# Superfluid density and two-component conductivity in hole-doped cuprates

Jake Ayres  H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8
1TL, United Kingdom  Mikhail I. Katsnelson  Institute for Molecules and
Materials, Radboud University, 6525ED Nijmegen, Netherlands  Nigel E. Hussey
H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United
Kingdom  Institute for Molecules and Materials, Radboud University, 6525ED
Nijmegen, Netherlands  HFML-FELIX, Radboud University, 6525ED Nijmegen,
Netherlands

###### Abstract

While the pseudogap dominates the phase diagram of hole-doped cuprates,
connecting the antiferromagnetic parent insulator at low doping to the strange
metal at higher doping, its origin and relation to superconductivity remains
unknown. In order to proceed, a complete understanding of how the single hole
– initially localized in the Mott state – becomes mobile and ultimately
evolves into a coherent quasiparticle at the end of the superconducting dome
is required. In order to affect this development, we examine recent transport
and spectroscopic studies of hole-doped cuprates across their phase diagram.
In the process, we highlight a set of empirical correlations between the
superfluid density and certain normal state properties of hole-doped cuprates
that offer fresh insights into the emergence of metallicity within the CuO2
plane and its influence on the robustness of the superconducting state. We
conclude by arguing that the overall behavior is best understood in terms of
two distinct current-carrying fluids, only one of which dominates the
superconducting condensate and is gapped out below the pseudogap endpoint at a
critical hole concentration $p^{\ast}$.

##  1 Introduction

At half-filling, the CuO4 square-planar motif generic to all cuprates is
insulating due to the strong on-site Coulomb repulsion that causes the single
charge carrier within each unit cell to localize. Upon hole doping, the
initial excess charge preferentially locates on the oxygen sites – giving rise
to a charge-transfer insulator – which then couples antiferromagnetically with
the localized charge on the copper site to create a band of Zhang-Rice
singlets. With further doping, the charge-transfer gap collapses and long-
range antiferromagnetic (AFM) order is destroyed, leading to a system of
highly-frustrated short-range spin correlations and ultimately $d$-wave
superconductivity. At the end of the superconducting (SC) dome, the charge
initially localized in the Mott state becomes fully itinerant and a correlated
Fermi liquid (FL) state with a large cylindrical Fermi surface containing the
total carrier density emerges.

Despite a decades-long investigation, the above summary is almost the extent
of our current understanding of the fate of the initially localized carrier as
further holes are introduced into the CuO2 plane. The starting point and
endpoint are well established – it is the transition from localization to
itinerancy as the number of doped holes ($p$) increases that is proving the
biggest challenge. The fact that this transition incorporates the most
enigmatic features of the cuprate problem; the normal state pseudogap, the
strange metal and high temperature superconductivity itself makes steps to
understand its evolution all the more pressing.

The aim of this Perspective is to highlight a number of intriguing
correlations that shed new light, not only on the fate of the localized hole –
as reflected in both single-particle probes like angle-resolved photoemission
spectroscopy (ARPES) and scanning tunneling microscopy (STM) and collective
responses like transport – but also on the impact of its enhanced mobility on
the superconductivity and in particular, on the superfluid density. The
totality of the transport behaviour across the cuprate phase diagram provides
compelling evidence for the presence of two current-carrying fluids, one
associated with coherent fermions, the other incoherent and possibly bosonic
in nature. The relative occupation of these two entities is found to show a
remarkably similar doping dependence to that of the superfluid density,
suggesting that the latter is dominated by only one of the former. Finally, in
order to motivate the development of a consistent theoretical description of
this dichotomy, we highlight systems that exhibit such a two-fluid response
before presenting results from a treatment of the Hubbard model on a 2 x 2
plaquette that captures a number of the essential elements seen in the
cuprates. The overall picture that emerges is one that departs significantly
from the standard BCS description.

