Nodal/Antinodal Dichotomy and the Two Gaps of a Superconducting Doped
  Mott Insulator
Nodal/Antinodal Dichotomy and the Two Gaps
of a Superconducting Doped Mott Insulator
M. Civelli1, M. Capone2, A. Georges3, K. Haule4, O. Parcollet5, T. D. Stanescu6 and G. Kotliar4
1 Theory Group, Institut Laue Langevin, Grenoble, France
2 SMC, CNR-INFM, and Physics Department, University of Rome “La Sapienza”,
Piazzale A. Moro 5, I-00185, Rome, Italy and ISC-CNR, Via dei Taurini 19, I-00185, Rome, Italy
3 Centre de Physique Théorique, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France
4 Physics Department and Center for Materials Theory, Rutgers University, Piscataway NJ USA
5 Service de Physique Théorique, CEA/DSM/SPhT-CNRS/SPM/URA 2306 CEA-Saclay, F-91191 Gif-sur-Yvette, France and
6 Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, Maryland 20742-4111, USA
(Dated: November 8, 2018)
We study the superconducting state of the hole-doped two-dimensional Hubbard model using
Cellular Dynamical Mean Field Theory, with the Lanczos method as impurity solver. In the under-
doped regime, we find a natural decomposition of the one-particle (photoemission) energy-gap into
two components. The gap in the nodal regions, stemming from the anomalous self-energy, decreases
with decreasing doping. The antinodal gap has an additional contribution from the normal com-
ponent of the self-energy, inherited from the normal-state pseudogap, and it increases as the Mott
insulating phase is approached.
PACS numbers: 71.10.-w,71.10.Fd,74.20.-z,74.72.-h
Superconductivity in strongly correlated materials
such as the high-Tc cuprates has been the subject of in-
tensive research for more than twenty years (for a re-
view see, e.g.,[1]). From the theoretical side, low-energy
descriptions in terms of quasiparticles interacting with
bosonic modes have been widely studied starting from
the weak correlation limit. A different approach views
the essence of the high-Tc phenomenon as deriving from
doping with holes a Mott insulator [2]. The strong corre-
lation viewpoint has not been yet developed into a fully
quantitative theory and whether the weak- and strong-
coupling pictures are qualitatively or only quantitatively
different is an important open issue.
The development of Dynamical Mean Field Theory
(DMFT) and its cluster extensions [3] provides a new
path to investigate strongly correlated systems. These
methods construct a mean-field theory for Hubbard-
like models using a cluster of sites embedded in a self-
consistent bath. Extensive investigations have been car-
ried out for intermediate interaction-strength using the
Dynamical Cluster Approximation on large clusters [4].
The strong coupling limit is more difficult, as only small
clusters can be employed. Many groups however have
identified interesting phenomena, such as the competi-
tion between superconductivity and antiferromagnetism
[5], the presence of a pseudogap (PG) [6], the forma-
tion of Fermi arcs [7, 8, 9, 10] and the existence of an
avoided critical point [11]. In this work we use Cellu-
lar DMFT (CDMFT) to explore the energy gap in the
one-particle spectra of the superconducting state when
correlations are strong. The goal is to identify qualita-
tive aspects of the approach to the Mott transition in the
light of recent experimental studies on superconducting
under-doped cuprates [12, 13, 14, 15, 16], which report
the presence of two distinct energy scales.
We consider the two-dimensional Hubbard Model:
H = −
i,j,σ
tij c
cj,σ + U
ni↑ni↓ (1)
ci,σ destroys a σ-spin electron on site i, niσ = c
iσciσ
is the number operator and tii ≡ µ is the chemical po-
tential. Only next-neighbor t and nearest-next-neighbor
t′ = −0.3t hoppings are considered. The on-site repulsion
is set U = 12t. We implement CDMFT on a 2×2 pla-
quette. Though this is the minimal configuration allow-
ing to study a d-wave superconducting state, it already
presents a rich physics and we think that its deep under-
standing is an essential step to be accomplished before
challenging bigger clusters (hardly accessible by the com-
putational methods presently available). H is mapped
onto a 2×2-cluster Anderson impurity model which is
solved using the Lanczos method [17]. The CDMFT self-
consistency condition [18] is then enforced via the Dyson
relations Σ̂(iωn) = Ĝ−1(iωn) − Ĝ−1[Σ̂](iωn), which de-
termines the cluster self-energy Σ̂. The hat denotes 8×8
matrices with cluster-site indices containing both normal
and anomalous components (Nambu notation). Ĝ is the
”Weiss field” describing the bath, Ĝ[Σ̂] is the one-particle
cluster Green’s function [18] and ωn = (2n + 1)π/β the
Matsubara frequencies, with βt = 300. The bath is de-
scribed by 8 energy levels determined through a fit on
the Matsubara axis (0 < ωn < 2U), which weights more
the low frequencies [8].
