# Planckian behaviour in the optical conductivity of the weakly coupled
Hubbard model

Maxence Grandadam  Department of Physics and Physical Oceanography, Memorial
University of Newfoundland, St. John’s, Newfoundland & Labrador, Canada A1B
3X7  J. P. F. LeBlanc<EMAIL_ADDRESS>Department of Physics and Physical
Oceanography, Memorial University of Newfoundland, St. John’s, Newfoundland &
Labrador, Canada A1B 3X7

###### Abstract

We study the frequency and temperature dependence of the optical conductivity
in the weakly coupled two-dimensional Hubbard model using a renormalized
perturbative expansion. The perturbative expansion is based on the skeleton
series for the current-current correlation function with a dressed Green‘s
function and the results are obtained directly on the real frequency axis
using Algorithmic Matsubara Integration (AMI). The resulting conductivity
shows a temperature-independent power law behaviour in the intermediate
frequency regime. Moreover, the associated transport scattering time and
renormalized mass exhibit a Planckian behaviour. We show that the self-energy
of the Hubbard model, however, is distinct from existing Planckian models. The
Planckian behaviour of the conductivity, observed in optimally doped cuprates
for example, can thus be obtained from a different form of self-energy than
the Planckian model, such as the weakly coupled Hubbard model at half-filling.

##  I Introduction

Strange metals are a key problem of interest in the study of strongly
correlated systems for the past years. The wide variety of experimental
observations of strange metallicity[1, 2, 3, 4] and the number of associated
theoretical descriptions[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] makes it
difficult to extract a unique underlying principle. The main characteristic of
the strange metal regime is the linear temperature dependence of the
longitudinal resistivity in contrast to the quadratic dependence expected from
Fermi liquid theory. More recently, a study of the optical conductivity in a
cuprate material has also shown signatures of this strange metal behaviour in
the frequency dependence of the longitudinal conductivity[16]. The
phenomenology was well reproduced by assuming a form for the self-energy of
the system associated with a “Planckian model” usually obtained from SYK or
Kondo-like models.

We present here a study of the optical conductivity in the weakly coupled 2D
Hubbard model from a normalized perturbative expansion of the current-current
correlation function. We use Algorithmic Matsubara Integration[17, 18] (AMI)
to compute the current-current correlation function directly on the real
frequency axis without the need for numerical analytic continuation and
extract from it the transport scattering time and mass renormalization by
using a generalized Drude model. The results for the conductivity present the
same behaviour in frequency and temperature as both the experimental and
theoretical works even though the self-energy of the Hubbard model does not
satisfy the constraint given in the Planckian model. We emphasize the
importance of the momentum dependence of the self-energy to the final result
and discuss the role of vertex corrections.

##  II Scaling of the optical conductivity

We study the half-filled, weakly coupled, Hubbard model

$\displaystyle
H=\sum_{ij\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow},$
(1)

where $t_{ij}$ is the hopping amplitude, $c_{i\sigma}^{(\dagger)}$
($c_{i\sigma}$) is the creation (annihilation) operator at site $i$,
$\sigma\in\\{\uparrow,\downarrow\\}$ is the spin, $U$ is the onsite Hubbard
interaction, $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is the number
operator. We restrict the sum over sites to nearest neighbours for a 2D square
lattice at half-filling and use $U=3$ unless stated otherwise in the
following. We use a skeleton series expansion for the current-current
correlation function $\chi_{jj}(i\omega_{n})=-\frac{1}{V}\int d\tau
e^{i\omega_{n}\tau}\left\langle
T_{\tau}j\left(\tau\right)j\left(0\right)\right\rangle$, where
$j=\sum_{k,\sigma}v_{k}c^{\dagger}_{k,\sigma}c_{k,\sigma}$ is the current
operator, from which we obtain the optical conductivity
$\sigma(\omega)=i\left(\chi_{jj}(0)-\chi_{jj}(\omega)\right)/\omega$ (see
supplemental material for details ). Using AMI we can perform the analytic
continuation symbolically without resorting to numerical techniques such as
maximum entropy inversion MAXENT[19, 20], Pade approximants[21] or other
modern methods[22, 23]. We thus obtain the optical conductivity at different
temperatures between $\beta=20$ and $\beta=2$.

