Full text of DOI:10.1007/s100510070163

Eur. Phys. J. B 17, 95–102 (2000)
THE EUROPEAN
PHYSICAL JOURNAL B
EDP Sciences
Societa` Italiana di Fisica
Springer-Verlag 2000
c
ꢀ
Gap and pseudogap evolution within the charge-ordering
scenario for superconducting cuprates
L. Benfattoa, S. Caprara, and C. Di Castro
Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy
and
Istituto Nazionale per la Fisica della Materia, Unit`a di Roma 1, Piazzale Aldo Moro 5, 00185 Roma, Italy
Received 17 February 2000
Abstract. We describe the spectral properties of underdoped cuprates as resulting from a momentum-
dependent pseudogap in the normal-state spectrum. Such a model accounts, within a BCS approach, for
the doping dependence of the critical temperature and for the two-parameter leading-edge shift observed
in the cuprates. By introducing a phenomenological temperature dependence of the pseudogap, which finds
a natural interpretation within the stripe quantum-critical-point scenario for high-Tc superconductors, we
reproduce also the Tc − T∗ bifurcation near optimum doping. Finally, we briefly discuss the different role
of the gap and the pseudogap in determining the spectral and thermodynamical properties of the model
at low temperatures.
PACS. 74.25.Dw Superconductivity phase diagrams – 71.10.Hf Non-Fermi-liquid ground states, electron
phase diagrams and phase transitions in model systems – 74.20.Fg BCS theory and its development
1 Introduction
LE is observed as a finite minimum distance of the quasi-
particle peak from the Fermi level in the non supercon-
ducting state, and leaves disconnected arcs of Fermi sur-
face. When the temperature is lowered, the LE regions
around the M points enlarge and the arcs of Fermi sur-
face reduce and shrink towards the nodal points of the
corresponding d-wave superconducting gap below Tc. At
the same time the doping dependence of the momentum
structure of the LE is not trivial. As the doping is in-
creased, the zero temperature LE at the M points, ∆0(0),
remains constant or decreases [2–4,7], while the LE around
the nodal points ∆0(π/4) seems to increase [4,6,7] and
follow the rising of the critical temperature. Penetration-
depth measurements of the superfluid density ρs(T) at low
temperature probe the low-energy excitations around the
nodal points in a d-wave superconductor, and therefore
∆0(π/4). The correspondence between ARPES measure-
ments of ∆0(π/4) and the slope of ρs(T) at T = 0 is
made however more involved by the presence of the Lan-
dau renormalization factors [4].
Many theoretical models have been proposed to obtain
a non-Fermi-liquid behaviour and to describe a pseudogap
state. A firm result is however that above one dimension
the Landau Fermi-liquid theory is generically stable and a
strongly singular effective potential is required to disrupt
it [8]. This result, together with the above phenomenol-
ogy, suggests that a consistent description of the cuprates
requires a strong momentum-, doping-, and temperature-
dependent effective interaction. This interaction should af-
fect the states near the M points of the Brillouin zone
The non-Fermi-liquid behaviour of the normal phase of
the cuprates has two major features: (i) nearby optimum
doping the in-plane resistivity is linear in T, signaling the
absence of any other energy scale besides the temperature;
(ii) in the underdoped regime photoemission and tunnel-
ing experiments show that a pseudogap persists well above
the critical temperature Tc up to a crossover tempera-
ture T∗ [1]. While Tc increases with increasing doping,
T∗ starts from much higher values and decreases. The
two temperatures merge around or slightly above opti-
mum doping. Angle-resolved photoemission spectroscopy
(ARPES) experiments indicate that the pseudogap and
the superconducting gap have the same momentum depen-
dence across Tc, which almost resembles a dx −y -wave,
namely ∆k = ∆0(φ) cos(2φ), with φ = arctan(ky/kx), and
that both are tied to the underlying Fermi surface [2–4].
