††thanks: These two authors contributed equally††thanks: These two authors
contributed equally††thanks: These two authors contributed equally

# Compartmentalizing the cuprate strange metal

M. Berben  High Field Magnet Laboratory (HFML-EMFL) and Institute for
Molecules and Materials, Radboud University, Nijmegen, Netherlands  J. Ayres
H. H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
C. Duffy  High Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules
and Materials, Radboud University, Nijmegen, Netherlands  R. D. H. Hinlopen
H. H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
Y.-T. Hsu  High Field Magnet Laboratory (HFML-EMFL) and Institute for
Molecules and Materials, Radboud University, Nijmegen, Netherlands  M. Leroux
LNCMI-EMFL, CNRS UPR3228, Univ. Grenoble Alpes, Univ. Toulouse, INSA-T,
Grenoble and Toulouse, France  I. Gilmutdinov  LNCMI-EMFL, CNRS UPR3228, Univ.
Grenoble Alpes, Univ. Toulouse, INSA-T, Grenoble and Toulouse, France  M.
Massoudzadegan  LNCMI-EMFL, CNRS UPR3228, Univ. Grenoble Alpes, Univ.
Toulouse, INSA-T, Grenoble and Toulouse, France  D. Vignolles  LNCMI-EMFL,
CNRS UPR3228, Univ. Grenoble Alpes, Univ. Toulouse, INSA-T, Grenoble and
Toulouse, France  Y. Huang  Van der Waals-Zeeman Institute, University of
Amsterdam, Amsterdam, Netherlands  T. Kondo  Institute for Solid State
Physics, University of Tokyo, Kashiwa, Japan  T. Takeuchi  Toyota
Technological Institute, Nagoya 468-8511, Japan  J. R. Cooper  Department of
Physics, University of Cambridge, Cambridge, United Kingdom  S. Friedemann  H.
H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
A. Carrington  H. H. Wills Physics Laboratory, University of Bristol, Bristol,
United Kingdom  C. Proust  LNCMI-EMFL, CNRS UPR3228, Univ. Grenoble Alpes,
Univ. Toulouse, INSA-T, Grenoble and Toulouse, France  Nigel E. Hussey  High
Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials,
Radboud University, Nijmegen, Netherlands  H. H. Wills Physics Laboratory,
University of Bristol, Bristol, United Kingdom

(August 28, 2024)

###### Abstract

Understanding the ‘anomalous normal’ state from which pairs form and condense
is key to unlocking the mystery of cuprate high-$T_{c}$ superconductivity.
While many of its signature transport properties – linear-in-temperature ($T$)
resistivity, linear-in-field ($H$) magnetoresistance (MR) at high field
strengths, quadratic-in-temperature inverse Hall angle and modified Kohler
scaling – have been identified, they are often considered either at a singular
doping level or across a narrow doping range. As a result, their relation to
each other and to the pseudogap, strange metal and non-superconducting regimes
that define the cuprate phase diagram has yet to be elucidated. Here, we
report an extensive series of high-field in-plane MR measurements on several
cuprate families spanning all three regimes that reveal a systematic evolution
of the MR, with each regime possessing its own distinct scaling behavior. In
the pseudogap regime, the MR exhibits pure $H/T^{2}$ scaling and $H$-linearity
at the highest field strengths. While the $H$-linearity persists inside the
strange metal regime, the scaling changes abruptly to $H/T$. The magnitude of
the $H$-linear slope is found to be correlated with both the $T$-linear
resistivity coefficient and $T_{c}$, strengthening the characterization of the
strange metal regime as a quantum critical phase. Finally, within the non-
superconducting, Fermi-liquid regime, we observe a recovery of conventional
Kohler scaling, highlighting the anomalous nature of the scaling in cuprates
that superconduct. This comprehensive study – over an unprecedented
temperature, field and doping range – enables us to compartmentalize the phase
diagram into distinct regimes, each with its own scaling behavior and
identifies power-law scaling of the normal state MR as a defining feature of
superconducting hole-doped cuprates. Phenomenology that obeys such simple
scaling laws suggests an equally simple organizing principle at play. The
incompatibility of such power-law scaling with any known variant of Boltzmann
transport theory, however, motivates the quest for an altogether new
theoretical framework, one in which the MR is entirely decoupled from all
elastic scattering processes.

August 28, 2024

##  I Introduction

The anomalous normal-state transport properties of high-$T_{c}$ cuprates [1,
2, 3, 4, 6, 6, 7, 4, 9, 10, 5, 12, 13] continue to challenge our understanding
of how metallicity and superconductivity emerge in a two-dimensional doped
Mott insulator [14, 15, 16, 17, 18]. Despite decades of concerted effort, a
consensus has yet to be reached on how the various phases/regimes that form
the cuprate phase diagram evolve or indeed relate to one another. At
sufficiently high (electron or hole) doping, it is now well established that
the system recovers a conventional, albeit highly correlated Fermi-liquid (FL)
ground state characterized by a pure $T^{2}$ zero-field resistivity
$\rho(0,T)$ at low temperatures [19, 20, 9]. Identifying the doping level $p$
at which the FL ground state is restored, however, has proved highly
controversial. In the case of hole-doped cuprates, this controversy centers on
two prominent, though distinct scenarios.

The first is a conventional quantum critical scenario in which the FL ground
state is restored beyond a specific hole doping $p^{\ast}\sim 0.2$ where the
normal state pseudogap (a partial gap in the electronic density of states)
closes [21, 22]. Resistivity curvature mapping – plots of the second
derivative d${}^{2}\rho$/d$T^{2}$ as a function of $T$ and $p$ – was used
early on to trace out the crossover line $T^{*}(p)$ (for $p<p^{*}$) below
which manifestations of pseudogap formation first appear in the dc transport
[23]. $T^{*}(p)$ also marks the temperature above which $\rho(0,T)$ adopts its
signature $T$-linearity extending up to the highest accessible temperatures
[24, 25, 26]. As $p$ increases, $T^{*}(p)$ drops monotonically before
vanishing abruptly in the vicinity of $p^{\ast}$ [27]. Beyond $p^{\ast}$, a
second temperature scale $T_{coh}$ emerges, above which $\rho(0,T)$ is again
proportional to $T$ [27, 28]. Together, these two scales define a fan-shaped
region of the phase diagram, centred on $p^{\ast}$ and hosting a robust
$T$-linear resistivity. Its resemblance to a quantum critical fan, coupled
with putative evidence for multiple broken symmetries below $T^{*}$ ([29] and
references therein), has motivated interpretations of $T^{*}$ as a second-
order phase transition line terminating at $p^{\ast}$ [21, 30, 22].

A key characteristic of the $T$-linear resistivity near $p^{*}$ is the
magnitude of its slope ($\sim$ 1-2 $\mu\Omega$cm/K) that can be parameterized
in terms of an effective scattering rate $\hbar/\tau\sim\zeta k_{B}T$ with
$\hbar$ Planck’s constant, $k_{B}$ Boltzmann’s constant and $\zeta$ a
dimensionless number 1 $<\zeta<\pi$ [31, 32, 33]. This rate is widely (though
not universally [34, 35]) attributed to Planckian dissipation [17], a notional
bound on the equilibration rate for charge carriers in correlated metals [36].
While there is as yet no microscopic theory for Planckian dissipation, it is
generally attributed to scattering off quantum critical fluctuations
associated with the end of the pseudogap regime.

One of the major challenges to any scenario based on conventional quantum
criticality is the persistence of a $T$-linear component in $\rho(T)$ (=
$a_{1}T$) far beyond $p^{\ast}$ and down to the lowest temperatures studied
(currently $\sim$ 0.02 K for electron-doped [9] and $\sim$ 0.1 K for hole-
doped cuprates [37, 38]). Such ‘strange metallic’ behavior [39] implies that a
FL ground state is not restored at $p^{\ast}$. Rather, a second scenario has
been proposed; that of a quantum critical phase extending from $p^{\ast}$ to
$p_{sc}\sim 0.3$ – the doping level corresponding to the edge of the
superconducting (SC) dome [4, 40, 32].

Deducing which scenario is valid is important as it may ultimately determine
whether the transition into the SC state beyond optimal doping involves
coherent quasiparticles $\grave{a}$ $la$ BCS (possibly with an effective mass
enhanced by critical fluctuations) or whether it requires a more exotic non-
BCS framework [41, 42, 43].

On a fundamental level, such limiting low-$T$ $T$-linear resistivity can arise
from scattering off bosonic fluctuations with a characteristic frequency
$\Omega$ that is lower than the temperature scale [44]. While charge and spin
fluctuations have been extensively probed in overdoped cuprates [45, 46, 47,
48, 49, 50], there is as yet no evidence that $\Omega\rightarrow$ 0 or that
the correlation length of the associated order parameter diverges, neither at
$p^{\ast}$ nor at any doping level up to $p_{sc}$. This lack of an
identifiable quasi-static bosonic mode has compelled theorists to look beyond
conventional many-body approaches [51, 52, 53] and experimentalists to look
beyond $\rho(0,T)$ [5, 54, 3, 12, 56] for additional clues as to the nature of
the strange metal.

In principle, magnetoresistance (MR) provides complementary information about
the nature of the scattering processes underlying $\rho(0,T)$. Moreover, even
in systems that appear highly complex, important insights can often be gained
through the observation of simple scaling laws. The low-field MR of many
conventional metals, for example, obeys Kohler scaling:

$\Delta\rho(H,T)/\rho(0,T)=f(H/\rho(0,T))$ (1)

where typically, $f(H/\rho(0,T))\propto(H/\rho(0,T))^{2}$ (the low-field MR
being quadratic in field strength). Adherence to Kohler scaling implies that
the $T$-dependence of the MR is (almost) entirely governed by changes in
$\hbar/\tau$ with temperature or purity. Implicit in this rule is the fact
that the scattering processes that determine $\rho(0,T)$ are the same as those
that set the size of the MR itself.

In cuprates, even those comprising a single, cylindrical Fermi sheet,
conventional Kohler scaling breaks down. Near optimal doping, for example, the
low-field MR is claimed to follow a ‘modified’ Kohler scaling [6]:

$\Delta\rho(H)/\rho(0)=s(\mathrm{tan^{2}}\theta_{\mathrm{H}})$ (2)

where the Hall angle
$\theta_{\mathrm{H}}\equiv\mathrm{tan}^{-1}(\rho_{xy}/\rho_{xx})$ and $s$ is a
prefactor. Since $\mathrm{cot}\theta_{\mathrm{H}}\sim A+BT^{2}$ [3] and
$\rho(T)\propto T$, modified Kohler scaling has often been interpreted as a
manifestation of the separation of transport and Hall lifetimes [15], the
$T^{-1}$/$T^{-2}$ lifetimes being associated with relaxation processes
normal/tangential to the Fermi surface in response to the application of an
electric/magnetic field, respectively. We note that similar behavior has now
been reported in other correlated electron systems [57, 58, 59].

Recent MR studies – carried out to much higher field strengths – have
uncovered an altogether different phenomenology however. The quadratic MR seen
at low fields is found to evolve into a $H$-linear MR (LMR) with a
$T$-dependent crossover field. Such behavior is best captured by the empirical
expression:

$\rho(H,T)=\mathcal{F}(T)+\sqrt{(\alpha
T)^{2}+(\gamma\mu_{0}H)^{2}}\leavevmode\nobreak\ ,$ (3)

where $\mathcal{F}(T)$ is a field-independent contribution to the resistivity
and $\alpha$ and $\gamma$ are fixed coefficients. This form of MR, first
identified [60] in BaFe2(As1-xPx)2 near its magnetic QCP (with
$\mathcal{F}(T)\simeq\rho_{0}$, the residual resistivity), has since been
observed in the iron chalcogenides [61] as well as in highly-doped cuprates
[12]. The coupling of $H$ and $T$ in quadrature leads to $H/T$ scaling, in
which a series of MR sweeps plotted as $\Delta\rho/T$ =
($\rho(H,T)-\mathcal{F}(T))/T$ vs. $H/T$ collapse onto a single curve. Eq.
((3)) also implies that at low-$T$, the LMR has a slope that is
$T$-independent and set by $\gamma$ and in zero-field, $\rho(0,T)$ possesses a
strictly $T$-linear component $\alpha T$.

There are several elements of Eq. ((3)) that stand out. The first is that the
residual (elastic scattering) contribution is contained only in
$\mathcal{F}(T))$ and not in the MR. This aspect of the scaling is wholly
incompatible with any form of Kohler scaling, conventional or otherwise, in
which the $T$-dependence of the MR is set by the total scattering rate. The
second is the implicit association of this form of MR scaling with $T$-linear
zero-field resistivity. In the pnictides and chalcogenides, this association
has tied the quadrature MR to their respective QCPs. In highly-doped cuprates,
however, it establishes a link between the $H/T$ scaling seen far beyond
$p^{\ast}$ [12] and the persistent $T$-linear component in $\rho(0,T)$ [4, 62,
32]. The final notable feature of Eq. ((3)) is that $\mathcal{F}(T))$ can also
include a $T$-dependent term. This feature, most evident in the overdoped
cuprates [12], implies that $\rho(H,T)$ can be decomposed into two distinct
components, only one of which has a non-negligible MR.

The origin of this dual character of the dc transport is not known at present.
Moreover, while that initial study [12] reaffirmed the notion that a FL ground
state is not restored at $p^{\ast}$, questions remained about the universality
of the phenomenology. The purpose of this article is to examine the evolution
of the in-plane MR across the entire phase diagram of hole-doped cuprates and
to establish the experimental framework out of which further theoretical
understanding can emerge. In so doing, we seek to address the following open
questions:

A: How does the magnitude of the high-field $H$-linear slope $\gamma$ evolve
across the SC dome? Specifically, how does it relate to $T_{c}$, $p$ or
$a_{1}$ – the $T$-linear component of the low-$T$ zero-field resistivity?
While the correlation of the latter with $T_{c}$ is well established [4, 13],
the inclusion of $\mathcal{F}(T)$ in Eq. ((3)) cautions that $\alpha$ and
$a_{1}$ are not necessarily equivalent. The evolution of $\gamma(p)$ should
indicate whether or not their origins are intimately linked.

B: How does the MR scaling evolve below $p^{\ast}$? If Kohler scaling is found
to emerge within the pseudogap regime, as previously suggested [8], one could
map how this FL component emerges out of the strange metal regime. If, on the
other hand, the power-law scaling is preserved, this would highlight a
distinct non-FL component to the remnant carriers below $p^{\ast}$.

C: What happens beyond the SC dome? More specifically, does the loss of
superconductivity on the overdoped side coincide with the disappearance of LMR
at high fields and/or the recovery of a more conventional Kohler scaling? This
question links directly to the controversy highlighted above surrounding the
exact location (in $p$) of the emergence of a FL ground state in overdoped
cuprates.

D: How can $H/T$ scaling be reconciled with the modified Kohler scaling seen
at low fields? While the distinct $T$-dependencies of the zero-field and low-
field or Hall resistivities have been an essential ingredient of many
phenomenological models of the cuprate normal state [15, 65, 66, 67, 68, 69,
70], all such models were proposed before the discovery of power-law MR
scaling.

