XeLatex=

# Comment on: “Crossover of Charge Fluctuations across the Strange Metal Phase
Diagram”

Jörg Fink  Leibniz Institute for Solid State and Materials Research Dresden,
Helmholtzstr. 20, D-01069 Dresden, Germany  Institut für Festkörperphysik,
Technische Universität Dresden, D-01062 Dresden, Germany

###### Abstract

In a recent paper by Husain et al. [PRX 9, 041062 (2019)], the two-particle
electronic excitations in Bi2Sr2CaCu2O8+x have been studied by Electron
Energy-Loss Spectroscopy in reflection (R-EELS) in the “strange metal” range
between underdoped and overdoped materials. The authors conclude that there
are no well defined plasmons. Rather they obtain a momentum-independent
continuum which they discuss in terms of holographic theories. In this Comment
it is pointed out that the experimental results are in stark contrast to
previous EELS in transmission (T-EELS), Resonant Inelastic X-ray Scattering
(RIXS), and optical studies. The differences can be probably explained by an
inaccurate momentum scale in the R-EELS experiments. Furthermore, it is shown,
that many material specific experimental results from T-EELS, R-EELS, RIXS,
and optical spectroscopy can be explained by a more traditional extended
Lindhard model. This model describes the energy, the width, and the dispersion
of normal and accustic plasmons in cuprates, as well as the continuum. The
latter is explained by electron-hole excitations inside a lifetime broadened
conduction band. This continuum is directly related to the scattering rates of
the charge carriers, which in turn, by a feed back process, lead to the
continuum.

##  I  I. INTRODUCTION

In a recent paper by Husain et al. [1], the two-particle electronic
excitations in Bi2Sr2CaCu2O8+x have been studied by Electron Energy-Loss
Spectroscopy in reflection (R-EELS) in the “strange metal” range between
underdoped and overdoped materials. The work is a continuation of previous
work by the group, which was focused on a more narrow doping range [2, 3]. The
main result of these investigations is that this high-$T_{c}$ superconductor
does not exhibit well-defined plasmon excitations anywhere in its phase
diagram. Rather they have detected a featureless, momentum-independent
continuum throughout the strange metal region. They come to the conclusion
that a new kind of theory of strongly interacting matter may be needed to
explain this continuum, which may also be connected to the phenomena of
high-$T_{c}$ superconductivity. Thus, the “overdamped” plasmon was discussed
in terms of holographic theories [4], which predicted, different from the
classical Landau damping, a strong damping even for the long-wavelength
plasmons due to “quantum critical fluctuations”.

The continuum, first detected by Raman spectroscopy [5, 6] provides a
reasonable explanation for the linear temperature dependence of the
resistivity in the normal state, one of the outstanding properties of the
strange metals. The reason for this is that the lifetime of the quasi-
particles is related to the energy integral over the constant susceptibility,
which leads to a linear energy ($\Omega$) or temperature ($T$) dependence of
the scattering rate $\Gamma(\Omega,T)$, or the imaginary part of the complex
self-energy $\Sigma(\Omega,T)$ [7]. This is the basis of the conjecture of the
marginal Fermi liquid [7].

Finally, we mention also that the existence of plasmons is interesting because
plasmons may mediate superconductivity as predicted by several theoretical
models [8, 9].

In this Comment, I first review the experimental methods connected with two-
particle electronic excitations: EELS in transmission (T-EELS), non-resonant
and Resonant Inelastic X-Ray Scattering (IXS, RIXS), optical spectroscopy, and
Raman spectroscopy. Then the R-EELS experimental results on cuprates are
compared with previous results. Later a model for the collective plasmon
excitations and the single electron-hole ($e-h$) excitations is described,
which is based on our knowledge in this field which was established during the
last 50 years: a Lindhard model [10, 11], extended by $e-h$ excitations
between the conduction band and a band caused by umklapp-scattering from the
second into the first Brillouin zone. A further extension comes from the
finite lifetime broadening of the single-particle excitations. Values for this
broadening are taken from Angle-Resolved Photoemission Spectroscopy (ARPES)
[12]. Finally, this model is compared to experimental results. The conclusion
is that the extended Lindhard model describes many details of the experimental
results on two-particle excitations in cuprates.

