#  Stranger than Metals

Philip W. Phillips  Department of Physics and Institute for Condensed Matter
Theory, University of Illinois, 1110 W. Green Street, Urbana, IL 61801  Nigel
E. Hussey  H. H. Wills Physics Laboratory, University of Bristol, Tyndall
Avenue, Bristol BS8 1TL, United Kingdom  High Field Magnet Laboratory (HFML-
EMFL) and Institute for Molecules and Materials, Radboud University,
Toernooiveld 7, 6525 ED Nijmegen, Netherlands  Peter Abbamonte  Department of
Physics, University of Illinois, 1110 W. Green Street, Urbana, IL 61801

###### Abstract

Although the resistivity in traditional metals increases with temperature, its
$T$ dependence vanishes at low or high temperature, albeit for different
reasons. Here, we review a class of materials, known as ‘strange’ metals, that
can violate both principles. In materials exhibiting such behavior, the change
in slope of the resistivity as the mean free path drops below the lattice
constant, or as $T\rightarrow 0$, can be imperceptible, suggesting complete
continuity between the charge carriers at low and high $T$. Since particles
cannot scatter at length scales shorter than the interatomic spacing, strange
metallicity calls into question the relevance of locality and a particle
picture of the underlying current. This review focuses on transport and
spectroscopic data on candidate strange metals with an eye to isolate and
identify a unifying physical principle. Special attention is paid to quantum
criticality, Planckian dissipation, Mottness, and whether a new gauge
principle, which has a clear experimental signature, is needed to account for
the non-local transport seen in strange metals. For the cuprates, strange
metallicity is shown to track the superfluid density, thereby making a theory
of this state the primary hurdle in solving the riddle of high-temperature
superconductivity.

To understand the essential tension between quantum mechanics and gravity,
simply imagine two electrons impinging on the event horizon of a black hole.
While classical gravity predicts that they meet at the center, quantum
mechanics forbids this should the electrons have the same spin. In essence,
classical gravity has no way of preserving Pauli exclusion. Of course
replacing classical general relativity with a quantum theory of gravity at
small enough scales resolves the problem, but what is this scale? In 1899,
Planck formulated a universal length now regarded as the scale below which a
quantum theory of gravity supplants its classical counterpart. The Planck
scale,

$\displaystyle\ell_{P}=\sqrt{\frac{\hbar G}{c^{3}}},$ (1)

is pure dimensional analysis on three fundamental constants: the speed of
light, $c$, Newton’s gravitational constant, $G$, and the quantum of
uncertainty, $\hbar$, Planck’s constant, $h$, divided by $2\pi$. This leads
naturally to a Planck time as the ratio of the Planck length to the speed of
light, $\ell_{P}/c$. Such a Planckian analysis can be extended equally to
many-body systems in contact with a heat bath. All that is necessary is to
include the temperature $T$. A similar dimensional analysis then leads to

$\displaystyle\tau_{P}=\frac{\hbar}{k_{B}T}$ (2)

as the shortest time for heat loss in a many-body system obeying quantum
mechanics with $k_{B}$, Boltzmann’s constant. As no system parameters enter
$\tau_{P}$, this quantity occupies a similar fundamental role in analogy with
the Planck length and is referred to as the Planckian dissipation time.
Although Eq. (2) has had previous incarnations matsubara ; chn89 , in the
realm of charge transport, it defines the time scale for scale-invariant or
Planckian dissipation zaanen04 . Scale-invariance follows because there is no
scale other than temperature appearing in $\tau_{P}$. Achieving such scale
invariance necessitates a highly entangled many-body state. Such a state would
lead to a breakdown of a local single-particle and the advent of new
collective non-local entities as the charge carriers. Precisely what the new
propagating degrees of freedom are is the key mystery of the strange metal.

While the Planck scale $\ell_{P}$ requires high-energy accelerators much
beyond anything now in use, such is not the case with physics at the Planckian
dissipation limit. Early table-top experiments on cuprate superconductors, for
example, revealed a ‘strange metal’ regime defined by a robust $T$-linear
resistivity extending to the highest temperatures measured gurvitch87 ;
martin90 ; takagi92  (see Fig. 1), a possible harbinger of Planckian
dissipation. Recall that in a Fermi liquid, the conductivity, can be well
described by a Drude formula,

$\displaystyle\sigma=\frac{n_{e}e^{2}}{m}\tau_{\rm tr}$ (3)

where $n_{e}$ is the charge carrier density, $e$ and $m$ the charge and mass
of an electron, respectively, and the transport lifetime

$\displaystyle\tau_{\rm tr}=\frac{\hbar
E_{F}}{(k_{B}T)^{2}}=\frac{E_{F}}{k_{B}T}\tau_{P},$ (4)

contains the Fermi energy $E_{F}$ of the quasiparticles. No such energy scale
appears in Eq. (2). If the scattering rate in cuprates is directly
proportional to the resistivity, as it is in simple metals, $T$-linear
resistivity is equivalent to scale-invariant Planckian dissipation only if
$\tau_{tr}=\alpha_{1}\tau_{P}$ with $\alpha_{1}\sim 1$. While this state of
affairs seems to be realized in a host of correlated metals, including the
cuprates marel03 ; cooper09 ; legros19 ; bruin13 , questions that deserve
further deliberation are how accurately is $\alpha_{1}$ known and what are the
assumptions that go into its determination? Regardless of the possible
relationship with Planckian dissipation, what makes $T$-linear resistivity in
the cuprates truly novel is its persistence – from mK temperatures (in both
the electron- and hole-doped cuprates) fournier98 ; mackenzie96b  up to 1000 K
(in the hole-doped cuprates) gurvitch87 ; takagi92  – and its omnipresence,
the strange metal regime dominating large swathes of the temperature vs.
doping phase diagram nagaosa92 . In normal metals iofferegel ; gurvitch81  as
well as some heavy fermions husseyMIR , the resistivity asymptotically
approaches a saturation value commensurate with the mean-free-path $\ell$
becoming comparable with the interatomic spacing $a$ – the minimum length over
which a Bloch wave and its associated Fermi velocity and wave vector can be
defined. In many correlated metals – collectively refered to as ‘bad metals’ –
$\ell<a$ at high $T$, thereby violating the so-called Mott-Ioffe-Regel (MIR)
limit iofferegel ; mott ; husseyMIR ; martin90 ; takagi92 ; hussey11 .
Remarkably, no saturation occurs in these bad metals across the MIR threshold,
implying that the whole notion of a Fermi velocity of quasiparticles breaks
down at high $T$. In certain cases, an example of which is shown in Fig. 1,
there is no discernible change in slope as the MIR limit is exceeded. While
this circumstance occurs only in a narrow doping window (in cuprates) hussey11
, such continuity does suggest that, even at low $T$, quasiparticles emkiv95
cannot be the effective propagating degrees of freedom. Evidently, in strongly
correlated electron matter, the current-carrying degrees of freedom in the IR
need not have a particle interpretation. Precisely what the charge carriers
are and the experimental delineation of the strange metal will be the subject
of this review.