In order to address these outstanding questions, we have carried out a
detailed study of the high-field MR of three different families of single-
layered cuprates: LSCO, (Pb/La)-doped Bi2Sr2CuO6+δ (Bi2201) – covering a far
wider doping range than previous studies [5, 12] – and Tl2Ba2CuO6+δ (Tl2201)
under hydrostatic pressure, combined with analysis of existing data on Bi2201,
LSCO and HgBa2CuO4+δ (Hg1201). The breadth of our study, over unprecedented
field, temperature and doping/pressure ranges, has enabled us to provide
definitive answers to all of the above questions, as we outline below. At the
same time, the abrupt changes observed in the form of MR scaling across the
phase diagram allow us to compartmentalize the magnetotransport of hole-doped
cuprates into three distinct doping regimes: the pseudogap, the strange metal
and the non-superconducting FL regime. Such compartmentalization supports
existing models characterizing the pseudogap as a doping-dependent energy
scale that collapses at $p^{\ast}$ [72, 71], rather than as a temperature
scale associated with a transition into some form of hidden order. The most
striking observation, however, is the crossover from conventional Kohler
scaling in the FL regime to pure power-law scaling for all superconducting
samples, the latter being wholly incompatible with Boltzmann transport theory.
Overall, these findings reveal that the normal state of high-$T_{c}$
superconductors has its own unique magnetic field response, one that is
seemingly decoupled from all elastic scattering processes.

Figure 1: Simplified phase diagram of hole-doped cuprates showing the doping
ranges (as double-headed arrows) of different families studied in this work
and elsewhere [7, 6, 5, 4, 8]. Dashed lines indicate the boundaries (in $p$ at
$T$ = 0) between the pseudogap, strange metal and FL regimes, as inferred from
the magnetotransport. The solid line represents the superconducting (SC) dome
for LSCO and Bi2201 (black line) and Tl2201 (red line), respectively,
normalized to $T_{c}^{\rm max}$ – the maximum value of $T_{c}$ – in each
family.

##  II Experimental Survey

Figure 2: (a)-(c) Representative $\rho(H,T)$ curves for cuprates within the
strange metal regime: (a) Bi2201 $p$ = 0.24, (b) Tl2201 $p$ = 0.275 and (c)
LSCO $p$ = 0.23. Note the crossover to $H$-linear MR at the highest fields
with a slope that is temperature independent. (d)-(f) Demonstration of
adherence to $H/T$ (quadrature) MR scaling after subtraction of the zero-field
resistivity, i.e. $\Delta\rho=\rho(H,T)-\rho(0,T)$, for the same three
samples. For MR curves below $T_{c}$, $\rho(0,T)$ was obtained from universal
fits to Eq. (3). (g)-(i) Corresponding scaling of the derivatives
d$\rho$/d$\mu_{0}H$. The data in panel (g) are reproduced from Ref. [12].
Dashed lines in all panels (d)-(i) are fits to the derivative of the
quadrature form given by Eq. (3).

In order to preempt our subsequent discussion, we introduce in Figure 1 a
simplified phase diagram of hole-doped cuprates in which the pseudogap,
strange metal and FL regimes are framed along the $T$ = 0 doping axis and
delineated by vertical dashed lines. In so doing, we define the strange metal
regime as the doping range ($p^{\ast}<p<p_{sc}$) over which a dominant low-$T$
$T$-linear resistivity prevails [4, 62, 32, 3]. For simplicity, only a
singular boundary line separating each regime has been drawn, though it is
anticipated that the precise location of each boundary will be slightly
material specific.

The double-headed arrows in Fig. 1 define the doping ranges of the various
families investigated in this study. The study itself includes new MR data on
LSCO crystals with $x$ = 0.20, 0.23, 0.30 and 0.33, as well as new analysis of
MR data reported elsewhere on LSCO ($x$ = 0.09 [7], 0.15 [6], 0.19 [5] and
0.23 [4]). In the case of Bi2201, we report new MR data for $p$ = 0.13, 0.16,
0.195, 0.205, 0.22 and 0.24 and 0.245 (much of it taken in pulsed fields up to
60 T) combined with further analysis of lower-field MR data reported in Ref.
[12] for the restricted doping range 0.24 $<p<$ 0.27. For each family, we have
performed field sweeps at multiple temperatures from below $T_{c}(0)$ up to
300 K. For Tl2201, we report MR data on a single sample whose $T_{c}$ (and
therefore $p$) is varied using hydrostatic pressure and whose ambient pressure
data were reported in Ref. [12]. Also included in Fig. 1 is a single Hg1201
sample reported in Ref. [8].

Figure 3: (a)-(e): $\rho(H,T)$ of a selection of Bi2201 crystals spanning the
doping range 0.13 $\leq p\leq$ 0.27. With decreasing $p$, a marked increase in
the magnitude of the magnetoresistance (at a constant temperature) is
observed, as highlighted within the red shaded regions for $T$ = 80 K. (f)
$\Delta\rho(H,80\mathrm{K})$ curves for all 8 dopings studied. (g)
Corresponding $\Delta\rho/\rho(0)$ curves. For $p>p^{\ast}\sim 0.2$, the
magnitude of $\Delta\rho/\rho(0)$ drops with increasing $p$, revealing that
within the strange metal regime ($p^{\ast}<p<p_{sc}$), the order-of-magnitude
enhancement in the MR is not a result of an increase in $\rho(0,T)$ (e.g. due
to a reduced carrier density).

Representative series of MR curves within the strange metal regime are shown
in panels (a)-(c) of Fig. 2 for Bi2201, Tl2201 and LSCO, respectively.
Adherence to $H/T$ scaling is evident through inspection of panels (d)-(f)
where plots of $\Delta\rho/T(=(\rho(H,T)-\rho(0,T)/T)$ vs. $H/T$ collapse
onto, or asymptotically approach (for $T<T_{c}(H$=0)), the dashed lines
defined by Eq. (3). Panels (g)-(i) of Fig. 2 show the derivatives
d$\rho$/d$H$, and highlight the tendency of all MR curves to become linear at
high $H/T$. The collapse of the d$\rho$/d$H$ curves at the highest fields
indicates that the LMR slope $\gamma$ is indeed $T$-independent. The dashed
lines in the lower panels are fits of the derivative of Eq. (3) to the
d$\rho$/d$H$ data. The key aspects of the MR: the crossover from $H^{2}$ to
$H$-linear MR with increasing $H/T$, the constancy of $\gamma$ over an
extended temperature range and the adherence to the $H/T$ quadrature scaling
form, are apparent in all three data sets and demonstrate universal scaling
behavior across different families within the strange metal regime.

Having confirmed the robustness of $H/T$ quadrature scaling in this region of
the phase diagram [12], we now turn to address each of the outstanding
questions (A-D) listed above. In the process, we reveal that while LMR is a
ubiquitous feature of the high-field MR, the scaling itself is distinct in
each regime, the reasons for which we will discuss in Section III.

###  II.1 Doping dependence of the $H$-linear MR

In both electron- and hole-doped cuprates, $a_{1}$ decreases with increasing
doping (beyond $p^{\ast}$) and vanishes around $p\sim p_{sc}$ [4, 9, 62, 32,
13], revealing a robust correlation between $a_{1}$ and $T_{c}$ [13]. While
previous cuprate studies [5, 74] have hinted at a similar trend in $\gamma$,
the doping ranges were too narrow (0.16 $\leq p\leq$ 0.19 [5] and 0.15 $\leq
p\leq 0.17$ [74]) to make any definitive claims. In this section, we explore
the evolution of $\gamma(p)$ across the SC dome and compare it with
$a_{1}(p)$.

####  II.1.1 Bi2201

Panels (a)-(e) of Figure 3 show $\rho(H,T)$ for a representative set of Bi2201
samples over a doping range (0.13 $\leq p\leq$ 0.27) that incorporates both
the pseudogap and strange metal regimes. At temperatures above $T_{F}$ (the
temperature below which paraconductivity contributions from SC fluctuations
become appreciable), the low-field MR is quadratic. In highly doped samples
with a low $T_{c}$ (and thus low $T_{F}$), the crossover from quadratic to LMR
is clearly visible within the accessible field range. This is in agreement
with the data reported in Ref. [12].

The red shaded regions of panels (a)-(e) highlight the doping evolution of the
MR at a fixed temperature $T$ = 80 K, i.e. well above $T_{F}$ for each sample.
Noting that the absolute span in $\rho$ is the same (= 140 $\mu\Omega$cm) in
all panels, we find that the magnitude of the MR decreases monotonically with
increasing $p$. The overall trend is summarized in Fig. 3f where
$\Delta\rho(H$, 80 K) ($=\rho(H$, 80 K) - $\rho$(0, 80 K)) is plotted for all
8 dopings studied. $\rho$(0, 80 K) itself decreases roughly by a factor of 2
across the series, reflecting changes in the scattering rate, the carrier
density or some combination of the two. $\Delta\rho(H$, 80 K), by contrast,
decreases by more than one order of magnitude. The corresponding
$\Delta\rho(H$, 80K)/$\rho$(0, 80 K) values are plotted in Fig. 3g.
Intriguingly, for the three pseudogapped samples with $p<p^{\ast}(\sim 0.2$),
$\Delta\rho(H$, 80K)/$\rho$(0, 80 K) is found to be unchanged, implying that
the increase in the magnitude of $\Delta\rho$ for these samples is a direct
consequence of the increase in $\rho(0)$ with underdoping, presumably due to
the loss of electronic states once the pseudogap opens.

For higher dopings, by contrast, even the fractional change in MR is found to
vary strongly with $p$, decreasing by an order of magnitude between $p$ =
0.205 and 0.27 (see Fig. 3g). In a typical metal, the magnitude of
$\Delta\rho/\rho(0)$ is set by the product $\omega_{c}\tau$ where
$\omega_{c}=e\mu_{0}H/m^{*}$ is the cyclotron frequency, $e$ is the electronic
charge and $m^{*}$ is the cyclotron mass and is thus largely independent of
carrier density. A monotonic decrease in $\tau$ as the system is doped away
from the Mott insulating state (and thus becomes increasingly more metallic)
seems unlikely; there is little variation in $\rho_{0}$ within this doping
range and no indication, e.g. from existing specific heat measurements [75],
for a large change in $m^{*}$. Thus, the marked decrease in
$\Delta\rho/\rho(0)$ with overdoping seems puzzling, though may arise from a
decrease in the anisotropy of the in-plane mean-free-path $\ell$ as $p$
approaches $p_{sc}$. Note that any decrease in the anisotropy of $\ell$ cannot
be caused by the Fermi energy $\epsilon_{F}$ moving away from a van Hove
singularity (vHs) since in Bi2201, the vHs crossing point is believed to be
located closer to $p_{sc}\sim 0.27$ than to $p^{\ast}\sim 0.2$ [76].

Figure 4: $T$-dependence of the derivative d$\rho$/d$\mu_{0}H$ at the highest
measured field range (specifically the last 3 T of each field sweep) – for the
MR curves displayed in panels (a)-(e) of Fig. 3. For all dopings,
d$\rho$/d$\mu_{0}H|_{H_{\rm max}}$ saturates at low $T$ at a value (=
$\gamma$) denoted by horizontal dashed lines. Shaded regions reflect the
uncertainty in $\gamma$ for each data set. Estimates for $\gamma$ obtained
from field sweeps for which the $H$-linear regime could not be reached are not
shown. For $p$ = 0.16, the upturn in d$\rho$/d$\mu_{0}H|_{H_{\rm max}}$ at the
lowest $T$ is due to paraconductivity effects. The origin of the low-$T$
downturn in d$\rho$/d$\mu_{0}H|_{H_{\rm max}}$ for $p$ = 0.13 is unknown.
Nevertheless, an estimate for $\gamma$ can still be made, albeit with greater
uncertainty.

At high $H/T$ (below $T_{c}$ in most samples) and for all dopings, the MR
becomes $H$-linear with a $T$-independent slope $\gamma$, consistent with the
quadrature expression in Eq. (3). Such behavior can also be inferred from the
high-$H$ convergence of the d$\rho$/d$H$ curves plotted in Fig. 2. In order to
obtain estimates for $\gamma(p)$, we study the low-$T$ limit of the highest
field data (where the normal state can be accessed). A summary of the results
is shown in Figure 4 where the $T$-dependence of d$\rho$/d$\mu_{0}H|_{H_{\rm
max}}$ – the MR slope at the highest measured field range (i.e. over the last
3 T of each field sweep) – is plotted for all the samples whose raw MR data
are included in panels (a)-(e) of Fig. 3. In agreement with Eq. (3) and with
previous measurements on LSCO [5], Tl2201 and Bi2201 [12],
d$\rho$/d$\mu_{0}H|_{H_{\rm max}}$ is found to increase with decreasing
temperature and to saturate at a constant value (= $\gamma$) at the lowest
temperatures. Consistent with our expectations from Fig. 3g, we observe a
marked monotonic increase in $\gamma(p)$ as $p$ is reduced.

####  II.1.2 Tl2201

Although the increase in $\gamma$ with decreasing $p$ shown in Fig. 4 is
significant, the above results involve measurements on different samples.
Inevitably, inter-sample variation and geometrical errors will introduce some
uncertainty in the determination and evolution of $\gamma(p)$. In order to
circumvent this, a second study was carried out on a single crystal of
overdoped Tl2201 ($T_{c}$ = 35 K at ambient pressure) as a function of
hydrostatic pressure. The $T_{c}$ (and inferred doping) of Tl2201 can be tuned
appreciably with the application of only modest pressures [77]. Panels (a)-(e)
of Figure 2 in the Supplementary Information (SI) show the full set of
$\rho(H,T)$ curves obtained across the pressure range 0 to 2 GPa, over which
$T_{c}$ decreases from 35 K to 26 K (26 %). Panels (f)-(i) of the same figure
show the corresponding derivatives plotted vs $H/T$. A gradual decrease in
$\gamma$ is observed with increasing pressure, consistent with the correlation
between $\gamma$ and $T_{c}$ (or $p$) seen in Bi2201 beyond $p^{\ast}$.

Figure 5: Doping dependence of $\gamma$ – the high $H/T$ limiting value of the
MR – and $a_{1}$ – the low-$T$ $T$-linear coefficient of the resistivity – for
(a,b) Bi2201, (c,d) LSCO and (e,f) Tl2201, respectively. Diamonds indicate
results obtained as part of this high-field study, while circles and squares
indicate results obtained from the literature. Circles and squares in panels
(c) and (d) are taken from Ref. [4] and [5], respectively. Circles in panel
(f) are from Ref. [62]. Note that $a_{1}$ is plotted only for samples with
$p>p^{\ast}$. In panel (e), $\gamma$ is plotted vs. $p$ as determined from the
$T_{c}$ value at each pressure using the correlation reported in Ref. [3]. All
dotted lines are guides to the eye. Arrowheads on the doping axes indicate
$p_{sc}$ for the respective material.

####  II.1.3 LSCO

As reported in Ref. [5], thin-film LSCO at a doping level ($x$ = 0.19) near
$p^{\ast}$ also exhibits LMR at the highest fields. In order to confirm this
behavior and to extend the measurements of the MR up to higher dopings (and
higher temperatures), we measured the MR of two superconducting LSCO single
crystals with $p$ = 0.20 and 0.23 up to 35 T. The derivatives
$\mathrm{d}\rho/\mathrm{d}H$ show that, just as in overdoped Bi2201 and
Tl2201, the MR becomes $H$-linear at high $H/T$ values (the data for $x=p$ =
0.23 are presented in Fig. 2). In an earlier study [4], high fields were
employed to obtain the limiting low-$T$ resistivity in overdoped LSCO over a
wide doping range, though there, the normal state MR was assumed to vary as
$H^{2}$ at all field strengths up to 55 T. Re-analysis of their MR data for
$x$ = 0.23, presented in Fig. 3 of the SI, combined with the new measurements,
confirms that the high-field MR in overdoped LSCO also shows a tendency
towards LMR at the lowest temperatures.

