# A Theoretical Study of Doping Evolution of Phonons in High-Temperature
Cuprate Superconductors

Saheli Sarkar  Division of Condensed Matter Physics and Materials Science,
Brookhaven National Laboratory, Upton, NY 11973-5000, USA

(August 28, 2024)

###### Abstract

Hole-doped high-temperature copper oxide-based superconductors (cuprates)
exhibit complex phase diagrams where electronic orders like a charge density
wave (CDW) and superconductivity (SC) appear at low temperatures. The origins
of these electronic orders are still open questions due to their complex
interplay and correlated nature. These electronic orders can modify the
phonons in the system, which has also been experimentally found in several
cuprates as a softening in the phonon frequency at the CDW vector. Recent
experiments have revealed that the softening in phonons in cuprates due to CDW
shows intriguing behavior with increasing hole doping. Hole doping can also
change the underlying Fermi surface. Therefore, it is an interesting question
whether the doping-induced change in the Fermi surface can affect the
softening of phonons, which in turn can reveal the nature of the electronic
orders present in the system. In this work, we investigate this question by
studying the softening of phonons in the presence of CDW and SC within a
perturbative approach developed in an earlier work. We compare the results
obtained within the working model to some experiments.

##  I Introduction

High-temperature copper oxide-based superconductors (cuprates) are
paradigmatic examples of a complex interplay between several orders like a
quasi-two-dimensional incommensurate charge density wave (CDW) Author4 (2002,
2012) and superconductivity (SC) Author4 (200), especially in the under-doped
regime of hole-doped cuprates Author1 (2015). Besides these orders, in the
under-doped regime, there exist a ‘pseudogap’ phase Author1 (1989, 1998);
Author2 (1998) and a variety of other complex orders Author4 (1999, 2006);
Author1 (2011); Author2 (2019). In addition to the observation of the CDW
order in the under-doped regime, recent X-ray scattering experiments have
observed CDW in over-doped samples of LSCO Author6 (2019, 2023), even going
well beyond the pseudogap region Author2 (2023), and in over-doped BSCCO
Author3 (2018). Furthermore, recent resonant X-ray scattering experiments have
observed charge density modulations with correlation length much shorter
compared to the CDW which are generally known as charge density fluctuations
(CDFs) Author7 (2023). These CDFs have a dynamical nature due to finite energy
and are present in an extremely large doping window starting from a highly
under-doped regime and going to a highly over-doped regime Author7 (2019,
2023). Additionally, these dynamical CDFs persist up to a very high
temperature, above the onset of CDW and even above the pseudogap temperature
TPG.

A change in doping not only tunes the electronic orders, but also changes the
electronic structure and the associated Fermi surface. Angle-resolved
photoemission spectroscopy (ARPES) experiments Author2 (1998, 2006); Author8
(2023) measured the evolution of the Fermi surface of La2-xSrxCuO4 (LSCO) for
a broad range of hole doping starting from the under-doped regime and going to
the over-doped regime. Another ARPES experiment Author1 (2006) measured the
hole doping evolution of the Fermi surface of Bi2Sr2CaCu2O8+δ (BSCCO). For
both the cases of LSCO and BSCCO, it is observed that the geometry of the
Fermi surface changes from an open ‘hole-like’ to a closed ‘electron-like’
geometry. This type of topological transition of the Fermi surface is known as
the Lifshitz transition.

Owing to such a complex phase diagram and Fermi surface evolution, the nature
of the interactions which induce the electronic orders still remains an open
question. A possible direction for disentangling various interactions is to
study the collective modes associated with the electronic orders Author1
(2019). It is well known that CDW can couple to the phonons and in metallic
systems give rise to the softening of the phonon frequency (we refer to it as
phonon softening) at the wave vector associated with the CDW ($\bm{Q}$) and
below the CDW transition temperature. This is known as the ‘Kohn anomaly’
Author1 (1963, 1973). Interestingly, experiments have found signatures of
phonon softening around $\bm{Q}$ but below the superconducting critical
temperature (Tc) in several under-doped cuprates like YBa2Cu3O7-x (YBCO),
Bi2Sr2CaCu2O8+δ, La2-xBaxCuO4 (LBCO) and La2-xSrxCuO4 Author1 (2014, 2021,
2006, 1999, 2020). In under-doped La-based 214 compounds Author1 (2020) and
BSCCO Author1 (2021); Author7 (2021), however, the phonon softening does not
completely vanish at Tc. Besides the temperature evolution, the phonon
softening in different cuprates also exhibits distinct features with a change
in doping. Resonant inelastic X-ray scattering (RIXS) and neutron scattering
experiments investigated phonons for different doping levels in
Bi2Sr2CaCu2O8+δ Author6 (2022), La2-xSrxCuO4 Author1 (2020) and La-based 214
compounds Author1 (2006), respectively. Remarkably, for doped BSCCO, the RIXS
experiment Author6 (2022) found a gradual suppression of phonon softening with
an increase in doping in a broad window of an under-doping to an over-doping
regime. In contrast to BSCCO, however, in doped LSCO Author1 (2020), it was
found that phonon softening related to charge density modulations remained
unchanged in a broad window of hole doping despite a continuous change in the
Fermi surface Author2 (2006), before vanishing discontinuously at a critical
doping.

To explain the phonon softening below Tc, in an earlier work Author1 (2021),
we investigated collective modes of phonons when they couple to both CDW and
SC orders within a perturbative approach. We showed, for prototypical under-
doped cuprates’ band structure, that a complex interplay between the CDW and
SC orders and their fluctuations Author5 (2020) can capture the temperature
evolution of the phonon softening, especially the anomalous phonon softening
below Tc. However, the evolution of the phonon softening with a change in
doping was not addressed. It is especially interesting to see if the change in
the Fermi surface due to doping can be captured through collective phonon
modes, which can also elucidate the nature of the electronic orders present in
the system.

Motivated by these, in this work we study, within our theoretical model
developed in Author1 (2021), the evolution of phonon softening for cuprates
with a change in the geometry of the Fermi surface. In particular, we focus on
two model cuprate bands replicating doped LSCO and BSCCO whose Fermi surfaces
pass through a topological Lifshitz transition, where a hole-like Fermi
surface transforms to an electron-like Fermi surface. We study the fate of the
phonon softening approaching this transition for both of the model cuprate
bands.