##  2 Experimental survey

Despite the diverse and complex chemistry of the cuprates, the evolution of
their physical (i.e. transport, thermodynamic and spectroscopic) properties
across the phase diagram is found to be strikingly similar for all well-
studied families. This remarkable uniformity suggests that Fermiology plays
only a minor role in defining the overall behaviour, though certain unique
aspects, such as the Lifshitz transition in La2-xSrxCuO4 (LSCO) or the CuO
chains in YBa2Cu3O7-δ (Y123) or YBa2Cu4O8 (Y124), need to be fully taken into
account before a complete picture can emerge. In this experimental survey, we
focus predominantly on the single-layered cuprates Tl2Ba2CuO6+δ (Tl2201) and
Bi2+z-yPbySr2-x-zLaxCuO6+δ (Bi2201) on which the bulk of our recent high-field
transport studies have been performed, though for completeness, we also
include complementary data obtained on LSCO, Y123, Y124 and single-layered
HgBa2CuO6+δ (Hg1201). Moreover, in order to aid subsequent discussions, we
redefine the phase diagram in Figure 1A in terms of the superfluid density
$n_{s}(0)$, rather than $T_{c}$. In this way, the SC dome can be divided
simply into two regimes – pseudogapped (PG) and strange metal (SM) – rather
than the usual three (underdoped, optimally doped and overdoped).

The growth of $n_{s}(0)$ with increasing $p$ inside the PG regime – known as
the Uemura relation – reflects the fact that the pseudogap itself is states-
non-conserving, i.e. that states lost through the opening of the pseudogap
(preferentially near the Brillouin zone edges) are not recovered below $T_{c}$
[1]. The reduction in $n_{s}(0)$ beyond $p^{*}$, on the other hand, has
commonly been attributed to pair breaking – the so-called dirty $d$-wave
scenario – though recent studies on LSCO thin films have challenged this
association [2, 3]. Robust correlations between $n_{s}(0)$ and certain key
transport coefficients – highlighted in Figure 1 and to be addressed in turn
in the following sections – provide further clues as to the origin of this
loss of condensate, with the nature of the carriers within the SM regime and
proximity to the Mott state at half-filling playing pivotal roles.

###  2.1 Zero-field Resistivity

In considering the in-plane resistivity $\rho_{ab}(T)$, we find it more
insightful to describe the doping evolution in reverse, i.e. with decreasing
$p$, starting from the far overdoped side. Beyond the SC dome ($p>p_{SC}$
where $p_{SC}$ $\sim 0.30$ is the doping at which superconductivity vanishes
on the overdoped side), $\rho_{ab}(T)$ displays a pure $T^{2}$ resistivity [4,
5] indicative of a correlated FL ground state. As $p$ is reduced below
$p_{SC}$, however, a finite $T$-linear term emerges with a coefficient $A_{1}$
that grows with decreasing $p$ [6]. The striking correlation between $A_{1}$
and $n_{s}(0)$ [7] within the SM regime – captured in panels A and B of Fig. 1
– is reminiscent of the correlation found in conventional BCS superconductors
between $T_{c}$ and $\lambda_{tr}$, the transport scattering rate linked to
the electron-phonon coupling strength $\lambda$ and the slope of the
$T$-linear resistivity (at intermediate temperatures). The fact that a similar
correlation exists in both hole- and electron-doped cuprates has motivated a
prolonged and intense search for a corresponding $\lambda$ in high-$T_{c}$
cuprates. There is an important distinction to be made here, however. In
cuprates, the $T$-linear resistivity extends down to anomalously low $T$ (in
marked contrast to expectations for a FL) and persists over a broad doping
range (in marked contrast to expectations for a conventional quantum critical
metal [8]). Moreover, the $T$-dependence of $\rho_{ab}(T)$ in Bi2201 and LSCO
was recently shown to be identical for samples with the same $T_{c}$ (and
therefore comparable $p$) values [9], despite the fact that the $T$\- and
$p$-dependencies of the spin and charge fluctuation spectra of both families
are markedly different [10, 11, 12, 13]. This simple result profoundly
challenges claims that charge and/or spin fluctuations are responsible for the
strange metallic behaviour seen in cuprates. Others have linked the ubiquitous
nature of the low-$T$ $T$-linear resistivity to the vague concept of Planckian
dissipation [14] and a gradual loss of quasiparticle (QP) integrity across the
SM regime [15], though how this QP integrity is lost remains an open question.