Our main result is the presence of two energy-scales
on the under-doped side of the phase diagram. We first
show that this can be established directly from an anal-
ysis of quantities inside the 2 × 2 cluster, which are the
output of the CDMFT procedure. In the left panel of
http://arxiv.org/abs/0704.1486v3
0 0.08 0.16
-0.15 0 0.15
0 0.5 1
0 0.1
12/t -Im G11/π
FIG. 1: (Color online). Left: ReΣano12 vs. ωn. In the inset,
the ωn → 0 value as a function of doping δ. Right: The
distance from the Fermi level (ω = 0) of the left (circle) and
right (square) edge-peaks in the local DOS − 1
ImG11 (see
inset) are displayed as a function of δ. The dashed line is the
average of the left and right values. In the inset G11 is shown
for δ = 0.06 using a broadening η ∼ 7×10−3t to display poles.
Fig. 1 we display the real part of the anomalous clus-
ter self-energy Σano on the Matsubara axis. Only the
nearest-neighbor component ReΣano
(iω) is appreciably
non-zero. The main observation is that at low energy
(0) presents a non-monotonic behavior with dop-
ing δ, as emphasized in the inset. A first characteris-
tic energy-scale, measuring the superconducting contri-
bution to the one-particle energy-gap, can be defined
as ZnodΣ
(0), where Znod is the quasiparticle spec-
tral weight at the nodal k-points, where quasiparticles
are well defined. As shown below, and as physically
expected, Znod decreases as the doping is reduced to-
wards the Mott insulator. Hence, ZnodΣ
(0) decreases
too due to the behavior of both Znod and Σ
(0). We
stress the sharp contrast of this result with resonating
valence bond mean-field (RVB-MF) theories [19], where
ZnodΣ
(0) corresponds to the spinon pairing amplitude
which is largest close to half-filling. In the right panel of
Fig. 1 we show that there is actually another energy-
scale, which increases when the doping level is reduced.
This is revealed by looking at the local density of state
(DOS) − 1
ImG11 in Ĝ[Σ̂]. In the Lanczos-CDMFT the
spectral function on the real axis is obtained as a discrete
set of poles (shown in the inset), which are displayed by
adding a small imaginary broadening iη. We extract rel-
evant energy scales by measuring the distance from the
Fermi level of the gap edge-peaks. While for δ > 0.08
the spectrum is symmetric, an asymmetry appears for
δ < 0.08. The total energy gap (dashed line in Fig. 1)
grows with decreasing doping δ, as in RVB-MF theories.
In order to make contact with experimental observ-
ables it is useful to obtain momentum-resolved quantities
from the local cluster quantities. For this we need a pe-
riodization procedure restoring the translational invari-
ance of the lattice. Several schemes have been proposed
[3]. Building on previous normal-state studies [7, 9] we
use a mixed scheme which is able to reconstruct the local
cluster Green’s function (upon integrating over k the lat-
tice Green’s function) in the nodal and antinodal points
better than uniform periodization schemes. Our method
is based on the idea that, when the self-energies are reg-
ular the, most suitable choice is to periodize the cluster
self-energy via the formula
Σσ(k, ω) =
e−ikµ Σµν,σ(ω) e
ikν (2)
(µ, ν label cluster sites). The anomalous self-energy
Σano and the normal self-energy Σnor in the nodal re-
gions, where we expect to find quasiparticles, are well
behaved quantities, therefore we extract them through
formula (2). In particular, the anomalous self-energy
acquires a standard dx2−y2 -wave form: Σ
ano(k, ω) =
(ω) (cos kx − cos ky). On the other hand, when the
self-energies develop singularities, the cluster self-energy
is not a good quantity to be periodized. In Ref. [9], it
has been shown that this takes place in the normal self-
energy Σnor in the antinodal regions, when the system
approaches the Mott insulator. In this case, a more suit-
able quantity to be periodized is the the irreducible two-
point cluster cumulant Mnorσ (ω) =
(ω + µ)1̂− Σ̂norσ
which is a more local and regular quantity. In the antin-
odal region, therefore, we can apply formula (2) to Mnor,
to obtain Mnor(k, ω) and finally extract the normal lat-
tice self-energy Σnor(k, ω) = ω + µ− 1/Mnor(k, ω). The
k-dependent Green’s function can be written as a matrix
in Nambu’s space.
