For $\theta_{1}$ and $p_{m}-\Sigma_{f}^{\prime\prime}>0$, we may write $\theta_{1}=2\pi-\delta\quad\text{with }0<\delta\ll 1$. We thus have

\begin{align}
\tan(2\pi-\delta)=\left(\frac{-(p_{m}-\Sigma_{f}^{\prime\prime})}{D}\right).
\end{align}

Using $\tan(A+B)=\left(\tan A+\tan B\right)/(1-\tan A\tan B)$, we reach $\tan(2\pi-\delta)=-\tan\delta\approx-\delta$, leading to $\delta=\frac{(p_{m}-\Sigma_{f}^{\prime\prime})}{D}$. We therefore obtain

\begin{align}
\theta_{1}=2\pi-\frac{(p_{m}-\Sigma_{f}^{\prime\prime})}{D},\quad\text{for \ensuremath{p_{m}}-\ensuremath{\Sigma_{f}^{\prime\prime}}}>0.
\end{align}

On the other hand, for $\theta_{1}$ and $p_{m}-\Sigma_{f}^{\prime\prime}<0$, we may write $\theta_{1}=\delta\quad\text{with }0<\delta\ll 1$. We find

\begin{align}
\theta_{1}=\delta=-\frac{(p_{m}-\Sigma_{f}^{\prime\prime})}{D},\quad\text{for }\ensuremath{p_{m}}-\Sigma_{f}^{\prime\prime} <0.
\end{align}

Gathering the above results, we have

\begin{align}
\theta_{1}\approx
\begin{cases}
2\pi-\frac{p_{m}-\Sigma_{f}^{\prime\prime}}{D}, & \text{for } p_{m}-\Sigma_{f}^{\prime\prime}>0\\
-\frac{p_{m}-\Sigma_{f}^{\prime\prime}}{D} & \text{for } p_{m}-\Sigma_{f}^{\prime\prime}<0
\end{cases}.
\end{align}

Following  a similar approach, to evaluate $\theta_2$, we can express

\begin{align}
\theta_{2}\approx
\begin{cases}
\pi+\delta, & p_{m}-\Sigma_{f}^{\prime\prime}>0,\\
\pi-\delta & p_{m}-\Sigma_{f}^{\prime\prime}<0.
\end{cases}
\end{align}

Applying a similar way to find $\delta$, we reach

\begin{align}
\theta_{2}=\pi+\frac{(p_{m}-\Sigma_{f}^{\prime\prime})}{D}.
\end{align}

Thus, to the leading order in $\frac{p_{m}-\Sigma_{f}^{\prime\prime}}{D},$

\begin{align}
\theta_{1}-\theta_{2}=\pi\text{sgn}(p_{m}-\Sigma_{f}^{\prime\prime})-\frac{2(p_{m}-\Sigma_{f}^{\prime\prime})}{D}.
\end{align}

The real part of $G_{f}$ can be approximated as

\begin{align}
G_{f}^{\prime}	(p_{m}) & =\left(-\rho_{0}\right)\ln\frac{\sqrt{\left(1+\Sigma_{f}^{\prime}(p_{m})/D\right)^{2}+\left(p_{m}-\Sigma_{f}^{\prime\prime}(p_{m})\right)^{2}/D^{2}}}{\sqrt{\left(1-\Sigma_{f}^{\prime}(p_{m})/D\right)^{2}+\left(p_{m}-\Sigma_{f}^{\prime\prime}(p_{m})\right)^{2}/D^{2}}} \nn
& \approx\left(-\rho_{0}\right)\ln\frac{\sqrt{1+2\Sigma_{f}^{\prime}(p_{m})/D}}{\sqrt{1-2\Sigma_{f}^{\prime}(p_{m})/D}} \nn
& \approx-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(p_{m}),
\end{align}

leading to

\begin{align}
G_{f}(p_{m})	& \approx -\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(p_{m})-i\pi \rho_{0}\text{sgn}\left[p_{m}-\Sigma_{f}^{\prime\prime}(p_{m})\right]+i\frac{2\rho_{0}}{D}\left[p_{m}-\Sigma_{f}^{\prime\prime}(p_{m})\right] \nn
& = -i\pi \rho_{0} \text{sgn} \left[p_{m}-\Sigma_{f}^{\prime\prime}(p_{m})\right]-\frac{2\rho_{0}}{D}\Sigma_{f}(p_{m})+\frac{2\rho_{0}}{D}ip_{m}.
\label{eq:Gf-final}
\end{align}

Utilizing the wide-band limit once more, where $|p_{m}|\ll D$, and the absence of an imaginary part for $\Sigma_{f}^{J}$, we have $\left(\Sigma_{f} (ip_{m})\right)^{\prime\prime} = \left(\Sigma_{f}^{g}(ip_{m})\right)^{\prime\prime}=\alpha-\varsigma\left|p_{m}\right|$, the sign function in $G_{f}(ip_{m})$ above can be approximated as

\begin{align}
\text{sgn}\left[p_{m}-\left(\Sigma_{f}^{g}(p_{m})\right)^{\prime\prime}\right]	\approx\text{sgn}\left(-\alpha\right)=-1,
\end{align}

since $\alpha \gg |p_{m}|>0$. Note that we do not include the contribution from the impurity scattering $A$ here in $\Sigma_{f}^{\prime\prime}$ since $A$ will be put by hand in $\Sigma_{f}^{\prime\prime}$ and thus will only be included at the end of the calculation when we want to evaluate the scattering rate for the $f$ spinon. Thus, $A$ should not be included in the subsequent calculation for $\Sigma_{\xi}$.  Applying $\rho_{0}=1/D$, $G_{f}$ in Eq. (<ref>) becomes

\begin{align}
G_{f}(p_{m})&	= i\pi \rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}(p_{m})+\frac{2\rho_{0}}{D}ip_{m} \nn
& =i\pi \rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(p_{m})-i\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime\prime}(p_{m})+\frac{2\rho_{0}}{D}ip_{m} \nn
& =i\pi \rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(p_{m})-i\frac{2\rho_{0}}{D}\left(\frac{D}{3}-\frac{|p_{m}|}{\pi}\right)+\frac{2\rho_{0}}{D}ip_{m} \nn
& \approx i\frac{5}{2}\rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(p_{m})+\frac{2\rho_{0}}{\pi D}\text{sgn(\ensuremath{p_{m}})}ip_{m}+\frac{2\rho_{0}}{D}ip_{m}.
\end{align}

where we have taken the approximation for $\pi-\frac{2}{3}\approx\frac{5}{2}$ in the last line of the above equation. Note that the last term in the above equation $\frac{2\rho_{0}}{D}ip_{m}$ is linearly proportional to frequency; thus we shall see that this contribution will vanish when we perform the integration over frequency.