We organize this paper in the following manner. In Section II, we briefly
discuss the theoretical model, where the effects of CDW and SC orders on
phonons are investigated. In Section III, we study the doping evolution of the
phonon softening for model Fermi surfaces representing doped LSCO and BSCCO
systems. In Section IV, we present a discussion of our work and compare with
experimental observations. In Section V, we present a summary of the work.

##  II Theoretical Model

In this section, we describe the theoretical model of a coupled
electron–phonon system in the presence of the CDW and SC orders. The total
Hamiltonian Author1 (2021) is given by $H_{tot}=H_{e}+H_{ph}+H_{e-ph}$, with

$\displaystyle H_{e}$
$\displaystyle=\sum_{\bm{k},\sigma}\xi_{\bm{k}}c_{\bm{k},\sigma}^{\dagger}c_{\bm{k},\sigma}+\sum_{\bm{k},\sigma}(\chi_{\bm{k}}c_{\bm{k}+\bm{Q},\sigma}^{\dagger}c_{\bm{k},\sigma}+h.c.)+\sum_{\bm{k}}(\Delta_{\bm{k}}c_{\bm{k},\uparrow}^{\dagger}c_{-\bm{k},\downarrow}^{\dagger}+h.c.),$
(1) $\displaystyle H_{ph}$
$\displaystyle=\sum_{\bm{q}}\omega_{\bm{q}}b_{\bm{q}}^{\dagger}b_{\bm{q}},$
$\displaystyle H_{e-ph}$
$\displaystyle=(1/\sqrt{N})\sum_{{\bm{q}}}\sum_{\bm{k},\sigma}g(\bm{k},\bm{q})c_{\bm{k}+\bm{q},\sigma}^{\dagger}c_{\bm{k},\sigma}(b_{-\bm{q}}^{\dagger}+b_{\bm{q}}).$

In Equation (1), $H_{e}$ depicts the electronic Hamiltonian in the presence of
CDW and SC orders at the mean-field level, where $c_{\bm{k}}$ and
$c_{\bm{k}}^{\dagger}$ are the electronic annihilation and creation operators.
$\xi_{k}$, $\chi_{k}$ and $\Delta_{k}$ are the tight-binding single-particle
dispersion, CDW gap and SC gap, respectively. The Hamiltonian for the phonons
is given by $H_{ph}$, where $b_{\bm{q}}$ and $b_{\bm{q}}^{\dagger}$ are the
phonon annihilation and creation operators and $\omega_{\bm{q}}$ is the bare
phonon frequency. The interaction between electrons and phonons is given by
the Hamiltonian $H_{e-ph}$, where $g(\bm{k},\bm{q})$ is the electron–phonon
(e-ph) coupling element and N is the number of lattice sites. In our model, we
consider $g(\bm{k},\bm{q})=g$ a constant number. We construct the matrix
$H_{e}$ in the extended Nambu basis
$\Psi^{\dagger}_{\bm{k}}=\left(c^{\dagger}_{\bm{k},\uparrow},c_{-\bm{k},\downarrow},c^{\dagger}_{\bm{k}+\bm{Q},\uparrow},c_{-\bm{k}-\bm{Q},\downarrow}\right)$,
where $\bm{Q}$ is the CDW wave vector. The Green’s function corresponding to
$H_{e}$ is given by
$\hat{G}^{-1}(i\omega_{n},\bm{k})=(i\omega_{n}-\hat{H}_{e})$ and has a matrix
form as follows:

$\displaystyle\tiny G^{-1}$
$\displaystyle=\begin{pmatrix}i\omega_{n}-\xi_{\bm{k}}&-\Delta_{\bm{k}}&-\chi_{\bm{k}}&0\\\
-\Delta^{*}_{\bm{k}}&i\omega_{n}+\xi_{\bm{k}}&0&\chi_{\bm{k}}\\\
-\chi^{*}_{\bm{k}}&0&i\omega_{n}-\xi_{\bm{k}+\bm{Q}}&-\Delta_{\bm{k}+\bm{Q}}\\\
0&\chi^{*}_{\bm{k}}&-\Delta^{*}_{\bm{k}+\bm{Q}}&i\omega_{n}+\xi_{\bm{k}+\bm{Q}}\end{pmatrix},$
(2)

where $\omega_{n}$ is the Matsubara frequency.

Now, we analyze the change in the phonon spectrum due to the presence of both
CDW and superconducting order. Because of the presence of the CDW with wave
vector Q, the new phonon propagator can be written in terms of a matrix
Green’s function given by
$D_{m,n}(\bm{q},\tau)=-\langle\mathcal{T}\phi_{\bm{q}+m\bm{Q}}(\tau)\phi^{\dagger}_{\bm{q}+n\bm{Q}}(0)\rangle$
Author1 (1993), where $\mathcal{T}$ is the time-ordering operator and
$\phi_{\bm{q}}$ is given by $b_{\bm{q}}^{\dagger}+b_{-\bm{q}}$ with $m,n=\pm$.
We also note that $D_{++}=D_{--}:=D_{1}(z,\bm{q})$ and
$D_{+-}=D_{-+}:=D_{2}(z,\bm{q})$. The modified propagators can be calculated
by treating the electron–phonon interaction perturbatively, as long as g is
small. For a single-dispersion-less phonon mode, an estimate for g can be
obtained by using the dimensionless coupling constant
$\lambda\sim\frac{g^{2}N(E_{F})}{\nu_{0}}$, where $N(E_{F})$ is the density of
states at the Fermi energy and $\nu_{0}$ is the bare phonon frequency. As
typically $\frac{\nu_{0}}{N(E_{F})}\ll 1$, the approximation remains valid
even for large values of $\lambda$. Following the results obtained in the
paper Author1 (2021), the self-energy corrections
[$\Sigma_{1,2,3,4}(z,\bm{q})$] for phonon propagators due to the presence of
both CDW and SC orders are

