# Planckian Behavior of Cuprate Superconductors: Reconciling the Scaling of
Optical Conductivity with Resistivity and Specific Heat

B. Michon  Department of Physics, City University of Hong Kong, 83 Tat Chee
Avenue, Kowloon, Hong Kong, China  Hong Kong Institute for Advanced Study,
City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China
Department of Quantum Matter Physics, University of Geneva, 24 quai Ernest-
Ansermet, 1211 Geneva, Switzerland  C. Berthod  Department of Quantum Matter
Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva,
Switzerland  C. W. Rischau  Department of Quantum Matter Physics, University
of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland  A. Ataei
Institut Quantique, Département de Physique & RQMP, Université de Sherbrooke,
Sherbrooke, Québec, Canada  L. Chen  Institut Quantique, Département de
Physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada  S.
Komiya  Central Research Institute of Electric Power Industry, Materials
Science Research Laboratory, 2-6-1 Nagasaka, Yokosuka, Kanagawa, Japan  S. Ono
Central Research Institute of Electric Power Industry, Materials Science
Research Laboratory, 2-6-1 Nagasaka, Yokosuka, Kanagawa, Japan  L. Taillefer
Institut Quantique, Département de Physique & RQMP, Université de Sherbrooke,
Sherbrooke, Québec, Canada  Canadian Institute for Advanced Research, Toronto,
Ontario, Canada  D. van der Marel  Department of Quantum Matter Physics,
University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland  A.
Georges  Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France
Center for Computational Quantum Physics, Flatiron Institute, New York, New
York 10010, USA  Department of Quantum Matter Physics, University of Geneva,
24 quai Ernest-Ansermet, 1211 Geneva, Switzerland  CPHT, CNRS, École
Polytechnique, IP Paris, F-91128 Palaiseau, France

(May 7, 2022)

###### Abstract

Materials tuned to a quantum critical point display universal scaling
properties as a function of temperature $T$ and frequency $\omega$. A long-
standing puzzle in the quantum critical behavior of cuprate superconductors
has been the observed power-law dependence of optical conductivity with an
exponent smaller than one, to be contrasted with the $T$-linear dependence of
the resistivity and the $\omega$-linear dependence of the optical scattering
rate. Here, we address this question by presenting and analyzing resistivity
and optical spectroscopy measurements on La2-xSrxCuO4 with $x=0.24$. We show
that the data display $\hbar\omega/k_{\mathrm{B}}T$ scaling over a wide range
of frequency and temperature, as well as $T$-linear resistivity and a $\sim\ln
T$ enhancement of the optical effective mass. The latter corroborates the
logarithmic enhancement of the specific heat coefficient previously reported
for this compound. We show that a simple $T,\omega$-linear scaling Ansatz for
the inelastic scattering rate leads to an excellent theoretical description of
the experimental data. We provide a solution to the aforementioned puzzle by
showing that the optical spectra obey an _apparent_ power-law behavior over an
intermediate range of frequencies, with an effective exponent depending
continuously on the inelastic scattering rate. This theoretical framework
yields a unified description of the optical spectra, resistivity and specific
heat, which provides new opportunities for understanding the unique properties
of quantum critical matter.

##  I Introduction

The linear-in-temperature electrical resistivity is one of the remarkable
properties of the cuprate high temperature superconductors [1, 2, 3, 4]. By
means of chemical doping it is possible to tune these materials to a carrier
concentration where $\rho(T)=\rho_{0}+AT$ in a broad temperature range. For
Bi2+xSr2-yCuO6±δ, it has been possible to demonstrate this from 7 to 700 K [5]
by virtue of the low $T_{c}$ of this material. For the underdoped cuprates the
linear-in-$T$ resistivity is ubiquitous for temperatures $T>T^{*}$, where
$T^{*}$ is a doping-dependent cross-over temperature that decreases as a
function of doping and vanishes at a critical doping $p^{*}$. From one cuprate
family to another the exact value of $p^{*}$ varies widely within the range
$0.19<p^{*}<0.40$ [6, 7, 8, 9, 10]. For doping levels $p<p^{*}$, many of the
physical properties indicate the presence of a pseudogap that vanishes at
$p^{*}$ [11, 12]. When $p$ is tuned exactly to $p^{*}$, the $T$-linear
resistivity persists down to $T=0$ K if superconductivity is suppressed e.g.
by applying a magnetic field [13, 14, 15]. The conundrum of the $T$-linear
resistivity has been associated to the idea that dissipation can not exceed
the “Planckian” limit $k_{\mathrm{B}}T/\hbar$ [16, 17, 18, 19], for which
there is now strong experimental support [20, 21].

As expected for a system tuned to a quantum critical point [22],
$\hbar\omega/k_{\mathrm{B}}T$ scaling has been observed in the optical
properties of high-$T_{c}$ cuprates [23, 24] over some range of doping. The
optical scattering rate obtained from an extended Drude fit to the data was
found to obey a $T$-linear dependence $1/\tau\sim k_{\mathrm{B}}T/\hbar$ in
the low-frequency regime ($\hbar\omega\lesssim 1.5k_{\mathrm{B}}T$) as well as
a linear dependence on energy over an extended frequency range [25, 26, 27,
28, 29, 24]. This observation is qualitatively consistent with the $T$-linear
dependence of the resistivity and ‘Planckian’ behavior. In contrast, by
analyzing the modulus and phase of the optical conductivity itself, a power-
law behavior $\sigma(\omega)=C/(-i\omega)^{\nu^{*}}$ with an exponent
$\nu^{*}<1$ was reported at higher frequencies $\hbar\omega\gtrsim
1.5k_{\mathrm{B}}T$ [28, 29, 30, 23, 24, 31]. The exponent was found to be in
the range $\nu^{*}\approx 0.65$ with some dependence on sample and doping
level [26, 28, 29, 23]. Hence, from these previous analyses, it would appear
that different power laws are needed to describe optical spectroscopy data:
one at low frequency consistent with $\hbar\omega/k_{\mathrm{B}}T$ scaling and
Planckian behavior ($\nu=1$) and another one with $\nu^{*}<1$ at higher
frequency, most apparent on the optical conductivity itself in contrast to
$1/\tau$. Despite a number of theoretical attempts to account for a power-law
dependence of the conductivity [32, 33, 34, 35, 36, 37, 38, 39, 40, 41], this
puzzle has not been resolved to this day, to the best of our knowledge.

Here we present systematic measurements of the optical spectra, as well as dc
resistivity, of a La2-xSrxCuO4 (LSCO) sample with $x=p=0.24$ close to the
pseudogap critical point, over a broad range of temperature and frequency. We
demonstrate that the data display Planckian quantum critical scaling over an
unprecedented range of $\hbar\omega/k_{\mathrm{B}}T$. Furthermore, a direct
analysis of the data reveals a logarithmic temperature dependence of the
optical effective mass. This establishes a direct connection to another
hallmark of Planckian behavior, namely the logarithmic enhancement of the
specific heat coefficient $C/T\sim\ln T$ previously observed for LSCO at
$p=0.24$ [42] as well as for other cuprate superconductors such as Eu-LSCO and
Nd-LSCO [43].

We introduce a theoretical framework which relies on a minimal Planckian
scaling Ansatz for the inelastic scattering rate. We show that this provides
an excellent description of the experimental data. Our theoretical analysis
offers, notably, a solution to the puzzle mentioned above. Indeed we show
that, despite the purely Planckian Ansatz which underlies our model, the
optical conductivity computed in this framework is well described by an
apparent power law with $\nu^{*}<1$ over an intermediate frequency regime, as
also observed in our experimental data. The effective exponent $\nu^{*}$ is
found to be non-universal and to depend on the inelastic coupling constant,
which we determine from several independent considerations. The proposed
theoretical analysis provides a unifying framework in which the behavior of
the $T$-linear resistivity, $\ln T$ behavior of $C/T$, and scaling properties
of the optical spectra can all be understood in a consistent manner.

##  II Experiments

###  II.1 Optical spectra and resistivity

We measured the optical properties and extracted the complex optical
conductivity $\sigma(\omega,T)$ of an LSCO single crystal with a-b orientation
(CuO2 planes). The hole doping is $p=x=0.24$, which places our sample above
and close to the pseudogap critical point of the LSCO family [7, 44, 14]. The
pseudogap state for $T<T^{*}$, $p<p^{*}$ is well characterized by transport
measurements [12] and ARPES [11]. The relatively low $T_{c}=19$ K of this
sample is interesting for extracting the normal-state properties in optics
down to low temperatures without using any external magnetic field. The LSCO
$p=0.24$ sample used for the present study was prepared as described in Refs.
45, 46. In particular, this sample is the same LSCO $p=0.24$ sample as in Ref.
46 where the evolution of optical spectral weights as a function of doping was
reported.

Figure 1:  (a) Real and (b) imaginary part of the optical conductivity
$\sigma$ deduced from the dielectric function $\epsilon$ (Fig. 6 in Appendix
A), using Eq. (1) and the value $\epsilon_{\infty}=2.76$. (c) Scattering rate
and (d) effective mass deduced from Eq. (2) using $K=211$ meV. The values of
$\epsilon_{\infty}$ and $K$ are discussed and justified in the text. Inset:
Temperature dependence of $m^{*}/m$ at $\hbar\omega=5k_{\mathrm{B}}T$ [see
dots in (d)].

We measured the infrared reflectivity from 2.5 meV to 0.5 eV using a Fourier-
transform spectrometer with a home-built UHV optical flow cryostat and in-situ
gold evaporation for calibrating the signal. In the energy range from 0.5 to 5
eV, we measured the real and imaginary parts of the dielectric function
$\epsilon(\omega)$ using a home-built UHV cryostat installed in a visible-UV
ellipsometer. Raw data for $\epsilon(\omega)$ are presented in Fig. 6 of
Appendix A. Combining the ellipsometry and reflectivity data and using the
Kramers–Kronig relations between the reflectivity amplitude and phase, we
obtained for each measured temperature the complex dielectric function in the
range from 2.5 meV to 5 eV (see Appendix A and Fig. 6). The complex optical
conductivity $\sigma(\omega)$ of low-energy transitions is directly linked to
$\epsilon(\omega)$ by 111We use international SI units, where
$\epsilon_{0}=8.85\times 10^{-5}$ kS/(cm THz). In the Gaussian CGS system,
$\epsilon_{0}=1/(4\pi)$.

$\sigma(\omega)=i\epsilon_{0}\omega\left[\epsilon_{\infty}-\epsilon(\omega)\right].$
(1)

In this expression, $\epsilon_{\infty}$ is the background relative
permittivity due to high-energy transitions. In Sec. II.2 we propose and
discuss in details a procedure to estimate the value of $\epsilon_{\infty}$.
Using the value $\epsilon_{\infty}=2.76$ determined there, we display in Figs.
1(a,b) the real and imaginary parts of the optical conductivity. In Fig. 1(a),
one observes a Drude-like behavior upon cooling from 300 K, characterized by a
sharpening of the Drude peak in $\mathrm{Re}\,\sigma$ and a maximum in
$\mathrm{Im}\,\sigma$ at a frequency that decreases with decreasing $T$. For
temperatures below 75 K, the Drude peak is narrower than the minimum photon
energy accessible with our spectrometer, 2.5 meV, which gives the impression
of a gap opening in $\mathrm{Re}\,\sigma$. Yet, the superconducting transition
only occurs at $T_{c}=19$ K. The conductivity decreases monotonically between
0.1 and 0.4 eV, before interband transitions gradually set in.