Plugging the result of $G_{f}$ into $\Sigma_{\xi}$ of Eq. (<ref>), we have

\begin{align}
\Sigma_{\xi}(k_{n})&=\frac{\bar{g}^{2}}{\beta}\sum_{ip_{m}}\frac{i\frac{5}{2}\rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(ip_{m})+\frac{2\rho_{0}}{\pi D}\text{sgn(\ensuremath{p_{m}})}ip_{m}+\frac{2\rho_{0}}{D}ip_{m}}{ik_{n}-ip_{m}-\lambda} \nn
& =-\frac{\bar{g}^{2}}{\beta}\sum_{ip_{m}}\frac{i\frac{5}{2}\rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(ip_{m})+\frac{2\rho_{0}}{D}(ip_{m})}{ip_{m}-ik_{n}+\lambda}-\frac{2\rho_{0}\bar{g}^{2}}{\pi D\beta}\sum_{ip_{m}}\frac{ (i p_{m}) \text{sgn}(p_{m})}{ip_{m}-ik_{n}+\lambda} \nn
&=-\bar{g}^{2}n_{F}(ik_{n}-\lambda)\left[i\frac{5}{2}\rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(ik_{n}-\lambda)+\frac{2\rho_{0}}{D}\left(ik_{n}-\lambda\right)\right]-\frac{2\bar{g}^{2}\rho_{0}}{\pi D}\left(\frac{1}{\beta}\sum_{ip_{m}}\frac{ (ip_{m}) \text{sgn}(p_{m}) }{ip_{m}-ik_{n}+\lambda}\right) \nn
& =-\bar{g}^{2}\left[i\frac{5}{2}\rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(ik_{n}-\lambda)+\frac{2\rho_{0}}{D}\left(ik_{n}-\lambda\right)\right]-\frac{2\bar{g}^{2}\rho_{0}}{\pi D}\left(\frac{1}{\beta}\sum_{p_{m}>0}\frac{ip_{m}}{ip_{m}-ik_{n}+\lambda}-\frac{1}{\beta}\sum_{p_{m}<0}\frac{ip_{m}}{ip_{m}-ik_{n}+\lambda}\right).
\end{align}

The summation over the $p_{m}>0$ and $p_{m}<0$ branches will be evaluated separately. For the $p_{m}>0$, we have

\begin{align}
\frac{1}{\beta}\sum_{p_{m}>0}\frac{ip_{m}}{ip_{m}-ik_{n}+\lambda}=(ik_{n}-\lambda)\Theta(k_{n})n_{F}(ik_{n}-\lambda)-\int_{-\infty}^{\infty}\frac{dx}{2\pi i}\cdot\frac{xn_{F}(x)}{x-ik_{n}+\lambda}.
\label{eq:p-greater}
\end{align}

Following a similar approach, we have, for the $p_{m}<0$ branch,

\begin{align}
-\frac{1}{\beta}\sum_{p_{m}<0}\frac{ip_{m}}{ip_{m}-ik_{n}+\lambda}=-(ik_{n}-\lambda)\Theta(-k_{n})n_{F}(ik_{n}-\lambda)+\int_{\infty}^{-\infty}\frac{dx}{2\pi i}\cdot\frac{xn_{F}(x)}{x-ik_{n}+\lambda}.
\label{eq:p-lesser}
\end{align}

Combining Eqs. (<ref>) and (<ref>), when $T\to0$, we obtain

\begin{align}
\frac{1}{\beta}\sum_{ip_{m}}\frac{\text{sgn(\ensuremath{p_{m}})}ip_{m}}{ip_{m}-ik_{n}+\lambda}=(ik_{n}-\lambda)\text{sgn}(k_{n})-\int_{-\infty}^{\infty}\frac{dx}{\pi i}\cdot\frac{xn_{F}(x)}{x-ik_{n}+\lambda},
\end{align}

leading to

\begin{align}
\Sigma_{\xi}(k_{n})	& =-\bar{g}^{2}\left[i\frac{5}{2}\rho_{0}-\frac{2\rho_{0}}{D}\Sigma_{f}^{\prime}(ik_{n}-\lambda)+\frac{2\rho_{0}}{D}\left(ik_{n}-\lambda\right)\right]+\frac{2\rho_{0}\bar{g}^{2}}{\pi D}\left[i|k_{n}|-\lambda\text{sgn}(k_{n})\right] \nn
& \qquad \qquad -\frac{2\rho_{0}\bar{g}^{2}}{\pi D}\int_{-\infty}^{\infty}\frac{dx}{\pi i}\cdot\frac{xn_{F}(x)}{x-ik_{n}+\lambda}.
\end{align}

In addition, using the results of $\left(\Sigma_{f}^{J} \right)^\prime=-J^{2}\rho_{0}D$ and $\left(\Sigma_{f}^{g}(k_{n}) \right)^\prime=\bar{g}^{2}\rho_{0}\ln\left(\frac{D}{k_{n}-\zeta^{-1}\varUpsilon}\right)$, we obtain the real and imaginary part for $\Sigma_{\xi}$, given by

\begin{align}
\Sigma_{\xi}^{\prime}(k_{n})&=\frac{2\rho_{0}^{2}\bar{g}^{4}}{D}\ln\left(\frac{D}{k_{n}-\zeta^{-1}\varUpsilon}\right)-\frac{2\rho_{0}\bar{g}^{2}}{D}J^{2}+\frac{2\rho_{0}\lambda\bar{g}^{2}}{D}+\frac{2\rho_{0}\lambda\bar{g}^{2}}{\pi D}\text{sgn}(k_{n})+\frac{2\rho_{0}\bar{g}^{2}}{\pi D}\text{Re}\left(\int_{-\infty}^{\infty}\frac{dx}{\pi i}\cdot\frac{xn_{F}(x)}{x-ik_{n}+\lambda}\right),
\end{align}

\begin{align}
\Sigma_{\xi}^{\prime\prime}(\omega+i\delta)=-\frac{5}{2}\bar{g}^{2}\rho_{0}-\frac{2\rho_{0}\bar{g}^{2}}{\pi^{2}D}\omega_{0}-\frac{2\rho_{0}\bar{g}^{2}}{\pi D}\left|\omega\right|-\frac{2\rho_{0}\bar{g}^{2}}{\pi^{2}D}\left(\omega-\lambda\right)\ln\frac{|\omega-\lambda|}{\omega_{0}}.
\end{align}

For $\lambda \gg \omega$, we may simply approximate $\ln\frac{|\omega-\lambda|}{\omega_{0}} \approx \ln\frac{\lambda}{\omega_{0}}$. In addition, the linear-in-$\omega$ term in the last term of $\Sigma_\xi^{\prime\prime}$ above can be shown to vanish when we calculate the electrical conductivity below (due to the symmetric boundary for the frequency integral). We therefore can neglect this term. Now,  $\Sigma_{\xi}^{\prime\prime}(\omega+i\delta)$ becomes

\begin{align}
\Sigma_{\xi}^{\prime\prime}(\omega+i\delta) = -A - \varsigma |\omega|
\label{eq:Im-Sigma-xi-final}
\end{align}

with $A = \frac{5}{2}\bar{g}^{2}\rho_{0} + \frac{2\rho_{0}\bar{g}^{2}}{\pi^{2}D}\omega_{0}-\frac{2\rho_{0}\bar{g}^{2}}{\pi^{2}D}\lambda \ln\frac{\lambda}{\omega_{0}} $ and $
\varsigma = \frac{2}{\pi}$ a constant here.

