# Interplay between the charge density wave phase and a pseudogap under
antiferromagnetic correlations

Leonardo Prauchner1∗, Eleonir Calegari2, Julian Faúndez1, Sergio Magalhaes1
1Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970,
Porto Alegre, RS, Brazil  2Departamento de Física - Universidade Federal de
Santa Maria, 97105-900, Santa Maria, RS, Brazil<EMAIL_ADDRESS>
###### Abstract

In this study, we explore the impact of short-range antiferromagnetic
correlations on the charge density wave (CDW) phase in strongly correlated
electron systems exhibiting the pseudogap phenomenon. Our investigation
employs a n-pole approximation to consider the repulsive Coulomb interaction
$(U)$ and antiferromagnetic correlations. Utilizing a two-dimensional Hubbard
model for the Coulomb interaction and a BCS-like model for the CDW order
parameter, we observe that an increase in $U$ enhances antiferromagnetic
fluctuations, resulting in a flattened re-normalized band around the anti-
nodal point $(\pi,0)$. The pseudogap manifests in the band structure and
density of states, prompting an exploration across various $U$ and occupation
number values. Our findings indicate that antiferromagnetic correlations
significantly influence the CDW state, as the Fermi surface is reconstructed
within the ordered phase. Furthermore, we find a Lifhsitz transition inside
both the CDW phase and the normal state, with the latter preceding the onset
of the pseudogap.

##  1 Introduction

The pseudogap (PG) is a very enigmatic, and yet, not fully understood
phenomenon. Characterized by suppression of states around the Fermi level
below a certain temperature $T^{*}$ [1, 2], as well as a topological
transition on the Fermi Surface (FS), which assumes a pocket-like shape [3, 4,
5, 6, 7, 8]. This phenomenon is generally observed in strongly correlated
electron systems, such as the cuprates [9] and pnictides [10]. However, we can
also highlight Selenium-based compounds, in particular, 1T - TaSe2 [11],
capable of showing both PG and charge density wave (CDW) [12] when the system
is quasi-2D. In some cases, these compounds also have a superconducting order
[13, 14, 15] displaying a phase diagram similar to the cuprates [16].
Additionally, we mention infinite-layered nickelates [17, 18], where the PG
phenomenon is accompanied by the emergence of superconductivity and,
eventually, CDW.

On the other hand, a plethora of mechanisms have been proposed to explain the
onset and nature of the PG phenomenon [19, 20, 21, 22, 23, 24, 25, 26, 27,
28]. In cuprates, the short-range magnetic correlations due to the vicinity of
the antiferromagnetic phase in the underdoped region can be of utmost
importance to clarify the entire PG phenomenon. For instance, there is a
change from the hole to electron-like FS topology, the so-called Lifshitz
transition (LT). This transition happens when the system is still doped by
holes. Moreover, this transition precedes the collapse of the FS to a pocket
shape, characterizing the onset of the PG [29, 30, 31, 32]. Surprisingly, for
temperatures below $T^{*}$, there is a CDW transition accompanied by a FS
reconstruction [33, 34]. There is some evidence that short-range
antiferromagnetic correlations may be present within the CDW [35, 36, 37, 38].
Thus, it is possible to speculate whether such correlations may be involved in
this FS reconstruction. It is also important to highlight that the role of
short-range antiferromagnetic correlations may not be exclusive to cuprates,
since the same origin of PG is considered in nickelates [39]. The same
scenario could be cogitated about the two-dimensional transition metal
dichalcogenides (TMDs) since a Mott insulator behavior is observed [40]. Thus,
a question arises: how short-range antiferromagnetic correlations can
influence an interplay between the CDW and the entire PG phenomenon?

As a proposal to answer the question above, we start from a BCS-like mean
field theory within the Green’s function equation of motion formalism to
describe the CDW instability. Then, we replace the normal state uncorrelated
Green’s function with a correlated one, obtained through the n-pole
approximation [41], applied to the single-band Hubbard model [42]. That is a
way to consider short-range antiferromagnetic correlations in the normal
state. The n-pole approximation was proposed as a correction to the Hubbard-I
approximation [43], which is unable to capture magnetic solutions and
antiferromagnetic correlations. The limitation of the Hubbard I approximation
is due to the absence of quantities such as the spin-spin correlation
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, and the double occupation
$\langle N_{i}N_{j}\rangle$. Those correlations compose the band shift
$(Y_{\vec{k},\sigma})$ which appears in the n-pole approximation. In our
theory, we highlight the importance of the next nearest neighbors hopping
$(t_{1})$. It plays a fundamental role along with the occupation number
$(n_{T})$ and the Coulomb interaction $(U)$, affecting $Y_{\vec{k},\sigma}$
through $\langle N_{i}N_{j}\rangle$ and mainly
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, distorting the bands. As a
consequence, there is a flattening of the band around the $(\pi,0)$ point,
which originates a Van Hove singularity (VHS) and can favor pair formation and
stabilize the CDW below a critical temperature $(k_{B}T_{C})$ [44, 45, 46].