####  II.1.4 Summary

The left and right panels of Figure 5 show, respectively, the doping/pressure
evolution of $\gamma$ and $a_{1}$ for all three families. Although an
equivalence between pressure and hole doping in Tl2201 has not yet been
definitively demonstrated, we adopt here the correlation between $T_{c}$ and
$p$ reported in Ref. [3] to arrive at the same conclusion. The striking
correlation between $\gamma$ and $a_{1}$, combined with the ubiquitous $H/T$
scaling beyond $p^{\ast}$, confirms that the two phenomena share a common
origin (though why the ratio $\gamma/a_{1}$ is higher in Tl2201 than in LSCO
or Bi2201 is not yet clear). Crucially, $\gamma(p)$ appears to extrapolate to
zero at a doping level $p\approx$ 0.3 for all three families, implying that
the emergence of LMR and $H/T$ scaling in hole-doped cuprates is closely tied
to the onset of superconductivity, as recently intimated for electron-doped
cuprates [74].

Figure 6: (a) $\Delta\rho(=\rho(T,H)-\rho(T,0)$) plotted versus $H^{2}$ in
Bi2201 $p$ = 0.205 highlighting the $H^{2}$ dependence of the MR at low
fields. The dashed lines are quadratic fits ($\Delta\rho=\beta(\mu_{0}H)^{2}$)
to the data in the low $H/T$ range from which the $\beta$ coefficients are
obtained. (b) Resultant $\beta^{-1}$ values plotted versus temperature on log-
log axes. The dashed, dash-dotted and dotted guides are shown for $m$ = 1, 1.5
and 2, respectively. (c) $\Delta\rho/T$ vs. $H/T$ plot for optimally doped
Bi2201 ($p$ = 0.16). Clearly, the data do not collapse. (d) $\beta^{-1}\times
T$ vs. $T$ for Bi2201 $p$ = 0.16 and 0.245. The dotted line represents a 1/$T$
dependence.

This correlation between $\gamma$ and $a_{1}$ within the strange metal regime
is robust and seemingly universal, having been observed in three different
families. Its preservation below $p^{\ast}$, however, is far less certain
since the low-$T$ $T$-linear component in $\rho(0,T)$ is either reduced [4,
5], or lost entirely [79, 8, 78, 80]. The observation of a continued increase
in $\gamma$ below $p^{\ast}$ is thus surprising as it suggests an apparent
decoupling of $\gamma$ and $a_{1}$ below $p^{\ast}$, an anomaly we will return
to in the Discussion section. But first, having established $H/T$ scaling as a
universal property of the strange metal regime, let us turn to examine how the
MR scaling itself evolves as the pseudogap develops.

###  II.2 MR scaling within the pseudogap regime

As the pseudogap opens, electronic states near the Fermi level become
depleted, initially near the zone boundary. At the same time, the $T$-linear
component in $\rho(0,T)$ (below $T^{\ast}$) becomes suppressed and $\rho(0,T)$
recovers a dominant $T^{2}$ dependence [78, 8]. In the clean cuprate Hg1201,
the MR is also reported to exhibit more conventional, Kohler-like scaling [8].
This, coupled with the observation of quantum oscillations, has led to the
conjecture that the remnant charge at the Fermi level in the low-$T$ pseudogap
regime forms a Fermi liquid. In order to differentiate between Kohler and
quadrature scaling, however, it is helpful to investigate the MR response of
underdoped cuprates with larger residual resistivities.

Until now, $H/T$ scaling has been revealed by collapsing plots of
$\Delta\rho/T$ or d$\rho/$d$H$ vs. $H/T$ onto a single curve. For certain
field sweeps, however, the maximum field strength is not high enough to
confirm the scaling relation. An alternative way to test for such scaling is
to focus on the low-field ($H\to 0$) regime where $\Delta\rho(H)$
asymptotically approaches a $H^{2}$-dependence. Rewriting Eq. (3) as:

$\displaystyle\Delta\rho(T,H)$ $\displaystyle=$ $\displaystyle\alpha
T\left(\sqrt{1+\left(\gamma\mu_{0}H/\alpha T\right)^{2}}-1\right)$ (4)

and performing a Taylor expansion, we find:

$\displaystyle\Delta\rho(T,H\rightarrow 0)$ $\displaystyle=$
$\displaystyle\frac{\gamma^{2}}{2\alpha T}(\mu_{0}H)^{2}=\beta(\mu_{0}H)^{2}$
(5)

Hence, for any material exhibiting quadrature MR with pure $H/T$ scaling, the
coefficient $\beta$ of the low-field $H^{2}$ MR should vary as $1/T$. More
generally, a quadrature form of the MR scaling as $H/T^{m}$ (as opposed to the
$H/T$ scaling form expressed through Eq. ((4))) will lead to $\beta\propto
1/T^{m}$. This is in marked contrast to the $\beta\propto$ 1/$\rho(0,T)$
dependence expected from Kohler scaling (recall that in LSCO and Bi2201,
$\rho_{0}$ is large relative to the $T$-dependent part of $\rho(0,T)$ – see,
e.g. Fig. 2). With this in mind, we now turn to examine the $T$-dependence of
$\beta^{-1}$ as we approach $p^{\ast}$ and as the pseudogap opens (see Fig. S4
and associated text in the SI for a discussion of the robustness of the
extraction of $\beta(T)$ through this procedure). Fig. 6a illustrates how
$\beta(T)$ is determined for a Bi2201 sample with $p$ = 0.205 (close to
$p^{\ast}$), while Fig. 6b shows resultant plots of $\beta^{-1}(T)$ for all
Bi2201 samples on a log-log scale.

The sequence of straight lines in Fig. 6b reveals two important features of
the low-field MR of Bi2201. Firstly, $\beta^{-1}(T)$ exhibits a pure power-law
$T$-dependence at all dopings and for all $T>T_{\mathrm{F}}$ at which the
normal state MR can be accessed down to $H$ = 0. Secondly, as indicated by the
dashed/dotted lines, the exponent $m$ of the power-law dependence is found to
change monotonically from 1 to 2 (see legend) as doping is reduced. It is
important to reiterate that the high residual resistivities of Bi2201 are such
that, were Kohler scaling to be obeyed, marked deviations from such power-law
fits would be evident at temperatures far higher than the base temperature of
these measurements. This particular feature of the data will become even more
evident in Section II.C when we consider highly overdoped, non-SC LSCO.

Figure 7: Scaling plots for selected Bi2201 samples where the appropriate
exponent $m$ for each sample is used to rescale both axes with $T^{m}$ and
collapse the MR data. (a) $p$ = 0.16: $\Delta\rho/T^{2}$ plotted versus
$\mu_{0}H/T^{2}$. (b) $p$ = 0.195: $\Delta\rho/T^{1.7}$ plotted versus
$\mu_{0}H/T^{1.7}$. (c) $p$ = 0.245: $\Delta\rho/T$ plotted versus
$\mu_{0}H/T$. For panels (a)-(c), only the MR above $T_{\mathrm{F}}$ is
plotted. (d)-(f): Same scaling plots as in panels (a)-(c) for temperatures
below $T_{\mathrm{F}}$ and over an extended $H/T^{m}$ range. The black dashed
lines indicate the quadrature form obtained from fits to the normal-state data
using Eq. (6).

The crossover from $H/T$ ($m=1$) scaling within the strange metal regime to
$H/T^{2}$ ($m=2$) scaling below $p^{\ast}$ is further highlighted in panels
(c) and (d) of Figure 6. Fig. 6c captures the loss of $H/T$ scaling for
$p=0.16$ (compare Fig. 6c with panels (c) and (f) of Fig. 7 for a Bi2201
sample with $p$ = 0.245, i.e. deep within the strange metal regime). Fig. 6d
shows the product $T\beta^{-1}$ plotted vs. $T$ for the same two samples. For
$p$ = 0.245, $T\beta^{-1}$ is a constant, consistent with $m$ = 1, while for
$p$ = 0.16, $T\beta^{-1}$ diverges (as $1/T$) as $T$ decreases.

Despite these marked changes in the power-law exponent of the low-field MR,
the field-dependence of the MR evolves in all cases from quadratic at low
fields to linear at high fields with a linear slope that saturates at high
fields (and/or low temperatures). This behavior can be captured by the
following generalized expression:

$\displaystyle\Delta\rho(H,T)$ $\displaystyle=$
$\displaystyle\alpha_{m}T^{m}\left(\sqrt{1+\frac{(\gamma\mu_{0}H)^{2}}{(\alpha_{m}T^{m})^{2}}}-1\right)\leavevmode\nobreak\
,$ (6)

where $\alpha_{m}$ is a prefactor independent of $T$ and $H$ with units
$\mu\Omega$cm/Km. The total resistivity is then expressed as:

$\displaystyle\rho(H,T)$ $\displaystyle=$
$\displaystyle\mathcal{F}(T)+\sqrt{(\alpha_{m}T^{m})^{2}+(\gamma\mu_{0}H)^{2}}\leavevmode\nobreak\
.$ (7)

To illustrate the goodness of fit to Eq. (6), Figure 7 shows MR curves for
three representative Bi2201 samples, where the appropriate exponent $m$ for
each sample is used to rescale both axes by $T^{m}$ and collapse the MR data.
Note that the data collapse occurs over a decade in temperature in all three
cases. Intriguingly, while $m$ has an integer value in the pseudogap and
strange metal regimes, in the vicinity of $p^{\ast}$, $m$ takes on a value
that is intermediate between the two.

Figure 8: (a) Log-log plot of $\beta^{-1}(T)$ for LSCO near $p^{\ast}$
($x=0.20$, filled diamonds) and beyond ($x=0.23$, filled squares). Dashed and
dotted lines represent power laws $T^{1.15}$ and $T^{1.55}$, respectively. (b)
Comparative plot for LSCO dopings beyond the SC dome ($x=0.30$, diamonds;
$x=0.33$, squares). The dashed line indicates that $\beta^{-1}(T)\sim
c+dT^{2}$.

Having established the ubiquity of power-law scaling in superconducting
Bi2201, we turn to investigate whether similar behavior is realized in other
cuprate families. In Figure 8a, we plot $\beta^{-1}$ vs. $T$ for two
superconducting LSCO samples, one close to $p^{\ast}$ ($x$ = 0.20) and one
inside the strange metal regime ($x$ = 0.23). For $x$ = 0.23, a pure power-law
dependence is observed from the highest temperature range studied down to 25 K
with $m=1.15\pm 0.1$, in accord with the collapse of the $\Delta\rho/T$ vs.
$H/T$ curves displayed in Fig. 2. For $x$ = 0.20, $\beta^{-1}(T)$ also follows
a pure power-law dependence, though in this case, $m=1.55\pm 0.15$, a value
that is similar to those found in Bi2201 in the vicinity of $p^{\ast}$.
Previously, it was shown that the MR data for LSCO $x$ = 0.19 [5] do not
exhibit $H/T$ quadrature scaling [64] even though LMR is observed at high
field strengths. Our own analysis of the data of Ref. [5] – shown in Figure 5
of the SI – reveals that the MR for $x$ = 0.19 actually exhibits $H/T^{2}$
scaling. Fig. 6 of the SI also shows that for optimally doped LSCO ($x$ = 0.17
[6]) and underdoped Hg1201 ($p=0.10$ [8]), $\beta^{-1}\propto T^{2}$. This
latter finding is consistent with the original conclusions of Chan et al. [8],
who claimed that the MR of underdoped Hg1201 obeys conventional Kohler
scaling. Being one of the cleanest cuprate systems, $\rho_{0}$ of underdoped
Hg1201 is extremely low and since $\rho(0,T)$ $\propto$ $T^{2}$ below around
200 K [8], a Kohler plot (with $\Delta\rho/\rho(0)$ vs. $H/\rho(0)$) does
indeed produce the same scaling. As a result, both descriptions are justified.
In underdoped LSCO or Bi2201, on the other hand, Kohler scaling clearly breaks
down due to the fact that their residual resistivities are much larger. Thus,
we conclude that $H/T^{2}$ scaling is the more appropriate way to categorize
the universal low-field MR response of hole-doped cuprates below $p^{\ast}$
and that the absence of $\rho_{0}$ in the MR scaling is a notable feature
which cannot be captured by Boltzmann theory, highlighting a distinctly non-FL
aspect to the MR within the pseudogap regime.

###  II.3 Scaling of the MR beyond the SC dome

As stated in the Introduction, for $p>p_{sc}$, $\rho(0,T)\propto T^{2}$ at
low-$T$, as expected for a correlated FL [19, 20]. Moreover, in electron-doped
cuprates, the field-dependence of the low-field MR switches from linear to
quadratic as superconductivity is suppressed [74]. The form of the MR in hole-
doped cuprates beyond $p_{sc}$, however, has not yet been investigated. Here,
we report a recovery of Kohler scaling in $p$-doped cuprates beyond the SC
dome.

Figure 9: Doping dependence of the power-law MR scaling in hole-doped
cuprates. Right axis: Normalized $T_{c}/T_{c}^{\rm max}$ for LSCO and Bi2201
(black line) and Tl2201 (red line). Left axis: The exponent $m$ as determined
from a power law fit to $\beta^{-1}$ ($\sim T^{m}$) for Bi2201, LSCO and
Hg1201. This plot combines data obtained in this work with published data on
LSCO [7, 6, 5] and Hg1201 [8]. As the temperature range to extract
$\beta^{-1}(T)$ from the low-field MR in Tl2201 is too narrow, the data points
for Tl2201 were deduced from the $H/T$ scaling first reported in Ref. [12] and
confirmed in Fig. S2 of the SI. As the pseudogap closes, $m$ transitions
rapidly from 2 to 1. In the FL regime beyond $p_{sc}$, conventional Kohler
scaling (with $\beta^{-1}\sim c+dT^{2}$) is restored (open squares).

Figure 8b shows log-log plots of $\beta^{-1}(T)$ for two highly-doped non-SC
LSCO crystals. It is immediately evident that in contrast to the SC samples,
$\beta^{-1}(T)$ does not follow a simple power-law, but instead tends towards
a constant value as $T\to$ 0\. (Note that the curvature in $\beta^{-1}(T)$ (on
a log-log scale) sets in at a temperature $T\sim$ 80 K that is significantly
higher than the lowest temperatures at which pure power-law scaling is found
to persist in SC samples.) As indicated by the dashed line in Fig. 8b, the
data can be fitted well to the expression $\beta^{-1}(T)=c+dT^{2}$, consistent
with the form of $\rho(0,T)$ at low $T$ (see Figs. S6 and S7 of the SI) and
thus with more conventional Kohler scaling.

The observed recovery of a finite residual component (= $c$ in Fig. 8b) in the
MR of non-SC samples reveals a third regime of distinct MR behavior in hole-
doped cuprates. The overall picture is summarized in Figure 9 where the
exponent $m$ (of $\beta^{-1}(T)$) is plotted versus $p$ for all three families
studied here together with the earlier result on Hg1201 [8]. The most striking
feature of Fig. 9 is the step-wise change in the exponent from $m\approx$ 2
throughout the pseudogap regime, to $m\approx$ 1 within the strange metal
regime over a crossover region $\Delta p$ that is much narrower than any of
the regimes themselves. The sharpness of the crossover is reminiscent of the
transition from coherent to incoherent anti-nodal ARPES spectra across
$p^{\ast}$ first observed in bilayer Bi2212 [81] and later confirmed in Bi2201
[80]. This summary plot is the central finding of this work whose significance
we shall discuss in more detail in Section III.

Figure 10: Comparison of the $T$\- and $p$-dependences of the normal state MR
and Hall angle in Bi2201. Top panels: Plots of $\sqrt{\rho(0)/\Delta\rho}$ vs.
$T^{n}$ for (a) $p$ = 0.13, (b) 0.195 and (c) 0.205. Bottom panels: Plots of
cot$\theta_{\rm H}$ vs. $T^{n}$ for (d) $p$ = 0.13, (e) 0.195 and (f) 0.24.
Note the evolution of $n$ with $p$. The Hall and MR data were obtained on
crystals from the same batches.

