Proximity of the superconducting dome and the quantum critical point in the two-dimensional Hubbard model.

Abstract

We use the dynamical cluster approximation to understand the proximity of the superconducting dome to the quantum critical point in the two-dimensional Hubbard model. In a BCS formalism, T(c) may be enhanced through an increase in the d-wave pairing interaction (V(d)) or the bare pairing susceptibility (χ(0d)). At optimal doping, where V(d) is revealed to be featureless, we find a power-law behavior of χ(0d)(ω=0), replacing the BCS log, and strongly enhanced T(c). We suggest experiments to verify our predictions.

Full text

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arXiv:1101.6050v2 [cond-mat.supr-con] 1 Feb 2011
Proximity of the Superconducting Dome and the Quantum Critical Point in the
Two-Dimensional Hubbard Model
S.-X. Yang1, H. Fotso1, S.-Q. Su1,2,∗, D. Galanakis1, E. Khatami3, J.-H. She4, J. Moreno1, J. Zaanen4, and M. Jarrell1
1Department of Physics and Astronomy,
Louisiana State University,
Baton Rouge, Louisiana 70803, USA
2Computer Science and Mathematics Division,
Center for Nanophase Materials Sciences,
Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831-6164, USA
3Department of Physics, Georgetown University,
Washington, District of Columbia, 20057, USA
4Instituut-Lorentz for Theoretical Physics,
Universiteit Leiden, P.O. Box 9506,
2300 RA Leiden, The Netherlands
∗shiquansu@hotmail.com
We use the dynamical cluster approximation to understand the proximity of the superconducting
dome to the quantum critical point in the two-dimensional Hubbard model. In a BCS formalism,
Tc may be enhanced through an increase in the d-wave pairing interaction (Vd) or the bare pairing
susceptibility (χ0d). At optimal doping, where Vd is revealed to be featureless, we find a power-law
behavior of χ0d(ω = 0), replacing the BCS log, and strongly enhanced Tc. We suggest experiments
to verify our predictions.
Introduction-
transition temperature of the cuprates remains an un-
solved puzzle, despite more than two decades of intense
theoretical and experimental research. Central to the ef-
forts to unravel this mystery is the idea that the high
critical temperature is due to the presence of a quan-
tum critical point (QCP) which is hidden under the su-
perconducting dome [1]. Numerical calculations in the
Hubbard model, which is accepted as the de-facto model
for the cuprates, strongly support the case of a finite-
doping QCP separating the low-doping region, found to
be a non-Fermi liquid (NFL), from a higher doping Fermi-
liquid (FL) region [2, 3]. Calculations also show that in
the vicinity of the QCP, and for a wide range of tem-
peratures, the doping and temperature dependence of
the single-particle properties, such as the quasi-particle
weight [2], as well as thermodynamic properties such as
the chemical potential and the entropy, are consistent
with marginal Fermi liquid (MFL) behavior [4].
QCP emerges by tuning the temperature of a second-
order critical point of charge separation transitions to
zero and is therefore intimately connected to q = 0 charge
fluctuations [5]. Finally, the critical doping seems to be in
close proximity to the optimal doping for superconduc-
tivity as found both in the context of the Hubbard [5]
and the t-J model [6]. Even though this proximity may
serve as an indication that the QCP enhances pairing,
the detailed mechanism is largely unknown.
In this Letter, we attempt to differentiate between two
incompatible scenarios for the role of the QCP in super-
conductivity. The first scenario is the quantum critical
BCS (QCBCS) formalism introduced by She and Zaanen
(She-Zaanen) [7]. According to this, the presence of the
The unusually high superconducting
This
QCP results in replacing the logarithmic divergence of
the BCS pairing bubble by an algebraic divergence. This
leads to a stronger pairing instability and higher critical
temperature compared to the BCS for the same pairing
interactions. The second scenario suggests that remnant
fluctuations around the QCP mediate the pairing inter-
action [8, 9]. In this case the strength of the pairing
interaction would be strongly enhanced in the vicinity
of the QCP, leading to the superconducting instability.
Here, we find that near the QCP, the pairing interac-
tion depends monotonically on the doping, but the bare
pairing susceptibility acquires an algebraic dependence
on the temperature, consistent with the first scenario.