Over time, the label ‘strange metal’ has seemingly become ubiquitous, used to
describe any metallic system whose transport properties display behavior that
is irreconcilable with conventional Fermi-liquid or Boltzmann transport
theory. This catch-all phraseology, however, is unhelpful as it fails to
differentiate between the various types of non-Fermi-liquid behavior observed,
some of which deserve special deliberation on their own. In this review, we
attempt to bring strange metal phenomenology into sharper focus, by addressing
a number of pertinent questions. Does the term refer to the resistive behavior
of correlated electron systems at high or low temperatures or both? Does it
describe any $T$-linear resistivity associated with the Planckian timescale,
or something unique? Does it describe the physics of a doped Mott insulator or
the physics associated with quantum criticality (whose underlying origins may
or may not include Mottness as a key ingredient)? Finally, does anything local
carry the current and if not, does explicating the propagating degrees of
freedom in the strange metal require a theory as novel as quantum gravity?

Figure 1:  In-plane resistivity of La2-xSrxCuO4 ($x$ = 0.21, adapted from Ref.
cooper09 ; hussey11 ). The dotted points are extrapolated from high-field
magnetoresistance data cooper09 . The shaded area shows the Mott-Ioffe-Regel
(MIR) boundary where the mean-free-path becomes comparable to the
interparticle scattering length.

##  I Is Strange Metallicity Ubiquitous?

Table 1: Summary of the dc transport properties of various strange metal
candidates. The first column identifies the candidate compound or family of
compounds. For the hole-doped cuprates, underdoped (UD), optimally doped (OP)
and overdoped (OD) compounds are treated separately, but individual compounds
within each sub-set are not listed as their transport properties are found to
be generic. For the electron-doped cuprates, only La2-xCexCuO4 is selected
since this is the material for which all relevant properties have been
studied, though the Pr- and Nd-based sister compounds do show similar
behavior. ‘MATBG’ stands for magic-angle twisted bilayer graphene. The second
column considers bad metallic behavior, though here a $\checkmark$ mark refers
only to those materials that exhibit $T$-linear resistivity beyond the Mott-
Ioffe-Regel (MIR) limit. Systems identified with a $\times$ show either a
tendency towards saturation or a marked reduction in slope near the MIR limit.
$\checkmark$ marks in the third column identify systems that at a singular
point in their respective phase diagram(s), exhibit $T$-linear resistivity
down to the lowest temperatures studied thus far. The $`$ extended
criticality’ heading for column 4 refers then to systems where a predominant
$T$-linear resistivity at low-$T$ extends over a finite region of the phase
diagram. Column 5 considers systems that exhibit a $T^{2}$ dependence of the
inverse Hall angle cot$\Theta_{\rm H}$ in the same temperature range where
$\rho(T)$ is $T$-linear. Compounds satifsying the ‘Modified Kohler’s’ label in
column 6 have a low-field magnetoresistance (MR), defined as
$\rho(H,T)-\rho(0,T)/\rho(0,T)$, that exhibits a similar $T$-dependence to
tan${}^{2}\Theta_{\rm H}$. The last two columns inspect the high-field MR
behavior of strange metal candidates. Note that the observation of a
$H$-linear MR at high fields does not imply the form of the MR over all fields
and temperatures exhibits quadrature scaling. La2-xSrxCuO4, for example,
displays simultaneous $H$\- and $T$-linearity but no quadrature scaling. The *
marks for FeSe1-xSx highlight the fact that the $H$-linear/quadrature MR seen
in this family coexists with a more conventional MR contribution, indicating
the presence of both strange metal and Fermi-liquid-like components in the dc
transport. The ** marks alongside YbAlB4 highlight the fact that while
$T$-linear resistivity is observed over a wide pressure range, its limiting
low-$T$ dependence is $T^{1.5}$. Finally, dash marks indicate where, as yet,
there have been no reports confirming or otherwise the considered behavior.
|  $\rho\propto T$ |  $\rho\propto T$ |  Extended |  cot$\Theta_{\rm H}\propto T^{2}$ |  Modified Kohler’s |  $H$-linear MR |  Quadrature
---|---|---|---|---|---|---|---
|  as $T$$\rightarrow\infty$ |  as $T$$\rightarrow$ 0 |  criticality |  (at low $H$) |  (at low $H$) |  (at high $H$) |  MR
UD $p$-cuprates |  $\checkmark$ takagi92  |  $\times$ proust16  |  $\times$ barisic13  |  $\checkmark$ carrington92  |  $\checkmark$ chan14  | - | -
OP $p$-cuprates |  $\checkmark$ gurvitch87  | - | - |  $\checkmark$ chien91  |  $\checkmark$ harris95  |  $\checkmark$ giraldo18  |  $\times$ boyd19
OD $p$-cuprates |  $\checkmark$ takagi92  |  $\checkmark$ cooper09  |  $\checkmark$ cooper09  |  $\checkmark$ manako92  |  $\times$ ayres20  |  $\checkmark$ ayres20  |  $\checkmark$ ayres20
La2-xCexCuO4 |  $\times$ poniatowski20  |  $\checkmark$ jin11  |  $\checkmark$ jin11  |  $\times$ li07  |  $\times$ poniatowski21  |  $\checkmark$ sarkar18  |  $\times$ sarkar18
Sr2RuO4 |  $\checkmark$ tyler98  |  $\times$ hussey98  |  $\times$ barber18  |  $\times$ mackenzie96  |  $\times$ hussey98  |  $\times$ hussey98  |  $\times$ hussey98
Sr3Ru2O7 |  $\checkmark$ bruin13  |  $\checkmark$ bruin13  |  $\times$ bruin13  | $\times$ | - | - | -
FeSe1-xSx |  $\times$ kasahara14  |  $\checkmark$ licci19a  |  $\times$ licci19a  |  $\checkmark$ huang20  |  $\checkmark$ huang20  |  $\checkmark$* licci19b  |  $\checkmark$* licci19b
BaFe2(As1-xPx)2 |  $\times$ hu18  |  $\checkmark$ analytis14  |  $\times$ analytis14  | - |  $\checkmark$ kasahara10  |  $\checkmark$ hayes16  |  $\checkmark$ hayes16
Ba(Fe1/3Co1/3Ni1/3)2As2 | - |  $\checkmark$ nakajima20  |  $\times$ nakajima20  | - | - |  $\checkmark$ nakajima20  |  $\checkmark$ nakajima20
YbRh2Si2 |  $\times$ trovarelli00  |  $\checkmark$ custers03  |  $\checkmark$ custers10  |  $\checkmark$ paschen04  | - | - | -
YbBAl4 |  $\times$ tomita15  |  $\checkmark^{**}$ tomita15  |  $\checkmark^{**}$ tomita15  | - | - | - | -
CeCoIn5 |  $\times$ nakajima07  |  $\checkmark$ bianchi03  |  $\times$ bianchi03  |  $\checkmark$ nakajima07  |  $\checkmark$ nakajima07  | - | -
CeRh6Ge4 |  $\times$ shen20  |  $\checkmark$ shen20  |  $\times$ shen20  | - | - | - | -
(TMTSF)2PF6 | - |  $\checkmark$ doiron09  |  $\checkmark$ doiron09  | - | - | - | -
MATBG |  $\checkmark$ polshyn19  |  $\checkmark$ cao20  |  $\checkmark$ cao20  |  $\checkmark$ lyu20  | - | - | -