In the BCS d-wave approach ∆0 is φ-independent and
is proportional to Tc. Here instead ∆0(φ) is angle de-
pendent and the ARPES spectra of the pseudogap state
evolve smoothly and continuously in the superconducting
ones across the critical temperature. A smooth evolution
is also observed in tunneling spectra [5]. Moreover, the
pseudogap at different k points opens at different tem-
peratures. At T∗ a leading-edge shift (LE) appears in the
ARPES spectra of underdoped Bi2212 for momenta near
the M ≡ (ꢀπ, 0); (0, ꢀπ) points of the Brillouin zone. The
2
2
a
e-mail: Lara.Benfatto@roma1.infn.it

96
The European Physical Journal B
more effectively than along the diagonals and should have ture Tc, by coupling to the stiffness of the pairing near the
the temperature as the only energy scale around optimum nodal points [11].
doping. It was shown that in strongly correlated systems,
in the presence of additional attractive interactions (e.g.
the Hubbard-Holstein model) and of long-range Coulomb
In this paper we elaborate the other possibility, that
the transition to the superconducting state takes place in
the presence of a normal-state pseudogap parameter ∆p
forces, the exchange of quasi-critical charge (and spin) resulting from interactions in the particle-hole channel.
fluctuations provides such an effective electron-electron in-
teraction both in the particle-particle and in the particle-
The issue arises of the interplay between the preformed
pseudogap in the p-h channel and the additional pairing
hole channel [9]. Non-Fermi-liquid behaviour and strong in the p-p channel. Having included most of the anoma-
pairing mechanism have in this way a common origin. lous effects in the pseudogap formation, we determine Tc
These fluctuations arise near a finite-temperature instabil-
ity line TCDW(δ) for charge-density wave or stripe-phase
via the BCS approach for the pairing in the p-p channel.
Our model originates as a simple schematization of a sys-
formation, which ends in a quantum critical point (QCP) tem interacting via the singular effective interaction (1)
at T = 0 and δ = δc near optimum doping [9,10]. As shown and is inspired to a similar model proposed by Nozi`eres
in reference [9], the effective electron-electron interaction
near the charge instability has the form
and Pistolesi [12], with the inclusion of some specific as-
pects of the phenomenology of the cuprates. In Section 2
we discuss the model for the normal-state spectrum in the
presence of a pseudogap which has a d-wave form with am-
plitude ∆p. Assuming at the beginning a constant ∆p, we
discuss in Section 3 the general properties of our model,
devoting a particular attention to the doping and/or tem-
perature dependence of Tc, of the LE and of the superfluid
density. In Section 4 we introduce a modulation for ∆p,
to take care of the δ-dependence of the new energy scale
set by the TCDW(δ) in the underdoped case. We assume
that the pseudogap opens nearby a mean-field tempera-
ture T0(δ) for the onset of CDW. T0(δ) should follow the
doping dependence of TCDW(δ) in the underdoped regime
and produce a variation in the density of states, as re-
vealed by NMR and resistivity measurements on several
compounds [1]. By a suitable fitting of T0(δ), we give at
the end a phenomenological description of the phase dia-
gram of the cuprates, together with some physical quan-
tities like the superfluid density, the specific heat and the
leading edge.
V
˜
Veff(q, ω) ' U −
,
(1)
κ2 + |q − qc| − iγω
2
both in the particle-hole and particle-particle channels.
Here q ≡ (qx, qy) and ω are the exchanged momenta and
˜
frequencies in the quasiparticle scattering, U is a resid-
ual repulsion, V is the strength of the attractive effec-
tive potential, qc is the critical wave-vector related to the
charge ordering periodicity (qc = 2π/λc). For physically
relevant values of the parameters of the Hubbard-Holstein
model qc turns out to be (ꢀ0.28, ꢀ0.86) or equivalently
(ꢀ0.86, ꢀ0.28) [9]. In this case, therefore, qc connects the
two branches of the FS around the M points and strongly
affects these states. The mass term κ2 = ξc−2 is the inverse
square of the correlation length of the charge order and
provides a measure of the distance from criticality. This
is given by δ − δc in the overdoped region, by T in the
quantum critical region around δc and by T − TCDW(δ)
in the underdoped region, where TCDW(δ) sets in a new
doping-dependent energy scale closely followed by T∗(δ).
The characteristic time scale of the critical fluctuations
is γ. The presence of a weak momentum-independent re-
2 The model
˜
pulsion U together with a strong attraction of the order
Within the above scenario, we describe the pseudogap in
the normal state by means of a simplified model where a
k-dependent separation is present between a valence band
and a conduction band, as a result of a k-dependent ef-
fective interaction in the particle-hole channel. Differently
from reference [12], we adopt a lattice electron model and
assume that the pseudogap vanishes at some points of
the Brillouin zone. Being interested to very qualitative as-
pects of the evolution of the pseudogap state, we shall
mainly concentrate on a two-dimensional system related
to the CuO2 planes, the third dimension being relevant
to establish the nature of the true transition and to cut
off the corresponding fluctuations. Accordingly, we model
the normal-state spectrum as
of −V/κ2 in the particle-particle channel (cf. Eq. (1))
favors d-wave superconductivity approaching optimum
doping from the overdoped regime, within direct BCS
calculations [10]. In the underdoped regime we expect
that precursor effects of charge ordering are relevant to
the pseudogap formation and extend up to a temperature
T0(δ), (the mean field temperature for CDW formation)
higher then TCDW ∼ T∗.
