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

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

(Dated: July 1, 2022)

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

I. Introduction

Elementary particles of a quantum fluid are indistin-guishable. Leggett [1, 2] argued that it is thanks tothis indistinguishibality that such fluids are governed byquantum statistics [and not only quantum mechanics].Trachenko and Zaconne [3, 4] recently highlighted thedynamical aspect of this indistinguishibality and used itas a departing point to explore the boundary betweenthe statistics-active and the statistics-inactive regimes ofquantum fluids.

The normal state of cuprate superconductors is nowa-days called a ‘strange metal’. The expression refers tothe puzzling temperature dependence of their electricalresistivity (for recent reviews, see [5] and [6]). The fo-cus of the present paper is a very specific point of thecuprate phase diagram. In hole-doped cuprates, the su-perconducting dome ends when doping level exceeds athreshold of p ≈ 0.3. Hussey and collaborators carriedout an extensive study of the evolution of resistivity inLa1−xSrxCuO4 [7]. They found that the superconduct-ing dome and strange metallicity emerge concomitantlywhen x < 0.3. La1.67Sr0.33CuO4 is not superconductingand its resistivity follows T2 [8], but La1.71Sr0.29CuO4

is a superconductor and its resistivity does not corre-spond to what is expected for a Fermi liquid. Instead,it contains a T-Linear term [7] (Figure 1). This obser-vation is not exclusive to overdoped cuprates. Taille-fer pointed out that the normal-state T-linear scatteringand the onset of superconducting instability are linkedin several other families of superconductors other thancuprates [9]. Greene and collaborators found that inelectron-doped La2−xCexCuO4(LCCO), T-square super-conductivity emerges only upon the destruction of super-conductivity by overdoping [10].

In this paper, I argue that the notion of dynamic distin-guishibality [4] illuminates the birth of a ’strange metal’at this locus of the phase diagram. The argument isbased on scrutinizing the amplitude of T-square resistiv-ity in the Fermi liquid La1.67Sr0.33CuO4, by comparingit with other Fermi liquids, and by recalling the fate of

p

Temperature

metal

superconductor

Psc~ 0.3

𝜌=𝜌0+𝐴1𝑇+𝐴2𝑇2

𝜌=𝜌0+𝐴2𝑇2

FIG. 1. The puzzle: Zoom on the cuprate phase diagramnear the end of the superconducting dome. Hussey and hisco-workers [7] found that below a threshold doping level, thesystem has a superconducting ground state and a resistivitywhich follows ρ = ρ0 + A1T + A2T

2. Above this threshold,the system is not a superconductor and resistivity can be fitwith a purely quadratic temperature-dependnet term: ρ =ρ0 +A2T

2.

fermion-fermion collisions when 3He solidifies [11–14].

II. Heavily-doped LSCO stands out among Fermiliquids

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

Let us pick up several exceptions. SrVO3 is a cor-related metal with vanadium in 3d1 configuration, whichremains 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,

arX

iv:2

202.

1012

8v5

[co

nd-m

at.s

tr-e

l] 2

9 Ju

n 20

22

2

a)b)b)

La1.7Sr0.3CuO4

10 100

1E-3

0.01

0.1A

(

cm

K-2)

(mJmol-1K

-2)

Sr3Ru

2O

7

Sr2RuO

4

SrVO3

La1.7

Sr0.3

CuO4

La1.95

Sr0.05

TiO3

FIG. 2. Standing out among Fermi liquids: a) The magnitude of the prefactor of T-square resistivity, A, vs. theSommerfeld coefficient, γ, 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 [7, 8] is fivetimes larger than what is expected given its γ. b) `quad, derived from the amplitude of A and fundamental constants (see text)in different Fermi liquids [20]. The large magnitude of `quad in La1.67Sr0.33CuO4 stands out.

Sr0.05La0.095TiO3 is a dense metal with almost one elec-tron per formula unit [19]. Sr2RuO4 is an unconventionalsuperconductor with a Fermi liquid normal state aboveits critical temperature [22]. Sr3Ru2O7 has a non-trivialelectronic instability at 7.8 T, but is a correlated Fermiliquid at zero magnetic field [17, 18]. The feature theyall share with La1.67Sr0.33CuO4 is being a dense metallicperovskyte. None of them, however, is a strange metalor become a high-temperature superconductor.