###  II.4 Link to modified Kohler scaling

Finally, we turn to address the last of the four outstanding questions;
namely, the link between the power-law MR scaling revealed here and modified
Kohler scaling. Harris et al. were the first to show that the $T$-dependence
of $\Delta\rho/\rho(0)$ in optimally doped and underdoped cuprates at low
fields scales as tan${}^{2}\theta_{\rm H}$ [6], behavior that has often been
interpreted as evidence for a separation of transport lifetimes [15]. A
similar separation of lifetimes has been reported in Bi2201 although here,
cot$\theta_{\rm H}=A+BT^{n}$ with $n$ varying from $\approx$ 2 below
$p^{\ast}$ to $\approx 1.6$ at higher dopings [82, 83].

Figure 10 shows plots of $\sqrt{\rho(0)/\Delta\rho}$ and cot$\theta_{\rm H}$
vs. $T^{n}$ for three different Bi2201 samples with comparable $p$ values: $p$
= 0.13 (panels (a) and (d)), 0.195 (panels (b) and (e)) and 0.205-0.24 (panels
(c) and (f)). In each case, $n$ is chosen to generate a straight line. When
plotted in this way, we find that $\sqrt{\rho(0)/\Delta\rho}$ can be equally
expressed as $A+BT^{n}$ with $n$ undergoing a similar evolution with doping as
cot$\theta_{\rm H}(T)$. Hence, modified Kohler scaling is also observed in
Bi2201. Crucially however, the residual term in the MR plots (panels (a)-(c))
has only appeared by dividing $\Delta\rho$ – that of itself exhibits pure
$1/T^{n}$ scaling – by $\rho(0,T)$ that contains a large $\rho_{0}$. In a
similar vein, the cot$\theta_{\rm H}(T)$ plots are ratios of $\rho(0,T)$ and
another quantity (the Hall coefficient $R_{\rm H}(T)$ in this case). Note that
the prefactors $A$ and $B$ in both sets of plots are not identical, reflecting
the fact that $R_{\rm H}(T)$ in Bi2201 does not exhibit a pure $1/T$
dependence, at least at low $T$ [82, 83, 3]. The key point, however, is that
while modified Kohler scaling is approximately realized in Bi2201, plotting
the data in this manner obscures the intrinsic power-law $T$-dependence of
$\Delta\rho(H,T)$ and its associated $H/T^{m}$ scaling. It will be interesting
to explore, in due course, whether the modified Kohler scaling reported for
YBCO [6] is also an artefact of their fitting protocol.

##  III Discussion

This comprehensive study has uncovered a number of new and surprising
relations in the normal state magnetotransport of hole-doped cuprates. The MR
in Bi2201, LSCO and Tl2201 exhibits $H/T$ scaling (Eq. 3)) throughout the
strange metal regime ($p^{\ast}<p<p_{sc}$) with a linear slope $\gamma$ that
correlates with the growth of $a_{1}$ – the low-$T$ $T$-linear component of
$\rho(0,T)$. Moreover, extrapolation of $\gamma(p)$ or $\gamma(T_{c})$ in all
three families suggests a close correlation between the magnitude of the
$H$-linear MR at high $H/T$ and superconductivity. $\gamma$ continues to grow
below $p^{\ast}$ while the relative MR $\Delta\rho(H)/\rho(0)$ is found to
have the same magnitude at all $p<p^{\ast}$, suggesting that the continued
growth in $\gamma$ with decreasing $p$ reflects the loss of carriers inside
the pseudogap regime. At the same time, the power-law scaling of
$\Delta\rho(H,T)$ transforms in the vicinity of $p^{\ast}$ from $H/T$ to
$H/T^{2}$. Only beyond the SC dome is conventional Kohler scaling restored.
Finally, despite the fact that $\sqrt{\rho(0)/\Delta\rho}$ obeys modified
Kohler scaling ($\propto$ cot$\theta_{\rm H}(T)$), it is the power-law scaling
of $\Delta\rho(H,T)$ that is found to be the most natural and robust way to
interpret the MR. In this section, we explore possible origins of the observed
phenomenology.

###  III.1 Origin of $H$-linear MR in $p$-doped cuprates

LMR and $H/T$ scaling have now been reported in several strange and/or quantum
critical metals [60, 74, 61, 84, 12]. Given their widely differing scattering
rates, Fermi surface (FS) geometries and magnetic field orientation
dependencies, it seems unlikely that mechanisms involving quantum MR [85],
sharp Fermi surface corners [86] or Zeeman splitting [87, 60, 5] are
responsible.

Other candidate proposals fall essentially into two categories. The first,
based on random resistor networks [88], attributes LMR to the presence of
disorder, through real space binary distributions [64], real space patches
[52] or doping inhomogeneity [89]. Given the large variations in disorder
levels and electronic inhomogeneity present in Bi2201, LSCO and Tl2201,
however, it is difficult for such models to account for the similarity in form
and magnitude of the MR seen in different families or the observed relation to
intrinsic properties such as $T_{c}$.

The second category centers on models involving momentum space
differentiation, specifically models where orbital motion is inhibited by some
form of anisotropic impedance; the stronger the impedance, the higher the
field range over which LMR is observed [90]. While certain systems, e.g. LSCO
and Nd-LSCO, appear to be consistent with this categorization due to a
specific feature of their band structure (to be outlined below), such
specificity once again conflicts with the universality of the MR observed
across various cuprate families.

In highly-doped non-SC LSCO ($x\approx$ 0.30), a sizeable (10-fold) anisotropy
in the low-$T$ quasiparticle mean-free-path $\ell_{0}(\phi)$ was inferred from
earlier ARPES [97] and low-field Hall effect [95] measurements. Such marked
anisotropy was attributed to the proximity of $\epsilon_{F}$ to the vHs at the
zone boundary and the predominance of small-angle (elastic) scattering induced
by out-of-plane substitutional disorder [98]. As shown in Fig. 6 of the SI,
the MR of non-SC LSCO also exhibits LMR at the highest fields. This, coupled
with adherence to Kohler scaling ($\beta^{-1}(T)\propto\rho(0,T)\propto
c+dT^{2}$ – see Fig. 8b) suggests that the LMR observed in overdoped non-SC
LSCO can be understood within a Boltzmann treatment incorporating strong
low-$T$ anisotropy in $\ell_{0}(\phi)$.

In a recent angle-dependent MR study, Grissonnanche et al. deduced an even
stronger, 100-fold in-plane anisotropy in $\ell_{0}(\phi)$ in SC samples of
Nd-LSCO (where $\epsilon_{F}$ is almost coincident with the vHs) [56].
Incorporating the deduced form of $\ell_{0}(\phi)$ into their Boltzmann
analysis, the authors were able to generate an in-plane MR response exhibiting
LMR at high fields [56], as found experimentally [99]. It is not yet clear,
however, whether the MR reported in Ref. [99] obeys power-law scaling, as
shown here for SC LSCO (Fig. 8a), or the Kohler-like scaling seen in non-SC
LSCO. The precise form of the observed scaling is actually an important,
though largely overlooked point, as we will argue below. It is also important
to stress that strong residual anisotropy in $\ell_{0}(\phi)$ has not been
reported in Bi2201 [100] nor in Tl2201 [101, 102] despite both systems
exhibiting an almost identical MR response to that seen in LSCO. Moreover, a
recent Boltzmann-based model invoking $T$-dependent anisotropy in $\ell(\phi)$
was shown to be incapable of reproducing simultaneously the $H,T$-dependencies
of the in-plane MR and Hall resistivity observed in Tl2201. Specifically, the
level of anisotropy required to account for the magnitude of the MR led to a
$T$\- and $H$-dependent Hall resistivity that was wholly inconsistent with the
experimental data. Finally, in both Bi2201 and Tl2201, the magnitude of the MR
is seen to grow as each system is doped further away from the vHs (with
reducing $p$) and thus any residual anisotropy from out-of-plane
substitutional disorder must diminish. Taken together, these inconsistencies
appear to rule out anisotropy in the elastic scattering channel as the
dominant source of LMR, at least for $p^{\ast}<p<p_{sc}$.

In fact, one can take this argument further and make a more general statement.
Adopting a modified Boltzmann theoretical treatment, Hinlopen et al. recently
showed that a high-field LMR of a similar magnitude to that observed can be a
robust feature of any system in which cyclotron motion is impeded at discrete
points or regions on the FS [90]. Impeded cyclotron motion can be simulated by
introducing an upper limit (bound) on the time integral of the expression for
the conductivity $\sigma_{xx}$ that makes the velocity-velocity correlation
terminate at a specific boundary on the FS, i.e.

$\sigma_{xx}=\frac{e^{2}}{4\pi^{3}\hbar}\int_{FS}d^{2}k\int_{0}^{bound}\frac{v_{i}(0)}{v_{F}(0)}v_{j}(-t)e^{-t/\tau}dt$
(8)

where the velocity $v=\nabla_{k}\epsilon/\hbar$ and $\epsilon$ is the band
energy. The model effectively covers multiple possible sources of impeded
cyclotron motion such as partial gaps, nesting fluctuations or peaks in the
density of states arising through hot spots [91], turning points [92],
magnetic breakdown [93], vHs [56] or states on the Fermi surface whose charge
dynamics have been rendered incoherent [3]. Although the model is able to
generate LMR, the absence of $\rho_{0}$ from the MR scaling could not be
captured. This failure is particularly troublesome in Bi2201 where $\rho_{0}$
can be as high as 150 $\mu\Omega$cm, equivalent to an impurity scattering rate
1/$\tau_{0}$ of order several hundred Kelvin. Evidently, the observation of
power-law scaling poses the most serious challenge to any semiclassical
treatment of the MR in strange metals. The generality of the above model
nevertheless compels us to consider impeded orbital motion as a possible
origin of LMR and to seek potential sources of impedance across the phase
diagram.

As the pseudogap opens, states are removed from the Fermi level initially near
the anti-nodal regions of the FS. This is believed to lead to the creation of
Fermi arcs, that may, below a certain doping level $p_{CO}<p^{\ast}$ and in
the presence of a large magnetic field, undergo FS reconstruction into a set
of diamond-shaped Fermi pockets located near the zone diagonals [109]. Both
the arcs (discontinuities in the FS) and the pockets (sharp turning points)
can create the necessary conditions for impeded cyclotron motion and thus LMR
at the highest field strengths. Moreover, the reduction in density of states
with further decrease in $p$ will lead to a shrinking of the arc lengths
and/or the pocket dimensions, both of which could drive the observed increase
in $\gamma$ with lowering $p$ inside the pseudogap regime.

The origin of the impedance within the strange metal regime, where the
underlying FS is believed to be intact, is arguably harder to identify. Given
that $H/T$ scaling is tied inextricably to $T$-linear resistivity, the answer
may lie in the emergence of non-quasiparticle states that have been rendered
highly dissipative or incoherent via scattering off critical fluctuations [18]
or through some other mechanism [53, 103]. According to Ref. [90], $\gamma$ is
determined by the size of the ‘impeded’ region(s), in which case, the increase
in both $\gamma$ and $a_{1}$ with underdoping could signify an increase in the
width (in $k$-space) of these incoherent sectors. It is important to recall
that the growth in $\gamma$ and $a_{1}$ occurs before the pseudogap opens and
tellingly, coincides with the $p$ to 1+$p$ crossover in the effective Hall
number across the strange metal regime [3].

Fluctuating charge order has been reported across the cuprate phase diagram
[104, 105], including in the three families studied here [106, 107, 50].
Likewise, spin fluctuations are known to extend to the edge of the SC dome at
$p=p_{sc}$ in LSCO [45]. Were fluctuations of either type to be of
sufficiently low-energy and sufficiently damped, scattering off such
fluctuations could account for the persistence of $T$-linear resistivity down
to low-$T$ [108]. To account simultaneously for LMR, however, they would have
to give rise to residual (i.e. $T$ = 0) anisotropic scattering – not foreseen
in models based on short-range charge fluctuations [108] – with an anisotropy
that grows markedly with decreasing $p$.

Finally, we turn to consider the field orientation dependence of the MR from
the perspective of impeded orbital motion. In overdoped Bi2201 and Tl2201 [12]
as well as in underdoped Y123 [6], the MR is relatively insensitive to field
orientation [12], prompting suggestions that Zeeman effects were in fact
playing a central role. Certainly, the quadrature expression given in Eq. (3)
implies the summation of thermal and magnetic energy scales. In P-Ba122 [110],
FeSe1-xSx [61], optimally doped Y123 [6] and LSCO (see Fig. 7 of the SI and
[99]), however, the quadrature MR is largely governed by the out-of-plane
component of the field. In looking at possible explanations for this
dichotomy, it is interesting to note that the resistive anisotropy in
underdoped Y123 and in Bi2201 and Tl2201 is orders of magnitude larger than in
P-Ba122, FeSe1-xSx, optimally doped Y123 and overdoped LSCO. Hence, while the
origin of this distinction is not known, one might speculate that in the
presence of an in-plane field, orbital motion in the more 2D cuprates is also
impeded along $k_{z}$, giving rise to quadrature MR of a similar magnitude,
while in more three-dimensional systems, such a tunneling barrier is absent.

In summary, while certain aspects of the MR response within the strange metal
regime, particularly those associated with the quadrature form of the MR,
appear to be consistent with a modified Boltzmann approach based on impeded
cyclotron motion, there are other aspects that cannot be explained within this
picture. Thus, as it stands, a global theory for $H/T$ scaling remains
elusive.

Figure 11: (a): Set of $\rho(H,T)$ curves for Bi2201 with $p$ = 0.195 (i.e.
just below $p\ast$) at representative temperatures. Dashed lines are fits of
the normal state data to Eq. (6) that allow for the extrapolation of the high-
field data to zero-field. (b): Measured zero-field resistivity (solid black
line) of the same sample. High-field resistivity $\rho(T$, 65 T) determined
from the end-points of the curves presented in (a) are denoted by blue
diamonds whilst the extrapolated zero-field resistivity is denoted by red
circles. The extrapolated $\rho(0,T)$ curve (red circles) obeys a $T^{2}$
dependence (dashed line fits). (c): Equivalently, a crude derivative
d$\rho$/d$T$ can be obtained from the extrapolated $\rho(0,T)$ (red circles in
panel (b)) yielding an extrapolated zero-field behavior below $T_{c}$ (red
circles). The obtained points (red circles) match the measured d$\rho$/d$T$
(solid black line) above $T_{c}$ and confirm pure $T^{2}$ behavior below
$T_{c}$ as indicated by a straight-line fit (dashed line). (d-f): The same
data and analysis as in panels (a-c) is presented for optimally doped Bi2201
($p$ = 0.16). Again, the low-$T$ dependence of $\rho(H=0,T)$ is found to be
quadratic.

###  III.2 Inside the pseudogap regime

The most profound change in the MR response below $p^{\ast}$ is the crossover
from $H/T$ to $H/T^{2}$ scaling. Intriguingly, a similar crossover from $H/T$
to $H/T^{2}$ scaling was also observed in P-Ba122. At its antiferromagnetic
QCP, the MR in P-Ba122 exhibits $H/T$ quadrature scaling [60], while inside
the magnetic phase, the MR scaling changes to $H/T^{2}$ coinciding with a
crossover to $T^{2}$ resistivity [96]. The LMR in P-Ba122 was attributed to
different origins within the two regimes: to scattering off hotspots on the FS
at the QCP [91] and to FS reconstruction and the creation of sharp turning
points inside the antiferromagnetic phase [92].