The two limiting cases when these precursors dominate
the pseudogap formation in a single channel only (either
particle-particle or particle-hole channel) are simpler to
analyze and each of them shows relevant aspects of the
physics of the cuprates. The interplay of the two channels
is an open problem still under investigation.
q
A first possibility is that the pseudogap opens due to
incoherent paring in the particle-particle channel, lead-
ing to a state where cooper pairs around the M points
ξηk = −µ + η ꢀ2k + ∆p2γk2,
(2)
are formed at T∗ & TCDW with strong long-wavelength where η = +1(−1) in the conduction (valence) band, ꢀk =
fluctuations. Phase coherence, which characterizes a real −2t(coskx + cos ky) is the tight-binding dispersion law in
superconducting state, is established at a lower tempera- the conventional metallic state (i.e. at ∆p = 0), ∆p is

L. Benfatto et al.: Gap and pseudogap evolution within the charge-ordering scenario
97
6
5
4
3
2
1
0
at ξ = −µ for ∆p = 0 is thus split into two singularities
∆ =160 meV
p
p
∆ =0
p
separated by 4t∆p/ (∆p/2)2 + 4t2. For ∆p ꢁ 2t, as we
shall assume in most of our calculation, the energy range
where the density of states in suppressed is of order 2∆p.
Having included most of the effects of the anisotropic
potential in the k-dependence of the pseudogap ∆p|γk|,
for simplicity we assume that the onset of superconduc-
tivity is produced within a BCS approach by a constant
pairing interaction among the carriers in the Cooper chan-
nel, apart again from a d-wave modulation. The relevance
of the superconducting fluctuations of the phase of the or-
der parameter will be discussed below, in parallel with the
analysis of the properties of the superfluid density.
-1.2 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
1.2
ξ (eV)
We introduce therefore an intraband interaction term
between time-reversed states
Fig. 1. Density of states at half-filling (µ = 0) in the presence
and in the absence of pseudogap for a hopping parameter t =
0.25 eV. Observe in the case ∆p = 160 meV the splitting of the
van Hove singularity which gives rise to two peaks at energies
ξ ≈ ꢀ∆p, with ∆p ꢁ 2t.
Z
d2k
X
HI = −V Ω
γkc+η,k,↑c+η,−k,↓
(2π)2
η
Z
d2k0
(2π)2
0
0
0
,
×
γk cη,−k ,↓cη,k ,↑
(3)
the pseudogap parameter, and µ is the chemical potential.
∆
p
governs together with the chemical potential the LE
opening at the M points. The modulation of the pseudogap where V is the pairing strength, Ω is the volume of
is given by the absolute value of γk = (cos kx − cos ky)/2, the system, c+η,k,σ is the creation operator of the elec-
which vanishes along the diagonals of the Brillouin zone.
0
trons in the η-band, and the factors γk, γk make the d-
Since ꢀk vanishes along the lines ky = ꢀ(π ꢀ kx), the
wave symmetry explicit. Introducing the order parameters
∆
= V hγkc+η,k,↑c+η,−k,↓i we obtain the self-consistency
two bands merge at the points kP = (ꢀπ/2, ꢀπ/2). In
the undoped system the LE has the k-modulation given
by the factor γk. As the system is doped with respect to
half-filling, a Fermi surface appears in the form of small
pockets around the kP points. The LE, which vanishes
along the Fermi surface, evolves thus continuously with
respect to the undoped case. This is in contrast with the
case of a k-independent pseudogap ∆p, which leads, in
the lattice model, to a large Fermi surface in the doped
system, with the chemical potential and the LE which
evolve discontinuously with respect to half-filling.
