Fluctuation Diagnostics of the Electron Self-Energy: Origin of the Pseudogap Physics
O. Gunnarsson,
1
T. Schäfer,
2
J. P. F. LeBlanc,
3,4
E. Gull,
4
J. Merino,
5
G. Sangiovanni,
6
G. Rohringer,
2
and A. Toschi
2
1
Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany
2
Institute of Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria
3
Max-Planck-Institute for the Physics of Complex Systems, D-01187 Dresden, Germany
4
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
5
Departamento de Física Teórica de la Materia Condensada, IFIMAC Universidad Autónoma de Madrid, Madrid 28049, Spai n
6
Institute of Physics and Astrophysics, University of Würzburg, D-97070 Würzburg, Germany
(Received 25 November 2014; published 10 June 2015)
We demonstrate how to identify which physical processe s dominate the low-energy spectral functions of
correlated electron systems. We obtain an unambiguous classification through an analysis of the equation
of motion for the electron self-energy in its charge, spin, and particle-particle representations. Our
procedure is then employed to clarify the controversial physics responsible for the appearance of the
pseudogap in correlated systems. We illustrate our method by examining the attractive and repulsive
Hubbard model in two dimensions. In the latter, spin fluctuations are identified as the origin of the
pseudogap, and we also explain why d-wave pairing fluctuations play a marginal role in suppressing the
low-energy spectral weight, independ ent of their actual strength.
DOI: 10.1103/PhysRevLett.114.236402 PACS numbers: 71.27.+a, 71.10.Fd
Introduction.—Correlated electron systems display some
of the most fascinating phenomena in condensed matter
physics, but their understanding still represents a formi-
dable challenge for theory and experiments. For photo-
emission [1] or STM [2,3] spectra, which measure
single-particle quantities, information about correlation is
encoded in the electronic self-energy Σ. However, due to
the intrinsically many-body nature of the problems, even an
exact knowledge of Σ is not sufficient for an unambiguous
identification of the underlying physics. A perfect example
of this is the pseudogap observed in the single-particle
spectral functions of underdoped cuprates [4], and, more
recently, of their nickelate analogues [5]. Although relying
on different assumptions, many theoretical approaches
provide self-energy results compatible with the experimen-
tal spectra. This explains the lack of a consensus about
the physical origin of the pseudogap: In the case of
cuprates, the pseudogap has been attributed to spin fluc-
tuations [6–10], preformed pairs [11–15], Mottness
[16,17], and, recently, to the interplay with charge fluctua-
tions [18–21] or to Fermi-liquid scenarios [22]. The
existence and the role of (d-wave) superconducting fluc-
tuations [11–15] in the pseudogap regime are still openly
debated for the basic model of correlated electrons, the
Hubbard model.
Experimentally, the clarification of many-body physics
is augmented by a simultaneous investigation at the
two-particle level, i.e., via neutron scattering [23], infrared
or optical [24] and pump-probe spectroscopy [25],
muon-spin relaxation [26], and correlation or coincidence
two-particle spectroscopies [27–29]. Analogously, theo-
retical studies of Σ can also be supplemented by a
corresponding analysis at the two-particle level. In this
Letter, we study the influence of the two-particle fluctua-
tions on Σ via its equation of motion. We apply this
method of “fluctuation diagnostics” to identify the role
played by different collective modes in the pseudogap
physics.
Self-energy decomposition.—We emphasize that all con-
cepts and equations below are applicable within any
theoretical approach in which the self-energy and the
two-particle scattering amplitude are calculated without
a priori assumptions of a predominant type of fluctuations.
This includes quantum Monte Carlo (QMC) methods (e.g.,
lattice QMC [30]), functional renormalization group [31],
parquet approximation [32–34], and cluster extensions [35]
of the dynamical mean field theory (DMFT) [36,37] such as
the cellular-DMFT [38,39] or the dynamical cluster
approximation (DCA) [40]. Within diagrammatic exten-
sions [41–47] of DMFT, our analysis is applicable if
parquet-like diagrams are included [48–50]. The outputs
of these techniques can be then post-processed by means of
the fluctuation diagnostics with a comparably lower
numerical effort [51].