While it is tempting to consider a similar scenario in the hole-doped
cuprates, the continuation of $H/T^{2}$ scaling up to room temperature appears
to rule out FS reconstruction (e.g. due to charge order) as its origin, since
the latter is believed to occur only at low $T$. The step-wise change in the
MR scaling is therefore more likely to be a consequence of the loss of states
at the anti-nodes, its sharpness reminiscent of the abrupt change in the
electronic specific heat [94] or the loss of coherent quasiparticle peaks in
the (single-particle) spectral functions across $p^{\ast}$ [81]. Given the
step-wise change in scaling, it is possible that the intermediate power-law
exponents recorded within the crossover region reflect a (real-space)
coexistence of gapped and non-gapped regions exhibiting $H/T^{2}$ and $H/T$
scaling, respectively, though more detailed studies around $p^{\ast}$ would be
needed to confirm whether this is indeed the case.

Given that the scaling is observed over an extended range of temperatures and
fields, we can use its form and the exponent $m$ to extrapolate the MR to
$H\to 0$ for $T<T_{c}$ and thus extract $\rho(0,T)$ in the absence of
superconductivity. In panels (a) and (d) of Figure 11, the MR data of two
Bi2201 crystals with $p$ = 0.195 and 0.16, respectively, are fitted to Eq. (6)
with $m$ fixed to the value found through the scaling plots described above
and $\alpha_{m}$ and $\gamma$ determined by fits to the scaled derivatives.
For the entire data set that can be ascribed to the normal-state, the fits are
excellent. In panels (b) and (e) of Fig. 11, the corresponding high-field (65
T) and extrapolated ($H\to 0$) values of the resistivity are plotted. At low
$T$, $\rho(0,T)$ clearly deviates from its higher temperature $T$-linear
behavior. Finally, in panels (c) and (f), the derivatives d$\rho(0,T)$/d$T$
are plotted. Around $T$ = 60 K, the zero-field d$\rho/$d$T$ diverges due to
the onset of SC fluctuations. In red, the derivatives of the extrapolated
resistivity reveal that d$\rho/$d$T\to 0$ in a linear fashion, indicative of a
crossover to a $T^{2}$ dependence of $\rho(0,T)$.

The analysis presented in Fig. 11 reveals that in optimally doped cuprates or
in samples with $p\lesssim p^{\ast}$, $\rho(0,T)$ is not $T$-linear, as is
widely believed. While low-$T$ $T^{2}$ resistivity has been reported at lower
dopings [79, 8, 78], we are not aware of any such report so close to
$p^{\ast}$ or at optimal doping. A note of caution is required here, however.
Firstly, the obtained quadratic behavior in Bi2201 contrasts with the
predominantly $T$-linear resistivity observed in LSCO just below $p^{\ast}$
(also extrapolated from high-field MR measurements) [5]. Moreover, small
resistive upturns are often seen in Bi2201 crystals at low $T$ [111, 112, 80]
that may mimic a crossover to $T^{2}$ resistivity. This seems unlikely however
given the fact that in both LSCO and Bi2201 the MR exhibits pure $H/T^{2}$
scaling below $p^{\ast}$ indicating a significant $T^{2}$ component in
$\rho(0,T)$ that is itself subject to impeded cyclotron motion in an applied
magnetic field. A conclusion to draw from this analysis is that the only
region of the phase diagram where a $T$-linear component to the resistivity
survives to the lowest temperatures is the strange metal regime
$p^{\ast}<p<p_{sc}$. This is also the only regime where $H/T$ scaling of the
MR is manifest.

The precise location of $p^{\ast}$ in Bi2201 has been the subject of much
debate [23, 113, 114, 112, 80]. NMR studies have suggested that $p^{\ast}$
lies close to the edge of the SC dome [113], though the distinction between
$T^{*}$ and $T_{\mathrm{F}}$ has not yet been fully resolved. A more recent
study involving transport and ARPES measurements found that the pseudogap
opens around $p=0.2$ [80]. The MR results presented above shed new light on
this issue by uncovering an abrupt change in the scaling of the MR from $H/T$
to $H/T^{2}$ at $p\sim 0.2$. Given that (a) $H/T^{2}$ scaling is seen in
different underdoped cuprates (such as LSCO, Hg1201 and Bi2201) and (b) a
similar crossover in the MR occurs in LSCO across $p^{\ast}$ ($\sim$ 0.2), the
crossover observed in Bi2201 at $p\sim$ 0.2 appears to confirm that
manifestations of the pseudogap in the charge transport end here.

Another key finding of this study is that for samples just below $p^{\ast}$,
no distinction in the scaling of the MR can be made for temperatures below and
above $T^{*}$. While the zero-field resistivity departs from a pure linear
form below $T^{*}$ (for $p$ = 0.195, for example, $T^{*}\sim$ 150-200 K), a
single power-law can describe the MR over the whole temperature range from 300
K to 4 K. These findings tend to support the notion that the pseudogap is
controlled by an energy scale rather than a phase transition [81, 72].

Finally, despite the observation of quantum oscillations [115], the
verification of the Wiedemann-Franz law [116, 117], the recovery of a $T^{2}$
resistivity [78, 8] and a $\omega^{2}$ scattering rate [118] inside the
pseudogap regime, the appearance of Fermi arcs challenges claims of a
conventional FL ground state below $p^{\ast}$. The extended range of $T^{2}$
resistivity is also anomalous. An in-plane resistivity of the form $c+dT^{2}$
is limited, for example, to temperatures $\leq$ 160 K [8], while the pure
$H/T^{2}$ scaling of the MR presented here is found to persist up to room
temperature. The manifestation of pure power-law scaling is also at odds with
both conventional and modified Boltzmann transport theory, suggesting that the
presence of a robust $T^{2}$ resistivity may require an explanation that is
beyond the realm of standard FL theory [119].

##  IV Conclusion

In the present study, we have carried out a universal analysis of the
temperature, field and doping dependence of the in-plane MR of hole-doped
cuprates to reveal unique scaling behavior within each distinct regime of the
phase diagram: $H/T^{2}$ scaling in the pseudogap regime, $H/T$ scaling in the
strange metal regime and Kohler scaling in the non-SC FL regime. In addition,
we have provided definitive answers to the four outstanding questions
highlighted in the Introduction:

A: $H$-linear MR is observed at the highest fields in all SC samples studied,
with a magnitude that falls monotonically with increasing $p$, extrapolating
to zero around $p_{sc}$ in precisely the same manner as $a_{1}$, the
$T$-linear coefficient of $\rho(0,T)$. The origin of the variation of $\gamma$
above and below $p^{\ast}$, however, is argued to be distinct. Nonetheless,
this correlation shows that the magnetotransport connects normal-state room
temperature physics with $T_{c}$, even when the two differ by two orders of
magnitude. Moreover, the ubiquity of the $H/T$ scaling throughout the entire
strange metal regime adds further weight to its interpretation as a quantum
critical phase rather than a response driven by proximity to a singular
critical point.

B: Across $p^{\ast}$, the form of scaling changes abruptly from $H/T$ to
$H/T^{2}$ scaling in the pseudogap regime signifying a loss of the $T$-linear
component in both $\rho(0,T)$ and $\Delta\rho(H,T)$ below $p^{\ast}$.
Nevetheless, the persistence of power-law scaling inside the pseudogap regime
suggests that the remnant states near the Fermi level still possess non-FL
character. As in the strange metal regime, $\rho_{0}$ and, by default,
$1/\tau_{0}$ are both absent from the scaling, even when $1/\tau_{0}$ is of
order 100-200 K. Data from four different cuprate families are found to
exhibit the same power-law scaling, demonstrating remarkable universality of
the phenomenon. Finally, the marked change in MR scaling mirrors that seen in
the single-particle response [94, 81]; the lack of signatures of $T^{\ast}$ in
the MR supporting interpretations of the pseudogap as a doping-dependent
energy scale that vanishes at $p^{\ast}$, rather than a phase transition to
some form of hidden order.

C: In overdoped non-SC LSCO, beyond $p_{sc}$, the low-field MR follows more
conventional Kohler scaling with $\beta^{-1}(T)=c+dT^{2}$. The recovery of a
finite residual component in $\beta^{-1}(T)$ across $p_{sc}$ appears to be
similarly abrupt. Despite the recovery of Kohler scaling (at low fields), the
MR still shows a tendency towards $H$-linearity at the highest fields. We have
argued here that the observation of LMR in LSCO reflects the presence of a
highly anisotropic elastic mean-free-path $\ell_{0}(\phi)$ [95]. In highly
doped, non-SC LSCO, the LMR is essentially ‘accidental’ and we expect
measurements to higher fields will show a tendency towards saturation as the
anisotropy in $\ell_{0}(\phi)$ is eventually washed out. At lower doping
levels, corresponding to where $\epsilon_{F}$ crosses the vHs, the anisotropy
in $\ell_{0}(\phi)$ may become so large [56] that the LMR will persist up to
the highest accessible field strengths. The observation of residual-free
scaling in SC samples, however, is striking and cannot be captured by any
modified Boltzmann theory and clearly requires further experimental and
theoretical study to be resolved.

D: While modified Kohler scaling (Eq. (2)) is observed in both the pseudogap
and strange metal regimes, the ubiquity of power-law scaling (Eq. (7)) and its
validity up to the highest fields clearly identifies the latter as the
intrinsic field-induced response of SC hole-doped cuprates. As a result,
modified Kohler scaling is relegated to an empirical finding generated by
dividing $\Delta\rho(H)$ by $\rho(0)$. How the Hall and MR responses can be
reconciled within the LMR regime, however, remains to be resolved.

While the generation of LMR at higher field strengths is compatible with a
scenario based on impeded cyclotron motion, the observed power-law scaling is
not. Nevertheless, one can speculate as to the origin of any such impedance
within the various doping regimes. As mentioned above, in highly-doped non-SC
LSCO, LMR may arise from a density of states effect amplified by the presence
of out-of-plane substitutional disorder. In the pseudogap regime, the
termination of the Fermi arc provides a natural ‘boundary’ to cyclotron
motion, one that is unlikely to be penetrated at higher field strengths. Thus,
we predict that the LMR in pseudogapped cuprates will not deviate up to the
highest fields available. The origin of LMR within the strange metal regime is
arguably the most difficult to discern. Nevertheless, the close correlation
between $\gamma$ and $a_{1}$ suggests that its origin is tied to the emergence
of non-FL transport and the accompanying loss of coherent quasiparticle
states. Within this picture, the growth in $\gamma$ across the strange metal
regime implies an expansion of this incoherent sector with decreasing $p$, as
inferred from our earlier Hall study [3].

When $p$ falls below $p^{\ast}$ (in Bi2201), $\gamma$ continues to increase in
a linear fashion, reflecting the reduction in Fermi arc length as the
pseudogap grows. In Bi2201, the extrapolated low-$T$ resistivity becomes
purely quadratic as $T\to 0$ just as the quadratic MR develops $H/T^{2}$
scaling, indicating a suppression of the $T$-linear component in $\rho(0,T)$
at low-$T$. Knowledge of the anisotropy of the pseudogap then leads us to
infer that the states removed from the Fermi level below $p^{\ast}$ were in
fact those exhibiting signatures of incoherent transport inside the strange
metal regime. Finally, the sharpness of the crossover in power-law scaling
from $H/T$ to $H/T^{2}$ is reminiscent of the abrupt changes seen in earlier
ARPES [81] and specific heat [94] studies.

In closing, we return to the central conundrum of our study – the fact that
the MR exhibits pure power-law scaling with no residual component (as would
have been expected from conventional or modified Kohler scaling). This robust
experimental result suggests that the magnetotransport properties of
superconducting cuprates require an altogether new theoretical framework, one
that is incompatible with conventional Boltzmann transport theory and entirely
decoupled from impurity scattering. The direct correlation between $H$-linear
MR, $T$-linear resistivity and $T_{c}$ suggests that developing such a
framework will surely pave the way for a more unified description, not only of
the anomalous normal state of hole-doped cuprates, but also of high-$T_{c}$
superconductivity itself.

##  V Methods

High quality crystals of Bi2201 and LSCO were grown in floating zone furnaces
located at different sites. To cover a large part of the phase diagram, some
of the Bi2201 crystals were doped with La and Pb, resulting in the chemical
formula: Bi2+z-yPbySr2-x-zLaxCuO6+δ. The Tl2201 crystal studied here was
synthesized using a self-flux technique [120]. The hole doping for Bi2201 was
estimated from the measured $T_{c}$ (defined as the temperature at which the
zero-field resistivity vanishes below the noise floor of our experiments)
using the Presland relation [1]: $1-T_{c}$/$T^{\rm max}_{c}$ = 82.6($p$ –
0.16)2 with $T^{\rm max}_{c}$ = 35 K. This relation was recently demonstrated
to hold well in Bi2201 [80]. For LSCO, the doping was estimated from the
measured $T_{c}$ using the same parabolic relation with $T^{\rm max}_{c}$ = 38
K and found to match closely the Sr content of each crystal. The
crystallographic axes of LSCO were oriented with a Laue camera. Typical sample
dimensions were 1000 $\times$ 250 $\times$ 10 $\mu$m3 for Bi2201 and Tl2201
and 1500 $\times$ 250 $\times$ 50 $\mu$m3 for LSCO. The $\rho(0,T)$ curves of
all the MR samples are shown in Figure 1 of the SI.

The MR was measured in DC and pulsed magnets up to 35 T and 70 T, respectively
with the current $I$ applied in-plane and H $\parallel c$. For the pulsed
field measurements, samples and wires were fully covered in GE varnish and/or
vacuum grease to reduce vibration. To increase the measurement signal, each
Bi2201 crystal was mechanically thinned to a thickness of 2 - 10 $\mu$m,
resulting in sample resistances of $\sim$ 1 $\Omega$, i.e. comparable to the
resistance of the current contacts. At each measurement temperature, the MR
curves were recorded for both polarities of the magnetic field. For Tl2201, a
single crystal with an ambient pressure $T_{c}$ = 35 K was selected and
prepared for transport measurements under the application of hydrostatic
pressure using a piston cylinder cell. The sample was oriented on a feed-
through such that the magnetic field H $\parallel c$. Daphne 7373 oil was used
as a pressure transmitting medium as it is known to remain hydrostatic at room
temperature (the temperature at which pressure was applied) up to 2.2 GPa
[122], beyond the pressures applied in this work.