§.§ The Planckian metal in DC-resistivity

Having the result of scattering rate, we further compute  the electrical resistivity of our model in the DC limit (zero frequency) directly via Boltzmann formula under the assumption of temperature-independent effective mass and carrier concentration. Using  Eq. (<ref>) in the above section and the the relation $\Sigma_c = (1/2)\Sigma_\xi$, the relaxation time for the conduction $c$ electron shows

\begin{align}
\frac{\hbar}{\tau_c(\omega)} = -2\Sigma_c^{\prime\prime} (\omega)=A + \varsigma |\omega|
\end{align}

The conductivity can be obtained as

\begin{align}
\sigma(T)	& =\left(-\frac{ne^{2}}{m^{\star}}\right)\int\tau_c(\omega)\frac{\partial n_{F}(\omega)}{\partial\omega}d\omega \nn
& \approx\left(\frac{ne^{2}\hbar}{m^{\star}}\right)\left[\int_{0}^{\infty}\frac{1}{A}\left(1+\frac{\varsigma\omega}{A}\right)\frac{\partial n_{F}(\omega)}{\partial\omega}d\omega  -\int_{-\infty}^{0}\frac{1}{A}\left(1+\frac{\varsigma\omega}{A}\right)\frac{\partial n_{F}(\omega)}{\partial\omega}d\omega\right]
\nn
& =\frac{ne^{2}\hbar}{m^{\star}A}-\left(\frac{ne^{2}\hbar\varsigma Y}{m^{\star}A^{2}}\right)k_{B}T,
\end{align}

where $Y\equiv \int_{-\infty}^{\infty}\frac{|x|e^{x}}{(e^{x}+1)^{2}}dx \approx 1.39$ is a constant. Thus, the resistivity at sufficiently low temperatures can be approximated as

\begin{align}
\rho(T)	& =\frac{1}{\sigma(T)}=\frac{m^{\star}A}{ne^{2}\hbar}\left(\frac{1}{1+\left(\frac{\varsigma Y}{A}\right)k_{B}T}\right)
\nn
& \approx\frac{m^{\star}A}{ne^{2}\hbar}+\frac{m^{\star}\varsigma Y}{ne^{2}\hbar}k_{B}T.
\end{align}

The linear-in-temperature term of resistivity $\rho_{L}(T)$ shows

\begin{align}
\rho_{L}(T)=\frac{m^{\star}}{ne^{2}\hbar}(\varsigma Yk_{B}T)\equiv\frac{m^{\star}}{ne^{2}\hbar}\left(\alpha_{P}k_{B}T\right),
\end{align}

where coefficient $\alpha_{P}$ defined above is $\alpha_{P}=\varsigma Y=1.39 \times \frac{2}{\pi}\approx 0.9$, indicating the Planckian scattering rate.

We find that the T-linear resistivity comes as a consequence of the isotropic electronic states. Though the self energy diagram, leading to the electron scattering rate, depends on $g^2$, this coupling constant dependence in electron scattering rate is eventually canceled by the same factor in the Green's function of the composite fermionic spinon-holon state. As a result, the system remains a Planckian state in transport in this phase without knowing that the coupling constant $t$ is flowing to an irrelevant fixed point at U(1) FL$^*$. Similar cancellation in transport rate has been found in the SYK model <cit.>.

§.§ Estimation of scattering rate with slightly anisotropic Fermi surface

In this section, we estimate the change of the scattering rate in the Planckian strange metal state if the Fermi surface includes a small anisotropic part, as shown in Supplementary Figure <ref>.  We find that the contribution to the scattering rate from the small anisotropic part of the Fermi surface is negligible.

For small anisotropy of the dispersion (or the Fermi surface), we assume that the anisotropic band structure starts to deviate significantly relative to the isotropic one at an energy $-\delta D < 0$ below the Fermi energy, where $\delta D$ is assumed to obey $\delta D/ D \ll 1$ to satisfy the small anisotropy condition. The small anisotropic part is also affected by a shift, $\delta k_F$, in the Fermi wave vector $k_F$.  Due to the additional anisotropic portions, the density of states at the Fermi surface is subjected to a small change,

\begin{align}
\rho_0 \to \rho^\prime  (\varepsilon) = \rho_0 + \delta \rho (\varepsilon).
\end{align}

Here, $\delta \rho (\varepsilon) \propto \delta k_F$ denotes the change in the density of states at the Fermi surface with respect to $\rho_0$.  $\Sigma_{f}^{>}$ with tiny anisotropic Fermi surface can then be expressed as

\begin{align}
\left[ \Sigma_{f}^{>}(\omega_{n}) \right]^\prime& =\frac{g^{2}}{\zeta} \sum_{\bm{p}} \frac{n_{F}(\varepsilon^{\prime}_{\bm{p}}+iD\pi)}{i\omega_{n}-\varepsilon^{\prime}_{\bm{p}}+\lambda-iD\pi} \nn
& = \frac{g^{2}}{\zeta}\int_{-D+\bar{\mu}_{\xi}}^{0}  \frac{\rho^\prime (\varepsilon)d\varepsilon}{i\omega_{n}-\varepsilon+\lambda-iD\pi},
\end{align}

where, in general, the density of states $\rho (\varepsilon)$ contains angular dependence,

\begin{align}
\rho (\varepsilon) = 4\times \int_{\varepsilon = \varepsilon_{\bm{k}}} \frac{d\theta_{\bm{k}}}{(2\pi)^2} \frac{1}{\left| \nabla_{\bm{k}}\varepsilon_{\bm{k}} \right|}
\end{align}

with $\theta_{\bm{k}}$ being the angle between $k_x$ and $k_y$ for the two-dimensional Brillouin zone. The change of $\Sigma_{f}^{>}$ in the presence of an anisotropic Fermi surface relative to the isotropic one takes the form,

\begin{align}
\delta \Sigma_{f}^{>} (\omega_n) & =\left[ \Sigma_{f}^{>}(\omega_{n}) \right]^\prime - \Sigma_{f}^{>}(\omega_{n})  \nn
& \approx  \frac{g^{2}}{\zeta}\int_{-\delta D}^{0}  \frac{\rho^\prime (\varepsilon) d\varepsilon }{i\omega_{n}-\varepsilon+\lambda-iD\pi} -  \frac{g^{2}\rho_0}{\zeta}\int_{-\delta D}^{0}  \frac{d\varepsilon}{i\omega_{n}-\varepsilon+\lambda-iD\pi} \nn
& = \frac{g^{2}}{\zeta}\int_{-\delta D}^{0}  \frac{\delta \rho (\varepsilon) d\varepsilon }{i\omega_{n}-\varepsilon+\lambda-iD\pi}.
\end{align}

For small $\delta D$ and performing the analytic continuation, we approximate $ \delta \Sigma_{f}^{>} (\omega +i\epsilon)$  as

\begin{align}
\delta \Sigma_{f}^{>} (\omega +i\epsilon) \approx  \frac{g^{2}}{\zeta}  \frac{\delta \rho (0) \delta D }{\omega+\lambda-iD\pi} ,
\end{align}

and its imaginary part is given by

\begin{align}
\left[\delta \Sigma_{f}^{>} (\omega +i\epsilon)\right]^{\prime\prime} =  \frac{g^{2} }{\zeta \pi}  \delta \rho(0) \frac{\delta D}{D}.
\end{align}

We find that the change of $\Sigma_{f}^{>}$ due to an anisotropic Fermi surface is proportional to $ \frac{\delta D}{D}$, thus negligibly small. Following a similar approach, however, it can be demonstrated that the correction for $\Sigma_{f}^{<}$ includes an additional minus sign compared to $\Sigma_{f}^{>}$. Therefore, the complete leading-order  correction to $\left[ \Sigma_{f}\right]^{\prime\prime}$ due to a slightly anisotropic Fermi surface vanishes.