Besides, we shall demonstrate that the short-range antiferromagnetic
fluctuations affecting $Y_{\vec{k},\sigma}$ can also favor the opening of a
PG. There is a displacement of accessible states to the VHS, causing
suppression of states in adjacent energy regions [47, 48]. Moreover, the
distortion of the bands eventually can cause a Lifshitz transition (LT) [49,
50, 51]. This scenario can be understood from the perspective of the density
of states (DOS). The position of the VHS relative to the Fermi level reflects
the change in the FS topology when it goes through it, changing from a hole-
like topology (from lower energy levels) to an electron-like one. Indeed, the
temperature effects are crucial in our theory since the shape of the DOS is
affected by the VHS peak structure, which depends if the system is in a normal
state, or within the ordered phase.

For this work, we chose a square lattice and set the CDW order form factor as
an unconventional d-wave symmetry, in contrast with the usual s-wave gap
symmetry that has been explored in some cuprates. In cuprates and TMDs, the
CDW coexists with another competing order [52, 53], such as the
superconductivity [54]. This choice of symmetry is not arbitrary, as both
s-wave and d-wave cases have been observed [55, 56], and can even compete,
making it difficult to determine the predominant form factor [57, 58].
However, we only investigated the $d-$wave symmetry case. Our theory does not
necessarily assume a phononic mechanism for electron-hole pair formation.
Indeed, several authors have suggested that strongly correlated electron
systems may have different origins for the phase transition from the normal
state to CDW [59, 60, 61, 62, 63].

This paper is organized as follows. The model, and the equations obtained with
the chosen method, are presented in section 2. The numerical results are
presented and discussed in section 3. Conclusions and further remarks are
shown at the end.

##  2 Model

We studied the effects of antiferromagnetic correlations on the CDW phase,
employing a BCS-like model [64]:

$\begin{split}\mathcal{H}=\sum_{\vec{k}}\xi_{\vec{k}}\bigg{[}c_{\vec{k},\uparrow}^{\dagger}c_{\vec{k},\uparrow}+c_{\vec{k},\downarrow}^{\dagger}c_{\vec{k},\downarrow}\bigg{]}+\sum_{\vec{k}}W_{\vec{k}}\bigg{[}c_{{\vec{k}+\vec{Q}},\uparrow}^{\dagger}c_{\vec{k},\uparrow}+c_{{\vec{k}}+{\vec{Q}},\downarrow}^{\dagger}c_{\vec{k},\downarrow}\bigg{]}+H_{0}\end{split},$
(1)

where $W_{\vec{k}}$ is the CDW order parameter,
$\xi_{\vec{k}}=\varepsilon_{\vec{k}}-\mu=2t_{0}(\cos{k_{x}}+\cos{k_{y}})+4t_{1}\cos{k_{x}}\cos{k_{y}}-\mu$
is the square lattice quasi-particle dispersion relation, $\mu$ is the
chemical potential and $H_{0}=\sum_{\vec{k}}|W_{\vec{k}}|^{2}/|V|$. The CDW
order parameter is obtained through the following relation:

$W_{\vec{k}}=|V|\gamma_{\vec{k}}\sum_{\vec{k^{\prime}}}\langle
c_{\vec{k}+\vec{Q}^{\prime},\sigma}^{\dagger}c_{\vec{k}^{\prime},\sigma}\rangle,$
(2)

$\gamma_{\vec{k}}$ is the $d$-wave symmetry factor and $|V|$ is the attractive
pairing potential absolute value. We propose that $W_{\vec{k}}$ can model the
CDW gap, as it represents the creation of an electron-hole pair, modulated by
the nesting vector $Q=(\pi,\pi)$. We do not consider the Debye cutoff energy
and a self-consistent non $\vec{k}$-dependent renormalizing term [64], since
the pairing mechanism is not defined as phononic. The Hamiltonian can be
written in the quadratic form as:

$\mathcal{H}=\Psi^{\dagger}\mathcal{H}_{\vec{k}}\Psi+H_{0},$ (3)

where $\Psi$ is a Nambu spinor, given by:

$\Psi^{\dagger}=\begin{pmatrix}c^{\dagger}_{\vec{k},\uparrow}&c_{-\vec{k},\downarrow}&c^{\dagger}_{\vec{k}+\vec{Q},\uparrow}&c_{-\vec{k}+\vec{Q},\downarrow}\end{pmatrix}.$
(4)

The coefficients matrix $\mathcal{H}_{\vec{k}}$ is needed, and was obtained
through the use of Green’s functions in the Zubarev’s formalism [65], whose
equation of motion is:

$\omega\langle\langle\hat{A};\hat{B}\rangle\rangle=\langle[\hat{A},\hat{B}]_{+}\rangle+\langle\langle[\hat{A},\mathcal{H}];\hat{B}\rangle\rangle.$
(5)

We can apply the last equation to the set of operators, starting with
$\langle\langle
c_{\vec{k},\uparrow};c^{\dagger}_{\vec{k},\uparrow}\rangle\rangle$:

