# On the dynamic distinguishability of nodal quasi-particles in overdoped
cuprates

Kamran Behnia  LPEM (CNRS-Sorbonne University), ESPCI Paris, PSL University,
75005 Paris, France

###### Abstract

La1.67Sr0.33CuO4 is not a superconductor and its resistivity follows a purely
T2 temperature dependence at very low temperatures. La1.71Sr0.29CuO4, on the
other hand, has a superconducting ground state together with a T-Linear term
in its resistivity. The concomitant emergence of these two features below a
critical doping is mystifying. Here, I notice that the electron-electron
collision rate in the Fermi liquid above the doping threshold is unusually
large. The scattering time of nodal quasi-particles expressed in a
dimensionless parameter $\zeta$ is very close to what has been found in liquid
3He at its melting pressure. In the latter case, fermionic particles become
dynamically distinguishable by excess of interaction. Ceasing to be
dynamically indistinguishable, nodal electrons will be excluded from the Fermi
sea. Such non-degenerate carriers will then scatter the degenerate ones within
a phase space growing linearly with temperature.

##  I Introduction

Elementary particles of a quantum fluid are indistinguishable. Leggett [1, 2]
argued that it is thanks to this indistinguishibality that such fluids are
governed by quantum statistics [and not only quantum mechanics]. Trachenko and
Zaconne [3, 4] recently highlighted the dynamical aspect of this
indistinguishibality and used it as a departing point to explore the boundary
between the statistics-active and the statistics-inactive regimes of quantum
fluids.

The normal state of cuprate superconductors is nowadays called a ‘strange
metal’. The expression refers to the puzzling temperature dependence of their
electrical resistivity (for recent reviews, see [5] and [6]). The focus of the
present paper is a very specific point of the cuprate phase diagram. In hole-
doped cuprates, the superconducting dome ends when doping level exceeds a
threshold of $p\approx 0.3$. Hussey and collaborators carried out an extensive
study of the evolution of resistivity in La1-xSrxCuO4 [7]. They found that the
superconducting dome and strange metallicity emerge concomitantly when
$x<0.3$. La1.67Sr0.33CuO4 is not superconducting and its resistivity follows
T2 [8], but La1.71Sr0.29CuO4 is a superconductor and its resistivity does not
correspond to what is expected for a Fermi liquid. Instead, it contains a
T-Linear term [7] (Figure 1). This observation is not exclusive to overdoped
cuprates. Taillefer pointed out that the normal-state T-linear scattering and
the onset of superconducting instability are linked in several other families
of superconductors other than cuprates [9]. Greene and collaborators found
that in electron-doped La2-xCexCuO4(LCCO), T-square superconductivity emerges
only upon the destruction of superconductivity by overdoping [10].

Figure 1: The puzzle: Zoom on the cuprate phase diagram near the end of the
superconducting dome. Hussey and his co-workers [7] found that below a
threshold doping level, the system has a superconducting ground state and a
resistivity which follows $\rho=\rho_{0}+A_{1}T+A_{2}T^{2}$. Above this
threshold, the system is not a superconductor and resistivity can be fit with
a purely quadratic temperature-dependnet term: $\rho=\rho_{0}+A_{2}T^{2}$.

In this paper, I argue that the notion of dynamic distinguishibality [4]
illuminates the birth of a ’strange metal’ at this locus of the phase diagram.
The argument is based on scrutinizing the amplitude of T-square resistivity in
the Fermi liquid La1.67Sr0.33CuO4, by comparing it with other Fermi liquids,
and by recalling the fate of fermion-fermion collisions when 3He solidifies
[11, 12, 13, 14].

Figure 2: Standing out among Fermi liquids: a) The magnitude of the prefactor
of T-square resistivity, $A$, vs. the Sommerfeld coefficient, $\gamma$, of
several metallic perovskites. SrVO3 [15], Sr2RuO4 [16], Sr3Ru2O7 (at zero
magnetic field [17, 18]), and La1.95Sr0.05TiO3 [19] all follow Kadowaki-Woods
scaling. In contrast, the magnitude of $A$ in La1.67Sr0.33CuO4 [8, 7] is five
times larger than what is expected given its $\gamma$. b) $\ell_{quad}$,
derived from the amplitude of $A$ and fundamental constants (see text) in
different Fermi liquids [20]. The large magnitude of $\ell_{quad}$ in
La1.67Sr0.33CuO4 stands out.

##  II Heavily-doped LSCO stands out among Fermi liquids

La1.67Sr0.33CuO4 is a Fermi liquid, but not a common one. This can be seen by
comparing its T-square resistivity with other metallic oxides. In perovskyte
family, a variety of instabilities lead to metal-insulator transitions [21]
and a metallic ground state is rare.

Let us pick up several exceptions. SrVO3 is a correlated metal with vanadium
in $3d^{1}$ configuration, which remains metallic when Sr is replaced by co-
valent Ca [15]. SrTiO3 is a band insulator and LaTiO3 a Mott insulator, but
Sr1-xLaxTiO3 alloys are metallic [19]. Specifically, Sr0.05La0.095TiO3 is a
dense metal with almost one electron per formula unit [19]. Sr2RuO4 is an
unconventional superconductor with a Fermi liquid normal state above its
critical temperature [22]. Sr3Ru2O7 has a non-trivial electronic instability
at 7.8 T, but is a correlated Fermi liquid at zero magnetic field [17, 18].
The feature they all share with La1.67Sr0.33CuO4 is being a dense metallic
perovskyte. None of them, however, is a strange metal or become a high-
temperature superconductor.