The modulus and phase of the optical conductivity are presented in Fig.1(a)
and (b) and we can clearly identify three different frequency regimes. At low
frequencies, the modulus goes to a temperature-dependent constant while for
frequencies in the intermediate regime $T\ll\omega\ll\Lambda$, the modulus of
the conductivity exhibits a power law dependence
$|\sigma\left(\omega\right)|\propto\omega^{-\nu^{*}}$ with a temperature-
independent exponent $\nu^{*}<1$. In the same frequency range, the phase of
the conductivity saturates at $\nu^{*}\pi/2$ lower than the expected $\pi/2$
which is only reached at higher frequencies above a high-energy cut-off
$\Lambda$. It is important to note that the dc conductivity
$\sigma\left(\omega\to 0\right)$ exhibits a $1/T$ temperature dependence
consistent with the linear-in-$T$ behaviour observed in other work on the
Hubbard model at strong[24] and at weak[25] coupling (see supplemental
material).

Figure 1: Frequency dependence of the modulus $\bm{(a)}$ and phase $\bm{(b)}$
of the conductivity at different temperature. The dashed line indicates the
power law form of the modulus $\omega^{\nu^{*}}$ and the saturation bound
$\nu^{*}\frac{\pi}{2}$ for the phase respectively. The exponent $\nu^{*}$ is
temperature independent. The high energy cut-off $\Lambda$ is taken as the
frequency at which the phase deviates from this bound. $\bm{(c)}$ inverse
scattering time and $\bm{(d)}$ effective mass extracted from the conductivity
when rewritten as an extended Drude model given in Eq.(2). The frequency axis
is scaled by the inverse temperature for each curve as well as the scattering
time to exhibit the linear scaling obeyed by the two quantities. An
approximate $m(0)$ has been subtracted from the effective mass but is subject
to the important noise in the data.

To understand the frequency dependence of the conductivity, it is useful to
rewrite it in a generalized Drude form with a frequency-dependent scattering
rate and optical mass

$\sigma\left(\omega\right)=\frac{\sigma_{0}}{1/\tau\left(\omega\right)-i\omega\frac{m^{*}\left(\omega\right)}{m_{0}}}.$
(2)

These two quantities can then be extracted from the conductivity data by
inverting Eq.(2) leading to

$\frac{1}{\tau\left(\omega\right)}=\text{Re}\left[\frac{1}{\sigma\left(\omega\right)}\right],\qquad\frac{m^{*}\left(\omega\right)}{m_{0}}=-\text{Im}\left[\frac{1}{\omega\sigma\left(\omega\right)}\right].$
(3)

Extracting the scattering time and the effective mass from the conductivity
and rescaling the former by the inverse temperature reveals the temperature
scaling of the two quantities as shown in Fig.1(c) and (d). The scaling of the
scattering time at low frequency with the inverse temperature is
representative of the linear dependence of the dc conductivity. The collapse
of curves at intermediate frequencies in Fig.1(c) and (d) is the hallmark of
Planckian behaviour.

It was pointed out recently[16] that both the power law behaviour of the
conductivity with $\nu^{*}<1$ and the scaling of the scattering time with
$\nu=1$ we obtained in the Hubbard model can be obtained from a single
“Planckian model” where one forces the conductivity to have Planckian
behaviour by a particular choice of self-energy. In that Planckian model,
there is a direct relation between the value of the interaction and the
exponent $\nu^{*}$ and one can show that this is also satisfied in the Hubbard
model (see supplemental material). We discussed this Planckian model ansatz
for the self-energy that leads to a Planckian behaviour of the conductivity
and compare it with the self-energy obtained in our calculation of the 2D
Hubbard model.

##  III Planckian model and local self-energy

###  III.1 Basics of the Planckian ansatz

One way of obtaining a linear temperature dependence for the resistivity is to
assume a form of the self-energy that has the right temperature dependence as
$\omega\to 0$. In order to give the observed scaling at intermediate
frequencies in the scattering time, the self-energy should also be a function
of the ratio $\omega/T$. These self-energy ansatzes are usually referred to as
(sub-)Planckian models and are given by[16]

$\displaystyle\text{Im}\left[\Sigma\left(\omega\right)\right]=-\frac{g\pi}{\beta^{\nu}}\
f\left(\beta\omega\right),$ (4)

where $f\left(x\right)$ is a function that goes to a constant at low
frequency, $f\left(x\to 0\right)\to f_{0}$, and has a power law form at high
frequency, $f\left(x\gg 1\right)\to|x|^{\nu}$. Note that this form of self-
energy is local, _i.e._ does not have any momentum dependence, which is
usually justified by the weak momentum dependence observed experimentally in
cuprates[26, 27].