$\displaystyle\Sigma_{1}(\omega,\bm{q})$
$\displaystyle=\frac{g^{2}}{N}\sum_{\bm{k},i\omega_{n}}\left[G_{11}(\bm{k},i\omega_{n})G_{33}(\bm{k}+\bm{q},i\omega_{n}+i\epsilon_{n})+(\bm{k}\rightarrow\bm{k}-\bm{q})\right]$
(3) $\displaystyle\Sigma_{2}(\omega,\bm{q})$
$\displaystyle=\frac{g^{2}}{N}\sum_{\bm{k},i\omega_{n}}\left[G_{12}(\bm{k},i\omega_{n})G_{34}(\bm{k}+\bm{q},i\omega_{n}+i\epsilon_{n})+(\bm{k}\rightarrow\bm{k}-\bm{q})\right]$
$\displaystyle\Sigma_{3}(\omega,\bm{q})$
$\displaystyle=\frac{g^{2}}{N}\sum_{\bm{k},i\omega_{n}}\left[G_{13}(\bm{k},i\omega_{n})G_{31}(\bm{k}+\bm{q},i\omega_{n}+i\epsilon_{n})+(\bm{k}\rightarrow\bm{k}-\bm{q})\right]$
$\displaystyle\Sigma_{4}(\omega,\bm{q})$
$\displaystyle=\frac{g^{2}}{N}\sum_{\bm{k},i\omega_{n}}\left[G_{14}(\bm{k},i\omega_{n})G_{32}(\bm{k}+\bm{q},i\omega_{n}+i\epsilon_{n})+(\bm{k}\rightarrow\bm{k}-\bm{q})\right],$

where $G_{ab}$ with $[a,b=1,2,3,4]$ are the various matrix elements of the
Green’s function matrix defined in Equation (2).

The renormalized phonon propagators can be written using the Dyson equations
obtained by using the above self-energies (Equation (3)). The Dyson equations
are as follows Author1 (1993) (also see Figure 1):

Figure 1: Diagrammatic representations of the Dyson equations for phonons in
the presence of charge density wave (CDW) and superconductivity (SC). The
double wavy lines represent the renormalized phonon modes. Each bubble diagram
corresponds to the self-energy corrections $\Sigma_{1}$, $\Sigma_{3}$,
$\Sigma_{2}$ and $\Sigma_{4}$, respectively, and their analytic expressions
are given in Equation (3).

$\displaystyle\tiny D_{1}(z,\bm{q})$
$\displaystyle=D_{0}(z,\bm{q}+\bm{Q})\bigg{[}1+\Sigma_{1}(z,q)D_{1}(z,\bm{q})+\Sigma_{2}(z,\bm{q})D_{1}(z,\bm{q})$
(4)
$\displaystyle+\Sigma_{3}(z,\bm{q})D_{2}(z,\bm{q})+\Sigma_{4}(z,\bm{q})D_{2}(z,\bm{q})\bigg{]},$
$\displaystyle D_{2}(z,\bm{q})$
$\displaystyle=D_{0}(z,\bm{q}-\bm{Q})\bigg{[}\Sigma_{1}(z,\bm{q})D_{2}(z,\bm{q})+\Sigma_{2}(z,\bm{q})D_{2}(z,\bm{q})$
$\displaystyle+\Sigma_{3}(z,\bm{q})D_{1}(z,\bm{q})+\Sigma_{4}(z,\bm{q})D_{1}(z,\bm{q})\bigg{]},$

where $D_{0}(z,\bm{q})=2\omega_{\bm{q}}/(z^{2}-\omega_{\bm{q}}^{2})$ is the
bare phonon propagator. We obtain the renormalized modes for phonons in the
presence of CDW and SC orders by decoupling Equation (4) through
$D_{\pm}(z,\bm{q})=D_{1}(z,\bm{q})\pm D_{2}(z,\bm{q})$ and then solving
$D_{\pm}(z,\bm{q})$ with the assumption that
$\omega_{\bm{Q}\pm\bm{q}}\approx\omega_{\bm{Q}}$ for small $\bm{q}$. Finally,
plugging in $D_{0}(z,\bm{q})$, we obtain the solutions for the renormalized
phonon propagators as

$\displaystyle
D_{\pm}(z,\bm{q})=\frac{2\omega_{\bm{Q}}}{z^{2}-\omega_{\bm{Q}}^{2}-2\omega_{\bm{Q}}\Sigma_{\pm}(z,\bm{q})},$
(5)

where $\Sigma_{+}=\Sigma_{1}+\Sigma_{2}+\Sigma_{3}+\Sigma_{4}$ and
$\Sigma_{-}=\Sigma_{1}+\Sigma_{2}-\Sigma_{3}-\Sigma_{4}$. The dispersion of
the renormalized phonon modes corresponds to the values of $z$ for which the
denominator of Equation (5) vanishes. Subsequently, keeping only $\bm{q}$
dependence in $\Sigma$, the frequency for each mode is given by

$\displaystyle\Omega_{\pm}^{2}(\bm{q})=\omega_{\bm{Q}}^{2}+2\omega_{\bm{Q}}\Sigma_{\pm}(\bm{q}).$
(6)

We notice, that in the ordered phase where CDW and SC have formed, there are
two normal phonon modes as given by Equation (6); however, we will only focus
on the $\Omega_{-}$ mode Author1 (2021). We also notice from Equation (6) that
the change in the phonon frequency of the $\Omega_{-}$ mode is depicted
through the self-energy correction $\Sigma_{-}(q)$, keeping $\omega_{Q}$
constant. Therefore, the softening of the phonon frequency of the $\Omega_{-}$
mode implies a suppression in the $\Sigma_{-}(q)$. Hence, it is sufficient to
plot the $\Sigma_{-}(q)\equiv\Sigma(q)$ instead of the full $\Omega(q)$, where
$\bm{q}$ measures the deviation from $\bm{Q}[\bm{q}=0]$.