$\omega\langle\langle
c_{\vec{k},\uparrow};c_{\vec{k},\uparrow}^{\dagger}\rangle\rangle=1+\xi_{\vec{k}}\langle\langle
c_{\vec{k},\uparrow};c_{\vec{k},\uparrow}^{\dagger}\rangle\rangle+W_{\vec{k}+\vec{Q}}\langle\langle
c_{\vec{k}+\vec{Q},\uparrow};c_{\vec{k},\uparrow}^{\dagger}\rangle\rangle,$
(6)

and repeating the procedure for the remaining operators, results in the
coefficients matrix $\mathcal{H}_{\vec{k}}$:

$\mathcal{H}_{\vec{k}}=\begin{pmatrix}\xi_{\vec{k}}&0&W_{\vec{k}}&0\\\
0&-\xi_{\vec{k}}&0&-W_{\vec{k}}\\\
W_{\vec{k}+\vec{Q}}&0&\xi_{\vec{k}+\vec{Q}}&0\\\
0&-W_{\vec{k}+\vec{Q}}&0&-\xi_{\vec{k}+\vec{Q}}\end{pmatrix}.$ (7)

Thus, the equation of motion for the Green’s function matrix
$\mathcal{G}(\vec{k},\omega)$, whose elements are given by
$\mathcal{G}_{l,s}=\langle\langle\hat{A}_{l};\hat{B}_{s}\rangle\rangle$, can
be solved, with $\mathcal{I}$ as a fourth-order identity matrix.

$(\omega\mathcal{I}-\mathcal{H}_{\vec{k}})\mathcal{G}(\vec{k},\omega)=\mathcal{I}.$
(8)

In the normal state, $W_{\vec{k}}=0$, so we define a new quantity,
corresponding to the ground state uncorrelated Green’s function
$G_{0}(\vec{k},\omega)$:

$G_{0}(\vec{k},\omega)=(\omega-\xi_{\vec{k}})^{-1}.$ (9)

This new quantity appears in the main diagonal of
$\mathcal{G}(\vec{k},\omega)$, so we rewrite it, which allows us to reevaluate
the coefficients matrix.

$\mathcal{G}(\vec{k},\omega)=\begin{pmatrix}G_{0}^{-1}(\vec{k},\omega)&0&-W_{\vec{k}}&0\\\
0&G_{0}^{-1}(\vec{k},\omega)&0&-W_{\vec{k}}\\\
-W_{\vec{k}+\vec{Q}}&0&G_{0}^{-1}(\vec{k}+\vec{Q},\omega)&0\\\
0&-W_{\vec{k}+\vec{Q}}&0&G_{0}^{-1}(\vec{k}+\vec{Q},\omega)\end{pmatrix}^{-1}.$
(10)

At this point, we replace the Green’s function $G_{0}(\vec{k},\omega)$ by a
normal state correlated Green’s function obtained through the n-pole
approximation applied to the single-band Hubbard model [42]. The single-band
Hubbard model is given as:

$\mathcal{H}=\sum_{\langle\langle
i,j,\sigma\rangle\rangle}t_{ij}c_{i,\sigma}^{\dagger}c_{j,\sigma}+U\sum_{i,\sigma}n_{i,\sigma}n_{i,-\sigma},$
(11)

where $t_{ij}$ is the hopping amplitude,
$n_{i,\sigma}=c_{i,\sigma}^{\dagger}c_{i,\sigma}$ is the occupation number
operator with spin $\sigma=\\{\uparrow,\downarrow\\}$, $U$ is the local
repulsive Coulomb interaction and the $\langle\langle...\rangle\rangle$ symbol
indicates a sum over first and second nearest neighbors. We highlight that
within the n-pole approximation, the second nearest neighbor hopping is
crucial to properly capture the short-range antiferromagnetic correlation
effects that give rise to the PG phenomenon. Additionally, to address the
infinite chain of Green’s functions given by the equation of motion applied to
the Hubbard model, a finite set of relevant operators
$\\{A_{n}\\}=\\{c_{i,\sigma},n_{i,-\sigma}c_{i,\sigma},c_{j,\sigma}^{\dagger},n_{j,-\sigma}c_{j,\sigma}^{\dagger}\\}$
is defined, capable of linearizing equation (5), giving rise to the correlated
Green’s functions matrix:

$\textbf{G}(\vec{k},\omega)=\textbf{N}(\omega\textbf{N}-\textbf{E}(\vec{k}))^{-1}\textbf{N},$
(12)

with $N_{nm}=\langle[\hat{A}_{n},\hat{A}^{\dagger}_{m}]_{+}\rangle$ and
$E_{nm}=\langle[[\hat{A}_{n},\mathcal{H}],\hat{A}^{\dagger}_{m}]_{+}\rangle$.
Solving the last equation results in the correlated Green’s function matrix.
One of the most important elements is given by:

$G_{11,\sigma}(\omega,\vec{k})=\frac{\omega-U(1-n_{\sigma})-Y_{\vec{k},\sigma}}{(\omega-\varepsilon_{\vec{k}})(\omega-
U-Y_{\vec{k},\sigma})-Un_{\sigma}(\varepsilon_{\vec{k}}-Y_{\vec{k},\sigma})},$
(13)

where $Y_{\vec{k},\sigma}$ is the band shift. This quantity is significant, as
it contains all of the antiferromagnetic correlation effects, responsible for
the opening of the PG. The shift causes the energy bands to be displaced
towards energy levels below the Fermi level. This effect is much stronger in
the region close to the nodal point $(\pi,\pi)$, but also affects the anti-
nodal points, such as $(\pi,0)$, where the PG is opening. Therefore,
$Y_{\vec{k},\sigma}$ is defined as:

$\langle n_{-\sigma}\rangle(1-\langle
n_{-\sigma}\rangle)Y_{\vec{k},\sigma}=\Gamma^{(0)}_{\sigma}+\sum_{j\neq
i}e^{i\vec{k}\cdot(\vec{R}_{i}-\vec{R}_{j})}t_{ij}\big{(}\Gamma^{(1)}_{i,j,\sigma}+\Gamma^{(2)}_{i,j,\sigma}+\Gamma^{(3)}_{i,j,\sigma}\big{)},$
(14)

with the components $\Gamma^{m}_{i,j,\sigma}$ given by the following terms:

$\Gamma^{(0)}_{\sigma}=-\sum_{j\neq i}t_{ij}\langle
c_{i,\sigma}^{\dagger}c_{j,\sigma}(1-n_{i,-\sigma}-n_{j,-\sigma})\rangle,$
(15)

$\Gamma^{(1)}_{i,j,\sigma}=\frac{1}{4}(\langle N_{i}N_{j}\rangle-\langle
N_{j}\rangle\langle N_{i}\rangle),$ (16)

$\Gamma^{(2)}_{i,j,\sigma}=\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle,$ (17)

and

$\Gamma^{(3)}_{i,j,\sigma}=-\langle
c_{j,\sigma}^{\dagger}c_{j,-\sigma}^{\dagger}c_{i,-\sigma}c_{i,\sigma}\rangle,$
(18)

where $N_{i}=n_{i,\sigma}+n_{i,-\sigma}$ being the total occupation operator
per site. When short-range antiferromagnetic correlations cause the PG, the
momentum dependence of $Y_{\vec{k},\sigma}$ is essential to capture this
phenomenon [66]. Then, the inverse Fourier transform of
$\Gamma^{m}_{i,j,\sigma}$ is taken to reintroduce this dependence into
$Y_{\vec{k},\sigma}$.

$\Gamma^{(m)}_{i,j,\sigma}=\frac{1}{L}\sum_{\vec{q}}e^{-i\vec{q}\cdot(\vec{R}_{j}-\vec{R}_{i})}\Gamma_{\vec{q},\sigma}^{(m)}.$
(19)

.

For this approach, we need the quasi-particle dispersion relation, given by:

$\varepsilon(\vec{k}-\vec{q})=\frac{1}{L}\sum_{j\neq
i}e^{i(\vec{k}-\vec{q})\cdot(\vec{R}_{j}-\vec{R}_{i})}t_{ij},$ (20)

now, it is possible to define a new compact form for $Y_{\vec{k},\sigma}$:

$\langle n_{-\sigma}\rangle(1-\langle
n_{-\sigma}\rangle)Y_{\vec{k},\sigma}=\Gamma^{(0)}_{\sigma}+\sum_{\vec{q},m=1,3}\varepsilon(\vec{k}-\vec{q})\Gamma^{(m)}_{\vec{q},\sigma}.$
(21)

With the last definition, $Y_{\vec{k},\sigma}$ has a non-uniform structure in
the momentum space, which implies a band distortion, leading to a PG similar
to that observed in cuprates. By solving equation (8), we obtain the
polynomial given by the determinant of the coefficients matrix, whose poles
are the energy bands. The polynomial is written as:

$\begin{split}P(\vec{k},\omega)=&\
G^{-1}_{11,\downarrow}(\vec{k},\omega)G^{-1}_{11,\uparrow}(\vec{k},\omega)G^{-1}_{11,\downarrow}(\vec{k}+\vec{Q},\omega)G^{-1}_{11,\uparrow}(\vec{k}+\vec{Q},\omega)\\\
&-\big{[}G^{-1}_{11,\downarrow}(\vec{k},\omega)G^{-1}_{11,\downarrow}(\vec{k}+\vec{Q},\omega)-G^{-1}_{11,\uparrow}(\vec{k},\omega)G^{-1}_{11,\uparrow}(\vec{k}+\vec{Q},\omega)\big{]}W^{2}_{\vec{k}}+W^{4}_{\vec{k}},\end{split}$
(22)

where we replaced $G_{0}(\vec{k},\omega)$ by the correlated normal state
Green’s function $G_{11,\sigma}(\vec{k},\omega)$ defined in equation (13).
Through such replacement, besides the correlations, an explicit spin
dependence is also included. For practicality, the Green’s functions are
separated in partial fractions, as it explicitly shows the form factors of the
spectral weights, which are extremely relevant for the study of the quantities
of interest. Thus, the general form for these Green’s functions is defined
below, with $z_{m,\sigma}$ as the poles of $P(\vec{k},\omega)$, which is the
determinant of the coefficients matrix.