The self-energy describes all scattering effects of one
added or removed electron, when propagating through the
lattice. In correlated electronic systems, these scattering
events originate from the Coulomb interaction among the
electrons themselves, rather than from the presence of an
external potential. Therefore, Σ is entirely determined by
the full two-particle scattering amplitude F. The formal
relation between F and Σ is known as the Dyson-
Schwinger equation of motion (EOM) [56]. In the
important case of a purely local interaction (as in the
Hubbard model [57–59]), this reads (in the paramagnetic
phase)
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ΣðkÞ¼
Un
2
−
U
β
2
N
X
k
0
;q
F
↑↓
ðk; k
0
;qÞgðk
0
Þgðk
0
þ qÞgðk þ qÞ;
ð1Þ
where U is the (bare) Hubbard interaction, n the electronic
density, g the electron Green’s function, β ¼ 1=T the
inverse temperature, and N the normalization of the
momentum summation [we adopt the notation k ¼
ðν; KÞ=q ¼ðω; QÞ for the fermionic/bosonic Matsubara
frequencies ν=ω and momenta K=Q; see the Supplemental
Material [51] for details]. Finally, F
↑↓
is the full scattering
amplitude (vertex) between electrons with opposite spins: It
consists of repeated two-particle scattering events in all
possible configurations compatible with energy, momen-
tum, or spin conservation. Therefore, it contains the
complete information of the two-particle correlations of
the system. Yet, much of the information encoded in F
↑↓
about the specific physical processes determining Σ is
washed out by averaging over all two-particle scattering
events, i.e., by the summations on the right-hand side of
Eq. (1). Hence, an unambiguous identification of the
physical role played by the underlying scattering or
fluctuation processes requires a “disentanglement” of the
EOM. The most obvious approach would be a direct
decomposition of the full scattering amplitude F
↑↓
of
Eq. (1) in all possible fluctuation channels, the so-called
parquet [32–34,60] decomposition. This approach works
well in the weakly correlated regime (small U, large
doping, high T), whereas for stronger correlations it suffers
from intrinsic divergences, recently discovered in the
Hubbard and Falicov-Kimball models [61–63]; see also
Ref. [64]. For example, in our DCA calculations for the
two-dimensional (2D) Hubbard model the breakdown of
the parquet decomposition of Σ occurs at lower values of U
(or larger values of doping) than those for which pseudogap
physics is numerically observed [61,65].
In this Letter we present an alternative route that can be
followed to circumvent this problem. Our idea exploits the
freedom of employing formally equivalent analytical
representations of the EOM. For instance, by means of
SU(2) symmetry and “crossing relations” (see, e.g.,
Refs. [66,67]), we can express F
↑↓
in Eq. (1) in terms
of the corresponding vertex functions of the spin or
magnetic F
sp
¼ F
↑↑
− F
↑↓
and charge or density F
ch
¼
F
↑↑
þ F
↑↓
sectors. Analogously, a rewriting in terms of the
particle-particle sector notation is done via F
pp
ðk; k
0
;qÞ¼
F
↑↓
ðk; k
0
;q− k − k
0
Þ. Inserting these results in Eq. (1) and
performing variable transformations, we recover Eq. (1),
with F
↑↓
replaced by F
sp
, F
ch
,orF
pp
[51]. These three
expressions,
ΣðkÞ − Σ
H
¼
U
β
2
N
X
k
0
;q
F
sp
ðk; k
0
; qÞgðk
0
Þgðk
0
þ qÞgðk þ qÞ;
ð2Þ
¼ −
U
β
2
N
X
k
0
;q
F
ch
ðk; k
0
; qÞgðk
0
Þgðk
0
þ qÞ gðk þ qÞ; ð3Þ
¼ −
U
β
2
N
X
k
0
;q
F
pp
ðk; k
0
; qÞgðk
0
Þgðq − k
0
Þgðq − kÞ; ð4Þ
yield the same result for Σ after all internal summations are
performed (Σ
H
denotes the constant Hartree term Un=2).