Figure 1: Sketch of a map of the loss function $\Im(1/\epsilon(\omega,q))$ as
a function of energy ($\omega$) and momentum transfer ($q$). Black/white
lines: weakly/heavily damped plasmon excitations. Red regions: intra-band
electron-hole excitations. Green regions: inter-band transitions due to
umklapp scattering. The latter are not shown in the range of intra-band
transitions. The size of the loss-function is indicated by the intensity of
the green and red color. (a) Extended Lindhard model for intra-, inter-band,
and plasmon excitations for a half-filled band without doping. (b) Extended
Lindhard model for intra-, inter-band and plasmon excitations for a half-
filled band with n-type (or p-type) doping. Acoustic plasmons appear for out-
of-plane momentum transfers in the “empty” region at low energy and low
momentum transfer. (c) Extended Lindhard model taking into account the
lifetime broadening of the conduction band. Only spectral weight for the
intra-band transitions at zero temperature is shown. Dashed magenta line:
maximum of the spectral weight of the continuum of intra-band transitions. (d)
Waterfall plots of the energy dependence of the intra-band part of the loss
functions for various momentum transfers ranging from $q=0.1$ Å-1 (lowest
curve) to $q=0.33~{}$Å-1 (uppermost curve). Black curve: Lindhard model. Red
curve: extended Lindhard model taking into account the lifetime broadening of
the conduction band. Inset in (d): the same as in the main panel but for low
energy and $q=0.4~{}$Å-1. In addition, the inset shows a calculation for
$T=300$ K (red dashed line).

##  II II. EXPERIMENTAL METHODS

Since the 60th of the last century, the most popular method for the direct
observation of plasmons was T-EELS using dedicated spectrometers. It measures
the energy ($\omega$) and momomentum ($q$) dependent loss-function
$\Im(1/\epsilon(\omega,q))$ which has a maximum at the energy ($\omega_{P}$)
of the plasmon and the width is related to the lifetime. The method is
documented in various review articles [13, 14, 15, 16, 17, 18]. From the very
beginning, T-EELS was momentum resolved. The momentum parallel and
perpendicular to the incident electron beam is well defined. The best energy
resolution is slightly below 0.1 eV. The momentum resolution $\Delta q_{||}$
is about 0.04 Å-1 and $\Delta q_{\perp}$ is very small for low energy
conduction band excitations. According to the Liouville theorem [19], the high
momentum resolution is directly related with the poor spacial resolution. The
cross section for inelastic electron scattering is proportional to $1/q^{2}$.
Therefore the method has its strength at low $q$ or long wavelengths.

T-EELS is very often performed using Transmission Electron Microscopes (TEM)
[20], which have a high spacial resolution and thus a poor momentum
resolution. At present there are some TEMs which now have a high energy
resolution up to 0.01 eV and a reasonable momentum resolution at the expense
of the spacial resolution [21, 22, 23, 24].

The loss-function can be also directly measured by IXS. Also in this case, the
cross section is well understood [25]. Because it increases with $q^{2}$, the
method is more useful at higher momentum transfer. The energy and the momentum
resolution is up to 0.04 eV and 0.04 Å-1, respectively.

RIXS was applied mostly for the investigation of phononic, magnetic, and non-
dipolar electronic excitations [26, 27]. Recently it was also possible to
study acoustic plasmon excitations with an energy resolution of about 0.04 eV
[28, 29].