In addressing this question, we must first acknowledge the many definitions of
strange metallic behavior that exist, the simplest being a material hosting a
metallic-like resistivity in the absence of quasiparticles. A more precise, if
empirical, definition centres on the $T$-linear resistivity, specifically one
that is distinguishable from that manifest in simple metals and attributed to
electron-phonon scattering. For a metal to be classified as strange, the
$T$-linearity must extend far beyond the typical bounds associated with
phonon-mediated resistivity. At low $T$, this is typically one third of the
Debye temperature, while at high $T$, it is once the magnitude of the
resistivity approaches (roughly 1/2) the value commensurate with the MIR
limit. A sub-set of correlated metals, such as SrRuO3 allen96  and Sr2RuO4
tyler98 , exhibit $T$-linear resistivity at high-$T$ with a magnitude that
clearly violates the MIR limit, but as the system cools down, conventional
Fermi-liquid behavior is restored mackenzie98 ; hussey98 . Hence, while they
are bona fide bad metals – exhibiting metallic resistivity beyond the MIR
limit – they do not classify as strange gunnarsson03 ; husseyMIR .

Another subset, identified here as quantum critical metals, exhibit $T$-linear
resistivity down to the lowest temperatures studied, but only at a singular
quantum critical point (QCP) in their phase diagram associated with a
continuous quantum phase transition to a symmetry broken phase that occurs at
$T$ = 0. In most cases, the phase transition in question is associated with
finite-Q antiferromagnetism (as in pure YbRh2Si2 trovarelli00 , CeCoIn5
bianchi03  and BaFe2(As1-xPx)2 analytis14 ) though recently, similar behavior
has also been reported in systems exhibiting zero-Q order, such as nematic
FeSe1-xSx licci19a  or ferromagnetic CeRh6Ge4 shen20 . Away from the QCP, the
low-$T$ resistivity recovers the canonical $T^{2}$ Fermi-liquid form, albeit
with a coefficient that is enhanced as the QCP is approached and the order
parameter fluctuations soften.

By contrast, in overdoped cuprates (both hole- cooper09 ; legros19  and
electron-doped jin11 ), Ge-doped YbRh2Si2 custers10 , YbBAl4 tomita15  and the
organic Bechgaard salts doiron09 , $\rho(T)$ is predominantly $T$-linear down
to low-$T$ not at a singular point in their respective phase diagrams but over
an extended range of the relevant tuning parameter. At first sight, this
‘extended criticality’ is difficult to reconcile with current theories of
quantum criticality, which predict a crossover to a purely $T^{2}$ resistivity
and thus a recovery of FL behavior at low $T$ everywhere except at the
(singular) QCP. Arguably, it is this feature – incompatibility with standard
Fermi-liquid and quantum critical scenarios – that distinguishes a geniune
strange metal from its aspirants. Intriguingly, in many of these systems
$\alpha_{1}$ – the coefficient of the $T$-linear resistivity – is found to
scale with the superconducting transition temperature $T_{c}$. Moreover, for
La2-xCexCuO4 jin11  and (TMTSF)2PF6 doiron09 , extended criticality emerges
beyond a spin density wave QCP, suggesting an intimate link between the
strange metal transport, superconductivity and the presence of critical or
long-wavelength spin fluctuations. In hole-doped cuprates, however, the
strange metal regime looks different, in the sense that the extended
criticality emerges beyond the end of the pseudogap regime that does not
coincide with a magnetic quantum phase transition hussey18 . Furthermore,
while the pseudogap plays host to a multitude of broken symmetry states, the
jury is still out as to whether any of these are responsible for pseudogap
formation or merely instabilities of it.

Besides $T$-linear resistivity, strange metals also exhibit anomalous behavior
in their magnetotransport, including 1) a quadratic temperature dependence of
the inverse Hall angle $\cot\Theta_{H}=\sigma_{\rm xy}/\sigma_{\rm xx}$, 2) a
transverse magnetoresistance (MR) that at low field exhibits modified Kohler’s
scaling ($\Delta\rho/\rho(0)\propto$ tan${}^{2}\Theta_{\rm
H}\propto(1/T^{2})^{2}$ or $(1/(A+BT^{2})^{2})$ harris95 ) and/or 3) a
$H$-linear MR at high fields that may or may not follow quadrature scaling
(whereby $\Delta\rho/T\propto$ $\sqrt{1+\gamma(H/T)^{2}}$) hayes16 ; licci19b
. A survey of the dc transport properties of several strange metal candidates
is presented in Table 1. The combination of a modified Kohler’s rule and
$T^{2}$ Hall angle has been interpreted to indicate the presence of distinct
relaxation times, either for different loci in momentum space carrington92  or
for relaxation processes normal and tangential to the underlying Fermi surface
chien91 . The $H$-linear MR, on the other hand, is inextricably tied to the
$T$-linear zero-field resistivity via its $H/T$ scaling relation, a relation
that can also extend over a broad range of the relevant tuning parameter
ayres20 . In some cases, this link can be obscured, either because $\rho(T)$
itself is not strictly $T$-linear ayres20  or because the quadrature MR co-
exists with a more conventional orbital MR licci19b . Both sets of behavior
highlight once again the possible coexistence of two relaxation times or two
distinct charge-carrying sectors in real materials. Curiously, quadrature
scaling does breaks down inside the pseudogap regime giraldo18 ; boyd19  while
modified Kohler’s scaling is recovered harris95 ; chan14 , suggesting that the
two phenomena may be mutually exclusive in single-band materials. In multiband
materials such as FeSe1-xSx, on the other hand, these different manifestations
of strange metallic transport appear side-by-side licci19b ; huang20 .
Irrespective of these caveats and complexities, what is striking about the
quadrature MR is that it occurs in systems with distinct Fermi surface
topologies, dominant interactions and energy scales, hinting at some
universal, but as yet unidentified, organizing principle.

Restricting the strange metal moniker, as done here, to materials that exhibit
low-$T$ $T$-linear resistivity over an extended region of phase space likewise
restricts strange metallicity to a select ‘club’. What shared feature binds
them together is the key question that will be explored in the coming
sections.

##  II Is it Quantum Critical?

Such scale-free $T$-linear resistivity is highly suggestive of some form of
underlying quantum criticality in which the only relevant scale is the
temperature governing collisions between excitations of the order parameter
damlesachdev97 . In fact, following the advent of marginal Fermi liquid (MFL)
phenomenology with its particular charge and spin fluctuation spectra and
associated ($T,\omega$)-linear self energies varma96 , the common
interpretation of such $T$-linear resistivity was and still remains the
nucleus of ideas centered on quantum criticality. The strict definition of
quantum criticality requires the divergence of a thermodynamic quantity. In
heavy fermion metals, the electronic heat capacity ratio $C_{\rm el}/T$ indeed
grows as $\ln(1/T)$ as the antiferromagnetic correlations diverge hf1 ; hf2 ;
bianchi03 . In certain hole-doped cuprates, $C_{\rm el}/T$ also scales as
$\ln(1/T)$ at doping levels close to the end of the pseudogap regime
MichonSheat  though here, evidence for a divergent length scale of an
associated order parameter is currently lacking tallon . Moreover,
photoemission suggests that at a $T$-independent critical doping $p_{c}\approx
0.19$, all signatures of incoherent spectral features that define the strange
metal cease, giving way to a more conventional coherent response chen19 . The
abruptness of the transition suggests that it is first-order, posing a
challenge to interpretations based solely on criticality.