Plotting the temperature derivative d$\rho_{ab}$/d$T$ reveals that the
evolution of $\rho_{ab}(T)$ across the SM regime is best interpreted as the
sum of two distinct ($\sim T$ and $T^{2}$) components [6, 9], rather than as a
single component with an exponent intermediate between 1 and 2, as found, for
example, in the heavy fermions. The $T^{2}$ component ($A_{2}$) – whose
evolution across the SC dome is also captured in Fig. 1B – is presumably a
continuation of the QP-QP scattering term found beyond $p_{SC}$. Hence, a
picture emerges in which the $T^{2}$ and $T$-linear components to
$\rho_{ab}(T)$ reflect, respectively, FL and non-FL contributions to the
charge transport whose relative ratio evolves smoothly across the SM regime.
Whether this duality reflects the presence of additive scattering rates, real-
space differentiated segments or independent conducting channels, remains to
be determined, though the former is difficult to reconcile with the evolution
of the in-plane magnetoresistance (MR) and low-$T$ Hall number as will be
outlined in the following sections.

Acknowledging the over-simplicity of a pure two-fluid picture, one can
nevertheless speculate as to which of these two fluids is the main contributor
to the SC condensate. The growth in $A_{1}$ with decreasing $p$ must reflect
either enhanced scattering from the non-FL component (the corresponding
$\lambda$) or an increased occupation of the non-QP sector. The striking
correlation shown in Fig. 1 between $A_{1}$ (a quantity determined both by
scattering rate and carrier number) and $n_{s}(0)$ (a carrier number) suggests
strongly that it is the latter. Upon entering the PG regime, $A_{1}$ drops
(almost precipitously) while $A_{2}$ increases sharply. This anti-correlation
of the two coefficients implies that the states responsible for the SM
component are the ones that are preferentially gapped out by the opening of
the pseudogap (at the zone edges). The growth in $A_{2}$ then reflects the
increasing dominance of the QP component to the total conductivity. The
overall evolution of the two coefficients indicates that upon crossing
$p^{\ast}$, only part of the ‘strange’ sector is gapped out initially. Upon
further reduction in $p$, the strange fluid shrinks further and faster than
the reduction of the residual ‘normal’ fluid, leaving the latter once again as
the dominant component (as it was for $p\sim p_{SC}$). The corresponding
reduction in $n_{s}(0)$ with decreasing $p$ across the PG regime appears then
to tie the superfluid density to that component of the normal state exhibiting
signatures of non-QP transport. In order to shed more light on the nature of
this non-QP sector, we turn now to consider the evolution of the in-plane MR
and Hall number across the phase diagram.

###  2.2 Magnetoresistance

The MR of most simple metals can be well described by conventional Boltzmann
transport theory and follows Kohler scaling, whereby $\Delta\rho/\rho(0)$ –
the fractional change of the resistivity in a magnetic field – is proportional
to $(H/\rho(0))^{2}$ and where $\rho(0)$ is the corresponding zero-field
resistivity. Adherence to this scaling indicates that $\Delta\rho(H,T)$ and
$\rho(0,T)$ are governed by the same scattering rates. In cuprates, Kohler
scaling is observed only beyond the SC dome [16]. Within the SM regime,
however, the in-plane MR adopts an entirely different form of scaling – so-
called quadrature or $H/T$ scaling – that has also been observed in certain
iron-based superconductors close to their respective quantum critical points
[17, 18]. Specifically, the $H^{2}$ dependence at low fields transforms into
an unsaturating $H$-linear dependence with a $T$-independent slope. At the
same time, the coefficient ($B$) of the quadratic, low-field $H^{2}$ MR is
found to exhibit a pure 1/$T$ power-law dependence, in marked contrast to the
1/$\rho(T)$ dependence one expects from Kohler scaling. Finally, in Tl2201 and
Bi2201, the MR response is found to be independent of both field-orientation
and impurity scattering [19]. While the $H^{2}$ to $H$-linear crossover can be
reproduced within a modified Boltzmann treatment through the introduction of
an impedance to cyclotron motion somewhere on the Fermi surface [20, 21, 22,
23], all other aspects of the MR response represent a radical departure from
the standard semi-classical picture.

