interacting model, Fig.~\ref{fig:7} (a) shows the bare boson Green's function generated from $H_b$ in Eq.~\eqref{eq:eq15} and the mass is $m_0=1$ and $\beta=16$. With such a mass term, the Green's function clearly exhibits exponential decay in imaginary time to $G_b(\tau=\beta/2,0)\approx0$. We can see it from the log-log plot: at the far left of the curve $\tau \rightarrow \beta$, which corresponds to the long-time limit, the value of $\ln(G_b(\tau,0))$ decays rapidly. However, once coupled with fermions in our model, as shown in Fig.~\ref{fig:7} (b) [$m_0=\omega_0=1$, averaged over 20 realizations] and (c) [$m_0=2\omega_0=2$, averaged over 20 realizations], with difference masses while keeping the $M=4, N=16$ and $\beta=16$, the boson Green's functions become critical. The Green's functions $G_b$ in imaginary time in  both cases do not decay exponentially, but instead are well consistent with the power-law form of Eq.~\eqref{eq:eq9}.  In (b) and (c), besides the QMC data, we plotted red dashed line which is large-$N$ result. The data in Fig.~\ref{fig:7} (c) turn out to be very close to the theoretical result. Remarkably, here we see that it does not require tuning the bare mass $m_0$ for the system to exhibit quantum-critical behavior, therefore exhibiting the self-tuned quantum criticality, consistent with analytical predictions at large-$N$.

\section{Discussion}
\label{sec:V}
In this work, we  performed unbiased sign-problem-free quantum Monte Carlo simulations of the Yukawa-SYK model, and reported direct evidence of self-tuned quantum-critical and NFL behaviors. We believe such SYK-like models provide a new venue to construct analytical solvable models for strange metals and unconventional superconductors. Our work serves as a starting point of further analyzing such models beyond the analytical large-$N$ limit, in a numerically unbiased manner. Further studies in several further directions are in order.

First, the numerical framework developed here allows one to incorporate the Hubbard $U$ interaction at half-filling without the fermion sign problem. From a theoretical point of view, such a generalized model likely exhibits a strange-metal to Mott insulator transition. It will also be interesting to study if a spin-glass phase can be realized in between, resembling the phase diagram of the underdoped cuprates. Second, recent works have revealed exotic quantum phase transitions between a strange metal and a trivial insulator as one varies the filling~\cite{wang-chubukov-2020}, but  analytical results have only been obtained in the weak-coupling limit. It is an open question whether more exotic phases exist at stronger coupling. Finally, the quantum dot model studied here can be generalized to a lattice model~\cite{XYSong2017,ZhenBi2017,Chowdhury2018}, in which more thermodynamical and transport properties can be examined.

In terms of numerical methodologies, the present work opens the directions of combining the randomness and all-connected models in the study of correlated electron systems, hence greatly broaden the scope of the correlated and itinerant systems. The Yukawa-SYK model and its QMC simulation provide a concrete example of NFL and give us the chance to have a systematic comparison with the large-$N$ analytical calculation. Therefore, one can certian foresee that more realistic and insightful NFL lattice models will eventually be solved with unbiased quantum many-body numerics as the one present here.

\section*{Acknowledgement}
We thank Yingfei Gu, Grigory Tarnopolsky, Subir Sachdev, Steven Kivelson for insightful discussions. G.P.P thanks Rui-Zhen Huang for helpful discussion on numerical calculations. We acknowledge the supports from the Ministry of Science and Technology of China through the National Key Research and Development Program (Grant No. 2016YFA0300502) and Research Grants Council of Hong Kong Special Administrative Region of China through 17303019. We thank the Center for Quantum Simulation Sciences in the Institute of Physics, Chinese Academy of Sciences, the Computational Initiative at the Faculty of Science and the Information Technology Services at the University of Hong Kong, the Platform for Data-Driven Computational Materials Discovery at the Songshan Lake Materials Laboratory, Guangdong, China and  the National Supercomputer Centers in Tianjin and Guangzhou for their technical support and generous allocation of CPU time. This research was initiated at the Aspen Center for Physics, supported by NSF PHY-1066293.

\begin{appendix}

	 \section{DQMC methodology}
	 \label{sec:iii}

	 The model described in Eq.~\eqref{eq:eq1} can be solved under the framework of determinant quantum Monte Carlo (DQMC)~\cite{BSS1981,Hirsch1983,Hirsch1985,AssaadEvertz2008,XiaoYanXuReview2019}. DQMC is the method of choice to study the interaction electron systems and has been used extensively in the past few decades in the addressing the problem such as Hubbard~\cite{Hirsch1985}, $t-J$~\cite{Brunner2000} and Kondo lattice~\cite{Assaad1999} models, and lately some great progress have been made in extending the DQMC scheme to interacting topological state of matter~\cite{Hohenadler2012,Meng2014}, duality and QCP beyond Landau-Ginzburg-Wilson paradigm~\cite{YYHe2016,YQQin2017}, and more relevant to this work, the designer Hamiltonians of critical boson coupled to fermions via Yukawa interactions~\cite{Berg2012,XiaoyuWang2017,XiaoYanXu2017,Gerlach2017,ZiHongLiu2018,ZiHongLiu2019,Berg2019,XiaoYanXuReview2019,YuzhiLiu2019,Bauer2020}. In this session, we will elucidate the DQMC setting for model in Eq.~\eqref{eq:eq1} in detail.

	 First, the partition function reads
	 \begin{eqnarray}
	 	&Z& =\operatorname{Tr}\left\{e^{-\beta \hat{H}}\right\} =\operatorname{Tr}\left\{\left( e^{-\Delta \tau \hat{H}} \right)^{L_{\tau}}\right\} \nonumber\\
	 	&=& \int (\prod_{\alpha \beta}  \mathrm{d \phi_{\alpha \beta}}) \operatorname{Tr}_{\mathbf{F}} \bra{\phi_{11}\cdots\phi_{NN}}( e^{-\Delta \tau \hat{H}})^{L_{\tau}}\ket{\phi_{11}\cdots\phi_{NN}}\nonumber\\
	 	\label{eq:eq14}
	 \end{eqnarray}
	 where we divide the imaginary time axis into $L_\tau$ slices, $\beta = L_{\tau} \times \Delta \tau$. Let the bosonic configuration at each time slice, $\vec{\Phi}_l=(\phi_{11,l},\phi_{12,l},\cdots,\phi_{N(N-1),l},\phi_{NN,l})$, serves as the complete basis of imaginary time propagation in the path-integral, then
	 \begin{equation}
	 	\begin{aligned} Z &=\int \left(\prod_{l=1}^{L_{\tau}} \mathrm{d} \vec{\Phi}_l\right)\\&\operatorname{Tr}_{\mathbf{F}} \bra{\vec{\Phi}_1}e^{-\Delta \tau \hat{H}}\ket{\vec{\Phi}_{L_{\tau}}} \bra{\vec{\Phi}_{L_{\tau}}}e^{-\Delta \tau \hat{H}}\ket{\vec{\Phi}_{L_{\tau}-1}}... \\
	 		& \;\; ...\bra{\vec{\Phi}_{2}}e^{-\Delta \tau \hat{H}}\ket{\vec{\Phi}_{1}}.
	 	\end{aligned}
	 \end{equation}
	 With the help of Suzuki–Trotter decomposition of the Hamiltonian in Eq.~\eqref{eq:eq1}, one has
	 \begin{equation}
	 	e^{-\Delta \tau \hat{H} }\approx e^{-\Delta \tau \hat{H}_{fb} } e^{-\Delta \tau \hat{H}_{b} }
	 \end{equation}
	 where
	 \begin{align}
	 	H_{fb}&=\sum_{i,j}^{M}\sum_{\alpha,\beta}^{N}\sum_{m,n}^{\uparrow\downarrow}\frac{i}{\sqrt{MN}} t_{i\alpha,j\beta}\phi_{\alpha\beta}c^\dagger_{\alpha i m}\sigma^z_{m,n}c_{\beta j n}\label{eq:eq14}\\
	 	H_{b}&=\sum_{\alpha , \beta=1}^N\(\frac{1}{2}\pi_{\alpha\beta}^2+\frac{m_0^2}{2}\phi_{\alpha\beta}^2\),
	 	\label{eq:eq15}
	 \end{align}
	 are the fermion-boson coupled term and the bosonic term, respectively.