## References

  * [1] Gurvitch, M. & Fiory, A. T. Resistivity of La1.825Sr0.175CuO4 and YBa2Cu3O7 to 1100 K: absence of saturation and its implications. Phys. Rev. Lett. 59 1337-1340 (1987)
  * [2] Martin, S., Fiory, A. T., Fleming, R. M., Schneemeyer, L. F. & Waszczak, J. V. Normal-state transport properties of Bi2+xSr2-yCuO6+δ crystals. Phys. Rev. B 41 846-849 (1990)
  * [3] Chien, T. R., Wang, Z. Z. & Ong, N. P. Effect of Zn impurities on the normal-state Hall angle in single-crystal YBa2Cu3-xZnxO7-δ. Phys. Rev. Lett. 67 2088-2091 (1991)
  * [4] Ando, Y., Boebinger, G. S., Passner, A., Kimura, T. & Kishio, K. Logarithmic divergence of the in-plane and out-of-plane normal state resistivities of superconducting La2-xSrxCuO4 in the zero-temperature Limit. Phys. Rev. Lett. 75 4662-4665 (1995)
  * [5] Harris, J. M., Yan, Y. F., Matl, P., Ong, N. P., Anderson, P. W., Kimura, T. & Kitazawa, K. Violation of Kohler’s rule in the normal-state magnetoresistance of YBa2Cu3O7-δ and La2-xSrxCuO4. Phys. Rev. Lett. 75, 1391-1394 (1995)
  * [6] Fournier, P., Mohanty, P., Maiser, E., Darzens, S., Venkatesan, T., Lobb, C. J., Czjzek, G., Webb, R. A. & Greene, R. L. Insulator-metal crossover near optimal doping in Pr2-xCexCuO4: anomalous normal-state low temperature resistivity. Phys. Rev. Lett. 81 4720-4723 (1998)
  * [7] van der Marel, D., Molegraaf, H. J. A., Zaanen, J., Nussinov, Z., Carbone, F., Damascelli, A., Eisaki, H., Greven, M., Kes, P. H. & Li, M. Quantum critical behaviour in a high-$T_{c}$ superconductor. Nature 425 271-274 (2003)
  * [8] Cooper, R. A., Wang, Y., Vignolle, B., Lipscombe, O. J., Hayden, S. M., Tanabe, Y., Adachi, T., Koike, Y., Nohara, M., Takagi, H., Proust, C. & Hussey, N. E. Anomalous criticality in the electrical resistivity of La2–xSrxCuO4. Science 323 603-607 (2009)
  * [9] Jin, K., Butch, N. P., Kirshenbaum, K., Paglione, J. P. & Greene, R. L. Link between spin fluctuations and electron pairing in copper oxide superconductors. Nature 476 73-75 (2011)
  * [10] Badoux, S., Tabis, W., Laliberté, F., Grissonnanche, G., Vignolle, B., Vignolles, D., Béard, J., Bonn, D. A., Hardy, W. N., Liang, N., Taillefer, L. & Proust, C. Change of carrier density at the pseudogap critical point of a cuprate superconductor. Nature 531 210-214 (2016)
  * [11] Giraldo-Gallo, P., Galvis, J. A., Stegen, Z., Modic, K. A., Balakirev, F. F., Betts, J. B., Lian, X., Moir, C., Riggs, S. C., Wu, J., Bollinger, A. T., He, X., Božović, I., Ramshaw, B. J., McDonald, R. D., Boebinger, G. S. & Shekhter, A. Scale-invariant magnetoresistance in a cuprate superconductor. Science 361 479-481 (2018)
  * [12] Ayres, J., Berben, M., Čulo, M., Hsu, Y., Heumen, E., Huang, Y., Zaanen, J., Kondo, T., Takeuchi, T., Cooper, J., Putzke, C., Friedemann, S., Carrington, A. & Hussey, N. E. Incoherent transport across the strange metal regime of highly overdoped cuprates. Nature 595 661-667 (2021)
  * [13] Yuan, J., Chen, Q., Jiang, K., Feng, Z., Lin, Z., Yu, H., He, J., Jiang, X., X., Z., Shi, Y., Zhang, Y., Cheng, Z., Tamura, N., Y., Y., Xiang, T., Hu, J., Takeuchi, I., Jin, K. & Zhao, Z. Scaling of the strange-metal scattering in unconventional superconductors. Nature 602 431 (2022)
  * [14] Varma, C. M., Littlewood, P. B., Schmitt-Rink, S., Abrahams, E. & Ruckenstein, A. E. Phenomenology of the normal state of Cu-O high-temperature superconductors. Phys. Rev. Lett. 63 1996-1999 (1989)
  * [15] Anderson, P. W. Hall effect in the two-dimensional Luttinger liquid. Phys. Rev. Lett. 67 2092-2094 (1991)
  * [16] Emery, V. J. & Kivelson, S. A. Superconductivity in bad metals. Phys. Rev. Lett. 74 3253-3256 (1995)
  * [17] Zaanen, J. Why the temperature is high. Nature 430 512-513 (2004)
  * [18] Wu, W., Wang, X. & Tremblay, A. M. Non-Fermi liquid phase and linear-in-temperature scattering rate in overdoped two dimensional Hubbard model. Proc. Natl. Acad. Sci. (USA) 119 e2115819119 (2022)
  * [19] Manako, T., Kubo, Y. & Shimakawa, Y. Transport and structural study of Tl2Ba2CuO6+δ single crystals prepared by the KCl flux method. Phys. Rev. B 46 11019-11024 (1992)
  * [20] Nakamae, S., Behnia, K., Mangkorntong, N., Nohara, M., Takagi, H., Yates, S. J. C. & Hussey, N. E. Electronic ground state of heavily overdoped nonsuperconducting La2-xSrxCuO4. Phys. Rev. B 68 100502(R) (2003)
  * [21] Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518 179-186 (2015)
  * [22] Varma, C. M. Colloquium: Linear in temperature resistivity and associated mysteries including high temperature superconductivity. Rev. Mod. Phys. 92 031001 (2020)
  * [23] Ando, Y., Komiya, S., Segawa, K., Ono, S. & Kurita, Y. Electronic phase diagram of high-$T_{c}$ cuprates from a mapping of the in-plane resistivity curvature. Phys. Rev. Lett. 93 267001 (2004)
  * [24] Gurvitch, M. & Fiory, A. T. Resistivity of La1.825Sr0.175CuO4 and YBa2Cu3O7 to 1100 K: absence of saturation and its implications. Phys. Rev. Lett. 59 1337-1340 (1987)
  * [25] Martin, S., Fiory, A. T., Fleming, R. M., Schneemeyer, L. F., & Waszczak, J. V. Normal-state transport properties of Bi2201. Phys. Rev. B 41 846–849 (1990)
  * [26] Bucher, B., Steiner, P., Karpinski, J., Kaldis, E. & Wachter, P. Influence of the spin gap on the normal state transport in YBa2Cu4O8. Phys. Rev. Lett. 70 2012-2015 (1993)
  * [27] Hussey, N. E., Cooper, R. A., Xu, X., Wang, Y., Mouzopoulou, I., Vignolle, B. & Proust, C. Dichotomy in the $T$-linear resistivity in hole-doped cuprates. Phil. Trans. R. Soc. A 369 1626-1639 (2011)
  * [28] Kaminski, A., Rosenkranz, S., Fretwell, H. M., Li, Z. Z., Raffy, H., Randeria, M., Norman, M. R. & Campuzano, J. C. Crossover from coherent to incoherent electronic excitations in the normal state of Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 90 207003 (2003)
  * [29] Zhao, L., Belvin, C. A., Liang, R., Bonn, D. A., Hardy, W. N., Armitage, N. P. & Hsieh, D. A global inversion-symmetry-broken phase inside the pseudogap region of YBa2Cu3Oy Nat. Phys. 567 218-222 (2019)
  * [30] Michon, B. Girod, C., Badoux, S., Kačmarčík, J., Ma, Q., Dragomir, M., Dabkowska, M. A., Gaulin, B. D., Zhou, J.-S., Pyon, S., Takayama, T., Takagi, H., Verret, S., Doiron-Leyraud, N., Marcenat, C., Taillefer, L. & Klein, T. Thermodynamic signatures of quantum criticality in cuprate superconductors. Nature 567 218-222 (2019)
  * [31] Bruin, J. A. N., Sakai, H., Perry, R. S. & Mackenzie, A. P. Similarity of scattering rates in metals showing $T$-linear resistivity. Science 339 804-807 (2013)
  * [32] Legros, A., Benhabib, S., Tabis, W., Laliberté, F., Dion, M., Lizaire, M., Vignolle, B., Vignolles, D., Raffy, H., Li, Z. Z., Auban-Senzier, P., Doiron-Leyraud, N., Fournier, P., Colson, D., Taillefer, L. & Proust, C. Universal $T$-linear resistivity and Planckian dissipation in overdoped cuprates. Nat. Phys. 15 142-147 (2019)
  * [33] Harada, K., Teramoto, Y., Usui, T., Itaka, K., Fujii, T., Noji, T., Taniguchi, H., Matsukawa, M., Ishikawa, H., Kindo, K., Dessau, D. S. & Watanabe, T. Revised phase diagram of a high-$T_{c}$ cuprate superconductor revealed by anisotropic transport measurements. Phys. Rev. B 105 085131 (2022)
  * [34] Sadovskii, M. Linear-in-T resistivity in dilute metals: A Fermi liquid perspective. Physics Uspekhi 64 175-190 (2021)
  * [35] Huang, E. & Das Sarma, S. Linear-in-$T$ resistivity in dilute metals: A Fermi liquid perspective. Phys. Rev. B 99 085105 (2019)
  * [36] Hartnoll, S. A. & Mackenzie, A. P. Planckian dissipation in metals. arXiv:2107.07802 (2021)
  * [37] Mackenzie, A. P., Julian, S. R., Sinclair, D. C. & Lin, C. Normal-state magnetotransport in superconducting Tl2Ba2CuO6+δ to millikelvin temperatures. Phys. Rev. B 53 5848-5855 (1996)
  * [38] Proust, C., Boaknin, E., Hill, R. W., Taillefer, L. & Mackenzie, A. P. Heat transport in a strongly overdoped cuprate: Fermi liquid and a pure $d$-wave BCS superconductor. Phys. Rev. Lett. 89 147003 (2002)
  * [39] Phillips, P. W., Hussey, N. E. & Abbamonte, P. Stranger than metals, Science 377 eabh4273 (2022)
  * [40] Hussey, N. E., Buhot, J. & Licciardello, S. A tale of two metals: contrasting criticalities in the pnictides and hole-doped cuprates. Rep. Prog. Phys. 81 052501 (2018)
  * [41] Pelc, D., Popčević, P., Požek1, M., Greven, M. & Barišić, N. Unusual behavior of cuprates explained by heterogeneous charge localization. Sci. Adv. 5 eaau4538 (2019)
  * [42] Čulo, M., Duffy, C., Ayres, J., Berben, M., Hsu, Y., Hinlopen, R. D. H., Bernáth, B. & Hussey, N. E. Possible superconductivity from incoherent carriers in overdoped cuprates. SciPost Phys. 11 012 (2021)
  * [43] Inkof, G.-A., Schalm, K. & Schmalian, J. Quantum critical Eliashberg theory, the Sachdev-Ye-Kitaev superconductor and their holographic duals. npj Quant. Mat. 7 56 (2022)
  * [44] Seibold, G., Arpaia, R., Peng, Y. Y., Fumagalli, R., Braicovich, C., Di Castro, C., Grilli, M., Ghiringhelli, G. C. & Caprara, S. Strange metal behaviour from charge density fluctuations in cuprates. Commun. Phys. 4 7 (2021)
  * [45] Wakimoto, S., Yamada, K., Tranquada, J. M., Frost, C. D., Birgeneau, R. J. & Zhang, H. Disappearance of antiferromagnetic spin excitations in overdoped La2-xSrxCuO4. Phys. Rev. Lett. 98 247003 (2007)
  * [46] Li, Y., Zhong, R., Stone, M. B., Kolesnikov, A. I., Gu, G. D., Zaliznyak, I. A. & Tranquada, J. M. Low-energy antiferromagnetic spin fluctuations limit the coherent superconducting gap in cuprates. Phys. Rev. B 98 224508 (2018)
  * [47] Arpaia, R., Caprara, S., Fumagalli, R., De Vecchi, G., Peng, Y. Y., Andersson, E., Betto, D., De Luca, G., Brookes, N. B., Lombardi, F., Salluzzo, M., Braicovich, C., Di Castro, C., Grilli, M. & Ghiringhelli, G. Dynamical charge density fluctuations pervading the phase diagram of a Cu-based high-$T_{c}$ superconductor. Science 365 906-910 (2019)
  * [48] Miao, H., Fabbris, G., Koch, R. J., Mazzone, D. G., Nelson, C. S., Acevedo-Esteves, R., Gu, G. D., Li, Y., Yilimaz, T., Kaznatcheev, K., Vescovo, E., Oda, M., Kurosawa, T., Momono, N., Assefa, M., Robinson, I. K., Bozin, E. S., Tranquada, J. M., Johnson, P. D. & Dean, M. P. M. Charge density waves in cuprate superconductors beyond the critical doping. npj Quant. Mat. 6 31 (2021)
  * [49] Li, X., Zou, C., Ding, Y., Yan, H., Ye, S., Li, H., Hao, Z., Zhao, L., Zhou, X. & Wang, Y. Evolution of charge and pair density modulations in overdoped Bi2Sr2CuO6+δ. Phys. Rev. X 11 011007 (2021)
  * [50] Tam, C. C., Zhu, M., Ayres, J., Kummer, K., Yakhou-Harris, F., Cooper, J. R., Carrington, A. & Hayden, S. M. Charge density waves and Fermi-surface reconstruction the clean overdoped cuprate superconductor Tl2Ba2CuO6+δ. Nat. Commun. 13 570 (2022)
  * [51] Blake, M. & Donos, A. Quantum critical transport and the Hall angle in holographic models. Phys. Rev. Lett. 114 021601 (2015)
  * [52] Patel, A. A., McGreevy, J., Arovas, D. P. & Sachdev, S. Magnetotransport in a model of a disordered strange metal. Phys. Rev. X 8 021049 (2018)
  * [53] Zaanen, J. Planckian dissipation, minimal viscosity and the transport in cuprate strange metals. SciPost Phys. 6 061 (2019)
  * [54] Husain, A. A., Mitrano, M., Rak, M., Rubeck, S., Uchoa, B., March, K., Dwyer, C., Schneeloch, J., Zhong, G., Gu, G. D. & Abbamonte, P. Crossover of charge fluctuations across the strange metal phase diagram. Phys. Rev. X 9 041062 (2019)
  * [55] Putzke, C., Benhabib, S., Tabis, W., Ayres, J., Wang, Z., Malone, L., Licciardello, S., Lu, J. M., Kondo, T., Takeuchi, T., Hussey, N. E., Cooper, J. R. & Carrington, A. Reduced Hall carrier density in the overdoped strange metal regime of cuprate superconductors. Nat. Phys. 17 826-831 (2021)
  * [56] Grissonnanche, G., Fang, Y., Legros, A., Verret, S., Laliberté, F., Collignon, C., Zhou, J., Graf, D., Goddard, P. A., Taillefer, L. & Ramshaw, B. J. Linear-in temperature resistivity from an isotropic Planckian scattering rate. Nature 595 667-672 (2021)
  * [57] Nakajima, Y., Shishido, H., Nakai, H., Shibauchi, T., Behnia, K., Izawa, K., Hedo, M., Uwatoko, Y., Matsumoto, T., Settai, R., Ōnuki, Y., Kontani, H. & Matsuda, Y. Non-Fermi liquid behavior in the magnetotransport of CeMIn5 (M: Co and Rh): striking similarity between quasi-two-dimensional heavy fermion and high-$T_{c}$ cuprates. J. Phys. Soc. Jpn. 76, 024703 (2007)
  * [58] Kasahara, S., Shibauchi, T., Hashimoto, K., Ikada, K., Tonegawa, S., Okazaki, R., Shishido, H., Ikeda, H., Takeya, H., Hirata, K., Terashima, T. & Matsuda, Y. Evolution from non-Fermi- to Fermi-liquid transport via isovalent doping in BaFe2(As1-xPx)2 superconductors. Phys. Rev. B 81 184519 (2010)
  * [59] Huang, W., Hosoi, S., Čulo, M., Kasahara, S., Sato, Y., Matsuura, K., Mizukami, Y., Berben, M., Hussey, N. E., Kontani, H., Shibauchi, T. & Matsuda, Y. Non-Fermi liquid transport in the vicinity of the nematic quantum critical point of superconducting FeSe1-xSx. Phys. Rev. Res. 2 033367 (2020)
  * [60] Hayes, I. M., McDonald, R. D., Breznay, N. P., Helm, T., Moll, P. J. W., Wartenbe, M., Shekhter, A. & Analytis, J. G. Scaling between magnetic field and temperature in the high-temperature superconductor BaFe2(As1-xPx)2. Nat. Phys. 12 916-919 (2016)
  * [61] Licciardello, S., Maksimovic, N., Ayres, J., Buhot, J., Čulo, M., Bryant, B., Kasahara, S., Matsuda, Y., Shibauchi, T., Nagarajan, V., Analytis, J. G. & Hussey, N. E. Coexistence of orbital and quantum critical magnetoresistance in FeSe1-xSx. Phys. Rev. Res. 1 023011 (2019)