##  III Doping-Induced Fermi Surface Evolution of Phonon Softening in Cuprates

In the last section, we described the general formalism for calculating the
renormalized phonon frequency in the presence of CDW and SC orders. Now, in
this section, we concentrate on the specific case for the hole-doped cuprates.
After decades of research on the phenomenology of the CDW phase, its
correlations with superconductivity and pseudogap phase are strongly debated,
and there is a common belief from theoretical Author1 (2013); Author2 (2014)
and experimental Author1 (2014); Author3 (2014) studies that the CDW wave
vector is incommensurate and supports a momentum space structure where the
Fermi surface plays crucial roles in defining the CDW wave vector $\bm{Q}$.
Within this picture Author1 (2013); Author2 (2014), the CDW wave vector
$\bm{Q}$ is measured by connecting the ‘hot-spots’ ($\bm{k}$-points where the
Antiferromagnetic Brillouin zone boundary intersects the Fermi surface) on the
Fermi surface, as indicated in Figures 2a and 3a by green arrows. In this
work, we consider CDW wave vector $\bm{Q}$, connecting the hot-spots and also
considering them to be bi-axial, following our previous work Author1 (2021)
for our model cuprate systems. Next, following the self-consistent results of
Author3 (2019); Author2 (2020), we consider that the CDW gap $\chi_{k}$ is
maximum ($\chi_{max}$) near the hot-spots and falls off exponentially away
from the hot-spots. The SC gap is chosen such that $\Delta_{k}$ gaps out the
rest of the Fermi surface; however, it vanishes along the nodal direction
$(\pi,\pi)$ of the Fermi surface dictated by a d-wave symmetric SC gap, given
by $\Delta_{k}=(\Delta_{max}/2)[\cos(k_{x})-\cos(k_{y})]$ Author2 (1995).
Below, we separately consider Fermi surfaces’ evolutions of phonon frequencies
for two tight-binding band structures which mimic Bi2Sr2CaCu2O8+δ (BSCCO) and
La2-xSrxCuO4 (LSCO) systems.

Table 1: Table for tight-binding parameters (in eV units) used in Equation
(III.1) and resulting in the six Fermi surfaces, as plotted with solid lines
in Figure 2a. Tight-binding parameters
---
Case: |  $t_{1}$ |  $t_{2}$ |  $t_{3}$ |  $t_{4}$ |  $t_{5}$ |  $\mu$
1 |  -0.7823 |  0.0740 |  -0.0587 |  -0.1398 |  -0.0174 |  0.0801
2 |  -0.7823 |  0.0740 |  -0.0487 |  -0.1398 |  -0.0074 |  0.080
3 |  -0.7823 |  0.0740 |  -0.0287 |  -0.1398 |  -0.0044 |  0.0795
4 |  -0.7823 |  0.0740 |  -0.0187 |  -0.1398 |  -0.0024 |  0.0793
5 |  -0.7823 |  0.0740 |  -0.0087 |  -0.1398 |  -0.0014 |  0.079
6 |  -0.7823 |  0.0740 |  -0.006 |  -0.1398 |  -0.00038 |  0.0789

###  III.1 Case of La2-xSrxCuO4

To capture the generic features of the Fermi surface evolution towards a
topological transition in La2-xSrxCuO4, we consider a six-parameter tight-
binding model band structure Author2 (2021) given by

$\displaystyle\xi_{k}$
$\displaystyle=t_{1}\frac{\cos(k_{x})+\cos(k_{y})}{2}+t_{2}\cos(k_{x})\cos(k_{y})+t_{3}\frac{\cos(2k_{x})+\cos(2k_{y})}{2}$
$\displaystyle+t_{4}\frac{(\cos(2k_{x})\cos(k_{y})+\cos(2k_{y})\cos(k_{x}))}{2}+t_{5}(\cos(2k_{x})\cos(2k_{y}))+\mu.$
(7)

To mimic the doping evolution of the Fermi surface on a qualitative level, we
perform minimal changes in the band structure (Equation (III.1)) by changing
only the hopping parameters $t_{3}$, $t_{5}$ and the chemical potential $\mu$.
A similar evolution of the Fermi surface is seen in the ARPES experiments
Author2 (1998, 2006) with a change in doping. The tight-binding parameters for
six different cases, replicating six different Fermi surfaces considered, are
given in Table 1. However, these tight-binding parameters are not fitted to
the experimentally observed Fermi surfaces. The resulting Fermi surfaces for
these six cases are shown in Figure 2a. We notice in Figure 2a that the Fermi
surface for case 1 is hole-like featuring Fermi arcs. The geometry of the
Fermi surface changes from hole-like to electron-like, going from case 1 to
case 6. We depict the closed electron-like Fermi surface with a dashed black
line in Figure 2a. The tight-binding parameters corresponding to the dashed
black line are $t_{1}=-0.7823$, $t_{2}=0.0740$, $t_{3}=-0.00387$,
$t_{4}=-0.1398$, $t_{5}=-0.00037$ and $\mu=0.0788$. To compute the
renormalized phonon frequency for the above six cases in Table 1, we choose
$\chi_{max}$ and $\Delta_{max}$ to be equal to 0.1 eV, a realistic value.

The computed phonon self-energies $\Sigma(q)$ as a function of $\bm{q}$ are
plotted in Figure 2b. First of all, we observe that $\Sigma(q)$ decreases
strongly around $\bm{q}=0$, implying a phonon softening around $\bm{Q}$. Also,
with going away from $\bm{q}=0$, suppression in $\Sigma(q)$ is diminished,
suggesting that phonon softening is reduced away from $\bm{Q}$. Next, we
closely look at the evolution of the $\Sigma(q)$ from case 1 to case 5. We
observe that the strength of phonon softening remains more or less equal, as
there are not many changes in $\Sigma(q)$. However, if we notice the Fermi
surfaces corresponding to cases 1 to 5 in Figure 2a, we observe that the hole-
like Fermi surface continuously changes towards an electronic Fermi surface.
Very interestingly, we notice that for case 6, where the Fermi surface (Figure
2a) is in close vicinity to the topological Lifshitz transition, the strength
of the phonon softening (Figure 2b) is significantly suppressed as compared to
the earlier five cases. Hence, in the case for the model LSCO band, the
strength of phonon softening changes discontinuously, remaining initially
unaffected by the change in the geometry of the Fermi surface and then
suddenly diminishing when the Fermi surface is very close to the Lifshitz
transition .

Figure 2: (a) Fermi surfaces obtained from band structure Equation (III.1)
with different tight-binding parameters as given in Table 1. The Fermi surface
evolves from hole-like to electron-like with change in the parameters. The
electron-like Fermi surface is plotted with black dashed line. The tight-
binding parameters corresponding to the black dashed line are given in the
main text. The CDW wave vector ($\bm{Q}$) is indicated with the green arrow.
(b) Phonon softening: plots for the evolution of the self-energy $\Sigma(q)$
as a function of $\bm{q}$ for the same sets of the tight-binding parameters
used in (a) and given in Table 1.