The self-energy in Eq.(4) with $\nu=1$ naturally leads to a linear temperature
dependence of the dc conductivity as well as the scaling observed in the
scattering time and in the effective mass. The ansatz for the local Planckian
self-energy thus seems to give a satisfactory description of the results
obtained both experimentally and from our study of the Hubbard model. One of
the advantages of the Planckian model is that the local nature of the self-
energy leads to a cancellation of the vertex correction in the Kubo formula
for the conductivity[28, 29] which would make the low-order expansion used in
this work sufficient for a full description of the conductivity. It is known
that such a form of the self-energy can be obtained from microscopic models
such as SYK-type models[30, 31] or Kondo-like models[32]. These models usually
rely on some large-N approach to obtain the linear scaling of the self-energy
in a non-perturbative scheme. The fact that the Hubbard model can exhibit such
properties at second-order in perturbation theory may thus seem surprising.

###  III.2 Breakdown of the local assumption

Each calculation of the conductivity required us to compute the self-energy on
a grid in momentum and frequency space from which we can directly check and
compare the Planckian ansatz presented in Eq.(4). One popular assumption is to
consider a local self-energy obtained from integrating the momentum-dependent
results at second-order in perturbation

$\Sigma_{\text{loc}}\left(\omega\right)=\frac{1}{N}\sum_{k}\Sigma\left(k,\omega\right).$
(5)

The local approximation, as is the case in methods such as dynamical mean-
field theory (DMFT), has been shown to lead to a linear-in-temperature
resistivity over a wide range of temperatures[33]. The resulting imaginary
part for the local self-energy in our calculation is shown in Fig.2(a) for
positive frequencies. The large energy cut-off $\Lambda$ identified previously
in the conductivity is related to the bandwidth after which the amplitude of
the self-energy decreases strongly.

Figure 2: $\bm{(a)}$ The Scaling of the imaginary part of the local self-
energy $\Sigma_{\text{loc}}$ at different temperatures. The absence of
collapse of the data shows that the local self-energy does not obey the same
scaling as the Planckian ansatz Eq.(4). $\bm{(b)}$ The Scaling of the
scattering time obtained using the local self-energy shown in panel $(a)$. In
contrast to Fig.1, there is no scaling of the scattering time with
temperature.

Scaling the imaginary part of the self-energy as indicated in Eq.(4) does not
produce any collapse of the calculations at each temperature as shown in
Fig.2(a). In fact, there is no value of $\nu$ that leads to a collapse of the
data (see supplemental material). This is an indication that the local form of
the self-energy obtained by integrating the momentum-dependent second-order
self-energy for the Hubbard model does not fall into the category of Planckian
models.

It is important to remember that the local self-energy presented in Fig.2(a)
is not the one that was used to obtain the conductivity in Fig.1 which
included the full momentum dependence. In fact, the scattering time extracted
from the calculation using the local form of the self-energy is presented in
Fig.2(b) and does not show the same scaling behaviour as the original results.
The resolution to this apparent paradox comes from the momentum dependence of
the self-energy in the Hubbard model with a peculiar differentiation between
the nodal and the antinodal point.

##  IV Nodal - Antinodal dichotomy

It was pointed out in previous work that the second-order self-energy of the
half-filled Hubbard model takes a “marginal Fermi liquid”(MFL) form at low
temperature[34, 35, 36]. The main reason for this seems to be the presence of
the van Hove singularity in the density of states that leads to an anomalous
frequency dependence of the self-energy on the Fermi surface. The typical form
for the imaginary part of the self-energy in the MFL case is given by
$\text{Im}\left[\Sigma_{\text{MFL}}\left(\omega\right)\right]=-g\ \pi\
\text{max}\left(|\omega|,T\right)$ which has the same scaling properties as
the self-energy from the Planckian model Eq.(4) as
$\Sigma_{\text{MFL}}\left(\omega\to 0\right)\sim T$ and
$\Sigma_{\text{MFL}}\left(\omega\gg T\right)\sim|\omega|$. The self-energy we
computed does in fact reproduce this behaviour at the nodal point
$k_{n}=\left(\frac{\pi}{2},\frac{\pi}{2}\right)$ as shown in Fig.3(a). Once
again we see that the imaginary part of the self-energy exhibit a linear power
law dependence in the intermediate frequency regime $T\ll\omega\ll\Lambda$ and
goes to a constant at low frequencies that scales as
$\Sigma\left(k_{n},\omega\to 0\right)\propto T$.