Formalism-
In a conventional BCS superconductor,
the superconducting transition temperature, Tc, is deter-
mined by the condition V χ′
the real part of the q = 0 bare pairing susceptibility, and
V is the strength of the pairing interaction. The transi-
tion is driven by the divergence of χ′
be related to the imaginary part of the susceptibility via
χ′
0(ω = 0) =
π
?dωχ′′
related to the spectral function, Ak(ω), through
0(ω = 0) = 1, where χ′
0is
0(ω = 0) which may
1
0(ω)/ω. And χ′′
0(ω) itself can be
χ′′
0(x) =π
N
?
ζ,k
?
dωAk(ω)Ak(ζx−ω)(f(ω − ζx) − f(ω))
(1)
where the summation of ζ
to anti-symmetrize χ′′
N(ω/2)tanh(ω/4T), and χ′
N(0) the single-particle density of states at the Fermi
surface and ωD the phonon Debye cutoff frequency.
This yields the well known BCS equation Tc
ωDexp[−1/(N(0)V )]. In the QCBCS formulation, the
∈
In a FL, χ′′
0(T) ∝ N(0)ln(ωD/T) with
{−1,+1} is used
0(ω).
0(ω)∝
=

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2
BCS equation is V χ′(ω = 0) = 1, where χ′is fully dressed
by both the self energy and vertices associated with the
interaction responsible for the QCP, but not by the pair-
ing interaction V . In the Hubbard model the Coulomb
interaction is responsible for both the QCP and the pair-
ing, so this deconstruction is not possible. Thus, we will
use the more common BCS Tccondition to analyze our
results with V χ′
the self energy but without vertex corrections. Since the
QCP is associated with MFL behavior, we do not expect
the bare bubble to display a FL logarithm divergence.
Here, we explore the possibility that χ′
The two-dimensional Hubbard model is expressed as:
0(ω = 0) = 1 where χ′
0is dressed by
0(ω = 0) ∼ 1/Tα.
H = Hk+ Hp=
?
kσ
ǫ0
kc†
kσckσ+ U
?
i
ni↑ni↓,(2)
where c†
electrons of wavevector k and spin σ, niσ = c†
the number operator, ǫ0
k= −2t(cos(kx) + cos(ky)) with
t being the hopping amplitude between nearest-neighbor
sites, and U is the on-site Coulomb repulsion.
We employ the dynamical cluster approximation
(DCA) [10] to study this model with a Quantum Monte
Carlo (QMC) algorithm as the cluster solver. The DCA
is a cluster mean-field theory which maps the original lat-
tice onto a periodic cluster of size Nc= L2
a self-consistent host. Spatial correlations up to a range
Lc are treated explicitly, while those at longer length
scales are described at the mean-field level. However the
correlations in time, essential for quantum criticality, are
treated explicitly for all cluster sizes. To solve the cluster
problem we use the Hirsch-Fye QMC method [11, 12] and
employ the maximum entropy method [13] to calculate
the real-frequency spectra.
We evaluate the results starting from the Bethe-
Salpeter equation in the pairing channel:
kσ(ckσ) is the creation (annihilation) operator for
iσciσ is
cembedded in
χ(Q)P,P′ = χ0(Q)PδP,P′
+
?
P′′
χ(Q)P,P′′Γ(Q)P′′,P′χ0(Q)P′ (3)
where χ is the dynamical susceptibility, χ0(Q)P [=
−G(P+Q)G(−P)] is the bare susceptibility, which is con-
structed from G, the dressed one-particle Green’s func-
tion, Γ is the vertex function, and indices P[...]and ex-
ternal index Q denote both momentum and frequency.
The instability of the Bethe-Salpeter equation is detected
by solving the eigenvalue equation Γχ0φ = λφ [14] for
fixed Q. By decreasing the temperature, the leading λ
increases to one at a temperature Tc where the system
undergoes a phase transition. To identify which part, χ0
or Γ, dominates at the phase transition, we project them
onto the d-wave pairing channel (which was found to be
dominant [3, 15]). For χ0, we apply the d-wave projection
as χ0d(ω) =?
gd(k) = (cos(kx) − cos(ky)) is the d-wave form factor.
kχ0(ω,q = 0)kgd(k)2/?
kgd(k)2, where
00.10.2
T
0.30.4
0
0.2
0.4
0.6
0.8
1
λ
Nc=12 magnetic q=(π,π)
Nc=16 magnetic q=(π,π)
Nc=12 charge q=(0,0)
Nc=16 charge q=(0,0)
Nc=12 d-wave pairing
Nc=16 d-wave pairing
0 0.05 0.1 0.15 0.2
δ
0
0.02
0.04
T
T*
TX
Tc
FIG. 1: (Color online) Plots of leading eigenvalues for differ-
ent channels at the critical doping for Nc = 12 and Nc = 16
site clusters. The inset shows the phase diagram with su-
perconducting dome, pseudogap T∗and FL TX temperatures
from Ref. [2]
As for the pairing strength, we employ the projection as
Vd =?
lowest Mastsubara frequency [16].