(a) A band with slightly anisotropic dispersion is composed of an isotropic region (blue), with the blue dashed line representing the isotropic Fermi surface,  and four crescent-shaped anisotropic portions (pink areas). (b) Schematic plots of the band with isotropic Fermi surface (blue, associated with the Fermi surface of blue region in (a)) and the slightly anistropic one (pink, associated with the Fermi surface of pink region in (a)). The band with slightly anisotropic Fermi surface starts to deviate from the blue one at energy $-\delta D$ below the Fermi energy $\varepsilon_F$.

§.§ Contribution from vertex correction to scattering rate

Diagrammatic representation of the first two leading terms for (a) the  current-current correlator and  (b) the current vertex for the $\xi$ fermion of our model. Right figure of (a) represents the bare current-current correlator of the $\xi$ fermion and the left figure shows its leading-order correction.

We estimate the significance of the vertex correction to the electrical conductivity (or equivalently the scattering rate) from the $\xi$ fermion (twice of the contribution of the conduction $c$ electron) in the Planckian strange metal state. We find that this contribution is negligible, consistent with the earlier studies in Refs. <cit.>. The leading-order contribution of vertex correction to the current-current correlator is diagrammatically depicted in Supplementary Figure (<ref>).

Without tackling the awkward diagrammatic calculation of the vertex correction $\Gamma_x$, here we use a simpler power-counting to evaluate how significant the contribution of $\Gamma_x$ is. The rule for the power-counting in the wide-band limit can be demonstrated as follows:

\begin{align}
\bar{g} = \frac{g}{\sqrt{\zeta}} \propto D, \quad \frac{1}{V}\sum_{\bm{k}} =\rho_0 \int d\varepsilon \propto D^0, \quad \frac{1}{\beta}\sum_{\omega_n}\propto D^0 ,\quad \mathcal{G}_f, \, \bar{\mathcal{G}}_\chi \propto D^{-1}, \quad \mathcal{G}_b \propto D^0.
\end{align}

The validity of the above set of rules for power-counting can be confirmed by examining $\Sigma_f^{\prime\prime} \sim \bar{g}^2 \mathcal{G}_\xi\mathcal{G}_b\propto D^2
\times \frac{1}{D} = D$ which is consistent with our calculation where the dominant term for $\Sigma_f^{\prime\prime}$ is proportional to $\alpha \propto D$ as shown in Eq. (<ref>).

When considering vertex correction, the dc-conductivity $\sigma_{\xi}$ from the $\xi$ fermion can be computed by [see, e.g., Ref.  <cit.>]

\begin{align}
\sigma_{\xi}&=2\times\frac{e^{2}}{2\pi}\frac{1}{V}\sum_{\bm{k}}\Gamma_{0}(\bm{k},\bm{k})\left[\bar{G}_{\xi}^{A}(\bm{k},0)\bar{G}_{\xi}^{R}(\bm{k},0)\Gamma^{RA}(\bm{k},\bm{k};0,0)-\bar{G}_{\xi}^{A}(\bm{k},0)\bar{G}_{\xi}^{R}(\bm{k},0)\Gamma^{AA}(\bm{k},\bm{k};0,0)\right],
\label{eq:conductivity-JJ-correlator}
\end{align}

where $\bar{G}_\xi$ denotes the dressed Green function for the $\xi$ field after normalization, as described in Section II. Here, $R \,(A)$ represents “retarded" (“advanced").

Powering counting for the leading-order vertex correction shown in Supplementary Figure (<ref>)  can be obtained as

\begin{align}
\Gamma  & \sim \bar{g}^4 \mathcal{G}_b^2 \, \mathcal{G}_f^2 \,  \bar{\mathcal{G}}_\xi^2 \nn
& \propto D^4 \times \frac{1}{D^2} \times \frac{1}{D^2} = D^0.
\end{align}

Substituting the above result into Eq. (<ref>), the conductivity from the contribution of vertex correction of Supplementary Figure (<ref>) can be estimated as

\begin{align}
\sigma_\xi \sim \frac{ n e^2 }{m_\xi} \left( \bar{G}_\xi^2  \times \Gamma \right)
\propto \frac{ n e^2 }{m_\xi} \left( \frac{1}{D^2}  \times D^0\right) ,
\end{align}

where $m_\xi$ coming from the bare current vertex of $\xi$, $\Gamma_0 \propto \frac{1}{m_\xi}$, of Eq. (<ref>) denotes the effective mass for $\xi$ and $n$ represents the particle density of $\xi$ which is estimated via $\sum_{\bm{k}}/V \propto n$. This leads to the following power-law-in-bandwidth dependence for the conductivity from the $\xi$ fermion,

\begin{align}
\sigma_\xi \sim \frac{ n e^2 }{m_\xi} \left( \frac{1}{D^2} \right),
\end{align}

indicating that the vertex correction of Supplementary Figure <ref>(b) contributes a term proportional to $1/D^2$ to the transport time.

Together with the contribution from the self-energy (lifetime) of the $\xi$ field, denoted as $\Sigma_\xi^{\prime\prime}$ in Eq. (<ref>), the conductivity in the wide-band limit can be approximated by

\begin{align}
\sigma_\xi & \sim \frac{ n e^2 }{m_\xi} \left( \frac{1}{D} + \frac{1}{D^2} \right) = \frac{ n e^2 }{m_\xi} \frac{1}{D} \left[1 + O\left(\frac{1}{D}\right)\right] \nn
&  \approx  \frac{ n e^2 }{m_\xi} \frac{1}{D}.
\end{align}

We thus conclude that, in the wide-band limit, the contribution of the vertex correction to the conductivity is vanishingly small compared to that of the self-energy and can be neglected, consistent with Refs. <cit.>.

§ UNIVERSAL SCALING

In this section, we will demonstrate the universal  scaling behavior of the electronic scattering  rate at $T=0$ and its generalization to the finite temperatures by referring to Refs. <cit.>. Notations below are mostly adopted from Refs. <cit.>.