Figure 3: $\bm{(a)}$ Scaling of the imaginary part of the self-energy at the
nodal point $k_{n}=\left(\frac{\pi}{2},\frac{\pi}{2}\right)$ that shows a
Planckian behaviour with $\nu=1$. $\bm{(b)}$ Scaling of the imaginary part of
the self-energy at the antinodal point $k_{an}=\left(\pi,0\right)$ that shows
a sub-Planckian behaviour with $\nu<1$.

Trying to find a similar scaling for the self-energy in the antinodal region,
$k_{an}=\left(\pi,0\right)$, however, leads to a different result as shown in
Fig.3(b). The self-energy now takes the form of a “sub-planckian” model from
Eq.(4) with an exponent $\nu<1$. This is the reason for the apparent breakdown
of the Planckian behaviour of the local self-energy as we showed in Fig.2.

In the case of the half-filled Hubbard model and without vertex corrections,
the band velocity $\partial\epsilon_{k}/\partial k_{x}$ that appears in the
conductivity formula is larger at the nodal point and vanishes at the
antinode. This could be an explanation for the apparent Planckian behaviour
observed in the conductivity despite the sub-Planckian behaviour observed at
the antinode.

##  V Effect of vertex corrections

In light of the strong momentum dependence of the scaling properties of the
self-energy presented in the previous section, it is important to take into
account the vertex corrections to the conductivity. These second-order
contributions to the conductivity are shown in Fig.4(a) for different
temperatures. The overall amplitude for the vertex correction is much smaller
than the one obtained at the leading order (bubble diagram).

Figure 4: $\bm{(a)}$ Relative contribution of the vertex corrections with
respect to the zeroth order contribution to the real part of the conductivity
at different temperatures. At these temperatures, the vertex corrections are
small and mainly affect the low-frequency regime. $\bm{(b)}$ Derivative of the
modulus of the conductivity without (full lines) and with (dashed lines) the
inclusion of vertex corrections. The main changes are shown to occur at low
frequencies and do not affect the power law scaling discussed previously in
the intermediate frequency regime $T\ll\omega<\Lambda$.

In fact, the second-order term accounts for less than $10\%$ of the zeroth-
order contribution to the total conductivity for the three temperatures
presented here. The importance of the vertex correction grows when lowering
the temperature and as such, they may become significant at lower
temperatures. Nonetheless, the main impact of the vertex correction is seen at
low frequency, outside of the intermediate frequency regime where the scaling
properties of the conductivity we discussed previously occur. This is apparent
in Fig.4(b) where the logarithmic derivative of the modulus of the
conductivity is shown for different temperatures. We thus expect our previous
conclusion to hold even when vertex corrections, related to the momentum
dependence of the self-energy, are included.

##  VI Conclusion

We studied the frequency dependence of the longitudinal conductivity using a
renormalized Green functions scheme. This allows us to obtain results directly
on the real-frequency axis without any numerical analytical continuation and
without the need for finite $\Gamma$ in the $i\omega_{n}\to\omega+i\Gamma$
analytic continuation of AMI integrands

At leading order, the conductivity for the weakly coupled Hubbard model shows
a power law behaviour at intermediate frequencies $T\ll\omega\ll\Lambda$ with
a temperature independent exponent $\nu^{*}<1$. This is reminiscent of the
behaviour of the optical conductivity in strange metals such as cuprates.
Moreover, it was recently shown that Planckian models can lead to such power
law scaling[16]. The frequency dependence of the scattering time and effective
mass obtained by fitting the conductivity to an extended Drude form do, in
fact, show Planckian behaviour with notably
$1/\tau\left(\omega\right)\propto\beta\omega$.