(ω) =
ω − tk − Σnorσ (k, ω) −Σano(k, ω)
−Σano(k, ω) ω + tk +Σnorσ (k,−ω)∗
The imaginary part of the diagonal entry yields the spec-
tral function A(k, ω) measured in photoemission.
In order to compare our results with experiments, it
is useful to disentangle the normal and superconducting
contributions to the spectral gap. To this end, we first
set Σano = 0 in Eq. (3). The results are shown in Fig.
2. The k-points along the nodal and antinodal directions
are chosen as those where the highest peak is observed in
A(k, ω), as done, e.g., in Ref. [13]. Their actual values are
shown in the inset of panel C of Fig. 2. Near the nodal
point (panel A) a quasiparticle peak is well defined at the
Fermi level (ω = 0) and decreases by decreasing doping.
In the antinodal region (panel B), a quasiparticle peak is
also found at the Fermi level for δ > 0.08. For δ < 0.08,
however, the spectral weight shifts to negative energies
signaling the opening of a PG, whose size increases as
δ → 0. The behaviour of the PG in the superconducting
solution smoothly connects to the PG previously found
in the normal state CDMFT study [9]. The approach to
the Mott transition is characterized by a strong reduction
in the area of the nodal spectral peak Znod, which is plot-
ted in panel C (green circles). We also plot the area of
-0.2 0 0.2
-0.2 0 0.2
δ=0.16
δ=0.13
δ=0.08
δ=0.05
δ=0.02
-0.2 0 0.2
0 0.08 0.16
antinodal
antinodal
nodal
nodal
(π,π)(0,π)
(0,0)
antinodal
0 0.08 0.16δ
FIG. 2: (Color online) Spectral function A(k, ω) for different
δ. Broadening η = 0.03t. Panel A: nodal quasiparticle peak;
Panel B, normal component (set Σano = 0 in Eq. (3)) of the
antinodal quasiparticle peaks; Panel C, nodal and antinodal
quasiparticle weights. The inset shows the k-positions of the
nodal and antinodal points; Panel D, spectra at the antinodes.
the antinodal peak Zanod, which shows a constant value
upon the opening of the PG (δ > 0.08). In panel D, we
restore Σano, and examine the actual superconducting so-
lution. The superconducting gap opens in the antinodal
region (the nodal region is practically unaffected). For
δ > 0.08 the spectra are almost symmetric around the
Fermi level, as in a standard BCS d-wave superconduc-
tor. In contrast, close to the Mott transition the PG,
which originates from the normal component, is super-
imposed to the superconducting gap, resulting in asym-
metric spectra. This reveals the origin of the left/right
asymmetry in the cluster DOS discussed in Fig. 1.
In the nodal region the quasiparticle peaks are
well defined at all dopings and we can expand the
self-energies at low frequencies. The quasiparti-
cle residue (1− ∂ωReΣk(ω))−1
(blue crosses in
panel C of Fig. 2) numerically coincides with the
area of the quasiparticle peak Znod. From Eq.
(3), we get A(k, ω) ≃ Znod δ
k2⊥ + v
where vnod = Znod|∇k (tk − Σnor(k, 0)) | and v∆ =√
2ZnodΣ
(0)| sin knod| are the normal and anomalous
velocities respectively perpendicular and parallel to the
Fermi surface. v∆ physically expresses the superconduct-
ing energy-scale discussed in the left panel of Fig. 1.
We display them as a function of doping δ on the left
side of Fig. 3. vnod does not show a special trend for
δ → 0 and it stays finite, consistently with experiments
[20]. The anomalous velocity, v∆ ≪ vnod presents a
0 0.05 0.1 0.15
v∆ /(aot)
0 0.05 0.1 0.15
γ/γδ=0.16
α/αδ=0.16
FIG. 3: (Color online) Left: vnod and v∆ as a function of
doping δ (ao is the lattice spacing). Right: low-frequency co-
efficients of local DOS γ and of the Raman B2g and superfluid
density response α, renormalized by the value at δ = 0.16.
dome-like shape. This behavior (confirmed by continuous
time quantum Monte Carlo (CTQMC) calculations [21])
is in agreement with recent experiments on under-doped
cuprates showing that, contrary to the antinodal gap, the
nodal gap decreases by reducing doping [12, 13, 14].