We start from the equation below for the spectral representation of correlation function <cit.>,

\begin{align}
G_{\psi}(\tau)=-\int_{-\infty}^{\infty}\frac{e^{-\tau \varepsilon}}{1+e^{-\beta\varepsilon}}A_{\psi}(\varepsilon)d\varepsilon,
\label{eq:seeback-spectral}
\end{align}

where $0\leq\tau\le\beta$ and $A_\psi (\varepsilon) = (-1/\pi) G^{\prime\prime}_\psi( \omega+i0^+)$ denotes the spectral function for the $\psi$ field. When a system has conformal invariance, any fermionic correlation function, such as the self-energy, in the imaginary-time domain can be expressed as <cit.>

\begin{align}
\Sigma(\tau)\propto e{}^{\alpha(\tau/\beta-1/2)}\left[\frac{\pi/\beta}{\sin\left(\pi\tau/\beta\right)}\right]^{1+\nu},
\label{eq:seeback-self-energy-1}
\end{align}

where $\alpha$ and $\nu$ are constants. Here, $\alpha$ is a measure of particle-hole asymmetry. Its spectral representation is given by

\begin{align}
\label{eq:seeback-self-energy-2}
\end{align}

\begin{align}
g_{\alpha,\nu}(x) & =\left|\Gamma\left(\frac{1+\nu}{2}+i\frac{x+\alpha}{2\pi}\right)\right|^2\frac{\cosh(x/2)}{\cosh(\alpha/2)\Gamma[(1+\nu)/2]^{2}}, \nn
C_{\alpha,\nu} & =\frac{(2\pi)^{\nu}\cosh(\alpha/2)\Gamma\left[\frac{1+\nu}{2}\right]^{2}}{\pi\Gamma[1+\nu]}.
\label{eq:seeback-g}
\end{align}

Following Eqs. (<ref>)-(<ref>), we will generalize $\Sigma_\xi$ that we obtain in the zero-temperature limit in the previous section  to the finite-temperature one. We will further derive its $\omega/T$ scaling behavior once the finite-temperature generalization  is acquired.

Applying Eq. (<ref>), we first evaluate the (imaginary) time dependence of $ \Sigma_{\xi}$ which shows a linear-in-frequency dependence in the frequency domain, $\Sigma_\xi \propto |\omega|$. It can be shown that $\Sigma_{\xi}$ exhibits a $\tau^{-2}$ dependence:

\begin{align}
\Sigma_{\xi}(\tau)	= & \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{|\varepsilon|e^{-\tau\varepsilon}}{1+e^{-\beta\varepsilon}}d\varepsilon \nn
= & \frac{1}{\pi}\left[\int_{0}^{\infty}|\varepsilon|e^{-\tau\varepsilon}+\int_{-\infty}^{0}|\varepsilon|e^{(\beta-\tau)\varepsilon}\right]d\varepsilon \nn
=& \frac{1}{\pi}\left[\frac{1}{\tau^{2}}\int_{0}^{\infty}|x|e^{-x}dx+\frac{1}{(\beta-\tau)^{2}}\int_{-\infty}^{0}|x|e^{x}dx\right] \nn
= & \frac{1}{\pi \tau^{2}}.
\end{align}

It implies that this situation corresponds to the case of $\alpha = 0$ (particle-hole symmetric) and $\nu=1$ (Planckian) as discussed in Ref <cit.>. The second term of the above equation vanishes when $T \to 0$.

Once $\alpha=0$ and $\nu = 1$  are decided, we can  generalize $\Sigma_{\xi}$ to the finite-temperature region by conformal transformation governed by Eqs. (<ref>)-(<ref>), leading to the following expression for $\Sigma_{\xi}(\omega,T)$,

\begin{align}
\Sigma_{\xi}^{\prime\prime}(\omega,T)=\lambda_{0}\beta^{-1}g_{1,0}(x)=\frac{\lambda_{0}}{2}\omega\coth\left(\frac{\omega}{2T}\right)
\label{eq:Sigma_xi-img-generalscaling}
\end{align}

with $\lambda_0$ being an unknown constant. $\lambda_0$ can be determined from its zero-temperature limit, where $\Sigma_{\xi}^{\prime\prime}(\omega,T=0) = -(2/\pi) |\omega| $. We thus find $\lambda_0 = -4/\pi$, leading to

\begin{align}
\Sigma_{\xi}^{\prime\prime}(\omega,T)=\lambda_{0}\beta^{-1}g_{1,0}(x)=-\frac{2}{\pi}\omega\coth\left(\frac{\omega}{2T}\right).
\label{eq:Sigma_xi-img}
\end{align}

Using the relation of $\Sigma_c = (1/2)\Sigma_\xi$, the  scattering rate for the conduction $c$ electron can be obtained by $\hbar/\tau_c = -2\Sigma_c^{\prime\prime} (\omega)$ (here we restore $\hbar$ and $k_B$), $\frac{\hbar}{\tau_c} =\frac{2}{\pi}\hbar \omega\coth\left(\frac{\hbar \omega}{2k_B T}\right)$. It shows the following frequency-to-temperature scaling  behavior

\begin{align}
\frac{\hbar/\tau_c}{k_B T} =\frac{2}{\pi}x\coth\left(\frac{x}{2}\right),
\end{align}

where $x\equiv \hbar \omega/k_B T$. The results of the above scaling relation for electronic scattering rate is plotted in Fig. 4(a) of the main text. In the high-frequency, low-temperature limit $x \gg 1$, the scattering rate divided by $k_B T$ shows a universal scaling behavior, $\frac{\hbar/\tau_c}{k_B T} \approx (2/\pi)x$. Conversely, in the DC-limit ($x \to 0$), the scattering rate manifests the Planckian scattering rate, revealing a universal feature that is insensitive to microscopic coupling constants:
$1/\tau_c \approx \alpha_P k_B T/\hbar$ with $\alpha_P  = 8/\pi \approx 2.55$.

§.§ Resistivity scaling and the equivalence of frequency and magnetic field in scaling regime

In this section, we observe that the scaling behavior of the resistivity (derivative) at the strange-metal region for cuprate superconductor Tl2201 <cit.> at zero frequency but finite applied magnetic field shows the same quantum critical scaling form as we find within our theory at a finite frequency but zero magnetic field. This implies an equivalent role played by frequency and magnetic field near quantum criticality associated with the strange metal state. The temperature-dependent AC-resistivity in the absence of magnetic fields is obtained by the inverse of AC-conductivity as: $\rho(\omega,T)=1/\sigma(\omega, T) = 1/\tau(\omega,T) \times (m^\star/ne^2)$. We assume $m^\star/n$ is independent of temperature, frequency and doping in the Planckian state though $m^\star$ and $n$, in general, depend on these variables. Using scaling form for electron scattering rate Eq. (3) of the main text, we obtain the scaling of the derivative of AC-resistivity $d\rho/d(\hbar \omega)$. The results are shown in Supplementary Figures <ref> and <ref>. The blue dashed lines in Supplementary Figures <ref> and <ref> are fitted to the scaling function for $d\rho(\hbar \omega/k_B T)/d (\hbar \omega)$ shown below <cit.>:

\begin{align}
\frac{ne^2}{m^\star}\times \frac{\pi}{2}\frac{\dd \rho \left(\frac{\hbar\omega}{4 k_B T}\right)}{\dd (\hbar\omega)}&=\coth \left( \frac{\hbar \omega}{4 k_B T}\right) -\frac{\hbar \omega}{4 k_B T}\rm{csch}^2 \left(\frac{\hbar\omega}{4 k_B T}\right) \;, \\
\frac{ne^2}{m^\star} \times \frac{\pi}{2}  \frac{\dd \rho(X)}{\dd (\hbar\omega)}&=\coth(X) -X\rm{csch}^2(X)
\label{eq:d_rho_SB_X_b}
\end{align}

with $X= \hbar \omega /4k_B T$, while the green ones are fitted to the following equations of the marginal Fermi liquid,

\begin{align}
\frac{ne^2}{m^\star} \times \frac{\pi}{2}\frac{\dd \rho\left(\frac{\hbar\omega}{4 k_B T}\right)}{\dd (\hbar\omega)}&=\frac{\frac{\hbar\omega}{4k_B T}}{\sqrt{1+(\frac{\hbar\omega}{4k_B T})^2}} \;, \\
\frac{ne^2}{m^\star} \times \frac{\pi}{2}\frac{\dd \rho[X]}{\dd (\hbar\omega)}&=\frac{X}{\sqrt{1+X^2}}\label{eq:d_rho_MFL_X_b} \np
\end{align}

In Ref. <cit.>, the magnetoresistivity of overdoped cuprates Bi2201 and Tl2201 is well described by the MFL form: $\rho(H,T)= F(T)+ \sqrt{(\alpha k_B T)^2+(\gamma \mu_B \mu_0 H)^2}$ where $\alpha$ and $\gamma$ are constants insensitive to field, temperature and doping. The authors in Ref. <cit.> discovered that the derivative of $\rho(H,T)$ with respect to the magnetic field (upon proper normalization) shows quantum critical $H/T$-scaling over a finite-range in doping, signature of a quantum critical strange metal “phase". As shown in Supplementary  Figures <ref> and <ref>, the data in Ref. <cit.> for $(1/\gamma) d\rho(\beta \mu_B H/T)/d(\mu_B H)$ exhibits a MFL scaling form $X/\sqrt{1+X^2}$ with $X=\beta\mu_B H/T(\beta = \gamma \mu_B/\alpha k_B)$ over a wide range in doping for two distinct cuprates (Bi2201 and Tl2201)  <cit.>. Strikingly, it is clear from Supplementary  Figures <ref> and <ref> as well as Eq. (<ref>) that this scaling function for  $(1/\gamma)d\rho(\beta \mu_B H/T)/d(\mu_B H)$ found in Ref. <cit.> is nicely reproduced  within our theory for the AC-resistivity (in the absence of magnetic fields) for $(\pi/2) d\rho(X)/d(\hbar \omega)$. This result indicates that in the scaling regime $\hbar \omega /4k_B T$ plays the role of $\beta \mu_B H/ T$, and prefactor $(m^\star/ne^2)\times (\pi/2)$ here plays the role of $\gamma$.

Derivative of the resistivity data for Tl2201 with $T_c=23~\rm{K}$ divided by $\gamma\mu_B$ plotted in function of $\frac{\beta\mu_0 H}{T}$. Here, $\beta \simeq 4.66$ and $\gamma \simeq 0.0049$ are fitting parameters. We also show the behavior of the Marginal Fermi-Liquid (MFL) form given in Eq. (<ref>) as well as the analytic form via our Kondo-Heisenberg approach to the slave-boson  t-J model (KHSB)  [Eq. (<ref>)]. We note that both curves nicely describe the experimental data.  The resistivity data shown here is reproduced from Ref. <cit.>.

Derivative of the resistivity data divided by $\gamma\mu_B$ plotted in function of $\frac{\beta\mu_0 H}{T}$. For the sample  Tl2201 with $T_c=26.5~\rm{K}$ <cit.>, for  $\beta \simeq 6.6$ and $\gamma \simeq 0.00562$. We note that the KHSB model [Eq. (<ref>)] nicely describes the experimental data.

§.§ Mass enhancement

According to Ref. <cit.> and its supplementary material,  the mass enhancement is given by

\begin{equation}
\frac{m^{\ast}(\omega, T)}{m}-1\simeq -\frac{2}{\hbar\omega}\left[\Sigma^{\prime}\left(\frac{\hbar\omega}{2}\right)-\Sigma^{\prime}(0,T)\right]\np
\end{equation}

This implies the knowledge of the real part of the electrons' self energy, which can be usually be obtained by using Kramers-Kroning relations

\begin{align}
\Sigma^{\prime}_{c}(\omega, T)&=\frac{1}{\pi}\mathcal{P}\int \frac{ \Sigma''_{c}(\omega_1)}{\omega_1-\omega} d \omega_1\label{eq:S10_KK}.
\end{align}

Unfortunately, within our KHSB model, the imaginary part of self energy obtained from Eq. (<ref>) and   yielding

\begin{equation}
\Sigma^{\prime\prime}_{c}(\omega, T)=-\frac{\omega}{\pi}\coth \left( \frac{\omega}{4 k_B T}\right) \label{eq:S2_def1}
\end{equation}

is not holomorphic. As s result, an analytic derivation of the real part of the self energy is not possible. As explained in Ref. <cit.>, this is overcome by considering a similar yet different scaling function $f(x)=\norm{x}+2\exp{\frac{\norm{x}}{2}}$ instead of  $f(x)=\frac{x}{2}\coth(\frac{x}{2})$ with $x=\frac{\hbar\omega}{k_BT}$ allows to evaluate the real part of self energy. The resulting real part of self energy  leads to a scaling of the mass enhancement $m^*/m-1 \propto g(x)$ with $g(x)$ taking the following form [see Eq. (S16) in Ref. <cit.>]:

\begin{equation}
g(x)=2g\left( 1-\gamma \ln(\frac{x}{4})+\frac{2}{x}\left[\exp{\frac{x}{4}}\textrm{Ei}(-\frac{x}{4})-\exp{-\frac{x}{4}}\textrm{Ei}(\frac{x}{4})\right]\right)\;,\label{eq:g_x}
\end{equation}

where $\gamma=0.577$ is Euler's constant, $g$ is a constant fitting parameter, and $\textrm{Ei}(x)$ is the exponential integral function.

Now, we compare our scaling function in the imaginary part of self energy with that shown in Ref. <cit.> and discuss the correspondence between  the fine-tuned prefactor $g$ of the scaling function in Ref. <cit.> and the universal prefactor in the scaling function of our approach. Following the notations of  Ref. <cit.>, the imaginary part of self energy takes the form

\begin{equation}
\Sigma^{\prime\prime}_{c}(\omega, T)=-\pi g k_BT f\left(\frac{\hbar\omega}{k_BT}\right) \; .\label{eq:S2_def2-2}
\end{equation}

A direct comparison of Eq. (<ref>) and  Eq. (<ref>) gives

\begin{equation}
\end{equation}

within our KHSB model, while within the work presented in Ref. <cit.> $g$ is a fitting parameter evaluated to $g=0.23$.

In Fig. 4b of the main text, we plotted the  mass enhancement scaling function $g(x)$ on top of the data set reproduced from Ref. <cit.>. The dashed red line corresponds to the fit of Ref. <cit.> with $g=0.23$, and the solid black line corresponds to the prediction for $g(x)$ within the KHSB model with $ g=\frac{2}{\pi^2}$.