The self-energy in the Hubbard model is however significantly different from
the local self-energy ansatz of the Planckian models. Even when getting rid of
the momentum dependence by averaging the self-energy over all momenta, the
resulting local quantity does not exhibit Planckian behaviour, and the
resulting conductivity also has a different temperature dependence. The
momentum dependence of the self-energy in the Hubbard model is thus crucial to
obtain the power law scaling observed in the scattering time and effective
mass. We showed that the vertex corrections that arise from this momentum
dependence do not affect our conclusion as they mostly affect the low-
frequency part of the conductivity.

The results presented here are based on a perturbative expansion and are thus
expected to be relevant in the weak coupling regime. It is surprising that
features such as Planckian behaviour of the conductivity, usually associated
with non-Fermi liquids and strongly correlated systems, can be recovered for
small $U$. The half-filled Hubbard model in itself is far from the complexity
that exists in real cuprates materials and as such, it would be interesting to
explore the doped case with next-nearest neighbours hoping. Previous studies
using Cellular Dynamical Mean Field Theory have in fact shown that an extended
region of linear-in-temperature resistivity[37] can be observed in the doped
Hubbard model.

We acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC) RGPIN-2022-03882 and support from the Simons
Collaboration on the Many Electron Problem.

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Supplementary material

##  VII Renormalized Perturbation expansion

We are interested in the half-filled Hubbard model on a 2D square lattice

$H=\sum_{ij\sigma}t_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow},$
(S1)

where $t_{ij}$ is the hopping amplitude, $c_{i\sigma}^{(\dagger)}$
($c_{i\sigma}$) is the creation (annihilation) operator at site $i$,
$\sigma\in\\{\uparrow,\downarrow\\}$ is the spin, $U$ is the onsite Hubbard
interaction, $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is the number
operator. The frequency-dependent longitudinal conductivity is related to the
current-current correlation function

$\sigma(\omega)=i\left(\chi_{jj}(0)-\chi_{jj}(\omega)\right)/\omega,$ (S2)

which is obtained in the linear response theory

$\chi_{jj}(i\omega_{n})=-\frac{1}{V}\int d\tau e^{i\omega_{n}\tau}\left\langle
T_{\tau}j\left(\tau\right)j\left(0\right)\right\rangle,$ (S3)

where $j=\sum_{k,\sigma}v_{k}c^{\dagger}_{k,\sigma}c_{k,\sigma}$ is the
current operator and $v_{k}=\frac{\partial\epsilon_{k}}{\partial k}$ is the
band velocity. We compute $\chi_{jj}(i\omega_{n})$ by performing a
perturbative expansion with respect to the electron-electron interaction $U$.
However, we do not start from the non-interacting
$G_{0}\left(k,i\nu_{n}\right)$ but instead, use a renormalized perturbation
scheme where the Green’s function is dressed by including self-energies
corrections

$G(k,i\nu_{n})=\frac{1}{i\nu_{n}-\epsilon_{k}-\Sigma(k,i\nu_{n})}.$ (S4)

In particular, we use here the second-order self-energy correction to include
a physical scattering time in the starting point of our expansion for the
conductivity. The dressed Green’s function is thus given by the Dyson equation

$\includegraphics[width=260.17464pt]{Figures/Sigma2},$ (S5)

where the single line is the non-interacting $G_{0}\left(k,i\nu_{n}\right)$,
the double line is the dressed $G(k,i\nu_{n})$ and the wavy line represent the
electron-electron interaction $U$. Note that there is no self-consistency in
the calculation of the dressed Green’s function in contrast to other ”bold
diagrammatic“ approaches.

The expansion for the current-current correlation function is thus given by
the skeleton series that excludes diagrams with self-energy insertions to
avoid double counting

(S6)

This diagrammatic expansion is treated with Algorithmic Matsubara Integration
(AMI) which allows for the symbolic summation over the internal Matsubara
frequencies. As we don’t have any analytic form for the self-energy that
appears in the dressed Green’s function, we use the spectral representation

$G(k,i\omega_{n})=\int\limits_{-\infty}^{\infty}dx\frac{A(k,x)}{i\omega_{n}-x},$
(S7)

where the spectral function $A(k,x)$ is directly related to the analytic
continuation of the self-energy

$A(k,x)=\frac{-1}{\pi}\frac{Im\Sigma(k,x)}{(x-\epsilon_{k}-Re\Sigma(k,x))^{2}+(Im\Sigma(k,x))^{2}}.$
(S8)

Here again, we use AMI to compute the self-energy on the real-frequency axis
without the need for numerical analytical continuation on a grid in momentum
space. In practice, the self-energy is computed using an analytic continuation
parameter $i\nu_{n}=\nu+i\Gamma$ with $\Gamma=1e-2$.