In the past, R-EELS was predominantly used for the study of excitations of
molecules on the surface of solids [30, 31]. Recently, the method was also
used to study momentum dependent electronic excitations in solids [32, 2]. A
very high energy resolution of the order of meV was achieved. On the other
hand, the scattering process is rather complicated: the dominant scattering
channel is supposed to be an elastic plus inelastic double scattering process
[33]. The large momentum perpendicular to the surface (for a primary energy of
50 eV $q_{\perp}\approx 4$ Å-1) may yield a finite $q_{||}$ for non-zero
energy losses [34].

Optical spectroscopy yields the reflectivity or by ellipsometry the complex
dielectric function [35]. From those data, using a Kramers-Kronig
transformation, the loss-function and electron-hole excitations can be derived
for zero momentum.

A complementary method is electronic Raman scattering, which provides
anisotropic and zero momentum information for electron-hole excitations [36].

##  III III Comparison of the R-EELS results with other results

Using EELS in transmission, well defined plasmons close to 1 eV were detected
in numerous 2D cuprates: Bi2Sr2CaCu2O8+x [37, 38, 39, 40, 21] , YBa2Cu2O7 [41,
42], YBa2Cu4O8 [43], and Ca1.9Na0.1CuO2Cl2 [44]. In compounds containing CuO
chains (YBa2Cu2O7 and YBa2Cu4O8), for momentum parallel to the chains, the
plasmon is shifted to higher energies. In all cases, the plasmon was not
overdamped. The full width at half maximum (FWHM) of the plasmon is about 0.5
eV, which is smaller than its energy, signaling a well defined quasi particle.
This width is not abnormal – the plasmon of the nearly free electron metals Na
and Al at 6 and 15 eV have a width of 0.25 eV and 1 eV, respectively [45, 13].
Also in the quasi-1D ladder compounds (La,Ca)xSr14-xCu24O41 well defined
plasmons have been detected for $q$ parallel to the ladders [46, 47].

In same cases, the plasmon dispersion has been studied. This dispersion was
always positive, typical of an electron liquid with a finite compressibility.
This plasmon dispersion could be well explained in terms of collective intra-
band transitions. The anisotropic dispersion provided detailed information on
the shape of the Fermi surface [40, 48, 47]. At higher momentum transfer near
$q=0.4$ Å-1 the plasmon is heavily damped and only a continuum is visible.
This was explained by Landau damping, i.e., the plasmon decays into intra-band
transitions [37, 38, 40].

On the other hand, EELS using a TEM with an energy resolution of 0.01 eV
showed in cuprates no plasmon, but just a continuum extending to $\approx 1.5$
eV [21]. This result can be explained by the poor momentum resolution in this
experiment which leads to a spectral weight predominantly stemming from the
high-$q$ loss function, where the plasmon has decayed by Landau damping.

Early optical spectroscopy on La2-xSrxCuO4 derived for $x$ close to optimal
doping a plasmonlike reflectivity edge near 0.8 eV [49]. More recent results
for the loss-function of Bi2Sr2CaCu2O8+x, derived from optical data, perfectly
agree with the EELS results for momentum transfer close to zero [50].

Another result which is in stark discrepancy with the recent R-EELS studies of
Ref. [1] and which also does not support theories for heavily damped plasmons
due to quantum fluctuations, results from recent RIXS studies on n- [28] and
p-type doped cuprates [29]. In these experiments acoustic plasmons with an
out-of-plane dispersion have been detected in the low-($\omega,q$) range with
a very small width of $\approx 0.1$ eV.

All these findings document that well defined plasmon excitations exist in
cuprate based systems. In the following, it is shown that they can be well
modeled based upon an extended Lindhard model.

