# Resonant Inelastic X-ray Scattering Study of Electron-Exciton Coupling in
High-$T_{c}$ Cuprates

F. Barantani  Department of Quantum Matter Physics, University of Geneva, 1211
Geneva, Switzerland  Institute of Physics, École Polytechnique Fédérale de
Lausanne, Lausanne, 1015, Switzerland  M. K. Tran  Department of Quantum
Matter Physics, University of Geneva, 1211 Geneva, Switzerland  I. Madan
Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne,
1015, Switzerland  I. Kapon  Department of Quantum Matter Physics, University
of Geneva, 1211 Geneva, Switzerland  N. Bachar  Department of Quantum Matter
Physics, University of Geneva, 1211 Geneva, Switzerland  Department of
Physics, Ariel University, Ariel, Israel  T. C. Asmara  Photon Science
Division, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  E. Paris
Photon Science Division, Paul Scherrer Institut, 5232 Villigen PSI,
Switzerland  Y. Tseng  Photon Science Division, Paul Scherrer Institut, 5232
Villigen PSI, Switzerland  W. Zhang  Photon Science Division, Paul Scherrer
Institut, 5232 Villigen PSI, Switzerland  Y. Hu  Department of Applied
Physics, Stanford University, CA 94305, USA  Stanford Institute for Materials
and Energy Sciences, SLAC,CA 94025, USA  E. Giannini  Department of Quantum
Matter Physics, University of Geneva, 1211 Geneva, Switzerland  G. Gu
Brookhaven National Laboratory, Upton, NY 11973 5000, USA  T. P. Devereaux
Stanford Institute for Materials and Energy Sciences, SLAC,CA 94025, USA
Department of Materials Science and Engineering, Stanford University,
Stanford, CA 94305, USA  Geballe Laboratory for Advanced Materials, Stanford
University, CA 94305, USA  C. Berthod  Department of Quantum Matter Physics,
University of Geneva, 1211 Geneva, Switzerland  F. Carbone  Institute of
Physics, École Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland
T. Schmitt  Photon Science Division, Paul Scherrer Institut, 5232 Villigen
PSI, Switzerland  D. van der Marel  Department of Quantum Matter Physics,
University of Geneva, 1211 Geneva, Switzerland

###### Abstract

Explaining the mechanism of superconductivity in the high-$T_{c}$ cuprates
requires an understanding of what causes electrons to form Cooper pairs.
Pairing can be mediated by phonons, the screened Coulomb force, spin or charge
fluctuations, excitons, or by a combination of these. An excitonic pairing
mechanism has been postulated, but experimental evidence for coupling between
conduction electrons and excitons in the cuprates is sporadic. Here we use
resonant inelastic x-ray scattering (RIXS) to monitor the temperature
dependence of the $\underline{d}d$ exciton spectrum of Bi2Sr2CaCu2O8-x
(Bi-2212) crystals with different charge carrier concentrations. We observe a
significant change of the $\underline{d}d$ exciton spectra when the materials
pass from the normal state into the superconductor state. Our observations
show that the $\underline{d}d$ excitons start to shift up (down) in the
overdoped (underdoped) sample when the material enters the superconducting
phase. We attribute the superconductivity-induced effect and its sign-reversal
from underdoped to overdoped to the exchange coupling of the site of the
$\underline{d}d$ exciton to the surrounding copper spins.

## Introduction

Ever since Bednorz and Muller’s Bednorz and Muller (1986) discovery of
high-$T_{c}$ superconductivity in cuprates the question as to what mediates
superconductivity in these materials has continued to occupy the scientific
community. The strong electron correlations of the Cu $3d$ states form a major
theoretical challenge Anderson (1987). In particular, in the insulating parent
compounds the on-site part of the Coulomb repulsion $U$ splits the Cu $3d$
band into a fully occupied lower Hubbard band (LHB) and an empty upper Hubbard
band (UHB). The LHB and UHB are $\sim 8$ eV apart and the O $2p$ band, which
is fully occupied, falls inside this gap with the top of the O $2p$ band about
$2$ eV below the UHB. The separation between the top of the O $2p$ band and
the UHB, the so-called charge transfer gap $\Delta_{CT}$ Zaanen  _et al._
(1985); Ghijsen  _et al._ (1988), constitutes the lower bound of the electron-
hole continuum.

Figure 1: (a) Experimental RIXS spectra for the three different dopings and
for different momentum transfers. To avoid clutter the spectra have been given
vertical offsets grouped according to the momentum values. Vertical bars
correspond to spin flip (red), $\underline{d}d$ excitons (blue) and charge
transfer excitations (orange) obtained from a cluster calculation (see
Appendix B). (b) Energy momentum dispersion of the fitted peak positions.
Error bars are smaller than the symbol size and therefore not indicated.
Momentum values are indicated on top.

Doping either electrons or holes induces an insulator-to-metal transition. It
has turned out to be difficult to obtain a simple analytical framework
describing simultaneously the localized charges of the correlated insulator
and the mobile charges responsible for the metallic conductivity. Even the
Hubbard model, describing a single band of charge carriers with on-site
Coulomb repulsion $U$, can not be solved analytically in 2 or 3 dimensions,
and the properties and predictions of this model are based on the numerical
solution of small clusters via density matrix renormalization group and
quantum Monte Carlo, or approximation schemes such as dynamical mean field
theory Arovas  _et al._ (2022). The present status of the Hubbard model is
that in 2 dimensions the superconducting state is unstable with respect to
charge stripes, and that stable superconductivity requires the suppression of
the charge stripes state by introducing next-nearest neighbor hopping, longer
range hoppings, dynamical lattice effects and other orbital contributions
Jiang and Devereaux (2019).

Theoretical studies of superconductivity in cuprates have mostly concentrated
on the aforementioned Hubbard model and the related $t$–$J$ model for which
the fundamental interaction is non-retarded Anderson (2007). When interacting
holes form Cooper pairs, the effective-mass change resulting from the pair
formation may stabilize the superconducting state Hirsch (2002). On a
microscopic scale a pairing-induced modification of the bare density response
function Leggett (1999), magnon mediated-interactions Eschrig (2006);
Scalapino (2012), plasmon-mediated interactions Bill  _et al._ (2003), pairing
mediated by the ionic electron polarizabilities Sawatzky  _et al._ (2009);
Yacoby and Girshberg (2016), and exciton-mediated pairing Weber  _et al._
(1988); Zaanen and Oleś (1993); Feiner  _et al._ (1992); Buda  _et al._
(1994); Holcomb  _et al._ (1994); Bucci  _et al._ (1995); Little  _et al._
(2007); Mazov (2012) derive from the Coulomb interaction between the electrons
(for a short review see Ref. Oleś  _et al._ (2019)), which does not involve
lattice degrees of freedom in contrast to phonon-mediated pairing Maksimov
_et al._ (2010). The concept of exciton-mediated pairing has been pioneered in
the context of organic materials Little (1964), and sandwich structures
Ginzburg (1970); Allender  _et al._ (1973); Mazov (2012). It has also been
considered for the cuprates Weber  _et al._ (1988); Zaanen and Oleś (1993);
Feiner  _et al._ (1992); Buda  _et al._ (1994); Holcomb  _et al._ (1994);
Bucci  _et al._ (1995); Little  _et al._ (2007); Mazov (2012), but
experimental information has remained scarce. Excitons are bound states of an
electron and a hole due to the attraction resulting from their opposite
charge. The strongest binding occurs when the electron and the hole occupy the
same site, namely, when an electron from one of the occupied $d_{xy}$,
$d_{yz}$, $d_{zx}$, or $d_{z^{2}}$ states is excited into the partially filled
$d_{x^{2}-y^{2}}$ state. Since this does not change the site occupancy there
is no energy cost of order $U$ and the only contribution comes from the level
splitting due to the crystal field Zaanen and Oleś (1993); Wang  _et al._
(2011). Such $\underline{d}d$ excitons are the most prominent features of
resonant inelastic x-ray scattering (RIXS) spectra at the Cu $L$-edge Kuiper
_et al._ (1998) and from numerous experimental Ghiringhelli  _et al._ (2004,
2007); Moretti Sala  _et al._ (2011); Dean  _et al._ (2013, 2014) and
theoretical Wang  _et al._ (2011) studies it has become clear that they are
robust for all experimentally achievable carrier concentrations.

It is known that phonons coupled to conduction electrons change energy and
lifetime when entering the superconducting phase Axe and Shirane (1973);
Shapiro  _et al._ (1975); Altendorf  _et al._ (1993). This behavior is
attributed to a change of the self-energy of the phonons below $T_{c}$ due to
the fact that they are coupled to the conduction electrons Zeyher and
Zwicknagl (1990). On the other hand, when a $\underline{d}d$ exciton is
created, there is an orbital flip from the $3d_{x^{2}-y^{2}}$ to one of the
other Cu $3d$ levels. Thus, the $\underline{d}d$ excitons are firmly tied to
the carriers at the Fermi level. Here we report the observation of the
renormalization of $\underline{d}d$ excitons with energies in the range of
$1$–$3$ eV. Using RIXS we observe the impact of the superconducting order on
the $\underline{d}d$ excitons in Bi2Sr2CaCu2O_8-x.

Figure 2: Top panels: Normalized RIXS spectra: the spectra for different
temperatures overlap except near the maximum of the $\underline{d}d$ peak.
Bottom panels: Normalized RIXS spectra for the same temperatures as the top
panels from which the spectra at $T_{c}$ have been subtracted. (a,c)
Underdoped Bi-2212 ($T_{c}=70$ K). (b,e) Optimally doped Bi-2212 ($T_{c}=91$
K). (c,f) Overdoped Bi-2212 ($T_{c}=70$ K). (a,b,c) Momentum value $(0.36,0)$.
(d,e,f) Momentum value $(0.25,0.25)$.

## Experimental results

We measured RIXS spectra of crystals from the Bi_2Sr_2CaCu_2O_8-x bi-layer
cuprate family across the superconducting dome: underdoped with carrier
concentration $p=0.11$ and $T_{c}=70$ K, optimally doped ($p=0.16$, $T_{c}=91$
K), and overdoped ($p=0.21$, $T_{c}=70$ K). The measurements, performed at the
ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institut (PSI)
Strocov  _et al._ (2010); Ghiringhelli  _et al._ (2006), were carried out
using $\sigma$-polarization of the incoming x-rays with energy resonant at the
Cu $L_{3}$ edge. We express the momentum vectors on the basis of the pseudo-
tetragonal structure with planar lattice parameter $a=3.83$ Å along the CuO
bonds and $c=30.9$ Å perpendicular to the planes. Each crystal was aligned on
the sample holder using Laue diffraction, and data were collected for selected
values of the momentum transfer $\mathbf{q}=(\mu,\nu,j)$ in units of the
reciprocal lattice, where $j$ is an integer number. Keeping in mind the
periodicity in reciprocal space, we will use the shorthand notation
$(\mu,\nu)$.

Figure 3: Temperature dependence of the first moment of the $\underline{d}d$
exciton peak and the magnon peak, as extracted from the experimental spectra
(considering respectively the energy intervals $1$–$4$ eV and $0.1$–$0.6$ eV).
The dashed lines are fits to a curve parametrized as Eq. (2). The error bars
represent the uncertainty on the elastic peak position computed as explained
in Appendix A. The vertical dotted lines indicate $T_{c}$. Top panels:
momentum value $(0.36,0)$. Bottom panels: momentum value $(0.25,0.25)$. (a,d)
Underdoped Bi-2212 ($T_{c}=70$ K). (b,e) Optimally doped Bi-2212 ($T_{c}=91$
K). (c,f) Overdoped Bi-2212 ($T_{c}=70$ K). Figure 4: Temperature dependence
of the integrated RIXS intensity $I_{j}(T)$ where the index $j=1,2$ refers to
the two energy ranges $1$–$2$ eV, and $2$–$3.5$ eV respectively. The dashed
lines indicate $T_{c}$. Top panels: momentum value $(0.36,0)$. Bottom panels:
momentum value $(0.25,0.25)$. (a,d) Underdoped Bi-2212 ($T_{c}=70$ K). (b,e)
Optimally doped Bi-2212 ($T_{c}=91$ K). (c,f) Overdoped Bi-2212 ($T_{c}=70$
K).

The samples were cleaved at $20$ K in ultra high vacuum (UHV) directly on the
manipulator for RIXS data acquisition.