	 \subsection{Bosonic Part}
	 Sine we use the space-time arrangement of the bosons $\{\vec{\Phi}_l\}$ to span the configuration space, we need to first express the canonical momentum $\pi_{\alpha\beta}$ in Eq.~\eqref{eq:eq15} in this configuration space. To this end, we first use the coherent state path integral
	 \begin{equation}
	 	\ket{\phi_{\alpha\beta}}=\frac{1}{\sqrt{2 \pi}} \int \mathrm{d \pi_{\alpha\beta}}\; e^{-i \pi_{\alpha\beta} \phi_{\alpha\beta}}\ket{\pi_{\alpha\beta}}
	 \end{equation}
	 then the momentum term in the partition function can be expressed as
	 \begin{equation}
	 	\begin{aligned}
	 		\bra{\phi'} e^{-\frac{1}{2} \Delta\tau \hat{\pi}^2} \ket{\phi}&=\frac{1}{2\pi} \int \mathrm{d \pi}\; e^{i \pi (\phi'-\phi)-\frac{1}{2}\pi^2 \Delta \tau}\\
	 		&\simeq C e^{-\frac{\left( \phi'-\phi\right)^2}{2 \Delta \tau}}
	 	\end{aligned}
	 \end{equation}
	 where $C$ is a constant, and $l$ and $l'$ are two consecutive time slices along the imaginary time axis, and the partition function then becomes
	 \begin{eqnarray}
	 	&Z&=\int \prod_{l=1}^{L_{\tau}} \mathrm{d} \vec{\Phi}_l \nonumber\\
	 	&&\underbrace{C^{L_{\tau}}( \prod_{l=1}^{L_{\tau}} \prod_{\alpha , \beta=1}^{N} e^{-\Delta \tau \frac{m_0^2}{2}\phi_{\alpha \beta,l}^2})(\prod_{\left<l,l'\right>} \prod_{\alpha , \beta=1}^{N} e^{-\frac{\left( \phi_{\alpha \beta,l}-\phi_{\alpha \beta,l'}\right)^2}{2 \Delta \tau}})}_{\mathcal{W}_b} \nonumber\\
	 	&&\underbrace{\operatorname{Tr}_{\mathbf{F}} \left\{ e^{-\Delta \tau \hat{H}_{fb}\left(\vec{\Phi}_{L_{\tau}}\right)}...\;e^{-\Delta \tau \hat{H}_{fb}\left(\vec{\Phi}_1\right)}\right\}}_{\mathcal{W}_{fb}}
	 	\label{eq:eq8}
	 \end{eqnarray}
	 where the first $\left( \ \right)$ in $\mathcal{W}_b$ contains the spatial boson interaction and the second $\left( \ \right)$ in $\mathcal{W}_b$ contains the temporal boson interaction with $\left<l,l'\right> $ stands for the nearest-neighbor interaction in imaginary time direction, and the $\operatorname{Tr}_{\mathbf{F}}$ in $\mathcal{W}_{fb}$ is the fermion trace we will deal with in Sec.2.B. It is now clear that the Monte Carlo sampling is performed in the bosonic field $\{\vec{\Phi}\}$ space of dimesion $N \times N \times L_{\tau}$ or $MN \times MN \times L_{\tau}$ if one consider the random hopping $t_{\alpha\beta}$ in $H_{fb}$, the configurational weight is comprised of the bosonic part $\mathcal{W}_b$ and the fermion determinant $\mathcal{W}_{fb}$.

	 \subsection{Fermion determinant}
	 \label{sec:IIIB}
	 For a specific bosonic configuration, the fermion determinant is of quadratic form and can be evaluated as that of the free system, following the standard expression
	 \begin{equation}
	 	\operatorname{Tr}_{\mathbf{F}} \left\{ e^{-\sum_{i,j}\hat{c}^\dagger_i A_{i,j}\hat{c}_j}e^{-\sum_{i,j}\hat{c}^\dagger_i B_{i,j}\hat{c}_j}\right\}=\mathrm{Det} \left(\mathbf{I}+e^{-\mathbf{A}}e^{-\mathbf{B}}\right).
	 \end{equation}
	 For the imaginary time propagation in the fermion trace in Eq.~\eqref{eq:eq8}, we define
	 \begin{equation}
	 	B\left(l_2 \Delta \tau, l_1 \Delta \tau \right)=\prod_{l=l_{1}+1}^{l_{2}} e^{-\Delta \tau V(\vec{\Phi}_{l})}
	 	\label{eq:eq23}
	 \end{equation}
	 where
	 \begin{equation}
	 	\begin{aligned}
	 		V(\vec{\Phi}_{l})&=\frac{i}{\sqrt{MN}}  \sigma^z_{2\times 2} \otimes \left( t_{i\alpha,j\beta}\phi_{\alpha \beta,l}\right)_{MN \times MN} .
	 	\end{aligned}
	 	\label{eq:eq11}
	 \end{equation}
	 It is interesting to note that in the conventional Hubbard-type model setting, there also exists a fermion hopping matrix on the exponential form, but since here we only have fermion Yukawa coupled with the bosonic field, that the hopping matrix is reduced to identical matrix, and the interaction matrix $V(\vec{\Phi}_l)$, which depends on the space-time configuration of the bosonic field $\{\vec{\Phi}_l\}$, contains both the randomness in hopping matrix $ \sigma^z_{2\times 2} \otimes (t_{i,j})_{M\times M}$ and the bosonic fluctuation matrix $(\phi_{\alpha \beta,l})_{N\times N}$. Such that after tracing out the fermion operators $c^{\dagger}_{\alpha i m}$ and $c_{\beta j n}$, the resulting fermion determinant is the determinant of matrices with size $MN \times MN$ and block diagonal in the fermion spin space of $\sigma^{z}$.

	 With these notations prepared, finally the partition function in Eq.~\eqref{eq:eq8} can now be written as
	 \begin{eqnarray}
	 	&Z&=\int \prod_{l=1}^{L_{\tau}} \mathrm{d} \vec{\Phi}_l \nonumber\\
	 	&&\underbrace{C^{L_{\tau}}(\prod_{l=1}^{L_{\tau}} \prod_{\alpha , \beta=1}^{N} e^{-\Delta \tau \frac{m_0^2}{2}\phi_{\alpha \beta,l}^2})(\prod_{\left<l,l'\right>} \prod_{\alpha , \beta=1}^{N} e^{-\frac{\left( \phi_{\alpha \beta,l}-\phi_{\alpha \beta,l'}\right)^2}{2 \Delta \tau}})}_{\mathcal{W}_{b}}\nonumber\\
	 	&&\underbrace{\operatorname{Det}[\mathbf{1}+B(L_\tau \Delta \tau,\(L_\tau-1\)\Delta \tau)\cdots B(\Delta \tau,0)]}_{\mathcal{W}_{fb}}
	 	\label{eq:eq25}
	 \end{eqnarray}
	 This is the partition function describing the SYK-Yukawa model in Eq.~\eqref{eq:eq1} and we can now simulate it in DQMC.

	 \subsection{Free from sign problem}
	 \label{sec:IIIC}
	 As aforementioned, the partition function in Eq.~\eqref{eq:eq25} is free from the minus-sign problem in the protection of a time-reversal symmetry~\cite{CJWu2005}, i.e., the Hamiltonian is invariant under such a symmetry operation, this can be easily demonstrated as follows.