η
equations
ꢀ
ꢁ
Z
V
∆η =
2
d2k γk2
βEηk
2
tanh
∆
(η = ꢀ1) (4)
η
(2π)2 Eηk
ꢂ
ꢀ
ꢁꢃ
βEηk
, (5)
2
Z
1
2
d2k
(2π)2
ξηk
X
δ = 1 −
1 −
tanh
Eηk
η
q
where Eηk
=
ξη2k + ∆η2γk2 is the quasiparticle energy
We are aware of the fact that the complicated band
structure of the cuprates is not simply fitted by the
form (2). For instance, the band structure can be improved
by extending the tight-binding expression beyond nearest
neighbours. However, this does not represent a severe lim-
itation to our approach, the generalization being straight-
forward. Moreover, as pointed out above, the presence of
the pseudogap leads, within our model, to the appearance
of small pockets in the weakly doped system. This issue is
experimentally controversial, and it seems definitely con-
firmed that the two branches of each pocket, even when
observed [13], are not equivalent [14]. However, the pres-
ence of pockets around the kP points, which result from
the oversimplification of our description, is not essential
to reproduce the finite LE observed in the single-particle
ARPES spectra near the M points, and does not play any
significant role in the forthcoming analysis. Therefore, we
shall not further discuss this aspect.
in the superconducting state. The last equation fixes the
chemical potential µ for any given doping δ with respect
to half-filling. As most of the cuprates become supercon-
ducting by doping with holes, we study the hole-doped
regime δ > 0, where the chemical potential at T = 0, in
the absence of pairing, falls in the valence band. There-
fore, at weak-coupling, we find the solution ∆η=−1 = ∆,
∆
= 0 for equation (4). We point out that, due to the
asηs=um+1ed band structure (2), the spectrum is particle-hole
symmetric, and doping with electrons leads to the same
self-consistent solution, provided the role of the two bands
is interchanged.
Adding to the Hamiltonian (3) an interband inter-
action term, as discussed in reference [12], induces the
∆
η=+1
to be different from zero even at weak-coupling.
The two order parameters are however generically dif-
ferent, contrary to the assumption of reference [12]. In-
The density of states corresponding to the band struc- deed, the solution with a single order parameter (∆η=+1
ture (2) vanishes only at ξ = −µ, and is finite elsewhere, as = ∆) has a higher free energy, which may become
shown in Figure 1. The van Hove singularity which exists equal under very specific conditions.
=
∆
η=−1

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The European Physical Journal B
0.035
0.03
3 General properties of the model
∆ =160 meV
p
∆=19 meV
In this section we shall analyze the main outcomes of our
model and their dependence on doping δ and on the tem-
perature T. The parameters ∆p and V at the beginning
will be assumed to be temperature and doping indepen-
dent, and not related to each other. This preliminary study
allows us to point out the consequences of the assumed k-
dependent normal-state pseudogap, and to single out the
regime of parameters suitable for a description of the phe-
nomenology of the cuprates.
By solving the equations (4, 5) we obtain the temper-
ature and doping dependence of the superconducting gap
∆ and of the chemical potential µ, and we determine the
phase diagram of the model in different physical regimes.
Before analyzing the two regimes of weak or strong
coupling at various doping, we investigate the existence
of a finite strength Vc of the interaction to have super-
µ=-130 meV
0.025
0.02
0.015
0.01
0.005
0
∆ =0
p
∆=12.6 meV
µ=-46 meV
0
5
10 15 20 25 30 35 40 45
ϕ (degrees)
Fig. 2. Zero-temperature LE vs. φ for t = 250 meV, δ = 0.1,
in the weak-coupling regime V = 87 meV, in the presence
(×) and in the absence (+) of the normal-state pseudogap ∆p.
conductivity in the undoped system, due to the fact that Observe, for ∆p = 160 meV, the crossover around φ ≈ 12◦ to
a regime dominated by the normal state pseudogap.
at half-filling the density of states vanishes at the Fermi
energy. In our model, due to the d-wave nature of the
preformed gap, the value of Vc is less than that found in
3.2 The leading edge
reference [12] in presence of a constant gap. The equa-
tion that defines Vc at T = 0, δ = 0 as a function of the
pseudogap amplitude ∆p is
The quasiparticle spectrum, both in the normal and in
the superconducting state, is characterized by a leading-
edge shift LE, which is defined as LE(φ) = minη,k Eηk
Z
2t
Vc
1
2
d2k
(2π)2
γk2
p
=
(6)
,
(cos kx + cos ky)2 + (∆p/2t)2γk2
where k = k(cos φ, sin φ)1. Due to the point symmetries
of the system our analysis can be limited to the quad-
rant 0 ≤ φ ≤ 45◦. In the strong coupling regime the
superconducting gap ∆ ꢂ ∆p, so the doping variation
of ∆ dominates the evolution of the zero-temperature LE
at the M points, which increases approaching optimum
doping, following Tc, in contrast with the behaviour ob-
served in the experiments in the underdoped regime. In
the weak-coupling regime instead the superconducting gap
is much smaller than the pseudogap. For δ < δopt the LE
of the superconducting state is controlled by two indepen-
dent parameters (see Fig. 2), whose existence is also sug-
gested by the combination of ARPES, penetration-depth
and Andreev experiments [2–4,6,7]. The amplitude at the
M points (LEM) is given essentially by the value of the LE
in the normal state, i.e.