##  IV IV The extended Lindhard model

In the original Lindhard model [10], the collective and the single $e-h$
excitations have been calculated for a free electron system. For a half-filled
band a sketch of the $(\omega,q)$ map for possible charge excitations is
presented in Fig. 1(a). The one-dimensional band structure $E(k)=-4tcos(ak)$
was used with $t=0.5$ eV and the lattice constant $a=3.8$ Å. This band
structure corresponds to that along the nodal direction in cuprates, were the
highest Fermi velocity ($v_{F}$) appears. Because both, the plasmon energy and
the single $e-h$ excitation region are related to the highest $v_{F}$, a two
dimensional cuprate band structure would not change very much the map shown in
Fig. 1(a). The $e-h$ intra-band excitations exists in a $2k_{F}$ wide stripe
which is underlayed by red color.

The collective plasmon excitations appear below a critical momentum $q_{c}$
showing a dispersion which can be explained in terms of the band structure
[40, 48]. Above $q_{c}\approx\omega_{P}/v_{F}$, the plasmon decays into single
$e-h$ excitations, termed Landau damping.

There should be no plasmon width for long wave-length caused by correlation
effects: the charge of the electrons is just shifted against the charge of the
ions and the restoring force is determined by the surface charges. This view
is supported by calculations of the plasmon broadening within a free-electron
model due to electron-electron interactions [51], which derived for the
plasmon width $\Gamma_{P}=Bq^{2}$. Thus the finite plasmon width at $q=0$ was
for many years a puzzle.

Finally, it was shown that in non-free-electron systems the plasmon width for
long wave lengths is determined by a decay of the plasmons into $e-h$ inter-
band transitions [52, 53]. The strength of the inter-band transitions at low
momentum transfer is related to the square of the Fourier transform of the
pseudo-potential, which leads to an umklapp scattering of the conduction band
from the second to the first Brillouin zone. This leads to a separate region
of inter-band transitions in the $(\omega,q)$ map, marked in green in Fig.
1(a).

Intra- and inter-band transitions have been detected in several EELS or IXS
experiments. Even for nearly-free electron systems such as alkali metals and
Be, both intra- and inter-band transitions are visible by EELS [45] and IXS
[54], respectively. In highly correlated systems the dielectric function and
thus the loss-function is enhanced by electronic local field effects [55, 56].

Upon doping, the Fermi level is shifted to lower or higher energies. This
leads in the $(\omega,q)$ map at low $q$ and low $\omega$ to a triangular
“empty” region without intra- or inter-band transitions. In Fig. 1(b) we show
this case for an electron doped cuprate. As demonstrated for p-type doped
graphite [57] and n-type doped graphite [58, 59], very narrow plasmons could
be detected in this region.

Another extension of the Lindhard model is related to the finite lifetime of
holes and electrons in the conduction band. Instead of bands, one should use
propagators for the occupied and unoccupied band, which depend on the complex
self-energy $\Sigma(\Omega)$ [60]. Its imaginary part is proportional to the
lifetime broadening $\Gamma(\Omega)$. A poor man’s version of the
quasiparticle lifetime model [61] is used here to estimate the influence of
the finite lifetime. The energy dependent loss function is convoluted with the
energy dependent lifetime broadening $\Gamma(\Omega)$ of the quasi particles.
The latter is taken from ARPES derived lifetime broadening of single particle
hole excitations in the conduction band of optimally doped cuprates [62, 63,
64, 65]: $\Gamma(\Omega)\approx\Omega$ typical of a marginal Fermi liquid [7].
This leads to a finite single-particle intra-band susceptibility also in the
region of the inter-band transitions, which is shown in Fig. 1(c). Fig. 1(d)
shows a waterfall plot of the spectral weight of the intra-band transition as
a function of energies for various momentum transfer. In the Lindhard model,
with increasing $q$, there is an increasing shark fin like structure. After
convolution with finite lifetime effects, there is a strong broadening of the
peak, the maximum is shifted to lower energies, and the $q$ dependence of the
maximum is strongly reduced (see Fig. 1(c)). For highly correlated systems,
even below $q_{c}$, there should be a finite plasmon broadening from a decay
into intra-band excitations. Thus the width of the plasmons is determined not
only by the decay into inter-band transition, the strength of which are
determined by the pseudo-potential, but also by a broadening of the intra-band
transitions due to the finite lifetime of the quasi particles of the
conduction band.