Fig. 2a shows the amplitude of the prefactor of T-square resistivity A as a function of the Sommerfeld co-efficient (the electronic T-linear specific heat), γ in thesemetals. This Kadowaki-Woods (KW) plot [23] reveals ananomaly. In correlated metals, the prefactor of T-squareresistivity scales with the square of γ over five ordersof magnitude [24]. This scaling is operative when thereis roughly one electron per formula unit [25]). As onecan see in Fig. 2a, heavily-doped LSCO does not followthe trend observed in other metallic perovskytes. Its T-square resistivity is more than five times larger than itshould be, given the magnitude of its γ. Interestingly,this is also the case of heavily overdoped electron-dopedcuprates [26]

The unusually large amplitude of A inLa1.67Sr0.33CuO4 betrays itself in a comparison ofall Fermi liquids. The Kadowaki-Woods scaling can beextended to dilute metals by plotting A as a function of

the Fermi energy, EF [20, 25, 27]. Due to Pauli exclusionprinciple, the phase space of scattering among fermionsis proportional to (kBTEF

)2. Dimensional considerations

imply [20]:

A =~e2

(kBEF

)2`quad (1)

Here ~ is the reduced Planck constant and e is the fun-damental charge. `quad is a phenomenological material-dependent length scale. A survey of available datashows that for all known Fermi liquids `quad is between1 to 50 nm [20, 27]. As one can see in Fig. 2b, inLa1.67Sr0.33CuO4, `quad ≈ 240 nm. Decidedly, this Fermiliquid is not a banal one. The unusually large A of thismetal, given its density of states and its degeneracy tem-perature is the first step for understanding its transfor-mation to a strange metal upon the removal of dopants.

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

III. Distinguishibality on the verge of solidification

in 3He

Under their own vapor pressure, the two isotopes ofhelium do not solidify down to zero temperature. These

3

0 1 2 30,0

0,2

0,4

0,6

Greywall 1984

Wheatley 1975

P(MPa)

0 1 2 30

2

4

6

m*/

m3

P(MPa)

0 1 2 30

1

2

EF/K

B (

K)

P (MPa)

a) b)

c) d)

𝜏 𝜅𝑇2(𝑝𝑠.𝐾2)

𝜁=(𝜏

𝜅𝑇2)−

1ℏ𝐸𝐹

𝑘𝐵2

0 1 2 30

20

40

60

P (MPa)

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

4

quantum liquids [1, 2] become quantum solids upon theapplication of pressure. With an odd number of pro-tons, neutrons and electrons, a 3He atom is a compositefermion. The molar volume changes from 36.84 cm3/molat zero pressure to 25.5 cm3/mol at p=3.4 MPa, whenit solidifies. This is twice larger than what is classicallyexpected (12 cm3/mol) and is a consequence of the largezero-point motion of the atoms in the crystal [14].

The temperature dependence of thermal conductivityand viscosity in normal liquid 3He at different pressureshave been carefully measured by several authors. Thequasi-particle scattering time extracted from these stud-ies, broadly consistent with each other, were reviewed indetail by Dobbs [14]. The scattering time derived fromthermal conductivity, τκ displays a T−2 behavior in thezero-temperature limit [11, 13], as expected for a Fermiliquid.

With increasing pressure, interaction between theatoms, consisting principally of a strong hard-core re-pulsion and a weak van der Waals attraction, intensi-fies. This leads to an amplification of the effective mass,quantified by measurements of specific heat [12] (Fig.3a). Fig. 3b shows the evolution of τκT

2 (in the zero-temperature limit) as a function of pressure reported byWheatley [11] and by Greywall [13]. The trend is sim-ilar, but Greywall’s values are about 25 percent lowerthan Wheatley’s. The highest pressure (3.44 MPa) cor-responds to the onset of solidification. By this pressure,the time between two fermion-fermion collisions has de-creased by a factor of almost 3.

Several theoretical studies have examined the evolu-tion of the effective mass and the Landau parameters bypressure. Vollhardt, Wolfle and Anderson [28] employeda Hubbard lattice-gas model with a variable density ofparticles to describe the pressure dependence of thermo-dynamic properties of normal liquid 3He. Pfitzner andWolfle [29, 30] gave a reasonable account of transportcoefficients under pressure. According to these studies,normal liquid 3He is a strongly interacting and almostlocalized Fermi liquid [31].

What will be scrutinized here are the amplitudes ofτκT2 and the Fermi energy, EF on the verge of solid-ification. According to Vollhardt and Wolfle [32], thequasi-particle lifetime on the Fermi surface is given by:

τ0N =64

π3

~EF(kBT )2

〈W 〉−1a (2)

〈W 〉a is the angular average of the transition prob-abilities between spin singlet and spin triplet states

[14, 32, 33]. Multiplying τκT2 by

k2B~EF

will yield a dimen-sionless number inversely proportional to the fermion-fermion collision strength.