Fig. 2a shows the amplitude of the prefactor of T-square resistivity $A$ as a
function of the Sommerfeld coefficient (the electronic T-linear specific
heat), $\gamma$ in these metals. This Kadowaki-Woods (KW) plot [23] reveals an
anomaly. In correlated metals, the prefactor of T-square resistivity scales
with the square of $\gamma$ over five orders of magnitude [24]. This scaling
is operative when there is roughly one electron per formula unit [25]). As one
can see in Fig. 2a, heavily-doped LSCO does not follow the trend observed in
other metallic perovskytes. Its T-square resistivity is more than five times
larger than it should be, given the magnitude of its $\gamma$. Interestingly,
this is also the case of heavily overdoped electron-doped cuprates 111R.L.
Greene, private communication.

The unusually large amplitude of $A$ in La1.67Sr0.33CuO4 betrays itself in a
comparison of all Fermi liquids. The Kadowaki-Woods scaling can be extended to
dilute metals by plotting $A$ as a function of the Fermi energy, $E_{F}$ [20,
27, 25]. Due to Pauli exclusion principle, the phase space of scattering among
fermions is proportional to $(\frac{k_{B}T}{E_{F}})^{2}$. Dimensional
considerations imply [20]:

$A=\frac{\hbar}{e^{2}}(\frac{k_{B}}{E_{F}})^{2}\ell_{quad}$ (1)

Here $\hbar$ is the reduced Planck constant and $e$ is the fundamental charge.
$\ell_{quad}$ is a phenomenological material-dependent length scale. A survey
of available data shows that for all known Fermi liquids $\ell_{quad}$ is
between 1 to 50 nm [20, 27]. As one can see in Fig. 2b, in La1.67Sr0.33CuO4,
$\ell_{quad}\approx 240$ nm. Decidedly, this Fermi liquid is not a banal one.
The unusually large $A$ of this metal, given its density of states and its
degeneracy temperature is the first step for understanding its transformation
to a strange metal upon the removal of dopants.

To see the significance of this, let us consider the case of normal liquid
3He.

##  III Distinguishibality on the verge of solidification in 3He

Figure 3: The case of normal liquid 3He : a) The pressure dependence of the
effective mass in 3He quantified by measuring the T-linear specific heat [12].
b) The quasi-particle scattering time extracted from thermal conductivity
multiplied by T2 as a function of pressure, according to Wheatley [11] and
Greywall [13]. c) The pressure dependence of the Fermi energy using the Fermi
momentum and the Fermi velocity given by Greywall [13]. d) The pressure
dependence of the inverse of the normalised quasiparticle lifetime using
Greywall’s data (see text). Black arrows show the pressure at which
solidification occurs.

Under their own vapor pressure, the two isotopes of helium do not solidify
down to zero temperature. These quantum liquids [1, 2] become quantum solids
upon the application of pressure. With an odd number of protons, neutrons and
electrons, a 3He atom is a composite fermion. The molar volume changes from
36.84 cm3/mol at zero pressure to 25.5 cm3/mol at p=3.4 MPa, when it
solidifies. This is twice larger than what is classically expected (12
cm3/mol) and is a consequence of the large zero-point motion of the atoms in
the crystal [14].

The temperature dependence of thermal conductivity and viscosity in normal
liquid 3He at different pressures have been carefully measured by several
authors. The quasi-particle scattering time extracted from these studies,
broadly consistent with each other, were reviewed in detail by Dobbs [14]. The
scattering time derived from thermal conductivity, $\tau_{\kappa}$ displays a
T-2 behavior in the zero-temperature limit [11, 13], as expected for a Fermi
liquid.

With increasing pressure, interaction between the atoms, consisting
principally of a strong hard-core repulsion and a weak van der Waals
attraction, intensifies. This leads to an amplification of the effective mass,
quantified by measurements of specific heat [12] (Fig. 3a). Fig. 3b shows the
evolution of $\tau_{\kappa}T^{2}$ (in the zero-temperature limit) as a
function of pressure reported by Wheatley [11] and by Greywall [13]. The trend
is similar, but Greywall’s values are about 25 percent lower than Wheatley’s.
The highest pressure (3.44 MPa) corresponds to the onset of solidification. By
this pressure, the time between two fermion-fermion collisions has decreased
by a factor of almost 3.

Several theoretical studies have examined the evolution of the effective mass
and the Landau parameters by pressure. Vollhardt, Wölfle and Anderson [28]
employed a Hubbard lattice-gas model with a variable density of particles to
describe the pressure dependence of thermodynamic properties of normal liquid
3He. Pfitzner and Wölfle [29, 30] gave a reasonable account of transport
coefficients under pressure. According to these studies, normal liquid 3He is
a strongly interacting and almost localized Fermi liquid [31].