##  V V Discussion

How can we explain the differences between the R-EELS and all the T-EELS and
optical data. The R-EELS at small momentum transfer are very similar to the
T-EELS data at higher $q$. As mentioned above, in R-EELS there is a very large
$q_{\perp}$. As long as it is perpendicular to the surface (specular
reflection) of a 2D electronic system, it has no influence on the measured
loss-function with a finite $q_{||}$. Normally, the samples are aligned into
the specular geometry by elastic Bragg reflections, i.e., without energy-loss.
However, for a finite energy-loss, the absolute momentum values of the
incident and scattered electrons differ, which leads to an off-specular
scattering process and thus to an additional $q_{||}$ [34]. The enormous
sensitivity of the data on the exact specular position was already discussed
in Ref. [32]. In this way, it is possible to explain the differences between
the T-EELS and the R-EELS data: the latter do not reach low momentum transfers
in case of finite energy losses. This may also explain, why there are only a
few R-EELS studies of plasmons and surface plasmons at higher energy [34].

Next, we point out that the more traditional model, discussed here, describes
many material specific experimental results. It describes the plasmon energy
and its positive and anisotropic dispersion on the basis of collective intra-
band transitions [40, 48]. It can explain the finite width of the plasmon at
low $q$ in terms of a decay into inter-band transitions. The model gives a
reasonable $q_{c}$ value which is connected to the Fermi velocity. The narrow
plasmon width of the acoustic plasmons [28, 29] is understandable by the
“empty” region at low $\omega$ and low $q_{||}$.

Furthermore, the continuum and its energy dependence can be explained by
intra-band transitions (see the red lines in Fig. 1(d)). At low $q$, which is
probably not reached in the R-EELS experiment, the electron-hole excitations
exhibit a strong $q$-dependence. On the other hand, at higher $q$, the
momentum dependence is strongly reduced (see Fig. 1(c) and (d)). Finally, the
strong reduction of the continuum at low energies, observed in Fig. 1 of Ref.
[1] when going from optimally doped compounds with a marginal Fermi liquid
($\Gamma\propto\omega$) to overdoped cuprates ($\Gamma\propto\omega^{2}$) can
be explained in the latter by the reduced lifetime broadening at low energies
[66].

Moreover, the strong temperature dependence for optimally and overdoped
compounds at low energies (see Fig. 1 of Ref. [1]) is related to the fact,
that there the broadening due to the thermal energy is much larger than the
broadening due to a finite energy (see inset in Fig. 1(d)).

Summarizing the discussion, the extended Lindhard model together with the
experimental results on the lifetime broadening of single particle excitations
from ARPES can explain the heavily discussed continuum by intra-band
transitions. In turn the continuum in cuprates is related by a feed-back
process to the large scattering rates caused by on-site Coulomb interactions.
Thus both the continuum and the finite lifetime broadening are related to the
resistivity in the “strange metal” range of cuprates.

##  VI VI Conclusion

Hopefully, this comment will motivate experimental groups to construct EELS
spectrometers with energy resolutions in the range of some meVs. Spectrometers
with a resolution of 4 meV were already built in the 70th of the last century
[67]. As mentioned above, there are also some more recent attempts to achieve
such resolutions. Possibly also R-EELS can improve the kinematics of the
scattering process to reach data at low momentum transfer. Then it will be
possible to bridge the gap with Raman data or compete with RIXS. Moreover, to
complement the single-particle excitations from ARPES with two particle
excitations from EELS will be an interesting task in future solid state
physics.

##  VII  ACKNOWLEDGMENTS

I thank Lara Benfatto, Bernd Büchner, Luis Craco, Stefan-Ludwig Drechsler,
Rudi Hackl, Martin Knupfer, Thomas Pichler, and Friedrich Roth for helpful
discussions.