Fig. 3c shows the evolution of the Fermi energy withpressure reconstructed from Greywall’s data [13]. Com-bining this with τκT

2 leads to Fig. 3d, which shows the

Body Centered Cubic 3He Normal liquid 3He

FIG. 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 makesthem distinguishable. In the solid near the melting transition,atoms wander around, thanks to their zero-point motion andcan exchange their places with their neighbors. On the otherhand, in the liquid, collisions in real space confine atoms to arestricted neighbourhood. The liquid solidifies when collisionsconfine each atom to a spatial neighborhood.

pressure dependence of the normalized scattering time:

ζ =~EF

τκ(kBT )2(3)

Solidification occurs when this number becomes aslarge as 60. This implies a huge collision rate betweenquasi-particles at the verge of solidification. Insertingthis number in Eq. 2 leads to the conclusion that atthe onset of solidification 〈W 〉a ≈ 140. Surprisingly, thisremarkably large value has not been hitherto explicitlynoticed, let alone commented.

The value of 〈W 〉a (or ζ) is set by the magnitudeof dimensionless Landau parameters of the Fermi liq-uid, which are denoted by F sl (spin symmetric) and F al(spin antisymmetric) [30, 32]. Practically, Landau pa-rameters with l < 2 are the ones which matter and thehigher order ones may be safely neglected [32, 34]. Voll-hardt and Wolfle [32] have used experimental data tocalculate the evolution of Landau parameters with pres-sure and have found that at the threshold of solidifica-tion (P = 3.4MPa) F s0 = 88.47, F a0 = −0.753, andF s1 = 14.56. These are large numbers. The Fermi liq-uid picture still holds, albeit restricted to a very narrowtemperature, at the onset of solidification and its Landauparameters are large, yet finite.

Let us first note that in standard theories of Motttransition [35], Landau parameters diverge. Let us alsonote that the mean-free-path of 3He atoms remains muchlonger than their wavelength even on the verge of solidifi-cation . According to Greywall’s data, at P=3.4MPa and

5

T=0.01 K, τκ = 1.43ns, vF = 32.4m/s, ` = 42.6nm andkF = 8.9nm−1. Since kF ` 1, Anderson localization isnot what drives confinement in space.

I conjecture that what drives solidification is the im-possibility for Landau parameters to become arbitrarilylarge. An infinite F s0 , for example, would make the liq-uid incompressible, which is implausible. Vollhardt andWolfle [32] highlighted the fact that normal liquid 3Heis much less compressible than a non-interacting Fermiliquid, thanks to its large F s0 . On the other hand, thisinteracting liquid is not more incompressible than solid3He. Indeed, according to the experiment [36], at themelting pressure, the compressibility of the two phasesare nearly equal. Now, the compressibility of solid isset by its phonon spectrum and, one may suspect, this iswhat sets the bound to F s0 in the liquid. After all, Landauparameters are two-particle correlators [35] and atoms ofa quantum liquid have a finite coordination number [37].Therefore, two-particle correlations in real space cannotattain an arbitrarily large magnitude.

Solid 3He is not subject to Fermi-Dirac statistics, be-cause each atom is confined to the neighborhood of a des-ignated site [38]. Near the melting pressure, in the solidstate, neighbouring atoms can frequently exchange theirssites thanks to their zero-point motion. Each atom fre-quently encounters its first neighbours far from its latticesite [39, 40], but there is still a one-to-one correspondencebetween an atom and its designated site. In contrast tothe solid, the atoms of the liquid are indistinguishable(See Fig. 4). Collisions allow an atom in the liquid stateto ‘observe’ its neighbours in real space [41]. As thesecollisions multiply, the atom cannot avoid being confinedto a specific and distinguishable volume of the real space.

The case of 3He illustrates that the time between twosuccessive fermion-fermion collisions can become unbear-ably short for the survival of a Fermi liquid, despite along mean-free-path and kF ` 1. Let us now return tocuprates.

IV. Nodal quasiparticles: the unbearable shortnessof being

What we saw in the case of 3He raises a question:Given the unusually large amplitude of the T-square re-sistivity in heavily doped LSCO, how short is the nor-malized quasi-particle lifetime?