What will be scrutinized here are the amplitudes of $\tau_{\kappa}$T2 and the
Fermi energy, $E_{F}$ on the verge of solidification. According to Vollhardt
and Wölfle [32], the quasi-particle lifetime on the Fermi surface is given by:

$\tau^{0}_{N}=\frac{64}{\pi^{3}}\frac{\hbar E_{F}}{(k_{B}T)^{2}}\langle
W\rangle^{-1}_{a}$ (2)

$\langle W\rangle_{a}$ is the angular average of the transition probabilities
between spin singlet and spin triplet states [33, 32, 14]. Multiplying
$\tau_{\kappa}T^{2}$ by $\frac{k_{B}^{2}}{\hbar E_{F}}$ will yield a
dimensionless number inversely proportional to the fermion-fermion collision
strength.

Figure 4: Discernibility and indiscernibility in 3He : In liquid 3He atoms are
indistinguishable, but in solid 3He, they are confined to specific sites in
real space. This makes them distinguishable. In the solid near the melting
transition, atoms wander around, thanks to their zero-point motion and can
exchange their places with their neighbors. On the other hand, in the liquid,
collisions in real space confine atoms to a restricted neighbourhood. The
liquid solidifies when collisions confine each atom to a spatial neighborhood.

Fig. 3c shows the evolution of the Fermi energy with pressure reconstructed
from Greywall’s data [13]. Combining this with $\tau_{\kappa}T^{2}$ leads to
Fig. 3d, which shows the pressure dependence of the normalized scattering
time:

$\zeta=\frac{\hbar E_{F}}{\tau_{\kappa}(k_{B}T)^{2}}$ (3)

Solidification occurs when this number becomes as large as 60. This implies a
huge collision rate between quasi-particles at the verge of solidification.
Inserting this number in Eq. 2 leads to the conclusion that at the onset of
solidification $\langle W\rangle_{a}\approx 140$. Surprisingly, this
remarkably large value has not been hitherto explicitly noticed, let alone
commented.

The value of $\langle W\rangle_{a}$ (or $\zeta$) is set by the magnitude of
dimensionless Landau parameters of the Fermi liquid, which are denoted by
$F_{l}^{s}$ (spin symmetric) and $F_{l}^{a}$ (spin antisymmetric) [30, 32].
Practically, Landau parameters with $l<2$ are the ones which matter and the
higher order ones may be safely neglected [34, 32]. Vollhardt and Wölfle [32]
have used experimental data to calculate the evolution of Landau parameters
with pressure and have found that at the threshold of solidification
($P=3.4MPa$) $F_{0}^{s}=88.47$, $F_{0}^{a}=-0.753$, and $F_{1}^{s}=14.56$.
These are large numbers. The Fermi liquid picture still holds, albeit
restricted to a very narrow temperature, at the onset of solidification and
its Landau parameters are large, yet finite.

Let us first note that in standard theories of Mott transition [35], Landau
parameters diverge. Let us also note that the mean-free-path of 3He atoms
remains much longer than their wavelength even on the verge of solidification
. According to Greywall’s data, at P=3.4MPa and T=0.01 K,
$\tau_{\kappa}=1.43ns$, $v_{F}=32.4m/s$, $\ell=42.6nm$ and $k_{F}=8.9nm^{-1}$.
Since $k_{F}\ell\gg 1$, Anderson localization is not what drives confinement
in space.

I conjecture that what drives solidification is the impossibility for Landau
parameters to become arbitrarily large. An infinite $F_{0}^{s}$, for example,
would make the liquid incompressible, which is implausible. Vollhardt and
Wölfle [32] highlighted the fact that normal liquid 3He is much less
compressible than a non-interacting Fermi liquid, thanks to its large
$F_{0}^{s}$. On the other hand, this interacting liquid is not more
incompressible than solid 3He. Indeed, according to the experiment [36], at
the melting pressure, the compressibility of the two phases are nearly equal.
Now, the compressibility of solid is set by its phonon spectrum and, one may
suspect, this is what sets the bound to $F_{0}^{s}$ in the liquid. After all,
Landau parameters are two-particle correlators [35] and atoms of a quantum
liquid have a finite coordination number [37]. Therefore, two-particle
correlations in real space cannot attain an arbitrarily large magnitude.

Solid 3He is not subject to Fermi-Dirac statistics, because each atom is
confined to the neighborhood of a designated site [38]. Near the melting
pressure, in the solid state, neighbouring atoms can frequently exchange
theirs sites thanks to their zero-point motion. Each atom frequently
encounters its first neighbours far from its lattice site [39, 40], but there
is still a one-to-one correspondence between an atom and its designated site.
In contrast to the solid, the atoms of the liquid are indistinguishable (See
Fig. 4). Collisions allow an atom in the liquid state to ‘observe’ its
neighbours in real space [41]. As these collisions multiply, the atom cannot
avoid being confined to a specific and distinguishable volume of the real
space.

The case of 3He illustrates that the time between two successive fermion-
fermion collisions can become unbearably short for the survival of a Fermi
liquid, despite a long mean-free-path and $k_{F}\ell\gg 1$. Let us now return
to cuprates.

##  IV Nodal quasiparticles: the unbearable shortness of being

What we saw in the case of 3He raises a question: Given the unusually large
amplitude of the T-square resistivity in heavily doped LSCO, how short is the
normalized quasi-particle lifetime?