In Fig. 1(a) we compare RIXS spectra of Bi-2212 for different dopings and
momentum values. Since as we will see below, the temperature dependence is too
weak to be noticeable on this scale, for each given doping $p$ and momentum
value $\mathbf{q}$ the spectra $I_{R}(p,\mathbf{q},\omega)$ were averaged in
the temperature range of approximately $50$ K above and below $T_{c}$. The
vertical bars are theoretical predictions of the excitation energies following
from a cluster calculation (see Appendix B). We see that the central peak at
$1.9$ eV corresponds to the $\underline{d}_{xz}d_{x^{2}-y^{2}}$ and
$\underline{d}_{yz}d_{x^{2}-y^{2}}$ excitons, the shoulder at $1.5$ eV to
$\underline{d}_{xy}d_{x^{2}-y^{2}}$ and the shoulder at $2.4$ eV to
$\underline{d}_{z^{2}}d_{x^{2}-y^{2}}$. We have fitted the spectra to a linear
superposition of Voigt profiles (see Appendix A). The $\underline{d}d$
manifold is described by the three aforementioned peaks, with Gaussian
profiles having a FWHM of around (0.5, 0.9, 0.4) eV respectively for (1.5,
1.9, 2.4) eV. This line shape has been shown by Lee et al. to originate in the
Franck-Condon principle, resulting in phonon overtones of the electronic mode
described by a Gaussian envelope Lee  _et al._ (2014). We observe that, when
the doping increases, the $\underline{d}d$ excitons shift to lower energy. In
particular, from the peak positions extracted by fitting, we can see that for
$\mathbf{q}=(0.36,0)$, where the effect is strongest, all excitons shift to
lower energy.

The momentum dispersion for the three different dopings is shown in Fig. 1(b).
We see that the peak positions have relatively small energy-momentum
dispersion, which confirms the observations of Moretti et al. Moretti Sala
_et al._ (2011). This finding underlines the strongly localized character of
these excitations. The peak energy below 0.5 eV has strong momentum
dependence, which is expected for a magnon.

In the top panels of Fig. 2 we present the RIXS data for different dopings,
momentum transfers and for different temperatures. The data in this figure are
averages spanning intervals of $15$ K (UD 70K and OD 70K) and $20$ K (OP 91K)
in order to minimize the statistical noise. In order to highlight the spectral
temperature dependence, we show in the bottom panels the spectra from which
the spectrum taken at $T_{c}$ is subtracted. The temperature dependence in the
paramagnon region between $0.2$ eV and $1$ eV is very weak, which is
consistent with the observations of Peng et al. Peng  _et al._ (2015) for
optimally doped Bi1.5Pb0.55Sr1.6La0.4CuO6+δ at $50$ K and $200$ K. Also the
charge transfer region between 3.5 and 8 eV has negligible temperature
dependence. The most significant temperature dependence occurs for the
underdoped and overdoped samples in the part of the spectrum corresponding to
the $\underline{d}d$ excitons. In particular the difference spectra show that
spectral weight is redistributed between different parts of the
$\underline{d}d$ manifold as a function of temperature. We calculated for each
spectrum the average energy

$E=\frac{\int_{\omega_{1}}^{\omega_{2}}I(\omega)\omega
d\omega}{\int_{\omega_{1}}^{\omega_{2}}I(\omega)d\omega}.$ (1)

In Fig. 3 the temperature dependence of the magnon and exciton energies are
displayed using Eq. (1). For the magnon we use the integration limits
$\omega_{1}=0.1$ eV and $\omega_{2}=0.6$ eV, for the exciton $\omega_{1}=1$ eV
and $\omega_{2}=4$ eV. In Appendix A we demonstrate that the temperature
evolution is essentially a uniform shift of $\underline{d}_{xy}d$ and
$\underline{d}_{xz,yz}d$, with a significantly smaller shift of
$\underline{d}_{z^{2}}d$. Since the latter component does not contribute very
strongly to the overall spectral weight of the $\underline{d}d$ manifold, the
average $\underline{d}d$ exciton energy provides a meaningful representation
of the thermal evolution of the spectrum. We see that this energy shifts as
function of temperature for different dopings. The temperature dependence
shows an upward change of slope at $T_{c}$ for the overdoped case [in Fig.
3(c) and (f)], and a downward change with smaller magnitude in the underdoped
sample [panels (a) and (d)]. In contrast the magnon peak shows a negligible,
if not absent, superconductivity induced effect. It is of interest to make a
connection to the doping dependence shown in Fig. 1, where we noticed that for
$\mathbf{q}=(0.36,0)$ the excitons soften with increasing doping. These trends
are consistent with the evolution of either antiferromagnetic or singlet
correlations as a function of doping and temperature. That said, for
$\mathbf{q}=(0.25,0.25)$ both doping and temperature dependence appear to
follow a more complex pattern.

These temperature dependencies are also clearly revealed by the intensities of
the left-hand side (1–2 eV) and right-hand side (2–3.5 eV) of the
$\underline{d}d$ manifold, confirming the change of slope near $T_{c}$ for the
underdoped and overdoped samples (see Fig. 4). To quantify these slope changes
we have fitted the temperature dependence of the average energies of the
magnon and the exciton manifold to

$\displaystyle T\geq T_{c}:E(T)=E_{c}+E_{n}\left(1-T/T_{c}\right)$
$\displaystyle T\leq
T_{c}:E(T)=E_{c}+\left(E_{n}+E_{sc}\right)\left(1-T/T_{c}\right)$ (2)

by adjusting the normal state parameters $E_{c}$, $E_{n}$, and the
superconductivity induced change of energy $E_{sc}$. The dashed curves in Fig.
3 are least-squares fits of this expression to the experimental data points.
In Fig. 4 we present the temperature dependence of the integrated intensity of
the spectra shown in Fig. 2

$I_{j}(p,\mathbf{q},T)=\int_{\omega_{j}}^{\omega_{j}^{\prime}}\left[I(p,\mathbf{q},\omega,T)-I_{R}(p,\mathbf{q},\omega)\right]d\omega$
(3)

in two regions defined by $(\omega_{j},\omega_{j}^{\prime})$: $1$–$2$ eV
(region 1), and $2$–$3.5$ eV (region 2). The split at 2 eV is motivated by the
fact that for all panels in Fig. 2 the intensity on either side has opposite
temperature dependence. It thus provides optimal signal-to-noise ratio of the
temperature dependence in the $\underline{d}d$ exciton region. Comparing the
intensity in regions 1 and 2, we see that, when entering the superconducting
phase, for the overdoped case spectral weight is transferred from the low
energy to the high energy side of the $\underline{d}d$ exciton spectrum and
vice versa for the underdoped case. As in Fig. 3, we observe a change of slope
in the $\underline{d}d$ temperature dependence around $T_{c}$, except for the
optimally doped sample. As can be seen in Fig. 3, for the OD 70K the
contribution of the superconducting state is to shift the $\underline{d}d$
exciton peak to higher energies. This conclusion is consistent with the
superconductivity-induced effect on the integrated intensity $I_{1,2}(T)$
which indicates a transfer of the spectral weight towards higher energies.

In Fig. 5 the superconductivity-induced energy changes $E_{sc}$ of the
$\underline{d}d$ excitons and of the magnon are compared for all samples and
momentum directions. At the overdoped side the effect of the superconducting
order is to shift the excitons from lower to higher energy, both along the
nodal and antinodal directions. We observe the opposite trend for the
underdoped sample, again for the nodal as well as the antinodal direction in
momentum space. For the optimally doped sample the superconductivity-induced
change is small along either direction. For the magnon, the influence of the
superconducting phase transition, if it exists, falls within the experimental
noise.

## Discussion

Figure 5: Doping dependence of the superconductivity-induced change of the
average $\underline{d}d$ exciton and magnon energies. The momentum values are
$(0.36,0)$ (green circles) and $(0.25,0.25)$ (blue squares). The error bars
are extracted from the fit procedure. Figure 6: (a) Phase diagram of the
cuprates indicating the antiferromagnetic phase (beige), the normal phase
(orange for underdoped and light blue for overdoped) and the superconducting
phase (green) with the corresponding short-range spin correlations and
$\underline{d}d$ exciton energies. (b,c) Sketch of the temperature dependence
of the $\underline{d}d$ exciton energy in the underdoped and overdoped cases.

The aforementioned electron-phonon coupling resulting in the Gaussian line
shape Lee  _et al._ (2014) can in principle lead to a finite temperature
dependence of the $\underline{d}d$ excitons, if the coupled vibrational modes
are sufficiently anharmonic. Such an effect has for example been observed in
the vibronic spectra of BEH-PPV conjugated polymer films Oliveira  _et al._
(2003), where the excitons have an Arrhenius type shift to higher energy with
increasing temperature. The observed temperature dependence in the present
case of the $\underline{d}d$ excitons may, at least in part, be due to the
electron-phonon coupling. A natural question is whether the observed change of
slope at the superconducting phase transition would also be a consequence of
the electron-phonon coupling. To have such a strong effect requires that the
phonon anharmonicity is strongly influenced by the superconducting phase
transition. Refs. Ahn and Nagaosa (2021) and Chen and Huang (2021) discuss
superconductivity induced change of orbital character and its impact on the
optical spectra. With some modifications, similar notions come into play into
the RIXS $\underline{d}d$ exciton spectra if, in conjunction with the SC phase
transition, there is a change of orbital character of the bands. The influence
on the excitonic spectrum is however very indirect and quantitatively much too
small compared to our observations. Little and Holcomb Holcomb  _et al._
(1994); Little  _et al._ (2007) analyzed the optical conductivity in the
superconducting and normal states and arrived at the conclusion that the
charge carriers are coupled to the $\underline{d}d$ excitons. Vice versa one
may expect a manifestation in the exciton spectrum of a mixing with the
electron-hole continuum, causing a renormalization of the exciton energy
$\omega_{\underline{d}d}$ by a self-energy
$g_{\mathbf{q}}^{2}\chi(\mathbf{q},\omega)$. Here $\chi(\mathbf{q},\omega)$ is
the charge susceptibility and $g_{\mathbf{q}}$ is a momentum dependent
coupling constant. Husain et al. Husain  _et al._ (2019) reported a strong
temperature dependence of the imaginary part of the charge susceptibility at
$\mathbf{q}=(0.17,0.17)$, with a change of sign from underdoped to overdoped.
In the weak coupling approach this phenomenology translates into a temperature
dependent exciton energy having a strong doping dependence of the temperature
coefficient. When the material passes from normal to superconducting, the
susceptibility changes, causing both an energy shift and a change of the
lifetime of the exciton, similar to what happens in the case of electron-
phonon coupling Axe and Shirane (1973); Shapiro  _et al._ (1975); Altendorf
_et al._ (1993); Zeyher and Zwicknagl (1990). The susceptibility function
satisfies
$V_{\mathbf{q}}\chi(\mathbf{q},\omega)=1/\epsilon(\mathbf{q},\omega)-1$, where
$V_{\mathbf{q}}$ is the Coulomb potential and $\epsilon(\mathbf{q},\omega)$
the dielectric function. For $q\approx 0$ the dielectric function
$\epsilon(\mathbf{q},\omega)$ can be measured with great precision using
optical techniques. Experimentally it was found that the electron-hole
susceptibility of the cuprates near $q=0$ is significantly impacted by the
superconducting order, even for energies higher than 1 eV Levallois  _et al._
(2016). By considering the renormalization due to the BCS charge
susceptibility, the temperature behavior calculated for the exciton is
qualitatively similar to the experimental observations at the overdoped side
(Appendix C). Quantitatively this weak-coupling model falls short in
describing the experimental data: even with the electron-exciton coupling
constant set to $g\sim 2$ eV, the calculated shift is an order of magnitude
smaller than in the experiment.

An explanation may be found in the fact that the superconducting carriers move
in the $3d_{x^{2}-y^{2}}$ bands, while a $\underline{d}d$ exciton is
effectively a localized orbital flip of a $3d_{x^{2}-y^{2}}$ hole to a
different $3d$ orbital. The circumstance that the electronic conductivity and
the excitons involve the same electrons calls for a more radical approach than
offered by the weak coupling scheme sketched above. In fact, if the material
is superconducting the orbital flip of one of the $3d_{x^{2}-y^{2}}$ holes
necessarily breaks up a Cooper’s pair. Consequently when passing from the
normal state to the superconducting state the exciton energy increases by the
pair breaking energy. The observation of a weaker effect of the opposite sign
for the underdoped sample appears to be, as we will argue below, a
manifestation of the physics underlying the pseudogap in the cuprates. A
deeper understanding is obtained by considering the following points:

  * •

The material is antiferromagnetic (AF) at zero doping. The lower bound of
inter-site exchange energy is therefor $-J/4$, where $J$ is the nearest-
neighbor exchange constant.

  * •

The $d$-wave superconducting state corresponds to a condensate of local
singlet pairs Kotliar and Liu (1988); Zhang  _et al._ (1988); Yokoyama and
Shiba (1988); Gros (1988). The exchange interaction of each singlet pair is
strongest if the two electrons or holes occupy nearest neighbor sites.
Consequently the lower bound of the exchange energy in the superconducting
state is $-3J/4$.

  * •

The antiferromagnetic state is suppressed by hole doping, but short-range (in
particular nearest-neighbor) spin-correlations persist and diminish gradually
as a function of increasing doping.

This scenario provides the formal basis for superconductivity by kinetic
energy lowering in the underdoped cuprates Scalapino and White (1998); Hirsch
(2002); Maier  _et al._ (2004); Toschi  _et al._ (2005); Haule and Kotliar
(2007); Gull and Millis (2012); Fratino  _et al._ (2016), and for
understanding why for the underdoped cuprates the Drude spectral weight
increases in the superconducting state Molegraaf  _et al._ (2002); Deutscher
_et al._ (2005); Gedik  _et al._ (2005); Carbone  _et al._ (2006); Giannetti
_et al._ (2011). It also explains why this effect flips sign at the overdoped
side, and why for overdoped cuprates the superconducting pairing is stabilized
by lowering the interaction energy Levallois  _et al._ (2016); Gull and Millis
(2012); Fratino  _et al._ (2016).