As is common for materials with strong electronic correlations, and well
documented for cuprates in particular [48, 49], the optical conductivity has a
richer frequency dependence than that of a simple Drude model. It is
convenient however to consider a generalized Drude parametrisation in terms of
a frequency-dependent scattering rate $1/\tau(\omega)$ and mass enhancement
$m^{*}(\omega)/m$:

$\sigma(\omega)=\frac{e^{2}K/(\hbar^{2}d_{c})}{1/{\tau}(\omega)-i\omega\,m^{*}(\omega)/m},$
(2a) so that the scattering rate and mass enhancement can be determined from
the optical conductivity according to: $\displaystyle\frac{1}{\tau(\omega)}$
$\displaystyle=\frac{e^{2}K}{\hbar^{2}d_{c}}\,\mathrm{Re}\,\left[\frac{1}{\sigma(\omega)}\right]$
(2b) $\displaystyle\frac{m^{*}(\omega)}{m}$
$\displaystyle=-\frac{e^{2}K}{\hbar^{2}d_{c}}\,\mathrm{Im}\,\left[\frac{1}{\omega\,\sigma(\omega)}\right].$
(2c)

In these expressions, $d_{c}=6.605$ Å is the distance between two CuO2 planes,
$m$ is the band mass and $K$ is the spectral weight for a single plane. The
determination of $m$ and $K$ is also discussed in Sec. II.2 along with that of
$\epsilon_{\infty}$. $K$ only affects the absolute magnitude of $1/\tau$ and
$m^{*}/m$, while the choice of $\epsilon_{\infty}$ has a more significant
influence.

The scattering rate $\hbar/\tau$ deduced using Eq. (2) and the values of $K$
and $\epsilon_{\infty}$ discussed below are displayed on Fig. 1(c). It depends
linearly on frequency for $k_{\mathrm{B}}T\ll\hbar\omega\lesssim 0.4$ eV and
approaches a constant value for $\hbar\omega<k_{\mathrm{B}}T$. This behavior
is similar to that reported for Bi2212 [23]. Above $0.4~{}\mathrm{eV}\approx
3000~{}\mathrm{cm}^{-1}$, the Bi2212 data show a crossover to a negative
curvature and a saturation of $\hbar/\tau$. The LSCO data indicate instead a
crossover to a positive curvature above 0.4 eV (not shown). The sign of the
curvature above 0.4 eV depends on $\epsilon_{\infty}$ and changes from
positive to negative near $\epsilon_{\infty}=4.5$. Our determination
$\epsilon_{\infty}=2.76$ presented in Sec. II.2 does not take into account
data for $\hbar\omega>0.4$ eV and may therefore yield unreliable values of
$\hbar/\tau$ in that range (see Appendix A).

This linear dependence of the scattering rate calls for a comparison with
resistivity. Hence we have also measured the temperature dependence of the
resistivity of our sample under two magnetic fields $H=0$ T and $H=16$ T. As
displayed on Fig. 2(a) the resistivity has a linear $T$-dependence
$\rho=\rho_{0}+AT$ over an extended range of temperature, with $A\approx
0.63\,\mu\Omega~{}\mathrm{cm/K}$. This is a hallmark of cuprates in this
regime of doping [13, 14, 20, 10, 50]. It is qualitatively consistent with the
observed linear frequency dependence of the scattering rate and, as discussed
later in this paper, also in good quantitative agreement with the
$\omega\rightarrow 0$ extrapolation of our optical data within experimental
uncertainties.

The optical mass enhancement $m^{*}(\omega)/m$ is displayed in Fig. 1(d). With
the chosen normalization, $m^{*}/m$ does not reach the asymptotic value of one
in the range $\hbar\omega<0.4$ eV, which means that intra- and interband
and/or mid-infrared transitions overlap above 0.4 eV. The inset of Fig. 1(d)
shows a semi-log plot of the mass enhancement evaluated at
$\hbar\omega=5k_{\mathrm{B}}T$, where the noise level is low for $T\geqslant
40$ K. Despite the larger uncertainties at low $T$, this plot clearly reveals
a logarithmic temperature dependence of $m^{*}/m$. This is a robust feature of
the data, independent of the choice of $\epsilon_{\infty}$ and $K$. We note
that the specific heat coefficient $C/T$ of LSCO at the same doping level was
previously reported to display a logarithmic dependence on temperature, see
Fig. 2(c) [43, 42]. We will further elaborate on this important finding of a
logarithmic dependence of the optical mass and discuss its relation to
specific heat in the next section.

###  II.2 Scaling analysis

In this section, we consider simultaneously the frequency and temperature
dependence of the optical properties and investigate whether
$\hbar\omega/k_{\mathrm{B}}T$ scaling holds for this sample close to the
pseudogap critical point. We propose a procedure to determine the three
parameters $\epsilon_{\infty}$, $K$, and $m$ introduced above.

####  II.2.1 Putting $\omega/T$ scaling to the test

Quantum systems close to a quantum critical point display scale invariance.
Temperature being the only relevant energy scale in the quantum critical
regime, this leads in many cases to $\omega/T$ scaling [22] (in most of the
discussion below, we set $\hbar=k_{\mathrm{B}}=1$ except when mentioned
explicitly). In such a system we expect the complex optical conductivity to
obey a scaling behavior $1/\sigma(\omega,T)\propto T^{\nu}F(\omega/T)$, with
$\nu\leqslant 1$ a critical exponent. More precisely, the scaling properties
of the optical scattering rate and effective mass read:

$\displaystyle 1/\tau(\omega,T)$ $\displaystyle=T^{\nu}f_{\tau}(\omega/T)$
(3a) $\displaystyle m^{*}(\omega,T)-m^{*}(0,T)$
$\displaystyle=T^{\nu-1}f_{m}(\omega/T).$ (3b)

with $f_{\tau}$ and $f_{m}$ two scaling functions. This behavior requires that
both $\hbar\omega$ and $k_{\mathrm{B}}T$ are smaller than a high-energy
electronic cutoff, but their ratio can be arbitrary. Furthermore, we note that
when $\nu=1$ (Planckian case) the scaling is violated by logarithmic terms,
which control in particular the zero-frequency value of the optical mass
$m^{*}(0,T)$. As shown in Sec. III within a simple theoretical model,
$\omega/T$ scaling nonetheless holds in this case to an excellent
approximation provided that $m^{*}(0,T)$ is subtracted, as in Eq. (3b). We
also note that in a Fermi liquid, the single-particle scattering rate
$\propto\omega^{2}+(\pi T)^{2}$ does obey $\omega/T$ scaling (with formally
$\nu=2$), but the optical conductivity does not. Indeed, it involves
$\omega/T^{2}$ terms violating scaling, and hence depends on two scaling
variables $\omega/T^{2}$ and $\omega/T$, as is already clear from an
(approximate) generalized Drude expression
$1/\sigma\approx-i\omega+\tau_{0}[\omega^{2}+(2\pi T)^{2}]$. For a detailed
discussion of this point, see Ref. 51. Such violations of scaling by
$\omega/T^{\nu}$ terms apply more generally to the case where the scattering
rate varies as $T^{\nu}$ with $\nu>1$. Hence, $\omega/T$ scaling for both the
optical scattering rate and optical effective mass are a hallmark of non Fermi
liquid behavior with $\nu\leqslant 1$. Previous work has indeed provided
evidence for $\omega/T$ scaling in the optical properties of cuprates [23,
24].

Here, we investigate whether our optical data obey $\omega/T$ scaling. We find
that the quality of the scaling depends sensitively on the chosen value of
$\epsilon_{\infty}$. Different prescriptions in the literature to fix
$\epsilon_{\infty}$ yield — independently of the method used — values ranging
from $\epsilon_{\infty}\approx 4.3$ for strongly underdoped Bi2212 to
$\epsilon_{\infty}\approx 5.6$ for strongly overdoped Bi2212 [52, 31]. Rather
than setting an a priori value for $\epsilon_{\infty}$, we follow here a
different route and we choose the value that yields the best scaling collapse
for a given value of the exponent $\nu$. This program is straightforwardly
implemented for $1/\tau$ and indicates that the best scaling collapse is
achieved with $\nu\approx 1$ and $\epsilon_{\infty}\approx 3$, see Fig. 2(b)
as well as Appendix A and Fig. 7. Turning to $m^{*}$, we found that
subtracting the dc value $m^{*}(\omega=0,T)$ is crucial when attempting to
collapse the data. Extrapolating optical data to zero frequency is hampered by
noise. Hence, instead of attempting an extrapolation, we consider $m^{*}(0,T)$
as adjustable values that we again tune such as to optimize the collapse of
the optical data. This analysis of $m^{*}/m$ confirms that the best scaling
collapse occurs for $\nu\approx 1$ but indicates a larger
$\epsilon_{\infty}\approx 7$ (Appendix A and Fig. 8). The determination of
$\epsilon_{\infty}$ from the mass data depends sensitively on the frequency
range tested for scaling and drops to value below $\epsilon_{\infty}=3$ when
focusing on lower frequencies. As a third step, we perform a simultaneous
optimization of the data collapse for $1/\tau$ and $m^{*}/m$, which yields the
values $\nu=1$, $\epsilon_{\infty}=2.76$ which we will adopt throughout the
following. Note that a determination of $\epsilon_{\infty}$ by separation of
the high-frequency modes in a Drude–Lorentz representation of
$\epsilon(\omega)$ yields a larger value $\epsilon_{\infty}=4.5\pm 0.5$, as
typically found in the cuprates [23, 53, 31]. Importantly, all our conclusions
hold if we use this latter value in the analysis, however the quality of the
scaling displayed in Figs. 2 and 5 is slightly degraded.

Figure 2:  (a) Temperature-dependent resistivity measured in zero field
(black) and at 16 teslas (red). The inset emphasizes the linearity of the 16 T
data at low temperature. The dashed line shows $\rho_{0}+AT$ with
$\rho_{0}=12.2~{}\mu\Omega~{}\mathrm{cm}$ and
$A=0.63~{}\mu\Omega~{}\mathrm{cm/K}$. (b) Scattering rate divided by
temperature plotted versus $\omega/T$; the collapse of the curves indicates a
behavior $1/\tau\sim Tf_{\tau}(\omega/T)$. (c) Effective quasiparticle mass
(in units of the indicated band mass $m$) deduced from the low-temperature
electronic specific heat [42]
[$m^{*}_{\mathrm{Cp}}=(3/\pi)(\hbar^{2}d_{c}/k_{\mathrm{B}}^{2})(C/T)$] and
zero-frequency optical mass enhancement; the dashed lines indicate $\ln T$
behavior. (d) Optical mass minus the zero-frequency mass shown in (c) plotted
versus $\omega/T$; the collapse of the curves indicates a behavior
$m^{*}(\omega)-m^{*}(0)\sim f_{m}(\omega/T)$. $\epsilon_{\infty}=2.76$ was
used here as in Fig. 1.