	 First, we note
	 \begin{equation}
	 	\begin{aligned}
	 		H_{fb}=\sum_{i,j=1}^{M}\sum_{\alpha,\beta=1}^{N}&\frac{i}{\sqrt{MN}} t_{i\alpha,j\beta}\phi_{\alpha \beta}c^\dagger_{\alpha i \uparrow}c_{\beta j \uparrow} \\
	 		-&\frac{i}{\sqrt{MN}} t_{i\alpha,j\beta}\phi_{\alpha \beta}c^\dagger_{\alpha i \downarrow}c_{\beta j \downarrow}
	 	\end{aligned}
	 \end{equation}
	 and time-reversal symmetry operator is $\mathcal{T}=i\sigma_y \mathcal{K}$. Its operation works as $\mathcal{T} c_{m} \mathcal{T}^{-1} = U_{mn} c_{n} $ , $\mathcal{T} c_{m}^\dagger \mathcal{T}^{-1} = U^{*}_{mn} c_{n}^\dagger  $ , $\mathcal{T} i \mathcal{T}^{-1} = -i $, where $m,n=\uparrow / \downarrow$, $U=i\sigma_y$, then
	 \begin{equation}
	 	\begin{aligned}
	 		\mathcal{T} H_{fb} \mathcal{T}^{-1}&=\sum_{i,j=1}^{M}\sum_{\alpha,\beta=1}^{N}-\frac{i}{\sqrt{MN}} t_{i\alpha,j\beta}\phi_{\alpha \beta}c^\dagger_{\alpha i \downarrow}c_{\beta j \downarrow}\\
	 		&\qquad  \qquad+\frac{i}{\sqrt{MN}} t_{i\alpha,j\beta}\phi_{\alpha \beta}c^\dagger_{\alpha i \uparrow}c_{\beta j \uparrow} \\
	 		&=H_{fb},
	 	\end{aligned}
	 \end{equation}
	 therefore $H_{fb}$ is invariant under $\mathcal{T}$.

	 Next, notice that $V\(\vec{\Phi}_l\)$ is block diagonal in the space of $m,n=\uparrow,\downarrow$, then the fermion determinant can be written as
	 \begin{equation}
	 	\begin{aligned}
	 		\operatorname{Det}&[\mathbf{1}+B(\beta, 0)]\\
	 		&=\operatorname{Det}[\mathbf{1}+B^{\uparrow}(\beta, 0)] \operatorname{Det}[\mathbf{1}+B^{\downarrow}(\beta, 0)]\\
	 		&=\operatorname{Det}[\mathbf{1}+B^{\uparrow}(\beta, 0)] \operatorname{Det}[\mathcal{T}\(\mathbf{1}+B^{\downarrow}(\beta, 0)\)\mathcal{T}^{-1}]^{*}\\
	 		&=\operatorname{Det}[\mathbf{1}+B^{\uparrow}(\beta, 0)] \operatorname{Det}[\mathbf{1}+B^{\uparrow}(\beta, 0)]^{*}\\
	 		&=\left | \operatorname{Det}[\mathbf{1}+B^{\uparrow}(\beta, 0)] \right |^2
	 	\end{aligned}
	 \end{equation}
	 and it is positive definite. Also note that the boson weight $\mathcal{W}_b$ is positive definite as the $\{\vec{\Phi}\}$ is the eigenstate of the $H_b$ in the space-time. So the entire configurational weight is positive definite and there is no sign-problem for the simulation.

	 \subsection{Update and measurement}
	 Another important ingredient in any Monte Carlo simulation is the update scheme between configurations, here since the bosonic fields are continuous variables, we have to adapt to local update with Metropolis-type acceptance rate.

	 The ensemble average of physical observable can be expressed as:
	 \begin{equation}
	 	\langle\hat{O}\rangle=\frac{\operatorname{Tr}\left\{e^{-\beta \hat{H}} \hat{O}\right\}}{\operatorname{Tr}\left\{e^{-\beta \hat{H}}\right\}}=\int \left(\prod_{l=1}^{L_{\tau}} \mathrm{d} \vec{\Phi}_l\right) \mathcal{P}_{\mathcal{C}}\langle\hat{O}\rangle_{\mathcal{C}}+O\left(\Delta \tau^{2}\right)
	 \end{equation}
	 where $\Delta\tau^2$ systematical error comes from the Trotter decomposition and the weight and expectation value for each bosonic field configuration $\mathcal{C}$ are
	 \begin{eqnarray}
	 	\mathcal{P}_{\mathcal{C}}&=& \frac{\mathcal{W}_{\mathcal{C}}^{fb} \operatorname{Det}[1+B_{\mathcal{C}}(\beta, 0)]}{\int \left(\prod_{l=1}^{L_{\tau}} \mathrm{d} \vec{\Phi}_l\right) \mathcal{W}_{\mathcal{C}}^{fb} \operatorname{Det}[1+B_{\mathcal{C}}(\beta, 0)]} \\\langle\hat{O}\rangle_{\mathcal{C}}&=& \frac{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, \tau) \hat{O} \hat{U}_{\mathcal{C}}(\tau, 0)\}}{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, 0)\}},
	 \end{eqnarray}
	 where
	 \begin{equation}
	 	\hat{U}\left(l_2 \Delta \tau, l_1 \Delta \tau \right)=\prod_{l=l_{1}+1}^{l_{2}} e^{-\Delta \tau \mathbf{\hat{c}^{\dagger}} V(\vec{\Phi}_{l}) \mathbf{\hat{c}}}
	 	\label{eq:eq32}
	 \end{equation}
	 here $\mathbf{\hat{c}}$ has $2\times M \times N $ components, so does the dimension of the matrix $V$. Once tracing out the quadratic fermions $\mathbf{\hat{c}}$ in Eq.~\eqref{eq:eq32}, one arrives at the $B(l_2\Delta\tau, l_1\Delta\tau)$ matrix in Eq.~\eqref{eq:eq23}, and the evaluation of fermion determinant follows from there down to Eq.~\eqref{eq:eq25}. The detailed derivation of physical observables, exemplified by the equal time and imaginary time displaced fermionic Green's functions, are given in Supplementary Information (SI).

	 Moreover, since the coupling matrix $t_{ij}$ in $H_{fb}$ is subject to randomness, the aforementioned Monte Carlo sample is performed for each disorder realization. Therefore, besides the Monte Carlo average over a fixed disorder configuration, the final physical observables such as the fermion and boson Green's functions are the disordered averaged quantities.

	\section{Monte Carlo measurements}
	\label{app:appA}

	The ensemble average of physical observables, in the DQMC formalism, can be calculated as,
	\begin{eqnarray}
		\langle\hat{O}\rangle_{\mathcal{C}} &=&\left.\frac{\partial}{\partial \eta} \ln \operatorname{Tr}\left[\hat{U}_{\mathcal{C}}(\beta, \tau) e^{\eta \hat{O}} \hat{U}_{\mathcal{C}}(\tau, 0)\right]\right|_{\eta=0} \nonumber\\
		&=&\left.\frac{\partial}{\partial \eta} \ln \operatorname{Det}\left[\mathbf{1}+B_{\mathcal{C}}(\beta, \tau) e^{\eta O} B_{\mathcal{C}}(\tau, 0)\right]\right|_{\eta=0} \nonumber\\
		&=&\left.\frac{\partial}{\partial \eta} \operatorname{Tr}\ln \left[\mathbf{1}+B_{\mathcal{C}}(\beta, \tau) e^{\eta O} B_{\mathcal{C}}(\tau, 0)\right]\right|_{\eta=0} \nonumber\\
		&=&\operatorname{Tr}\left[B_{\mathcal{C}}(\tau, 0)\left(1+B_{\mathcal{C}}(\beta, 0)\right)^{-1} B_{\mathcal{C}}(\beta, \tau) O\right] \nonumber\\ &=&\operatorname{Tr}\left[\left(1-(1+B_{\mathcal{C}}(\tau, 0) B_{\mathcal{C}}(\beta, \tau))^{-1}\right) O\right]
		\label{eq:eqA1}
	\end{eqnarray}
	in the case of equal time fermionic Green's function, $\hat{O}=\mathbf{\hat{c}}^{\dagger} O \mathbf{\hat{c}}$. $\hat{U}_{\mathcal{C}}$ and $B_{\mathcal{C}}$ are defined in Eq.(23) and Eq.(32) in the main text, respectively.