As touched upon in the previous section, another major hurdle for the standard
criticality scenario is that the $T$-linear resistivity persists over a wide
range of the relevant tuneable parameter, be it doping as is the case for
cuprates cooper09 ; jin11 ; hussey13 ; legros19  and MATBG cao20 , pressure
for YbBAl4 tomita15  and the organics doiron09  or magnetic field for Ge-doped
YbRh2Si2 custers10 . If quantum criticality is the cause, then it is difficult
to fathom how a thermodynamic quantity can be fashioned to diverge over an
entire phase.

Despite these difficulties, it is worth exploring the connection $T$-linear
resistivity has with continuous quantum critical phenomena, which for the sake
of argument we presume to be tied to a singular point in the phase diagram.
Regardless of the origin of the QCP, universality allows us to answer a simple
question: What constraints does quantum criticality place on the
$T$-dependence of the resistivity? The answer to this question should just be
governed by the fundamental length scale for the correlations. The simplest
formulation of quantum criticality is single-parameter scaling in which the
spatial and temporal correlations are governed by the same diverging length
(see Fig. (2)). Making the additional assumption that the relevant charge
carriers are formed from the quantum critical fluctuations, a simple scaling
analysis on the singular part of the free energy results in the scaling law
chamon05

$\displaystyle\sigma(\omega=0,T)=\frac{q^{2}}{\hbar}f(\omega=0)\left(\frac{k_{B}T}{\hbar
c}\right)^{(d-2)/z}$ (5)

for the $T$-dependence of the conductivity where $f(\omega=0)$ is a non-zero
constant, $q$ is the charge and $z$ is the dynamical exponent, which from
causality must obey the inequality $z\geq 1$. Absent from this expression is
any dependence on an ancillary energy scale for example $E_{F}$ or the plasma
frequency $\omega_{p}$ as the only assumption is scale-invariant transport
irrespective of the details of the system. The analogous expression for the
optical conductivity is wen92

$\displaystyle\sigma(\omega,T=0)\propto\omega^{(d-2)/z}.$ (6)

In pure YbRh2Si2, for example, $\sigma^{-1}(\omega)$ follows an
$\omega$-linear dependence at low frequencies in the same region of the
($T,H$) phase diagram – the quantum critical ‘fan’ – where $\rho(T)$ is also
linear, consistent with this notion of single-parameter scaling prochaska . In
cuprates, on the other hand, the situation is more nuanced. At intermediate
frequencies – sometimes referred to as the mid-infrared response –
$\sigma(\omega$) exhibits a ubiquitous $\omega^{-2/3}$ dependence marel03 .
While this feature in $\sigma(\omega)$ has been interpreted in terms of
quantum critical scaling marel03 , it is inconsistent with the single-
parameter scaling described above. At any doping level, $\sigma(\omega)$ in
the cuprates exhibits a minimum at roughly the charge transfer scale of 1 eV.
This is traditionally marelcolorchange ; CooperUVIR  used as the energy scale
demarcating the separation between intraband and interband transitions and
hence serves to separate the low-energy from the high-energy continua. It has
long been debated whether the broad sub-eV $\sigma(\omega)$ response in
cuprates is best analysed in terms of one or two components tannerdrude ;
CooperUVIR . In the former, the $\omega^{-2/3}$ tail is simply a consequence
of the strong $\omega$-linear dependence in 1/$\tau_{tr}(\omega)$ – à la MFL –
while in the latter, it forms part of an incoherent response that is distinct
from the coherent Drude weight centred at $\omega=0$ which itself is described
with either a constant or $\omega$-dependent scattering rate.

Returning to the dc resistivity, we find that in cuprates, where $d$ = 3, an
exponent $z=-1$ is required, a value that is strictly forbidden by causality
chamon05 . For $d$ = 2, as in the case of MATBG, the $T$-dependence vanishes.
This is of course fixed with the replacement of $d\rightarrow d^{\ast}=1$ for
both materials. While $d^{\ast}$ can be construed as the number of dimensions
shl  transverse to the Fermi surface, it is difficult to justify such a
procedure here as the persistence of $T$-linearity with no change in slope
above and below the MIR requires a theory that does not rely on FL concepts
such as a Fermi velocity or energy. Furthermore, it is well known that
introducing $d^{\ast}$ yields a power law for the heat capacity, $C\propto
T^{3/2}$ which is not seen experimentally loramSH . On dimensional grounds,
the $z=-1$ result in the context of the Drude formula is a consequence of
compensating the square power of the plasma frequency with powers of $T$ so
that the scaling form Eq. (5) is maintained. A distinct possibility is that
perhaps some other form of quantum criticality beyond single-parameter
scaling, such as a non-critical form of the entropy suggested recently
zaanenentropy , is at work here. We shall return to this idea in section V.

Figure 2: Single-parameter scaling hypothesis in which all length and time
scales are governed by the same diverging length scale, $\xi$ which diverges
at the quantum critical point as $(g-g_{c})^{-\nu}$, $g$ the coupling constant
for the critical interaction. $z$ is the dynamical critical exponent.

Another critical feature of the conductivity is its behavior at finite wave
vector $k$ which may be quantified by the dynamic charge susceptibility,

$\chi^{\prime\prime}(k,\omega)=-\frac{k^{2}}{\omega
e^{2}}\Re\sigma(k,\omega),$ (7)

determined from electron energy-loss spectroscopy (EELS). A restriction on
EELS is that it measures the longitudinal charge response while optics yields
the transverse. At vanishing momentum both are expected to be equal. As optics
has no momentum resolution, comparison with EELS can only be made as
$k\rightarrow 0$. The primary charge excitation in strange metals is a 1 eV
plasmon that was long believed to exhibit the same behavior as in a normal
Fermi liquid nucker1989 ; nucker1991 . Recent high-resolution M-EELS
measurements have called this belief into question, showing that the large-$k$
response is dominated by a continuum which remains flat to high energies,
roughly 2 eV vig17 ; mitrano18 ; husain19 . Such behavior is reminiscent of
the MFL varma96  scenario except in that picture, the continuum persists up to
a cut-off scale determined by the temperature not the Mott scale of 2 eV. In
addition, the continuum exhibits scale-invariant features but with a dynamical
critical exponent, $z\sim\infty$, not possible from a simple QCP.