## References

  * Husain  _et al._ [2019] A. A. Husain, M. Mitrano, M. S. Rak, S. Rubeck, B. Uchoa, K. March, C. Dwyer, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, PRX 9, 041062 (2019).
  * Vig  _et al._ [2017] S. Vig, A. Kogar, M. Mitrano, A. Husain, L. Venema, M. Rak, V. Mishra, P. Johnson, G. Gu, E. Fradkin, M. Norman, and P. Abbamonte, SciPost Phys. 3, 026 (2017).
  * Mitrano  _et al._ [2018] M. Mitrano, A. A. Husain, S. Vig, A. Kogar, M. S. Rak, S. I. Rubeck, J. Schmalian, B. Uchoa, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, Proc Natl Acad Sci USA 115, 5392 (2018).
  * Romero-Bermúdez  _et al._ [2019] A. Romero-Bermúdez, A. Krikun, K. Schalm, and J. Zaanen, PRB 99, 235149 (2019).
  * Bozovic  _et al._ [1987] I. Bozovic, D. Kirillov, A. Kapitulnik, K. Char, M. R. Hahn, M. R. Beasley, T. H. Geballe, Y. H. Kim, and A. J. Heeger, PRL 59, 2219 (1987).
  * Staufer  _et al._ [1990] T. Staufer, R. Hackl, and P. Müller, Solid State Communications 75, 975 (1990).
  * Varma  _et al._ [1989] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 1996 (1989).
  * Ruvalds [1987] J. Ruvalds, PRB 35, 8869 (1987).
  * Kresin and Morawitz [1988] V. Z. Kresin and H. Morawitz, PRB 37, 7854 (1988).
  * Lindhard [1954] J. Lindhard, Kgl. Danske Videnskab. Selskab, Mat-phs. Medd 28, 8 (1954).
  * Pines and Nozieres [1966] D. Pines and P. Nozieres,  _The Theory of Quantum Liquids_ (Perseus Books, 1966).
  * Damascelli  _et al._ [2003] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473 (2003).
  * Raether [1965] H. Raether, Springer Tracts in Modern Physics 38, 85 (1965).
  * Daniels  _et al._ [1970] J. Daniels, C. von Festenberg, H. Raether, and K. Zeppenfeld, Springer Tracts in Modern Physics 54, 77 (1970).
  * Schnatterly [1979] J. Schnatterly, Solid State Phys. 34, 275 (1979).
  * Schattschneider [1986] P. Schattschneider,  _Fundamentals of Inelastic Electron Scattering_ (Springer Verlag, Wien, 1986).
  * Fink [1989] J. Fink, Adv. Electr. Electr. Phys. 75, 121 (1989).
  * Roth  _et al._ [2014] F. Roth, A. König, J. Fink, B. Büchner, and M. Knupfer, Journal of Electron Spectroscopy and Related Phenomena 195, 85 (2014).
  * Liouville [1838] J. Liouville, Journ. de Math. 3, 342 (1838).
  * Egerton [1996] R. Egerton,  _Electron Energy-Loss Spectroscopy in the Electron Microscope_ (Plenum Press, New York, 1996).
  * TERAUCHI  _et al._ [1999] M. TERAUCHI, M. TANAKA, K. TSUNO, and M. ISHIDA, Journal of Microscopy 194, 203 (1999).
  * Krivanek  _et al._ [2014] O. L. Krivanek, T. C. Lovejoy, N. Dellby, T. Aoki, R. W. Carpenter, P. Rez, E. Soignard, J. Zhu, P. E. Batson, M. J. Lagos, R. F. Egerton, and P. A. Crozier, Nature 514, 209 (2014).