so that Vc ≈ π2∆p/2, for ∆p ꢂ 2t, and Vc ∼ 2t/ ln(4t/∆p)
for ∆p ꢁ 2t. The former limiting case is however not
realistic: henceforth we shall definitely assume ∆p ꢁ 2t.
Moreover, in the present scenario, the pseudogap is asso-
ciated with scattering in the particle-hole channel. Pairing
must therefore be absent at half-filling and we assume as
the relevant regime for the cuprates V < Vc.
3.1 Critical temperature
Within the BCS approach, the variation of the critical
temperature with doping is controlled by the density of
states at the Fermi level. At ∆p = 0, we recover the spec-
trum of a normal metal with nearest-neighbours hopping:
at half-filling the Fermi energy is at the van Hove sin-
gularity, and Tc is maximum. By increasing doping, the
density of states at the Fermi energy decreases, and so
does Tc. This behaviour is excluded by the experiments.
On the contrary, when ∆p > 0, the critical temperature is
zero at half-filling (if V < Vc). At fixed finite doping, the
critical temperature increases with increasing ∆p, reaches
a maximum, and then decreases. By doping, the critical
temperature follows the evolution of the density of states
at the Fermi energy, reaching a maximum when the Fermi
energy passes through the boundary of the pseudogap re-
gion in the density of states, i.e. at the doping δopt such
that µ ≈ −∆p. For δ > δopt, Tc decreases with increasing
doping. The resulting bell-shaped Tc vs. δ curve captures
the main features of the experimental results, as it will
be more accurately discussed in Section 4 with a doping
dependent ∆p.
2t∆p
(∆p/2)2 + 4t2
p
LEM
=
− |µ| ≈ ∆p − |µ|
(∆p ꢁ 2t)
(7)
independent of Tc, whereas the slope at the nodes is
v∆ = ∆0(φ = π/4) ∝ ∆ ∝ Tc, in agreement with the
experimental finding [4,6,7]. Note instead that LEM ≈ ∆
for ∆p = 0.
1
Within our mean-field treatment the excitations have a
well defined energy and this quantity should be more cor-
rectly called “one-particle excitation gap”. However, by using
the term LE we want to put more emphasis on the distribu-
tion of spectral weight than on its coherent character, which
is an artifact of our description of the cuprates in terms of a
semiconducting band structure.

L. Benfatto et al.: Gap and pseudogap evolution within the charge-ordering scenario
99
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
3.3 The superfluid density
δ=0.10
δ=0.14
δ=0.20
δ=0.24
In the superconducting phase the thermodynamic proper-
ties at low temperature are essentially determined by the
quasiparticles near the nodes, i.e. by the superconduct-
ing gap ∆, whereas the pseudogap amplitude ∆p plays no
role. The superfluid density for instance can be evaluated
according to the simple BCS formula:
ꢄ
ꢀ
ꢁ
Z
X
1
2
d2k
(2π)2
∂2ξηk
∂kl2
ꢀ
ξηk
βEηk
ρs =
1 −
tanh
Eηk
2
η,l=x,y
)
ꢁ
2
∂ξk
∂f(Eηk)
∂Eηk
0
5
10
15
20
25
30
35
40
45
+2
,
(8)
∂kl
ϕ (degrees)
Fig. 3. Doping dependence of the zero temperature LE for
t = 250 meV, ∆p = 160 meV and V = 87 meV. The
LEM = LE(φ = 0) decreases with increasing doping, due to
the reduction of the normal-state LEM given by equation (7).
At δ > δopt ≈ 0.2 the normal state LE closes and LE(φ, T = 0)
is that of a standard d-wave superconductor (see also Fig. 2).