In the absence of interactions with the surrounding sites, the energies of the
$\underline{d}d$ excitons are $D_{j}$ ($1\leq j\leq 4$). In what follows we
assume for simplicity that $D_{j}$ is independent of doping and temperature.
The excitons can be visualized as a flip of a hole in a $d_{x^{2}-y^{2}}$
orbital to one of the other orbitals. Since creating an exciton breaks the
spin-correlations with the neighboring sites, the exciton energy is given by
$E_{j}=D_{j}+X^{e}-X^{g}$ where $X^{g}$ ($X^{e}$) is the exchange interaction
with the neighboring sites of the ground (excited) state. For $d_{xy}$,
$d_{xz}$ and $d_{yz}$ the symmetry is such that the hopping matrix-element to
a nearest-neighbor $d_{x^{2}-y^{2}}$ orbital vanishes. Consequently the
exchange interaction with a hole on a neighboring $d_{x^{2}-y^{2}}$ orbital is
zero, i.e. $X^{e}=0$ so that $E_{j}=D_{j}-X^{g}$.

For zero doping $X^{g}\geq-J$ due to the anti-ferromagnetic order and taking
into account that there are 4 nearest neighbors, and the lower bound
$X^{g}=-J$ corresponds to full spin polarization. The following expressions
for $X^{g}$ refer to the extremal value for each of the given phases. At
optimal doping $X^{g}=-3J/4$ due to the d-wave superconducting order. At very
high doping $X^{g}=0$. Increasing temperature at the underdoped side changes
the exchange energy $X^{g}$ from $-3J/4$ in the superconducting state to $-J$
in the normal state. Correspondingly the exciton energy $E_{j}=D_{j}-X^{g}$
decreases by an amount $J/4$. At the overdoped side $X^{g}$ changes from
$-3J/4$ in the superconducting state to $0$ in the normal state, so that the
exciton energy increases by an amount $3J/4$. Since at high temperature
$X^{g}$ gradually diminishes as a function of doping, having $-J$ and $0$ as
the extremal values at zero doping and high doping respectively, a doping
level exists where $X^{g}$ has the same value in the normal and
superconducting phase. For this doping there is no superconductivity induced
shift of the exciton energy. In practice, taking into account the experimental
limitations on resolution, the superconductivity induced shift of $X^{g}$
falls below the resolution limit not only at a single doping level but in a
range of dopings which, in view of Fig. 5, overlaps with optimal doping. This
state of affairs is depicted in Fig. 6. These effects are strongest for the
$\underline{d}_{xy,xz,yz}d_{x^{2}-y^{2}}$ excitons. For the
$\underline{d}_{z^{2}}d_{x^{2}-y^{2}}$ exciton all these effects are smaller
due to mixing with the oxygen-copper charge transfer excitations.

We now return to the question to what extent coupling to excitons plays a role
in the pairing mechanism of the cuprates. From the above discussion
illustrated in Fig. 6 we can conclude that our observations neither exclude
nor favor any of the pairing mechanisms described in the introduction. That
said, our experiments expose a strong coupling between charge carriers and
$\underline{d}d$ excitons. Therefore, it is reasonable to expect that
electron-exciton coupling contributes to the energy balance of the different
phases, including the part of the phase diagram where superconductivity is
observed.

## Conclusions

Using RIXS we have observed clear evidence for the coupling of the low-energy
conduction electrons in Bi-2212 to $\underline{d}d$ excitons. The coupling is
visible as a distinct change of the $\underline{d}d$ exciton peak at 2 eV when
the material becomes superconducting. On the basis of cluster calculations the
2 eV peak is predominantly a $\underline{d}_{yz/xz}d_{x^{2}-y^{2}}$ exciton.
While this doesn’t exclude that there exists coupling to the other excitons,
their weak intensity makes those couplings harder to detect. By applying
perturbation theory, we can qualitatively understand the experimental
temperature dependence at the overdoped side as a renormalization of the
$\underline{d}d$ excitons, but this model does not explain the change of sign
of this effect as a function of doping. In an approach that takes into
consideration that the superconducting pairs and the excitons involve the same
electrons, we expect a blueshift of the exciton energy smaller than $3J/4$
when the material turns superconducting. The change of sign of the
superconductivity induced changes from underdoped to overdoped is a natural
consequence of the presence (absence) of short-range AF correlations in normal
state of the former (latter), which make way for d-wave singlet correlations
in the superconducting phase.

## Acknowledgments

We thank A. Georges, G. Ghiringelli and X. X. Huang for stimulating
discussions. The RIXS experiments have been performed at the ADRESS beamline
of the Swiss Light Source at the Paul Scherrer Institut (PSI). We acknowledge
funding support from the Swiss National Science Foundation through projects
No. 179157 (D.v.d.M.), 178867 (T.S.), and 160765 (T.S.). T.P.D. was supported
by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering, under Contract No. DE-
AC02-76SF00515. T.C.A. acknowledges funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 701647 (PSI-FELLOW-II-3i program). The work at BNL was supported
by the US Department of Energy, office of Basic Energy Sciences, contract No.
DOE-sc0012704. Data and materials availability: The datasets generated and
analyzed during the current study are available in Ref. Barantani  _et al._
(2022). These will be preserved for 10 years.

##  Appendix A Experimental methods

Single crystals of Bi2Sr2CaCu2O8-x (Bi-2212) were prepared as described in
Ref. Wen  _et al._ (2008) with carrier concentrations $p=0.11$ (underdoped,
$T_{c}=70$ K), $p=0.16$ (optimally doped, $T_{c}=91$ K), and $p=0.21$
(overdoped, $T_{c}=70$ K). An additional optimally doped sample ($p=0.16$,
$T_{c}=91$ K) was prepared as described in Ref. Weyeneth  _et al._ (2009).

RIXS spectra were measured at the ADRESS beamline of the Swiss Light Source
(Paul Scherrer Institut, Villigen PSI), using $\sigma$ polarization of the
incoming x-rays with incoming energy resonant at the Cu $L_{3}$ edge, $E\simeq
931.2$ eV. The scattering angle between the incoming and outgoing x-rays beams
was set at 130 degrees. The sample manipulator allowed adjustment of azimutal
and polar angles to a momentum transfer of $\mathbf{q}=(0.36,0,2)$,
$\mathbf{q}=(0.25,0.25,2)$ such that the out-of-plane momentum transfer was a
multiple of the reciprocal lattice vector $c^{*}$. The outgoing x-rays were
detected without resolving the polarization. The energy resolution was $\Delta
E=0.13$ eV (as obtained by measuring the elastic line on a tape containing
graphite), which is sufficiently precise for measuring the $\underline{d}d$
exciton at 2 eV while maintaining good statistics. In order to define the
nodal and antinodal directions, the samples were aligned using aa Laue setup.
The samples were cleaved at the base temperature ($20$ K) at a pressure of
$10^{-10}$ mbar.

For each doping and momentum direction, the spectra were acquired in the
temperature range of approximately $50$ K below $T_{c}$ to $50$ K above
$T_{c}$. The temperature was swept at a fixed slow rate of $0.2$ K/min ($0.3$
K/min for the optimally doped) while continuously sampling the spectra,
resulting in 8 to 10 hours temperature scans. We binned the data in intervals
of approximately $5$ K in temperature. To avoid radiation damage on the focus
position of the x-ray beam, the beam was slowly scanned along a line of
50$~{}\mathrm{\mu m}$ during the data acquisition. We normalized the spectra
to the total counts in the energy range from $-1$ to $10$ eV. The zero energy
position $\omega_{0}$ was obtained using the following procedure: For each
temperature the spectra are fitted to the expression

$\displaystyle I(T,\omega)$ $\displaystyle=$ $\displaystyle
I_{0}F(\omega_{0},\gamma_{0},\sigma_{0},\omega)+$ (4) $\displaystyle+$
$\displaystyle\sum_{n=1}^{6}I_{n}F(\omega_{n}+\omega_{0},\gamma_{n},\sigma_{n},\omega)$

where $\omega_{0}$ describes the zero-energy position of the quasi-elastic
peak, $\omega_{n}$ the relative energy of all other peaks, and $\gamma_{n}$
and $\sigma_{n}$ are the Lorentzian and Gaussian widths of the Voigt profile
$F(\Omega,\gamma,\sigma,\omega)$ respectively. An additional weak peak at
twice the magnon energy was needed in the fitting procedure to reproduce the
asymmetric line-shape in the magnon region.

We used the following method to determine the error bar of the zero energy
position $\omega_{0}$. We have simulated the error in fitting $\omega_{0}$ due
to the statistical noise. To do so, we consider an ideal spectrum
$I_{\text{fit}}(\omega)$ given by the fit at low temperature, such that it
reproduces the experimental spectrum for a particular doping and momentum
transfer. We show the case of the OD 70K, which is representative for other
dopings and momenta, all of which have similar features. To simulate the
statistical noise we add a random Poissonian noise $\delta
I_{\text{Pn}}(\omega)$ to the spectrum of equal magnitude as in the
experimental data (i.e. we simulated the noise level of $N$ counts per channel
as $\delta N\sim\sqrt{N}$). We then fitted the noisy spectrum
$I_{\text{fit}}(\omega)+\delta I_{\text{Pn}}(\omega)$ with the previous
starting parameters and repeated the procedure multiple times to generate a
statistically significant sample. We considered 10000 realizations, the result
of which corresponds to the distribution shown in Fig 7. For this particular
example the distribution has a full width at half maximum $3.4$ meV
corresponding to $2\sigma=2.9$ meV where $\sigma$ is the standard deviation.
This procedure was repeated for each momentum and doping giving the error bars
of length $2\sigma$ in Figs. 3, and 8.

Figure 7: (a) Fitted spectrum for the OD70K $\mathbf{q}=(0.36,0)$ at 30 K; the
shaded area indicates the Poissonian noise. The shaded Voigt peaks represent
the decomposition of the spectrum. (b) Distribution of the elastic peak
position extracted by fitting $10^{4}$ spectra with random Poissonian noise.

With incident $\sigma$ polarization, due to direct re-emission the spectra
present a reasonably intense elastic peak, which is convenient since all
energies are measured relative to the position of the zero-loss peak. A
possible complication arises if, given the finite resolution (in the present
case around 130 meV) low energy inelastic features such as phonons also
contribute to what we should now interpret as a quasi-elastic peak. If these
contributions depend on temperature, this imports a temperature dependence in
the quasi-elastic peak, which carries over to a spurious temperature
dependence of the energy loss of e.g. the magnon peak and the $\underline{d}d$
excitons. Suzuki et al. Suzuki  _et al._ (2018) have reported temperature
dependence of the low energy continuum of Bi-2212 at small momentum transfer.
On the other hand such contributions were not present in their data for high
momentum transfer similar to the momentum values subject of the present study.
In fact, if present, a spurious temperature dependence of the quasi-elastic
peak will affect both the magnon and the $\underline{d}d$ exciton peak in
exactly the same way. Here we observe that the temperature dependencies of
magnon and exciton energies are completely different (see Fig. 3) and the
temperature dependence of the magnon is extremely weak. Together, it indicates
the absence of a spurious temperature dependence of the quasi-elastic peak for
the high momentum transfers considered here, in accordance with the findings
of Suzuki et al. Suzuki  _et al._ (2018).

Figure 8: Average energy and of the $\underline{d}d$ exciton peak of overdoped
Bi-2212 ($p=0.21$,$T_{c}=70$ K) for momentum $(0.36,0)$ measured with 3
different detectors.

The RIXS spectra are collected by 3 separate CCD detectors. The spectra
collected with each detector were analyzed separately. The zero-energy
position of each spectrum was determined as detailed before. In Fig. 8 we
report the average energy $E$ of the $\underline{d}d$ exciton region
($\omega_{1}=1$ eV and $\omega_{2}=4$ eV) for the overdoped sample and
momentum transfer $(0.36,0)$, measured with the three detectors. Above $\sim
70$ K the average energy is almost temperature independent, and below $\sim
70$ K it gradually shifts to $20$ meV higher energy. The observation with
three detectors of the same temperature dependence of energy indicates that
these features are intrinsic properties of this material. For other momentum
transfers and dopings we also observe temperature dependencies of the
$\underline{d}d$ energy that are qualitatively similar, but the effects are
weaker. The focus in this paper is therefore on the temperature shift of the
average energy of the $\underline{d}d$ manifold.

Figure 9: Integral function $G(\omega)$ defined in Eq. (5) (solid) and the
spectrum $I(\omega,T)$ (dashed) normalized to their maximum, for the OD 70 K
sample and $\mathbf{q}=(0.36,0)$.