$\mathcal{G}_{ls,\sigma}(\vec{k},\omega)=\sum_{m=1,4}\frac{Z^{ls}_{m,\sigma}(\vec{k})}{(\omega-
z_{m,\sigma})},$ (23)

where $Z^{ls}_{m,\sigma}$ are the spectral weights of the Green’s function.
Moreover, we define $\Lambda^{ls}_{\sigma}(\vec{k})$ as the numerator of the
Green’s function $\mathcal{G}_{ls,\sigma}(\vec{k},\omega)$, resulting in:

$Z^{ls}_{n,\sigma}(\vec{k})=\frac{\Lambda^{ls}_{\sigma}(\vec{k})}{\prod^{m=1,4}_{m\neq
n}(z_{n,\sigma}-z_{m,\sigma})}.$ (24)

The choice of this representation for Green’s functions facilitates the
calculation of correlation functions. The most relevant Green’s functions for
this work is $\mathcal{G}_{11,\sigma}(\vec{k},\omega)$, associated with the
occupation number, the DOS, and the spectral function. Furthermore,
$\mathcal{G}_{13,\sigma}(\vec{k},\omega)$ enables the order parameter
calculation.

To evaluate the correlation functions, we use the standard integration, where
the contour encircles the real axis without enclosing the poles of the Fermi
function $\eta_{F}(\omega)$. Thus, the correlation functions are given by:

$\langle\hat{B}\hat{A}\rangle=\oint\langle\langle\hat{A};\hat{B}\rangle\rangle\eta_{F}(\omega)d\omega,$
(25)

and applying that representation to $\mathcal{G}_{11,\sigma}(\vec{k},\omega)$,
results in the occupation number:

$\langle
n_{\sigma}\rangle=\frac{1}{N}\sum_{\vec{k}}\sum_{m=1,4}Z^{11}_{m,\sigma}(\vec{k})\eta_{F}(z_{m,\sigma}).$
(26)

Repeating the process for $\mathcal{G}_{13,\sigma}(\vec{k},\omega)$:

$W_{\vec{k}}=\frac{1}{N}|V|\gamma_{\vec{k}}\sum_{\vec{k^{\prime}}}\sum_{m=1,4}Z^{13}_{m,\sigma}(\vec{k}^{\prime})\eta_{F}(z_{m,\sigma}).$
(27)

##  3 Numerical Results

To ensure all of the results are consistent, we use $|t_{0}|=1.0$ eV as the
energy unit, and $|V|=1.2|t_{0}|$.

The primary goal of this study is to demonstrate the connection between the
impact of short-range antiferromagnetic correlations and the reconstruction of
the FS within the CDW phase. We aim to elucidate the connection between this
phenomenon and the emergence of the PG. The band shift $Y_{\vec{k},\sigma}$
having a structure on the momentum space leads to a distortion on the nodal
point $(\pi,\pi)$, as well as in the anti-nodal point $(\pi,0)$. The
distortions are enhanced by second nearest neighbors hopping $t_{1}$ and
characterized as a displacement of the bands towards lower energies, that when
sufficiently intense, allow the reconstruction of the FS, and can lead to the
opening of a PG. To analyze the relevance of antiferromagnetic correlations in
the CDW phase, energy bands were obtained through the spectral function,
calculated along the high-symmetry regions of the first Brillouin zone.
Analyzing the bands from this perspective allows for visualizing the
contribution of each band component to the system.

Figure 1: Fermi surfaces in the normal state with $U=9.0|t_{0}|$,
$n_{T}=0.90$, $k_{B}T=0.05|t_{0}|$ for (a) $t_{1}=0.05|t_{0}|$, (b)
$t_{1}=0.10|t_{0}|$ and (c) $t_{1}=0.20|t_{0}|$. In panel (c), the Fermi
surface collapses into a pocket, indicating the opening of the PG. Figure 2:
Energy bands for the normal state with $U=9.0|t_{0}|$, $n_{T}=0.90$,
$k_{B}T=0.05|t_{0}|$ for $t_{1}=0.05|t_{0}|$, $t_{1}=0.10|t_{0}|$ and
$t_{1}=0.20|t_{0}|$. The dotted black line indicates the position of the
chemical potential.