Fig. 5a shows the Fermi surface of of La1.68Sr0.32CuO4 according to a tight
binding model with nearest-neighbor hopping parameters chosen to fit the Fermi
surface seen by Angle-Resolved Photoemission Spectroscopy (ARPES). [42, 43,
44]. The radius of this Fermi surface has a modest angular variation. In
comparison, the angular variation of the Fermi velocity, $v_{F}$, is much
larger (see Fig. 5b). This is a consequence of the proximity of Fermi surface
and the Brillouin zone boundary along the anti-nodal direction. Because of the
van hove singularity, the evolution of the Fermi surface with doping differs
along nodal and anti-nodal orientations. As a consequence, the derivative of
Fermi energy in momentum space ($v_{F}$) has a strong angular dependence.

Figure 5: Angular variation of collision time in heavily overdoped LSCO: a)
Fermi surface and the Brillouin zone of La1.68Sr0.32CuO4; b) Angular variation
of the Fermi velocity; c) Angular variation of the scattering time derived
from resistivity times T2; d) Angular dependence of dimensionless $\zeta$
extracted from $\tau_{\rho}$T2 and $\frac{k_{B}^{2}}{\hbar E_{F}}$. Compare
the absolute value of the maximum for nodal orientations with what was seen in
the case of 3He on the verge of solidification.

Solving the Boltzmann equation, one finds a general expression for electric
conductivity [45]:

$\sigma=\frac{1}{4\pi^{3}}\frac{e^{2}}{\hbar}\int\tau\bm{v_{k}}\frac{\bm{v_{k}}}{v_{k}}\bm{dS_{F}}$
(4)

In the case of cubic symmetry, $\sigma$ is identical for the whole solid angle
and therefore:

$\sigma_{cub.}=\frac{1}{3\pi^{2}}\frac{e^{2}}{\hbar}\tau v_{F}k_{F}^{2}$ (5)

Note that $\tau$, $v_{F}$ and $k_{F}$ can be anisotropic. However, cubic
symmetry, by constraining $\sigma_{cub.}$ to be isotropic, restricts possible
profiles for $\tau(\theta,\phi)$, $v_{F}(\theta,\phi$) and
$k_{F}(\theta,\phi$).

Heavily overdoped LSCO has a tetragonal symmetry and a quasi-cylindrical Fermi
surface extending vertically along the whole Brillouin zone. In this case, the
in-plane electrical conductivity is constrained to be isotropic and equal to :

$\sigma_{tet.}=\frac{1}{2\pi c}\frac{e^{2}}{\hbar}\tau v_{F}k_{F}$ (6)

Here, $c$ is the lattice parameter along the c-axis. The experimentally
measured prefactor of in-plane T-square resistivity ($A=2.5n\Omega cmK^{-2}$
[8, 7]) is indeed isotropic in the basal plane. Using Eq. 6 and
$AT^{2}\equiv\sigma_{ee}^{-1}$, to quantify $\tau_{\rho}$ (the index refers to
the fact that the experimental probe used to extract this time scale is
electrical resistivity):

$\tau{{}_{\rho}}(\theta)T^{2}=\frac{\hbar}{e^{2}}\frac{2\pi
c}{Ak_{F}(\theta)v_{F}(\theta)}$ (7)

The Fermi velocity, $v_{F}$, the Fermi wave-vector, $k_{F}$ and $\tau_{\rho}$
have all their angular dependence. Since the anisotropies of $k_{F}$ and
$v_{F}$ do not cancel out, one expects a significant angle dependence of
$\tau_{\rho}T^{2}$. As seen in Fig. 5c, this is indeed the case. It is more
than twice smaller along the nodal orientation. Note that, because of the
large anisotropy of the Fermi velocity, the anisotropy of the mean-free-path
is the inverse of the anisotropy of the scattering time. Nodal quasi-particles
have a longer mean-free-path yet a shorter scattering time.

Combining $\tau_{\rho}T^{2}$ and the Fermi energy $E_{F}=5900K$ (extracted
from the magnitude of the T-linear electronic specific heat $\gamma$=6.9
mJ.mol-1.K-2 [8]) leads to the quantification of the dimensionless $\zeta$ and
its angular dependence (Fig. 5d). Along the nodal orientation, it becomes as
large as $\approx$60, close to what was found above in 3He on the verge of
solidification.

System | $k_{F}(nm^{-1})$ | m⋆/m0 |  $E_{F}$ (K) | $\zeta$
---|---|---|---|---
3He (p=0) | 7.9 | 2.8 | 1.8 | 35
3He (p=3.4MPa) | 8.9 | 5.8 | 1.1 | 60
La1.67Sr0.33CuO4 | 5.6 | 5 | 5900 | 24-61
UPt3 | 5 | 16-130 | 90 | $\approx 10$
Sr2RuO4 | 5 | 3.3-16 | 1800 | $\approx 16$
Sb | 0.8 | 0.07-1 | 1100 | $\approx 0.1$
SrTiO3-δ (n=4$\times 10^{17}cm^{-3}$) | 0.23 | 1.8 | 18 | $\approx 0.1$
Table 1: 3He and heavily-doped LSCO compared to heavy-fermion UPt3 [46, 47,
48], correlated oxide Sr2RuO4 [22], semi-metallic antimony [49, 50] and dilute
metallic strontium titanate [20]. Note the exceedingly large normalized
amplitude of $\zeta$ in LSCO. UPt3 and Sr2RuO4 despite their larger mass
enhancements have smaller $\zeta$s.