We conclude then that no form of traditional quantum criticality can account
easily for the power laws seen in strange metallic transport (though we
recognize that $T$-linear resistivity is observed above what appear to be
genuine singular QCPs). The photoemission experiments chen19  indicating a
first-order transition pose an additional problem exacerbated by the
possibility that the criticality might be relevant to a whole region cooper09
; legros19 ; greene ; dessau ; cao20 ; tomita15 ; doiron09 ; custers10 ;
hussey18  rather than a point. Such criticality over an extended region is
reminiscent of critical charged matter kiritsis2 ; kiritsis1  arising from
dilatonic models in gauge-gravity duality. We will revisit aspects of these
ideas in a later section as they have been the most successful (see Table 2)
thus far in reproducing the various characteristics of strange metal physics.

##  III Is it Planckian?

While the electrical resistivity in metals can be measured directly, the
scattering rate is entirely an inferred quantity. Herein lies the catch with
Planckian dissipation. Angle-resolved photoemission (ARPES) experiments on
cuprates as early as 1999 reported that the width of the momentum distribution
curves (MDCs) at optimal doping along the nodal direction ($(0,0)$ to
$(\pi,\pi)$) scale as a linear function of temperature and $a_{0}$ \+
0.75$\omega$ for frequencies that exceed $2.5k_{B}T$ valla99 . The momentum
linewidth, which in photoemission enters as Im$\Sigma$ – the imaginary part of
the self energy – can be used to define a lifetime through

$\displaystyle\hbar v_{k}\Delta k={\rm Im}\Sigma(\bf
k,\omega)=2\frac{\hbar}{\tau},$ (8)

with $v_{k}$ the group velocity for momentum state $k$. Extracting the slope
from the data in Figure (2) of Ref. valla99  and using the experimentally
reported Fermi velocity $v_{F}$ = 1.1
eV/$\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, we find that the single-particle
scattering rate $\hbar/\tau\sim 1.7k_{B}T$, i.e. of order the Planckian limit.
Similar results were obtained in subsequent ARPES studies kaminski05 ; bok10 ;
dama03  with a key extension added by Reber et al. dessau  whereby the width
of nodal states was observed to obey the quadrature form indicative of a
power-law liquid, $((\hbar\omega)^{2}+(\beta k_{B}T)^{2})^{\lambda}$ where
$\lambda$ is a doping-dependent parameter equal to $1/2$ at optimal doping.

This extraction of the scattering rate from ARPES, however, is not entirely
problem-free as $v_{F}$ is hard to define in ARPES experiments at energies
close to the Fermi level and where, for the most part, the width of the state
exceeds its energy. Indeed, the integral of the density of states using as
input the $v_{F}$ extracted from APRES measurements is found to account for
only half of the as-measured electronic specific heat coefficient yoshida07 .
Furthermore, this reliance on Fermiology leaves open the precise meaning of
Fig. (2) of Ref. bruin13  in which $\tau$ is plotted versus $v_{F}$ for a
series of materials that violate the MIR limit at intermediate to high
temperatures. Despite this, a similar extraction by Legros and colleagues
legros19 , again using Fermiology but focusing on the low-$T$ resistivity,
also found a transport scattering rate close to the Planckian bound. This
consistency between the two analyses reflects the curious fact that the
$T$-linear slope of the dc resistivity does not vary markedly as the MIR
threshold is crossed. It does not, however, necessarily justify either
approach in validating $T$-linear scattering at the Planckian limit. Finally,
while $T$-linearity and Planckian dissipation appear synonymous in the
cuprates, this is not universally the case. In YbRh2Si2 prochaska , for
example, the $T$-linear scattering rate is found to deviate strongly from the
Planckian limit with $\tau_{tr}\sim 0.1\tau_{P}$ paschen04 , while in the
electron-doped cuprates, the notion of a Planckian limit to the scattering
rate has recently been challenged poniatowski21b . This certainly adds to the
intrigue regarding quantum criticality as the underlying cause of Planckian
dissipation.

In principle, the optical conductivity permits an extraction of $\tau$ without
recourse to Fermiology. Within a Drude model, the optical conductivity,

$\displaystyle\sigma(\omega)=\frac{1}{4\pi}\frac{\omega_{p}^{2}\tau_{\rm
tr}}{1+i\omega\tau_{\rm tr}},$ (9)

contains only $\tau_{\rm tr}$ and $\omega_{p}=\sqrt{4\pi n_{e}e^{2}/m}$. At
zero frequency, the Drude formula naturally yields the dc conductivity
$\sigma_{\rm dc}$ while an estimate for the relaxation rate can be extracted
from the width at half maximum of the full Drude response. However, there is
an important caveat: $\tau_{\rm tr}$ is frequency dependent in the cuprates, a
condition that is consistent with various physical models including both the
Fermi liquid and MFL scenarios as well as the large body dessau ; valla99  of
MDC analysis performed on the cuprates. While this prevents a clean separation
of the conductivity into coherent and incoherent parts, van der Marel and
colleagues marel03  were able to show that in the low-frequency limit,
$\omega<1.5k_{B}T/\hbar$, $\tau_{\rm tr}\sim 0.8\tau_{P}$, in agreement with
the dc analysis of Legros legros19 .

A second key issue remains, namely; how can such Drude analysis be justified
for those strange metals in which the MIR limit is violated and the Drude peak
shifts to finite frequencies husseyMIR ? Indeed, in the high-$T$ limit, ‘bad
metallicity’ can be ascribed to a transfer of spectral weight from low- to
high-$\omega$, rather than from an ever-increasing scattering rate (that
within a Drude picture results in a continuous broadening of the Lorentzian
fixed at zero frequency). Given the marked crossover in the form of
$\sigma(\omega)$ at low frequencies, it is indeed remarkable and mysterious
that the slope of the $T$-linear resistivity continues unabated with no
discernible change.

##  IV Is it Mottness?

Table 1 encompasses a series of ground states from which $T$-linear
resistivity emerges. In some of these materials, such as the heavy fermions,
the high and low-energy features of the spectrum are relatively distinct in
the sense that spectral weight transfer from the UV to the IR is absent. On
the other hand, hole or electron doping of the parent cuprate induces a marked
transfer of spectral weight of roughly 1-2 eV. As a result, the low-energy
spectral weight grows ctchen ; meinders ; eskes ; PhillipsRMP ; CooperUVIR  at
the expense of the degrees of freedom at high energy, a trend that persists
marelcolorchange  even inside the superconducting state. This is an intrinsic
feature of Mott systems, namely that the number of low-energy degrees of
freedom is derived from the high-energy spectral weight. As this physics is
distinct from that of a Fermi liquid and intrinsic to Mott physics, it is
termed ‘Mottness’PhillipsRMP . Notably, the mid-infrared response with its
characteristic $\omega^{-2/3}$ scaling is absent from the parent Mott
insulating state. Hence, it must reflect the doping-induced spectral weight
transfer across the Mott gap. It is perhaps not a surprise then that no
low-$T_{c}$ material exhibits such a significant midinfrared feature. In fact,
some theories of cuprate superconductivity Leggett  credit its origin to the
mid-infrared scaling. We can quantify the total number of low-energy degrees
of freedom that arise from the UV-IR mixing across the Mott gap by integrating
the optical conductivity,