	For the imaginary time displaced fermionic Green's function, $G_{f,ij}(\tau,0)= \langle c_i(\tau) c_j^\dagger (0) \rangle $ where $i,j$ encapsulate the dot, flavor and spin indices in the Hamiltonian in Eq.(1) in the main text and the imaginary time $\tau\in[0,\beta]$, it can be evaluated in DQMC as
	\begin{eqnarray}
		\langle c_i(\tau) c_j^\dagger (0) \rangle&=&\frac{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, \tau)\; \hat{c}_i\; \hat{U}_{\mathcal{C}}(\tau, 0)\hat{c}_j^\dagger\}}{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, 0)\}}\nonumber\\
		&=&\frac{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, 0)\;[\hat{U}^{-1}_{\mathcal{C}}(\tau, 0) \hat{c}_i\; \hat{U}_{\mathcal{C}}(\tau, 0)]\hat{c}_j^\dagger\}}{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, 0)\}} \nonumber\\
		&=&\sum_k B_{\mathcal{C}}(\tau, 0)_{ik} \frac{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, 0)\; \hat{c}_k \hat{c}_j^\dagger\}}{\operatorname{Tr}\{\hat{U}_{\mathcal{C}}(\beta, 0)\}} \nonumber\\
		&=&[B_{\mathcal{C}}(\tau, 0)(1+B_{\mathcal{C}}(\beta, 0))^{-1}]_{ij}
		\label{eq:eqA2}
	\end{eqnarray}
	where the intermediate steps in Eq.~\eqref{eq:eqA2} are given explicitly in Ref.~\cite{Hirsch1985}.

		   \begin{figure}[!htp]
	   	\centering
	   	\includegraphics[width=\columnwidth]{figure_8.pdf}
	   	\caption{DQMC results of Matsubara Green function $G_b$ at $M=N$, $m_0=\omega_0=1$, $\beta=16$. We plot the zero frequency component and its errorbar. Galssy behaviors are seen.}
	   	\label{fig:8}
	   \end{figure}
	\section{Glass behavior in a less-random model}
	\label{sec:appC}
	We construct a model in a similar form in which the random coupling is of a lower rank,
	 \begin{equation}
	 	\begin{aligned}
	 		H =& \sum_{i,j=1}^{M}\sum_{\alpha,\beta=1}^{N} \sum_{m,n}^{\uparrow,\downarrow}\(\frac{i}{\sqrt{MN}} t_{i,j}\phi_{\alpha\beta}c^\dagger_{i \alpha;m}\sigma^z_{m,n}c_{j\beta ;n}
	 		\) \\
	 		&+\sum_{\alpha , \beta =1}^N\(\frac{1}{2}\pi_{\alpha\beta}^2+\frac{m_0^2}{2}\phi_{\alpha\beta}^2\),
	 	\end{aligned}
	 \end{equation}
where the random coupling between fermion and boson is realized as $\left\langle t_{i j}\right\rangle=0,\left\langle t_{i j} t_{k l}\right\rangle=\left(\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right) \omega_{0}^{3},$ . Still for the sake of simplicity we set $\omega_{0}=1$ as the energy unit throughout the paper, and the temperature scale is then $T \equiv \omega_{0} / \beta . \pi_{\alpha \beta}$ is the canonical momentum of $\phi_{\alpha \beta}$. Hermiticity of the first term requires $\phi_{\alpha \beta}=-\phi_{\beta \alpha}$.

{In Fig.~\ref{fig:8} , we show the static component (with $\omega_n=0$) for bosonic Green's function  $G_b(\omega_n)$. In the large-$N$ limit, this component can be regarded as an Edwards-Anderson order parameter of the spin glass phase~\cite{fu-sachdev-2016}. As $N$ increases, the static component, along with its variance for different disorder realizations,  increases with the increase of $N$ at $M=N, \beta=16, m_0=\omega_0=1$, which is indicative of a spin glass behavior. }

\end{appendix}

\bibliographystyle{apsrev4-1}

redpajama_set_name: RedPajamaArXiv


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established of old; thou art from everlasting.
have lifted up their voice; the rivers are flowing with purity.
the noise of many waters, yea, than the mighty waves of the sea.
holiness becomes thy house, O Mar-Yah, for ever.
O come, let us sing to Mar-Yah; let us make a joyful noise to the God of our salvation.
come before his presence with thanksgiving, and make a joyful noise to him with psalms.
Mar-Yah is a great God and a great King above all gods.
earth; the strength of the hills is his also.
O come, let us worship and bow down; let us kneel before Mar-Yah our Maker.
I have set Mar-Yah always before me; because he is at my right hand, I shall not be moved.
Therefore my heart is glad and my glory rejoices; my flesh also shall rest in hope.
left not my soul in Sheol; neither hast thou suffered thy Holy One to see corruption.
victory of thy right hand.
arm have gotten him the victory.
he revealed in the sight of the heathen.
house of Israel; all the ends of the earth have seen the salvation of our God.
Praise Mar-Yah, all the earth; make ye merry and sing and give praise.
the harp, with the harp and the voice of a choir.
comes to judge the earth; with righteousness shall he judge the world, and the people with equity.
Praise Mar-Yah. Praise God in his sanctuary; praise him in the firmament of his power.
the sound of the trumpet; praise him with the psaltery and harp.
pipe; praise him with sweet stringed instruments.
Let every thing that has breath praise Mar-Yah. Praise Mar-Yah.
Our Father in heaven, may your name be sanctified.
10 Your Kingdom come. Your will be done, as in heaven, so on earth.
11 Give us the bread of our need this day.
12 Forgive us our offences, as we have forgiven those who have offended us.


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Play Senior Futsal (Indoor Soccer)
Adult's futsal is an established competition at the Copper Coast Sports and Leisure Centre. It is played throughout the whole year, with a maximum of three weeks rest between competitions. Futsal is played Monday night during winter and Thursday night during summer.
The first game begins at 6:30pm and, depending on the amount of teams registered, the last game can finish at 9:30pm.
The cost is $40 per team, and payment must been made prior to the game commencing.
If you're looking to develop fitness and have fun, while having strong competition simply call up five friends, and register before next season!
Download your team nomination form and email the form to coppercoast@belgravialeisure.com.au and one of our friendly staff will get in touch with you.
Alternatively, team nominations forms can be collected from reception.
The next intake for Futsal begins in September, ensure that you nominate your team for the next competition!
Download Form Code of Conduct

redpajama_set_name: RedPajamaCommonCrawl


SBR: Unlike in the case of Vietnam and Sri Lanka, the book I picked up to get an overview of French history turned out to be suitable for this purpose. A Brief History of France is readable with clear chronology and just the right amount of information. If you are not familiar with European history in general, you would have to do an Internet search once in a while for related events or people. But that is bound to happen while reading any book on history. There is no way an author can account for what all their readers do or do not know already. Don't expect thoroughness, but everything major seems to have been covered. It does lean a bit too heavily towards modern history and compresses the ancient one too much. But that is a minor complain when the aim to get a quick and brief overview.
To read or not to read: Yes – as a good starting point for or as a quick overview of French history.