####  II.2.2 Scaling of the optical scattering rate and connection to
resistivity

The scaling properties of the scattering rate obtained from our optical data
according to the procedure described above is illustrated on Fig. 2(b), which
displays $\hbar/\tau$ divided by $k_{\mathrm{B}}T$ and plotted versus
$\hbar\omega/k_{\mathrm{B}}T$ for temperatures above the superconducting
transition. The collapse of the curves at different temperatures reveals the
behavior $\hbar/\tau\propto Tf_{\tau}(\omega/T)$. The function $f_{\tau}(x)$
reaches a constant $f_{\tau}(0)>0$ at small values of the argument, and
behaves for large arguments as $f_{\tau}(x\gg 1)\propto x$. This is consistent
with the typical quantum critical behavior $\hbar/\tau\sim\max(T,\omega)$.
When inserted in the $\omega=0$ limit of Eq. (2), the value
$f_{\tau}(0)\approx 5$ indicated by Fig. 2(b) yields $1/\sigma(0)=AT$ with
$A=0.55~{}\mu\Omega~{}\mathrm{cm/K}$, in fairly good agreement with the
measured resistivity [Fig. 2(a)]. Hence the resistivity and optical-
spectroscopy data are fully consistent, both of them supporting a Planckian
dissipation scenario with $\nu=1$ for LSCO at $p=0.24$.

####  II.2.3 Spectral weight, effective mass and connection to specific heat

The dc mass enhancement values $m^{*}(0,T)/m$ resulting from the procedure
described above are displayed in Fig. 2(c) [54]. Remarkably, as seen on this
figure, the scaling analysis delivers an almost perfectly logarithmic
temperature dependence of $m^{*}(0,T)$, consistent with a Planckian behavior
$\nu=1$. As mentioned above, this logarithmic behavior can actually be
identified in the unprocessed optical data, (see inset of Fig. 1). In order to
compare this behavior to the corresponding logarithmic behavior reported for
the specific heat, we note that the scaling analysis provides $m^{*}(0,T)$ up
to a multiplicative constant $Km$, where $m$ is the band mass. In contrast,
the electronic specific heat yields the quasiparticle mass in units of the
bare electron mass $m_{e}$. We expect that the logarithmic $T$-variation of
$m^{*}(0,T)$ and $m^{*}_{\mathrm{qp}}\propto C/T$ are both due to the critical
inelastic scattering and that the $\ln T$ term in each quantity should
therefore have identical prefactors. Imposing this identity provides a
relationship between $Km$ and $m_{e}$, namely $(m/m_{e})K=583$ meV.

Remarkably, we have found that this condition is obeyed within less than a
percent by a square-lattice tight-binding model with parameters appropriate
for LSCO at $p=0.24$ (Appendix D). This model has nearest and next-nearest
neighbor hopping amplitudes $t=0.3$ eV and $t^{\prime}/t=-0.17$ [55],
respectively, and an electronic density $n=0.76/a^{2}$. The Fermi-level
density of states is $1.646/(\mathrm{eV}a^{2})$, which corresponds to a band
mass $m/m_{e}=2.76$ using the LSCO lattice parameter $a=3.78$ Å. The spectral
weight is $K=211$ meV, such that the prediction of this tight-binding model is
$(m/m_{e})K=582$ meV, in perfect agreement with the previously determined
value. In view of this agreement, we use the tight-binding model in order to
fix the remaining two system parameters: $m=2.76\,m_{e}$ and $K=211$ meV.

Figure 2(c) compares the mass enhancement inferred from the low-temperature
specific heat and from the scaling analysis of the optical data. The tight-
binding value of the product $Km$ insures that both data sets have the same
slope on a semi-log plot. However, the resulting optical mass enhancement is
larger than the quasiparticle mass enhancement by $\approx 0.75$, which is
also the amount by which the infrared mass enhancement exceeds unity in Fig.
1(d). This points to a mixture of mid-infrared transitions into the intraband
signal, which prevents us from accessing the absolute value of the genuine
intraband mass by optical means. Figure 2(d) shows the collapse of the
frequency-dependent change of the mass enhancement, confirming the behavior
$m^{*}(\omega)-m^{*}(0)\approx T^{\nu-1}f_{m}(\omega/T)$ with $\nu=1$. The
shape of the scaling function $f_{m}(x)$ agrees remarkably well with the
theoretical prediction derived in Sec. III below.

####  II.2.4 Apparent power-law behavior: a puzzle

The above scaling analysis has led us to the following conclusions. (i) The
optical scattering rate and optical mass enhancement of LSCO at $p=0.24$
exhibit $\omega/T$ scaling over two decades for the chosen value
$\epsilon_{\infty}=2.76$. (ii) The best collapse of the data is achieved for
an exponent $\nu=1$ corresponding to Planckian dissipation. This behavior is
consistent with the measured $T$-linear resistivity. (iii) The temperature
dependence of $m^{*}(0,T)$ that produces the best data collapse is
logarithmic, consistently with the temperature dependence of the electronic
specific heat.

Hence, the data presented in Fig. 2 provide compelling evidence that the low-
energy carriers in LSCO at the doping $p=0.24$ experience linear-in-energy and
linear-in-temperature inelastic scattering processes, as expected in a scale-
invariant quantum critical system characterized by Planckian dissipation. It
is therefore at first sight surprising that the infrared conductivity exhibits
as a function of frequency a power law with an exponent that is clearly
smaller than unity, as highlighted in Figs. 3(a) and 3(b). These figures show
that the modulus and phase of $\sigma$ are both to a good accuracy consistent
with the behavior
$\sigma\propto(-i\omega)^{-\nu^{*}}=\omega^{-\nu^{*}}e^{i\frac{\pi}{2}\nu^{*}}$
with an exponent $\nu^{*}=0.8$. A similar behavior with exponent
$\nu^{*}\approx 0.6$ was reported for optimally- and overdoped Bi2212 [23],
while earlier optical investigations of YBCO and Bi2212 have also reported
power law behavior of $\mathrm{Re}\,\sigma(\omega)$ [26, 28, 29]. We now
address this question by considering a theoretical model presented in the
following section. As derived there, and illustrated on Figs. 3(c) and 3(d),
we show that an apparent exponent $\nu^{*}<1$ is actually _predicted by
theory_ for Planckian systems with single-particle self-energy exponent
$\nu=1$, over an intermediate range of values of $\omega/T$. This is one of
the central claims of our work.

Figure 3:  (a) Modulus and (b) phase of the complex conductivity shown in Fig.
1(a) and 1(b); the modulus decays with an exponent $\nu^{*}\approx 0.8$ and
the phase approaches a value slightly lower than $(\pi/2)\nu^{*}$. (c) and
(d): same quantities calculated using a Planckian model with linear-in-energy
scattering rate, Eqs. (5) and (8). The model and parameters are discussed in
the text.

##  III Theory

In this section, we consider a simple theoretical model and explore its
implications for the optical conductivity. Our central assumption is that the
inelastic scattering rate (imaginary part of the self-energy) obeys the
following scaling property:

$-\mathrm{Im}\,\Sigma(\varepsilon)=g\pi
k_{\mathrm{B}}TS\left(\frac{\varepsilon}{k_{\mathrm{B}}T}\right).$ (4)

In this expression $g$ is a dimensionless inelastic coupling constant and
$\varepsilon=\hbar\omega$. This $\hbar\omega/k_{\mathrm{B}}T$ scaling form is
assumed to apply when both $\hbar\omega$ and $k_{\mathrm{B}}T$ are smaller
than a high-energy cutoff $\Lambda$ but their ratio can be arbitrary. The
detailed form of the scaling function $S$ is not essential, except for the
requirements that $S(0)$ is finite and $S(x\gg 1)\propto|x|$. These properties
ensure that the low-frequency inelastic scattering rate depends linearly on
$T$ for $\hbar\omega\ll k_{\mathrm{B}}T$ and that dissipation is linear in
energy for $\hbar\omega\gg k_{\mathrm{B}}T$, which are hallmarks of
‘Planckian’ behavior. We note that such a scaling form appears in the context
of microscopic models such as overscreened non-Fermi liquid Kondo models [56]
and the doped SYK model close to a quantum critical point [57, 58, 59, 60,
61]. In such models, conformal invariance applies and dictates the form of the
scaling function to be $S(x)=x\coth(x/2)$ (with possible modifications
accounting for a particle-hole spectral asymmetry parameter, see Refs. 56, 62
and Appendix E. We have assumed that the inelastic scattering rate is momentum
independent (spatially local) i.e. uniform along the Fermi surface. This
assumption is supported by recent angular-dependent magnetoresistance
experiments on Nd-LSCO at a doping close to the pseudogap quantum critical
point [21] — see also Ref. 63. In contrast, the elastic part of the scattering
rate (not included in our theoretical model) was found to be strongly
anisotropic (angular dependent).

The real part of the self-energy is obtained from the Kramers–Kronig relation
which reads, substituting the scaling form above:

$\Sigma(z)=gk_{\mathrm{B}}T\int_{\Lambda}dx\,\frac{S(x)}{z/k_{\mathrm{B}}T-x}.$
(5)

We note that this expression is only defined provided the integral is bounded
at high-frequency by the cutoff $\Lambda$, as detailed in Appendix B. This
reflects into a logarithmic temperature dependence at low energy:

$\mathrm{Re}\,[\Sigma(\varepsilon)-\Sigma(0)]=-2g\varepsilon\ln(a\Lambda/k_{\mathrm{B}}T)$
(6)

with $a=0.770542$ a numerical constant (Appendix B). Correspondingly, the
effective mass of quasiparticles, as well as the specific heat, is
logarithmically divergent at low temperature:

$\frac{m^{*}_{\mathrm{qp}}}{m}=\frac{1}{Z}=1+2g\ln\left(a\frac{\Lambda}{k_{\mathrm{B}}T}\right)$
(7)

with $1/Z=1-d\mathrm{Re}\,\Sigma(\varepsilon)/d\varepsilon|_{\varepsilon=0}$.
Importantly, the coefficient of the dominant $\ln T$ term depends only on the
value of the inelastic coupling $g$.