$\displaystyle N_{\rm eff}(\Omega)=\frac{2mV_{\rm cell}}{\pi
e^{2}}\int_{0}^{\Omega}\sigma(\omega)d\omega,$ (10)

up to the optical gap $\Omega\approx$ 1.2 eV where $V_{\rm cell}$ is the unit-
cell volume. The energy scale of 1.2 eV corresponds to the minimum of the
optical conductivity as mentioned in the previous section. In a rigid-band
semiconductor model in which such spectral weight transfer is absent, $N_{\rm
eff}=x$, where $x$ is the number of holes. In the cuprates, however, $N_{\rm
eff}$ exceeds $x$ as shown in Fig. (3). This is the defining feature of
Mottness ctchen ; meinders ; eskes ; PhillipsRMP  since it is ubiquitous in
Mott systems and strictly absent in weakly correlated metals. Even in many of
the strange or quantum critical metals described in Table 1, there is little
or no evidence that Mottness is playing any significant role. Such a
distinction may thus offer a hint to the source of the uniqueness of the
cuprate strange metal. In bad metals, on the other hand, a gradual transfer of
low-frequency spectral weight out to energies of order the Mott scale is
almost universally observed with increasing temperature husseyMIR  suggesting
that Mottness is one of the key components of bad metallic transport.

The optical response in cuprates tells us that there are degrees of freedom
that couple to electromagnetism that have no interpretation in terms of doped
holes. That is, they are not local entities as they arise from the mixing of
both UV and IR degrees of freedom. It is such mixing that could account for
the lack of any distinctive energy scalePhillipsRMP , that is scale
invariance, underlying the strange metal. Additionally, Lee et al. showed,
also from optical conductivity studies LeeEM , that throughout the underdoped
regime of the cuprate phase diagram, the effective mass remains constant. As a
result, the Mott transition proceeds by a vanishing of the carrier number
rather than the mass divergence of the Brinkman-Rice scenario BRice . (Note
that while quantum oscillation experiments on underdoped cuprates show
evidence for mass enhancement ramshaw15 , this is thought to be tied to the
charge order centred around 1/8 doping). Such dynamical mixing between the UV
and IR scales in Mott systems is well known to give rise to spectral weight in
the lower Hubbard band meinders ; eskes ; PhillipsRMP  that exceeds the number
of electrons, strictly $1+x$, that the band can hold. Consequently, part of
the electromagnetic response of the strange metal at low energies has no
interpretation in terms of electron quasiparticles as it arises strictly from
UV-IR mixing. Precisely how such mixing leads to scale-invariant $T-$linear
resistivity remains open.

Figure 3: Integrated optical conductivity for electron-doped Pr2-xCexCuO4-δ
(triangles) and hole-doped La2-xSrxCuO4-δ (circles) The dashed line indicates
what is expected for a doping a semiconductor. The expectation is that each Ce
or Sr atom contributes just a single charge carrier. Reprinted from Ref.
CooperUVIR .

##  V Is it about Gravity?

Table 2: Snapshot of current theoretical modeling of the strange metal based
on consistency with $T-$ linear resistivity, $\omega^{-2/3}$ scaling of the
mid-infrared optical conductivity, quadrature magnetoresistance, extended
quantum criticality, and what predictions are made in terms of experimental
observables. The transcription of the abbreviations is as follows: MFL =
marginal Fermi liqiud, EFL = ersatz Fermi liqiud, SYK = Sachdev-Ye-Kitaev,
AdS/CFT = anti de Sitter space/conformal field theory conjecture, AD/EMD =
anomalous dimensions/Einstein-Maxwell-dilaton, HM = Hubbard model, QMC =
quantum Monte Carlo, ED = exact diagonalization, CA = cold atoms, DMFT/EDMFT =
dynamical mean-field theory/embedded dynamical mean-field theory, A-B =
Aharonov-Bohm effect, ECFL = extremely correlated Fermi liquid, and QCP =
quantum critical point scenarios.

111Apologies to anyone whose work we did not cite.

|  $\rho\propto T$ |  $\rho\propto T$ |  $\sigma\propto\omega^{-2/3}$ |  Quadrature |  Extended |  Experimental
---|---|---|---|---|---|---
|  as $T\rightarrow 0$ |  as $T\rightarrow\infty$ |  | MR |  criticality | Prediction
Phenomenological |  |  |  |  |  |
MFL |  $\checkmark$ varma96  |  $\times$varma96  | $\times$ | $\times$ | $\times$ | loop currents loopvarma
EFL | \- 222$T$-linear resistivity is an input. | - | - | $\times$ | $\times$ | loop currents else1
Numerical |  |  |  |  |  |
ECFL | $\times$ | ✓maishastry  | - | - | $\times$ | $\times$
HM (QMC/ED/CA) | \- Huang987  |  $\checkmark$ Huang987 ; ED ; CA ; ED1 ; ED2  | $\times$ | - | - | -
DMFT/EDMFT |  $\checkmark$ Cha18341  |  $\checkmark$kotliarDMFT ; tremblay  | $\times$ | - |  $\checkmark$ tremblay  | -
QCP | ✓INem  | - | - | - | $\times$ | -
Gravity-based |  |  |  |  |  |
SYK |  $\checkmark$ patelPM ; syk2  |  $\checkmark$333A slope change occurs through the MIR. syk2  | $\times$ |  $\checkmark$444Quadrature scaling obtained only for a bi-valued random resistor model syk1  with equal weights boyd19 . syk1  | - | $\times$
AdS/CFT |  $\checkmark$ adscftstrange  |  $\checkmark$ adscftstrange  |  $\checkmark$555While this scaling was thought to arise in pure AdS with an inhomogenous charge density horowitz , later studies langley ; donos  found otherwise. kiritsis ; kiritsis2  | $\times$ | $\times$ | $\times$
AD/EMD |  $\checkmark$ hk ; gl1 ; limtra  |  $\checkmark$ hk ; kiritsis ; kiritsis2 ; limtra ; karch2  |  $\checkmark$ karch2 ; kiritsis ; kiritsis2  | $\times$ |  $\checkmark$kiritsis  | Fractional A-B limtra

To frame the theoretical modeling of strange metallicity tabulated in Table 2,
we group the work into three principal categories: 1) phenomenological, 2)
numerical and 3) gravity-related. While both phenomenological models
considered here require (EFL) or predict (MFL) loop currents, they do so for
fundamentally different reasons. (For an explanation of the various acronyms,
please refer to the caption in Table 2). On the EFL account else1 , such
current order is needed to obtain a finite resistivity in the absence of
momentum relaxation (certainly not a natural choice given the Drude fit to the
optical conductivity discussed previously), while in MFL, loop currents
loopvarma  are thought to underpin the local fluctuation spectrum varma96 .
ECFL maishastry  predicts a resistivity that interpolates between Fermi-
liquid-like $T^{2}$ at low $T$ to $T$-linear for $T\gg T_{\rm FL}$. QMC
Huang987 ; ED ; ED1 ; ED2  as well as cold atom (CA) experiments CA  on the
Hubbard model (HM) have established that at high temperatures, the resistivity
is indeed $T$-linear. The Fermion-sign problem, however, prevents any
definitive statement about the low-$T$ behavior in the presence of Mott
physics. Non-Fermi liquid transport in SYK models SY ; Kitaev ; K  is achieved
by an all-to-all random interaction. While such interactions might seem
initially unphysical, SYK models are nevertheless natural candidates to
destroy Fermi liquids which, by their nature, permit a purely local
description in momentum space. As a result, they are impervious to repulsive
local-in-space interactions polchinski . Coupling a Fermi liquid to an array
of disordered SYK islands, however, leads syk1 ; syk2  to a non-trivial change
in the electron Green function across the MIR and hence a change in slope of
the resistivity is unavoidable syk1  though it can be minimized through fine
tuning syk2 .