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REIT News: MFA Financial, Inc. (NYSE: $MFA) Provides Status Update on Status of Financing Arrangements as of March 23, 2020
Alert - Stock trading at $0.983+0.623 (+173.0555%) As of 11:56AM EDT
NEW YORK - March 25, 2020 (Investorideas.com Newswire) MFA Financial, Inc. (NYSE: MFA) (the "Company") announced that in recent weeks, due to the turmoil in the financial markets resulting from the global pandemic of the COVID-19 virus, the Company and its subsidiaries have received an unusually high number of margin calls from financing counterparties, and have also experienced higher funding costs in respect of its repurchase agreements. At the close of business on March 23, 2020, the Company did not meet its margin calls. Further, on March 23, 2020, the Company notified its financing counterparties that it does not expect to be in a position to fund the anticipated volume of future margin calls under its financing arrangements in the near term as a result of market disruptions created by the COVID-19 pandemic. The Company estimates that, as of March 20, 2020, the Company's aggregate obligations under its various financing arrangements is approximately $9.5 billion.
In light of the events and conditions described above, the Company is engaged in discussions with its financing counterparties with regard to entering into forbearance agreements pursuant to which each counterparty would agree to forbear from exercising its rights and remedies with respect to an event of default under the applicable financing arrangement for an agreed-upon period. The Company cannot predict whether its financing counterparties will enter into a forbearance agreement, the timing of any such agreement, or the terms thereof.
Under the terms of the Company's financing arrangements, if the Company fails to deliver additional collateral or otherwise meet margin calls when due, the counterparties may demand immediate payment by the Company of its aggregate outstanding financing obligations and/or take ownership of the securities securing the Company's financing obligations.
The Company has engaged Hunton Andrews Kurth LLP and Quinn Emanuel Urquhart & Sullivan, LLP as legal counsel and FTI LLC as financial advisor in connection with its financings and related matters.
MFA Financial, Inc. is a real estate investment trust primarily engaged in the business of investing, on a leveraged basis, in residential mortgage assets, including residential mortgage-backed securities and residential whole loans.
When used in this report or other written or oral communications, statements which are not historical in nature, including those containing words such as "will," "believe," "expect," "anticipate," "estimate," "plan," "continue," "intend," "should," "could," "would," "may" or similar expressions, are intended to identify "forward-looking statements" within the meaning of Section 27A of the Securities Act and Section 21E of the Exchange Act, and, as such, may involve known and unknown risks, uncertainties and assumptions. Statements regarding the following subjects, among others, may be forward-looking: the Company's estimates of its outstanding borrowings under its financing arrangements, uncertainties related to negotiations with the Company's financing counterparties including with respect to any forbearance agreement and the timing of any such agreement or the terms thereof. Forward-looking statements are based on estimates, projections, beliefs and assumptions of management of the Company at the time of such statements and are not guarantees of future performance. Forward-looking statements involve risks and uncertainties in predicting future results and conditions. Actual results and outcomes could differ materially from those projected in these forward-looking statements due to a variety of factors, including, without limitation, the Company's ability to accurately estimate its outstanding borrowings under its financing arrangements, uncertainties related to negotiations with the Company's financing counterparties including with respect to any forbearance agreement, the timing of any such agreement or the terms thereof, changes in interest rates, changes in default rates, changes in the yield curve, changes in prepayment rates, the availability and terms of financing, changes in the market value of our assets, general economic conditions, conditions in the market for our residential mortgage backed securities, residential whole loans, residential mortgage securities, MSR-related assets and other assets, the timing and amount of distributions to our stockholders, our ability to maintain our exemption from registration under the Investment Company Act of 1940, as amended, conditions in the real estate market, legislative and regulatory changes that could adversely affect the business of the Company and the ongoing spread and economic effects of the novel coronavirus (COVID-19). Additional information concerning these and other risk factors are contained in the Company's filings with the Securities and Exchange Commission, including our most recent Annual Report on Form 10-K and subsequent filings. All information in this press release is as of March 23, 2020. The Company undertakes no duty to update any forward-looking statements to reflect any change in its expectations or any change in events, conditions or circumstances on which any such statement is based.
MFA Investor Relations
www.mfafinancial.com
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I've been tagged by Alan to post 5 Things I Dig About Jesus. At first I thought, "Wow, this will be so fun!" but then I thought, "How in the world am I going to limit this to just five things!" :) (I tend to labor too much over simple things).
Well, I'll just have to prove myself wrong!
2. God did not send Jesus into the world to condemn the world, but that the world through Him might be saved.
Wow... that was easy and fun!
Now... I gotta tag 5 peeps.
Here is my list. I posted this on my message board, hopefully I will get a few responses.
1. It's finished. I'm forgiven and there is no going back.
2. He has overcome the World. No matter what I am going through He has already won the battle for me.
3. Because of His gift of grace, I am who I am.
4. I find peace and rest in Jesus.
5. By Him I cry, "Abba, Father".
Thanks for starting this thing! It's been wonderful to follow the trails and see what other people have written.
Thanks for sharing your list here and on your message board! I hope others will take the time to read and participate.
Thanks. I've enjoyed following this too!


@thebiglybestestdoggie: Just sitting around, chilling with my boy. Or a bird. Not really sure. Smells like a human, but looks like a mighty fine dinner prospect… Wonder if this is one of those goofy sphinxes I heard about? Hope he takes his head off soon so I know for sure.
Oh my goodness that dog is me LOL I get super anxious around people in those giant heads/full body costumes, it's always been one of my more irrational fears.


The Coalition of Progressive Electors is calling for Vancouver City Council to embrace specific targets that will make elections in Vancouver more transparent, more equitable, and less dominated by big money. While supporting the spirit of Councilor Reimer's motion calling for reform, COPE says that the request to the BC Government should be more specific.
COPE is proposing that the Provincial Government amend the Vancouver Charter by imposing a $1.50 per registered voter spending limit for electoral organizations along with a per person contribution limit of $500.
However, Woodsworth noted that the Provincial government has ignored all three Vancouver parties when they called for changes. "This government ignored COPE in 2005, the NPA in 2007, and Vision in 2010. The issue isn't Vancouver, the issue is the Province." Woodsworth suggested that if the Province turns down the city again, it might be time to have major parties enter into a voluntary arrangement based on city by-laws.


Comments: Why did so many people leave northern NM for the San Luis Valley?
Comments: I can only go by what my parents told me. They left Las Vegas/Sapello in 1924 because there was no work for the men folk. Many were hired by Great Western sugar beet company in northern Colorado to plant, tap, and harvest the sugar beet crops there. They stayed on after that work because there was an abundance of farm labor work. I believe the same holds true for The Valley.


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Franklin Graham and the Billy Graham Evangelistic Association have consistently used current technology to spread the Gospel to the ends of the earth. The Billy Graham TV Specials are half-hour, ministry-focused programs covering various initiatives. Each program highlights how the Gospel is impacting people around the world. The Billy Graham TV Specials are broadcast on many Christian networks. See recent episodes and get involved online at www.BillyGraham.tv.
Billy Graham TV Specials Billy Graham Evangelistic Association
Ready for Opportunity
A chaotic year has left the world reeling in its wake. In this time of uncertainty how should we respond? Follow the unfolding action as Franklin Graham and other believers wrestle with how to react in these difficult days.
Canvas Cathedral
Will Graham goes behind the scenes of a Hollywood film as he portrays his grandfather, Billy Graham. Follow Will, as he investigates a crisis of faith that nearly sidelined his grandfather's early ministry.
A skeptic, a struggling teen, and a woman hiding from her past reach a moment of truth. Watch their stories, interwoven with a powerful message from Franklin Graham on life's most important decision.
The Burning Search
All mankind is searching for truth and happiness. Explore the powerful tool that reaches people at their moment of need.
CHANGE: NEEDING MORE
We all want change for the better. But what if you can't get the change you need most? Follow three lives, on the verge of disaster, that desperately needed something more.
The Cross (2020)
Billy Graham, in one of his final recordings, brings a timeless message of hope and redemption in Jesus Christ. "With all my heart, I want to leave you with the truth that He loves you and wants to forgive you of your sins." Featuring illusionist Jim Munroe, and Platinum recording artist Lacey Sturm.
© BGEA
The Bold Moms Podcast
Brianna Chapman
GPS: God. People. Stories.
Anne Graham Lotz - Living in the Light
In Touch Ministries Daily Devotions
CBN.com - The 700 Club - Video Podcast
The Christian Broadcasting Network
Christian Quotes | Encouragement for Christians
Ryan Maher
PowerPoint on Oneplace.com