In Fig. 9 we show the integral function defined as

$G(\omega)=\int_{\omega_{0}}^{\omega}\left[I(\omega^{\prime},T_{c})-I(\omega^{\prime},T)\right]d\omega^{\prime}$
(5)

for the overdoped sample at two temperatures $T$, using the difference spectra
reported in Fig. 2 and considering $\omega_{0}=-0.5$ eV. The shape matches up
to about 2 eV the RIXS spectra at the same temperature $T$. Above 2 eV
$G(\omega)$ falls well below $I(\omega)$. Taken together this indicates that
the temperature evolution is a shift of $\underline{d}_{xy}d_{x^{2}-y^{2}}$
and $\underline{d}_{xz,yz}d_{x^{2}-y^{2}}$ while the shift of
$\underline{d}_{z^{2}}d_{x^{2}-y^{2}}$ is significantly smaller.

##  Appendix B Cluster calculations of the $\underline{d}d$ exciton multiplets

The expression for the RIXS intensity is given by the Kramers-Heisenberg
formula

$\displaystyle
R(\mathbf{e}_{i},\mathbf{e}_{f},\mathbf{k}_{i},\mathbf{k}_{f},\omega_{i},\omega_{f})=\frac{1}{Z}\sum_{i,f}e^{-\beta
E_{i}}$ (6) $\displaystyle\times\left|\sum_{\nu}\frac{\langle
f\mid\hat{D}_{\mathbf{k}_{f}}^{*}(\mathbf{e}_{f})\mid\nu\rangle\langle\nu\mid\hat{D}_{\mathbf{k}_{i}}({\mathbf{e}_{i}})\mid
i\rangle}{\omega_{i}-(E_{\nu}-E_{i})-i\Gamma}\right|^{2}\times\delta\bigl{(}\Omega-(E_{f}-E_{i})\bigr{)}$
,

which is the largest resonant diagram at lowest order in the fine structure
constant $e^{2}/\hbar c$. Here, $Z$ is the partition function, $E_{i,f}$,
$|i,f\rangle$ are the initial and final state energies and eigenstates,
respectively, $E_{\nu}$, $|\nu\rangle$ are the RIXS intermediate core hole
states, $\Gamma$ is the core hole decay rate, $\mathbf{e}_{i,f}$,
$\mathbf{k}_{i,f}$, $\omega_{i,f}$ are the incident, scattered x-ray
polarization vectors, momenta, and energies, respectively, and
$\Omega=\omega_{i}-\omega_{f}$. The dipole operator $\hat{D}$ is evaluated in
the Coulomb or “transverse” gauge $\nabla\cdot\mathbf{A}=0$, where
$\mathbf{A}$ is the light vector potential, and the dipole approximation is
used.

RIXS is obtained via full matrix diagonalization of the generalized
Hamiltonian

$\hat{H}_{\mathrm{full}}=\hat{H}_{\mathrm{int}}+\hat{H}_{\mathrm{CEF}}+\hat{H}_{\mathrm{SO}}+\hat{H}_{\mathrm{KE}}$
(7)

where the terms in the Hamiltonian are given in terms of second quantized
operators $c^{\phantom{\dagger}}_{\mu,i,\sigma},c^{\dagger}_{\mu,i,\sigma}$
that remove, create electrons in orbital $\mu$ in unit cell $i$ with spin
$\sigma$ as follows:

The Coulomb interaction

$\hat{H}_{\mathrm{int}}=\frac{1}{2}\sum_{i}\sum_{\alpha,\beta,\gamma,\delta}\sum_{\sigma,\sigma^{\prime}}U_{\alpha,\beta,\gamma,\delta}c^{\dagger}_{i,\alpha,\sigma}c^{\dagger}_{i,\beta,\sigma^{\prime}}c^{\phantom{\dagger}}_{i,\delta,\sigma^{\prime}}c^{\phantom{\dagger}}_{i,\gamma,\sigma}$
(8)

represent generalized local two-body interactions that can include both core
and valence electrons, while the sum of single particle terms can be grouped
together as

$\hat{H}_{\mathrm{CEF}}+\hat{H}_{\mathrm{SO}}+\hat{H}_{\mathrm{KE}}=\sum_{i,j}\sum_{\sigma,\sigma^{\prime}}\sum_{\mu,\nu}t_{i,j,\sigma,\sigma^{\prime}}^{\mu,\nu}c^{\dagger}_{i,\mu,\sigma}c^{\phantom{\dagger}}_{j,\nu,\sigma^{\prime}}$
(9)

with the tensor $t$ having site energies, crystal fields and spin-orbit
contributions (for $i=j$ terms), and hybridization among orbitals ($i\neq j$).

We utilize a unit cell consisting of five copper $3d$ orbitals, three copper
$2p$ core orbitals, and three oxygen $2p$ ligand orbitals along the $x$ and
$y$ bond directions, giving 14 orbitals per unit cell. We consider a 2-site
cluster (28 orbitals) to yield momentum transfer $(\frac{1}{2},0)$ in
reciprocal lattice units. The Hilbert space is restricted to contain at most
one core hole in the cluster.

The following set of parameters have been used (hole language is employed).
All energies are in units of eV. Cu $3d$ site energies: $(0,1.8,0.9,1.15)$ for
$d_{x^{2}-y^{2}}$, $d_{z^{2}}$, $d_{xy}$, $d_{xz,yz}$ orbitals, respectively.
O $2p$ site energies: 2.8 for $p_{xx}$, $p_{yy}$, $p_{xy}$, $p_{yx}$ and 0.8
for $p_{xz}$, $p_{yz}$ orbitals, where $p_{i,j}$ denotes ligand orbital
$2p_{j}$ for site $i$ in the unit cell. Slater parameters for Cu $3d$:
$F_{0}=6$, $F_{2}=0.13$, $F_{4}=0.025$ to approximately give Hubbard $U=7.4$
and Hund’s $J=0.9$. Slater parameters for core Cu $2p$–$3d$ interaction:
$F_{0}=6$, $F_{2}=0.15$, $G^{1}=4.63$, $G^{3}=2.63$. Slater parameters for O
$2p$: $F_{0}=1.0$, $F_{2}=0.1$. Core spin-orbit coupling: 10.5.
Hybridizations: $3d_{x^{2}-y^{2}}$–$2p_{x,y}=1$, $3d_{z^{2}}$–$2p_{x,y}=0.1$,
$3d_{xy}$–$2p_{y,x}=0.225$, $3d_{xz,yz}$–$2p_{z}=0.1$. The core hole decay
rate $\Gamma=0.1$.

Finally, the incident photon polarization direction is taken to be along $y$
direction ($\sigma$ polarization in the scattering plane), and the outgoing
photon is unpolarized. In Fig. 1 of the main text we show the peaks resulting
from the calculations and the relative intensity normalized to the $yz$
exciton.

##  Appendix C Weak-coupling model for the temperature dependence of the
$\underline{d}d$ excitons

For a two-dimensional single-band superconductor, the temperature-dependent
BCS charge susceptibility at wave-vector $\mathbf{q}$ and complex energy $z$
is

$\chi(\mathbf{q},z)=2\int\frac{d^{2}k}{(2\pi)^{2}}\int_{-\infty}^{\infty}d\varepsilon_{1}\int_{-\infty}^{\infty}d\varepsilon_{2}\left[A\left(\mathbf{k},\varepsilon_{1}\right)A\left(\mathbf{k}+\mathbf{q},\varepsilon_{2}\right)+B\left(\mathbf{k},\varepsilon_{1}\right)B\left(\mathbf{k}+\mathbf{q},\varepsilon_{2}\right)\right]\frac{f\left(\varepsilon_{1}\right)-f\left(\varepsilon_{2}\right)}{z+\varepsilon_{1}-\varepsilon_{2}}.$
(10)

The wave-vectors $\mathbf{k}$ span the first Brillouin zone,
$f(\varepsilon)=1/\left[\exp\left(\varepsilon/k_{\mathrm{B}}T\right)+1\right]$
is the Fermi distribution at temperature $T$, and $A$, $B$ are the normal and
anomalous spectral functions, respectively. The spectral functions depend on
the band dispersion measured from the temperature-dependent chemical
potential, $\xi_{\mathbf{k}}=\varepsilon_{\mathbf{k}}-\mu$, and on the
temperature-dependent superconducting gap function $\Delta_{\mathbf{k}}$. They
are best expressed in terms of the Bogoliubov excitation energies
$E_{\mathbf{k}}=(\xi_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2})^{1/2}$ and
coherence factors
$u_{\mathbf{k}}^{2}=\frac{1}{2}(1+\xi_{\mathbf{k}}/E_{\mathbf{k}})$ and
$v_{\mathbf{k}}^{2}=1-u_{\mathbf{k}}^{2}$, according to
$A(\mathbf{k},\varepsilon)=u_{\mathbf{k}}^{2}\delta\left(\varepsilon-
E_{\mathbf{k}}\right)+v_{\mathbf{k}}^{2}\delta\left(\varepsilon+E_{\mathbf{k}}\right)$
and
$B(\mathbf{k},\varepsilon)=\frac{\Delta_{\mathbf{k}}}{2E_{\mathbf{k}}}\left[\delta\left(\varepsilon-
E_{\mathbf{k}}\right)-\delta\left(\varepsilon+E_{\mathbf{k}}\right)\right]$.

For the band dispersion, we use a square-lattice tight-binding model with
hopping amplitudes up to the fifth neighbors,

$\varepsilon_{\mathbf{k}}=2t_{1}\left[\cos\left(k_{x}a\right)+\cos\left(k_{y}a\right)\right]+4t_{2}\cos\left(k_{x}a\right)\cos\left(k_{y}a\right)\\\
+2t_{3}\left[\cos\left(2k_{x}a\right)+\cos\left(2k_{y}a\right)\right]\\\
+4t_{4}\left[\cos\left(2k_{x}a\right)\cos\left(k_{y}a\right)+\cos\left(k_{x}a\right)\cos\left(2k_{y}a\right)\right]\\\
+4t_{5}\cos\left(2k_{x}a\right)\cos\left(2k_{y}a\right),$ (11)

where $a$ is the lattice spacing. The parameters are
$t_{1-5}=(-148.8,40.9,-13,-14,12.8)$ meV, as determined in Norman  _et al._
(1995). At each temperature, we adjust the chemical potential $\mu$ such that
the hole density is $1-n\equiv\rho$, where the electron density $n$ is

$n=2\int\frac{d^{2}k}{(2\pi)^{2}}\int_{-\infty}^{\infty}d\varepsilon\,A(\mathbf{k},\varepsilon)f(\varepsilon).$
(12)

The gap function has $d_{x^{2}-y^{2}}$ symmetry with an empirical temperature
dependence that follows closely the numerical result of the BCS gap equation
for a two-dimensional $d$-wave superconductor:

$\Delta_{\mathbf{k}}(T)=\Delta_{\mathbf{k}}(0)\sqrt{1-x^{2}}\left(1+\frac{x^{2}}{2}e^{-\frac{4}{5}x^{2}}\right)$
(13)

where $x={T}/{T_{c}}$ and the $d$-wave gap reads
$\Delta_{\mathbf{k}}(0)={\Delta_{0}}\left[\cos\left(k_{x}a\right)-\cos\left(k_{y}a\right)\right]/2$.
We set $\rho=0.2$, $T_{c}=70$ K, and $\Delta_{0}=30$ meV to represent the
overdoped sample.

Figure 10: Temperature dependence of the $\underline{d}d$ exciton average
energy, from the spectral function computed in Eq. (14) for the two direction
in momentum space $\mathbf{q}=(0.36,0)$ (green circles) and
$\mathbf{q}=(0.25,0.25)$ (blue squares).

For the exciton spectral function the susceptibility enters via the self-
energy correction

$I_{\underline{d}d}(\mathbf{q},\omega)\propto-\mathrm{Im}\,\frac{1}{\omega-\omega_{\underline{d}d}-g_{\mathbf{q}}^{2}\chi(\mathbf{q},\omega)+i\Gamma},$
(14)

where $\omega_{\underline{d}d}$ is the unrenormalized energy of the exciton
and $\Gamma$ represents the exciton intrinsic linewidth–mostly due to
interaction with the phonons–that we represent phenomenologically with a
constant scattering rate $\Gamma=0.15$ eV.

We substitute the BCS susceptibility in Eq. (14) with $g_{\mathbf{q}}=2$ eV
and we sett $\omega_{\underline{d}d}=1.973$ eV and
$\omega_{\underline{d}d}=1.87$ eV for $\mathbf{q}=(0.36,0)$ and
$\mathbf{q}=(0.25,0.25)$, such that the peak position at $T_{c}$ is 2.174 eV
and 2.17 eV, respectively (see Fig. 3). Fig. 10 shows the resulting
temperature dependence of the exciton energy.