To further explore the different contributions to the
pairing vertex, we employ the formally exact parquet
equations to decompose it into different components [16,
17]. Namely, the fully irreducible vertex Λ, the charge
(S=0) particle-hole contribution, Φc, and the spin (S=1)
particle-hole contribution, Φs, through: Γ = Λ+Φc+Φs.
Similar to the previous expression, one can write Vd =
VΛ
d, where each term is the d-wave component
of the corresponding term. Using this scheme, we will be
able to identify which component contributes the most
to the d-wave pairing interaction.
Results-
We use the BCS-like approximation, dis-
cussed above, to study the proximity of the supercon-
ducting dome to the QCP. We take U = 6t (4t = 1) on
12 and 16 site clusters large enough to see strong evidence
for a QCP near doping δ ≈ 0.15 [2, 4, 5]. We explore the
physics down to T ≈ 0.11J on the 16 site cluster and
T ≈ 0.07J on the 12-site cluster, where J ≈ 0.11 [18] is
the antiferromagnetic exchange energy. The fermion sign
problem prevents access to lower T.
Fig. 1 displays the eigenvalues of different channels
(pair, charge, magnetic) at the QC filling. The results for
the two cluster sizes are nearly identical, and the pairing
channel eigenvalue approaches one at low T, indicating a
superconducting d-wave transition at roughly Tc= 0.007.
However, in contrast to what was found previously [16],
the q = 0 charge eigenvalue is also strongly enhanced,
particularly for the larger Nc = 16 cluster, as it is ex-
pected from a QCP emerging as the terminus of a line of
second-order critical points of charge separation transi-
tions [5]. The inset shows the phase diagram, including
the superconducting dome and the pseudogap T∗and FL
k,k′gd(k)Γk,k′gd(k′)/?
kgd(k)2, using Γ at the
d+Vc
d+Vm

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3
0.05
0.1
δc
0.2
0.25
0.3
δ
0
10
20
30
40
50
Vd
T=0.100
T=0.045
T=0.029
T=0.021
0
0.05
0.1
0.15
0.2
T
-5
0
5
10
Vd
VdΛ
Vdc
Vds
δ=0.15
FIG. 2: (Color online) Plots of Vd, the strength of the d-wave
pairing interaction for various temperatures with U = 1.5
(4t = 1) and Nc = 16.Vd decreases monotonically with
doping, and shows no feature at the critical doping. In the
inset are plots of the contributions to Vd from the charge
Vc
vertex VΛ
is lowered, T ≪ J ≈ 0.11, the contribution to the pairing
interaction from the spin channel is clearly dominant.
dand spin Vs
d versus T at the critical doping. As the temperature
dcross channels and from the fully irreducible
TXtemperatures.
In Fig. 2, we show the strength of the d-wave pair-
ing vertex Vdversus doping for a range of temperatures.
Consistent with previous studies [19], we find that Vdfalls
monotonically with increasing doping.
doping, δc= 0.15, Vdshows no feature, invalidating the
second scenario described above. The different compo-
nents of Vdat the critical doping versus temperature are
shown in the inset of Fig. 2. As the QCP is approached,
the pairing originates predominantly from the spin chan-
nel. This is similar to the result of Ref. [16] where the
pairing interaction was studied away from quantum crit-
icality.
In contrast, the bare d-wave pairing susceptibility χ0d
exhibits significantly different features near and away
from the QCP. As shown in Fig. 3, in the underdoped
region (typically δ = 0.05), the bare d-wave pairing sus-
ceptibility χ′
0d(ω = 0) saturates at low temperatures.
However, at the critical doping, it diverges quickly with
decreasing temperature, roughly following the power-law
behavior 1/√T, while in the overdoped or FL region it
displays a log divergence.