Let us recall that the microscopic interaction between fermions are very
different in the two cases. In metals, point-like electrons repulse each other
through screened Coulomb repulsion. In contrast, neutral 3He atoms interact
over a short range comparable to their hard-sphere radius and the interaction
has both attractive and repulsive components. Nevertheless, the strength of
fermion-fermion scattering rate ($\tau^{-1}$) can be quantified in both cases
by the dimensionless $\zeta$.

Only when there is a single Fermi surface and a unique Fermi temperature, the
amplitude of $\zeta$ is unambiguous. This is the case of heavily overdoped
LSCO, but not other Fermi liquids. Most often, they have multiple anisotropic
Fermi pockets. Assuming a single average Fermi energy, one can directly
extract from the T-square prefactor, the phenomenological length scale
$\ell_{quad}$, introduced first in ref.[20] and defined by Eq. 1, which is
proportional to $\zeta$ through a dimension-dependent length scale.

In a simplified one-band approximation, the total carrier density and the
measured $T$-linear specific heat yield an average Fermi energy for a given
system. In that case, for a three-dimensional metal with a spherical Fermi
surface, one has:

$\zeta_{3d}=\frac{2}{3\pi^{2}}\frac{e^{2}}{\hbar}\frac{A}{k_{B}^{2}}E_{F}^{2}k_{F}$
(8)

For a two-dimensional metal with a cylindrical Fermi surface, the expression
becomes:

$\zeta_{2d}=\frac{1}{3\pi c}\frac{e^{2}}{\hbar}\frac{A}{k_{B}^{2}}E_{F}^{2}$
(9)

These expressions, which neglect anisotropy and multiplicity of Fermi pockets,
can be cautiously used to extract the rough magnitude of $\zeta$ in various
metals.

##  V Comparison with other metals

Table 1 compares 3He and in heavily-doped LSCO with two weakly-interacting and
two strongly-interacting Fermi liquids. The list includes antimony [50],
dilute oxygen-reduced strontium titanate [51], the heavy-fermion metal UPt3
[46] and the correlated oxide Sr2RuO4 [22].

UPt3 displays a quadratic resistivity below 1.5 K with A=1.55$\mu\Omega$ cm
K-2 for in-plane charge flow [46]. It has five different Fermi surface pockets
with different sizes and effective masses [46, 47, 48]. The largest and the
most relevant sheet of the Fermi surface is the band 3 (‘Oysters and urchins’
[46]) giving rise to the $\omega$ orbit seen by quantum oscillations [48].
Table 1 uses the effective mass and the average radius of this sheet for rough
estimates of $k_{F}$ and $E_{F}$. The Fermi energy of 90 K is compatible with
an alternative estimation using the magnitude of the Sommerfeld coefficient
($\gamma=450J/K^{2}.mol$) and carrier density (1.4 $\times 10^{28}m^{-3}$) in
UPt3.

Resistivity in Sr2RuO4 is quadratic below 25 K with A=6 n$\Omega$ cm K-2 for
in-plane charge flow [16]. Its Fermi surface consists of three warped
cylinders [22] with radii ranging from 3.04 to 7.53 nm-1 and the effective
mass from 3.3 to 16 bare electron masses [22]. The table uses average values
for $k_{F}$ and $E_{F}$ for Sr2RuO4.

Antimony (Sb) displays a quadratic resitivity below 10 K with a prefactor of
0.8 n$\Omega$ cm K-2 [50]. Its Fermi surface has three electron pockets and a
single interconnected Fermi surface. The Fermi wave-vector along different
orientations varies by one order of magnitude [49]. The table gives an average
value for $k_{F}$ in antimony. The Fermi energy is almost the same for
electrons and holes.

SrTiO3 is a band insulator. It becomes a dilute metal with a very low carrier
density when a small amount of oxygen atoms are removed [52]. The resistivity
of this dilute metal follows a T-square resistivity with A=9 $\mu\Omega$ cm
K-2 when n=4$\times 10^{17}cm^{-3}$ [20]. At this carrier density, only a
single band is occupied and the Fermi surface seen by quantum oscillations can
be roughly approximated to a spherical one.

The experimentally resolved $A$ in set by the overall contribution of the
sheets of a multi-component Fermi surface. Therefore, the average values yield
only a rough estimate of $\zeta$. It can have different values for different
pockets. As seen in the table, according to this rough estimate, $\zeta>1$ in
strongly-correlated systems and $\zeta<1$ in the weakly correlated ones. The
maximum $\zeta$ is lower in Sr2RuO4 and in UPt3 than in LSCO (or in 3He),
despite their larger mass enhancement. Note also that while the magnitude of
$A$ in SrTiO3-δ and in Sb differ by four orders of magnitude, their $\zeta$ is
roughly similar.

Thus, $\zeta$ is unusually large in heavily overdoped LSCO. Given the angular
variation of $\zeta$, an eventual upper boundary will be encountered by nodal
quasi-particles before other electrons of the Fermi sea. Let us assume that an
exceedingly large fermion-fermion collision rate freezes the nodal quasi-
particles out of the Fermi sea.