Figure S1:  Comparison of the DC conductivity when changing the analytic
continuation parameter $\Gamma$ between the renormalized (left) and the bare
(right) expansion for the current-current correlation function. We used $U=$,
$\beta=$

An example of the type of expressions generated by AMI can be obtained by
looking at the lowest order diagram in Eq.(S6)

$\displaystyle\chi_{jj}^{(0)}\left(i\omega_{n}\right)$
$\displaystyle=\frac{-2}{\beta
N}\sum_{k,i\nu_{n}}G(k,i\nu_{n})G(k,i\nu_{n}+i\omega_{n})$
$\displaystyle=\frac{-2}{\beta
N}\sum_{k,i\nu_{n}}\int\limits_{-\infty}^{\infty}dxdy\frac{A(k,x)A(k,y)}{\left(i\nu_{n}-x\right)\left(i\nu_{n}+i\omega_{n}-y\right)}$
$\displaystyle=\frac{-2}{N}\sum_{k}\int\limits_{-\infty}^{\infty}dxdy\frac{A(k,x)A(k,y)\left(n_{f}\left(x\right)-n_{f}\left(y\right)\right)}{i\omega_{n}+x-y},$
(S9)

where the remaining integrations are done using standard Monte-Carlo
integration schemes.

This renormalized perturbation scheme has the advantage of drastically
reducing the dependence of the result on the analytic continuation parameter
used as the electronic self-energy acts as a regulator itself. An example is
given in Fig.S1 where we compare the imaginary part of the current-current
correlation obtained using the renormalized expansion (left) to the one
obtained with the standard bare expansion (right) in the dc ($\omega\to 0$)
limit as we change the value of the analytic continuation parameter $\Gamma$.
While the bare expansion diverges as $\Gamma\to 0$, the renormalized expansion
converges to a finite value which is the dc conductivity.

Note that we can also get rid of the analytic continuation parameter entirely
as the additional integrations can be used to resolve the $\delta$-function
arising when taking the limit $i\omega_{n}=\omega+i0^{+}$

$\displaystyle\text{Im}\left[\chi_{jj}^{(0)}\left(\omega+i0^{+}\right)\right]=\frac{2\pi}{N}\sum_{k}\int\limits_{-\infty}^{\infty}$
$\displaystyle dxA(k,x)A(k,x+\omega)$
$\displaystyle\times\left(n_{f}\left(x\right)-n_{f}\left(x+\omega\right)\right).$
(S10)

This is the method we use for all frequency-dependent results at leading order
while we keep an analytic continuation parameter $\Gamma=1e-2$ in the
calculation of the second-order results (vertex corrections) for numerical
purposes.

Figure S2:  Frequency dependence of the imaginary (top) and real (bottom) part
of the current-current correlation function at leading order. We used $U=3$
and $\Gamma\to 0$ (see text).

##  VIII Temperature dependence of the current-current correlation function

We present here the direct result of the calculation of the current-current
calculation for the different temperatures $\beta=2,3,5,8.33$ and $20$. The
frequency dependence of the real and imaginary part of the current-current
correlation function at leading ($0^{\text{th}}$ order in interaction) are
shown in Fig.S2, error bars are smaller than the line width if not shown. As
mentioned previously, the analytic continuation parameter $\Gamma=0$ is used.
The calculation has been done up to $\omega=10$ but is not shown here for
clarity.

The frequency-dependent longitudinal conductivity can then directly be
obtained using Eq.(S2) for $\omega\neq 0$. The results for the real part of
the conductivity are shown in Fig.S3. The conductivity has a Drude-like form
with a temperature-dependent width and height. These are the results presented
in Fig.1 of the main text although using the modulus and phase of the
conductivity instead.