Instead, the slope of the LE at the node v∆ ∝ ∆ increases be-
fore reaching δopt and then decreases. The critical temperature
(proportional to ∆(T = 0) in the BCS approach) follows the
same doping dependence.
where f(x) is the Fermi function. ρs decreases linearly in T
at low temperature, as expected in a d-wave superconduc-
tor, with a slope determined only by the superconducting
gap. For the assumed band dispersion (2) it can be shown
that
T 16t ln2
ρs(T) − ρs(0) ' −
·
(9)
∆
π
The slope of the superfluid density at low temperature,
α = dρs/dT(T = 0) is estimated in reference [4] by means
of direct ARPES measurements of the slope v∆ of the su-
perconducting gap near the node. Because the observed
v∆ decreases in underdoped region (contrary to the gap
at the M points), the doping dependence of α ∝ 1/v∆ is
found to increase by decreasing doping with respect to its
optimum value. This behaviour is confirmed by our model:
the slope of dρs/dT given in equation (9) is controlled by
the superconducting gap ∆, which follows the doping de-
pendence of Tc in the underdoped regime. However, as ob-
served in reference [4], the general trend of α(δ), obtained
by direct measurements of penetration depth, seems to be
opposite, giving a decrease in the underdoped compounds
of about 40% respect to its value at optimum doping.
The correspondence between ARPES estimates and ex-
perimental data on α(δ) requires the inclusion of doping
dependent Landau renormalization factors, as discussed
in reference [4] using the expression for ρs with the inclu-
sion of quasiparticle interaction (see also Ref. [15]). The
presence of a doping-dependent Landau factor is plausible
within our scenario if we attribute the origin of quasiparti-
cle scattering to quasi-critical charge and spin fluctuations
0.035
T=0
T=8.0 meV
0.03
0.025
0.02
0.015
0.01
0.005
0
T=8.5 meV
T=9.5 meV
0
5
10
15
20
25
30
35
40
45
ϕ (degrees)
Fig. 4. Temperature dependence of the LE for the same set
of parameters of Figure 3 (increasing temperature from top to
bottom). At T > Tc = 9 meV the superconducting gap closes,
leaving a finite LEM ≈ ∆p − |µ|.
The variation of the chemical potential accounts now as in equation (1). The same holds for a correspondence
for the main dependence of the LE on doping. By increas- between the slope of the density of states N(ξ) near ξ = 0,
ing doping the normal-state LEM, controlled by the pseu- where N(ξ) ≈ ηξ with η ∝ 1/v∆, and the values of η ex-
dogap parameter ∆p according to equation (7), decreases tracted by specific heat measurements [16].
and vanishes at a doping δc. If ∆p is doping independent
As the temperature is increased, the system crosses
(as we are assuming in this preliminary discussion), the over to a regime of higher energy excitations, which sample
δc for which LEM = 0 coincides with the doping δopt for regions where LE > ∆, and the slope is reduced. This
the highest Tc. At δ > δc the LEM in the superconduct- can be seen in Figure 5, where the temperature variation
ing state is then given only by the superconducting order of ρs(T) in weak and strong coupling is reported. The
parameter ∆, as shown in Figure 3. At fixed doping, the variation of the slope with respect to its low-temperature
temperature variation of the LE depends on the closure value is less pronounced in the strong-coupling case, where
of the superconducting gap at Tc, and eventually on the almost the same parameter controls the LEM and the slope
temperature dependence of ∆p, if any (see Fig. 4).
of the superconducting gap at the node.

100
The European Physical Journal B
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Phase fluctuations effects have also been invoked to ex-
δ=0.065
δ=0.075
δ=0.1
δ=0.05
δ=0.1
δ=0.15
δ=0.2
8T/π
plain the linear-T behaviour of ρs(T) [17]. Within a model
with pseudogap formation in the p-p channel [11] these ef-
fects deserve a careful analysis.
δ=0.15
δ=0.2
8T/π
0.25
0.2
4 Conclusions and discussion
0.15
0.1
In conclusion we want to summarize the previous re-
sults in closer connection with the phenomenology of the
cuprates. The discussion of the previous section has given
some preliminary indications on the set of parameters suit-
able to reproduce the general shape of the phase diagram
observed in the cuprates. Nevertheless, to be consistent
with the experiments, we have to introduce the tempera-
ture and doping dependence of the pseudogap parameter
∆p(T, δ).
The mechanism for pseudogap formation has not been
the main issue in this paper, which focuses on the prop-
erties of superconductivity arising in the pseudogap state.