## References

  * Bednorz and Muller (1986) J. G. Bednorz and K. A. Muller, “Possible high $T_{c}$ superconductivity in the Ba-La-Cu-O system,” Z. Phys. B: Condens. Matter 64, 189–193 (1986).
  * Anderson (1987) Philip W. Anderson, “The Resonating Valence Bond State in La2CuO4 and Superconductivity,” Science 235, 1196–1198 (1987).
  * Zaanen  _et al._ (1985) J. Zaanen, G. A. Sawatzky, and J. W. Allen, “Band gaps and electronic structure of transition-metal compounds,” Phys. Rev. Lett. 55, 418–421 (1985).
  * Ghijsen  _et al._ (1988) J. Ghijsen, L. H. Tjeng, J. van Elp, H. Eskes, J. Westerink, G. A. Sawatzky, and M. T. Czyzyk, “Electronic structure of Cu2O and CuO,” Phys. Rev. B 38, 11322–11330 (1988).
  * Arovas  _et al._ (2022) Daniel P. Arovas, Erez Berg, Steven A. Kivelson, and Srinivas Raghu, “The Hubbard Model,” Annual Review of Condensed Matter Physics 13, 239–274 (2022).
  * Jiang and Devereaux (2019) Hong-Chen Jiang and Thomas P. Devereaux, “Superconductivity in the doped Hubbard model and its interplay with next-nearest hopping $t^{\prime}$,” Science 365, 1424–1428 (2019).
  * Anderson (2007) Philip W. Anderson, “Is there glue in cuprate superconductors?” Science 316, 1705–1707 (2007).
  * Hirsch (2002) J. E. Hirsch, “Why holes are not like electrons: A microscopic analysis of the differences between holes and electrons in condensed matter,” Phys. Rev. B 65, 184502 (2002).
  * Leggett (1999) A. J. Leggett, “A “midinfrared” scenario for cuprate superconductivity,” Proc. Natl. Acad. Sci. U.S.A 96, 8365–8372 (1999).
  * Eschrig (2006) Matthias Eschrig, “The effect of collective spin-1 excitations on electronic spectra in high-Tc superconductors,” Adv. Phys. 55, 47–183 (2006).
  * Scalapino (2012) D. J. Scalapino, “A common thread: The pairing interaction for unconventional superconductors,” Rev. Mod. Phys. 84, 1383–1417 (2012).
  * Bill  _et al._ (2003) A. Bill, H. Morawitz, and V. Z. Kresin, “Electronic collective modes and superconductivity in layered conductors,” Phys. Rev. B 68, 144519 (2003).
  * Sawatzky  _et al._ (2009) G. A. Sawatzky, I. S. Elfimov, J. van den Brink, and J. Zaanen, “Heavy-anion solvation of polarity fluctuations in pnictides,” Europhys. Lett. 86, 17006 (2009).
  * Yacoby and Girshberg (2016) Y. Yacoby and Y. Girshberg, “Cooper-like Pairing and Energy Gap Induced by Ion Electronic Polarizability,” J. Supercond. Novel Magn. 30, 379–386 (2016).
  * Weber  _et al._ (1988) W. Weber, A. L. Shelankov, and X. Zotos, “A Cu d-d excitation model for the pairing in the high-Tc cuprates,” Physica C 153-155, 1305–1306 (1988).
  * Zaanen and Oleś (1993) J. Zaanen and A. M. Oleś, “Carriers binding to excitons: Crystal-field excitations in doped Mott-Hubbard insulators,” Phys. Rev. B 48, 7197–7215 (1993).
  * Feiner  _et al._ (1992) L. F. Feiner, M. Grilli, and C. Di Castro, “Apical oxygen ions and the electronic structure of the high-Tc cuprates,” Phys. Rev. B 45, 10647–10669 (1992).
  * Buda  _et al._ (1994) F. Buda, D.L. Cox, and M. Jarrell, “Random-phase-approximation analysis of orbital- and magnetic-fluctuation- mediated superconductivity in a two-band Hubbard model,” Phys. Rev. B 49, 1255–1268 (1994).
  * Holcomb  _et al._ (1994) M. J. Holcomb, J. P. Collman, and W. A. Little, “Optical evidence of an electronic contribution to the pairing interaction in superconducting Tl2Ba2Ca2 Cu3O10,” Phys. Rev. Lett. 73, 2360–2363 (1994).
  * Bucci  _et al._ (1995) F. Bucci, C. Castellani, C. Di Castro, and M. Grilli, “Charge fluctuations in the four-band extended Hubbard model,” Phys. Rev. B 52, 6880–6893 (1995).
  * Little  _et al._ (2007) W. A. Little, M. J. Holcomb, G. Ghiringhelli, L. Braicovich, C. Dallera, A. Piazzalunga, A. Tagliaferri, and N. B. Brookes, “A determination of the pairing interaction in the high $T_{c}$ cuprate superconductor Tl2Ba2CaCu2O8 (Tl2212),” Physica C 460-462 I, 40–43 (2007).
  * Mazov (2012) L.S. Mazov, “Possible Room-Tc Superconductor with in-plane Ginzburg Sandwich in Cuprate-and Pnictide-like family of Compounds,” Physics Procedia 36, 735–740 (2012).
  * Oleś  _et al._ (2019) Andrzej M. Oleś, Krzysztof Wohlfeld, and Giniyat Khaliullin, “Orbital Symmetry and Orbital Excitations in High-Tc Superconductors,” Condens. Matter 4, 46 (2019).
  * Maksimov  _et al._ (2010) E. G. Maksimov, M. L. Kulić, and O. V. Dolgov, “Bosonic Spectral Function and the Electron-Phonon Interaction in HTSC Cuprates,” Adv. Condens. Matter Phys. 2010, 1–64 (2010).
  * Little (1964) W. A. Little, “Possibility of Synthesizing an Organic Superconductor,” Phys. Rev. 134, A1416–A1424 (1964).
  * Ginzburg (1970) V. L. Ginzburg, “The problem of high-temperature superconductivity. II,” Sov. Phys. Usp. 13, 335–352 (1970).
  * Allender  _et al._ (1973) David Allender, James Bray, and John Bardeen, “Model for an Exciton Mechanism of Superconductivity,” Phys. Rev. B 7, 1020–1029 (1973).
  * Wang  _et al._ (2011) Xin Wang, Hung The Dang, and Andrew J. Millis, “${d}_{3{z}^{2}-{r}^{2}}$ orbital in high-${T}_{c}$ cuprates: Excitonic spectrum, metal-insulator phase diagram, optical conductivity, and orbital character of doped holes,” Phys. Rev. B 84, 014530 (2011).
  * Kuiper  _et al._ (1998) Pieter Kuiper, J.-H. Guo, Conny Såthe, L.-C. Duda, Joseph Nordgren, J. J. M. Pothuizen, F. M. F. de Groot, and G. A. Sawatzky, “Resonant X-Ray Raman Spectra of Cu $\mathit{dd}$ Excitations in ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{2}{\mathrm{Cl}}_{2}$,” Phys. Rev. Lett. 80, 5204–5207 (1998).
  * Ghiringhelli  _et al._ (2004) G. Ghiringhelli, N. B. Brookes, E. Annese, H. Berger, C. Dallera, M. Grioni, L. Perfetti, A. Tagliaferri, and L. Braicovich, “Low Energy Electronic Excitations in the Layered Cuprates Studied by Copper $L_{3}$ Resonant Inelastic X-Ray Scattering,” Phys. Rev. Lett. 92, 117406 (2004).
  * Ghiringhelli  _et al._ (2007) G. Ghiringhelli, N. B. Brookes, C. Dallera, A. Tagliaferri, and L. Braicovich, “Sensitivity to hole doping of Cu ${L}_{3}$ resonant spectroscopies: Inelastic x-ray scattering and photoemission of LaSrCuO4,” Phys. Rev. B 76, 085116 (2007).
  * Moretti Sala  _et al._ (2011) M. Moretti Sala, V. Bisogni, C. Aruta, G. Balestrino, H. Berger, N. B. Brookes, G. M. De Luca, D. Di Castro, M. Grioni, M. Guarise, P. G. Medaglia, F. Miletto Granozio, M. Minola, P. Perna, M. Radovic, M. Salluzzo, T. Schmitt, K. J. Zhou, L. Braicovich, and G. Ghiringhelli, “Energy and symmetry of dd excitations in undoped layered cuprates measured By Cu $L_{3}$ resonant inelastic x-ray scattering,” New J. Phys. 13, 043026 (2011).
  * Dean  _et al._ (2013) M. P. M. Dean, A. J. A. James, R. S. Springell, X. Liu, C. Monney, K. J. Zhou, R. M. Konik, J. S. Wen, Z. J. Xu, G. D. Gu, V. N. Strocov, T. Schmitt, and J. P. Hill, “High-Energy Magnetic Excitations in the Cuprate Superconductor Bi2Sr2CaCuO8+δ: Towards a Unified Description of Its Electronic and Magnetic Degrees of Freedom,” Phys. Rev. Lett. 110, 147001 (2013).
  * Dean  _et al._ (2014) M. P. M. Dean, A. J. A. James, A. C. Walters, V. Bisogni, I. Jarrige, M. Hücker, E. Giannini, M. Fujita, J. Pelliciari, Y. B. Huang, R. M. Konik, T. Schmitt, and J. P. Hill, “Itinerant effects and enhanced magnetic interactions in Bi-based multilayer cuprates,” Phys. Rev. B 90, 220506(R) (2014).
  * Axe and Shirane (1973) J. D. Axe and G. Shirane, “Influence of the Superconducting Energy Gap on Phonon Linewidths in Nb3Sn,” Phys. Rev. Lett. 30, 214–216 (1973).
  * Shapiro  _et al._ (1975) S. M. Shapiro, G. Shirane, and J. D. Axe, “Measurements of the electron-phonon interaction in Nb by inelastic neutron scattering,” Phys. Rev. B 12, 4899–4908 (1975).
  * Altendorf  _et al._ (1993) E. Altendorf, X. K. Chen, J. C. Irwin, R. Liang, and W. N. Hardy, “Temperature dependences of the 340-, 440-, and 500 cm-1 Raman modes of YBa2Cu3Oy for $6.7\leq y\leq 7.0$,” Phys. Rev. B 47, 8140–8150 (1993).
  * Zeyher and Zwicknagl (1990) R. Zeyher and G. Zwicknagl, “Superconductivity-induced phonon self-energy effects in high $T_{c}$ superconductors,” Z. Phys. B: Condens. Matter 78, 175–190 (1990).
  * Strocov  _et al._ (2010) V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17, 631–643 (2010).
  * Ghiringhelli  _et al._ (2006) G. Ghiringhelli, A. Piazzalunga, C. Dallera, G. Trezzi, L. Braicovich, T. Schmitt, V. N. Strocov, R. Betemps, L. Patthey, X. Wang, and M. Grioni, “SAXES, a high resolution spectrometer for resonant x-ray emission in the 400–1600 eV energy range,” Rev. Sci. Instrum. 77, 113108 (2006).
  * Lee  _et al._ (2014) J. J. Lee, B. Moritz, W. S. Lee, M. Yi, C. J. Jia, A. P. Sorini, K. Kudo, Y. Koike, K. J. Zhou, C. Monney, V. Strocov, L. Patthey, T. Schmitt, T. P. Devereaux, and Z. X. Shen, “Charge-orbital-lattice coupling effects in the $dd$ excitation profile of one-dimensional cuprates,” Phys. Rev. B 89, 041104(R) (2014).
  * Peng  _et al._ (2015) Y. Y. Peng, M. Hashimoto, M. M. Sala, A. Amorese, N. B. Brookes, G. Dellea, W.-S. Lee, M. Minola, T. Schmitt, Y. Yoshida, K.-J. Zhou, H. Eisaki, T. P. Devereaux, Z.-X. Shen, L. Braicovich, and G. Ghiringhelli, “Magnetic excitations and phonons simultaneously studied by resonant inelastic x-ray scattering in optimally doped ${\mathrm{Bi}}_{1.5}{\mathrm{Pb}}_{0.55}{\mathrm{Sr}}_{1.6}{\mathrm{La}}_{0.4}{\mathrm{CuO}}_{6+\delta}$,” Phys. Rev. B 92, 064517 (2015).
  * Oliveira  _et al._ (2003) F. A. C. Oliveira, L. A. Cury, A. Righi, R. L. Moreira, P. S. S. Guimaraes, F. M. Matinaga, M. A. Pimenta, and R. A. Nogueira, “Temperature effects on the vibronic spectra of BEH-PPV conjugated polymer films,” The Journal of Chemical Physics 119, 9777–9782 (2003).
  * Ahn and Nagaosa (2021) Junyeong Ahn and Naoto Nagaosa, “Superconductivity-induced spectral weight transfer due to quantum geometry,” Phys. Rev. B 104, L100501 (2021).
  * Chen and Huang (2021) Weipeng Chen and Wen Huang, “Quantum-geometry-induced intrinsic optical anomaly in multiorbital superconductors,” Phys. Rev. Research 3, L042018 (2021).
  * Husain  _et al._ (2019) Ali A. Husain, Matteo Mitrano, Melinda S. Rak, Samantha Rubeck, Bruno Uchoa, Katia March, Christian Dwyer, John Schneeloch, Ruidan Zhong, G. D. Gu, and Peter Abbamonte, “Crossover of Charge Fluctuations across the Strange Metal Phase Diagram,” Phys. Rev. X 9, 041062 (2019).
  * 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, “Temperature-Dependent Ellipsometry Measurements of Partial Coulomb Energy in Superconducting Cuprates,” Phys. Rev. X 6, 031027 (2016).
  * Kotliar and Liu (1988) Gabriel Kotliar and Jialin Liu, “Superexchange mechanism and d-wave superconductivity,” Phys. Rev. B 38, 5142–5145 (1988).
  * Zhang  _et al._ (1988) F. C. Zhang, C. Gros, T. M. Rice, and H. Shiba, “A renormalised Hamiltonian approach to a resonant valence bond wavefunction,” Superconductor Science and Technology 1, 36–46 (1988).
  * Yokoyama and Shiba (1988) Hisatoshi Yokoyama and Hiroyuki Shiba, “Variational Monte-Carlo Studies of Superconductivity in Strongly Correlated Electron Systems,” Journal of the Physical Society of Japan 57, 2482–2493 (1988).
  * Gros (1988) Claudius Gros, “Superconductivity in correlated wave functions,” Phys. Rev. B 38, 931–934 (1988).
  * Scalapino and White (1998) D. J. Scalapino and S. R. White, “Superconducting condensation energy and an antiferromagnetic exchange-based pairing mechanism,” Phys. Rev. B 58, 8222–8224 (1998).
  * Maier  _et al._ (2004) Th. A. Maier, M. Jarrell, A. Macridin, and C. Slezak, “Kinetic Energy Driven Pairing in Cuprate Superconductors,” Phys. Rev. Lett. 92, 027005 (2004).
  * Toschi  _et al._ (2005) A. Toschi, M. Capone, and C. Castellani, “Energetic balance of the superconducting transition across the BCS—Bose Einstein crossover in the attractive Hubbard model,” Phys. Rev. B 72, 235118 (2005).
  * Haule and Kotliar (2007) K. Haule and G. Kotliar, “Optical conductivity and kinetic energy of the superconducting state: A cluster dynamical mean field study,” Europhysics Letters 77, 27007 (2007).
  * Gull and Millis (2012) E. Gull and A. J. Millis, “Energetics of superconductivity in the two-dimensional Hubbard model,” Phys. Rev. B 86, 241106(R) (2012).
  * Fratino  _et al._ (2016) L. Fratino, P. Sémon, G. Sordi, and A.-M. S. Tremblay, “An organizing principle for two-dimensional strongly correlated superconductivity,” Scientific Reports 6, 22715 (2016).
  * Molegraaf  _et al._ (2002) H. J.A. Molegraaf, C Presura, D. Van der Marel, P H Kes, and M Li, “Superconductivity-induced transfer of in-plane spectral weight in Bi2Sr2CaCu2O8+δ,” Science 295, 2239–2241 (2002).
  * Deutscher  _et al._ (2005) Guy Deutscher, Andrés Felipe Santander-Syro, and Nicole Bontemps, “Kinetic energy change with doping upon superfluid condensation in high-temperature superconductors,” Phys. Rev. B 72, 092504 (2005).
  * Gedik  _et al._ (2005) N. Gedik, M. Langner, J. Orenstein, S. Ono, Yasushi Abe, and Yoichi Ando, “Abrupt Transition in Quasiparticle Dynamics at Optimal Doping in a Cuprate Superconductor System,” Phys. Rev. Lett. 95, 117005 (2005).
  * Carbone  _et al._ (2006) F. Carbone, A. B. Kuzmenko, H. J. A. Molegraaf, E. van Heumen, V. Lukovac, F. Marsiglio, D. van der Marel, K. Haule, G. Kotliar, H. Berger, S. Courjault, P. H. Kes, and M. Li, “Doping dependence of the redistribution of optical spectral weight in Bi2Sr2CaCuO8+δ,” Phys. Rev. B 74, 064510 (2006).
  * Giannetti  _et al._ (2011) Claudio Giannetti, Federico Cilento, Stefano Dal Conte, Giacomo Coslovich, Gabriele Ferrini, Hajo Molegraaf, Markus Raichle, Ruixing Liang, Hiroshi Eisaki, Martin Greven, Andrea Damascelli, Dirk van der Marel, and Fulvio Parmigiani, “Revealing the high-energy electronic excitations underlying the onset of high-temperature superconductivity in cuprates,” Nat. Commun. 2, 353 (2011).
  * Barantani  _et al._ (2022) F. Barantani, M. K. Tran, I. Madan, I. Kapon, N. Bachar, T. C. Asmara, E. Paris, Y. Tseng, W. Zhang, Y. Hu, E. Giannini, G. Gu, T. P. Devereaux, C. Berthod, F. Carbone, T. Schmitt, and D. van der Marel, “Open Data,” doi:10.26037/yareta:e6a76wuxsfa3vpcluyei5k2kru (2022).
  * Wen  _et al._ (2008) J. S. Wen, Z. J. Xu, G. Y. Xu, M. Hucker, J. M. Tranquada, and G. D. Gu, “Large Bi-2212 single crystal growth by the floating-zone technique,” J. Cryst. Growth 310, 1401–1404 (2008).
  * Weyeneth  _et al._ (2009) S. Weyeneth, T. Schneider, and E. Giannini, “Evidence for Kosterlitz-Thouless and three-dimensional $XY$ critical behavior in ${\text{Bi}}_{2}{\text{Sr}}_{2}{\text{CaCu}}_{2}{\text{O}}_{8+\delta}$,” Phys. Rev. B 79, 214504 (2009).
  * Suzuki  _et al._ (2018) Hakuto Suzuki, Matteo Minola, Yi Lu, Yingying Peng, Roberto Fumagalli, Emilie Lefrançois, Toshinao Loew, Juan Porras, Kurt Kummer, Davide Betto, Shigeyuki Ishida, Hiroshi Eisaki, Cheng Hu, Xingjiang Zhou, Maurits W. Haverkort, Nicholas B. Brookes, Lucio Braicovich, Giacomo Ghiringhelli, Matthieu Le Tacon, and Bernhard Keimer, “Probing the energy gap of high-temperature cuprate superconductors by resonant inelastic x-ray scattering,” npj Quantum Materials 2018 3:1 3, 1–8 (2018).
  * Norman  _et al._ (1995) M. R. Norman, M. Randeria, H. Ding, and J. C. Campuzano, “Phenomenological models for the gap anisotropy of Bi2Sr2CaCu2O8 as measured by angle-resolved photoemission spectroscopy,” Phys. Rev. B 52, 615–622 (1995).