###  III.2  Case of Bi2Sr2CaCu2O8+δ

In order to capture the broad features of the evolution of the Fermi surface
from hole-like to electron-like for Bi2Sr2CaCu2O8+δ, we consider a six-
parameter tight-binding model band structure Author1 (2003) given by

$\displaystyle\xi_{k}$
$\displaystyle=t_{1}\frac{\cos(k_{x})+\cos(k_{y})}{2}+t_{2}\cos(k_{x})\cos(k_{y})+t_{3}\frac{\cos(2k_{x})+\cos(2k_{y})}{2}$
$\displaystyle+t_{4}\frac{(\cos(2k_{x})\cos(k_{y})+\cos(2k_{y})\cos(k_{x}))}{2}+t_{5}(\cos(2k_{x})\cos(2k_{y}))+\mu,$
(8)

with $t_{1}=-0.5908$, $t_{2}=0.0962$, $t_{3}=-0.1306$, $t_{4}=-0.0507$ and
$t_{5}=0.0939$. Here, for simplicity, we only change the chemical potential
$\mu$ to capture the qualitative behavior of the topological transition of the
Fermi surface, as observed in the ARPES experiment Author1 (2006); thus, the
model band is not expected to exactly fit the ARPES bands Author1 (2006). The
tight-binding parameters for the band are in eV units. The resulting Fermi
surfaces for different $\mu$ values are displayed in Figure 3a. We notice that
the geometry of the Fermi surface continuously evolves from hole-like to
electron-like with increasing $\mu$ continuously from $0.0789$ to $0.1189$. On
further increasing $\mu$, the Fermi surface goes through a topological
Lifshitz transition. The closed electron-like Fermi surface corresponding to a
$\mu$ value of $0.1329$ is plotted with a dashed black line in Figure 3a.

To study the renormalized phonon frequency, we choose both $\Delta_{max}$ and
$\chi_{max}$ to be 0.02 eV, a value realistically obtained. In Figure 3b, we
plot the self-energy $\Sigma(q)$ as a function of $\bm{q}$ for each of the
$\mu$ values in Figure 3a. We notice that for a $\mu$ value of $0.0789$, the
value of the $\Sigma(q)$ is suppressed around $\bm{q}=0$, and the suppression
is strongest at $\bm{q}=0$, that is, at the CDW wave vector. This implies that
there is a phonon softening around the CDW wave vector. Once we go away from
$\bm{q}=0$, the phonon softening is significantly reduced as the suppression
in $\Sigma(q)$ is diminished. Next, we see that with an increase in the value
of $\mu$, the phonon softening becomes diminished continuously as the
suppression in $\Sigma(q)$ around $\bm{q}=0$ becomes smaller with an increase
in $\mu$ values. This implies that for the case of the model BSCCO band, the
phonon softening continuously diminishes as the hole-like Fermi surface
evolves continuously towards the topological Lifshitz transition. This
continuous evolution is in stark contrast to the case for the model LSCO band,
where the phonon softening changes discontinuously as the hole-like Fermi
surface evolves towards the topological transition.

Figure 3: (a) Tight-binding Fermi surfaces obtained from Equation (III.2) for
different values of $\mu$. A continuous evolution from hole-like to electron-
like Fermi surface can be seen with increase in $\mu$. Electron-like Fermi
surface is shown in black dashed line for $\mu=0.1329$. The CDW wave vector
($\bm{Q}$) is indicated with the green arrow. (b) Phonon softening: plots of
the self-energy $\Sigma(q)$ of phonons as a function of $\bm{q}$ and for the
same values of $\mu$ used in (a).

##  IV Discussion

In this work, applying a perturbative approach, we study the Fermi surface
evolution of the phonon softening in model cuprate bands, specially focusing
on the parameter regimes where the Fermi surface approaches a topological
Lifshitz transition. In particular, we focus on two cuprate systems LSCO and
BSCCO, where ARPES experiments also observed a topological transition from a
hole-like Fermi surface to an electron-like Fermi surface due to a change in
doping.

In this work, to only focus on the possible effects of the change in the
geometry of the Fermi surface due to doping on phonon softening, we have
excluded the doping dependencies of the CDW and SC gaps. Furthermore, as the
theoretical tight-binding parameters are not fitted to experimental Fermi
surfaces, the doping levels ($x$) (as shown in Figure 4a,b) and number of
holes ($p=1+x$) per Cu atom associated with the parameters chosen in this work
might differ from the experimental doping values where phonon softening
occurs.

For the case of the model system representing BSCCO, we have found that the
phonon softening is continuously suppressed (Figure 3b) with the continuous
evolution of the Fermi surface from a hole-like to an electron-like geometry,
and it is significantly suppressed close to the topological transition. For
the model system of LSCO, we have found two features in the evolution of the
phonon softening. First, the phonon softening remains initially unaffected,
even when the Fermi surface continuously changes from a hole-like towards an
electron-like geometry, which is in sharp contrast to the case of BSCCO.
Second, only close to the topological transition of the Fermi surface, the
phonon softening becomes significantly suppressed. This difference in the
nature of evolution of the phonon softening between these two materials can be
mainly attributed to the difference in the band structure and associated
differences in the nesting properties of the Fermi surfaces in these systems.

Figure 4: (a) Plot of the hole doping ($x$) versus chemical potential ($\mu$)
for the model tight-binding band in Equation (III.2) imitating BSCCO. The last
point represents the doping around the topological Lifshitz transition in the
theoretical model. (b) Plot of the hole doping ($x$) versus tight-binding
parameters (shown in Table 1) for the model band in Equation (III.1) imitating
LSCO. The last point represents the doping around the topological Lifshitz
transition in the theoretical model.