In a local (momentum-independent) theory, vertex corrections are absent [64,
65] and the optical conductivity can thus be directly computed from the
knowledge of the self-energy as [66]:

$\sigma(\omega)=\frac{i\Phi(0)}{\omega}\int_{-\infty}^{\infty}d\varepsilon\,\frac{f(\varepsilon)-f(\varepsilon+\hbar\omega)}{\hbar\omega+\Sigma^{*}(\varepsilon)-\Sigma(\varepsilon+\hbar\omega)}$
(8)

where $f(\varepsilon)=(e^{\varepsilon/k_{\mathrm{B}}T}+1)^{-1}$ is the Fermi
function and $\Sigma^{*}$ denotes complex conjugation. In this expression
$\Phi(\varepsilon)=2(e/\hbar)^{2}\int_{\mathrm{BZ}}\frac{d^{2}k}{(2\pi)^{2}}\left(\partial\varepsilon_{\bm{k}}/\partial
k_{x}\right)^{2}\delta(\varepsilon+\mu_{0}-\varepsilon_{\bm{k}})$ is the
transport function associated with the bare bandstructure. We have assumed
that its energy dependence can be neglected so that only the value $\Phi(0)$
at the Fermi level matters (we set $\mu_{0}=0$ by convention). Using a tight-
binding model for the band dispersion, $\Phi(0)$ can be related to the
spectral weight $K$ discussed in the previous section as:
$(\hbar/e)^{2}\Phi_{\mathrm{2D}}(0)=K=211$ meV, i.e.
$\Phi(0)=\Phi_{\mathrm{2D}}(0)/d_{c}=1.33\times
10^{7}\epsilon_{0}~{}\mathrm{THz}^{2}$ (see Appendix D).

Within our model, the behavior of the optical conductivity relies on three
parameters: the cutoff $\Lambda$, the Drude weight related to $\Phi(0)$ and,
importantly, the dimensionless inelastic coupling $g$. An analysis of Eq. (8),
detailed in Appendix B, yields the following behavior in the different
frequency regimes:

  * •

$\hbar\omega\lesssim k_{\mathrm{B}}T$. The optical conductivity in this regime
takes a Drude-like form Eq. (2) with $\hbar/\tau=4\pi gk_{\mathrm{B}}T$. The
numerically computed zero-frequency optical mass enhancement $m^{*}(0)/m$
agrees very well with $m^{*}_{\mathrm{qp}}/m=1/Z$ as given by Eq. (7), see
Fig. 11. Fitting Eq. (7) to the $m^{*}(0)/m$ data in Fig. 2(c) provides the
values $g=0.23$ and $\Lambda=0.4$ eV.

  * •

$\hbar\omega\gtrsim\Lambda$. In this high-frequency regime, the asymptotic
behavior is fixed by causality and reads $|\sigma|\sim 1/\omega$,
$\mathrm{arg}(\sigma)\to\pi/2$ (see Figs. 9 and 10).

  * •

$k_{\mathrm{B}}T\lesssim\hbar\omega\lesssim\Lambda$. In this regime, which is
the most important in practice when considering our experimental data, one can
derive the following expression:

$\sigma(\omega)\approx\frac{\Phi(0)}{-i\omega}\frac{1}{1+2g\left[1-\ln\left(\frac{\hbar\omega}{2\Lambda}\right)\right]+i\pi
g}.$ (9)

Remarkably, as shown in Fig. 4, the theoretical optical conductivity is very
well approximated in this regime by an apparent power-law dependence
$|\sigma|\sim|\omega|^{-\nu^{*}}$, over at least a decade in frequency. The
effective exponent $\nu^{*}<1$ depends continuously on the inelastic coupling
constant $g$ and can be estimated as:

$\displaystyle\nu^{*}$
$\displaystyle\equiv-\left.\frac{d\ln|\sigma|}{d\ln\omega}\right|_{\hbar\omega=\Lambda/2}$
$\displaystyle=1-\frac{2g[1+2g(1+\ln 4)]}{\pi^{2}g^{2}+[1+2g(1+\ln 4)]^{2}}.$
(10)

Correspondingly, $\mathrm{arg}(\sigma)$ has a plateau at
$\mathrm{arg}(\sigma)\approx\pi\nu^{*}/2$ before reaching its eventual
asymptotic value $\pi/2$ (Fig. 10). Using the value $g=0.23$ deduced above
from $m^{*}(0)/m$ yields $\nu^{*}=0.8$, in very good agreement with
experiment, as shown in Fig. 3.

Figure 4:  Emergence of an apparent sub-linear power-law in a pure Planckian
model. (a) Apparent exponent given by Eq. (10) versus interaction strength
$g$. (b–d) Modulus of the optical conductivity on a log-log scale showing the
apparent power law at energies between $k_{\mathrm{B}}T$ and the cutoff
$\Lambda=0.4$ eV. Data are shown for three values of $g$ [dots in (a)] and a
range of temperatures. Both horizontal and vertical axes cover exactly two
decades, such that a $1/\omega$ behavior would correspond to a slope of $-1$
(dotted line).

In the dc limit $\omega\rightarrow 0$, Eq. (8) together with our ansatz for
the scattering rate, yields a $T$-linear resistivity:

$\rho=AT,\quad
A=\frac{4\pi^{3}k_{\mathrm{B}}}{7\zeta(3)\hbar}\frac{g}{\Phi(0)}=\frac{4\pi^{3}\hbar
k_{\mathrm{B}}d_{c}}{7\zeta(3)e^{2}}\frac{g}{K}.$ (11)

Using the values of $g$ and $\Phi(0)$ determined above, we obtain:
$A=0.38~{}\mu\Omega~{}\mathrm{cm/K}$, to be compared to the experimental value
$A=0.63~{}\mu\Omega~{}\mathrm{cm/K}$. It is reassuring that a reasonable order
of magnitude is obtained (at the 60% level) for the $A$-coefficient, while
obviously a precise quantitative agreement cannot be expected from such a
simple model.

Finally, we present in Fig. 5 an $\omega/T$ scaling plot of $1/\tau$ and
$m^{*}/m-m^{*}(0)/m$ for our model, as well as a direct comparison to
experimental data. We emphasize that $\omega/T$ scaling does not hold exactly
for either of these quantities within our Planckian model. This is due to the
fact that the real part of the self-energy behaves logarithmically at low $T$
and thus leads to violations of scaling, as also clear from the need to retain
a finite cutoff $\Lambda$. However, approximate $\omega/T$ scaling is obeyed
to a rather high accuracy, as shown in panels (a) and (b) of Fig. 5 and
discussed in more details analytically in Appendix B. Panels (c) and (d) allow
for a direct comparison between the scaling properties of the theoretical
model and the experimental data, including analytical expressions of the
approximate scaling functions derived in Appendix B. These functions stem from
an approximate expression for the conductivity, Eq. (B.2), that displays exact
$\omega/T$ scaling. The approximation made in deriving them explains why the
scaling functions differ slightly from the numerical data in Figs. 5(a) and
5(b). Note the similar difference with the experimental data in Fig. 5(d).

Figure 5:  (a) Approximate collapse of the theoretical scattering rate and (b)
mass enhancement; the dashed lines show $2\pi gS(x/2)$ in (a) and Eq. (20) in
(b). (c) Same data as in Fig. 2(b). (d) Same data as in Fig. 2(d) on a
logarithmic scale (not displayed here because of excessive noise:
$\hbar\omega/k_{\mathrm{B}}T<10$ for $T<T_{c}$); the dashed line is Eq. (20).

##  IV Discussion and Conclusion

In this article, we have shown that our experimental optical data for LSCO at
$p=0.24$ display scaling properties as a function of
$\hbar\omega/k_{\mathrm{B}}T$ which are consistent with ‘Planckian’ behavior
corresponding to a scaling exponent $\nu=1$. We found that the accuracy of the
data scaling depends on the choice of the parameter $\epsilon_{\infty}$
relating the optical conductivity to the measured dielectric permittivity, and
that optimal scaling is achieved for a specific range of values of this
parameter.

From both a direct analysis of the optical data and by requiring optimal
scaling, we demonstrated that the low-frequency optical effective mass
$m^{*}(\omega\approx 0,T)/m$ displays a logarithmic dependence on temperature.
This dependence, also a hallmark of Planckian behavior, is qualitatively
consistent with that reported for the specific heat (quasiparticle effective
mass) [43, 42]. We showed that the coefficient of the logarithmic term can be
made quantitatively consistent between these two measurements if a specific
relation exists between the spectral weight $K$ and the ratio $m/m_{e}$ of the
band mass to the bare electron mass. Interestingly, we found that a realistic
tight-binding model satisfies this relation. The low-frequency optical
scattering rate $1/\tau$ extracted from our scaling analysis displays a linear
dependence on temperature, consistent with the $T$-linear dependence of the
resistivity that we measured on the same sample, with a quite good
quantitative agreement found between the $T$-linear slopes of these two
measurements.

We have introduced a simple theoretical model which relies on the assumption
that the single-particle inelastic scattering rate (imaginary part of the
self-energy) displays $\hbar\omega/k_{\mathrm{B}}T$ scaling properties with
$\nu=1$ and that its angular dependence along the Fermi surface can be
neglected. These assumptions are consistent with angular dependent
magnetoresistance measurements [21]. The model involves a dimensionless
inelastic coupling constant $g$ as a key parameter. We calculated the optical
conductivity based on this model and showed that it accounts very well for the
frequency dependence (Fig. 3) and $\omega/T$ scaling properties (Fig. 5) of
our experimental data.

A key finding of our analysis is that the calculated optical conductivity
displays an _apparent_ power-law behavior with an effective exponent
$\nu^{*}<1$ over an extended frequency range relevant to experiments (Figs. 3
and 4). We were able to establish that $\nu^{*}$ depends continuously on the
inelastic coupling constant $g$ [Eq. 10 and Fig. 4(a)]. This apparent power
law is also clear in the experimental data, especially when displaying the
data for $|\sigma|$ and $\mathrm{arg}(\sigma)$ as a function of frequency.
Hence, our analysis solves a long-standing puzzle in the field, namely the
seemingly contradictory observations of Planckian behavior with $\nu=1$ for
the resistivity and optical scattering rate versus a power law $\nu^{*}<1$
observed for $|\sigma|$ and $\mathrm{arg}(\sigma)$. We note that the apparent
exponent $\nu^{*}$ reported in previous optical spectroscopy literature varies
from one compound to another, which is consistent with our finding that
$\nu^{*}$ depends on $g$ and is hence not universal. For our LSCO sample, the
measured value of $\nu^{*}$ leads to the value $g\approx 0.23$.