##  VI Consequences of nodal freeze-out

What happens to the nodal quasi-particles once are excluded from the Fermi sea
is not known. However, let us assume that this happens below x=0.3 and
generates two categories of electrons 222For an alternative scenario referring
to two distinct (coherent and incoherent) charge sectors see [68]. Note the
very different nature of electron dichotomy in the two scenarios.: Those for
which quantum statistics is non-operative and others remaining in the Fermi
sea. Such a hypothesis will provide new possible solutions to a number of
longstanding puzzles. They are listed below:

  * •

Planckian dissipation: Bruin and co-workers noticed that in many metals with a
T-linear resistivity, the amplitude of scattering time is of the order of
$\tau_{P}=\frac{\hbar}{k_{B}T}$ [54]. Legros and co-workers [55] have reported
that this is indeed the case of overdoped cuprates. This so-called ‘Planckian
dissipation’ is encountered in a variety of contexts in both conventional and
unconventional metals [56]. A scattering time of the order of
$\sim\frac{\hbar}{k_{B}T}$ is expected when degenerate electrons are scattered
off classical objects. Two examples are such scattering centers are phonons
above their Debye temperature or electrons above their Fermi temperature.

The presence of classical electrons can account for the strange metallicity of
Sr3Ru2O7 [57]. Mousatov and co-workers showed that in this system, $T$-linear
resistivity and its Planckian prefactor [54] can be explained by invoking the
scattering of degenerate electrons in a large pocket by classical electrons in
a small pocket. Such a scenario, which may be relevant to other correlated
metals, does not directly apply to cuprates, which have a single Fermi pocket.
On the other hand, if nodal quasi-particles become classical, i.e. if they get
excluded from the Fermi sea forming a distinct liquid with a distinct
degeneracy temperature, then a scenario similar to the one invoked for
Sr3Ru2O7 [57] can work for cuprates.

  * •

Isotropic $T^{-1}$ scattering rate: Grissonnanche and co-workers [58] have
recently reported that the T-linear scattering is independent of direction.
Let us consider the angular dependence of the $T$-linear scattering time when
degenerate electrons are scattered by non-degenerate nodal electrons. In this
case, the angular distribution of scattering centers peaks along two
orthogonal nodal orientations and this will dictate the angular dependence of
the scattering rate. Assuming a square cosine variation for each nodal
orientation, since $cos^{2}(\phi)+cos^{2}(\phi+\pi/2)$ does not vary with
$\theta$, one finds a flat scattering rate.

  * •

Saturation of the amplitude of the T-square prefactor: Cooper and co-workers
[7] found that the amplitude of the prefactor of T-square resistivity does not
increase with doping when it falls below the threshold of strange metallicity.
This behavior contrasts with what has been seen in quantum-critical metals
[59, 60], where $T$-square resistivity diverges to a $T$-linear one. On the
other hand, it is compatible with a ceiling for electron-electron scattering
met by a subset of electrons.

  * •

The evolution of carrier density with doping: Experiments [61, 62] have found
that the Hall carrier density gradually decreases from $1+p$ at high doping to
$p$ at low doping. On the other hand, the superfluid density does not show any
sharp feature as a function of doping and shows a dome-like structure similar
to the critical temperature [63], as found in a conventional superconductor
[52]. In the present picture, with decreasing carrier density, a larger
fraction of electrons is peeled off the Fermi surface. On the other hand, the
superfluid density, which keeps to be zero along the nodal direction is not
affected by this peeling. If the nodal electrons cease to participate in
charge transport in the normal state, the discrepancy between the doping
dependencies of the normal-state density and the superfluid density may find
an explanation.

  * •

Nodal excitons: Being extracted off the Fermi sea, nodal electrons and nodal
holes can pair up and form excitons, plausible candidates for playing the role
of pair-forming Bosons. Such a mechanism for the formation of pairs has been
already proposed in other contexts [64], but not in cuprates. Since the nodal
electrons do not participate in the Fermi sea, the superconducting order
parameter would therefore vanish along the nodal orientations, in conformity
with the d-wave symmetry of cuprates [65]. If this happens to be the case,
then the the charge order [66] competing with superconductivity is also
eventually the one driving the superconducting instability through its
fluctuations.

##  VII Concluding remarks

The present paper reports on two observations and on a speculation.

The observations are about the amplitude of fermion-fermion scattering in two
different strongly correlated fermioinc systems: 3He atoms near the melting
pressure and nodal quasi-particles in cuprates on the verge of
superconductivity. Their dimensionless amplitude is strikingly similar and
fermion-fermion collision rate in liquid 3He is near the threshold of
distinguishability.

The speculation is that the large collision rate in the cuprate would render
some carriers distinguishable. The exclusion of this subset of fermions from
the Fermi can lead to plausible explanations for the sudden emergence of
T-linear resistivity and a robust superconducting ground state.

The present approach may also prove relevant to deciphering the passage from
T-square to T-linear resistivity in magic-angle twisted bilayer graphene [67],
where T-linear and T-square resistivity emerge in close proximity of each
other.

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#  Requiring negative probabilities from ”the thing” researched, else that
thing doesn’t exist, is insufficient ground for any conclusion

Han Geurdes, Pensioner, Netherlands
<EMAIL_ADDRESS>
###### Abstract

It is demonstrated that the statistical method of the famous Aspect - Bell
experiment requires negative densities and negative probabilities from ”the
thing” researched, else that thing doesn’t exist. The thing refers here to
Einstein hidden variables. This requirement in the experiment is absurd and so
the results from such experiment are meaningless.