§ THE SPECTRAL WEIGHT

In this section, we present our results on single-electron spectral weight via RG renormalized perturbative approach to our Kondo-Heisenberg formulated slave-boson t-J model. We further compare our results with the recent ARPES measurements on the overdoped cuprates in Ref. <cit.>. Excellent agreement between our theoretical predictions and the experimental observations is achieved.

§.§ Imaginary part of the self energy at zero temperature

We assume the dispersion of the electrons to be given by a tight-binding model with nearest $t$ and second–nearest $t'$ hopping parameters on the square lattice, same as that used in Ref. <cit.>:
ε(k)=-2t(cos(k_x)+cos(k_y))-4t'cos(k_x)cos(k_y) .

The zero-temperature imaginary part of the self energy for the $\xi$ field is given by Eq. (<ref>), using $\Sigma_c^{\prime\prime}(\omega) = \Sigma_\xi^{\prime\prime}/2$, we obtain
Σ^''_c(ω)=-1/πω+Γ  .
The real part of the self energy is given by the Kramers-Kroning relations

\begin{align}
\Sigma^{\prime}_c(\omega) =\frac{1}{\pi}\mathcal{P}\int \frac{ \Sigma''_c(\omega_1)}{\omega_1-\omega}\dd \omega_1
=\frac{2}{\pi^2}\omega\ln \left(\frac{\norm{\omega}}{D}\right)\; ,\label{eq:S10_def}
\end{align}

where $D\gg\omega$ is a large cutoff.

§.§ Imaginary part of the self energy at finite temperatures

At finite temperature, the imaginary part of the self energy is given by Ref. <cit.> [cf. Eqs. (<ref>) and (<ref>)]
Σ^_c(ω,T)=-πg k_B T f(x/2)+Γ,

where $f(\frac{x}{2})\equiv \frac{x}{2}\coth(\frac{x}{4})$ and $x=\frac{\hbar \omega}{k_B T}$. In our case of the slave-boson approach to the t-J model, the parameter $g$ corresponds to $g=\frac{2}{\pi^2}$, leading to
Σ^''_c(ω, T)=-ω/π(ω/4 k_B T)+Γ,

which is equivalent to Eq. (<ref>) supplemented by a constant term $\Gamma$.

The real part is given by the  Kramers–Kronig  relations
Σ^'_c(ω,T)=1/π𝒫∫ Σ^''_c(ω_1,T)/ω_1-ωω_1  .
Because $f(x)$ is not a holomorphic function, the exact analytical integration in Eq. (<ref>) is not possible. However, one can evaluate it numerically. Meanwhile, we note that $\Sigma''_c(\omega)$ in Eq. (<ref>) is well approximated by marginal Fermi liquid-like (MFL) form
Σ_MFL^''(ω,T)=-gπ/2√(ω^2+(πk_B T)^2)+Γ  ,
but with different prefactors:
Σ_c^''(ω,T)≈-1/π√(ω^2+(αk_B T)^2)+Γ  ,
where the prefactor $g$ in the MFL form becomes a universal constant in our case: $g=2/\pi^2$ and $\alpha = 4$, as will be explained below.

One can  now use the  Kramers–Kronig relations with the approximate expression in the marginal Fermi-liquid form to find an approximate form of the real part of the self-energy

\begin{align}
\Sigma_{c}^{\prime}(\omega,T)&=\frac{1}{\pi}\mathcal{P}\int \frac{ \Sigma_{c}^{\prime\prime}(\omega_1,T)}{\omega_1-\omega}\dd \omega_1 \nn
&=\frac{2}{\pi^2}\omega\ln \left(\frac{\sqrt{\omega^2+(\alpha  k_B T)^2}}{D} \right)\; , \label{eq:S1_MFL}
\end{align}

where again $D\gg\omega$ is a large energy cutoff.

From the marginal Fermi–liquid approximations, Eq. (<ref>) and Eq. (<ref>), we note that in the limit $T \to 0$, we recover the zero temperature expressions in Eqs. (<ref>) and (<ref>) of our KHSB approach , and that at finite temperature the limit $\omega \to 0$ of the marginal Fermi–liquid imaginary part of the self-energy [Eq. (<ref>)] recovers the slave-boson expression [Eq. (<ref>)]  for $\alpha=4$:

\begin{align}
\lim_{T \to 0}	\Sigma_{c}^{\prime\prime}(\omega,T)&= -\frac{\norm{\omega}}{\pi}=\lim_{T \to 0}\Sigma^{\prime\prime}_c(\omega, T)=\Sigma''_c(\omega)\; ,\label{eq:LimSigmaIm}\\
\lim_{T \to 0}	\Sigma_{c,MFL}^{\prime}(\omega,T)& =\frac{2}{\pi^2}\omega\ln \left(\frac{\norm{\omega}}{D}\right)=\Sigma^{\prime}_c(\omega)\; , \\
\lim_{\omega  \to 0}\Sigma_{c,MFL}^{\prime\prime}(\omega,T)&=	-\frac{\alpha  k_B T}{\pi}=\lim_{\omega  \to 0}\Sigma^{\prime\prime}_c(\omega, T)=	-\frac{4 k_B T}{\pi}\; ,
\end{align}

hence $\alpha=4$. This is represented in Supplementary Figure <ref> where we compare the imaginary part of the self-energy for the marginal Fermi-liquid and our KHSB model .

Comparison of the imaginary part of the self-energy between Eq. (<ref>) and the approximated marginal Fermi-liquid-like form in Eq. (<ref>). Within the physically relevant range in $\omega$ and $T$, the marginal Fermi-liquid-like form Eq. (<ref>) is a good approximation to Eq. (<ref>).

In Supplementary Figure <ref>, we also see that the numerical evaluation of Eq. (<ref>) is almost identical to the  marginal Fermi-liquid form in Eq. (<ref>). The numerical integration of Eq. (<ref>) is performed between $-D$ and $D$. We indeed note that  for a sufficiently large cutoff $D\geq1000$ eV, and a not too high temperature $T<10^{5}~\rm{K}$ (i.e. $k_B T\simeq 1~\rm{eV}$) the numerically evaluated  Eq. (<ref>) and the  marginal Fermi-liquid form in Eq. (<ref>) lie very close to each other.

Indeed, from inspection of the  Kramers–Kronig relation of Eq. (<ref>), it is clear that, if the integration boundary $D$ is too close to the energy $\omega$ of interest, the pole lies very close to the integration boundary leading to fallacious results. We also note that for relatively “low" temperature $k_B T\ll \hbar\omega$, i.e in the limit $\frac{ \hbar \omega}{k_B T}\to \infty$ or in the limit $T\to 0$, the value of the imaginary part of self-energy approaches to $-\frac{1}{\pi}\norm{\omega}$ [see Eq. (<ref>)] for both marginal Fermi-liquid-like form  [Eq. (<ref>)] and the form via our KHSB model.  We expect this agreement to also appear for the real part of the self energy.