A 21 year-old Gisborne boy, Thomas Kini, is a member of the Maori "Hi Quins", a sister group to the highly successful Maori "Hi Fives".
Lead guitarist for the group, which performs both instrumental and vocal items, Thomas is a brother of Martin Kini and Barney Taihuka of local "Kini Quartet" fame.
Started in 1960 by Jimmy Rivers, the group worked in Wellington for three months before travelling to Surfers Paradise for an engagement at the Chevron Hotel.
Since then, they have not looked back.
So successful were they on their first overseas engagement, that they remained at the Chevron for two years.
From there, they went to Sydney, appearing on many notable shows, and later touring Australia.
Next stop was further afield, to London, where the boys were an immediate hit performing in the smash season of "Tropical Paradise". It was here that visiting American celebrity Sammy Davis jnr. saw their show and arranged to meet the quintet backstage afterwards.
He invited them to his "Mr Wonderful" show, and recommended their act to the manager or the "Sands Hotel" in Las Vegas, where they will open in 1965.
The "Hi Quins" can feel justly proud of Davis's recommendation, as it is the first the star has made for any group in ten years.
From London they played to the U.S. Forces in Germany, and also toured Spain and Sweden, where they cut six recordings for release throughout Scandinavia.
They are at present playing in Blackpool, the entertainment capital of Europe, for the summer season.
After their stay there, they will move on to Canada, New York, and Las Vegas, and then back home to New Zealand.
Thomas Kini (left) with Lonnie Donnegan and Jimmy Rivers.
Lyn Rogers and the Maori "Hi Quins". Thomas Kini is at left.


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What are the 3 parts of the axillary artery?
1 What are the 3 parts of the axillary artery?
2 What muscles does the axillary artery supply?
3 Where does axillary artery begin and end?
4 Is there a major artery under your armpit?
5 What separates axillary artery?
6 How do you approach an axillary artery?
7 How deep is the axillary artery?
8 What organ is under left armpit?
9 What is the function of the axillary artery?
10 What does the axillary artery turn into?
11 When should I worry about left side pain?
12 What are the signs that you have a cancerous lymph node?
13 What is meant by axilla?
14 Where is the axillary artery located?
15 How can you tell the difference between an artery and a vein?
16 How do you know if you have an axillary vein?
17 Where is main artery in left arm?
18 Where is the axillary artery located in the human body?
19 Which is a synonym for arteria axillaris [ Ta ]?
20 What are the hallmarks of an axillary artery injury?
21 What is the treatment for an axillary artery aneurysm?
It has three branches that are variable in their order: the subscapular trunk, the anterior humeral circumflex artery, and the posterior humeral circumflex artery. The subscapular trunk is typically the largest branch of the axillary artery.
What muscles does the axillary artery supply?
trapezius muscle.
latissimus dorsi muscle.
levator scapulae muscle.
rhomboid minor muscle.
rhomboid major muscle.
pectoralis major muscle.
pectoralis minor muscle.
serratus anterior muscle.
Where does axillary artery begin and end?
The axillary artery is a continuation of the subclavian artery that begins at the outer border of the first rib. It then courses through the axilla while being bordered by the lateral (superiorly), posterior (posteriorly), medial (inferiorly) cords of the brachial plexus and the ansa pectoralis (anteriorly).
Is there a major artery under your armpit?
Axillary Artery: This is a continuation of the subclavian artery. This artery travels deep in the arm pit, feeding muscles and bones around the shoulder with its branches. It eventually turns into the brachial artery.
What separates axillary artery?
The pectoralis minor muscle is used as a landmark for dividing the axillary artery into three parts. In human anatomy, the axillary artery is a large blood vessel that conveys oxygenated blood to the lateral aspect of the thorax, the axilla (armpit) and the upper limb.
How do you approach an axillary artery?
The approach to the axillary artery is made via an infraclavicular incision that can easily be extended down the medial aspect of the arm if exposure of the brachial artery is necessary.
How deep is the axillary artery?
The axillary artery and vein are exposed using a longitudinal incision positioned approximately one fingerbreadth below the inferior border of the clavicle. The fibers of the pectoralis major muscle are split. The axillary vein lies below the muscle. The vein is mobilized for a distance of approximately 5 or 6 cm.
What organ is under left armpit?
The cells within your lymph nodes help to fight infection. If you develop a significant viral or bacterial infection in your body, your lymph nodes can swell and cause pain under your armpit.
What is the function of the axillary artery?
In human anatomy, the axillary artery is a large blood vessel that conveys oxygenated blood to the lateral aspect of the thorax, the axilla (armpit) and the upper limb.
What does the axillary artery turn into?
In human anatomy, the axillary artery is a large blood vessel that conveys oxygenated blood to the lateral aspect of the thorax, the axilla (armpit) and the upper limb. Its origin is at the lateral margin of the first rib, before which it is called the subclavian artery….
Axillary artery
FMA 22654
Anatomical terminology
When should I worry about left side pain?
Even if it isn't anything dangerous, it's still better to be sure." Importantly, if you notice you're having severe pain, fever, swelling and tenderness of the abdomen, bloody stools, yellowing of the skin or persistent nausea and vomiting, see a doctor immediately.
What are the signs that you have a cancerous lymph node?
What Are Signs and Symptoms of Cancerous Lymph Nodes?
Lump(s) under the skin, such as in the neck, under the arm, or in the groin.
Fever (may come and go over several weeks) without an infection.
Drenching night sweats.
Weight loss without trying.
Itching skin.
Feeling tired.
Loss of appetite.
What is meant by axilla?
: the cavity beneath the junction of a forelimb and the body especially : armpit.
Where is the axillary artery located?
How can you tell the difference between an artery and a vein?
The main difference between nerve and vein is that nerve is an axon bundle of neurons in the peripheral nervous system, which carries nerve impulses whereas veins are blood vessels, which return deoxygenated blood towards the heart. Both nerves and veins vary in size.
How do you know if you have an axillary vein?
The axillary vein can be visualized simply as a bluish cord while dissecting yellowish fat globules around it after opening the fibrous fascia. The axillary vein is then dissected downwards and laterally along its course towards the arm.
Where is main artery in left arm?
The brachial artery is a major blood vessel located in the upper arm and is the main supplier of blood to the arm and hand. The brachial artery continues from the axillary artery at the shoulder and travels down the underside of the arm.
Where is the axillary artery located in the human body?
[edit on Wikidata] In human anatomy, the axillary artery is a large blood vessel that conveys oxygenated blood to the lateral aspect of the thorax, the axilla (armpit) and the upper limb. Its origin is at the lateral margin of the first rib, before which it is called the subclavian artery.
Which is a synonym for arteria axillaris [ Ta ]?
Synonym (s): arteria axillaris [TA] . The continuation of the subclavian artery beyond the lateral border of the first rib; its name changes to the brachial artery when it passes the lower border of the tendon of the teres major muscle.
What are the hallmarks of an axillary artery injury?
However, in contrast to isolated subclavian artery injuries—in which the patient will present in shock—isolated injuries to the axillary artery rarely present with hemodynamic collapse. More common hallmarks include absent distal pulse or reduced injured extremity index (<0.9), pulsatile bleeding, and/or an expanding hematoma.
What is the treatment for an axillary artery aneurysm?
The definitive treatment of an axillary artery aneurysm is surgical. It involves excising the aneurysm and reconstructing the vessel wall using a vascular graft. The brachial artery is a continuation of the axillary artery past the lower border of the teres major. It is the main supply of blood for the arm.
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Helping those who might make help difficult
Clinical Practice | Gerald Masley
Erin Wasinger
Gerald Masley's patients aren't always appreciative.
For the last 11 years, Masley's been a registered nurse at the largest men's prison in the state. Central Michigan Correctional Facility has two on-site clinics that employ about a dozen nurses. The clinics provide emergency treatments and preventative care to the roughly 2,280 men housed there. His dedication to the job has merited him a Salute to Nurses award for Clinical Nursing.
A typical shift starts with blood work, then sick calls for complaints such as rashes, colds or chest pain. Staff perform annual exams on patients daily, too. Within those parameters, the number of inmates ensures his days are varied.
"You get to do and see a lot of different things," he said.
Having worked in other healthcare settings since entering the field in 1993, Masley said he finds the prison clinic requires more autonomy; a doctor isn't always on site.
"You have to make decisions for yourself and sometimes you're the only person working," such as on weekends or holidays. "Nurses are the front line. We see everything first. A lot of times, we can handle it ourselves." A doctor is called when the problem is more complicated.
Masley said his experience in a variety of other areas, from cardiac to psychiatry – including a supervisory role – prepared him for his job at the correctional facility. This has influenced his coworkers, too: "Jerry inspires confidence in our staff," the nominator writes. He's tasked with training new employees and he's been an American Heart Association-certified CPR instructor since 1997.
The setting is stressful, though. His nomination letter speaks to Masley's ability to keep a positive attitude where others find it difficult.
"I have the most seniority at 11 years. It can be a hard job," Masley said. "Prisoners aren't always appreciative. They're not always very compliant for what's in their best interest. That can be frustrating."
Finding fulfillment on the job comes from doing his job well, he said. Staff are told not to get "over-familiar" with prisoners, but Masley said it's no excuse not to be polite and thorough.
"Sometimes you can get in the trap of seeing (patients) as prisoners and not seeing them as patients, because they are in that setting," he said. "What I found what works best for me, I try not to know why they're there. I want to treat everyone the same. I try to give the same level of care that I would anyone else."
Masley said he and the clinic staff spend a lot of time educating patients who may not have ever had a routine checkup by a doctor or dentist. "We're providing care to people have never gotten care before."
Helping them understand their bodies and how choices they make impact their health sounds basic, but could prevent them from chronic illnesses later on, Masley said. Information is tailored for prisoners and explained simply.
"It's rewarding knowing I'm helping out people, teaching them how to take care of themselves," he said.
Return to LSJ's Salute to Nurses