##  1  Introduction

###  1.1 Kolmogorov axioms

Let us start with the presentation of some relevant Kolmogorov axioms [1] that
are the foundation of probability laws. A probability is a function of a set.
We have three relevant axioms,

  * •

Positivity: A probability, $P$, is _never_ negative

  * •

Certain event: The probability $P$ of the universe set $U$ is unity

  * •

Additivity: If sets $Q$ and $R$ are disjoint then $P(Q\cup R)=P(Q)+P(R)$.

Here, $Q$ and $R$ and $Q\cup R$ are subsets of $U$. Sets are connected to
events via an appropriate random variable.

###  1.2 Bell’s experiment

With Bell’s formula for correlation [2] Einstein hidden parameters [3, page
320, ..unvolständig..] are modeled as classical probability random variables.
The value(s) of those random variables, in a universe set $\Lambda$, are
$\lambda$. The density is $\rho(\lambda)\geq 0$ for all $\lambda\in\Lambda$.
The density is normalized, i.e.
$\int_{\lambda\in\Lambda}\rho(\lambda)d\lambda=1$. Bell’s correlation formula

$\displaystyle
E(a,b)=\int_{\lambda\in\Lambda}\rho(\lambda)A(a,\lambda)B(b,\lambda)d\lambda$
(1)

is therefore surely embedded in Kolmogorovian classical probabiliy. If one
argues that a density somehow is associated with quantum variables, then $E$
in (1), based on complex $\lambda$, may theoretically either violate or
reproduce the quantum result. Introducing e.g. a quantum density in Bell’s
formula is petitio principi. See also [4] for a classical probability based
reproduction of quantum results. In addition in the early days of quantum
mechanics, Einstein already noticed the difference between classical
probability, leading to Wien’s law, and a kind of quantum probability leading
to Planck’s law [5], [6]. Moreover, there is the problem of an association of
a quantum (1) with an experiment, ruled by definition by classical probability
laws.

Returning to a classical probability based Bell formula. In a more formal way
we can state that Bell supposed that Einstein’s extra parameters are somehow
represented by a classical probability space $(\Lambda,\Sigma,P)$. The
$\Lambda$ is the universal set, the $\Sigma$ is the associated sigma-algebra
[7] and $P$ is the probability measure. The $P$ projects a set $S\in\Sigma$ in
the interval $[0,1]$. Note, $dP(\lambda)=\rho(\lambda)d\lambda$. Sometimes one
also writes, $P(d\lambda)$.

In Bell’s formula (1), $\boldsymbol{a}$ is the unit length setting parameter
vector of Alice’s instrument and $\boldsymbol{b}$ is the unit length setting
parameter vector of Bob’s instrument. Alice doesn’t know Bob’s setting and Bob
doesn’t know Alice’s setting. We have,
$A(\boldsymbol{a},\lambda)\in\\{-1,1\\}$ Alice’s spin measurement function
$B(\boldsymbol{b},\lambda)\in\\{-1,1\\}$ Bob’s spin measurement function.
There is a sufficiently large distance (see [3]) between Alice and Bob.

For photons it is sufficient to consider the angle
$x=\angle(\boldsymbol{a},\boldsymbol{b})$ between the vectors $\boldsymbol{a}$
and $\boldsymbol{b}$. This angle is a continuous variable in $0\leq x<2\pi$
and is determined in the plane orthogonal to the direction of propagation. The
expectation value $E(\boldsymbol{a},\boldsymbol{b})$ in (1) then reduces to
$E(x)$.

In experiment use is made of what can be called a raw product moment (rpm)
correlation [11]. This rpm correlation is of course embedded in classical
probability theory.

Furthermore, an excellent example of an experiment can be found in the
literature e.g. [8]. The rpm correlation for photons is,

$\displaystyle R(x)=\frac{N(x,\neq)-N(x,=)}{N(x,\neq)+N(x,=)}$ (2)

With, $N=N(x,\neq)+N(x,=)$ the total number of entangled photon pairs measured
under angle x. Here, given $0\leq x<2\pi$, $N(x,\neq)$ represents the number
of unequal spin measurements by Alice and Bob, i.e. $(+,-),~{}(-,+)$ and
$N(x,=)$ represents the number of equal spin measurements by Alice and Bob,
i.e. $(-,-),~{}(+,+)$. This enables to rewrite $R(x)$ in (2) as

$\displaystyle R(x)=1-2P(x,=)$ (3)

Here, $P(x,=)$ equals $N(x,=)/N$. It represents the (estimate) classical
probability to find ”$=$” spin under angle $x\in[0,2\pi)$. This is a
statistical frequency of an event divided by a total, therefore a probability
estimate. The $N$ can be large if needed.

The set structure behind the probability space of the experiment is
$(U,\Phi,P)$ with $U=[0,2\pi)$ and $\Phi$ the to $U$ associated sigma-algebra.
The $P$ is the probability measure.

The event ”$x,=$” is represented by a random variable $X$. A random variable
connects the probability set structure with what can be found in measurement
(events). Here it associates the $\Phi$ set for the angle $x$, from the
universe, $U=[0,2\pi)$ to the real numbers, i.e.
$X:\Phi\rightarrow\mathbb{R}$. It does that in such a way that it connects a
continuous variable to a set, viz. [9, page 117]. A probability is
$P:\Phi\rightarrow[0,1]$.