Learning-graph-based quantum algorithm for $k$-distinctness.
In Proc. of 53rd IEEE FOCS, pages
http://dx.doi.org/10.1109/FOCS.2012.18207–216, 2012.

[11]
A. Belovs.
Span programs for functions with constant-sized 1-certificates.
In Proc. of 44th ACM STOC, pages
http://dx.doi.org/10.1145/2213977.221398577–84, 2012.

[12]
A. Belovs.
Quantum walks and electric networks.
http://arxiv.org/abs/1302.3143arXiv:1302.3143, 2013.

[13]
A. Belovs.
Applications of the Adversary Method in Quantum Query
PhD thesis, University of Latvia, 2014.

[14]
A. Belovs.
Variations on quantum adversary.
http://arxiv.org/abs/1504.06943arXiv:1504.06943, 2015.

[15]
A. Belovs and B. W. Reichardt.
Span programs and quantum algorithms for $st$-connectivity and claw
In Proc. of 20th ESA, volume 7501 of LNCS, pages
http://dx.doi.org/10.1007/978-3-642-33090-2_18193–204. Springer,

[16]
A. Belovs and D. Yolcu.
One-way ticket to Las Vegas and the quantum adversary.
http://arxiv.org/abs/2301.02003arXiv:2301.02003, 2023.

[17]
E. Bernstein and U. Vazirani.
Quantum complexity theory.
SIAM Journal on Computing,
26(5):http://dx.doi.org/10.1137/S00975397963009211411–1473, 1997.
Earlier: http://dx.doi.org/10.1145/167088.167097STOC'93.

[18]
G. Brassard, P. Høyer, M. Mosca, and A. Tapp.
Quantum amplitude amplification and estimation.
In Quantum Computation and Quantum Information: A Millennium
Volume, volume 305 of AMS Contemporary Mathematics Series, pages
53–74, 2002.

[19]
H. Buhrman and R. de Wolf.
Complexity measures and decision tree complexity: a survey.
Theoretical Computer Science,
288:http://dx.doi.org/10.1016/S0304-3975(01)00144-X21–43, 2002.

[20]
R. Cleve.
An introduction to quantum complexity theory.
http://arxiv.org/abs/quant-ph/9906111arXiv:quant-ph/9906111, 1999.

[21]
A. Cornelissen, S. Jeffery, M. Ozols, and A. Piedrafita.
Span programs and quantum time complexity.
In Proc. of 45th MFCS, pages
http://dx.doi.org/10.4230/LIPIcs.MFCS.2020.2626:1–26:14, 2020.

[22]
E. Farhi, J. Goldstone, and S. Gutmann.
A quantum algorithm for the Hamiltonian NAND tree.
Theory of Computing, 4:169–190, 2008.

[23]
L. K. Grover.
A fast quantum mechanical algorithm for database search.
In Proc. of 28th ACM STOC, pages
http://dx.doi.org/10.1145/237814.237866212–219, 1996.

[24]
S. Jeffery.
Quantum subroutine composition.
http://arxiv.org/abs/2209.14146arXiv:2209.14146, 2022.

[25]
S. Jeffery.
Span programs and quantum space complexity.
Theory of Computing,
18(11):http://dx.doi.org/10.4086/toc.2022.v018a0111–49, 2022.

[26]
S. Jeffery and S. Kimmel.
Quantum algorithms for graph connectivity and formula evaluation.
Quantum, 1(26), 2017.

[27]
S. Jeffery and S. Zur.
Multidimensional quantum walks, with application to k-distinctness.
In Proc. of 55th ACM STOC, pages
http://dx.doi.org/10.1145/3564246.35851581125–1130, 2023.

[28]
T. Lee, R. Mittal, B. W. Reichardt, R. Špalek, and M. Szegedy.
Quantum query complexity of state conversion.
In Proc. of 52nd IEEE FOCS, pages
http://dx.doi.org/10.1109/FOCS.2011.75344–353, 2011.

[29]
N. Leonardos.
An improved lower bound for the randomized decision tree complexity
of recursive majority.
In Proc. of 40th ICALP, Part I, volume 7965 of LNCS,
pages http://dx.doi.org/10.1007/978-3-642-39206-1_59303–314.
Springer, 2013.

[30]
F. Magniez, A. Nayak, M. Santha, and D. Xiao.
Improved bounds for the randomized decision tree complexity of
recursive majority.
In Proc. of 38th ICALP, volume 6755 of LNCS, pages
http://dx.doi.org/10.1007/978-3-642-22006-7_27317–329. Springer,

[31]
B. W. Reichardt.
Span programs and quantum query complexity: The general adversary
bound is nearly tight for every Boolean function.
In Proc. of 50th IEEE FOCS, pages
http://dx.doi.org/10.1109/FOCS.2009.55544–551, 2009.

[32]
B. W. Reichardt.
Reflections for quantum query algorithms.
In Proc. of 22nd ACM-SIAM SODA, pages
http://dx.doi.org/10.1137/1.9781611973082.44560–569, 2011.

[33]
B. W. Reichardt.
Span-program-based quantum algorithm for evaluating unbalanced
In Proc. of 6th TQC, volume 6745 of LNCS, pages
http://dx.doi.org/10.1007/978-3-642-54429-3_673–103. Springer,

[34]
B. W. Reichardt and R. Špalek.
Span-program-based quantum algorithm for evaluating formulas.
Theory of Computing,
8:http://dx.doi.org/10.4086/toc.2012.v008a013291–319, 2012.
Earlier: http://dx.doi.org/10.1145/1374376.1374394STOC'08,  http://arxiv.org/abs/0710.2630arXiv:0710.2630.

[35]
M. Saks and A. Wigderson.
Probabilistic Boolean decision trees and the complexity of
evaluating game trees.
In Proc. of 27th IEEE FOCS, pages
http://dx.doi.org/10.1109/SFCS.1986.4429–38, 1986.

[36]
M. Santha.
On the Monte Carlo boolean decision tree complexity of read-once
Random Structures and Algorithms,
6(1):http://dx.doi.org/10.1002/rsa.324006010875–87, 1995.

[37]
M. Santha.
Quantum walk based search algorithms.
In Proc. of 5th TAMC, volume 4978 of LNCS, pages 31–46.
Springer, 2008.

[38]
M. Snir.
Lower bounds for probabilistic linear decision trees.
Theoretical Computer Science,
38:http://dx.doi.org/10.1016/0304-3975(85)90210-569–82, 1985.

[39]
M. Szegedy.
Quantum speed-up of Markov chain based algorithms.
In Proc. of 45th IEEE FOCS, pages 32–41, 2004.

[40]
B. Zhan, S. Kimmel, and A. Hassidim.
Super-polynomial quantum speed-ups for Boolean evaluation trees
with hidden structure.
In Proc. of 3rd ACM ITCS, pages
http://dx.doi.org/10.1145/2090236.2090258249–265, 2012.