Crucial physical insight can be gained at this stage, by
performing partial summations. We can, e.g., perform all
summations, except for the one over the transfer momen-
tum Q. This gives
~
Σ
Q
ðkÞ, i.e., the contribution to Σ for
fixed Q, so that ΣðkÞ¼
P
Q
~
Σ
Q
ðkÞ. The vector Q corre-
sponds to a specific spatial pattern given by the Fourier
factor e
iQR
i
. For a given representation such a spatial
structure is associated to a specific collective mode, e.g.,
Q ¼ðπ; πÞ for antiferromagnetic or charge-density wave
(CDW) and Q ¼ð0; 0Þ for superconducting or ferromag-
netic fluctuations. Hence, if one of these contributions
dominates,
~
Σ
Q
ðkÞ is strongly peaked at the Q vector of that
collective mode, provided that the corresponding repre-
sentation of the EOM is used. On the other hand, in a
different representation, not appropriate for the dominant
mode
~
Σ
Q
ðkÞ will display a weak Q dependence. These
heuristic considerations can be formalized by expressing F
through its main momentum and frequency structures
[51,66]. Hence, in cases where the impact of the different
fluctuation channels on Σ is not known a priori, the
analysis of the Q dependence of
~
Σ
Q
ðkÞ in the alternative
representations of the EOM will provide the desired
diagnostics. Below, we show that this procedure works
well for the 2D (attractive and repulsive) Hubbard models,
allowing for an interpretation of the origin of the pseudogap
phases observed there.
Results for the attractive Hubbard model.—To demon-
strate the applicability of the fluctuation diagnostics, we
start from a case where the underlying, dominant physics is
well understood, namely, the attractive Hubbard model,
U<0. This model captures the basic mechanisms of the
BCS/Bose-Einstein crossover [68–72] and has been inten-
sively studied both analytically and numerically, e.g., with
QMC [72–74] and DMFT [75–78]. Because of the local
attractive interaction, the dominant collective modes are
necessarily s-wave pairing fluctuations [Q ¼ð0; 0Þ] in the
particle channel, and, for filling n ∼ 1, CDW fluctuations
[Q ¼ðπ; πÞ ] in the charge channel. As we show in the
following, this underlying physics is well captured by our
fluctuation diagnostics.
We present here our DCA results computed on a
cluster with N
c
¼ 8 sites for a 2D Hubbard model with
the following parameter set: t ¼ −0.25 eV (t
0
¼ 0),
U ¼ −1 eV, μ ¼ −0.53 eV and β ¼ 40 eV
−1
[51]. This
leads to the occupancy n ¼ 0.87, for which, at this T,no
superconducting long-range order is observed in DCA, and
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to the self-energy shown in Fig. 1 (upper panel) which
exhibits a metallic behavior with weak K dependence. The
lower panels of Fig. 1 show the fluctuation diagnostics for
Σ. The histogram depicts the different contributions to Im
Σ½K; ν for K ¼ð0; πÞ and ðπ=2; π=2Þ (upper panel of
Fig. 1) at the lowest Matsubara frequency (ν ¼ π=β)asa
function of the momentum transfer Q within the three
representations [spin, charge, and particle, i.e., via Eqs. (2),
(3), (4)]. We observe large contributions for Q ¼ðπ; πÞ in
the charge representation (blue bars) and for Q ¼ð0; 0Þ in
the particle-particle one (green bars). At the same time, no
Q dominates in the spin picture. Hence, the fluctuation
diagnostics correctly identifies the key role of CDW and
s-wave pairing fluctuations in this system. This outcome is
supported by a complementary analysis in frequency space
(pie chart in Fig. 1 ): Defining
~
Σ
ω
ðK; νÞ as a contribution to
the self-energy, where in Eq. (1) all summations, except the
one over the transfer frequency ω are performed, we
observe a largely dominant contribution at ω ¼ 0
(∼70%) both in the charge and particle-particle pictures.
This proves that the corresponding fluctuations are well
defined and long lived.
Results for the repulsive Hubbard model.—We now
apply the fluctuation diagnostics to the much more debated
physics of the repulsive Hubbard model in two dimensions,
focusing on the analysis of the pseudogap regime. As
before, we use DCA calculations with a cluster of N
c
¼ 8
sites. Σ and F have been calculated using the Hirsch-Fye
[79] and continuous time [80,81] QMC methods, accu-
rately cross-checking the results. In the view of a crude
modelization of the cuprate pseudogap regime, we
consider the parameter set t ¼ −0.25 eV, t
0
¼ 0.0375,
U ¼ 1.75 eV, μ ¼ 0.6 eV (corresponding to n ¼ 0.94)
and β ¼ 60 eV
−1
[51]. For these parameters, the self-
energy (see upper panels of Fig. 2) displays strong
momentum differentiation between the “antinodal”
[K ¼ð0; πÞ] and the “nodal” [K ¼ðπ=2; π=2Þ] momen-
tum, with a pseudogaplike behavior at the antinode [82,83].