It would also be interesting to compare the results obtained from the
theoretical models to the experiments. However, these comparisons must be
considered at a qualitative level. A recent RIXS experiment Author6 (2022)
measured phonon softening in BSCCO with a change in doping at Tc and below Tc.
They found a gradual suppression of the phonon softening with an increase in
doping from an under-doped to an over-doped regime at Tc. At a low
temperature, the experiment again found a gradual suppression of phonon
softening with an increase in doping in a broad window between the under-doped
and over-doped regime, with one exception in the highly under-doped regime.
From the calculations performed in this work for BSCCO bands, we have found a
gradual suppression of phonon softening with an increase in doping, which is
in general agreement to the above experimental findings. Noticeably, we have
not found any enhancement of phonon softening in the under-doped regime within
the parameter regime explored in this work. Therefore, it would be interesting
to investigate whether incorporating a more accurate description of the band
and doping dependence of the CDW and the SC orders and also the effects of
CDFs can result in such an enhancement, as seen in the extreme under-doped
case in the experiment.

Another recent RIXS experiment Author1 (2020) measured the evolution of the
phonon softening in LSCO in a broad doping regime across the topological
Lifshitz transition. The experimental results suggest that the strength of the
phonon softening associated with the CDW remains almost the same with a
gradual increase in the doping, even in the vicinity of the Lifshitz
transition, but it abruptly vanishes at a critical doping, where eventually
the CDW order disappears. By taking into account a theoretical model of a
tight-binding band for LSCO and coupling between phonons with bi-axial CDW and
SC, we have found that the strength of phonon softening remains approximately
the same while the Fermi surface evolves continuously towards the Lifshitz
transition. This bears a resemblance to the experimental observations.
Moreover, we have also found that close to the topological Lifshitz
transition, the phonon softening is abruptly suppressed. This feature is in
contrast to the experimental observations.

In the context of LSCO, we now discuss some aspects which need further
investigation. The RIXS experiment Author1 (2020) in LSCO suggests a CDW-
driven phonon softening, implying that in the doping regime considered in the
experiment, there might be no signature of the magnetic order and only CDW
correlations survive, which is also in accordance with previous experimental
observations Author6 (2019, 2023). Hence, our theoretical model which involves
coupling between phonons, CDW and SC is a consistent model for describing
phonon softening in the vicinity of the topological transition in this
material. However, the effects of uni-axial stripy CDW Author4 (1999); Author6
(1995) pertinent to LSCO, which are not currently considered in our work, can
play crucial roles in describing the differences. Additionally, more accurate
descriptions of the doping dependence of bands and the CDW and the SC orders
can play unique roles. It should also be noted that while for LSCO, the
experimental Author1 (2020) wave vector corresponding to the strongest phonon
softening slightly differs from the CDW wave vector ($\bm{Q}$), within our
theoretical framework, the strongest phonon softening always occurs at the CDW
wave vector $\bm{Q}$. This can be related to the fact that for simplicity we
have considered in our model an electron–phonon coupling constant which is
independent of both fermion and phonon momenta, while for the breathing modes
in cuprates, the electron–phonon coupling can in general depend on the momenta
Author9 (2023) and can result in such an offset of the wave vector.

Besides the doping dependence, another vital aspect of phonon softening in
cuprates is their temperature evolution. The RIXS experiment in BSCCO Author6
(2022) observed an enhancement in the phonon softening between Tc and low
temperature for some doping regime. The RIXS experiment Author1 (2020) in LSCO
measured phonon softening at Tc and above Tc. Curiously, for both BSCCO
Author1 (2021) and LSCO Author1 (2020), the phonon softening still survives
above Tc. To theoretically account for these behaviors at Tc and above, one
needs to include the effects of thermal fluctuations, the short-range nature
of the CDW and also the possible coupling between CDFs and the lattice, which
all can play important roles. In an earlier work Author1 (2021), it was indeed
found that fluctuation-driven damping effects and the interplay between the
temperature dependence of CDW, SC and the damping can capture some crucial
aspects of the temperature evolution of the phonon softening in the under-
doped cuprates. As in this work, we work at the mean-field level, and the
effects of higher temperatures cannot be properly accounted for and can also
be partially responsible for the differences. Therefore, it will be an
interesting future direction to theoretically study the combined temperature
and doping evolution of the phonon softening in both BSCCO and LSCO.

##  V Conclusions

To summarize, in this work, by employing a perturbative technique, we have
theoretically studied the effect of Fermi surface evolution close to the
topological Lifshitz transition on phonon softening for two simple model
tight-binding band structures broadly capturing doped LSCO and doped BSCCO
systems in the presence of a charge density wave and superconductivity. We
have found interesting features of phonon softening in these two materials and
have compared them with experimental observations.

###### Acknowledgements.

This work was supported by the Office of Basic Energy Sciences, Material
Sciences and Engineering Division, U. S. Department of Energy, under Contract
No. DE-SC0012704.