The fact that the MR scales with $1/T$ (and not with
$1/\rho(T)=1/(\rho_{0}+\alpha_{1}T+\alpha_{2}T^{2})$) within the SM regime
favours the two-fluid description introduced above. While $\rho(T)$ is
composed of two distinct $T$-dependent components, the MR response appears to
be associated only with those carriers undergoing linear-in-$T$ scattering or
dissipation. The observed correlation [16] between $\alpha_{1}$ and the
magnitude of the $H$-linear MR further supports this singular association. The
carriers responsible for the QP-like ($\rho_{0}+\alpha_{2}T^{2}$) part of the
zero-field resistivity, on the other hand, appear to play no role in the MR
scaling. Of course, one cannot exclude a small orbital MR of the form
predicted by Boltzmann transport in Bi2201 or Tl2201 [19], but this appears to
be dwarfed completely by the quadrature component.

Across $p^{\ast}$, the $H$-dependence of the MR remains qualitatively
unchanged. The magnitude of the $H$-linear slope continues to increase with
decreasing $p$, though now its continued rise appears to be tracking the loss
of carriers inside the PG regime, as reflected in the growth of $\rho_{0}$
and, by correspondence, $\alpha_{2}$ (Fig. 1C). Single-power law scaling is
still observed, albeit with an exponent that changes abruptly across
$p^{\ast}$ – from $H/T$ to $H/T^{2}$ [16] – mirroring the crossover to a
dominant quadratic $T$-dependence in $\rho_{ab}(T)$ [9]. The sharpness of this
crossover is reminiscent of the discontinuous collapse of the anti-nodal QP
peak across $p^{\ast}$ seen by ARPES [24, 25]. It should be noted that
although a return to quadratic resistivity suggests a recovery of conventional
FL behaviour within the PG regime, the power-law scaling and in particular the
absence of the residual resistivity component $\rho_{0}$ in the scaling of the
MR is highly anomalous. It should also be stressed that such scaling is wholly
incompatible with the notion of a single fluid with multiple scattering rates.

###  2.3 Hall Number

The Hall number $n_{\rm H}$, derived from measurements of the in-plane Hall
coefficient $R_{\rm H}$, serves as a convenient and informative quantifier of
the mobile carrier density. For an isotropic, single-band metal $R_{\rm
H}=V/n_{\rm H}e$, where $V$ is the unit cell volume and $e$ the electronic
charge. In lightly-doped LSCO ($x<0.08$) [26], $R_{\rm H}$ is approximately
$T$-independent below 300 K with a value set by $n_{\rm H}=p=x$ (see Fig. 1B),
indicating that initially, only the doped holes are mobile. In single particle
probes, the opening of the pseudogap at $p^{*}$ is characterised by the
formation of discontinuous Fermi arcs, preferentially located near the zone
diagonals, whose arc-length also scales with $p$. Remarkably, the arcs
themselves continue to host coherent quasiparticles, as evidenced by the
observation of quantum oscillations [27]). At the other end of the SC dome,
$n_{\rm H}(0)=1+p$, where $n_{\rm H}(0)$ is the effective carrier number
obtained from $R_{\rm H}$ in the high-field, low-$T$ limit once
superconductivity has been suppressed and any $T$-dependent in-plane
anisotropy [28] washed out [15]. The value of $n_{\rm H}(0)$ in highly
overdoped Tl2201 is consistent with quantum oscillations [29] and ARPES [30].
Thus, at this end of the SM regime, the hole that was originally localized in
the Mott parent state becomes fully itinerant.