Fig. 5a shows the Fermi surface of of La1.68Sr0.32CuO4

according to a tight binding model with nearest-neighborhopping parameters chosen to fit the Fermi surfaceseen by Angle-Resolved Photoemission Spectroscopy(ARPES). [42–44]. The radius of this Fermi surface hasa modest angular variation. In comparison, the angularvariation of the Fermi velocity, vF , is much larger (seeFig. 5b). This is a consequence of the proximity of Fermisurface and the Brillouin zone boundary along the anti-nodal direction. Because of the van hove singularity, theevolution of the Fermi surface with doping differs along

nodal and anti-nodal orientations. As a consequence, thederivative of Fermi energy in momentum space (vF ) hasa strong angular dependence.

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

σ =1

4π3

e2

~

∫τvk

vk

vkdSF (4)

In the case of cubic symmetry, σ is identical for thewhole solid angle and therefore:

σcub. =1

3π2

e2

~τvF k

2F (5)

Note that τ , vF and kF can be anisotropic. However,cubic symmetry, by constraining σcub. to be isotropic, re-stricts possible profiles for τ(θ, φ), vF (θ, φ) and kF (θ, φ).

Heavily overdoped LSCO has a tetragonal symmetryand a quasi-cylindrical Fermi surface extending verticallyalong the whole Brillouin zone. In this case, the in-planeelectrical conductivity is constrained to be isotropic andequal to :

σtet. =1

2πc

e2

~τvF kF (6)

Here, c is the lattice parameter along the c-axis. Theexperimentally measured prefactor of in-plane T-squareresistivity (A = 2.5nΩcmK−2 [7, 8]) is indeed isotropicin the basal plane. Using Eq. 6 and AT 2 ≡ σ−1

ee , toquantify τρ (the index refers to the fact that the experi-mental probe used to extract this time scale is electricalresistivity):

τρ(θ)T2 =

~e2

2πc

AkF (θ)vF (θ)(7)

The Fermi velocity, vF , the Fermi wave-vector, kFand τρ have all their angular dependence. Since theanisotropies of kF and vF do not cancel out, one expectsa significant angle dependence of τρT

2. As seen in Fig.5c, this is indeed the case. It is more than twice smalleralong the nodal orientation. Note that, because of thelarge anisotropy of the Fermi velocity, the anisotropy ofthe mean-free-path is the inverse of the anisotropy ofthe scattering time. Nodal quasi-particles have a longermean-free-path yet a shorter scattering time.

Combining τρT2 and the Fermi energy EF = 5900K

(extracted from the magnitude of the T-linear electronicspecific heat γ=6.9 mJ.mol−1.K−2 [8]) leads to the quan-tification of the dimensionless ζ and its angular depen-dence (Fig. 5d). Along the nodal orientation, it becomesas large as ≈60, close to what was found above in 3He onthe verge of solidification.

Let us recall that the microscopic interaction betweenfermions are very different in the two cases. In metals,point-like electrons repulse each other through screened

6

kx

ky

f

0 90 180 270 360100

200

300

400

Ferm

i velo

city (

km

/s)

f (degrees)

0 90 180 270 3600,0

0,5

1,0

1,5

2,0

f (degrees)

𝜁=(𝜏

𝜌𝑇2)−

1ℏ𝐸𝐹

𝑘𝐵2

𝜏 𝜌𝑇2(𝑛𝑠.𝐾2)

a) b)

c) d)

0 90 180 270 3600

20

40

60

f (degrees)

FIG. 5. Angular variation of collision time in heavily overdoped LSCO: a) Fermi surface and the Brillouin zoneof 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 ζ extracted from τρT2 and

k2B~EF

. Compare the absolute value of

the maximum for nodal orientations with what was seen in the case of 3He on the verge of solidification.

Coulomb repulsion. In contrast, neutral 3He atoms inter-act over a short range comparable to their hard-sphereradius and the interaction has both attractive and repul-sive components. Nevertheless, the strength of fermion-fermion scattering rate (τ−1) can be quantified in bothcases by the dimensionless ζ.