An added feature of these disordered models is that in certain limits, they
have a gravity dual Kitaev ; SY1 ; K ; dual . This state of affairs arises
because the basic propagator K ; SY1 ; Kitaev  in the SYK model in imaginary
time describes the motion of fermions, with appropriate boundary conditions,
between two points of the asymptotic boundary of a hyperbolic plane. In real
time, simply replacing the hyperbolic plane with the space-time equivalent,
namely two-dimensional anti de Sitter (AdS) space (a maximally symmetric
Lorentzian manifold with constant negative curvature), accurately describes
all the correlators. It is from this realization that the dual description
between a random spin model and gravity in AdS2 lies sachdev ; Kitaev ; K .
Hence, although the origins of SYK were independent of gravity, its
correlators can be deduced from the asymptotics of the corresponding
spacetime. At the asymptote, only the time coordinate survives and hence
ultimately, SYK dynamics is ultra-local in space with only diverging
correlations in time, an instantiation of local quantum criticality.

Such local quantum criticality is not a new concept in condensed matter
systems and indeed lies at the heart of MFL phenomenology varma96 , DMFT
kotliarDMFT , and is consistent with the momentum-independent continuum found
in the M-EELS data discussed earlier mitrano18 . The deeper question is why
does gravity have anything to do with a spin problem with non-local
interactions? The issue comes down to criticality and to the structure of
general relativity. The second equivalence principle on which general
relativity is based states that no local measurement can detect a uniform
gravitational field. A global measurement is required. Ditto for a critical
system since no local measurement can discern criticality. Observables tied to
the diverging correlation length are required. Hence, at least conceptually,
it is not unreasonable to expect a link between critical matter and gravity.
The modern mathematical machinery which makes it possible to relate the two is
the gauge-gravity duality or the AdS/CFT (conformal field theory) conjecture.
The key claim of this duality maldacena ; witten ; gubser  is that some
strongly interacting quantum theories, namely ones which are at least
conformally invariant in $d$-dimensions, are dual to a theory of gravity in a
$d+1$ spacetime that is asymptotically AdS. The radial direction represents
the energy with the quantum theory residing at the UV boundary and the IR
limit deep in the interior at the black hole horizon. Hence, intrinsic to this
construction is a separation between bulk (gravitational) and boundary
(quantum mechanical) degrees of freedom. That the boundary of a gravitational
object has features distinct from the bulk dates back to the observations of
Beckenstein beckenstein  and Hawking hawking ; hawkingarea  that the
information content of a black hole scales with the area, not the volume. The
requirement that the boundary theory be strongly coupled then arises by
maintaining that the AdS radius exceeds the Planck length $\ell_{P}$. More
explicitly, because the AdS radius and the coupling constant of the boundary
theory are proportional, the requirement $R\gg\ell_{P}$ translates into a
boundary theory that is strongly coupled.

The first incarnation Faulkner ; fireball ; schalm  of this duality in the
context of fermion correlators involved modeling fermions at finite density in
$2+1$ dimensions. From the duality, the conformally invariant vacuum of such a
system corresponds to gravity in AdS4, the extra dimension representing the
radial direction along which identical copies of the boundary CFT lie albeit
with differing energy scales. Surprisingly, what was shown Faulkner  is that
the low-energy (IR) properties of such a system in the presence of a charge
density are determined by an emergent AdS${}_{2}\times R^{2}$ (with $R^{2}$
representing a plane) spacetime at the black hole horizon. The actual symmetry
includes scale invariance and is denoted by $SL(2,R)$ (a special Lie group of
real $2\times 2$ matrices with a unit determinant). Once again, the
criticality of boundary fermions is determined entirely by the fluctuations in
time, that is, local quantum criticality as seen in SYK. The temperature and
frequency dependence of the conductivity are then determined by the same
exponent Faulkner  as expected from Eqs. (5) and (6) and as a result, a
simultaneous description of $T$-linearity and $\omega^{-2/3}$ is not possible,
as noted in Table 2.

This particular hurdle is overcome by the AD/EMD theories Anomdim0 ; Anomdim01
; Anomdim02 ; kiritsis ; kiritsis1 ; kiritsis2 ; cremonini ; Anomdim1  which
as indicated in Table 2, have been the most successful to date in describing
the range of physics observed in strange metals. What is new here is the
introduction of extra fields, dilatons for example, which permit hyperscaling
violationshl  and anomalous dimensionsAnomdim0 ; Anomdim01 ; Anomdim02 ;
kiritsis ; kiritsis1 ; kiritsis2 ; cremonini ; Anomdim1  for all operators.
Consequently, under a scale change of the coordinates, the metric is no longer
unscathed. That is, the manifold is not fixed and it is the matter fields that
determine the geometry. Such systems have a covariance, rather than scale
invariance indicative of pure AdS metrics. A consequence of this covariance is
that even the currents acquire anomalous dimensions. But how is this possible
given that a tenet of field theory is that no amount of renormalization can
change the dimension of the current gross  from $d-1$? What makes this
possible is that in EMD theories, the extra radial dimension allows physics
beyond Maxwellian electro-magnetism. For example, the standard Maxwell action,
$S=\int dV_{d}F^{2}$ where $F=dA$, requires that the dimension of the gauge
field be fixed to unity, $[A]=1$666What is really required is that $[qA]=1$,
with $q$ the charge. In insisting that $[A]=1$, we are setting $q=1$ but still
all of our statements refer to the product $qA$.. EMD theories use instead an
action of the form $S=\int dV_{d}dyy^{a}F^{2}$ where $y$ is the radial
coordinate of the $d+1$ AdS spacetime. Comparing these two actions leads
immediately to the conclusion that the dimension of $A$ now acquires the value
$[A]=1-a/2$. Hence, even in the bulk of the geometry, the dimension of the
gauge field is not unity. Depending on the value of $a$, $a<0$ at the UV
conformal boundary or $a>0$ at the IR at the black hole horizon, the equations
of motion are non-standard and obey fractional electromagnetism gl1 ; gl2
consistent with a non-traditional dimension for the gauge field. In EMD
theories, it is precisely the anomalous dimensionkiritsis ; kiritsis1 ;
kiritsis2 ; cremonini ; Anomdim0 ; Anomdim01 ; Anomdim02  for conserved
quantities that gives rise to the added freedom for extended quantum
criticality to occur, the simultaneous fitting karch2  of $T-$linearity and
$\omega^{-2/3}$ of the optical conductivity, and the basis for a proposal for
the strange metal based on $[A]=5/3$ hk .