Figure 1 shows the FS for different values of $t_{1}$ in the normal state.
Panels (a) and (b) show two competing topologies for the FS, originating from
the superposition of two bands with $\mu$. Panel (c) illustrates the FS
collapsed into a pocket, indicating that the PG is open. The band shift
$Y_{\vec{k},\sigma}$ is affected in a more significant manner by $t_{1}$,
enhancing the short-range antiferromagnetic correlations on the nodal point
$(\pi,\pi)$ and on the anti-nodal point $(\pi,0)$. This is evident in figure
2, as a band displacement to lower energy levels characterizes the distortion.
The collapse into a pocket indicates that an increase in the short-range
antiferromagnetic correlations caused by $t_{1}$ is responsible for the
opening of the PG. The mechanism is characterized by a distortion of the bands
on the anti-nodal point $(\pi,0)$, which gets further displaced to lower
energies, until both bands are below $\mu$, thus, opening the PG, this is
evident in the insert, which is a close-up view of the bands close to $\mu$ on
the $(\pi,0)$ point. Furthermore, $t_{1}$ also distorts the bands on the nodal
point $(\pi,\pi)$ in a similar manner, in this case, one of the bands gets
displaced to further lower energies, while the other approaches $\mu$.

Figure 3: (a) Reconstructed Fermi surface in the CDW phase due to
antiferromagnetic correlations for $k_{B}T=0.05|t_{0}|$. Normal state Fermi
surfaces, (b) for $k_{B}T=0.10$ and (c) $k_{B}T=0.15$, with $U=16|t_{0}|$,
$n_{T}=0.90$, $t_{1}=0.12|t_{0}|$. In panel (c), the FS collapses into a
pocket, indicating the opening of the PG. The dotted black line indicates the
position of the chemical potential.

Figure 3 shows the FS evolution with $k_{B}T$. We have a fully-fledged
reconstructed electron-like surface in the CDW phase on panel (a). On panel
(b), the thickness of the lines in the boundary regions of the Brillouin zone
predicts a change in the Fermi surface topology, in this case, the collapse
into a pocket. Indeed, this is observed in panel (c), where an increase in
$k_{B}T$ leads to the opening of the PG, characterized by the collapse of the
FS into a pocket. The electron-like topology presented here comes from the
strong antiferromagnetic correlations, differing from the expected behavior of
a total surface collapse. This reconstruction suggests that even within the
CDW phase, the correlations play a relevant role.

Figure 4: (a) Energy bands for $k_{B}T=0.05|t_{0}|$, $k_{B}T=0.10|t_{0}|$ and
$k_{B}T=0.15|t_{0}|$, with $U=16.0|t_{0}|$, $n_{T}=0.90$, $t_{1}=0.12|t_{0}|$.
(b) Close-up view of the bands close to the chemical potential level on the
anti-nodal point $(\pi,0)$, (c) Close-up view of the CDW gap opening on the
$(\pi,0)$ point, for $k_{B}T=0.05|t_{0}|$. (d) Close-up view of the pseudogap,
on the $(\pi,0)$ point, for $k_{B}T=0.15|t_{0}|$. The dotted black line
indicates the position of the chemical potential.

Figure 4 displays the energy bands corresponding to the FS in figure 3. The
opening of the CDW gap happens on the anti-nodal point $(\pi,0)$, for
$k_{B}T=0.05|t_{0}|$. There is a degeneracy lifting, that splits the bands on
this point, opening the gap. $Y_{\vec{k},\sigma}$ heavily distorted the bands
in the nodal points, displacing those to regions below $\mu$. The band region
around the nodal point $(\pi,0)$ is distorted in the same way as in the nodal
point $(\pi,\pi)$, this was caused by the effects of $Y_{\vec{k},\sigma}$. The
PG is observed when $k_{B}T=0.15|t_{0}|$, as the bands do not cross $\mu$ in
$(\pi,0)$. However, intersections still occur at $(\pi/2,\pi/2)$, meaning the
gap is partial. In figure 4 panel (c), $k_{B}T=0.05|t_{0}|$. The CDW gap
opening happens in the same region as the PG. The shift of the upper band to
an energy region above $\mu$ breaks the degeneracy introduced by the
periodicity of the lattice. The occurrence of both effects in the same point
of the first Brillouin zone complicates the separate analysis, as within the
ordered phase, the effects of the PG are no longer as clear. Figure 4 panel
(d) represents the case when $k_{B}T=0.15|t_{0}|$. The difficulty in analyzing
the relation between CDW and the PG lies in the superposition of their effects
on the electronic structure of the system. The PG and the CDW gap open on the
anti-nodal point $(\pi,0)$. Therefore, any indicative information about the PG
is lost with the gap opening. However, the band shift causes the two lower
bands to be completely localized below $\mu$ at the point $(\pi,0)$. Thus, the
gap opening causes a degeneracy breaking at this point, separating the bands
and giving rise to the CDW. The gap is the distance between the two band
values at this $\vec{k}$ point. Nevertheless, the flatness of this region is
kept after the splitting. This suggests that there are still localized
electronic states, reminiscent of the normal state. We can see that this
degeneracy breaking leads to an intersection between $\mu$ and the upper band,
reconstructing the FS.

Figure 5: (a) Fermi surface reconstructed by antiferromagnetic correlations in
the CDW phase. (b) Energy bands near the chemical potential level are
calculated using the spectral function along the high-symmetry path of the
reduced first Brillouin zone, with $U=16.0|t_{0}|$, $n_{T}=0.85$ and
$t_{1}=0.1|t_{0}|$. The dotted white line indicates the position of the
chemical potential.