The fluctuation diagnostics is performed in Fig. 2, where
we show the contributions to Im Σ½K; π=β for K ¼ð0; πÞ
FIG. 1 (color online). Fluctuation diagnostics of Im ΣðK; νÞ
(first row) for the attractive Hubbard model. The histogram shows
the contributions of Im
~
Σ
Q
ðK; π=βÞ from different values of Q in
the spin, charge, and particle-particle representations for the
attractive 2D Hubbard model (see text). The pie charts display the
relative magnitudes of jIm
~
Σ
ω
ðK; π=βÞj for the first eight Matsu-
bara frequencies jωj in the charge and particle-particle picture,
respectively.
FIG. 2 (color online). As for Fig. 1: Fluctuation diagno stics of
the electronic self-energy, for the case of the repulsive Hubbard
model (see text and Supplemental Material [51] ).
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and ð π=2; π=2Þ (upper panels) as a function of the transfer
momentum Q in the three representations. This illustrates
clearly the underlying physics of the pseudogap. In the spin
representation (red bars in the histogram), the Q ¼ðπ; πÞ
contribution dominates, and contributes more than 85% and
80% of the result for K ¼ð0; πÞ and K ¼ðπ=2; π=2Þ,
respectively. Conversely, all the contributions at other
transfer momenta Q ≠ ðπ; πÞ are about an order of magni-
tude smaller. The dominant Q ¼ðπ; πÞ contribution is also
responsible for the large momentum differentiation, being
almost twice as large for the antinodal self-energy.
Performing the same analysis in the charge (blue bars)
or particle-particle (green bars) representation, we get a
completely different shape of the histogram. In both cases,
the contributions to Σ are almost uniformly distributed
among all transfer momenta Q.
Hence, we do not find important contributions to Σ from
charge or pairing modes, while the histogram in the spin-
representation marks the strong impact of antiferromag-
netic fluctuations [6–10,84–86]. This picture is further
supported by the complementary frequency analysis. The
pie chart in Fig. 2 is dominated by the ω ¼ 0 contribution
in the spin picture, reflecting the long-lived nature of well-
defined spin fluctuations. At the same time, in the particle
(and charge, not shown) representation, the contributions
are more uniformly distributed among all ω’s, which
corresponds to short-lived pairing (charge) fluctuations.
We note, eventually, that the fluctuation diagnostics of
Fig. 2 qualitatively resembles the results obtained for the
unfrustrated model at half-filling (not shown). In fact, a
significant weakening of the spin dominance is found,
moving away from the doping or interaction level, where a
pseudogap is found in DCA.
Physical interpretation of the pseudogap.—We are now
in the position to draw some general conclusions on the
physics underlying a pseudogap. These considerations are
relevant for the underdoped cuprates, up to the extent their
low-energy physics is captured by the 8-site DCA for the
repulsive 2D Hubbard model. We focus here on our data of
Fig. 2: By means of fluctuations diagnostics we identify a
well-defined [Q ¼ðπ; πÞ] collective spin mode to be
responsible (on the 80% level) both for the momentum
differentiation of Σ and for its pseudogap behavior at the
antinode: The large values of
~
Σ
Q
at Q ¼ðπ; πÞ and
~
Σ
ω
at
ω ¼ 0 are the distinctive hallmark of long-lived and
extended (antiferromagnetic) spin fluctuations. At the same
time, the rather uniform Q and ω distribution of
~
Σ
Q
and
~
Σ
ω
in the charge/particle pictures shows that the well-defined
spin mode can be also viewed as short-lived and short-
range charge/pair fluctuations. The latter cannot be inter-
preted, hence, in terms of preformed pairs. This scenario
matches very well the different estimates of fluctuation
strengths in previous DCA studies [83,86,87]. We also
emphasize the general applicability of our result (see
Supplemental Material [51]): A well-defined mode in
one channel appears as short-lived fluctuations in other
channels. This dichotomy is not visible in Σ, which makes
our fluctuations diagnostics a powerful tool for identifying
the most convenient viewpoint to understand the physics
responsible of the observed spectral properties.