  * [62] Hussey, N. E., Gordon-Moys, H., Kokalj, J. & McKenzie, R. H. Generic strange-metal behaviour of overdoped cuprates. J. Phys. Conf. Series. 449 012004 (2013)
  * [63] Chan, M. K., Veit, M. J., Dorow, C. J., Ge, Y., Li, Y., Tabis, W., Tang, Y., Zhao, X., Barišić, N. & Greven, M. In-plane magnetoresistance obeys Kohler’s rule in the pseudogap phase of cuprate superconductors. Phys. Rev. Lett. 113 177005 (2014)
  * [64] Boyd, C. & Phillips, P. W. Single-parameter scaling in the magnetoresistance of optimally doped La2-xSrxCuO4. Phys. Rev. B 100 155139 (2019)
  * [65] Carrington, A., Mackenzie, A. P., Lin, C. T.& Cooper, J. R. Temperature dependence of the Hall angle in single-crystal YBa2(Cu1-xCox)3O7-δ. Phys. Rev. Lett. 69 2855-2858 (1992)
  * [66] Stojkovic, B. P. & Pines, D. Theory of the longitudinal and Hall conductivities of the cuprate superconductors. Phys. Rev. B 55 8576-8595 (1997)
  * [67] Ioffe, L. B. & Millis, A. J. Zone-diagonal-dominated transport in high-$T_{c}$ cuprates. Phys. Rev. B 58 11631-11637 (1998,11)
  * [68] Hussey, N. E. The normal state scattering rate in high-$T_{c}$ cuprates. Eur. Phys. J. B. 31 495-507 (2003)
  * [69] Barišić, N., Chan, M. K., Li, Y., Yu, G., Zhao, X., Dressel, M., Smontara, A. & Greven, M. Universal sheet resistance and revised phase diagram of the cuprate high-temperature superconductors. Proc. Nat. Acad. Sci. (USA). 110 12235-12240 (2013)
  * [70] Rice, T. M., Robinson, N. & Tsvelik, A. Umklapp scattering as the origin of $T$-linear resistivity in the normal state of high-$T_{c}$ cuprate superconductors. Phys. Rev. B 96 220502(R) (2017)
  * [71] Nachtigal, J., Avramovska, M., Erb, A., Pavicevic, D., Guehne, R. & Haase, J. Temperature independent cuprate pseudogap from planar oxygen NMR. Condens. Matter 5 66 (2020)
  * [72] Tallon, J. L., Storey, J. G., Cooper, J. R. & Loram, J. W. Locating the pseudogap closing point in cuprate superconductors: Absence of entrant or reentrant behavior. Phys. Rev. B 101 174512 (2020)
  * [73] Kimura, T., Miyasaka, S., Takagi, H., Tamasaku, K., Eisaki, H., Uchida, S., Kitazawa, K., Hiroi, M., Sera, M. & Kobayashi, N. In-plane and out-of-plane magnetoresistance in La2-xSrxCuO4 single crystals. Phys. Rev. B 53 8733-8742 (1996)
  * [74] Sarkar, T., Mandal, P. R., Poniatowski, N. R., Chan, M. K. & Greene, R. L. Correlation between scale-invariant normal-state resistivity and superconductivity in an electron-doped cuprate. Sci. Adv. 5 eaav6753 (2019)
  * [75] Girod, C. Chaleur spécifique à basse temperature dans l’état normal des cuprates superconducteurs. (Université Grenoble Alpes, 2021)
  * [76] Ding, Y., Zhao, L., Yan, H., Gao, Q., Liu, J., Hu, C., Huang, J., Li, C., Xu, Y., Cai, Y., Rong, H., Wu, D., Song, C., Zhou, H., Dong, X., Liu, G., Wang, Q., Zhang, S., Wang, Z., Zhang, F., Yang, F., Peng, Q., Xu, Z., Chen, C. & Zhou, X. J. Disappearance of superconductivity and a concomitant Lifshitz transition in heavily overdoped Bi2Sr2CuO6 superconductor revealed by angle-resolved photoemission spectroscopy. Chin. Phys. Lett. 36 017402 (2018)
  * [77] Looney, C. W., Kline, J. E. Mascarenhas, F., Schilling, J. S., & Hermann, A. M. Influence of oxygen content on the activation energy for oxygen ordering in Tl2Ba2CuO6+δ. Physica C, 289, 203-210 (1997)
  * [78] Proust, C., Vignolle, B., Levallois, J., Adachi, S. & Hussey, N. E. Fermi-liquid behavior of the in-plane resistivity in the pseudogap state of YBa2Cu4O8. Proc. Nat. Acad. Sci. (USA). 113 13654-13659 (2016)
  * [79] Rullier-Albenque, F., Alloul, H., Balakirev, F. & Proust, C. Disorder, Metal-insulator crossover and phase diagram in high-$T_{c}$ cuprates. EPL (Europhys. Lett.). 81 37008 (2008)
  * [80] Berben, M., Smit, S., Duffy, C., Hsu, Y., Bawden, L., Heringa, F., Gerritsen, F., Cassanelli, S., Feng, X., Bron, S., Heumen, E., Huang, Y., Bertran, F., Kim, T., Cacho, C., Carrington, A., Golden, M. & Hussey, N. E. On the superconducting dome and pseudogap endpoint in Bi2201. Phys. Rev. Mater. 6 044804 (2022)
  * [81] Chen, S., Hashimoto, M., He, Y., Song, D., Xu, K., He, J., Devereaux, T. P., Eisaki, H., Lu, D., Zaanen, J. & Shen, Z. X. Incoherent strange metal sharply bounded by a critical doping in Bi2212. Science 366 1099-1102 (2019)
  * [82] Ando, Y. & Murayama, T. Nonuniversal power law of the Hall scattering rate in a single-layer cuprate Bi2Sr2-xLaxCuO6+δ. Phys. Rev. B 60 R6991-R6994 (1999)
  * [83] Konstantinović, Z., Li, Z. Z. & Raffy, H. Temperature dependence of the Hall effect in single-layer and bilayer Bi2Sr2Can-1CunOy thin films at various oxygen contents. Phys. Rev. B 62 R11989-R11992 (2000)
  * [84] Nakajima, Y., Metz, T., Eckberg, C., Kirshenbaum, K., Hughes, A., Wang, R., Wang, L., Saha, S. R., Liu, I-L., Butch, N. P., Campbell, D., Eo, Y. S., Graf, D., Liu, Z., Borisenko, S. V., Zavalij, P. V. & Paglione, J., Quantum-critical scale invariance in a transition metal alloy. Commun. Phys. 3 181 (2020)
  * [85] Abrikosov, A. Quantum magnetoresistance. Phys. Rev. B 58 2788-2794 (1998)
  * [86] Pippard, A. Magnetoresistance in metals. (Cambridge University Press,1989)
  * [87] Ando, Y. & Segawa, K. Magnetoresistance of untwinned YBa2Cu3Oy single crystals in a wide range of doping: anomalous hole-doping dependence of the coherence length. Phys. Rev. Lett. 88 167005 (2002)
  * [88] Parish, M. J. & Littlewood, P. B. Non-saturating magnetoresistance in heavily disordered semiconductors. Nature 426 162-165 (2003)
  * [89] Singleton, J. Temperature scaling behavior of the linear magnetoresistance observed in high-temperature superconductors. Phys. Rev. Mat. 4 061801(R) (2020)
  * [90] Hinlopen, R. D. H., Ayres, J., Hinlopen, F. A. & Hussey, N. E. $B^{2}$ to $B$-linear magnetoresistance due to impeded orbital motion. Phys. Rev. Res. 4 033195 (2022)
  * [91] Koshelev, A. E. Magnetotransport of multiple-band nearly antiferromagnetic metals due to hot-spot scattering. Phys. Rev. B 94 125154 (2016)
  * [92] Koshelev, A. E. Linear magnetoconductivity in multiband spin-density-wave metals with nonideal nesting. Phys. Rev. B 88 060412(R) (2013)
  * [93] Naito, M. & Tanaka, S. Galvanomagnetic effects in the charge-density-wave state of 2H-NbSe2 and 2H-TaSe2. J. Phys. Soc. Japan. 51 228-232 (1982)
  * [94] Loram, J. W., Luo, J., Cooper, J. R., Liang, W. Y. & Tallon, J. Evidence on the pseudogap and condensate from the electronic specific heat. J. Phys. Chem. Solids. 62, 59-64 (2001)
  * [95] Narduzzo, A., Albert, G., French, M. M. J., Mangkorntong, N., Nohara, M., Takagi, H. & Hussey, N. E. Violation of the isotropic mean free path approximation for overdoped La2-xSrxCuO4. Phys. Rev. B 77 220502(R) (2008)
  * [96] Maksimovic, N., Hayes, I. M., Nagarajan, V., Analytis, J. G., Koshelev, A. E., Singleton, J., Lee, Y. & Schenkel, T. Magnetoresistance scaling and the origin of $H$-linear resistivity in BaFe2(As1-xPx)2. Phys. Rev. X 10 041062 (2020)
  * [97] Yoshida, T., Zhou, X. J., Tanaka, K., Yang, W., Hussain, Z., Shen, Z. X., Fujimori, A., Sahrakorpi, S., Lindroos, M., Markiewicz, R. S., Bansil, A., Komiya, S., Ando, Y., Eisaki, H., Kakeshita, T. & Uchida, S. Systematic doping evolution of the underlying Fermi surface of La2-xSrxCuO4. Phys. Rev. B 74 224510 (2006)
  * [98] Abrahams, E. & Varma, C. M. What angle-resolved photoemission experiments tell about the microscopic theory for high-temperature superconductors. Proc. Natl. Acad. Sci. (USA). 97 5714-5716 (2000)
  * [99] Ataei, A., Gourgout, A., Grissonnanche, G., Chen, L., Baglo, J., Boulanger, M.-E., Laliberté, F., Badoux, S., Doiron-Leyraud, N., Oliviero, V., Benhabib, S., Vignolles, D., Zhou, J.-S., Ono, S., Takagi, H., Proust, C. & Taillefer, L. Electrons with Planckian scattering obey standard orbital motion in a magnetic field. Nat. Phys AOP (2022)
  * [100] Kondo, T., Takeuchi, T., Tsuda, S. & Shin, S. Electrical resistivity and scattering processes in (Bi,Pb)2(Sr,La)2CuO6+δ studied by angle-resolved photoemission spectroscopy. Phys. Rev. B 74 224511 (2006)
  * [101] Abdel-Jawad, M., Kennett, M. P., Balicas, L., Carrington, A., Mackenzie, A. P., McKenzie, R. H. & Hussey, N. E. Anisotropic scattering and anomalous normal-state transport in a high-temperature superconductor. Nat. Phys. 2 821-825 (2006)
  * [102] Peets, D. C., Mottershead, J. D. F., Wu, B., Elfimov, I. S., Liang, R., Hardy, W. N., Bonn, D. A., Raudsepp, M., Ingle, N. J. C. & Damascelli, A. Tl2Ba2CuO6+δ brings spectroscopic probes deep into the overdoped regime of the high-$T_{c}$ cuprates. New J. Phys. 9 28-28 (2007)
  * [103] Banerjee, A., Grandadam, M., Freire, H. & Pépin, C. Strange metal from incoherent bosons. Phys. Rev. B 104 054513 (2021)
  * [104] Arpaia, R. & Ghiringhelli, G. Charge order at high temperature in cuprate superconductors. J. Phys. Soc. Jpn. 90 111005 (2021)
  * [105] Uchida, S. Ubiquitous Charge order correlations in high-temperature superconducting cuprates. J. Phys. Soc. Jpn. 90 111001 (2021)
  * [106] Lin, J., Miao, H., Mazzone, D., Gu, G. D., Nag, A., Walters, A., Garcıá-Fernández, M., Barbour, A., Pelliciari, J., Jarrige, I., Oda, M., Kurosawa, K., Momono, N., Zhou, K., Bisogni, V., Liu, X. & Dean, M. P. M. Strongly correlated charge density wave in La2-xSrxCuO4 evidenced by doping-dependent phonon anomaly. Phys. Rev. Lett. 124 207005 (2020)
  * [107] Peng, Y., Fumagalli, R., Ding, Y., Minola, M., Caprara, S., Betto, D., Bluschke, M., De Luca, G., Kummer, K., Lefrançois, E., Salluzzo, M., Suzuki, H., Le Tacon, M., Zhou, X., Brookes, N., Keimer, B., Braicovich, L., Grilli, M. & Ghiringhelli, G. Re-entrant charge order in overdoped (Bi,Pb)2.12Sr1.88CuO6+δ outside the pseudogap regime. Nat. Mater. 17 697 (2018)
  * [108] Caprara, S., Di Castro, C., Mirarchi, G., Seibold, G. & Grilli, M. Dissipation-driven strange metal behavior. Commun. Phys. 5 10 (2022)
  * [109] Harrison, N. & Sebastian, S. Protected nodal electron pocket from multiple-$\mathbf{Q}$ ordering in underdoped high temperature superconductors. Phys. Rev. Lett. 106 226402 (2011)
  * [110] Hayes, I. M., Hao, Z., Maksimovic, N., Lewin, S. P., Chan, M. K., McDonald, R. D., Ramshaw, B. J., Moore, J. E. & Analytis, J. G. Magnetoresistance scaling reveals symmetries of the strongly correlated dynamics in BaFe2(As1-xPx)2. Phys. Rev. Lett. 121 197002 (2018)
  * [111] Ono, S., Ando, Y., Murayama, T., Balakirev, F. F., Betts, J. & Boebinger, G. S. Metal-to-insulator crossover in the low-temperature normal state of Bi2Sr2-xLaxCuO6+δ. Phys. Rev. Lett. 85 638-641 (2000)
  * [112] Lizaire, M., Legros, A., Gourgout, A., Benhabib, S., Badoux, S., Laliberté, F., Boulanger, M., Ataei, A., Grissonnanche, G., LeBoeuf, D., Licciardello, S., Wiedmann, S., Ono, S., Kawasaki, S., Zheng, G., Doiron-Leyraud, N., Proust, C. & Taillefer, L. Transport signatures of the pseudogap critical point in the cuprate superconductor Bi2Sr2-xLaxCuO6+δ. Phys. Rev. B 104 014515 (2021)
  * [113] Zheng, G., Kuhns, P. L., Reyes, A. P., Liang, B. & Lin, C. T. Critical point and the nature of the pseudogap of single-layered copper-oxide Bi2Sr2-xLaxCuO6+δ superconductors. Phys. Rev. Lett. 94 047006 (2005)
  * [114] Kondo, T., Hamaya, Y., Palczewski, A., Takeuchi, T., Wen, J., Xu, Z., Gu, G. D., Schmalian, J. & Kaminski, A. Disentangling Cooper-pair formation above the transition temperature from the pseudogap state in the cuprates. Nat. Phys. 7 21-25 (2011)
  * [115] Doiron-Leyraud, N., Proust, C., LeBoeuf, D., Levallois, J., Bonnemaison, J., Liang, R., Bonn, D., Hardy, W. & Taillefer, L. Quantum oscillations and the Fermi surface in an underdoped high-$T_{c}$ superconductor. Nature 447 565-568 (2007)
  * [116] Grissonnanche, G., Laliberté, F., Dufour-Beauséjour, S., Matusiak, M., Badoux, S., Tafti, F., Michon, B., Riopel, A., Cyr-Choinière, O., Baglo, J., Ramshaw, B. J., Liang, R., Bonn, D. A., Hardy, W. N., Krämer, S., LeBoeuf, D., Graf, D., Doiron-Leyraud, N. & Taillefer, L. Wiedemann-Franz law in the underdoped cuprate superconductor YBa2Cu3Oy. Phys. Rev. B 93 064513 (2016)
  * [117] Michon, B., Ataei, A., Bourgeois-Hope, P., Collignon, C., Li, S., Badoux, S., Gourgout, A., Laliberté, F., Zhou, J., Doiron-Leyraud, N. & Taillefer, L. Wiedemann-Franz Law and abrupt change in conductivity across the pseudogap critical point of a cuprate superconductor. Phys. Rev. X 8 041010 (2018)
  * [118] Mirzaei, S. I., Stricker, D., Hancock, J. N., Berthod, C., Georges, A., Heumen, E., Chan, M. K., Zhao, X., Li, Y., Greven, M., Barišić, N. & van der Marel, D. Spectroscopic evidence for Fermi liquid-like energy and temperature dependence of the relaxation rate in the pseudogap phase of the cuprates. Proc. Nat. Acad. Sci. (USA). 110 5774-5778 (2013)
  * [119] Reber, T., Zhou, X., Plumb, N., Parham, S., Waugh, J., Cao, Y., Sun, Z., Li, H., Wang, Q., Wen, J., Xu, Z., Gu, G., Yoshida, Y., Eisaki, H., Arnold, G. & Dessau, D. A unified form of low-energy nodal electronic interactions in hole-doped cuprate superconductors. Nat. Commun. 10 5737 (2019)
  * [120] Tyler, A. W. J. An investigation into the magnetotransport properties of layered superconducting perovskites. (University of Cambridge, 1997)
  * [121] Presland, M. R., Tallon, J. L., Buckley, R. G., Liu, R. S. & Flower, N. E. General trends in oxygen stoichiometry effects on Tc in Bi and Tl superconductors . Physica. 176C 95-105 (1991)
  * [122] Yokogawa, K., Murata, K., Yoshino, H. & Aoyama, S. Pressure transmitting medium Daphne 7474 solidifying at 3.7 GPa at room temperature. Jpn. J. Appl. Phys. 46, 3636-3639 (2007)