Nevertheless, according to the discussion of Section 1,
we can imagine that ∆p(T, δ)|γk| schematizes the whole
complication of the strong scattering due to quasicriti-
cal charge fluctuations in the p-h channel, which arises in
the stripe-QCP scenario [9]. We assume that the doping
dependence of ∆p follows the doping variation of the tem-
perature T0 (T0 > T∗) at which a first variation in the
density of states occurs, as revealed by NMR and resistiv-
ity measurements [1]. We adopt the simple relation:
0.05
0
0
0.004 0.008 0.012 0.016
T (eV)
0 0.05 0.1 0.15 0.2 0.25 0.3
T (eV)
Fig. 5. ρs(T) at different doping for ∆p = 0.16 eV, t = 0.25 eV
in weak coupling (V = 0.09 eV, left) and strong coupling (V =
0.9 eV, right). The temperature at which the numerical curves
intersect the line 8T/π is the TKT, as explained in the text.
Phase fluctuations are important between TKT and TBCS
.
Finally, we consider the doping dependence of the su-
perfluid density. The variation of the ρs(0) with doping is
related to the variation of the number of carriers with dop-
ing: in a lattice model it does not increases monotonically,
but reaches a maximum at an intermediate 0 < δ < 1, and
then decreases. Nevertheless, the doping for the maximum
ρs(0) does not necessarily coincide with δopt
.
The temperature dependence of the superfluid density
allows us to give an estimate of the range of tempera-
tures in which the phase fluctuations of the supercon-
ducting order parameter are relevant. Indeed, the criti-
cal temperature for phase coherence in a two-dimensional
system, TKT, is related to the phase stiffness ρs via the
Kosterlitz-Thouless relation ρs = 8TKT/π. Assuming the
BCS temperature dependence for ρs(T), the above relation
becomes a self-consistency condition ρs(TKT) = 8TKT/π,
and the phase fluctuations play an important role in the
range of temperatures between TKT and TBCS. As it is
naturally, in our model this region is very narrow in weak
coupling, and becomes larger and larger as the strength
of the pairing interaction increases. Indeed while ρs(0)
mainly depends on the doping δ, TBCS rapidly increases
with increasing the coupling, and becomes much larger
than TKT. In reference [12], due to the discontinuity of
the density of states at the band edges, at low doping
a regime Tc ꢂ TKT is found. In such a case the transi-
tion is expected to be Kosterlitz-Thouless. In our model
the density of states at the Fermi level vanishes smoothly
and continuosly as half-filling is approached. Therefore,
in the weak-coupling limit, the critical temperature in-
creases slowly with doping, and the Kosterlitz-Thouless
regime is never found. Consistently with our assumptions
within this model, in the regime of interest for cuprates,
i.e. at weak coupling, the superconducting transition is
essentially BCS-like, even in the underdoped region. The
main part of the non classical behaviour is enforced by the
presence of ∆p.
∆p(T, δ) = cT0(δ)g (T/T0(δ)) ,
(10)
where g(1) = 0, g(0) = 1. g(x) interpolates smoothly be-
tween these two limits2 and c is a constant which we use
as a fitting parameter. Since the identification of T0(δ)
via NMR and resistivity measurements leaves large un-
certainty, we decide to refer to the ARPES experiments,
which however only provide the temperature T∗ of LE
closure. We proceed in the following way. In the heavily
underdoped regime the main temperature dependence of
the LEM (at fixed doping) is due to the decreasing of ∆p
with increasing the temperature, rather than to the tem-
perature variation of the chemical potential. Then T∗ and
T0 coincide at low doping in our model, whereas the ex-
perimental data seem to indicate that T0 stays larger then
T∗. This is an artifact of our approach, which is not de-
voted to establish the connection between the pseudogap
state and the Mott insulator and/or the antiferromagnetic
phase. This is a relevant open problem still under inves-
tigation. Meanwhile we put T0(δ = 0) = T∗(δ = 0) and
extrapolating to δ = 0 the T∗(δ) dependence reported in
reference [2] for Bi2212 we obtain T0(δ = 0) = 40 meV.
The constant c and the coupling V are adjusted to fix the
values of the LE in the underdoped regime and the maxi-
mum Tc. The temperature T0 is assumed to decrease with
√
Here we assume g(x) = (1 − x4/3) 1 − x4, which repro-
2
duces a mean-field-like behaviour near T = 0 and T0. However,
the specific form of g(x) does not play a crucial role in deter-
mining the general shape of the phase diagram.