The crossover in $n_{\rm H}(0)$ from $p$ to 1 + $p$ was originally observed in
Y123 and reported to be sharp and confined to a narrow doping region around
$p^{\ast}$ [31]. In Bi2201 and Tl2201, however, this crossover was found to
occur over a much broader doping range between $p^{*}$ and $p_{SC}$ (see Fig.
1B). While the role of the conducting CuO chains in modifying this crossover
in Y123 remains to be clarified, the claim that the crossover in Tl2201 occurs
beyond $p^{\ast}$ is strengthened by the observation that it coincides with a
reduction in $n_{s}(0)$ with increasing not decreasing $p$ [32]. The fact that
$n_{\rm H}(0)$ decreases upon lowering $p$ before the PG itself opens suggests
a profound disconnect between single-particle and particle-particle probes
within the SM regime, possibly reflecting an increasing influence of current
vertex corrections, though the coincidence of the reduction in $n_{\rm H}(0)$
and the growth in $A_{1}$ appears to support the aforementioned picture of a
gradual and preemptive loss of QP integrity with underdoping.

The anti-correlation between $n_{\rm H}(0)$ and $n_{s}(0)$ across the SM
regime (panels A and C of Fig. 1) is particularly striking and non-intuitive.
In a clean BCS superconductor, all carriers are expected to participate in the
SC condensate. As mentioned in the Introduction, the prevailing explanation
for the reduction in $n_{s}(0)$ is pair breaking induced by disorder coupled
with a rapidly diminishing pairing amplitude $\Delta$. This conventional
viewpoint – based on the Landau BCS paradigm – fails to take into account,
however, the highly anomalous nature of the normal state [7]. Moreover, the
dirty $d$-wave scenario predicts a rapid reduction in $T_{c}$ as the normal
state scattering rate increases [33], in marked contrast with recent findings
[34]. Last but not least, a similar correlation, shown here in Fig. 1C for the
first time, is also found between $n_{\rm H}(0)$ and $F$ – the fraction of
residual (zero-energy) states in the SC state deduced from STM [35]. The
quantitative agreement between the two parameters in the same cuprate family
(Bi2201) suggests that the states contributing to $n_{\rm H}(0)$ in the field-
induced normal state do not form part of the SC condensate. Conversely, those
states that emerge from the SC condensate upon application of a large magnetic
field do not appear to exhibit an intrinsic Hall response of their own, due to
them being localized [36], bosonic (particle-hole symmetric) or, if still
fermionic, fundamentally incoherent.

##  3 Theoretical considerations

The picture emerging from this survey of recent high-field studies on hole-
doped cuprates is of a transport current comprising two electron fluids – one
coherent and FL-like (albeit correlated), the other non-FL-like and of unknown
origin – whose relative ratio evolves smoothly across the phase diagram.
Before discussing peculiarities in cuprates from a theoretical perspective, it
is worthwhile to start with a general remark on the coexistence of two types
of electronic liquids for interacting electrons in crystals. FL theory,
together with its microscopic justification [37, 38, 39] has long been
considered the ‘default’ many-body theory of conducting condensed matter.
Recently, the concept of a ‘non-particle’ state was developed, mostly within
the language of holographic duality [40, 41], which is argued to be a more
appropriate framework to describe cuprates and other strange metals [7]. In
both cases, the normal state of the system is described as a single liquid,
either as a FL or a non-FL. It is not obvious therefore whether the hypothesis
of a two-liquid state is consistent with general ideas of contemporary quantum
many-body theory. Being very far from a complete theoretical description of
the experimental situation, we are nevertheless able to give a positive answer
to this question. To this aim, we will first make a general remark and then
give two examples where the coexistence of two liquids in the same many-body
system is reliably established.