A Glendale police officer was patrolling near a construction zone near 43rd Avenue and Grand Avenue on Wednesday morning when he heard a motorcycle approaching. The officer aimed his radar gun at the bike and was shocked to realize that the motorcyclist was traveling at 114 miles per hour, more than twice the legal limit in the area.
The officer pulled over the motorcyclist who is being described as a 21-year-old male after he caught up with the biker at Grand Avenue and 35th Street. The motorist reportedly did not know the legal speed limit in the area or how fast he had been traveling. According to the police officer, the biker did not seem to care that he had been traveling at more than twice the legal limit and did not seem surprised by the information.
The biker reportedly used a lane that is currently closed for construction to speed past cars that were waiting for a red light, enabling him to reach a high speed without traffic blocking his path. When questioned about his speed, the man informed the officer that he was late for work and therefore in a hurry.
The motorcyclist was arrested at the scene and charged with numerous misdemeanor traffic violations, though the specific charges are not clear at this time. Police have also stated that the man did not have a motorcycle endorsement on his driver's license, a requirement to operate a motorcycle in the state of Arizona.
Fortunately this motorcyclist was stopped before he had a chance to cause an accident, but traveling at 114 miles per hour through a construction zone and in an closed lane was a serious and imminent threat to all other drivers in the area. Often, motorcyclists are the victims of accidents because drivers in cars and trucks do not see a biker and cause an accident. However, as this incident illustrates, motorcyclists can also be responsible for reckless conduct on the roads, and this can lead to serious or even fatal accidents. In some crashes, a motorcycle receives the majority of the force of the impact, leading most of the damage to be inflicted on the motorcyclists and the bike. However, when a biker hits a pedestrian or a bicyclist, the opposite is often true. If the negligent conduct of a motorcyclist causes injuries to another, the biker can be held financially liable for any damage caused.
If you have been injured in an accident with a motorcyclist, called the Phoenix personal injury lawyers at Abels & Annes, P.C. today for a free case consultation. We understand that you may need answers outside of normal business hours so we have an attorney standing by 24 hours a day to take your call at (602) 819-5191. At Abels & Annes, P.C., we put our clients first and we fight for victims' rights. Call us today and let us help you recover after an accident.
Driver May Have Been on Drugs When He Caused Fatal Mesa Crash, Phoenix Injury Lawyer Blog, published August 13, 2013.
Motorcyclist arrested for traveling 114 mph on Grand Ave, by Julie Rose, Fox 10 News, published August 14, 2013.


Certified Organic Eucalyptus Essential Oil.
Mr Retour Eucalyptus Essential Oil is 100% pure and Australian Certified Organic. Intensely aromatic, this oil feels cool when applied to the skin, yet conversely feels warm internally to help relax muscles. It has antiseptic, anti-inflammatory, antibacterial and antiviral properties and can be used in vaporizers. Additionally, it can help alleviate the symptoms associated with fevers.
Provides respiratory support e.g. clears sinuses, breaks down mucus during colds and flu.
Can assist in relieving stress, insomnia, migraines and nervous unrests.
It's a great Insect repellant.
Promotes the healing of wounds.


Standard shipping cost $15 + tax.
Orders must be received by 3:00 pm EST Monday – Friday to be shipped the same day. Orders placed after this time or on weekends and statutory holidays will be shipped the next business day.
No P.O. Boxes please. Physical Addresses Only!
We do not ship to other countries.
The information that you give us and information about your order may be combined with other personally identifiable information (such as demographic information and past purchase history) available from our records and other sources. This information will be used to make our future marketing efforts more efficient. .
Where do we collect the information?
What do you do with the information?
Once we have collected the above information, it is used to process orders, to communicate with you about your order, and to provide future notification to you of special offers or promotions. We also use it to improve our website. We may provide statistical information about buying patterns and traffic patterns to others, but this data does not include any personal information.
How and why do we use "Cookies"?
"Cookies" are a type of information that is stored on your hard drive via your web browser. The "cookies" that Compex.info uses do not contain any personal data such as credit card information, passwords, or even your name or address. Cookies simply help us to identify your session on the site. They also help us maintain specific visit information about you without forcing you to register or to give personal information.
However, you should have the option of disabling "cookies" through your browser. This is a function of your computer, so the manner in which to turn the "cookies" off is quite individual and thus, cannot be addressed here. Please realize, if you choose to disable the "cookies" we will not be able to personalize your experience with us at compex.info nor can we guarantee a complete login, registration or shopping experience.
We periodically send emails to our customers to help keep you informed of new products, features and/or special promotions. In every email that we distribute, we include instructions on how to remove [unsubscribe] your email address from the distribution list.
Our web site uses Secure Sockets Layer (SSL) encryption to transmit your credit card information from a secure web page to a secure application on our server. Once on our server, your credit card number is saved only long enough for us to process and approve the transaction. During the final stages of ordering online, you will see an unbroken key or locked padlock at the bottom of your browser window and https:// in the location window at the top of your browser. Both of these indicate that the page you are on is secured using SSL encryption to communicate with our secure web server. Because SSL encryption slows down a web site, it is only used in the final stage of ordering, before and after you enter your credit card information. You will not see these security indicators until you begin the checkout process and indicate that you prefer to order online with a credit card.
Quite simply, we want your purchase to work for you. Our no-risk, unconditional 90-Day Money-Back Guarantee allows you ample time to see if your investment in your health is doing the job you expect and deserve. If, after trying the product within this time frame, you are not completely satisfied, you may exchange the item or return the product for a refund of the full purchase price (minus shipping & handling).
If you would like to return or exchange an item you ordered, please email canada.orders@djoglobal.com for RA# and return instructions or check our after sales policy here. If you return any item(s), please ship your items postage-paid, insured, with a tracking number. Allow 2-3 business days for processing, once your item is received. The RA# (Return Authorization Number) must be marked clearly on the exterior of the package in order for the refund to be issued promptly. Unfortunately, we cannot refund shipping and handling charges or return postage.
If you exchange your item(s), please return the item(s) using an insured and/or traceable service. Your exchange will be processed within 7-10 business days. If the exchange is due to an incorrect ordering procedure for the customer, the customer is responsible for shipping charges for the exchanged item. If the exchange is due to an incorrect ordering procedure by Copex.info, there will be no shipping charge required.
The Compex electrostimulation devices are covered by a 2 year warranty. For extended warranty 2+1, please register here.