In Figure 5 panel (a), there is a shadow band effect happening, characterized
by a much smaller intensity FS, originating from the proximity of the
antiferromagnetic state [67]. In panel (b), a luminosity maximum is observed
in the regions around the $(0,0)$ point, and around the CDW gap region, near
the $(\pi,0)$ region. This indicates the effects of the short-range
antiferromagnetic correlations within the CDW phase, suggesting a connection
between the PG phenomenon and CDW. Similar behavior is illustrated in figure
6, but the system is now in a regime of stronger correlations, as
$n_{T}=0.90$. However, the luminosity distribution in the bands follows the
same pattern as in the case of $n_{T}=0.85$. The reconstructed FS undergoes a
LT, indicating that the antiferromagnetic correlations are intense enough to
cause it within the CDW phase.

Figure 6: (a) Fermi surface reconstructed by antiferromagnetic correlations in
the CDW phase. (b) Energy bands near the chemical potential level are
calculated using the spectral function along the high-symmetry path of the
reduced first Brillouin zone, with $U=16.0|t_{0}|$, $n_{T}=0.90$ and
$t_{1}=0.1|t_{0}|$. The dotted white line indicates the position of the
chemical potential. Figure 7: The density of states in the CDW phase for
$U=10|t_{0}|$, $U=12|t_{0}|$ and $U=16|t_{0}|$, shown under (a) $n_{T}=0.85$
and (b) $n_{T}=0.90$, at $k_{B}T=0.05|t_{0}|$ and $t_{1}=0.1|t_{0}|$. The
dotted black line indicates the position of the chemical potential.

To investigate the consequences of this LT, the DOS for $n_{T}=0.85$ and
$n_{T}=0.90$ is analyzed in Figure 7. A VHS, shifted from $\mu$, is
illustrated. In other words, there is a reduction in accessible states for the
stability of the CDW phase, as the peak of the VHS does not align with $\mu$.
In Figure 4, the PG and the CDW gap occur in the same first Brillouin zone
region, indicating an intertwining between the phenomena, caused by a regime
of strong correlations, since $U=16.0|t_{0}|$ corresponds to twice the
system’s bandwidth. Indeed, the effects of $k_{B}T$ are pivotal in our theory.
For $k_{B}T^{*}\geq k_{B}T>k_{B}T_{C}$, the DOS assumes a one-peak structure,
which can get displaced by $t_{1}$. For $k_{B}T<k_{B}T_{C}$ there is a rise of
a second, smaller VHS-like peak, separated by a partial gap in the DOS. This
indicates the opening of the CDW, as there is a gap between both peaks. The
effects of the short-range antiferromagnetic correlations cause a piling of
states on $\mu$, further emphasizing a connection with the PG, as the expected
behavior would be a total gap.

The order parameter $W_{\vec{k}}$ is calculated as a function of $n_{T}$ for
different values of $U$, see figure 8 panel (a). An initial favoring is
followed by suppression in the order parameter amplitude, caused by increasing
$n_{T}$. This implies an optimal correlation regime for $0.75\gtrapprox
n_{T}\gtrapprox 0.85$, where the antiferromagnetic correlations favor CDW
stability. For the specific case with $U=16.0|t_{0}|$, the LT precedes the
onset of the CDW collapse. In this result, $k_{B}T$ was kept fixed; however,
at higher $k_{B}T$, the PG opens. Thus, assuming the change in the density of
states structure, the suppression of $W_{\vec{k}}$, and the change in the FS
topology, it is inferred that at low $k_{B}T$ and strong antiferromagnetic
correlations, there is an intertwining between CDW and the PG.

Figure 8: (a) CDW order parameter and (b) critical temperature as a function
of $n_{T}$ for $U=10|t_{0}|$, $U=12|t_{0}|$ and $U=16|t_{0}|$, with
$t_{1}=0.1|t_{0}|$ and $k_{B}T=0.05|t_{0}|$.

To further investigate the CDW regime, $k_{B}T_{C}$ as a function of $n_{T}$
is analyzed for various values of $U$ in Figure 8, panel (b). The increase in
$n_{T}$ causes a suppression in $k_{B}T_{C}$, particularly pronounced for more
intense correlation regimes. Notably, for $U=16.0|t_{0}|$, the decay of
$k_{B}T_{C}$ is more pronounced, emphasizing the connection between CDW and
the PG. Supporting the previously mentioned optimal correlations regime, the
behavior of $k_{B}T_{C}$ reflects what was stated in panel (a). This optimal
regime happens in the $0.75\gtrapprox n_{T}\gtrapprox 0.85$ interval, before
the LT, at $n_{T}=0.90$. The collapse of the CDW occurring in the vicinity of
the LT, supports the intertwining of the PG and the CDW, as it was previously
stated that the onset of the PG is related to the LT.