We proceed by discussing the emerging physics in terms of the single-particle
electron Green’s function [37, 38, 39]. To avoid possible misunderstanding,
one should emphasise that this formalism contains full information on the
many-particle system. In particular, it is sufficient to reproduce rigorously
all equilibrium [42, 43] and non-equilibrium [44] properties of interacting
fermionic systems. The simplest analysis can be done in terms of single-
particle density matrix (let us stress again that, despite its ‘single-
particle’ name, it characterizes the whole many-particle system). One can
derive the following general expression [45] for the accelerating action of a
constant uniform electric field $\mathbf{E}$:

$\frac{dj_{a}}{dt}=e^{2}E_{b}\sum_{\mathbf{k},n}m^{-1}_{ab}(\mathbf{k},n)\rho_{n}(\mathbf{k}),$
(1)

where $\mathbf{j}$ is the electric current, $t$ the time, $e$ the electron
charge and $a,b$ are Cartesian indices,

$m^{-1}_{ab}(\mathbf{k},n)=\hbar^{-2}\frac{\partial^{2}}{\partial
k_{a}\partial
k_{b}}\left\langle\mathbf{k}n\left|\hat{H}_{0}\right|\mathbf{k}n\right\rangle$
(2)

plays the role of inverse effective mass tensor,
$\left|\mathbf{k}n\right\rangle$ and $\rho_{n}(\mathbf{k})$ are eigenfunctions
and eigenvalues of the single-particle density matrix $\hat{\rho}$ dependent
on the wave vector $\mathbf{k}$, $n$ labels different eigenstates and plays
the role of a ‘band index’ in the many-body description of crystalline solids
[45], $\hat{H}_{0}$ is the single-particle part of the Hamiltonian (we assume
that the interaction Hamiltonian depends on coordinates only), and the
$\mathbf{k}$-summation in Eq. (1) is taken over the Brillouin zone. The
equations (1), (2) can be considered as a real-time equivalent of the optical
f-sum rule frequently used to analyse strongly correlated systems [46].

The single-particle density matrix is nothing but the equal-time single-
particle Green’s function $\hat{G}$, and the latter, for energies close enough
to the Fermi surface, can be decomposed into QP (that is, pole) and non-QP
(incoherent) parts [37]. These two contributions to the density matrix
corresponds to two contributions to Eq. (1). Rigorously speaking, the QP and
non-QP parts of $\hat{\rho}$ do not necessarily commute one with one another
and thus do not share a common set of eigenfunctions, but at the level of the
operator $\hat{\rho}$, this decomposition is always possible and accurate.

Half-metallic ferromagnets [47] provide a neat example of a physical system in
which two types of electron liquids coexist naturally and rigorously. Whereas
for one spin projection, the electron subsystem is metallic and FL-like, for
the other, non-QP states in the gap dominate close to the Fermi surface, their
density of states being non-zero above the Fermi energy only for the minority-
spin gap and below it for the majority-spin gap [47]. Importantly, their
contribution to the occupation number $\rho_{n}(\mathbf{k})$ is almost
$\mathbf{k}$-independent and thus these states are effectively currentless
[45]. Nevertheless, they may contribute to photoemission and scanning probe
spectra, tunneling, etc. [47], as confirmed experimentally [48, 49].