Hi everyone, I've just started doing Opencart , and I'm trying to link Shipstation to the Opencart.
How about searching OpenCart Marketplace by "shipstation" first?
Check this help article: How do I connect my OpenCart store to ShipStation?
3) Modify the permissions (chmod) on your config.php file to 777.
Well, what a foolish and highly dangerous information, given to people on this Site !


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Utility Management Group
Senior decision-makers come together to connect around strategies and business trends affecting utilities.
So Goes the Build Back Better Plan, So Goes Offshore Drilling in Florida?
Todd Carney 15,175
Writer Freelance
Todd Carney is a graduate of Harvard Law School. He holds a Bachelor's degree in Political Science and Public Communications. He writes on many different aspects of energy, in particular how it...
Dec 30, 2021Dec 30, 2021 8:05 pm GMT
Over the last few weeks, the Build Back Better (BBB) legislation has dominated much of the political news cycle. But the coverage has largely focused on the overall size of the bill, and whether President Joe Biden and his allies have the votes needed to pass the BBB. Although it could take a while to sort out the legislative process around the bill, it is important to focus on the substance of the bill. Biden is using a lot of political capital to try to pass this bill, so it could be his biggest achievement, if it passes. While the BBB covers many areas, in terms of energy related policies, perhaps the most consequential one has been the proposal to ban offshore drilling in federal waters. In particular how this policy could impact Florida.
The Contentious Issue of Offshore Drilling in Florida
The debate over the BBB is not the first time the ban on offshore drilling in FL has come up. From the 1940s through 1970s, offshore drilling occurred in FL's waters. In the 1980s, voters started opposing offshore drilling nationwide, including in FL. Voters were concerned about impact on their environment, along with the threats to tourism.
This opposition led to FL banning offshore drilling in waters that they controlled in 1992. Voters codified this ban in 2018, when they passed a ballot initiative that made the ban part of FL's constitution.
Despite these actions from FL, offshore drilling remains an issue in FL because three nautical miles off the coast, the area is considered "federal waters", so the federal government controls drilling there. Despite these waters' distance from FL's coast, if a drilling accident occurred, it could impact FL's waters, and then their tourism industry.
Starting in the 1980s, the federal government started prohibiting offshore drilling in most places, including Florida. Congress would pass a moratorium each year. Additionally in 1990, President George HW Bush enacted an executive moratorium on offshore drilling. When President Bill Clinton took office, he extended the moratorium through 2012.
In 2006, Congress overrode part of the overall moratorium by passing the Gulf of Mexico Energy Security Act (GOMESA). The bill opened up drilling in parts of the Gulf of Mexico, but did put a moratorium in federal waters off the coast of FL through 2022.
President Barack Obama considered allowing offshore drilling in FL, among other places, but reversed his position amid the British Petroleum spill. During President Donald Trump's administration, Trump considered expanding offshore oil drilling in Florida twice. Trump first got rebuffed in 2018 by FL Republicans. Then in 2020, Trump seemingly backed off due to FL's status as a swing state. Trump even went as far as issuing an executive order that kept the moratorium in place until 2032.
President Joe Biden signed an executive order early in his administration that backed up Trump's moratorium on offshore drilling in FL.
Florida Offshore Drilling and the Build Back Better Plan
The BBB has many ambitious overarching policies. Given how big Biden and his allies were looking to go in the bill, Florida Democrats moved to build support for including a ban on offshore drilling. Congresswoman Kathy Castor helped lead the effort.
Regarding FL, this push might seem redundant because there already is a moratorium on federal offshore drilling in FL. But FL advocates see this as an opportunity to put the issue to rest once and for all. If the ban passes, FL would not be at the whim of whoever is President, nor would they need to repeatedly lobby Congress on extending the moratorium.
The politics of the BBB puts the policy of banning offshore drilling in an interesting position. Although the leading politicians of FL, such as Senator Marco Rubio, Senator Rick Scott, and Governor Ron DeSantis are all opposed to offshore drilling for FL, they also oppose the BBB. In terms of FL's members in the US House of Representatives, it is a party line split in terms of Democrats supporting the BBB and Republicans opposing it.
In the days before West Virginia Democrat Senator Joe Manchin came out against the BBB, in a last ditch effort to woo Manchin, Senate proponents of the BBB cut the ban on offshore drilling. Castor has urged the Senate to restore it, but so far the plea has gone unanswered.
Despite the fact that Republicans will not support the BBB overall, they may vote for some amendments to it. So some hope that Rubio could at least support an amendment to ban offshore drilling, since he is against it in FL. However, Rubio has said he will only support an amendment that would ban offshore drilling in FL alone.
Other Relevant Legislation
As Congress wrestles with the BBB, members on both sides have proposed similar limitations on offshore drilling in FL. This past July, Rubio, Scott, and FL Republican US House member, Michael Waltz, proposed a bill to legislatively back Trump and Biden's moratorium through 2032. Castor also worked with 19 US House members from FL to introduce a separate bill to ban offshore drilling in FL. Such a bill passed the last session of Congress, but the Senate did not take it up. Rubio also supports this legislation.
Additionally, Castor is behind two bills that would make offshore drilling more expensive and restrictive in the future. These pieces of legislation protect environmental interests everywhere, but they also defend FL in case there is future offshore drilling in the federal waters off its coast.
For the time being, any legislation to curb offshore drilling is in a precarious position due to the opposition of leading politicians, such as Manchin. Given that Manchin is not only a swing vote in the US Senate, but is also the Chairman of the Senate Energy and Natural Resource Committee, any ban will likely need his support.
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Texas Student Dies of Bacterial Meningitis
HOUSTON – Two students at Texas Southern University have contracted bacterial meningitis.
A spokesperson for the school told Fox station KRIV in Houston that one student died on Thursday, March 29.
The other student remained hospitalized on Monday.
It is unknown if either student lived on the school's campus.
TSU says the school is collaborating with the Houston Department of Health and Human Services to identify those who may have been close to the infected students.
The school is offering preventive antibiotics to students on Monday and Tuesday in the TSU Health Center.
Bacterial meningitis, also known as acute meningitis, mostly occurs in children, although anyone can contract it. If caught early, it can be treated with antibiotics.
The National Institute of Health estimates there are 17,500 cases of bacterial meningitis every year.


Crafted with realistic hammered detailing, the arrowhead detail on this Pamela Love ring elevates the beauty of a natural relic.
Disappointed by the ring in person..
I never thought that I would be disappointed in any Pamela Love jewelry, but this ring ended up being SO ordinary in person. For the price, the quality is alright and the antique finish is nice. But, firstly, the ring sizing is way off, as I usually wear size 7 rings for my middle/pointy finger, but the ring fit like an 8 or even 9, and secondly, it looked so cheap and ordinary on my finger and definitely unflattering. I tried it on several times to see if I would change my mind about the way it looked, but finally decided it was not worth the money in the end, so I returned it.


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