In the last column of Supplementary Figure <ref>, we observe that, when the condition $k_B T \ll \hbar\omega$ is no longer satisfied, a discrepancy begins to emerge in the real part of the self-energy between the marginal Fermi liquid-like form and the form derived via our KHSB approach.

The main difference between the analytical form of our self-energy [Eq. (<ref>)] and the approximated form by the marginal Fermi liquid-like expression [Eq.  (<ref>)] is that, at finite temperature, the real part of the self-energy is well defined at $\omega=0$ in the approximated marginal Fermi-liquid-like form shown in Eq. (<ref>), but it is not well-defined via the analytic form in Eq. (<ref>).  Indeed, we note from Eq. (<ref>) that, for $\omega=0$, the integrand becomes simply $\coth(\frac{\hbar\omega_1}{4 k_B T})$, which has a discontinuity in $\omega_1=0$ lying inside the integration interval $\left[-D, D\right]$.

Comparison of the real part of the self-energy for the marginal Fermi-liquid [Eq. (<ref>)] (dashed orange line) and Eq. (<ref>) (blue dots) obtained by numerical integration of imaginary part according to the Kramers-Kronig relations for different value of the cutoff parameter $D=100 \, \rm{eV}, \,  200\, \rm{eV}, \, 500\, \rm{eV},$ and $ 1000 \, \rm{eV}$  and different temperatures $T=10^{-5} \rm{K}, 250 \rm{K},$ and $ 10^{5} \rm{K}$. We note that for a large enough cutoff $D\geq1000$ eV, the numerically obtained results by Eq. (<ref>) is well-fitted by the marginal Fermi-liquid-like form in Eq. (<ref>).

§.§ Single-particle spectral function

The spectral function of the single-particle's Green's function is given by
A(k,ω)=-1/πIm[1/ω-(ε(k)-μ)-Σ(ω,T)]  .

Our aim is to compare spectral function $A(\bm{k},\omega)$ via our KHSB approach to the ARPES data measured in Ref. <cit.> for Bi2212 $\rm{[(Bi, Pb)}_2 \rm{Sr}_2 \rm{CaCu}_2 \rm{O}_{8+\delta}]$ in the strange metal phase with hole doping $p=0.196$ at $T=250$K. This is done in the strange metal phase of this compound (doping $p=0.196$) at 250 K. This material features bilayer Cu-O planes and shows  two parabolic bands (bonding and anti-bonding bands) in the  heavily overdoped case which corresponds to the strange metal phase. Therefore, the total spectral function is the  sum of the two spectral functions associated with each band

\begin{align}
A(\bm{k},\omega)=\sum_{i=1}^{2}\frac{-C_i}{\pi}\rm{Im}\left[ \frac{1}{\omega-\varepsilon_i(\bm{k})+\mu_i-\Sigma_{c,i}(\omega,T)}\right] ,
\label{eq:A2_def}
\end{align}

where $\rm{Im} \left[ \Sigma_{c,i} \right]$ is given by Eq. (<ref>) and it is well-approximated by the marginal Fermi-liquid-like form  Eq. (<ref>) (see the comparison between these two forms in Supplementary Figures <ref> and <ref>), and $C_i$ are the normalization weights of the experimental data. We also assume the two dispersion bands to be given by Eq. (<ref>) with different parameter for each bands
ε_i(k)=-2t_i [cos(k_x)+cos(k_y)]-4t_i'cos(k_x)cos(k_y) ,

with $t_i $ and $t_i^\prime$ as defined in Eq. (<ref>). In the following, the self-energy $\Sigma(\omega,T)$ is treated by the marginal Fermi-liquid approximation, since it is a faithful representation of the scaling form in Eqs. (<ref>) and (<ref>) [with $(x/2)\coth\left(x/4\right)$], and it is a smooth and well-defined function over the entire regions of $\omega$ and $T$, convenient for our calculations:

\begin{align}
\Sigma_i(\omega,T)&=	\Sigma_{i,c}^{\prime}(\omega,T)+i\Sigma_{i,c}^{\prime\prime}(\omega,T)\\
&=\frac{2}{\pi^2}\omega\ln \left(\frac{\sqrt{\omega^2+(\alpha  k_B T)^2}}{D_i} \right)
-i\frac{1}{\pi}\sqrt{\omega^2+(\alpha  k_B T)^2}+i\Gamma_i\; , \label{eq:Stot_MFL}
\end{align}

with $\alpha=4$.

We fit our results for the spectral weight [Eqs. (<ref>)-(<ref>)] to the experimental data from Ref. <cit.> in the following two cases. The first one is the energy density curve along $\omega$ at the anti-nodal point $(k_x=\frac{-\pi}{a_0}, k_y\approx0)$, and the second one is the momentum density curve along $k_y$ at zero-energy $\omega\approx0$ and $k_x=\frac{-\pi}{a_0}$. These are shown in Figs. 5(a) and 5(b) of the main text, respectively, and correspond to the blue lines which are associated to a doping value of $\delta=0.196$. The fitting of our model is done by simultaneously fitting both curves together. Our model consist of a total of 12 parameters that need to be found in order to reproduce the experimental data. Namely the parameters are

μ_1,     Γ_1    D_1,   t_1,    t_1',   C_1,

μ_2,    Γ_2,    D_2,   t_2,    t_2',   C_2 ,

where the subscripts refers to the two bands.

We first set $T=250$ K, same as that taken in experiment in Ref. <cit.>. The energies $\omega$ are measured from the chemical potential $\mu = \mu_1 =\mu_2$. Here, we set $\omega \to \omega-\mu$ since experimentally the energies are measured from the chemical potential $\mu_1 , \, \mu_2$. We also a large bandwidth cutoff for the two bands, $D_1=D_2=100$ eV, since for the energy region $\omega\in [-0.25\, \rm{eV}, 0.05\, \rm{eV}]$ that we are interested in, these are sufficiently big enough cutoff for the marginal Fermi-liquid version of the real part of the self-energy in Eq. (<ref>) to reliably reproduce the features of Eq. (<ref>) from our KHSB approach, as depicted in Supplementary Figure <ref>. We find that the set of parameters that reproduces the best the experimental data (blue lines in Fig. 5 of the main text) are given by

Γ_1=-0.1 eV,    D_1=1000  eV,   t_1=2.2 eV,    t_1'=-0.005 eV,   C_1 =5.8,

Γ_2=-0.2 eV,    D_2=1000  eV,   t_2=6.5 eV,    t_2'=-0.07  eV,   C_2=8 .

The predictions from our approach plotted for the parameter set given in Eq. (<ref>) are depicted in Figs. 5(a) and 5(b) of the main text (green lines) on top the experimental data (blue lines) for easy comparison. In Fig. 5(a) of the main text, we show the energy density curve of the spectral function  applied to Bi2212, predicted by our theoretical approach  [Eqs. (<ref>) - (<ref>)] and plotted for the parameter set given in Eq. (<ref>). In Fig. 5(b) of the main text, we plotted  the momentum density curve, and in Fig. 5(c), we show the single-particle spectral function for the same parameters as used in  Fig. 5(a) of the main text.