The logarithmic temperature dependence of both the optical effective mass and
the quasiparticle effective mass is directly proportional to the inelastic
coupling constant $g$. We emphasize that this is profoundly different from
what happens in a Fermi liquid. There, using the Kramers–Kronig relation, one
sees that the effective mass enhancement (related to the low-frequency
behavior of the real part of the self-energy) depends on the whole high-
frequency behavior of the imaginary part of the self-energy. In contrast, in a
Planckian metal obeying $\omega/T$ scaling, the dominant $\ln T$ dependence of
the mass is entirely determined by the low-energy behavior of the imaginary
part of the self-energy, see Eq. (7). Based on this observation, we found that
the slope of the $\ln T$ term in the effective mass and specific heat is
consistent with the value $g\approx 0.23$ independently determined from the
effective exponent $\nu^{*}$. Using that same value of $g$ within our simple
theory leads to a value of the prefactor $A$ of the $T$-linear term in the
resistivity which accounts for 60% of the experimentally measured value.
Quantitative agreement would require $g\approx 0.38$, corresponding to a value
of $\nu^{*}\approx 0.77$ also quite close to the experimentally observed value
$\nu^{*}\approx 0.8$. It is also conceivable that electron-phonon coupling
contributes to the experimental value of $A$. In view of the extreme
simplification of the theoretical model for transport used in the present
work, it is satisfying that overall consistency between optics, specific heat
and resistivity can be achieved with comparable values of the coupling $g$.

In recent works [62, 67], Planckian behavior has also been put forward as an
explanation for the observed unconventional temperature dependence of the in-
plane and $c$-axis Seebeck coefficient of Nd-LSCO. In these works, the same
scaling form of the inelastic scattering rate than the one used here was used,
modified by a particle-hole asymmetry parameter. For simplicity, this
asymmetry parameter was set to zero in the present article. We have checked,
as detailed in Appendix E, that our results and analysis are unchanged if this
asymmetry parameter is included, as is expected from the fact that optical
spectroscopy measures particle-hole excitations and is thus rather insensitive
to the value of the particle-hole asymmetry parameter.

Finally, we note for completeness that a power-law behavior of the optical
conductivity has also been observed in other materials, including quasi one-
dimensional conductors [68, 69, 70, 71] with $\nu^{*}\sim 1.5$, and three-
dimensional conductors [72, 73, 74, 75] with $\nu^{*}\sim 0.5$. In the former
case, Luttinger liquid behavior provides an explanation for the observed
behavior at intermediate frequencies [68], while the interpretation of the
power-law behavior for materials such as Sr/CaRuO3 is complicated by a high
density of low-energy interband transitions [76].

Summarizing, our results demonstrate a rather remarkable consistency between
experimental observations based on optical spectroscopy, resistivity and
specific heat, all being consistent with $\nu=1$ Planckian behavior and
$\omega/T$ scaling. We have explained the long-standing puzzle of an apparent
power law of the optical spectrum over an intermediate frequency range and
related the non-universal apparent exponent to the inelastic coupling
constant. Looking forward, it would be valuable to extend our measurements and
analysis to other cuprate compounds at doping levels close to the pseudogap
quantum critical point. Our findings provide compelling evidence for the
quantum critical behavior of electrons in cuprate superconductors. This raises
the fundamental question of what is the nature of the associated quantum
critical point, and its relation to the enigmatic pseudogap phase.

## Data and materials availability

The datasets generated and analyzed during the current study are available in
Ref. 54. These will be preserved for 10 years.

###### Acknowledgements.

A.G. acknowledges useful discussions with Andrew J. Millis, Jernej Mravlje and
Subir Sachdev. We acknowledge support from the Swiss National Science
Foundation under Division II through project No. 179157 (D.v.d.M.) and support
through the JSPS KAKENHI grant 20H05304 (S.O.). The Flatiron Institute is a
division of the Simons Foundation. L.T. acknowledges support from the Canadian
Institute for Advanced Research (CIFAR) as a CIFAR Fellow and funding from the
Institut Quantique, the Natural Sciences and Engineering Research Council of
Canada (PIN:123817), the Fonds de Recherche du Québec-Nature et Technologies
(FRQNT), the Canada Foundation for Innovation (CFI), and a Canada Research
Chair.

##  Appendix A Frequency-temperature scaling of the optical data

In this appendix, we show that the frequency and temperature dependencies of
the optical scattering rate and optical mass enhancement indicate a Planckian
dissipation (with single-particle self-energy exponent $\nu=1$) — despite the
fact that the modulus of the optical conductivity displays an exponent
$\nu^{*}<1$ as seen in Fig. 3. Based on the local models considered in the
present work and discussed in detail in Appendices B and C, we expect that the
scattering rate and mass enhancement scale according to $1/\tau(\omega)\approx
T^{\nu}f_{\tau}(\omega/T)$ and $m^{*}(\omega)-m^{*}(0)\approx
T^{\nu-1}f_{m}(\omega/T)$. If not for the approximate signs and the shift by
$m^{*}(0)$, these scaling laws would imply that the conductivity behaves
ideally as $1/\sigma(\omega)=T^{\nu}F(\omega/T)$ with
$F(x)=f_{\tau}(x)-ixf_{m}(x)$ [see Eq. (2)]. The numerical simulations show
that this property is not obeyed by Eqs. (8), (5), and (22), while the
approximate scaling laws with the mass shift by $m^{*}(0)$ are numerically
well obeyed (see Figs. 5 and 14). The subtraction of $m^{*}(0)$ in order to
observe $\omega/T$ scaling is mandatory for $\nu=1$, because $m^{*}(0)$ varies
as $\ln T$ in that case, while for $\nu<1$ it is optional, because
$m^{*}(0)\sim T^{\nu-1}$.

Figure 6:  Complex dielectric function $\epsilon$ of LSCO at $p=0.24$. These
data are obtained from a mix of two techniques, namely the infrared
reflectivity between 2.5 meV and 0.5 eV and ellipsometry from 0.5 to 5 eV. The
part 0.5–5 eV is using a fit to the ellipsometry in order to extrapolate the
reflectivity data at higher photon energy and extract $\epsilon$ through
Kramers–Kronig relations.  Figure 7:  (a) Optimal $\omega/T$ scaling of the
scattering rate for the values of $\nu$ indicated in each panel; the value of
$\epsilon_{\infty}$ is the one ensuring the best collapse of the curves. (b)
Sensitivity of the scaling collapse to the value of $\epsilon_{\infty}$ for
$\nu=1$. The vertical axis starts at zero in each panel and is otherwise
arbitrary, unless a value of the spectral weight $K$ is specified.  Figure 8:
(a) Optimal $\omega/T$ scaling of the mass enhancement for the values of $\nu$
indicated in each panel; the value of $\epsilon_{\infty}$ is the one ensuring
the best collapse of the curves. (b) Values of $m^{*}(0)$ that give the best
collapse.

The measured infrared dielectric function of LSCO at doping $p=0.24$ is
displayed in Fig. 6. We emphasize in the main text that the frequency
dependence of $1/\tau$ and $m^{*}/m$ extracted from the dielectric function
depends on the value chosen for the background dielectric constant. Here, in
order to test the presence of scaling laws in the data, we optimize the value
of $\epsilon_{\infty}$ to achieve the best scaling collapse. The results of
this procedure for $1/\tau$ are displayed in Fig. 7(a). For each value of
$\nu$, we minimize a cost function representing the quality of the collapse
[54] and we deduce the optimal $\epsilon_{\infty}$ indicated in each panel. We
use the optical data in the range $T>30$ K and $\hbar\omega<0.4~{}\mathrm{eV}$
for this analysis. The data for $T=30$ K and below are strongly affected by
the loss of information due to the sharpening of the Drude peak below our
observation limit of 2.5 meV and the opening of the superconducting gap below
$T_{c}=19$ K. Furthermore, interband transitions come into play above
$0.4~{}\mathrm{eV}$. The figure shows that the collapse improves upon
increasing $\nu$ towards $\nu=1$. For $\nu\lesssim 0.8$, the optimal
$\epsilon_{\infty}$ is larger than the largest measured infrared dielectric
function (Fig. 6) and the collapse is poor. Figure 7(b) illustrates how the
scaling collapse for $\nu=1$ depends on the value of $\epsilon_{\infty}$.

In order to test the scaling laws for $m^{*}$, we must determine $m^{*}(0)$.
The low-frequency noise prevents us from extracting this information directly
from the data. We therefore treat $m^{*}(0)$ for each temperature as a
variable that we adjust, like $\epsilon_{\infty}$, for optimal collapse. The
results are displayed in Fig. 8. Fig. 8(a) shows a systematic improvement of
the collapse with $\nu$ approaching unity. All temperatures were included in
this analysis, which allows us to get the full $T$-dependence of $m^{*}(0)$,
as shown in Fig. 8(b). Note that this procedure does not set an absolute scale
for $m^{*}(0)$, unless the scale of $m^{*}$ is fixed by a choice of the
spectral weight $K$. For $\nu=1$, the optimal $m^{*}(0)$ values align on a
straight line, indicating a precise $\ln T$ behavior consistent with the
behavior of the quasiparticle mass revealed by the electronic specific heat.

The optimizations presented so far establish that the self-energy exponent
takes the value $\nu=1$. However, the optimal $\epsilon_{\infty}$ determined
from $1/\tau$ and $m^{*}$ differ. One sees in Fig. 7(b) that the collapse of
$1/\tau$ quickly deteriorates as $\epsilon_{\infty}$ increases beyond 3 and
would be badly broken for $\epsilon_{\infty}=6.66$, as required for optimal
collapse of the mass. In fact, the large values of $\epsilon_{\infty}$ stem
from the $m^{*}$ data above $\sim 0.2$ eV: the optimal $\epsilon_{\infty}$
drops if those data are ignored. If we restrict to $\hbar\omega<0.25$ eV, for
instance, the optimum is $\epsilon_{\infty}=3.24$ for $\nu=1$; for the data
below 0.2 eV, the optimal $\epsilon_{\infty}$ is even smaller than unity. It
is seen in Fig. 1(d) that a change of behavior occurs around 0.22 eV in the
mass enhancement data. We therefore adopt the following pragmatic approach: We
perform a common optimization for both $1/\tau$ and $m^{*}$ by minimizing the
sum of the two cost functions (normalized to the value at their respective
minimum), considering energies up to 0.4 eV and temperatures larger than 30 K
for $1/\tau$, while keeping all temperatures but only energies up to 0.22 eV
for $m^{*}$. The resulting optimum is $\epsilon_{\infty}=2.76$, very close to
the optimum for $1/\tau$, and the resulting zero-frequency masses and scaling
collapses are displayed in Fig. 2 of the main text.

##  Appendix B Study of the Planckian model ($\nu=1$)

In this appendix, we study the Planckian model defined by Eqs. (5) and (8). We
calculate the quasiparticle mass, derive approximate analytical expressions
for the self-energy and the frequency-dependent optical conductivity, and
check these approximations against the numerically computed conductivity. We
finally discuss the $\omega/T$ scaling properties of the model.