To better understand the temperature-dependence
of χ′
0d(ω = 0) at the QC doping, we looked into
T1.5χ′′
0d(ω)/ω and plotted it versus ω/T in Fig. 4. When
scaled this way, the curves from different temperatures
fall on each other such that T1.5χ′′
(ω/T)−1.5for ω/T>
∼9 ≈ 4t/J. For 0 < ω/T < 4t/J, the
curves deviate from the scaling function H(x) and show
nearly BCS behavior, with χ′′
sublinear in 1/T as shown in the inset. The curves away
At the critical
0d(ω)/ω = H(ω/T) ≈
0d(ω)/ω|ω=0which is weakly
0
0.05
0.1
0.15
0.2
T
1
2
3
4
5
6
χ′0d(T)
δ=0.05, Nc=12
δ=0.05, Nc=16
δ=0.15, Nc=12
δ=0.15, Nc=16
δ=0.25, Nc=16
FIG. 3: (Color online) Plots of χ′
the bare d-wave pairing susceptibility, at zero frequency vs.
temperature at three characteristic dopings. The solid lines
are fits to χ′
the underdoped case (δ = 0.05), χ′
with decreasing temperature. At the critical doping (δ = δc =
0.15), χ′
0d(ω = 0) shows power-law behavior with B = 0.04
for the 12 site, and B = 0.09 for the 16-site clusters (in both
A = 1.04 and ωc = 0.5). In the overdoped region (δ = 0.25),
a log divergence is found, with B = 0 obtained from the fit.
0d(ω = 0), the real part of
0d(ω = 0) = B/√T + Aln(ωc/T) for T < J. In
0d(ω = 0) does not grow
02040
60
80 100
1/T
0
10
20
30
40
50
60
χ"0d(ω)/ω|ω=0
δ=0.25
δ=0.15
δ=0.05
020
ω/T
40
0
0.1
0.2
0.3
T1.5χ"0d(ω)/ω
0.200
0.125
0.083
0.056
0.036
0.025
0.017
0.012
(ω/T)-1.5
T=
Ts/T
FIG. 4: (Color online) Plots of T1.5χ′′
the QC doping (δ = 0.15) for Nc = 16. The arrow denotes the
direction of decreasing temperature. The curves coincide for
ω/T > 9 ≈ (4t/J) defining a scaling function H(ω/T), cor-
responding to a contribution to χ′
1/√T as found in Fig. 3. For ω/T > 9 ≈ (4t/J), H(ω/T) ≈
(ω/T)−1.5(dashed line). On the x-axis, we add the label
Ts/T ≈ (4t/J), where Ts represents the energy scale where
curves start deviating from H.
scaled zero-frequency result χ′′
verse temperature.
0d(ω)/ω versus ω/T at
0d(T) =
1
π
?
dωχ′′
0d(w)/w ∝
The inset shows the un-
0d(ω)/ω|ω=0plotted versus in-
from the critical doping (not displayed) do not show such
a collapse. In the underdoped region (δ = 0.05) at low
frequencies, χ′′
0d(ω)/ω goes to zero with decreasing tem-
perature (inset). In the FL region (δ = 0.25) χ′′
develops a narrow peak at low ω of width ω ≈ TX and
height ∝ 1/T as shown in the inset.
0d(ω)/ω

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4
Discussion-
instability takes place.
at the QC filling contribute a term T−1.5H(ω/T) to
χ′′
is also a component which does not scale, especially at
low frequencies. In fact, χ′′
0d(ω)/ω at zero frequency in-
creases more slowly than 1/T as expected for a FL. From
this sublinear character, we infer that the contribution
of the non-scaling part of χ′′
0d(ω)/ω to the divergence of
χ′
0d(T) is weaker than BCS and may cause us to overes-
timate A and underestimate B in the fits performed at
the critical doping in Fig. 3. In addition, if H(0) is finite,
it would contribute a term to χ′
1/T1.5, so H(0) = 0. From Eq. 1 we see that the con-
tribution to χ′′
0d(ω)/ω at small ω comes only from states
near the Fermi surface. H(0) = 0 would indicate that
the enhanced pairing associated with χ′
due to higher energy states. The vanishing of χ′′
in the pseudogap region (δ = 0.05) for small frequency
when T → 0 indicates that around the Fermi surface, the
dressed particles do not respond to a pair field. Or, per-
haps more correctly, none are available for pairing due
to the pseudogap depletion of electron states around the
Fermi surface. Thus, even the strong d-wave interaction,
seen in Fig. 2, is unable to drive the system into a su-
perconducting phase. In the overdoped region, χ′′
displays conventional FL behavior for T < TX, and the
vanishing Vdsuppresses Tc.