Within these holographic systems, a Drude-like peak in the optical
conductivity can emerge both from the coherent (quasiparticle-like) sector
Davison_15  as well as the incoherent (‘un-particle unparticle ’) sector
hartnoll10 ; kiritsis15 ; chen_17 ; Davison_19 . Application of EMD theory has
also provided fresh insights into the phenomenon of ‘lifetime separation’ seen
in the dc and Hall conductivities of hole-doped cuprates carrington92 ;
chien91 ; manako92  as well as in other candidate strange metals paschen04 ;
lyu20 . For a system with broken translational invariance, the finite density
conductivity comprises two distinct components blake_donos , with the dc
resistivity being dominated by the particle-hole symmetric term – with a
vanishing Hall conductivity – and one from an explicit charge density governed
by more conventional (Umklapp) momentum relaxation that sets the
$T$-dependence of the Hall angle.

The success of EMD theories in the context of strange metal physics raises a
philosophical question: Is all of this just a game? That is, is the
construction of bulk theories with funky electromagnetism fundamental? The
answer here lies in Nöther’s Second Theorem (NST) gl1 ; gl2 ; PhillipsRMP , a
theorem far less known than her ubiquitous first theorem but ultimately of
more importance as it identifies a shortcoming. To illustrate her first
theorem, consider Maxwellian electromagnetism which is invariant under the
transformation $A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\Lambda$. This
theorem states that there must be a conservation law with the same number of
derivatives as in the gauge principle. Hence the conservation law only
involves a single derivative, namely $\partial_{\mu}J_{\mu}=0$. This is
Nöther’s First Theorem N  in practice.

What Nöther N  spent the second half of her famous paper trying to rectify is
that the form of the gauge transformation is not unique; hence the
conservation law is arbitrary. It is for this reason that in the second half N
of her foundational paper, she retained all possible higher-order integer
derivatives in the gauge principle. These higher-order derivatives both add
constraints to and change the dimension of the current. Stated succinctly, NST
N  dictates that the full family of generators of U(1) invariance determines
the dimension of the current. It is easy to see how this works. Suppose we can
find a quantity, $\hat{Y}$ that commutes with $\partial_{\mu}$. That is,
$\partial_{\mu}\hat{Y}=\hat{Y}\partial_{\mu}$. If this is so, then we can
insert this into the conservation law with impunity. What this does is
redefine the current:
$\partial_{\mu}\hat{Y}J^{\mu}=\partial_{\mu}\tilde{J}^{\mu}$. The new current
$\tilde{J}^{\mu}$ acquires whatever dimensions $\hat{Y}$ has such that
$[\tilde{J}^{\mu}]=d-1-d_{Y}$. But because of the first theorem, $\hat{Y}$
must have come from the gauge transformation and hence must ultimately be a
differential operator itself. That is, there is an equally valid class of
electromagnetisms with gauge transformations of the form $A_{\mu}\rightarrow
A_{\mu}+\partial_{\mu}\hat{Y}\Lambda$. For EMD theories gl1 ; gl2 ;
PhillipsRMP , $\hat{Y}$ is given by the fractional Laplacian,
$\Delta^{(\gamma-1)/2}$ with $[A_{\mu}]=\gamma$ (with $\gamma=1-a/2$ to make
contact with the EMD theories introduced earlier). For most matter as we know
it, $\gamma=1$. The success of EMD theories raises the possibility that the
strangeness of the strange metal hinges on the fact that $\gamma\neq 1$. This
can be tested experimentally using the standard Aharonov-Bohm geometry limtra
; gl1  in which a hole of radius $r$ is punched into a cuprate strange metal.
Because $[A]$ is no longer unity, the integral of $A\cdot d\ell$ is no longer
the dimensionless flux. For physically realizable gauges, this ultimately
provides an obstruction to charge quantization. As a result, deviations limtra
; gl1  from the standard $\pi r^{2}\times B$ dependence for the flux would be
the key experimental feature that a non-local gauge principle is operative in
the strange metal. An alternative would be, as Anderson anderson  advocated,
the use of fractional or unparticle propagators with the standard gauge
principle. However, in the end, it all comes down to gauge invariance. The
standard gauge-invariant condition prevents the power laws in unparticle stuff
from influencing the algebraic fall-off of the optical conductivity limtragool
; karch2  as they offer just a prefactor to the polarizations Liao2008 . The
escape route, an anomalous dimension for the underlying gauge field, offers a
viable solution but the price is abandoning locality bora  of the action.

Figure 4:  Correlation between the superfluid density $n_{s}(0)$ and the
coefficient $\alpha_{1}$ of the $T$-linear resistivity in Tl2Ba2CuO6+δ
(Tl2201) across the strange metal regime (adapted from Refs. culo21 ; putzke21
).

##  VI Is it Important?

Given the immense difficulty in constructing a theory of the strange metal,
one might ask why bother? To gauge the importance of the strange metal, look
no further than Fig. (4). This figure shows that the coefficient $\alpha_{1}$
of the $T$-linear resistivity component in the strange metal regime of
overdoped hole-doped cuprates tracks the doping dependence of the $T=0$
superfluid density $n_{s}(0)$. As mentioned earlier, a similar correlation
exists between $\alpha_{1}$ and $T_{c}$ in electron-doped cuprates jin11 , the
Bechgaard salts doiron09  as well as the iron pnictides doiron09 ,
establishing a fundamental link between high-temperature superconductivity and
the strange metal.

For a long time, the drop in $n_{s}(0)$ with doping in cuprates was attributed
to pair breaking, a symptom of the demise of the dominant pairing interaction
within a disordered lattice. Recent mutual inductance measurements, however,
have challenged this view, arguing that the limiting low-$T$ behavior of
$n_{s}(T)$ was incompatible with conventional pair breaking scenarios
bozovic16 . Certainly, the correlation between $\alpha_{1}$ and $n_{s}(0)$ is
unforeseen in such models. Moreover, if the strange metal regime is indeed
populated with non-quasiparticle states, then Fig. (4) indicates a pivotal
role for these states in the pairing condensate culo21 . On more general
grounds, this result informs us that the door to unlocking cuprate
superconductivity is through the strange metal and any theory which divorces
superconductivity from the strange metal is a non-starter. To conclude,
solving the strange metal kills two birds with one stone. Perhaps there is
some justice here. After all, we know from Pippard’s pippard  work, which can
be reformulated gl1 ; gl2  in terms of fractional Laplacians, that even
explaining superconductivity in elemental metals necessitates a non-local
relationship between the current and the gauge field. What seems to be
potentially new about the cuprates is that now the normal state as a result of
the strange metal also requires non-locality.

Acknowledgements

PA and PWP acknowledge support from Center for Emergent Superconductivity, a
DOE Energy Frontier Research Center, Grant No. DE-AC0298CH1088. NEH is funded
by Netherlands Organisation for Scientific Research (NWO) (‘Strange Metals’
16METL01), the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (No. 835279-Catch-22) and EPSRC
(EP/V02986X/1). The work on fractional electromagnetism was funded through
DMR-2111379.

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