  * Senga  _et al._ [2019] R. Senga, K. Suenaga, P. Barone, S. Morishita, F. Mauri, and T. Pichler, Nature 573, 247 (2019).
  * Hong  _et al._ [2020] J. Hong, R. Senga, T. Pichler, and K. Suenaga, PRL 124, 087401 (2020).
  * Schülke [2007] W. Schülke,  _Electron Dynamics by Inelastic X-Ray Scattering_ (Oxford University Press, USA, 2007).
  * Kotani and Shin [2001] A. Kotani and S. Shin, RMP 73, 203 (2001).
  * Ament  _et al._ [2011] L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, RMP 83, 705 (2011).
  * Hepting  _et al._ [2018] M. Hepting, L. Chaix, E. W. Huang, R. Fumagalli, Y. Y. Peng, B. Moritz, K. Kummer, N. B. Brookes, W. C. Lee, M. Hashimoto, T. Sarkar, J.-F. He, C. R. Rotundu, Y. S. Lee, R. L. Greene, L. Braicovich, G. Ghiringhelli, Z. X. Shen, T. P. Devereaux, and W. S. Lee, Nature 563, 374 (2018).
  * Nag  _et al._ [2020] A. Nag, M. Zhu, M. Bejas, J. Li, H. C. Robarts, H. Yamase, A. N. Petsch, D. Song, H. Eisaki, A. C. Walters, M. García-Fernández, A. Greco, S. M. Hayden, and K.-J. Zhou, PRL 125, 257002 (2020).
  * Ibach and Mills [1982] H. Ibach and D. Mills,  _Electron Energy Loss Spectroscopy and Surface Vibrations_ (Academic, New York, 1982).
  * Lüth [1993] Lüth,  _Surfaces and Interfaces of Solids_ (Springer Verlag Berlin, 1993).
  * Schulte [2002] K. H. G. Schulte,  _The interplay of spectroscopy and correlated materials_ , Ph.D. thesis, University of Groningen (2002).
  * Mills [1975] D. L. Mills, Surface Science 48, 59 (1975).
  * Chiarello  _et al._ [2000] G. Chiarello, V. Formoso, A. Santaniello, E. Colavita, and L. Papagno, PRB 62, 12676 (2000).
  * Wooten [1972] F. Wooten,  _Optical properties of solids_ (Academic Press, New York, 1972).
  * Devereaux and Hackl [2007] T. P. Devereaux and R. Hackl, RMP 79, 175 (2007).
  * Nücker  _et al._ [1989] N. Nücker, H. Romberg, S. Nakai, B. Scheerer, J. Fink, Y. F. Yan, and Z. X. Zhao, PRB 39, 12379 (1989).
  * Wang  _et al._ [1990] Y.-Y. Wang, G. Feng, and A. L. Ritter, PRB 42, 420 (1990).
  * Nakai  _et al._ [1990] S. Nakai, N. Nücker, H. Romberg, M. Alexander, and J. Fink, Physica Scripta 41, 596 (1990).
  * Nücker  _et al._ [1991] N. Nücker, U. Eckern, J. Fink, and P. Müller, Phys. Rev. B 44, 7155 (1991).
  * Tarrio and Schnatterly [1988] C. Tarrio and S. E. Schnatterly, PRB 38, 921 (1988).
  * Romberg  _et al._ [1990] H. Romberg, N. Nücker, J. Fink, T. Wolf, X. X. Xi, B. Koch, H. P. Geserich, M. Dürrler, W. Assmus, and B. Gegenheimer, Zeitschrift für Physik B Condensed Matter 78, 367 (1990).
  * Knupfer  _et al._ [1994] M. Knupfer, G. Roth, J. Fink, J. Karpinski, and E. Kaldis, Physica C: Superconductivity 230, 121 (1994).
  * Schuster  _et al._ [2012] R. Schuster, S. Pyon, M. Knupfer, M. Azuma, M. Takano, H. Takagi, and B. Büchner, PRB 86, 245112 (2012).