Only when there is a single Fermi surface and a unique

Fermi temperature, the amplitude of ζ is unambiguous.This is the case of heavily overdoped LSCO, but not otherFermi liquids. Most often, they have multiple anisotropicFermi pockets. Assuming a single average Fermi energy,one can directly extract from the T-square prefactor, thephenomenological length scale `quad, introduced first inref.[20] and defined by Eq. 1, which is proportional to ζ

7

System kF (nm−1) m?/m0 EF (K) ζ3He (p=0) 7.9 2.8 1.8 353He (p=3.4MPa) 8.9 5.8 1.1 60

La1.67Sr0.33CuO4 5.6 5 5900 24-61

UPt3 5 16-130 90 ≈ 10

Sr2RuO4 5 3.3-16 1800 ≈ 16

Sb 0.8 0.07-1 1100 ≈ 0.1

SrTiO3−δ (n=4×1017cm−3) 0.23 1.8 18 ≈ 0.1

TABLE I. 3He and heavily-doped LSCO compared to heavy-fermion UPt3 [46–48], correlated oxide Sr2RuO4 [22], semi-metallicantimony [49, 50] and dilute metallic strontium titanate [20]. Note the exceedingly large normalized amplitude of ζ in LSCO.UPt3 and Sr2RuO4 despite their larger mass enhancements have smaller ζs.

through a dimension-dependent length scale.In a simplified one-band approximation, the total car-

rier density and the measured T -linear specific heat yieldan average Fermi energy for a given system. In that case,for a three-dimensional metal with a spherical Fermi sur-face, one has:

ζ3d =2

3π2

e2

~A

k2BE2F kF (8)

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

ζ2d =1

3πc

e2

~A

k2BE2F (9)

These expressions, which neglect anisotropy and multi-plicity of Fermi pockets, can be cautiously used to extractthe rough magnitude of ζ in various metals.

V. Comparison with other metals

Table I compares 3He and in heavily-doped LSCOwith two weakly-interacting and two strongly-interactingFermi liquids. The list includes antimony [50], di-lute oxygen-reduced strontium titanate [51], the heavy-fermion metal UPt3 [46] and the correlated oxideSr2RuO4 [22].

UPt3 displays a quadratic resistivity below 1.5 K withA=1.55µΩ cm K−2 for in-plane charge flow [46]. It hasfive different Fermi surface pockets with different sizesand effective masses [46–48]. The largest and the mostrelevant sheet of the Fermi surface is the band 3 (‘Oys-ters and urchins’ [46]) giving rise to the ω orbit seen byquantum oscillations [48]. Table I uses the effective massand the average radius of this sheet for rough estimatesof kF and EF . The Fermi energy of 90 K is compati-ble with an alternative estimation using the magnitudeof the Sommerfeld coefficient (γ = 450J/K2.mol) andcarrier density (1.4 ×1028m−3) in UPt3.

Resistivity in Sr2RuO4 is quadratic below 25 K withA=6 nΩ cm K−2 for in-plane charge flow [16]. Its Fermi

surface consists of three warped cylinders [22] with radiiranging from 3.04 to 7.53 nm−1 and the effective massfrom 3.3 to 16 bare electron masses [22]. The table usesaverage values for kF and EF for Sr2RuO4.

Antimony (Sb) displays a quadratic resitivity below10 K with a prefactor of 0.8 nΩ cm K−2 [50]. Its Fermisurface has three electron pockets and a single intercon-nected Fermi surface. The Fermi wave-vector along dif-ferent orientations varies by one order of magnitude [49].The table gives an average value for kF in antimony. TheFermi energy is almost the same for electrons and holes.

SrTiO3 is a band insulator. It becomes a dilute metalwith a very low carrier density when a small amount ofoxygen atoms are removed [52]. The resistivity of this di-lute metal follows a T-square resistivity with A=9 µΩ cmK−2 when n=4×1017cm−3 [20]. At this carrier density,only a single band is occupied and the Fermi surface seenby quantum oscillations can be roughly approximated toa spherical one.

The experimentally resolved A in set by the overallcontribution of the sheets of a multi-component Fermisurface. Therefore, the average values yield only a roughestimate of ζ. It can have different values for differentpockets. As seen in the table, according to this roughestimate, ζ > 1 in strongly-correlated systems and ζ < 1in the weakly correlated ones. The maximum ζ is lower inSr2RuO4 and in UPt3 than in LSCO (or in 3He), despitetheir larger mass enhancement. Note also that while themagnitude of A in SrTiO3−δ and in Sb differ by fourorders of magnitude, their ζ is roughly similar.

Thus, ζ is unusually large in heavily overdoped LSCO.Given the angular variation of ζ, an eventual upperboundary will be encountered by nodal quasi-particlesbefore other electrons of the Fermi sea. Let us assumethat an exceedingly large fermion-fermion collision ratefreezes the nodal quasi-particles out of the Fermi sea.