## References

  * Author1 (2021) Sarkar, S.; Grandadam, M.; Pepin, C. Anomalous softening of phonon dispersion in cuprate superconductors. Phys. Rev. Res. 2021, 3, 013162.
  * Author4 (2002) Hoffman, J.E.; Hudson, E.W.; Lang, K.M.; Madhavan, V.; Eisaki, H.; Uchida, S.I.; Davis, J.C. A four unit cell periodic pattern of quasi-particle states surrounding vortex cores in Bi2Sr2CaCu2O8+δ. Science 2002, 295, 466–469.
  * Author4 (2012) Ghiringhelli, G.; Le Tacon, M.; Minola, M.; Blanco-Canosa, S.; Mazzoli, C.; Brookes, N.B.; De Luca, G.M.; Frano, A.; Hawthorn, D.G.; He, F.; et al. Long-range incommensurate charge fluctuations in (Y, Nd) Ba2Cu3O6+x. Science 2012, 337, 821–825.
  * Author4 (200) Tsuei, C.C.; Kirtley, J.R. Pairing symmetry in cuprate superconductors. Rev. Mod. Phys. 2000, 72, 969.
  * Author1 (2015) Fradkin, E.; Kivelson, S.A.; Tranquada, J.M. Colloquium: Theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 2015, 87, 457.
  * Author1 (1989) Alloul, H.; Ohno, T.; Mendels, P. 89Y NMR Evidence for a Fermi-Liquid Behavior in YBa2Cu3O6+x. Phys. Rev. Lett. 1989, 63, 1700.
  * Author1 (1998) Renner, C.; Revaz, B.; Genoud, J.-Y.; Kadowaki, K.; Fischer, O. Pseudogap Precursor of the Superconducting Gap in Under and Overdoped Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 1998, 80, 149.
  * Author2 (1998) Ino, A.; Mizokawa, T.; Kobayashi, K.; Fujimori, A.; Sasagawa, T.; Kimura, T.; Kishio, K.; Tamasaku, K.; Eisaki, H.; Uchida, S. Doping Dependent Density of States and Pseudogap Behavior in La2-xSrxCuO4. Phys. Rev. Lett. 1998, 81, 2124.
  * Author4 (1999) Emery, V.J.; Kivelson, S.A.; Tranquada, J.M. Stripe phases in high-temperature superconductors. Proc. Natl. Acad. Sci. USA 1999, 96, 8814–8817.
  * Author4 (2006) Fauque, B.; Sidis, Y.; Hinkov, V.; Pailhes, S.; Lin, C.T.; Chaud, X.; Bourges, P. Magnetic order in the pseudogap phase of high-Tc superconductors. Phys. Rev. Lett. 2006, 96, 197001.
  * Author2 (2019) Sarkar, S.; Chakraborty, D.; Pepin, C. Incipient loop-current order in the underdoped cuprate superconductors. Phys. Rev. B 2019, 100, 214519.
  * Author1 (2011) He, R.H.; Hashimoto, M.; Karapetyan, H.; Koralek, J.D.; Hinton, J.P.; Testaud, J.P.; Nathan, V.; Yoshida, Y.; Yao, H.; Tanaka, K.; et al. From a single-band metal to a high-temperature superconductor via two thermal phase transitions. Science 2011, 331, 1579–1583.
  * Author6 (2019) Wen, J.J.; Huang, H.; Lee, S.J.; Jang, H.; Knight, J.; Lee, Y.S.; Fujita, M.; Suzuki, K.M.; Asano, S.; Kivelson, S.A.; et al. Observation of two types of charge-density-wave orders in superconducting La2-xSrxCuO4. Nat. Commun. 2019, 10, 3269.
  * Author6 (2023) von Arx, K.; Wang, Q.; Mustafi, S.; Mazzone, D.G.; Horio, M.; Mukkattukavil, D.J.; Pomjakushina, E.; Pyon, S.; Takayama, T.; Takagi, H.; et al. Fate of charge order in overdoped La-based cuprates. NPJ Quantum Mater. 2023, 8, 7.
  * Author2 (2023) Li, Q.; Huang, H.Y.; Ren, T.; Weschke, E.; Ju, L.; Zou, C.; Zhang, S.; Qiu, Q.; Liu, J.; Ding, S.; et al. Prevailing Charge Order in Overdoped La2-xSrxCuO4 beyond the Superconducting Dome. Phys. Rev. Lett. 2023, 131, 116002.
  * Author3 (2018) Peng, Y.Y.; Fumagalli, R.; Ding, Y.; Minola, M.; Caprara, S.; Betto, D.; Bluschke, M.; De Luca, G.M.; Kummer, K.; Lefrançois, E.; et al. Re-entrant charge order in overdoped (Bi,Pb)2.12Sr1.88CuO6+δ outside the pseudogap regime. Nat. Mater. 2018, 17, 697–702.
  * Author7 (2023) Arpaia, R.; Martinelli, L.; Sala, M.M.; Caprara, S.; Nag, A.; Brookes, N.B.; Camisa, P.; Li, Q.; Gao, Q.; Zhou, X.; et al. Signature of quantum criticality in cuprates by charge density fluctuations. Nat. Commun. 2023, 14, 7198.
  * Author7 (2019) Arpaia, R.; Caprara, S.; Fumagalli, R.; De Vecchi, G.; Peng, Y.Y.; Andersson, E.; Betto, D.; De Luca, G.M.; Brookes, N.B.; Lombardi, F.; et al. Dynamical charge density fluctuations pervading the phase diagram of a Cu-based high-Tc superconductor. Science 2019, 365, 906–910.
  * Author2 (2006) Yoshida, T.; Zhou, X.J.; Tanaka, K.; Yang, W.L.; Hussain, Z.; Shen, Z.X.; Fujimori, A.; Sahrakorpi, S.; Lindroos, M.; Markiewicz, R.S.; et al. Systematic doping evolution of the underlying Fermi surface of La2-xSrxCuO4. Phys. Rev. B 2006, 74, 224510.
  * Author2 (1998) Fujimori, A.; Ino, A.; Mizokawa, T.; Kim, C.; Shen, Z.X.; Sasagawa, T.; Kimura, T.; Kishio, K.; Takaba, M.; Tamasaku, K.; et al. Chemical potential shift, density of states and Fermi surfaces in overdoped and underdoped La2-xSrxCuO4. J. Phys. Chem. Solids 1998, 59, 1892–1896.
  * Author8 (2023) Zhong, Y.; Chen, Z.; Chen, S.D.; Xu, K.J.; Hashimoto, M.; He, Y.; Uchida, S.I.; Lu, D.; Mo, S.K.; Shen, Z.X.; et al. Differentiated roles of Lifshitz transition on thermodynamics and superconductivity in La2-xSrxCuO4. Proc. Natl. Acad. Sci. USA 2023, 119, e2204630119.
  * Author1 (2006) Kaminski, A.; Rosenkranz, S.; Fretwell, H.M.; Norman, M.R.; Randeria, M.; Campuzano, J.C.; Park, J.M.; Li, Z.Z.; Raffy, H. Change of Fermi-surface topology in Bi2Sr2CaCu2O8+δ with doping. Phys. Rev. B 2006, 73, 174511.