Let us finally turn our attention to the still open question
about the impact of superconducting d-wave fluctuations
on the normal-state spectra in the pseudogap regime of the
Hubbard model. The instantaneous fluctuations are defined
as hΔ
†
d
Δ
d
i, with Δ
†
d
¼
P
K
fðKÞc
†
K↑
c
†
−K↓
and fðKÞ¼
cos K
x
− cos K
y
. These Q ¼ 0 fluctuations are certainly
strong in proximity of the superconducting phase, but they
were also found [83] to be significant over short distances
in the pseudogap regime. Their intensity gets stronger as U
is increased, beyond the values where superconductivity
exists. The expression for
~
Σ
Q¼ð0;0Þ
in the particle picture is
closely related to hΔ
†
d
Δ
d
i, except that the factor fðKÞ is
missing in
~
Σ
Q
[51]. One might therefore have expected that
large Q ¼ 0 pair fluctuations, irrespectively of their life-
time, would have contributed strongly to Σ. For unconven-
tional superconductivity, e.g., d wave, this does not happen.
The reason is the angular variation of fðKÞ. For strong pair
fluctuations, the variations of fðKÞ make the contributions
to the fluctuations add up, while the contributions to Σ then
tend to cancel. This explains why suppressing supercon-
ductivity fluctuations [42,44,83,87–91] does not affect the
description of the pseudogap of the Hubbard model. In the
case of a purely local interaction such as in the EOM like
Eq. (1), enhanced hΔ
†
d
Δ
d
i fluctuations are mostly averaged
out by the momentum summation (see Supplemental
Material [51]).
Our diagnosis of dominant spin fluctuations in the DCA
self-energy in the underdoped 2D-Hubbard model does not
represent per se the conclusive scenario for the cuprate
pseudogap. However, important information about the
realistic modeling of cuprates can be already extracted:
If definitive experimental evidence for an impact of
supposedly “secondary” (e.g., charge) fluctuations on the
pseudogap is found, extensions of the modelization will be
unavoidable for a correct pseudogap theory: Nonlocal
interactions (e.g., extended Hubbard model) or explicit
inclusion of the oxygen orbitals (e.g., Emery model) might
be required. In fact, such extensions represent in itself an
intriguing playground for future fluctuation diagnostics
applications.
Conclusions.—We have shown that if a simultaneous
calculation of the self-energy and the vertex functions is
performed, it is possible to identify the impact of the
different collective modes on the spectra of correlated
systems (fluctuation diagnostics). This is achieved by
expressing the equation of motion for Σ in different
representations (e.g., spin, charge, or particle), which
avoids all the intrinsic instabilities of parquet decomposi-
tions. We apply this procedure to the U<0 and U>0 2D
Hubbard model. In the attractive case we have confirmed
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the dominant role of pair fluctuations, supporting the
validity of our approach. For the repulsive model, relevant
for the physics of the underdoped cuprates, spin fluctua-
tions emerged as mainly responsible for the spectral
function results, in agreement with other studies
[6–10,86]. The same well-defined spin modes might
appear, on a different perspective, as strong, but rapidly
decaying, pair fluctuations. Finally, for a purely local
interaction, d-wave pairing fluctuations will only weakly
affect the pseudogap spectral properties even on the verge
of the superconducting transition.
These results, as well as the insight on the pseudogap
physics, suggest that fluctuation diagnostics can be broadly
used in future studies. The progress in calculating vertex
functions [66,92,93] will allow its applicability also to
other, more complex, multiorbital models [94–100]: Here,
due to the increased number of degrees of freedom, the
identification of the dominant fluctuation mode(s) will
be of the utmost importance for a correct physical
understanding.
We thank A. Tagliavini, C. Taranto, S. Andergassen, M.
Sing, and M. Capone for insightful discussions. We
acknowledge support from FWF through the PhD
School “Building Solids for Function” (T. S., Project
No. W1243) and the project I 610 (G. R., A. T.), from
the research unit FOR 1346 of the DFG (G. S.), from
MINECO: MAT2012-37263-C02-01 (J. M.), and from the
Simons Foundation (J. P. F. L., E. G.). G. S. and A. T. also
acknowledge the hospitality in Campello sul Clitunno.
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![FIG. 2 (color online). As for Fig. 1: Fluctuation diagnostics of the electronic self-energy, for the case of the repulsive Hubbard model (see text and Supplemental Material [51]).](/figures/fig-2-color-online-as-for-fig-1-fluctuation-diagnostics-of-1q770cih.webp)