###  B.1 Single-particle self-energy

We implement the ultraviolet cutoff in Eq. (5) by limiting the imaginary part
of the self-energy to the constant value $\mathrm{Im}\,\Sigma(\Lambda)=-g\pi
k_{\mathrm{B}}TS(\Lambda/k_{\mathrm{B}}T)$ for all energies
$|\varepsilon|>\Lambda$. The real part contains a constant term
$\mathrm{Re}\,\Sigma(0)$ that is compensated by a shift of chemical potential
and can therefore be subtracted from $\Sigma(\varepsilon)$. The remaining real
part describes the renormalization of the single-particle dispersion by
inelastic scattering processes, which at leading order amounts to a
renormalization of the mass. The renormalized single-particle mass or
“quasiparticle mass” is

$\frac{m^{*}_{\mathrm{qp}}}{m}=1-\frac{d}{d\varepsilon}\mathrm{Re}\left[\Sigma(\varepsilon)-\Sigma(0)\right]_{\varepsilon=0}\equiv\frac{1}{Z},$
(12)

where $Z$ is the quasiparticle residue. After evaluating the derivative of
$\Sigma(\varepsilon)$ using Eq. (5), performing an integration by parts, and
separating the even and odd terms in the resulting integral, we arrive at the
expression:

$\displaystyle\frac{m^{*}_{\mathrm{qp}}}{m}$
$\displaystyle=1+g\int_{0}^{\frac{\Lambda}{k_{\mathrm{B}}T}}dx\,\frac{S^{\prime}(x)-S^{\prime}(-x)}{x}$
$\displaystyle=1+2g\int_{0}^{\frac{\Lambda}{k_{\mathrm{B}}T}}dx\,\frac{\sinh
x-x}{x(\cosh x-1)}.$ (13)

The function to integrate approaches $1/x$ at large $x$ and the integral
therefore approaches $\ln(\Lambda/k_{\mathrm{B}}T)+\mathrm{cste}$ if
$\Lambda\gg k_{\mathrm{B}}T$. We write this as

$\frac{m^{*}_{\mathrm{qp}}}{m}=1+2g\ln\left(a\frac{\Lambda}{k_{\mathrm{B}}T}\right)\qquad(\Lambda\gg
k_{\mathrm{B}}T),$ (14)

where the constant $a$ is given by

$a=\lim_{x_{c}\to\infty}\frac{1}{x_{c}}\exp\left[\int_{0}^{x_{c}}dx\,\frac{\sinh
x-x}{x(\cosh x-1)}\right]=0.770542.$

If $\varepsilon\gg k_{\mathrm{B}}T$, the integral giving $\Sigma(\varepsilon)$
is dominated by the region $|x|\gg 1$ and we can replace the scaling function
by its asymptotic form $S(|x|\gg 1)=|x|$. We then find

$\mathrm{Re}\left[\Sigma(\varepsilon)-\Sigma(0)\right]\approx
g\varepsilon\ln\left|\frac{\varepsilon^{2}}{\Lambda^{2}-\varepsilon^{2}}\right|+g\Lambda\ln\left|\frac{\Lambda-\varepsilon}{\Lambda+\varepsilon}\right|$
(15)

in this regime, which will be used below in building approximations for the
conductivity.

###  B.2 Optical conductivity

Figure 9 shows the optical conductivity calculated for weak coupling
($g=0.04$) and strong coupling ($g=4$) at a temperature
$k_{\mathrm{B}}T=10^{-3}\Lambda$, and plotted using the cutoff $\Lambda$ as
the unit of energy. One can distinguish three regimes of frequency, as pointed
out in Ref. 23.

Figure 9:  Real (blue) and imaginary (red) parts of the optical conductivity,
Eqs. (8) and (5), for $g=0.04$ (dashed) and $g=4$ (dotted) at temperature
$k_{\mathrm{B}}T=10^{-3}\Lambda$. The thick lines show Eqs. (16), (19), and
(18) in the corresponding regime of frequency.

In the regime (I), the frequency is lower than the temperature. The scattering
rate approaches a frequency-independent value proportional to $T$ and the
conductivity approaches a Drude form with renormalized spectral weight
$Z\Phi(0)$. To obtain the asymptotic expression of the conductivity, we expand
numerator and denominator in Eq. (8) around $\omega=0$ to get

$\sigma_{\mathrm{(I)}}(\omega)=i\hbar\Phi(0)\int_{-\infty}^{\infty}d\varepsilon\,\frac{-df(\varepsilon)/d\varepsilon}{\hbar\omega\left[1-d\Sigma(\varepsilon)/d\varepsilon\right]-2i\mathrm{Im}\,\Sigma(\varepsilon)}.$

At low enough temperature, the derivative of the Fermi function is sharply
peaked at $\varepsilon=0$ and the denominator in the function to integrate can
be approximated by its value at $\varepsilon=0$. Since the imaginary part of
the self-energy is an even function of $\varepsilon$, the term in square
brackets becomes
$1-d\mathrm{Re}\,\Sigma(\varepsilon)/d\varepsilon|_{\varepsilon=0}=1/Z$.
Furthermore, Eq. (5) gives $\mathrm{Im}\,\Sigma(0)=-2\pi gk_{\mathrm{B}}T$,
leading to

$\sigma_{\mathrm{(I)}}(\omega)\approx\frac{Z\Phi(0)}{-i\omega+4\pi
gZk_{\mathrm{B}}T/\hbar}.$ (16)

This has the usual dissipative ($\mathrm{Re}\,\sigma>\mathrm{Im}\,\sigma$)
Drude structure with a scattering time $\tau=\hbar/(4\pi gk_{\mathrm{B}}T)$,
corresponding to a $T$-linear resistivity. The coefficient $A$ of the
resistivity deduced from Eq. (16) is smaller than the exact result Eq. (11) by
a factor $7\zeta(3)/\pi^{2}=0.853$. The predictions of Eq. (16) with $1/Z$
given by Eq. (14) are displayed in Fig. 9 as the thick lines in the regime
(I).

In the regimes (II) and (III), the frequency is large compared with
temperature and we can therefore replace the Fermi functions by their
expression at $T=0$:

$\displaystyle\sigma(\hbar\omega\gg k_{\mathrm{B}}T)$
$\displaystyle=\frac{i\Phi(0)}{\omega}\int_{-\hbar\omega}^{0}\frac{d\varepsilon}{\hbar\omega+\Sigma^{*}(\varepsilon)-\Sigma(\varepsilon+\hbar\omega)}$
$\displaystyle\approx\frac{i\hbar\Phi(0)}{\hbar\omega-2\Sigma(\hbar\omega/2)}.$
(17)

At the second line, we have approximated the function to integrate by its
value in the middle of the integration window, noting that the real
(imaginary) part of $\Sigma(\varepsilon)$ is odd (even) in $\varepsilon$ —
provided that the constant $\mathrm{Re}\,\Sigma(0)$ is subtracted from
$\Sigma(\varepsilon)$, which is implicit in Eq. (B.2). The condition
$\hbar\omega\gg k_{\mathrm{B}}T$ allows us to use Eq. (15) and the asymptotic
form $S(x)=|x|$ when evaluating $\mathrm{Im}\,[-2\Sigma(\hbar\omega/2)]=\pi
g\min(\hbar\omega,2\Lambda)$. In the regime (III), the real part of
$\Sigma(\hbar\omega/2)$ disappears from the conductivity because the right-
hand side of Eq. (15) drops as $\Lambda/\varepsilon$, such that we arrive at
the following approximation:

$\sigma_{\mathrm{(III)}}(\omega)\approx\frac{\Phi(0)}{-i\omega+\pi
g\min\left(\omega,2\Lambda/\hbar\right)}.$ (18)

This is asymptotically an inductive regime
($\mathrm{Im}\,\sigma>\mathrm{Re}\,\sigma$), where the real part decreases as
$1/\omega^{2}$ and the imaginary part as $1/\omega$, such that
$|\sigma_{\mathrm{(III)}}|\sim 1/\omega$. As the asymptotic conductivity is
purely imaginary, $\mathrm{arg}(\sigma_{\mathrm{(III)}})$ approaches the value
$\pi/2$. This is illustrated in Fig. 10. Equation (18) agrees with the
numerics, as the thick lines show in the regime (III) of Fig. 9.

For the regime (II), we expand Eq. (15) for $\hbar\omega\ll\Lambda$ and arrive
at the expression

$\sigma_{\mathrm{(II)}}(\omega)\approx\frac{\Phi(0)}{-i\omega}\frac{1}{1+2g\left[1-\ln\left(\frac{\hbar\omega}{2\Lambda}\right)\right]+i\pi
g},$ (19)

which is also in good agreement with the numerics [thick lines in the regime
(II) of Fig. 9]. Without the cutoff-dependent logarithmic correction,
$\sigma_{\mathrm{(II)}}(\omega)$ would display pure
$\sigma\propto(-i\omega)^{-1}$ behavior, as may be expected in a quantum
critical system with linear-in-energy single-particle scattering rate [23].
This is realized for $g\to 0$, as shown by the dashed lines in Fig. 9. For
finite $g$, however, the logarithmic correction reduces the decay rate of both
real and imaginary parts (dotted lines).

Figure 10:  Crossover from the regime of effective exponent $\nu^{*}<1$ for
$\hbar\omega<\Lambda$ to the asymptotic regime showing $\nu=1$ and
$\arg(\sigma)=\pi/2$ for $\hbar\omega>\Lambda$. The model parameters are
$g=0.23$ and $\Lambda=0.4$ eV.

We see in Fig. 9 that the approximation Eq. (19) captures the change of
behavior of the conductivity with increasing $g$ in the intermediate frequency
regime $k_{\mathrm{B}}T<\hbar\omega<\Lambda$. We therefore use this
approximation in order to extract the apparent power-law exponent of
$|\sigma|$ by computing the logarithmic derivative at $\hbar\omega=\Lambda/2$,
near the middle of domain (II). We thus arrive at Eq. (10) of the main text.

###  B.3 $\omega/T$ scaling

As the approximations presented so far take one of the two limits $\omega\ll
T$ or $\omega\gg T$, they cannot predict the form of the $\omega/T$ scaling
expected when $\omega$ and $T$ are comparable. If the self-energy were obeying
the scaling property $\Sigma(\varepsilon)=T\mathscr{F}(\varepsilon/T)$, then
also the conductivity, Eq. (8), would obviously scale exactly like
$1/\sigma(\omega)=TF(\omega/T)$. The cutoff that must be introduced in Eq. (5)
breaks this property, since the self-energy is rather of the form
$\Sigma(\varepsilon)=T\mathscr{F}(\varepsilon/T,\Lambda/T)$. The perfect
scaling of $1/\sigma$ is therefore lost as well. In the following, we obtain
closed formula for the approximate scaling of $1/\sigma$.

At low energy and temperature, $\mathrm{Im}\,\Sigma$ is not influenced by the
cutoff, such that the ideal scaling $1/\sigma\propto-i\omega
m^{*}/m+1/\tau=TF(\omega/T)$ is expected to hold reasonably well for the real
part, i.e., $1/\tau\approx Tf_{\tau}(\omega/T)$. This expectation is confirmed
by the numerical simulations [Fig. 5(a)]. Equation (16) indicates that the
function $f_{\tau}(x)$ is close to $4\pi g$ at $x=0$, which is confirmed as
well by the numerics. Since this corresponds to
$-2\mathrm{Im}\,\Sigma(0)/k_{\mathrm{B}}T$, it is tempting to infer that
$1/\tau$ is close to twice the single-particle scattering rate, which would
give $2\pi gS(x)$ for the scaling function. However, that function goes to
$2\pi gx$ at large $x$, while the simulations go to $\pi gx$. We therefore try
$f_{\tau}(x)=2\pi gS(x/2)$, which works well despite small deviations at low
$x$, as seen in Fig. 5(a). This result could have been directly guessed from
Eq. (B.2).