Optimizing the Placement and Beamforming of RIS in Cellular Networks: A System-Level Modeling Perspective

Pavan Reddy M ., SaiDhiraj Amuru, and Kiran Kuchi

Pavan Reddy M. is with WiSig Networks, Hyderabad, India. Kiran Kuchi and SaiDhiraj Amuru are with the Department of Electrical Engineering, Indian Institute of Technology Hyderabad, Telangana, India.  (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>kkuchi@ee.iith.ac.in).

In this letter, we present in detail the system-level modeling of reconfigurable intelligent surface (RIS)-assisted cellular systems by considering a 3-dimensional channel model between base station, RIS, and user. We prove that the optimal placement of RIS to achieve wider coverage is exactly opposite to the base station, under the constraint of single RIS in each sector. We propose a novel beamforming design for RIS-assisted cellular systems  and derive the achievable sum rate in the presence of ideal, discrete, and random phase shifters at RIS. Through extensive system-level evaluations, we then show that the proposed beamforming design achieves significant improvements as compared to the state-of-the-art algorithms.

Beamforming, phase shifters, reconfigurable intelligent surfaces (RIS), and spectral efficiency.

§ INTRODUCTION

RIS is  considered a promising technology for the next generation cellular communications to improve the achievable network capacity and cellular coverage <cit.>. RIS consists of a large number of passive antenna elements (or meta-surfaces) which can reflect the incident ray toward the desired direction. By controlling the impedance of the  meta-surfaces through a passive electronic circuit, an additional phase shift is introduced into the reflected signal, and thus, the signal is steered in the desired direction <cit.>. This way, RIS helps in achieving improved signal reception for the users and also providing coverage to the users who are affected by the signal blockages.

In the existing literature, RIS has been extensively analyzed and significant improvements are demonstrated for single cell scenario <cit.>.  In <cit.>, the authors propose a joint beamforming algorithm to maximize the network capacity.
In <cit.>, the authors propose power efficient and sum-rate maximizing algorithms for RIS-assisted systems. In <cit.>, the authors present an initial access protocol for RIS aided cellular systems. However, these works do not consider the multi-cell analysis while quantifying the achievable gains. Note that only in the multi-cell analysis, the impact of the inter-cell interference from the beamformed RIS is captured, and thus, realistically achievable gains can be understood. Additionally, while performing such multi-cell analysis, a 3-dimension channel modeling between the BS, RIS, and user has to be considered to quantify the achievable gains. However, very few works in the literature have considered the 3-dimension channel model and carried out system-level evaluations <cit.>.

Further, the beamforming at RIS has a significant impact on the achievable gains with RIS-assisted systems <cit.>.   In <cit.>, the authors consider deploying multiple RIS and using random phase shifters at each RIS. This is a low-complex way to enhance the network performance but only a few users whose channel coefficients are aligned with those assigned phase shifters will observe spectral enhancements. In <cit.>, the authors consider random phase shift allocation in each time slot and assume that the users decode the pilot signals transmitted at the beginning of each slot.
Then, the users' feedback the channel quality reports to the BS, and the users with the best instantaneous channel conditions will be scheduled for data transmission within the rest of the slot. With this approach, the effective channel observed by each user changes from slot to slot and BS schedules the data transmissions according to the channel quality reports to maximize the achievable capacities. However, this procedure requires a large number of active users to realize the desired gains, and also, the feedback from the users within the same time slots is difficult to realize in practice. Hence, there is a need to consider all the aforementioned details and design a  practically feasible, low-complex, and yet optimal way of  beamforming at RIS.

\begin{align}
\gls{H}&=\left[\gls{au}\left(\gls{thetaUAB},\gls{phiUAB}\right) \right]^T.\  \bold{\chi_{\scriptscriptstyle B,k}}. \ \gls{aB}\left(\gls{thetaBDk},\gls{phiBDk}\right). \exp\left(\dfrac{j2\pi(\gls{rbk}\gls{dbk})}{\lambda}\right).
\exp\left(\dfrac{j2\pi(\gls{rkb}\gls{dkb})}{\lambda}\right).
\exp\left(\dfrac{j2\pi\gls{rbk}}{\lambda}\bold{\nu_{B,k}} t\right)\label{eqn:HFD1}\\
\gls{G}&=\left[\gls{aR}\left(\gls{thetaRA},\gls{phiRA}\right) \right]^T.\  \bold{\chi_{\scriptscriptstyle B,R}}. \ \gls{aB}\left(\gls{thetaBDR},\gls{phiBDR}\right). \exp\left(\dfrac{j2\pi(\gls{rbr}\gls{dbr})}{\lambda}\right).
\exp\left(\dfrac{j2\pi(\gls{rrb}\gls{drb})}{\lambda}\right)\label{eqn:HFDH3}\\%.\exp\left(\dfrac{j2\pi(\gls{rR})}{\lambda}\right)
\gls{F}&=\left[\gls{au}\left(\gls{thetaUAR},\gls{phiUAR}\right) \right]^T  \hspace{-0.2cm}. \ \bold{\chi_{\scriptscriptstyle R,k}}. \ \gls{aR}\left(\gls{thetaRD},\gls{phiRD}\right). \exp\left(\dfrac{j2\pi(\gls{rrk}\gls{drk})}{\lambda}\right).
\exp\left(\dfrac{j2\pi(\gls{rkr}\gls{dkr})}{\lambda}\right).
\exp\left(\dfrac{j2\pi\gls{rrk}}{\lambda}\bold{\nu_{R,k}} t\right) \label{eqn:H3}
\end{align}

Motivated by these facts, we present the following key contributions in this letter.

* We prove that optimal RIS placement (under the constraint of single RIS in each sector) to achieve a wider coverage is exactly opposite to the base station with boresight of RIS facing the boresight of the base station.

* We propose a novel beamforming design for RIS-assisted cellular systems and  analyze the impact of the selection of phase shifters on the proposed design. We derive the achievable sum rates in the presence of ideal, discrete, and random phase shifters.

* We present the system-level modeling of RIS-assisted cellular systems by
considering the 3-dimensional channel modeling between the BS, RIS, and
user. We perform extensive system-level simulations and show that the proposed beamforming design significantly outperforms the state-of-the-art algorithms.

§ SYSTEM MODEL

§.§ Scenario Description

We consider a cellular network with BS, RIS, and users as shown in Fig. <ref>. We assume M, N, and U as the number of antenna elements at the BS, RIS, and user, respectively, and formulate the equivalent channel $\gls{h}\in \mathbb{C}^{U \times M}$ between the BS and user $k$ as follows.

\begin{align}
\gls{h}=\gls{H}+\gls{F}\gls{Theta}\gls{G},
\label{eqn:heq}
\end{align}

where, $[\cdot]^H$ represents Hermitian operation, and $\gls{G} \in \mathbb{C}^{N \times M}$, $\bold{F}\in \mathbb{C}^{U \times N}$, and $\bold{H}\in \mathbb{C}^{U \times M}$ denote the channel coefficients between the BS to RIS, RIS to the user $k$, and the direct link between BS and user $k$, respectively. Further, $\gls{Theta}\in \mathbb{C}^{N \times N}$ denotes the phase shift matrix at the RIS formulated as follows <cit.>

\begin{align}
\gls{Theta}=&\ \diag(e^{j\theta_{1}},e^{j\theta_{2}}, \ldots,e^{j\theta_{N}}), \forall \ \theta_{n} \in \gls{Fst},\nonumber\\
\gls{Fst}\triangleq&\ \Big\{\theta_{n}\Bigl\vert |e^{j\theta_{n}}|\leq 1 \Big\}, \forall\ 1\leq n\leq N.\nonumber
\end{align}

Scenario Depiction

RIS Modelling

System Model.

Note that ideal phase shifters are difficult to realize in practice. Thus, we formulate the discrete phase shifters as follows <cit.>

\begin{align}
\mathcal{F}_D\triangleq \bigg\{\theta_{n} \Bigl \vert \theta_{n} \in \Big\{0, \tfrac{2\pi}{D}, \ldots, \tfrac{2\pi (D-1)}{D}\Big\} \bigg\}, \label{eqn:FD}
\end{align}

where $D$ denotes the number of discrete-levels at RIS.
We consider the RIS-assisted downlink transmission for the user $k$ as follows.

\begin{align}
\bold{x}_k&=\sqrt{\gls{Pmax}} s_k,\nonumber
\end{align}

where $s_k$ is the transmitted symbol corresponding to the user $k$ with unit power i.e., $\mathbb{E}\{s_k s_k^H\}=1$ and Pmax is the maximum power the BS can transmit.
Thus, we formulate the received signal at the user $k$ as follows.

\begin{align}
\bold{y}_k=&\gls{h} \bold{x}_k+\bold{z}_k,\nonumber\\
=& \Big(\gls{H}+ \bold{F}_k^H\bold{\Theta}\bold{G}\Big)\bold{x}_k+\bold{z}_k.\label{eqn:hsmpl}\end{align}

where  $\bold{z}_k$ represents the additive white Gaussian noise at the user $k$ with zero mean and co-variance $\bold{\Xi}_k=\sigma^2\bold{I}$.

§.§ 3-dimensional Channel Model

Typically, the physical size of the RIS structure is comparatively much smaller than the actual distance of the RIS from the BS and user. Hence, we assume the far-field region while formulating the channel model <cit.>. Further, considering the various existing evaluations in the literature <cit.>, we adapt the 3GPP based 3-dimension channel between the BS, RIS, and user <cit.>. We define the clusters, sub-clusters, rays,  angle of arrivals, angle of departures, path loss, shadow-fading, and small-scale fading as per the channel modeling formulated in <cit.>. The formulation of channel coefficients between the BS, RIS, and user (excluding the large scale fading parameters such as path loss and shadow fading) is summarized in (<ref>)-(<ref>), where subscripts $\scriptstyle \bold{B}$, $\scriptstyle \bold{R} $, and $\scriptstyle \bold{k}$ represent base stations, RIS, and user, respectively.
$\bold{\phi_{AOA}}, \ \bold{\theta_{AOA}},\   \bold{\phi_{AOD}},$ and $\bold{\theta_{AOD}}$ represent the angle of arrivals in azimuth and elevation, and angle of departures in azimuth and elevation, respectively. Based on the cellular environment (such as urban, rural, etc.), these angles are assigned considering various sub-rays corresponding to various clusters between the transmitter and receiver as per <cit.>.
au, aR, and aB represent the antenna gains at the user, RIS, and BS, respectively. We consider the sectoral antenna pattern as per <cit.> at the BS and omni directional antenna at the user. We model the transmissions from RIS as just reflections (passive) i.e., with zero additional antenna gain ($\gls{aR}(\cdot)\leq1$). $\bold{\chi}$ captures the cross polarisation powers, $\bold{d}$ represents the location vector, and $\bold{r}$ represents the spherical unit vector for corresponding antenna elements, respectively. For the planar array shown in Fig. <ref>, the spherical unit vector is formulated as shown below <cit.>.

\begin{align}
\bold{r}=\begin{bmatrix} \sin \boldsymbol{\theta} \cos \boldsymbol{\phi} \\ \sin \boldsymbol{\theta} \sin \boldsymbol{\phi}\\ \cos \boldsymbol{\theta} \end{bmatrix}.\label{eqn:rm}
\end{align}

$\nu$ represents the speed of the receiver and the corresponding exponential in (<ref>) and (<ref>) captures the Doppler effect. Note that no Doppler term is considered in (<ref>), as we consider RIS is stationary. Further, we also consider the path loss and shadow fading for each cellular environment as per <cit.>.

Proposed RIS organization and beamforming

§ PROPOSED RIS  ORGANIZATION AND BEAMFORMING

§.§ RIS organization

For the RIS-assisted cellular networks, we propose the RIS organization as shown in Fig. <ref>.  The boresights of the sectors are considered to be in the directions of $[30^\circ, 150^\circ, 270^\circ]$ and the RIS is placed at the edge of each sector with the boresight of RIS elements facing the boresight of the BS. This organization of the BS orientations and positioning of the RIS helps in ensuring RIS access to all the users in the sector. When compared to the scenario with boresight of the sectors in the direction of $[0^\circ, 120^\circ, 240^\circ ]$ and RIS placed at cell-edge with its elements facing the boresight of the BS,   the proposed organization has two key benefits as follows. With the proposed organization, the BS to RIS distance is $\frac{ISD}{2}$ instead of $\frac{ISD}{\sqrt{3}}$, where, $ISD$ indicates the inter-site distance. Note that the achievable gains with RIS-assisted systems are heavily dependent on the path loss observed in the BS-RIS-user link. Thus, with the proposed organization, the path loss observed will be minimal, and hence, enhanced signal reception can be achieved by the user. Further, with the proposed organization, the interference caused by the beamforming of the RIS to the other sectors is also comparatively minimum, as the nearest neighbor sectors are not in the boresight direction of the RIS.

The optimal location for the RIS to achieve a wider coverage is at the opposite end of the sector, with the boresight of the RIS facing the boresight of the BS antenna.

Optimal RIS location analysis for proof of Lemma <ref>.