The dc conductivity $\sigma\left(\omega\to 0\right)$ is obtained by taking
symbolically the derivative of the current-current correlation function before
the integration and evaluating it at $\omega=0$,

$\sigma\left(\omega\to
0\right)=-i\left.\frac{d\chi_{jj}\left(\omega\right)}{d\omega}\right|_{\omega=0}.$
(S11)

The results are shown by the corresponding crosses at $\omega=0$ and the inset
shows the $1/T$ temperature dependence of the dc conductivity which is
equivalent to the linear-in-$T$ resistivity reported in other numerical
studies in the Hubbard model. This temperature dependence is also recovered in
the scaling of the scattering time at $\omega=0$ shown in the main text
(Fig.1).

Figure S3:  Real part of the conductivity obtained from $\chi_{jj}$ shown in
Fig.S2 for $\omega\neq 0$. The crosses indicate the value for
$\sigma_{DC}=\sigma\left(\omega\to 0\right)$ of the corresponding colours are
obtained using Eq.(S11). The inset shows the temperature dependence of
$\sigma_{DC}$.

##  IX Scaling of the conductivity with $U=4$

Studies of the Planckian model have shown that the power law exponent observed
in the modulus of the conductivity can be obtained analytically[16]. Using the
Planckian model (Eq.4 of the main text) with $\nu=1$,

$\displaystyle\text{Im}\left[\Sigma\left(\omega\right)\right]=-\frac{g\pi}{\beta}\
f\left(\beta\omega\right),$ (S12)

the exponent $\nu^{*}$ can be shown[16] to only depends on the coupling
constant $g$,

$\nu^{*}=1-\frac{2g\left[1+2g\left(1+\ln
4\right)\right]}{\pi^{2}g^{2}+\left[1+2g\left(1+\ln 4\right)\right]^{2}}.$
(S13)

This relation between the frequency dependence of the conductivity and the
coupling constant can be verified in our study of the Hubbard model as shown
in Fig.S4.

We can extract the exponent $\nu^{*}$ by identifying the plateau reached by
the phase of the conductivity for intermediate frequencies. Changing the value
of the Hubbard interaction that enters in the expression of the second-order
self-energy $\Sigma=U^{2}\Sigma^{(2)}$ leads to a reduction of the effective
exponent. We can extract the effective coupling constant from the data
obtained at $U=3$, giving $g\left(U=3\right)\sim 0.047$. If we take the
coupling constant $g$ to be proportional to the Hubbard interaction we then
expect $g\left(U=4\right)=0.047\times\left(\frac{4}{3}\right)^{2}$ when taking
the Hubbard interaction to be $U=4$. If the relation Eq.(S13) is respected we
should then find $\nu^{*}\left(U=4\right)\sim 0.88$ which is what we observe
in Fig.S4.

Figure S4: Frequency dependence of the phase of the conductivity at a fixed
temperature $\beta=20$ obtained for two different values of the Hubbard
interaction $U$. The value of the effective exponent $\nu^{*}$ decreases with
the interaction strength according to Eq.(S13). The inset shows the
corresponding modulus of the conductivity that has a power law behaviour with
an exponent $\nu^{*}$ similar to the one extracted from the phase plateau.

We can also check that the scaling properties described in the main text
remain when we change the value of the Hubbard interaction. The scattering
time $\tau\left(\omega\right)$ and effective mass $m\left(\omega\right)/m_{0}$
extracted form the conductivity with $U=4$ when using a generalized Drude
model,

$\frac{1}{\tau\left(\omega\right)}=\text{Re}\left[\frac{1}{\sigma\left(\omega\right)}\right],\qquad\frac{m^{*}\left(\omega\right)}{m_{0}}=-\text{Im}\left[\frac{1}{\omega\sigma\left(\omega\right)}\right],$
(S14)

are shown in Fig.S5 for the lowest temperatures. We can see that we recover
the temperature scaling observed at $U=3$ which we described in the main text.

Figure S5: $\bm{(c)}$ inverse scattering time and $\bm{(d)}$ effective mass
extracted from the conductivity when rewritten as an extended Drude model
given in Eq.(S14) for $U=4$.

Note that because we restricted our calculation to the second-order self-
energy and to the leading order for the current-current calculation we do not
expect it to hold in the moderate to strong coupling regime. The results
presented here should however be exact in the weak coupling regime when the
higher order terms we neglected are suppressed and the fact that the scaling
properties discussed are independent of the interaction strength means that we
expect the Planckian behaviour to occur in the weakly coupled Hubbard model.