Figure 11:  Temperature-dependent optical mass enhancement predicted by Eqs.
(8) and (5) for $g=0.23$, $\Lambda=0.4$ eV and various frequencies, compared
with the quasiparticle mass from Eq. (14) in green. The dots indicate
$\hbar\omega=k_{\mathrm{B}}T$. For $k_{\mathrm{B}}T>\hbar\omega$, we have
$m^{*}(\omega)\approx m_{\mathrm{qp}}$.

Since Eq. (B.2) correctly predicts the approximate scaling of $1/\tau$, we use
this same relation for $m^{*}$, which leads to $m^{*}/m\approx
1-(2/\hbar\omega)\mathrm{Re}[\Sigma(\hbar\omega/2)-\Sigma(0)]$. This
expression immediately gives $m^{*}(0)/m\approx
1-\Sigma^{\prime}(0)=m_{\mathrm{qp}}/m$, a result confirmed by the numerics
(see Fig. 11). If the function $S(x)$ is replaced by the very similar but
simpler function $|x|+2e^{-|x|/2}$, which has the same behavior at low and
high values of $x$, a complete evaluation of
$\mathrm{Re}[\Sigma(\varepsilon)-\Sigma(0)]$ becomes possible. We thus find
that $m^{*}(\omega)-m^{*}(0)$ scales as $f_{m}(\omega/T)$ with

$f_{m}(x)=2g\left\\{1\\!-\\!\gamma\\!-\\!\ln\left(\frac{x}{4}\right)\\!+\\!\frac{2}{x}\left[e^{\frac{x}{4}}\mathrm{Ei}\left(-\frac{x}{4}\right)\\!-\\!e^{-\frac{x}{4}}\mathrm{Ei}\left(\frac{x}{4}\right)\right]\right\\}.$
(20)

$\gamma=0.577$ is Euler’s constant and $\mathrm{Ei}$ is the exponential
integral function. The function $f_{m}$ is compared in Fig. 5(b) with the
numerical simulations. The slight difference is a consequence of using the
approximation Eq. (B.2) for computing the optical mass, not a consequence of
approximating the function $S(x)$.

Figure 12:  Absence of $\omega/T$ scaling in
$\hbar/(k_{\mathrm{B}}T\,\mathrm{Re}\,\sigma)$ evaluated numerically in the
Planckian model over (a) a wide and (b) a narrow range of $\omega/T$. The
dotted lines in (b) show the approximation Eq. (21).

To conclude this section, we note that the presence of $\omega/T$ scaling has
been previously tested in Bi2212 by considering the quantity
$\hbar/(k_{\mathrm{B}}T\mathrm{Re}\,\sigma)$ [23]. Equation (1) shows that
$\mathrm{Re}\,\sigma$ is independent of $\epsilon_{\infty}$ and $K$, making it
the ideal quantity for testing the presence of $\omega/T$ scaling in the raw
data. In our model, however, this quantity does not scale with $\omega/T$, as
seen in Fig. 12. Using the form $\sigma\propto 1/(-i\omega\,m^{*}/m+1/\tau)$
and our observations that, to a very good accuracy,
$\hbar/\tau=k_{\mathrm{B}}Tf_{\tau}(\omega/T)$ and
$m^{*}/m=m^{*}_{\mathrm{qp}}/m+f_{m}(\omega/T)$, we deduce

$\frac{\hbar}{k_{\mathrm{B}}T\mathrm{Re}\,\sigma}\sim
f_{\tau}\left(\frac{\omega}{T}\right)+\frac{(\hbar\omega/k_{\mathrm{B}}T)^{2}}{f_{\tau}(\omega/T)}\left[\frac{m^{*}_{\mathrm{qp}}}{m}+f_{m}\left(\frac{\omega}{T}\right)\right]^{2}.$

Due to the logarithmic variation of $m^{*}_{\mathrm{qp}}$ with temperature,
the second term on the right-hand side is not a function of $\omega/T$.
Despite the conspicuous absence of scaling, the behavior of the model in a
narrow range of $\omega/T$ [Fig. 12(b)] and for temperatures above 100 K is
strikingly similar to that reported in Ref. 23, suggesting that the model
could be useful to analyze Bi2212 data as well. In particular,
$\hbar/(k_{\mathrm{B}}T\,\mathrm{Re}\,\sigma)$ increases as $(\omega/T)^{2}$
as pointed out in Ref. 23, albeit with a $T$-dependent curvature. Since this
curvature is in principle accessible in the raw optical data, it is
interesting to relate it with the parameters of our Planckian model. Using our
approximate scaling functions, we find for $\omega\to 0$:

$\frac{\hbar}{k_{\mathrm{B}}T\mathrm{Re}\,\sigma}\approx\frac{4\pi
g}{\Phi(0)}\left\\{1+\left[\frac{1}{48}+\left(\frac{m_{\mathrm{qp}}/m}{4\pi
g}\right)^{2}\right]\left(\frac{\hbar\omega}{k_{\mathrm{B}}T}\right)^{2}\right\\}.$
(21)

This expression is in reasonable agreement with the exact numerical result
[see dotted lines in Fig. 12(b)]. Equation (21) can in principle be used to
extract the three parameters of the theory all at once from the raw optical
data. This endeavor is somewhat risky, though, because the approximate
validity of Eq. (21) is limited to a very narrow low-frequency range, where
optical data is usually noisy. Furthermore, the sensitivity to the cutoff
$\Lambda$ is only logarithmic.

##  Appendix C Study of the “Sub-Planckian” model ($\nu<1$)

Here, we introduce and study a model with sub-linear energy and temperature
dependencies of the self-energy. We follow the same outline as for the
Planckian model, first discussing the self-energy, then the optical
conductivity, and finally the $\omega/T$ scaling. Beside presenting the model,
our main goal is to argue that it gives predictions that are in disagreement
with the experimental observations in LSCO at doping $p^{*}$, _including_ for
the apparent exponent $\nu^{*}<1$ of the optical conductivity.

###  C.1 Single-particle self-energy

The central property of the class of theories discussed in this paper is that
the single-particle scattering rate is local (independent of momentum) and
assumes the form $-\mathrm{Im}\,\Sigma(\varepsilon)=\pi
g(k_{\mathrm{B}}T)^{\nu}S_{\nu}(\varepsilon/k_{\mathrm{B}}T)$, where
$S_{\nu}(0)$ is finite and $S_{\nu}(|x|\gg 1)\propto|x|^{\nu}$. These two
conditions ensure that the dc resistivity behaves as $T^{\nu}$ and the ac
scattering rate approaches $\omega^{\nu}$ for $\hbar\omega>k_{\mathrm{B}}T$,
while at intermediate frequencies the scattering rate divided by $T^{\nu}$ is
a function of $\omega/T$. The case $\nu=1$ is the Planckian model described in
Appendix B, while here we consider the case $\nu<1$. An important aspect of
the sub-Planckian model is that it does not require an ultraviolet cutoff,
because the Kramers–Kronig integral giving the real part of the self-energy is
now convergent. Thus the theory is completely specified in terms of the low-
energy properties of the carriers, which then determine the thermodynamic,
transport, and spectroscopic properties. This is an important difference
relative to the case $\nu=1$, where the high-energy structure of the theory,
represented by the ultraviolet cutoff, controls the thermodynamics and the
optical mass.

Certain microscopic models with conformal invariance realize exactly the
scaling form of the single-particle scattering rate with $\nu<1$ and provide
an explicit expression for the function $S_{\nu}(x)$ [77, 78, 56, 60]. We
borrow our self-energy model from those:

$\displaystyle\Sigma(z)$
$\displaystyle=g(k_{\mathrm{B}}T)^{\nu}\int_{-\infty}^{\infty}dx\,\frac{S_{\nu}(x)}{z/k_{\mathrm{B}}T-x}$
(22a) $\displaystyle S_{\nu}(x)$
$\displaystyle=\frac{(2\pi)^{\nu}}{\pi\,\Gamma(1+\nu)}\cosh(x/2)\left|\,\Gamma\left(\frac{1+\nu}{2}+i\frac{x}{2\pi}\right)\right|^{2},$
(22b)

where $\Gamma$ denotes the Euler gamma function. Note that $g$ is
dimensionfull with the unit of energy to the power $1-\nu$. Proceeding like in
the case $\nu=1$, we find that the quasiparticle mass diverges at low
temperature like $T^{\nu-1}$:

$\frac{m^{*}_{\mathrm{qp}}}{m}=1+g(k_{\mathrm{B}}T)^{\nu-1}c_{\nu},\quad
c_{\nu}=\int_{0}^{\infty}dx\,\frac{S_{\nu}^{\prime}(x)-S_{\nu}^{\prime}(-x)}{x}.$

This is a robust property of the sub-Planckian model — it does not depend on
the particular scaling function $S_{\nu}(x)$ — that disagrees with the
logarithmic temperature dependence observed experimentally in LSCO at doping
$p=0.24$. Arguably, for $\nu$ close to unity it is difficult to distinguish
the power law $T^{\nu-1}$ from a logarithmic temperature dependence.

In order to derive approximations for the conductivity, it is helpful to
estimate the self-energy at energies larger than $k_{\mathrm{B}}T$. For
$\varepsilon\gg k_{\mathrm{B}}T$, we replace $S_{\nu}(x)$ by its asymptotic
value, which is $|x|^{\nu}/\Gamma(1+\nu)$, and we obtain for the real part:

$\mathrm{Re}[\Sigma(\varepsilon)-\Sigma(0)]=-g\frac{\pi\tan(\pi\nu/2)}{\Gamma(1+\nu)}\mathrm{sign}(\varepsilon)|\varepsilon|^{\nu}\quad(\varepsilon\gg
k_{\mathrm{B}}T).$ (23)

###  C.2 Optical conductivity and $\omega/T$ scaling

Figure 13:  Real (blue) and imaginary (red) parts of the optical conductivity,
Eqs. (8) and (22), for $\nu=0.5$ (dashed) and $\nu=0.8$ (dotted) at
temperature $k_{\mathrm{B}}T=10^{-3}\,\hbar\omega^{*}$. The thick lines show
Eqs. (25) and (26). Inset: effective exponent as given by Eq. (27).