  * vom Felde  _et al._ [1989] A. vom Felde, J. Sprösser-Prou, and J. Fink, Physical Review B 40, 10181 (1989).
  * Roth  _et al._ [2010] F. Roth, C. Hess, B. Büchner, U. Ammerahl, A. Revcolevschi, and M. Knupfer, PRB 82, 245110 (2010).
  * Roth  _et al._ [2020] F. Roth, A. Revcolevschi, B. Büchner, M. Knupfer, and J. Fink, PRB 101, 195132 (2020).
  * Grigoryan  _et al._ [1999] V. G. Grigoryan, G. Paasch, and S.-L. Drechsler, PRB 60, 1340 (1999).
  * Uchida  _et al._ [1991] S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Phys. Rev. B 43, 7942 (1991).
  * Levallois  _et al._ [2016] J. Levallois, M. K. Tran, D. Pouliot, C. N. Presura, L. H. Greene, J. N. Eckstein, J. Uccelli, E. Giannini, G. D. Gu, A. J. Leggett, and D. van der Marel, PRX 6, 031027 (2016).
  * DuBois and Kivelson [1969] D. F. DuBois and M. G. Kivelson, Physical Review 186, 409 (1969).
  * Paasch [1970] G. Paasch, phys. stat. sol. 38, K123 (1970).
  * Gibbons and Schnatterly [1977] P. C. Gibbons and S. E. Schnatterly, PRB 15, 2420 (1977).
  * Schülke  _et al._ [1987] W. Schülke, U. Bonse, H. Nagasawa, S. Mourikis, and A. Kaprolat, PRL 59, 1361 (1987).
  * Hubbard [1957] J. Hubbard, Proc. R. Soc.,london 243, 336 (1957).
  * Mahan [2000] G. D. Mahan,  _Many-Particle Physics_ (Kluwer Academic/Plenum Publishers, New York, 2000).
  * Ritsko and Mele [1980] J. J. Ritsko and E. J. Mele, PRB 21, 730 (1980).
  * Grunes and Ritsko [1983] L. A. Grunes and J. J. Ritsko, PRB 28, 3439 (1983).
  * Roth  _et al._ [2013] F. Roth, A. König, C. Kramberger, T. Pichler, B. Büchner, and M. Knupfer, EPL (Europhysics Letters) 102, 17001 (2013).
  * Varma [2017] C. M. Varma, PRB 96, 075122 (2017).
  * Green  _et al._ [1985] F. Green, D. Neilson, and J. Szymański, PRB 31, 2779 (1985).
  * Valla  _et al._ [1999] T. Valla, A. V. Fedorov, P. D. Johnson, B. O. Wells, S. L. Hulbert, Q. Li, G. D. Gu, and N. Koshizuka, Science 285, 2110 (1999).
  * Kaminski  _et al._ [2001] A. Kaminski, M. Randeria, J. C. Campuzano, M. R. Norman, H. Fretwell, J. Mesot, T. Sato, T. Takahashi, and K. Kadowaki, Phys. Rev. Lett. 86, 1070 (2001).
  * Koitzsch  _et al._ [2004] A. Koitzsch, S. V. Borisenko, A. A. Kordyuk, T. K. Kim, M. Knupfer, J. Fink, H. Berger, and R. Follath, Phys. Rev. B 69, 140507 (2004).
  * Reber  _et al._ [2019] T. J. Reber, X. Zhou, N. C. Plumb, S. Parham, J. A. Waugh, Y. Cao, Z. Sun, H. Li, Q. Wang, J. S. Wen, Z. J. Xu, G. Gu, Y. Yoshida, H. Eisaki, G. B. Arnold, and D. S. Dessau, Nature Communications 10, 5737 (2019).
  * Fink [2016] J. Fink, EPL (Europhysics Letters) 113, 27002 (2016).
  * Schröder and Geiger [1972] B. Schröder and J. Geiger, PRL 28, 301 (1972).