Absence of a BCS-BEC crossover in the cuprate superconductors John Sous1Yu He2and Steven A. Kivelson1 1Department of Physics Stanford University Stanford California 94305 USA

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Absence of a BCS-BEC crossover in the cuprate superconductors

John Sous,1, ∗Yu He,2, †and Steven A. Kivelson1, ‡

1Department of Physics, Stanford University, Stanford, California 94305, USA

2Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA

(Dated: October 16, 2023)

We examine key aspects of the theory of the Bardeen-Cooper-Schrieﬀer (BCS) to Bose-Einstein

condensation (BEC) crossover, focusing on the temperature dependence of the chemical potential,

µ. We identify an accurate method of determining the change of µin the cuprate high temperature

superconductors from angle-resolved-photoemission data (along the ‘nodal’ direction), and show that

µvaries by less than a few percent of the Fermi energy over a range of temperatures from far below

to several times above the superconducting transition temperature, Tc. This shows, unambiguously,

that not only are these materials always on the BCS side of the crossover (which is a phase transition

in the d-wave case), but are nowhere near the point of the crossover (where the chemical potential

approaches the band bottom).

INTRODUCTION

The zero temperature (T) superﬂuid density, ns(0),

of the cuprate high temperature superconductors is sev-

eral orders of magnitude smaller than that of conven-

tional superconductors [1–3]. Indeed (when translated

into energy units) it is comparable to the critical tran-

sition temperature (Tc) [3]. This has led to the proba-

bly inescapable inference that Tc, itself, is determined,

at least to a signiﬁcant degree, by the condensation scale

(i.e. the phase ordering temperature, Tθ∝ns(0)), rather

than by the pairing scale (∼∆0/2), in contrast to the

case in the Bardeen-Cooper-Schrieﬀer (BCS) theory of

conventional superconductors. There is also compelling

evidence that some degree of clearly identiﬁable super-

conducting ﬂuctuations - colloquially referred to as ‘pair-

ing without phase coherence’ - persists for a substantial

range (at least 20% or so) above Tc[4–14]. This has

been known for some time for the underdoped cuprates,

but it has recently become increasingly clear that the

same is true for many or all overdoped cuprates as well

(Fig. 1) [8, 13, 15]. Indeed, as a function of doping, the

onset temperature (however deﬁned) of superconducting

ﬂuctuations more closely parallels Tcthan it does the

conventionally deﬁned pseudo-gap crossover.

However, what is unclear is why this occurs, and what

we should learn from this. One proposal is that this

should be taken as evidence that the system is approach-

ing a strong pairing situation, referred to as the Bose-

Einstein condensation (BEC) limit, in which the elec-

trons form non-overlapping charge 2ebosons at a scale

far above Tc[16–27]. However, as we will discuss be-

low, there are other conceptually distinct, yet equally

well understood circumstances in which Tcis determined

by phase ordering and in which Cooper pairing persists

above Tc. The purpose of this work is to analyze the

behavior that would be expected of a system either in

the BEC limit or approaching the BCS to BEC crossover

from the BCS side, and to present direct experimental

evidence that this is not the case for the cuprates.

FRAMING THE ISSUE

BCS theory is a weak coupling theory that is built on a

starting point that is the electronic structure from band

theory. The BEC limit invokes electronic bound states.

In the former case, the pairing is highly collective and

the chemical potential, µ, is only weakly aﬀected by the

advent of pairing. In the latter, the chemical potential

- by the deﬁnition of a bound state - must approach a

value that lies below the band bottom as T→0. These

diﬀerences do not refer to subtle low-energy phenomena

but rather to entirely diﬀerent regimes of microscopic

physics on energy scales of the order of the Fermi energy,

EF, or larger [32].

From this perspective, the fact that the Fermi surface

and general features of the electron dispersion seen in

ARPES experiments across the superconducting dome

of the cuprates are more or less in agreement with expec-

tations from band-structure calculations appears to be

inconsistent with any large excursions toward the BEC

limit. (This is illustrated in Fig. 2.) Emergent features

of the low energy physics, such as a normal state pseudo-

gap that competes with superconductivity (apparent be-

low some generally relatively ill-deﬁned T⋆) [7, 33–39]

and various low energy kinks in the dispersion relations

are certainly interesting and important, but occur on en-

ergy scales small compared to EF[40–43]. The fact that

the application of magnetic ﬁelds large enough to quench

superconductivity produces quantum oscillations [44] is

further evidence that pairing is a collective property of

the superconducting state rather than a microscopic fea-

ture associated with bound-state formation [45].