Together, the results for χ0dand Vdshed light on the
shape of the superconducting dome in the phase diagram
found previously [5]. With increasing doping, the pair-
ing vertex Vd falls monotonically. On the other hand,
χ′
0d(T) is strongly suppressed in the low doping or pseu-
dogap region and enhanced at the critical and higher dop-
ing. These facts alone could lead to a superconducting
dome. Futhermore, the additional algebraic divergence
of χ′
0d(T) seen in Fig. 3 causes the superconductivity to
be enhanced even more strongly near the QCP where one
might expect Tc∝ (VdB)2, with B =1
pared to the conventional BCS form in the FL region.
Similar to the scenario for cuprate superconductivity
suggested by Castellani et al. [8], we find that the super-
conducting dome is due to charge fluctuations adjacent
to the QCP related to charge ordering. However, we dif-
fer in that we find the pairing in this region is due to an
algebraic temperature dependence of the bare suscepti-
bility χ0drather than an enhanced d-wave pairing vertex
Vd, and that this pairing interaction is dominated by the
spin channel.
Our observation in the Hubbard model offers an exper-
imental accessible variant of She-Zaanen’s QCBCS. We
use the bare pairing susceptibility χ0while She-Zaanen
use the full χ, which includes all the effects of quan-
tum criticality but not the correction from the pairing
vertex (the pairing glue is added separately). This de-
composition is not possible in numerical calculations or
χ′′
0d(ω)/ω reveals details about how the
The overlapping curves found
0d(w)/w or χ′
0d(T) ∝ 1/√T as found in Fig. 3. There
0d(T) that increases like
0d(T) ∝ 1/√T is
0d(ω)/ω
0d(ω)/ω
π
?dxH(x), com-
experiments since both quantum criticality and pairing
originate from the Coulomb interaction. However, the
effect of quantum criticality already shows up in the one-
particle quantities, and the spectra have different be-
haviors for the three regions around the superconduct-
ing dome. She-Zaanen assume that χ′′(ω) ∝ 1/ωαfor
Ts < ω < ωc, where ωc is an upper cutoff, and that
it is irrelevant (α < 0), marginal (α = 0), or relevant
(α > 0), respectively in the pseudo gap region, FL re-
gion and QCP vincity. We find the same behavior in χ0
and we have the further observation that near the QCP
Ts≈ (4t/J)T and α = 0.5.
Experiments combining angle-resolved photo emission
(ARPES) and inverse photo emission results, with an
energy resolution of roughly J, could be used to construct
χ0dand explore power law scaling at the critical doping.
Since the energy resolution of ARPES is much better
than inverse photo emission, it is also interesting to study
χ′′
0d(ω)/ω|ω=0, which only requires ARPES data, but not
inverse photo emission.
Conclusion-
Using the DCA, we investigate the d-
wave pairing instability in the two-dimensional Hubbard
model near critical doping. We find that the pairing in-
teraction remains dominated by the spin channel and is
not enhanced near the critical doping. However, we find a
power-law divergence of the bare pairing susceptibility at
the critical doping, replacing the conventional BCS log-
arithmic behavior. We interpret this behavior by study-
ing the dynamic bare pairing susceptibility which has a
part that scales like χ′′
H(ω/T) is a universal function. Apparently, the NFL
character of the QCP yields an electronic system that is
far more susceptible to d-wave pairing than the FL and
pseudogap regions. We also suggest possible experimen-
tal approaches to exploit this interesting behavior.
Acknowledgments-
We would like to thank F. Assaad,
I. Vekhter and E. W. Plummer for useful conversations.
This research was supported by NSF DMR-0706379 and
OISE-0952300. This research used resources of the Na-
tional Center for Computational Sciences (Oak Ridge Na-
tional Lab), which is supported by the DOE Office of
Science under Contract No. DE-AC05-00OR22725. J.-H.
She and J. Zaanen are supported by the Nederlandse Or-
ganisatie voor Wetenschappelijk Onderzoek (NWO) via
a Spinoza grant.
0d(ω)/ω ∼ T−1.5H(ω/T), where
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