As seen earlier, $t_{1}$ can also lead to the opening of the PG. To analyze
its effect, the order parameter is calculated as a function of $t_{1}$ for
different $n_{T}$, as shown in Figure 9 panel (a). For $n_{T}=0.80$ and
$n_{T}=0.85$, there is a steady increase in the order parameter, emphasizing
the existence of two regimes, mediated by the intensity of the short-range
antiferromagnetic correlations. This is associated with the displacement of
states to regions near $\mu$, which cease to contribute to the CDW phase. For
$n_{T}=0.90$, the CDW suppressive effects of increasing $t_{1}$ are more
relevant, indicating that the number of accessible states near $\mu$ has
decreased, as $t_{1}$ distorts not only the bands but also the position of the
VHS peak. This is reinforced by the difference in the gap amplitude by the
decay for $t_{1}/|t_{0}|\gtrapprox 0.07$, indicating the possible opening of
the PG for slightly more intense correlations or $k_{B}T$. Finally,
$k_{B}T_{C}$ as a function of $t_{1}/|t_{0}|$ is analyzed in Figure 9 panel
(b). For higher values of $t_{1}$, there is an inversion in the amplitude of
$T_{C}$, reinforcing the existence of an optimal correlation regime. For
$U=16.0|t_{0}|$, suppressive effects occur for smaller values of $t_{1}$ in a
more pronounced behavior. The PG and CDW are connected through
antiferromagnetic correlations. These initially favor the ordered phase, but
when sufficiently intense, tend to suppress the CDW and change the topology of
the FS, which after the LT, collapses into a pocket, showing that the PG is
open.

Figure 9: (a) CDW order parameter with $k_{B}T=0.05|t_{0}|$ and $U=12|t_{0}|$,
for $n_{T}=0.80$, $n_{T}=0.85$ and $n_{T}=0.90$. (b) CDW critical temperature
at $n_{T}=0.90$, as a function of $t_{1}$ for $U=10|t_{0}|$, $U=12|t_{0}|$ and
$U=16|t_{0}|$.

##  4 Conclusions

We employed a BCS-like mean field approximation, within the Green’s function
equation of motion formalism to describe the CDW instability. We replaced the
normal state uncorrelated Green’s functions with a correlated one, obtained
through the n-pole approximation applied to the single-band Hubbard model.
This has been done considering next-nearest neighbors $t_{1}$, that along
$n_{T}$ and $U$, act on the spin-spin correlation
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, strongly affecting the band shift
$Y_{\vec{k},\sigma}$. We emphasize that $Y_{\vec{k},\sigma}$ acquires a
structure in the reciprocal space that enhances such correlations. Remarkably,
the effect becomes relevant at anti-nodal point $(\pi,0)$ as discussed below.
It is important to mention that generally, the correlation functions in the
band shift do not have an explicit dependence on the reciprocal space [41,
44]. This dependence displaces the bands in the nodal point $(\pi,\pi)$ and
the anti-nodal point $(\pi,0)$ to lower energy regions. For $t_{1}$
sufficiently large, the entire region around $(\pi,0)$ is distorted, which is
responsible for the opening of the PG.

The VHS peak can be displaced by $n_{T}$ and $t_{1}$, which triggers a LT,
favoring the onset of the PG since the DOS presents a change in shape. This
deformation reorganizes states, resulting in their suppression in the region
around $\mu$ and accumulation near the VHS peak. In other words, both the LT
and the PG can be induced by short-range antiferromagnetic correlations [68,
69]. When $k_{B}T$ is lowered, the DOS assumes a two-peak structure. The
superposition of the main VHS peak with $\mu$, creates an abundance of
accessible states, further stabilizing the CDW as given by the order parameter
behavior. Nevertheless, the FS is reconstructed due to the opening of the CDW
gap, which deforms the bands, allowing the intersection of $\mu$ with the
upper band at this anti-nodal point. Ultimately, this deformation comes from
the dependence on $\vec{k}$ of the band shift which enhances short-range
antiferromagnetic correlations at $(\pi,0)$. For $k_{B}T>>k_{B}T_{C}$, within
a specific range of $n_{T}$ and $U$, the displacement of the VHS peak along
the energy axis caused by $Y_{\vec{k},\sigma}$ can be either beneficial or
harmful to the electron-hole pair formation. The states displaced away from
$\mu$ stop contributing to pair formation. We highlight the presence of a LT
within the ordered phase, triggered by an increase in $n_{T}$, further
emphasizing the importance of this analysis, since the VHS peak position was
displaced to higher energies.

In conclusion, we have proposed a theory, where the short-range
antiferromagnetic correlations constitute its cornerstone, to describe the
connection between CDW and the PG. We explored the connections between the PG
and the FS reconstruction within the CDW phase through the band distortions
caused by such correlations. The VHS originated from the flat region on the
anti-nodal point $(\pi,0)$ exerts a fundamental role, by triggering a LT, but
also being able to suppress and benefit the ordered phase. The role of the VHS
is a direct consequence of the short-range antiferromagnetic correlations [70,
71]. We mention results from cuprates where a LT precedes the onset of the PG
[72, 73, 74] and the FS reconstruction coexisting with Fermi arcs [75, 76]. In
the latter case, we have a fully formed Fermi surface, which coexists with an
unstable Fermi arc spread on the anti-nodal line.

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