L. Benfatto et al.: Gap and pseudogap evolution within the charge-ordering scenario
101
400
350
300
250
200
150
100
50
400
350
300
250
200
150
100
50
1
0.8
0.6
0.4
0.2
0
δ=0.12
δ=0.16
δ=0.18
δ=0.22
T
0
LE/2
*
T
T
∆/2
c
0
0.2
0.4
0.6
0.8
1
0
0
T/T
0.08 0.1 0.120.140.160.18 0.2 0.22
0.08 0.1 0.120.140.160.18 0.2 0.22
c
δ
δ
Fig. 7. Doping dependence of the normalized superfluid den-
sity ρs(T)/ρs(0) as a function of T/Tc. We used the same set
of parameters of Figure 4. According to the phase diagram of
Figure 4, the optimum doping is at δopt ≈ 0.16. Observe that
in the overdoped regime δ > δopt the superfluid density has
the standard d-wave mean field behaviour.
Fig. 6. Left panel: doping dependence of T∗, T0 and of the
critical temperature Tc. Right panel: doping dependence of the
superconducting gap ∆(T = 0) and of the zero temperature
LEM. The set of parameters used is t = 250 meV, V = 85 meV,
c = 7, δ0 = 0.2, and we assumed T0(δ) = 40[1 − (δ/δ0)4] meV.
The value of T∗(δ = 0) = T0(δ = 0) has been extrapolated
from reference [2].
LEM closes faster than it would be under the only effect
of the variation of the chemical potential with doping.
Finally, we report in Figure 7 the normalized super-
fluid density ρs(T)/ρs(0) as a function of T/Tc at var-
ious doping. In the overdoped regime, where the pseu-
dogap closes, we recover the standard d-wave mean
field result, which seems in a good agreement with experi-
mental data [16] (see the curve for δ = 0.22 in Fig. 7).
However, in the underdoped region the dependence on
doping of dρs/dT(T = 0) does not agree with the experi-
ments as already discussed in Section 3, where the possible
improvement due to Landau factors has been adressed. We
observe here another discrepancy: the slope of ρs(T) near
Tc in our model reduces by increasing the temperature
contrary to the experimental findings [16]. As already in-
dicated in Section 3, within our model the superconduct-
ing fluctuations are only important in the short region
between TKT and TBCS. The inclusion of these fluctua-
tions, by reducing ρs near Tc, would at least partially take
care of this discrepancy.
increasing doping and to vanish at δ0 = 0.2, slightly above
the optimum doping, as expected in the QCP scenario [9].
Finally, we interpolate between δ = 0 and δ0 with the ex-
pression T0(δ) = 40[1 − (δ/δ0)4] meV, which allows us to
reproduce the shape of the curve T∗(δ) observed in the
underdoped regime.
The resulting phase diagram is shown in Figure 6. Even
within a simplified description of the doping and tem-
perature dependence of ∆p, we recover the bifurcation
of T∗ and Tc observed in the underdoped regime, while
they merge around optimum doping. Tc follows the typ-
ical bell-shaped curve as a function of δ. Our approach
is still lacking of the temperature and doping dependence
of the superconducting coupling, which plays an impor-
tant role in determining the Tc(δ) variation around the
QCP, as it has been shown elsewhere [9,10]. Even though
the introduction of this further complication would defini-
tively improve the agreement with the experimental data,
with a faster decay of Tc at high doping, it would not
change the main features of the pseudogap state obtained
here. We reproduce the variation of the zero temperature
LEM in all the phase diagram: according to the general
discussion of the previous section, the LEM is determined
by the normal state pseudogap in the underdoped region,
and by the superconducting gap in the overdoped regime.
As a consequence, in the underdoped regime the LEM is
uncorrelated to the characteristic energy scale of low tem-
perature quasiparticle excitations, which probe the value
of the superconducting order parameter around the nodal
points. Having now a doping and temperature dependent
∆p, the doping for the maximum Tc does not coincide any-
more with the doping at which the normal state leading
edge closes, as we had in the presence of a constant ∆p.
Indeed, approaching optimum doping ∆p itself closes at a
lower temperature, and as a consequence the normal state
Even though we did not reach yet the point to exploit
fully the momentum and doping dependence of a quasi-
singular effective interaction among quasiparticles arising
nearby an instability in the p-p and p-h channels, alto-
gether within the present simplified approach we produce
a behaviour of Tc(δ), of T∗(δ) and of the LEM(0, δ) in
reasonable agreement with the experimental findings.
We acknowledge A. Perali, C. Castellani, and M. Grilli for
helpful discussions.
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