The case of half-metallic ferromagnets is useful as it provides us with a
unique opportunity to prove the separation of electron liquids into QP and
non-QP (or incoherent) components. Moreover, this separation is rigorous due
to its connection to conserving quantities such as the total spin (we assume
that spin-orbit effects are negligible). Unsurprisingly, the situation in
cuprates is much more complicated however, and we need to find a closer
analog, even if its consideration cannot be done with the same rigour. In this
regard, the heavy fermion (or ‘Kondo lattice’) systems [50, 51] provide a
useful bridge. In these systems, the 4f or 5f electrons are mostly atomic-like
and form local magnetic moments but in some regime they become itinerant,
contributing to the shape of the Fermi surface, charge transport etc. In some
sense, these states can be considered as ‘Abrikosov pseudofermions’ – usually
a purely mathematical construction introduced as a way to represent local spin
operators – that hybridize with the true conduction-band fermions and become
real [52]. Therefore one needs to distinguish between large and small Fermi
surface constructions with and without their contribution [51].

With this in mind, let us now return to the cuprate problem. The minimal model
that is supposed to keep the essential elements of cuprate physics is the
single-band $t-t^{\prime}$ Hubbard model derived via mapping of density-
functional computational results for various representatives of this family
[53, 54]. The corresponding Hamiltonian has the form

$H=-\sum_{ij}t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+\sum_{i}Un_{i\uparrow}n_{i\downarrow}$
(3)

where $i,j$ are sites belonging to the simple square lattice, $t_{ij}$ is an
effective hopping that is supposed to be non-zero only for nearest- ($t$) and
next-nearest-neighbours ($t^{\prime}$) and $U$ is the on-site Coulomb
interaction parameter. Operators $c_{i\sigma}^{\dagger}$ ($c_{i\sigma}$)
create (annihilate) fermions at site $i$ with spin
$\sigma=\uparrow,\downarrow$ and
$n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$. The ratio $t^{\prime}/t$ is
about 0.3 for the most of cuprates, with the exception of LSCO where it is
closer to 0.15. Empirically, $T_{c}$ grows with the value of $t^{\prime}/t$
[54]. The Hubbard parameter $U$ is comparable with the bandwidth, so we deal
with the most theoretically demanding case of intermediately strong
correlations.

Arguably the simplest approach to correlated electrons is dynamical mean-field
theory (DMFT) where the lattice problem is mapped onto a single-site effective
impurity problem with self-consistent determined characteristics [55]. For
cuprates, however, the minimal DMFT model should start with a 2 x 2 plaquette
rather than a single site since the $d$-wave SC order parameter lives on bonds
rather than on sites [56]. There have been an enormous number of cluster DMFT
calculations, the corresponding references for which can be found in Ref. [57,
58]. Rather than discussing numbers, we focus here on an interesting
observation [59]; that for the values of $t^{\prime}/t$ near 0.3 and a
realistic value of $U$, the ground state of the minimal plaquette is
degenerate, i.e. the ground states for two, three and four electrons on the
plaquette coincide. As a result, the plaquette Green’s function has poles at
energies close to zero which may be a physical realization of the soft hidden
fermion mode suggested earlier phenomenologically [60]. This makes the system
in some sense analogous to the Kondo lattice, and this degeneracy plays a role
similar to spin degeneracy in the conventional Kondo problem. In particular,
making the lattice from such resonant centers naturally explains the
appearance of the pseudogap [59, 57].

In the context of the two-fluid concept discussed here, the most important
theoretical observation was made in Ref. [58] at the numerically exact
solution of the many-body problem for a larger, $4\times 4$ size plaquette. It
turns out that the binding energy of two holes per plaquette is very strongly
dependent on the ratio $t^{\prime}/t$ and can reach values about $0.8t\approx
0.3$ eV which is more than an order of magnitude larger than $T_{c}$, as shown
explicitly in Fig. 2. This means that, at least, for some range of doping,
holes build incoherent pairs already in the normal phase. Despite the holes
being initially expected to form something similar to a charged Bose liquid,
one can suppose that the interaction with the thermal bath makes this liquid
rather more classical than of Bose-Einstein character [61].