Therefore, a random variable $X$ is associated to the event ”$=$” spin under
angle $x$ and $x\in[0,2\pi)$. Furthermore, we are allowed here to assume ideal
(no loss) measurements.

The random variable $X$ is a continuous random variable. The random variable
$X$ attains continuous values because the angle is a continuous variable. It
is a mistake to think that $P(x,=)$ is the probability associated to a
discrete random variable. The point of coarse graining and discreteness is
dealt with in a separate subsection.

Subsequently, note that for a continuous random variable, the probability in a
point is zero [10]. The probability for a continuous random variable is
computed [12] like e.g.

$\displaystyle P(0\leq X<x)=\int_{0}^{x}f(y)dy$ (4)

In Riemannian integration, inclusion of limits doesn’t make a difference in
outcome.

###  1.3 Hypothesis

In the experiment we ask if it is possible in principle that $R(x)$ can be
equal to the quantum correlation $\cos(x)$. Because, we have
$\cos(x)=1-2\sin^{2}(x/2)$, this leads to the simple testing of the hypotheses

$\displaystyle H_{0}:P(0\leq X<x)=\sin^{2}(x/2)$ (5) $\displaystyle
H_{1}:\text{The hypothesis~{}~{}}H_{0}\text{~{}~{}is false }$

Therefore note, the probability $P(x,=)$ is in fact $P(0\leq X<x)$. In this
way we can via the random variable $X$ have $\sin^{2}(x/2)$ associated to the
set structure. However, the following things immediately catches the eye.

  * •

The function $\sin^{2}(x/2)$ isn’t monotone on $x\in[0,2\pi)$,

  * •

$P(0\leq X<2\pi)=0$ instead of $1$,

  * •

The probability density, $f(x)=(1/2)\sin(x)$ in (4) is not positive definite
for $x\in[0,2\pi)$.

###  1.4 Continuity & negative probabilities

With the fact that $\sin^{2}(x/2)$ is not monotone on $x\in[0,2\pi)$, negative
probabilities are required to let $P(0\leq X<x)$ meet $\sin^{2}(x/2)$.
Suppose, e.g. $S_{1}=[0,\pi)$ and $S_{2}=[\pi,3\pi/2)$. Then, $S_{1}\cap
S_{2}=\emptyset$ and when $\sin^{2}(x/2)$ is the probability function,
$P(S_{1}\cup S_{2})=P([0,3\pi/2))=1/2$. Because of the additivity in
Kolmogorovian axioms, we also must have $P(S_{1}\cup
S_{2})=P(S_{1})+P(S_{2})=1/2$. Again, when $\sin^{2}(x/2)$ is the probability
function it follows, $P(S_{1})=1$. But this leads, via
$P(S_{1})+P(S_{2})=1/2$, to $P(S_{2})=-1/2$ which is outside $[0,1]$. Hence,
$\sin^{2}(x/2)$ cannot be a probability function for a continuous random
variable. This is because negative probabilities arise from the additivity
which is a basic part of probability theory [12].

###  1.5 Discreteness

Special attention again is given to the coarse graining and discreteness point
above. Here we might use $P(x,=)$ in discrete points $x$. With coarse graining
and/or discreteness the sum of discrete probabilities $\sin^{2}(x/2)$ over $x$
in nontrivial partitioning $\mathcal{X}$, cardinality e.g. $>5$ that contains
$x=\pi$, always is larger than 1. The $\mathcal{X}$ is the discretisation of
$U=[0,2\pi)$. There is a discrete sigma-algebra $\Phi^{\prime}$ associated. We
have,

$\displaystyle\sum_{x\in\mathcal{X}\cup\\{\pi\\}}\sin^{2}(x/2)>1$ (6)

This means there are sets $A$ with $P(A)>1$ and $A\in\Phi^{\prime}$. For
$P(U^{\prime})$ we also might find $>1$. For comparison, the discrete Poisson
probability distribution is an example of a probability distribution of a
discrete random variable with a summation to unity. And please note that the
Einstein hidden variables concept, [3, page 320, ..unvolständig..], wasn’t
rejected for a finite number of $x$. The claim was that for every possible
$x\in[0,2\pi)$, we have no go Einstein variables.

##  2 Conclusion

A classical probability is a function that projects a set into the interval
$[0,1]$. Bell’s experiment requires a Kolmogorovian probability to be not
Kolmogorovian in order to meet the quantum correlation. This remains true,
even if one claims that Bell’s formula is quantum mechanics. The only thing in
the latter case is that there is a disconnect between experiment probability
and the (supposed quantum) Bell theoretical correlation formula. Note also, we
only may know the quantum world through the use of classical probability.

The points raised in the paper demonstrate that we always will find $H_{1}$ in
(5), in an experiment where the rpm correlation (2) is employed. This is not
because Einstein variables are impossible or that inequalities demonstrated
that Einstein variables do not exist. It is because it is _not possible_ in
the data to find $H_{0}$ is true. Data based on negative probability doesn’t
exist.

We conclude that the statistics of Bell’s experiment doesn’t allow any
sensible conclusion about go, or, no go Einstein.

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