We consider the circular approximation for a cell site and assume the 3-sector division as shown in Fig. <ref> <cit.>, where $OA, OB,$ and $OC$ are the sector divisions. We prove that $D$ is the optimal location of the RIS for the sector $OAB$, and the same can be extended to the other sectors. Note that the reflections from the RIS are observed on only one side of the RIS, and hence, to ensure RIS access to the entire sector, RIS has to be placed only on the circumference of the sector $OAB$. Further, the reflection amplitude of the RIS is typically approximated as $\cos(\theta)$, where $\theta$ is the angle of incidence on RIS <cit.>. Thus, when RIS is placed along the lines $OA$ and $OB$ but facing the sector, the signal from the base station is received at an angle of incidence of $90^\circ$ resulting in minimal gains. Also, note that typical BS cellular antenna patterns have the least gains in the direction of sector-edges <cit.>. Thus, the optimal location of the RIS can only be on the arc $AB$.

As shown in Fig. <ref>, consider 3 points on the arc $AB$ which are at $0^\circ, >0^\circ,$ and $<0^\circ$ from the BS denoted as $D, E,$ and $F$, respectively. Note that all three points are at the same distance from BS (or $O$ in Fig. <ref>). Thus, the distance-dependent path loss observed by the RIS at these locations would be similar.
Further, when we place RIS at $D$, the maximum distance of any user in the sector from $D$ does not exceed the radius of the sector $r$. For example, $A$ is the farthest location from $D$.  Since, $OA=OD=r$ and $\angle AOD = 60^\circ$, $AD=r$.
However, this is not the case when the RIS is placed at either $E$ or $F$. The RIS will be closer to some users in the sector, but the farthest locations from $E$ and $F$ are

$EB>r$ and $FA>r$. This increase in the distance from RIS to the users will result in decreased signal reception, and thus provides less coverage to these users when compared with the case of RIS at $D$. Hence, the optimal location of RIS for wider coverage is $D$. This ends the proof of Lemma <ref>.

§.§ Beamforming design

To perform optimal beamforming at RIS for each user, the BS has to know the complete channel state information between RIS and each user. However, this is not always achievable in practice. Hence, we propose a practically feasible beamforming design as follows. We configure B number of predefined digital beams at the RIS such that they cover the entire sector, as shown in Fig. <ref>. The number of beams B can be decided by the network operator based on the number of antenna elements at the RIS. Given B beams, the BS can tune the phase shifters at RIS and activate one of the configured beams over various time slots. At the receiver, the user calculates the effective received signal strength from the configured beam and also the direct link from the BS. The user then reports back the best beam index to the BS in the uplink. For example, the best beam for users 1 and 2 in Fig. <ref> are beam-B and beam-3, respectively. This way, the BS can configure the desired beams at the RIS while scheduling the data for each user.  Note that a similar approach can also be adapted for the initial user attach. While transmitting the downlink synchronization signals, the BS can tune the phase shifters at RIS and transmit different beams over various time slots such that the user can lock to the beam from which it receives maximum signal strength.

System-level evaluation of achievable coupling loss, SINR, and spectral efficiency with various algorithms

The maximum achievable data rate for user $k$ in the presence of ideal phase shifters at RIS is given by

\begin{align}
\log_2\left(1+\gls{Pt}\dfrac{\Big||\gls{H}|+|\gls{F}||\gls{G}|\Big|^2}{\bold{\Xi}_k}\right) \label{eqn:l1}
\end{align}

The optimal values of phase shifters that maximize the achievable data rate are given by

\begin{align}
\gls{ThetaO}=\arg \max_{\gls{Theta}} ||\gls{h}||^2 =\arg \max_{\gls{Theta}} ||\gls{H}+\gls{F}\gls{Theta}\gls{G} ||^2\nonumber\end{align}

\begin{align}
\hspace{-1.5cm} \text{From \cite{Chandra}},
\gls{ThetaO}= \exp\left(j \left(\angle \gls{H}-\angle(\gls{F}+\gls{G})\right)\right) \label{eqn:ThetaOpt}
\end{align}

Using (<ref>), we formulate the achievable SNR as

\begin{align}
\gls{gamma}
=\gls{Pt}\dfrac{\Big|\Big| |\gls{H}|\angle \gls{H}+|\gls{F}| |\gls{G}|\angle \gls{H}\Big|\Big|}{\bold{\Xi}_k}
\end{align}

Thus, the maximum achievable data rate at user $k$ in the presence of ideal phase shifters at RIS is as shown in (<ref>).

As $N\rightarrow\infty$, the maximum achievable data rate at user $k$ with discrete phase shifters at RIS is given by

\begin{align} \log_2\left(1+ \gls{Pt}\dfrac{\Big||\gls{H}|+\mathbb{E}[r_i]\sinc(\frac{\pi}{D})\Big|^2}{\bold{\Xi}_k}\right)\end{align}

where $D$ represents the number of discrete-levels of the phase shifters at RIS and $r_i$ represents the diagonal elements of $\diag(\gls{F}\gls{ThetaO}\gls{G})$.

Using the optimal beamforming derived in (<ref>) at RIS with discrete phase shifters, we get

\begin{align}
\gls{ThetaD}=\gls{ThetaO}\odot\gls{ThetaE}\label{eqn:thetaD}
\end{align}

where, $\odot$ denotes the element-wise (Hadamard) product and $\gls{ThetaE}=\diag\{e^{j\theta^e_1},\ldots, e^{j\theta^e_N}\}$ captures the quantization error. From (<ref>), we obtain $\theta_i^e\in \left[-\tfrac{\pi}{D},\tfrac{\pi}{D} \right]$. Further, using (<ref>), we get

\begin{align}
\gls{F}\gls{ThetaD}\gls{G}=\gls{F}\left(\gls{ThetaO}\odot\gls{ThetaE}\right)\gls{G}\overset{(a)}{=}(\gls{ThetaE}\bold{1}_N)^H\diag(\gls{F}\gls{ThetaO}\gls{G})\nonumber
\end{align}

where $\bold{1}_N$ is a $N\times 1$ array and (a) is obtained by using Hadamard-product properties. By denoting $r_i$ as the diagonal elements of $\gls{F}\gls{ThetaO}\gls{G}$, we get

\begin{align}
\gls{F}\gls{ThetaD}\gls{G}&=\sum_{n=1}^{N} e^{j\theta_i^e}r_i\overset{(b)}{=}N\mathbb{E}\big[e^{j\theta_i^e}r_i\big]\nonumber\\
&\overset{(c)}{=} N\mathbb{E}\big[e^{j\theta_i^e}\big] \mathbb{E}\big[r_i\big]\overset{(d)}{=}N\mathbb{E}\big[\cos(\theta^e_i)\big]\mathbb{E}\big[r_i\big]\nonumber\\
\nonumber
\end{align}

(b) is obtained by considering the law of large numbers <cit.>, (c) is obtained assuming the independent nature of the random variables, (d) is obtained after integration of the odd symmetrical function $\sin(\theta_i^e)$ over $[-\tfrac{\pi}{D}, \tfrac{\pi}{D}]$, and (e) is based on the assumption of the uniform distribution of $\theta_i^e$ over $[-\tfrac{\pi}{D}, \tfrac{\pi}{D}]$

Note that based on (<ref>), $\mathbb{E}\big[r_i\big]$ also has the phase in the direction of $\angle \gls{H}$, and thus, we get

\begin{align}
||\gls{H}+\gls{F}\gls{ThetaD}\gls{G}||^2=\big||\gls{H}|+N\sinc(\tfrac{\pi}{D})\mathbb{E}[r_i]\big|^2 \label{eqn:l2proof}
\end{align}

This completes the proof of Lemma <ref>.

As $N\rightarrow \infty$, the achievable data rates with the random phase shifters at RIS are similar to that of the case with no RIS.

Note that with random phase shifters at RIS, $\theta_i^e \in [-\pi, \pi]$, which is same as considering $D=1$.
Thus, substituting $D=1$ in (<ref>), we get

\begin{align} ||\gls{H}+\gls{F}\gls{ThetaD}\gls{G}||^2=||\gls{H}||^2
\end{align}

This completes the proof of Lemma <ref>.

§.§ Baseline algorithms considered for the evaluation

§.§.§ Cellular system without any RIS

We consider cellular system without any RIS as the baseline system for the comparison to understand the achievable gains with the introduction of RIS. Further, to ensure the correctness of the system-level modelling along with channel formulation, we have considered the antenna configuration and simulation parameters of the 3GPP Phase-2 Calibration <cit.> as the baseline and aligned the results with <cit.>.

§.§.§ Random phase shifters at the RIS <cit.>

We evaluate the performance of the RIS-assisted cellular systems while applying random phase shifters (i.e., values of $\theta_{n}$ are assigned from complex random distribution with $|\theta_n |\leq 1$) at the RIS in each time slot <cit.>.

§ SIMULATION RESULTS AND DISCUSSION

Simulation parameters

Parameter   Values
Cellular environment     Urban macro (UMa) <cit.>
Carrier frequency     2 GHz
Path loss model     UMa <cit.>
System Bandwidth     10 MHz
User drop     10 per sector
BS transmit power $P_t$    43 dBm
Wraparound     Geographical distance based
RIS drop     1 per sector
User attachment     Based on received signal power
Small scale model     3GPP FD-MIMO <cit.>
BS antenna elements, $M$    $10 \times 2 \times 2 = 40$ elements
BS antenna pattern     3GPP sectoral pattern
User antenna elements, $U$     1 element
User antenna pattern     omni directional
RIS antenna elements, $N$     $16 \times 16 =256$ elements
RIS antenna pattern     passive (only reflections modelled)
Thermal Noise     -174 dBm/Hz
Inter-cell interference     explicitly modelled from both RISs and BSs
Other general parameters      as per <cit.>

The simulation parameters considered for the system-level evaluation of the proposed algorithms are presented in Table <ref>.

We have performed Monte-Carlo simulations in MATLAB and presented the achievable coupling loss, SINR, and spectral efficiency with all the considered algorithms in Fig. <ref>.

The coupling loss is calculated as the difference between the transmitted power and received power at each user.
In Fig. <ref>, for the case without RIS, the coupling loss  is aligned with the 3GPP Phase 2 Calibration curves presented in <cit.>, thus, ensuring the correctness of our implementation. As compared to the scenario with no RIS, the proposed design with ideal phase shifters achieves a 4 dB gain in the received signal power. The gains achievable with random phase shifters while using single RIS are very minimal. Further, the proposed design with 4-bit discrete phase shifters also has significant improvement as compared to the scenario without any RIS.  With the 12-bit discrete phase shifters, the coupling loss with the proposed design is similar to the coupling loss with ideal phase shifters.

Comparison of data rates with varying $D$ and $N$

We have calculated the SINR by explicitly capturing the interference from the neighboring BS and RIS as follows.

\begin{align}
\text{\gls{SINR}}=\gls{Pmax}\frac{\bold{h}_k\gls{h}}{\text{IN}_k+\bold{\Xi}_k},
\end{align}

where $\text{IN}_k$ represents the interference captured from the neighbouring BS and RIS transmissions.
In Fig. <ref>, the SINR for the scenario without RIS is aligned with 3GPP Phase 2 Calibration curves presented in <cit.>, and thus, calibrating the  SINR calculation. As compared to the scenario with no RIS, the proposed design with ideal phase shifters achieves close to 2 dB improvement in the SINR. Note that despite the maximum of 4 dB gain in the coupling loss, we achieve only 2 dB improvements in the SINR because of the increase in the inter-cell interference caused by the addition of RIS in the neighboring sectors. This reduction in SINR also highlights the need for carrying out multi-cell analysis to quantify the practical gains with the inclusion of RIS. Further, for the same reason, all the cell-edge users do not observe similar interference, and hence, the pattern of achievable gains in coupling loss and SINR differ. Note that the performance of the random phase shifters is close to the scenario without any RIS and the performance of the proposed RIS with 4-bit discrete phase shifters is also significantly better than the baseline algorithms. With the 12-bit discrete phase shifters, the SINRs of the proposed design are similar to the SINRs of the proposed design with ideal phase shifters. Further, the gains in spectral efficiency follow a similar pattern as that of the SINRs. As shown in Fig. <ref>, the proposed design achieves 30% improvement in spectral efficiency as compared to the baseline algorithms.

In Fig. <ref>, we present the comparison of achievable data rates for varying $D$ and $N$. We show that the simulated results completely align with the theoretical analysis derived in Lemma <ref> and <ref>.
Note that with an increase in $N$, the system performance with the random phase shifts does not change and is same as that of the case with no RIS. Further,
with the increase in $D$, the performance of the proposed design with discrete phase shifters converges to that of ideal phase shifters.

§ CONCLUSION

In this letter, we have presented in detail the system-level modeling of the RIS-assisted cellular systems. Our numerical evaluations outline that only with the system-level evaluations the actual gains achievable with RIS can be understood.
We have proposed a novel cellular layout and RIS organization which ensures uniform accessibility of the RIS to all the active users in the desired sector and creates minimal interference to users in the neighboring cells. We have then proposed a beamforming design for the RIS-assisted cellular systems and analyzed the achievable gains with the proposed design by considering ideal and discrete phase shifters.
Through extensive system-level analysis, we have shown that the proposed design achieves significant improvements in network performance when compared to the baseline algorithms.
In the future, we plan to evaluate the proposed design on the hardware testbed.