One important feature of the superconducting state in

the cuprates is that it has d-wave symmetry and gap-

less, nodal quasi-particle excitations [46]. There can

be no nodal quasi-particles in the BEC limit. (See

Ref. [47].) Thus, for this d-wave case, the BCS to BEC

crossover [24, 26] would constitute a (Lifshitz) phase

transition from a nodal to a nodeless state [47, 48]. The

existence of well deﬁned nodes is, of itself, proof that

arXiv:2210.13478v2 [cond-mat.supr-con] 12 Oct 2023

120

100

Temperature (K)

0.250.200.150.100.05

Doping (x)

Torque magne�za�on

THz conduc�vity onset

Tc LSCO ﬁlms

Speciﬁc heat

Tc LSCO crystals

Paraconduc�vity

THz conduc�ivty onset

0.250.200.150.100.05

Doping (p)

Speciﬁc heat

Nernst

Torque magne�za�on

ARPES AN

ARPES NN

100

Temperature (K)

200

300

La2-xSrxCuO4Bi2Sr2CaCu2O8+δ

(a) (b)

Paraconduc�vity

“universal” parabola

FIG. 1. Cuprates phase diagram. Phase diagram of two representative cuprates, (a) La2−xSrxCuO4(LSCO) and (b)

Bi2Sr2CaCu2O8+δ(Bi-2212), as a function of doped hole concentration and temperature. Open black markers indicate Tcfor

bulk crystals, while the solid black line is the same for crystalline ﬁlms from Ref. [10]. The other symbols indicate crossover

lines below which the existence of signiﬁcant superconducting ﬂuctuations are inferred from various diﬀerent experiments that

are directly sensitive to Cooper pair formation. Data, including error bars where applicable, are reproduced from Refs. [4–

10, 28–31]. The quantitative identiﬁcation of any crossover depends on the criterion used, and moreover distinct probes should

have diﬀerent sensitivity to superconducting correlations, so it is reasonable that the various lines do not coincide.

the cuprates are on the BCS side of the transition. This

leaves only the question of how far on the BCS side they

are from the point at which a BCS to BEC transition

would have occurred [49]. In the language of eﬀective

ﬁeld theories, the question we consider is not one con-

cerning the correct infra-red description (i.e. phases of

matter) but rather concerns the ultra-violet (high energy

‘microscopic’) description consistent with experimental

data.

EXPERIMENTAL PERSPECTIVE

We will focus our attention on the behavior of the

chemical potential, µ, as this is a fundamental thermo-

dynamic quantity that exhibits qualitatively diﬀerent be-

havior in the two limits. Since by deﬁnition, in the BEC

limit the chemical potential is below the band bottom,

on approach to the BEC limit from the BCS side one

should see that the chemical potential is signiﬁcantly de-

pressed from its band theory value toward the bottom of

the band. Moreover, it should show strong Tdependen-

cies for temperatures of order Tc.

A number of fortuitous features of the electronic struc-

ture of the cuprates make it possible to stringently bound

the evolution of the chemical potential from the electron

dispersion measured in ARPES along the ‘nodal direc-

tion’ in the Brillouin zone. Speciﬁcally, it is possible to

determine the value of the Fermi momentum, kF=|⃗

kF|,

as a function of temperature with a high degree of pre-

cision. In the superconducting state, since the gap van-

-1.0

-0.5

0.0

E - E

(eV)

-1.5

-2.0

Bi-2212

∆

ΓΜ

-0.2

0.0

E - EF (eV)

ΓΜX

p = 0.22

(a)

(b)

DFT

x = 0.23 p = 0.22

Bi-2212

FIG. 2. Electronic structure of Bi-based cuprates

along high symmetry directions. (a) Schematic Fermi

surface and momentum cut trajectory in the tetragonal Bril-

louin zone of a CuO2plane. (b) Low energy electronic struc-

ture near (π,0) in Bi-2212 (p = 0.22, Tc= 66 K) in the

normal state. Data are reproduced from Ref. [8]. (c) Elec-

tronic structure along Γ −Xdirection in LSCO (x= 0.23,

Tc= 24 K) and Bi-2212. Light green lines are density func-

tional theory (DFT) calculated band structure. For Bi-2212

only the antibonding band is shown. Deviations from the ﬁrst

principles dispersion apparent at low energies represent mass

renormalization due to additional interaction eﬀects, see Sup-

plementary Note 9. Data are adapted from Refs. [8, 50–52].

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AbsenceofaBCS-BECcrossoverinthecupratesuperconductorsJohnSous,1,∗YuHe,2,†andStevenA.Kivelson1,‡1DepartmentofPhysics,StanfordUniversity,Stanford,California94305,USA2DepartmentofAppliedPhysics,YaleUniversity,NewHaven,Connecticut06511,USA(Dated:October16,2023)WeexaminekeyaspectsofthetheoryoftheBardeen-...

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Absence of a BCS-BEC crossover in the cuprate superconductors John Sous1Yu He2and Steven A. Kivelson1 1Department of Physics Stanford University Stanford California 94305 USA

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