The frequency-dependent conductivity crosses over from the dissipative Drude
regime to the asymptotic inductive regime via an intermediate regime whose
extension is controlled by $g$ and $\nu$ (while for $\nu=1$, it is controlled
by the cutoff $\Lambda$). We define a crossover frequency $\omega^{*}$ by the
condition $\hbar\omega^{*}=2\mathrm{Re}[\Sigma(\hbar\omega^{*})-\Sigma(0)]$:
for $\omega<\omega^{*}$, the self-energy dominates in Eq. (8) while for
$\omega>\omega^{*}$, one approaches the regime where $\mathrm{Re}\,\Sigma$
becomes irrelevant. Using Eq. (23), we find

$\hbar\omega^{*}=\left[\frac{2\pi
g\tan(\pi\nu/2)}{\Gamma(1+\nu)}\right]^{\frac{1}{1-\nu}}.$ (24)

If the temperature and the frequency are both measured in units of
$\omega^{*}$, the conductivity multiplied by $\omega^{*}$ becomes a function
of $\omega/\omega^{*}$ and $T/\omega^{*}$ that depends on $\nu$, but no longer
on $g$. This function is displayed in Fig. 13 for two values of $\nu$. The
change of behavior around $\omega^{*}$ is clearly visible.

Figure 14:  (a) Modulus and (b) phase of the optical conductivity given by
Eqs. (8) and (22) with the parameters indicated in (a). (c) Approximate
collapse of the scattering rate and (d) mass enhancement.  Figure 15:  (a)
Temperature dependence of the noninteracting (orange) and interacting (blue)
chemical potential. (b) Noninteracting Fermi surface (dotted) and zero-energy
spectral function at $T=300$ K [same color scale as in (d)]. (c)
Noninteracting (orange) and interacting density of states at $T=10$ K (red)
and $T=300$ K (blue). (d) Tight-binding band (yellow) and spectral function at
$T=300$ K. (e) Transport function. The energies in (c), (d), and (e) are
measured relative to the corresponding chemical potential.

In the Drude regime, we proceed like for $\nu=1$ and get

$\sigma_{\mathrm{(I)}}(\omega)\approx\frac{Z\Phi(0)}{-i\omega+2gZ\frac{\left[\Gamma\left(\frac{1+\nu}{2}\right)\right]^{2}}{\Gamma(1+\nu)}(2\pi
k_{\mathrm{B}}T)^{\nu}/\hbar},$ (25)

as indicated by the thick lines in the regime (I) of Fig. 13. This shows that
the resistivity is proportional to $T^{\nu}$, another robust property of the
sub-Planckian model that disagrees with the linear resistivity observed in
LSCO at $p=0.24$.

In the regimes (II) and (III), we use the approximation Eq. (B.2), together
with the asymptotic self-energy given by Eq. (23) for the real part and by
$-\pi g|\varepsilon|^{\nu}/\Gamma(1+\nu)$ for the imaginary part, yielding

$\sigma_{\mathrm{(II,III)}}(\omega)\approx\frac{\Phi(0)}{-i\omega+2^{1-\nu}\pi\hbar^{\nu-1}g\frac{1-i\tan(\pi\nu/2)}{\Gamma(1+\nu)}\omega^{\nu}}.$
(26)

This expression reproduces the behavior proportional to $i/\omega$ required by
causality in the limit $\omega\to\infty$, irrespective of the value of $\nu$,
and interpolates well across $\omega^{*}$, as shown by the thick lines in Fig.
13. In the regime (II), a nontrivial power law emerges with an apparent
exponent $\nu_{\mathrm{eff}}>\nu$. The logarithmic derivative of
$|\sigma_{\mathrm{(II,III)}}|$ evaluated at $\omega^{*}/2$ gives the exponent

$\nu_{\mathrm{eff}}=\nu+\frac{\left(2^{2\nu-1}+2^{4\nu-2}\right)(1-\nu)}{2^{2\nu}+2^{4\nu-2}+1/\sin^{2}(\pi\nu/2)}.$
(27)

This function is displayed as an inset in Fig. 13. An effective exponent of
$0.8$, as observed experimentally in LSCO, requires $\nu=0.61$. Note that the
apparent exponent in Eq. (27) is independent of the coupling $g$, in contrast
to the Planckian case, Eq. (10), where it depends only on $g$.

Figures 14(a) and 14(b) show the modulus and argument of the optical
conductivity calculated numerically using Eqs. (8) and (22) with $\nu=0.61$.
The coupling constant $g$ is fixed such that the magnitude of the conductivity
is similar to that in Fig. 3. This calculation reproduces the power law
observed experimentally in the modulus, but predicts a behavior of the
argument that agrees less well with experiment than the model with $\nu=1$,
especially at low frequency [compare Figs. 3(b), 3(d), and 14(b)]. Figures
14(c) and 14(d) shows that $1/\tau$ and $m^{*}$, after being properly scaled
(and divided by $g$ to produce dimensionless quantities), display a good
collapse as a function of $\omega/T$. This justifies the scaling laws employed
in Appendix A for $\nu<1$. The most striking disagreement with experiment is
seen in the scattering rate. Indeed, the scaling requires to normalize
$1/\tau$ by $T^{\nu}$, which does not lead to a good collapse of the
experimental data for $\nu\approx 0.6$ (see Fig. 7).

##  Appendix D Tight-binding model and one-particle properties

Here, we present the tight-binding model that is used in the main text for
fixing the values of the band mass $m$ and spectral weight $K$. We discuss the
properties of the model and show how these properties are modified by the
self-energy.

The one-particle properties of the model are displayed in Fig. 15. The tight-
binding dispersion with up to second-neighbor hopping amplitudes is given by
$\varepsilon_{\bm{k}}=-2t[\cos(k_{x}a)+\cos(k_{y}a)]-4t^{\prime}\cos(k_{x}a)\cos(k_{y}a)$
with $t=0.3$ eV, $t^{\prime}/t=-0.17$, and $a=2.78$ Å. All properties are
shown for an electron density $n=0.76/a^{2}$, i.e., a hole doping $p=0.24$.
The noninteracting chemical potential $\mu_{0}$ varies from $-253$ meV at
$T=0$ to $-257$ meV at $T=300$ K [Fig. 15(a)]. The corresponding tight-binding
band with energies measured relative to $\mu_{0}(T=0)$ is shown in Fig. 15(d),
and the associated density of states in Fig. 15(c). The Fermi-level DOS is
$1.646$ eV${}^{-1}a^{-2}$, which corresponds to a band mass $m=2.76m_{e}$. The
Fermi surface is closed around the Brillouin-zone center, as seen in Fig.
15(b). The transport function

$\Phi(\varepsilon)=2e^{2}\int_{\mathrm{BZ}}\frac{d^{2}k}{(2\pi)^{2}}\left(\frac{1}{\hbar}\frac{d\varepsilon_{\bm{k}}}{dk_{x}}\right)^{2}\delta(\varepsilon+\mu_{0}-\varepsilon_{\bm{k}})$
(28)

is displayed in Fig. 15(e); the $T=0$ value of $(\hbar/e)^{2}\Phi(0)$ is 211
meV.

The self-energy Eq. (5) renormalizes the dispersion and consequently all one-
particle properties. The chemical potential calculated for $g=0.23$ and
$\Lambda=0.4$ eV is shown in Fig. 15(a) and varies from $-290$ meV at $T=0$ to
$-297$ meV at $T=300$ K. We emphasize that the high-energy details of the
self-energy influence the value of $\mu$, such that this model calculation,
although internally consistent, is not expected to be realistic for LSCO as it
misses those high-energy non-universal aspects. The downward renormalization
of the chemical potential by the interaction implies a renormalization of the
“Fermi surface” (maximum of the zero-energy spectral function), as seen in
Fig. 15(b). A change of the Fermi-surface volume at fixed density seems to
violate Luttinger’s theorem. The theorem is not applicable here, however,
because it requires a true Fermi surface — a discontinuity of the momentum
distribution. Since the self-energy does not vanish on the Fermi surface,
there is no such discontinuity in the model. The color map in Fig. 15(d) shows
the renormalization of the tight-binding band, which, since the self-energy is
local, is the same at all wave-vectors. The corresponding interacting DOS is
plotted in Fig. 15(c) for two temperatures.

##  Appendix E Effect of particle-hole asymmetry

In the main text, we compare the optical spectra of LSCO with microscopic
models that possess particle-hole (p-h) symmetry. The cuprate materials break
p-h symmetry at the level of the band structure and, consequently, also at the
level of the self-energy. The band-structure effects enter the optical spectra
via the energy-dependent transport function $\Phi(\varepsilon)$ replacing
$\Phi(0)$ in Eq. (8). The transport function is smooth [see Fig. 15(e)],
unlike the DOS that varies rapidly due to the van Hove singularity [Fig.
15(c)]. As $\Phi(\varepsilon)$ is almost featureless, we do not expect a
significant effect of the band structure on the frequency dependence of the
optical spectra. These spectra represent transitions from occupied to empty
states and thus mix scattering effects at positive and negative energies. For
that reason, they should also not be particularly sensitive to the p-h
asymmetry of the self-energy. It is indeed clear that the self-energy is to a
large extent averaged between positive and negative energies in Eq. (8). In
order to check this expectation and justify our use of p-h symmetric models
for data analysis, we consider here a more general model including p-h
asymmetry in the self-energy. This model was proposed recently to investigate
the thermopower of non-Fermi liquids [62], which is sensitive to p-h asymmetry
because, unlike the optical spectra, it vanishes identically for p-h symmetric
systems.

Figure 16:  (a) Modulus and (b) phase of the optical conductivity given by
Eqs. (8) and (29) with $\nu=1$ and the other parameters as indicated in (a);
the dependence on $\alpha$ is shown for low and room temperature in
blue–yellow and blue–red shades, respectively. (c) Corresponding $\alpha$
dependence of the scattering rate and (d) the mass enhancement.

Particle-hole asymmetry is introduced in the self-energy by replacing
$S_{\nu}(x)$ in Eq. (22) by the function [62]

$S_{\nu}^{\alpha}(x)=\frac{(2\pi)^{\nu}}{\pi\,\Gamma(1+\nu)}\frac{\cosh(x/2)}{\cosh(\alpha/2)}\,\left|\,\Gamma\left(\frac{1+\nu}{2}+i\frac{x+\alpha}{2\pi}\right)\right|^{2}.$
(29)

A positive value of the dimensionless parameter $\alpha$ “skews” the
scattering by enhancing the scattering rate for holes relative to that for
particles. In Fig. 16, we illustrate the effect of $\alpha$ on the optical
conductivity. Results are shown for the Planckian case $\nu=1$. Similar
results are found for $\nu<1$. Figures 16(a) and 16(b) show that upon
increasing p-h asymmetry from zero to $\alpha=2$, the apparent exponent of the
conductivity hardly changes, both at low and room temperatures. Note that
$\alpha=2$ represents a strong p-h symmetry violation: with this value, the
scattering rate is $\sim 7$ times larger for holes at $\varepsilon\ll-
k_{\mathrm{B}}T$ than for electrons at $\varepsilon\gg k_{\mathrm{B}}T$.
Figures 16(c) and 16(d) show that the $\omega/T$ scaling laws of the optical
scattering rate and mass enhancement also only change in a minor quantitative
way upon increasing p-h asymmetry.

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