The low-energy behaviour of several physical observ-
ables in the superconducting state is controlled by nodal-
quasiparticle properties and hence can be related to
vnod, v∆ and Znod. Two specific ratios are particu-
larly significant, namely: γ = Znod/(vnodv∆) and α =
/(vnodv∆). The first one is associated with the low-
energy behaviour of the local DOS measured in tun-
neling experiments: N(ω) =
A(k, ω) ∼ γ ω (for
ω → 0). Neglecting vertex corrections [12], the sec-
ond one determines the low-energy B2g Raman response
function χ′′(ω) ∝ αω and the low-temperature (T → 0)
behaviour of the penetration depth (superfluid density)
ρs(T ) − ρs(0) ∝ αT . We display α and γ in the right
panel of Fig. 3 as a function of δ. α is monotonically
decreasing (see also CTQMC results [21]) and, on the
under-doped side δ < 0.08, it saturates to a constant
value, in agreement with Raman spectroscopy [12] and
penetration depth measurements [22]. Also γ neatly de-
creases in going from the over-doped to the under-doped
side, but it presents a weak upturn for low doping. The
low-frequency linear behavior of N(ω) is well established
in scanning tunneling experiments on the cuprates [23].
However, it is not currently possible to determine the ab-
solute values of the tunneling slope α from experiments,
hence the behavior we find is a theoretical prediction.
We finally turn to the one-electron spectra in the antin-
odal region, shown in Fig. 4, physically interpreting the
cluster energy-scales observed in Fig. 1. We evaluate
the antinodal gap in the superconducting state ∆tot by
measuring the distance from the Fermi level (ω = 0) at
which spectral peaks are located (panel D of Fig. 2). ∆tot
monotonically increases by reducing doping, as observed
in experiments. The data of panel B in Fig. 2, where
Σano = 0, allow us to extract the normal contribution
∆nor. We notice that the peaks found there at negative
0.05 0.1 0.15
/t= ( ∆
/t= Z
FIG. 4: (Color online) Antinodal energy gap ∆tot (circles),
obtained from the spectra of panel D in Fig. 2, as a func-
tion of doping δ, and decomposed in a normal contribution
∆nor (squares), obtained from panel B in Fig. 2, and in a
superconducting contribution ∆sc (diamonds).
frequency ωpg do not represent Landau quasiparticles in a
strict sense, but we can estimate the PG as |ωpg|. We also
display the anomalous contribution to the antinodal gap
∆sc =
∆2tot −∆2nor, and find that, within numerical
precision, ∆sc ≃ Zanod|Σano(kanod, ωpg)|. The appear-
ance of ∆nor signs a downturn in ∆sc. We interpret ∆tot
as the monotonically increasing antinodal gap observed
in cuprate superconductors, while the superconducting
gap ∆sc, detectable as the nodal-slope v∆ (Fig. 3), is
decreasing in approaching the Mott transition.
The concept of two energy gaps with distinct doping
dependence in the cuprates has recently been brought
into focus from an analysis of Raman spectroscopy [12],
and photoemission experiments [13, 14], which have re-
vived experimental and theoretical debate [16]. Our the-
oretical dynamical mean-field study of superconductivity
near the Mott transition establishes the remarkable co-
existence of a superconducting gap, stemming from the
anomalous self-energy, with a PG stemming from the
normal self-energy. This is reminiscent of slave-boson
RVB-MF of the t − J model [19, 24], which uses or-
der parameters defined on a link and includes the pos-
sibility of pairing in both the particle-particle and the
particle-hole channels. Compared to the self-energy of
the RVB-MF, the CDMFT lattice-self-energy has con-
siderably stronger variations on the Fermi surface [9] and
additional frequency dependence, which makes the elec-
tron states near the antinodes very incoherent even in
the superconducting state. Furthermore, in the RVB-MF
theory the anomalous self-energy monotonically increases
by decreasing doping, in contrast to our CDMFT results
which reveal a second energy scale associated with su-
perconductivity, distinct from the PG, which decreases
with decreasing doping. Whether this feature survives
in larger clusters, representing a property of the real
ground-state, or it requires some further ingredient to
be stabilized against competing instabilities (above all
antiferromagnetism at low doping [5]) remains an impor-
tant open question addressed to future developments. We
think however that the assumption of a d-wave super-
conducting ground-state is a reasonable starting point,
and the importance of our 2×2-plaquette-CDMFT result
stands in the natural explanation it provides of the prop-
erties of under-doped cuprates.
We thank E.Kats, P.Nozières, P.Phillips, C.Castellani,
A.-M. Tremblay, B.Kyung, S.S. Kancharla, A. Sacuto
and M. Le Tacon for useful discussions. M.Ca. was sup-
ported by MIUR PRIN05 Prot. 200522492, G.K. by the
NSF under Grant No. DMR 0528969.
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http://arxiv.org/abs/cond-mat/0508205