  * Author1 (2019) Loret, B.; Auvray, N.; Gallais, Y.; Cazayous, M.; Forget, A.; Colson, D.; Julien, M.-H.; Paul, I.; Civelli, M.; Sacuto, A. Intimate link between charge density wave, pseudogap and superconducting energy scales in cuprates. Nat. Phys. 2019, 15, 771–775.
  * Author1 (1963) Woll, E.J., Jr.; Kohn, W. Images of the fermi surface in phonon spectra of metals. Phys. Rev. 1962, 126, 1693.
  * Author1 (1973) Renker, B.; Rietschel, H.; Pintschovius, L.; Gläser, W.; Brüesch, P.; Kuse, D.; Rice, M.J. Observation of Giant Kohn Anomaly in the One-Dimensional Conductor K2Pt(CN)4 Br0.3·3H2O. Phys. Rev. Lett. 1973, 30, 1144.
  * Author1 (2014) Le Tacon, M.; Bosak, A.; Souliou, S.M.; Dellea, G.; Loew, T.; Heid, R.; Bohnen, K.P.; Ghiringhelli, G.; Krisch, M.; Keimer, B. Inelastic X-ray scattering in YBa2Cu3O6.6 reveals giant phonon anomalies and elastic central peak due to charge-density-wave formation. Nat. Phys. 2014, 10, 52–58.
  * Author1 (2021) Lee, W.S.; Zhou, K.J.; Hepting, M.; Li, J.; Nag, A.; Walters, A.C.; Garcia-Fernandez, M.; Robarts, H.C.; Hashimoto, M.; Lu, H.; et al. Spectroscopic fingerprint of charge order melting driven by quantum fluctuations in a cuprate. Nat. Phys. 2021, 17, 53–57.
  * Author1 (2006) Reznik, D.; Pintschovius, L.; Ito, M.; Iikubo, S.; Sato, M.; Goka, H.; Fujita, M.; Yamada, K.; Gu, G.D; Tranquada, J.M. Electron–phonon coupling reflecting dynamic charge inhomogeneity in copper oxide superconductors. Nature 2006, 440, 1170–1173.
  * Author1 (1999) McQueeney, R.J.; Petrov, Y.; Egami, T.; Yethiraj, M.; Shirane, G.; Endoh, Y. Anomalous dispersion of LO phonons in La1.85Sr0.15CuO4 at low temperatures. Phys. Rev. Lett. 1999, 82, 628.
  * Author1 (2020) Lin, J.Q.; Miao, H.; Mazzone, D.G.; Gu, G.D.; Nag, A.; Walters, A.C.; García-Fernández, M.; Barbour, A.; Pelliciari, J.; Jarrige, I.; et al. Strongly correlated charge density wave in La2-xSrxCuO4 evidenced by doping-dependent phonon anomaly. Phys. Rev. Lett. 2020, 124, 207005.
  * Author7 (2021) Wang, Q.; von Arx, K.; Horio, M.; Mukkattukavil, D.J.; Küspert, J.; Sassa, Y.; Schmitt, T.; Nag, A.; Pyon, S.; Takayama, T.; et al. Charge order lock-in by electron-phonon coupling in La1.675Eu0.2Sr0.125CuO4. Sci. Adv. 2021, 7, eabg7394.
  * Author6 (2022) Lu, H.; Hashimoto, M.; Chen, S.D.; Ishida, S.; Song, D.; Eisaki, H.; Nag, A.; Garcia-Fernandez, M.; Arpaia, R.; Ghiringhelli, G.; et al. Identification of a characteristic doping for charge order phenomena in Bi-2212 cuprates via RIXS. Phys. Rev. B 2022, 106, 155109.
  * Author5 (2020) Pepin, C.; Chakraborty, D.; Grandadam, M.; Sarkar, S. Fluctuations and the Higgs mechanism in underdoped cuprates. Annu. Rev. Condens. Matter Phys. 2020, 11, 301–323.
  * Author1 (1993) Lee, P.A.; Rice, T.M.; Anderson, P.W. Conductivity from charge or spin density waves. Solid State Commun. 1993, 88, 1001.
  * Author1 (2013) Efetov, K.B., Meier, H. and Pepin, C. Pseudogap state near a quantum critical point. Nat. Phys. 2013, 9, 442–446.
  * Author2 (2014) Wang, Y.; Chubukov, A. Charge-density-wave order with momentum (2 Q, 0) and (0, 2 Q) within the spin-fermion model: Continuous and discrete symmetry breaking, preemptive composite order, and relation to pseudogap in hole-doped cuprates. Phys. Rev. B 2014, 90, 035149.
  * Author1 (2014) da Silva Neto, E.H.; Aynajian, P.; Frano, A.; Comin, R.; Schierle, E.; Weschke, E.; Gyenis, A.; Wen, J.; Schneeloch, J.; Xu, Z.; et al. Ubiquitous interplay between charge ordering and high-temperature superconductivity in cuprates. Science 1993, 343, 393–396.
  * Author3 (2014) Comin, R.; Frano, A.; Yee, M.M.; Yoshida, Y.; Eisaki, H.; Schierle, E.; Weschke, E.; Sutarto, R.; He, F.; Soumyanarayanan, A.; He, Y. Charge order driven by Fermi-arc instability in Bi2Sr2-xLaxCuO6+δ. Science 2014, 342, 390–392.
  * Author2 (2020) Grandadam, M.; Chakraborty, D.; Montiel, X.; Pépin, C. Electronic spectral function in the fractionalized pair density wave scenario. Phys. Rev. B 2020, 102, 121104(R).
  * Author3 (2019) Chakraborty, D.; Grandadam, M.; Hamidian, M.H.; Davis, J.C.S. Sidis, Y.; Pépin, C. Fractionalized pair density wave in the pseudogap phase of cuprate superconductors. Phys. Rev. B 2019, 100, 224511.
  * Author2 (1995) Scalapino, D.J. The case for d${}_{x^{2}-y^{2}}$ pairing in the cuprate superconductors. Phys. Rep. 1995, 250, 329–365.
  * Author2 (2021) Chakraborty, D.; Black-Schaffer, A.M. Odd-frequency pair density wave correlations in underdoped cuprates. New J. Phys. 2021, 23, 033001.
  * Author1 (2003) Eschrig, M.; Norman, M.R. Effect of the magnetic resonance on the electronic spectra of high Tc superconductors. Phys. Rev. B 2003, 67, 144503.
  * Author9 (2023) Devereaux, T.P.; Cuk, T.; Shen, Z.X.; Nagaosa, N. Anisotropic electron-phonon interaction in the cuprates. Phys. Rev. Lett. 2004, 93, 117004.
  * Author6 (1995) Tranquada, J.M.; Sternlieb, B.J.; Axe, J.D.; Nakamura, Y.; Uchida, S.I. Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 1995, 375, 561–563.