Thus, in general one can propose the following phenomenological picture. The
coherent or QP part of the fermionic spectrum is mostly responsible for
electron transport in the normal state and completely responsible for its
coherent manifestations such as quantum oscillations. The incoherent part, on
the other hand, may be associated with the formation of two-hole bound states
(one could call them bipolarons but we prefer to avoid any associations with
phonons since they do not appear to be relevant in this simplified picture).
At high enough temperatures, the Fermi liquid of coherent electron excitations
acts as a dissipative environment for the ‘bipolarons’ preventing their Bose-
Einstein condensation. The latter can nevertheless occur as the temperature
decreases leading ultimately to superconductivity, while the coherent part
remains largely untouched. While this picture provides a natural explanation
for the presence of the two fluids, it remains to be explored how this
incoherent part can play such a prominent role in the magnetic field response
[19]. This will be the subject of future studies.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

All authors listed have made a substantial, direct, and intellectual
contribution to the work and approved it for publication.

## Funding

This work was supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme, grant
agreement nos 835279-CATCH-22 (N.E.H.) and 854843-FASTCORR (M.I.K). NEH also
acknowledges the support of EPSRC (grant ref. EP/V02986X/1). Finally, J.A.
also acknowledges the support of an EPSRC Doctoral Prize Fellowship (Ref.
EP/T517872/1).

## Acknowledgments

We thank Milan Allan and Alexander Lichtenstein for inspiring discussions.

## Data Availability Statement

The datasets presented in this study are tabulated in the Supplementary
Information. Further inquiries can be directed to the corresponding author.

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Figure 1: Correlations between superfluid density $n_{s}(0)$ and various
transport properties of hole-doped cuprates. (A) Evolution of $n_{s}(0)$ (per
Cu) with doping $p$ in Y123 (orange crosses) and Tl2201 (red squares) obtained
from measurements of the zero-temperature in-plane penetration depth
$\lambda_{ab}(0)$ (summarized in Ref. [62, 32]. In order to deduce $n_{s}(0)$
in Y123, we used $m^{\ast}$ values obtained from high-field quantum
oscillation studies over the doping range 0.10 $\leq p\leq$ 0.15 [63] and
extrapolated to $p^{\ast}$ = 0.19 assuming that $n_{s}(0)$ approaches unity
there. For a description of how $n_{s}(0)$ is obtained for Tl2201, please
refer to Appendix A in Ref. [32]. All parameters used in these determinations
are listed in the Supplementary Material. Note that the peak in $n_{s}(0)$ at
the pseudogap endpoint $p^{\ast}$ (vertical dashed line) delineates the
pseudogapped (PG) regime, shaded in orange, from the strange metal (SM)
regime, shaded in blue. (B) $p$-dependence of $A_{1}=N\alpha_{1}/d$ (right
axis, solid points) and $A_{2}=N\alpha_{2}/d$ (left axis, open points), where
$d$ is the $c$-axis lattice constant, $N$ is the number of CuO2 planes per
unit cell and $\alpha_{1}$, $\alpha_{2}$ are coefficients of the low-$T$ zero-
field resistivity $\rho(T)=\alpha_{1}T+\alpha_{2}T^{2}$. Open crosses
(plusses) indicate data for Y123 (Y124), respectively. See Supplementary
Material for the precise values and references related to these data points.
(C) Comparison of $n_{\rm{H}}(0)-p$, the $p$-dependence of the as-measured
low-$T$ Hall carrier number of Tl2201 and Bi2201 minus its corresponding $p$
value (left axis, closed points [15]) and $F$, the fractional filling of the
superconducting gap as measured recently by STM on Bi2201 (right axis, open
points [35]). Shaded columns are histograms representing the local variations
in the measured filling (as in Ref. [35]). In each panel, coloured dashed
lines are guides to the eye. Figure 2: Pairing energy $\Delta_{2h}$ of two
holes in a 4$\times$4 cluster with periodic boundary condition as a function
of $U$ and $t^{\prime}$. All energies are in units of $t\approx 0.35$ eV.
Reproduced with permission from Ref. [58]