VI. Consequences of nodal freeze-out

What happens to the nodal quasi-particles once areexcluded from the Fermi sea is not known. However, letus assume that this happens below x=0.3 and generates

8

two categories of electrons [53]: Those for which quantumstatistics is non-operative and others remaining in theFermi sea. Such a hypothesis will provide new possiblesolutions to a number of longstanding puzzles. They arelisted below:

• Planckian dissipation: Bruin and co-workers no-ticed that in many metals with a T-linear resistiv-ity, the amplitude of scattering time is of the orderof τP = ~

kBT[54]. Legros and co-workers [55] have

reported that this is indeed the case of overdopedcuprates. This so-called ‘Planckian dissipation’ isencountered in a variety of contexts in both con-ventional and unconventional metals [56]. A scat-tering time of the order of ∼ ~

kBTis expected when

degenerate electrons are scattered off classical ob-jects. Two examples are such scattering centersare phonons above their Debye temperature or elec-trons above their Fermi temperature.

The presence of classical electrons can account forthe strange metallicity of Sr3Ru2O7 [57]. Mousatovand co-workers showed that in this system, T -linearresistivity and its Planckian prefactor [54] can beexplained by invoking the scattering of degenerateelectrons in a large pocket by classical electrons ina small pocket. Such a scenario, which may berelevant to other correlated metals, does not di-rectly apply to cuprates, which have a single Fermipocket. On the other hand, if nodal quasi-particlesbecome classical, i.e. if they get excluded from theFermi sea forming a distinct liquid with a distinctdegeneracy temperature, then a scenario similar tothe one invoked for Sr3Ru2O7 [57] can work forcuprates.

• Isotropic T−1 scattering rate: Grissonnancheand co-workers [58] have recently reported that theT-linear scattering is independent of direction. Letus consider the angular dependence of the T -linearscattering time when degenerate electrons are scat-tered by non-degenerate nodal electrons. In thiscase, the angular distribution of scattering centerspeaks along two orthogonal nodal orientations andthis will dictate the angular dependence of the scat-tering rate. Assuming a square cosine variation foreach nodal orientation, since cos2(φ)+cos2(φ+π/2)does not vary with θ, one finds a flat scattering rate.

• Saturation of the amplitude of the T-squareprefactor: Cooper and co-workers [7] found thatthe amplitude of the prefactor of T-square resistiv-ity does not increase with doping when it falls belowthe threshold of strange metallicity. This behav-ior contrasts with what has been seen in quantum-critical metals [59, 60], where T -square resistivitydiverges to a T -linear one. On the other hand, it is

compatible with a ceiling for electron-electron scat-tering met by a subset of electrons.

• The evolution of carrier density with doping:Experiments [61, 62] have found that the Hall car-rier density gradually decreases from 1 + p at highdoping to p at low doping. On the other hand,the superfluid density does not show any sharp fea-ture as a function of doping and shows a dome-likestructure similar to the critical temperature [63],as found in a conventional superconductor [52]. Inthe present picture, with decreasing carrier density,a larger fraction of electrons is peeled off the Fermisurface. On the other hand, the superfluid density,which keeps to be zero along the nodal directionis not affected by this peeling. If the nodal elec-trons cease to participate in charge transport in thenormal state, the discrepancy between the dopingdependencies of the normal-state density and thesuperfluid density may find an explanation.

• Nodal excitons: Being extracted off the Fermisea, nodal electrons and nodal holes can pair up andform excitons, plausible candidates for playing therole of pair-forming Bosons. Such a mechanism forthe formation of pairs has been already proposed inother contexts [64], but not in cuprates. Since thenodal electrons do not participate in the Fermi sea,the superconducting order parameter would there-fore vanish along the nodal orientations, in confor-mity with the d-wave symmetry of cuprates [65].If this happens to be the case, then the the chargeorder [66] competing with superconductivity is alsoeventually the one driving the superconducting in-stability through its fluctuations.

VII. Concluding remarks

The present paper reports on two observations and ona speculation.

The observations are about the amplitude of fermion-fermion scattering in two different strongly correlatedfermioinc systems: 3He atoms near the melting pressureand nodal quasi-particles in cuprates on the verge of su-perconductivity. Their dimensionless amplitude is strik-ingly similar and fermion-fermion collision rate in liquid3He is near the threshold of distinguishability.

The speculation is that the large collision rate inthe cuprate would render some carriers distinguishable.The exclusion of this subset of fermions from the Fermican lead to plausible explanations for the sudden emer-gence of T-linear resistivity and a robust superconductingground state.

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

9

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