Supplementary Information

##  I Description and characterization of all samples under study

Bi2201 |  |  |  |  |  |
---|---|---|---|---|---|---
Bi |  Pb |  Sr |  La |  Annealing |  Doping |  $T_{c}$
1.72 |  0.38 |  1.85 |  0 |  400C, 96h in 2.5 atm. O2 |  0.270 |  $<$1 K
1.72 |  0.38 |  1.85 |  0 |  750C, 24h in Air |  0.260 |  7 K
1.72 |  0.38 |  1.85 |  0 |  550C, 72h in N2 |  0.245 |  13 K
1.72 |  0.38 |  1.85 |  0 |  550C, 72h in N2 |  0.240 |  17 K
1.72 |  0.38 |  1.45 |  0.4 |  As grown |  0.22 |  24 K
1.35 |  0.85 |  1.47 |  0.38 |  600C, 24h in Air |  0.205 |  28 K
1.2 |  0.9 |  1.3 |  0.55 |  As grown |  0.195 |  30 K
1.35 |  0.85 |  1.47 |  0.38 |  650C, 72h in N2 |  0.16 |  35 K
1.2 |  0.9 |  1.3 |  0.55 |  650C, 72h in N2 |  0.13 |  31 K
Table 1: Details of the Bi2201 samples measured as part of this study. The $T_{c}$ values are defined as the temperature below which the zero-field resistivity falls below the noise floor. The doping levels are then determined from the $T_{c}$ values using the Presland formula [1] with $T_{c}^{\rm max}$ = 35 K. The Bi2201 sample labelled with a $T_{c}<$ 1 K was measured down to 1.4 K, and although it shows an onset of superconductivity, it does not become fully superconducting. We have therefore specified its doping level as $p=0.27$ in line with the parabolic Presland formula. LSCO |  |
---|---|---
Sr content |  $T_{c}$ |  Inferred Doping
0.20 |  30 K |  0.205
0.23 |  22.4 K |  0.230
0.30 |  no SC |  /
0.33 |  no SC |  /
Table 2: Details of the LSCO samples studied. The $T_{c}$ values are defined as the temperature below which the zero-field resistivity falls below the noise floor. The listed Sr content values are either the nominal ones or, for $x$ = 0.33, have been measured with electron-probe micro-analysis [2]. The samples with $x$ = 0.30 and 0.33 do not show any sign of superconductivity down to $T$ = 0.45 K. Tl2201 |
---|---
$T_{c}$ |  Inferred Doping
35 K |  0.26
Table 3: Tl2201 sample studied as part of this work. The $T_{c}$ value is
defined as the temperature below which the zero-field resistivity falls below
the noise floor at ambient pressure. The same is true also for all data sets
recorded under applied hydrostatic pressure. The doping level is inferred from
$T_{c}$ as described in Ref. [3]. Figure 1: Zero-field resistivity curves for
the (a) Bi2201, (b) LSCO and (c) Tl2201 samples whose magnetoresistance was
investigated as part of this study. The error in the absolute values of
$\rho(0,T)$ (due to geometrical uncertainties) is estimated to be $\sim$ 15%.

##  II Magnetoresistance of overdoped Tl2201 under pressure

Figure 2: (a)-(e) Evolution with pressure of $\rho(H,T)$ of the Tl2201 single
crystal (ambient pressure $T_{c}$ = 35K). Dashed lines are simultaneous fits
to the MR to Eq. (3) of the main report. (f)-(i) Derivatives
d$\rho$/d($\mu_{0}H$) of the MR presented in panels (a)-(e). The derivatives
presented in panels (f)-(i) are plotted against $H/T$. The dashed lines are
derived from a single fit to the full set of MR derivatives (in the normal
state) to Eq. (3) of the main article. $\gamma$ – the limiting low-$T$,
high-$H$ slope of the $H$-linear MR – can be read off from the plateau region
of the individual dashed line fits.

##  III Quadrature scaling of the MR in overdoped LSCO ($x$ = 0.23)

Figure 3: (a) Reproduction of the set of MR curves for LSCO $x$ = 0.23
reported in Ref. [4]. Dashed lines are fits to the quadrature MR expression
given in Eq. (3) of the main report. (b) Corresponding plot of
d$\rho$/d$\mu_{0}H$ vs. $H/T$ indicating that the form of the MR does indeed
follow $H/T$ scaling. Dashed line is the fit to the derivative of Eq. (4) of
the main article. (c) Corresponding plot of $\Delta\rho/T$ vs. $H/T$. Again
dashed line is the fit to Eq. (3) of the main article. (d) Resultant
$\rho(0,T)$ obtained by extrapolating the quadrature fits to the MR curves in
panel (a) to $H$ = 0. Here, the dashed line is simply a guide to the eye.

##  IV Simulation of quadrature scaling

Figure 4: (a) Simulation of a set of MR curves obeying strict quadrature
($H/T$) scaling. Solid lines are the simulated dataset. The dashed lines
indicate the fitting range used to fit each curve with a quadratic function.
(b) $T$-dependence of the estimated coefficient $\beta$ of the low-field
$H^{2}$ MR. Dashed black line indicates the analytical $H\to 0$ value of
$\beta$. The circles indicate $\beta$ as found from the fitting procedure. The
inset shows the offset of the analytical $\beta$ value in percent as described
in the text.

Since the hyperbolic form of the MR (Eq. (4) in the main article) is never
purely quadratic except in the $H\rightarrow$ 0 limit, fitting the data with a
quadratic function is merely an approximation. In order to check whether our
fitting procedure can capture a similar quadratic component to the analytical
zero-field limit, we simulate in Figure 4 a dataset that obeys pure quadrature
scaling, where the following parameters have been used: $\rho_{0}$ = 10
$\mu\Omega$cm, $\alpha$ = 0.1 $\mu\Omega$cm/K and $\gamma$ = 0.1
$\mu\Omega$cm/T. From Eq. (5) of the main article, an analytical quadratic
component $B$ = 0.05 $\mu\Omega$cm/K2·$T$is found, as indicated by the black
dashed line in Fig. 4(b).

In our approach, the simulated curves are first fitted with a quadratic
function in a field range of 0 - 20 T and 0 - 10 T. While at high temperatures
this gives a similar quadratic component to the analytical value, both start
to deviate at low $T$. As expected, with a decrease of the fitting range, the
deviation for the analytical value decreases at low-$T$. Given the $H/T$
scaling form of the quadrature MR and using the fact that $\gamma/\alpha$
represents the turnover field from quadratic to linear, we now use a fitting
range that is determined in terms of $H/T$. With a range from 0 - ($\alpha
T/\gamma$) Tesla (for the simulated data $\gamma/\alpha$ = 1 K/T) a constant
offset of $\sim$ 18% is found. While this is non-negligible, the key point is
that the temperature dependence of the extracted $\beta$ coefficient remains
$\sim 1/T$, and in the main article, it is the $T$-dependence that we are most
interested in. Upon decreasing the $H/T$ range the offset quickly drops,
reaching $\sim$ 1% with a range of $\alpha T/5\gamma$, as shown in the inset
of Fig. 4(b).

##  V Power-law scaling of the MR in other cuprates

Figure 5: (a) 1/$B$ vs. $T$ derived from the LSCO sample with $p$ = 0.19
reported by Giraldo-Gallo et al. [5]. (b) Corresponding plot of
$\Delta\rho/T^{2}$ vs. $\mu_{0}H/T^{2}$ for the $p$ = 0.19 LSCO sample from
Ref. [5]. (c) $1/B(T)$ for optimally doped LSCO from Harris et al. [6]. Here
the zero-field resistivity curve needed to extract the $B$-term was taken from
Kimura et al.. [7]. (d) $1/B(T)$ for underdoped Hg1201 ($p$ = 0.14) from Chan
et al. [8]. The dotted lines in panels (a), (c) and (d) are all guides to the
eye.

##  VI Resistivity and MR in highly overdoped non-SC LSCO

Figure 6: (a) Derivatives d$\rho$/d$T$ of the zero-field resistivity curves
for the two highly overdoped non-superconducting LSCO samples shown in Fig. 1
of the SI. The dashed lines are guides to the eye, highlighting the fact that
$\alpha_{1}(0)$ \- the low-$T$ $T$-linear resistivity coefficients (the
intercepts on these derivative plots) is negligible and thus $\rho(T)$ shows
purely quadratic behavior at low-$T$ consistent with correlated Fermi-liquid
behavior. (b) Evolution of $\rho(H,T)$ for $p$ = 0.33 up to 35 T between 100 K
and 4.2 K. At low-$T$, a crossover to $H$-linear MR can be seen at the highest
fields.

##  VII Comparison of Kohler and $H/T^{m}$ scaling in LSCO

Figure 7: (a, b, c) A direct comparison of (a) Kohler, (b) $H/T$ and (c)
$H/T^{2}$ scaling for LSCO with $x=0.23$ within the strange metallic regime.
In this doping regime, $H/T$ scaling is observed whilst both Kohler and
$H/T^{2}$ scaling can be seen to fail. (d, e, f) The same scaling plots are
also presented for non-superconducting LSCO with $x=0.30$, a doping regime in
which Kohler scaling is most closely adhered to.

##  VIII Anisotropic MR in overdoped LSCO ($x$ = 0.23)

Figure 8: (a) Evolution of $\rho(H,T)$ for $p$ = 0.23 between 100 K and 4.2 K
in magnetic fields up to 35 T oriented H$\|$$c$. (b) Evolution of $\rho(H,T)$
over the same range of temperatures and fields H$\|$$ab$. (c) Corresponding
derivatives d$\rho$/d$\mu_{0}H$ for H$\|$$c$. (d) Corresponding
d$\rho$/d$\mu_{0}H$ curves for H$\|$$ab$. The strong anisotropy in the two
field orientations is in contrast to what was found for overdoped Bi2201 and
Tl2201 [9].

## References

  * [1] Presland, M. R. Tallon, J. L. Buckley, R. G. Liu, R. S. & Flower, N. E. General trends in oxygen stoichiometry effects on $T_{c}$ in Bi and Tl superconductors. _Physica C_ 176, 95 (1991)
  * [2] Nakamae, S. Behnia, K. Mangkorntong, N. Nohara, M. Takagi, H. Yates, S. J. C. & Hussey, N. E. Erratum: Electronic ground state of heavily overdoped nonsuperconducting La2-xSrxCuO4 [Phys. Rev. B 68, 100502 (2003)]. _Phys. Rev. B_ 79, 219904(E) (2009)
  * [3] Putzke, C., Benhabib, S., Tabis, W., Ayres, J., Wang, Z., Malone, L., Licciardello, S., Lu, J. M., Kondo, T., Takeuchi, T., Hussey, N. E., Cooper, J. R. & Carrington, A. Reduced Hall carrier density in the overdoped strange metal regime of cuprate superconductors. Nat. Phys. 17 826-831 (2021)
  * [4] Cooper, R. A., Wang, Y., Vignolle, B., Lipscombe, O. J., Hayden, S. M., Tanabe, Y., Adachi, T., Koike, Y., Nohara, M., Takagi, H., Proust, C. & Hussey, N. E. Anomalous criticality in the electrical resistivity of La2–xSrxCuO4. Science. 323, 603-607 (2009)
  * [5] Giraldo-Gallo, P., Galvis, J. A., Stegen, Z., Modic, K. A., Balakirev, F. F., Betts, J. B., Lian, X., Moir, C., Riggs, S. C., Wu, J., Bollinger, A. T., He, X., Božović, I., Ramshaw, B. J., McDonald, R. D., Boebinger, G. S. & Shekhter, A. Scale-invariant magnetoresistance in a cuprate superconductor. Science. 361, 479-481 (2018)
  * [6] Harris, J. M., Yan, Y. F., Matl, P., Ong, N. P., Anderson, P. W., Kimura, T. & Kitazawa, K. Violation of Kohler’s rule in the normal-state magnetoresistance of YBa2Cu3O7-δ and La2-xSrxCuO4. Phys. Rev. Lett. 75, 1391-1394 (1995)
  * [7] Kimura, T., Miyasaka, S., Takagi, H., Tamasaku, K., Eisaki, H., Uchida, S., Kitazawa, K., Hiroi, M., Sera, M. & Kobayashi, N. In-plane and out-of-plane magnetoresistance in La2-xSrxCuO4 single crystals. Phys. Rev. B. 53 8733-8742 (1996)
  * [8] Chan, M., Veit, M., Dorow, C., Ge, Y., Li, Y., Tabis, W., Tang, Y., Zhao, X., Barišić, N. & Greven, M. In-plane magnetoresistance obeys Kohler’s rule in the pseudogap phase of cuprate superconductors. Phys. Rev. Lett. 113 177005 (2014)
  * [9] Ayres, J., Berben, M., Čulo, M., Hsu, Y., Heumen, E., Huang, Y., Zaanen, J., Kondo, T., Takeuchi, T., Cooper, J. R., Putzke, C., Friedemann, S., Carrington, A. & Hussey, N. E. Incoherent transport across the strange metal regime of highly overdoped cuprates. Nature. 595 661-667 (2021)