# Formation of spin and charge ordering in the extended Hubbard model during a
finite-time quantum quench

Isaac M. Carvalho  Departamento de Física, Universidade Federal de Minas
Gerais, C.P. 702, 30123-970, Belo Horizonte, MG, Brazil  Helena Bragança
Instituto de Física and International Center for Physics, Universidade de
Brasília, Brasília 70919-970, DF, Brazil  Walber H. Brito  Departamento de
Física, Universidade Federal de Minas Gerais, C.P. 702, 30123-970, Belo
Horizonte, MG, Brazil  Maria C. O. Aguiar  Departamento de Física,
Universidade Federal de Minas Gerais, C.P. 702, 30123-970, Belo Horizonte, MG,
Brazil

###### Abstract

We investigate the formation of charge and spin ordering by starting from a
non-interacting state and studying how it evolves in time under a Hamiltonian
with finite electronic interactions. We consider the one-dimensional, half-
filled extended Hubbard model, which we solve within time-dependent density
matrix renormalization group. By employing linear finite-time quenches in the
onsite and nearest-neighbor interactions, we find the existence of impulse,
intermediate, and adiabatic regimes of time evolution. For the quenches we
analyze, we observe that the adiabatic regime is reached with distinct ramping
time scales depending on whether the charge density wave (CDW) or the spin
density wave (SDW) is formed. The former needs to be slower than the latter to
prevent entangled excited states from being accessed during the quench. More
interestingly, in the intermediate regime, we observe an enhancement of the
entanglement entropy with respect to its initial value, which precedes the
formation of the CDW ordering; a similar enhancement is not seen in the quench
towards SDW. Our findings also show that the breaking of the system
integrability, by turning on the nearest-neighbor interactions, does not give
rise to significant changes in the non-equilibrium behavior within the
adiabatic approximation.

##  I Introduction

The dynamics of closed many-body systems following a quantum quench has become
an active topic of research. In solid-state experimental setups, realizing a
platform that is, at the same time, sufficiently well isolated from the
environment and accessible for the experimental probe is a hampering factor to
study the coherent evolution after a perturbation of the system. Ultracold
atoms trapped in optical lattices [1, 2, 3, 4, 5, 6], on the other hand,
provide an unprecedented opportunity to explore non-equilibrium phenomena due
to the large set of available methods to isolate, manipulate and measure these
systems [7]. In this experimental framework, the system parameters can be
tuned either abruptly (sudden quench) [8] or by a finite-time protocol [9,
10]. Furthermore, ultracold atoms give access to novel observables and extreme
parameter regimes, that go beyond the ones accessible in solid-state systems.
Alkali atoms (such as potassium 40K and lithium 6Li) in optical lattices, for
instance, can be used as quantum simulators of the Fermi-Hubbard model with
highly controllable local interaction [11]. Recently, nearest-neighbor
interactions have also been realized using Rydberg dressing of 6Li [12]. From
the theoretical side, the investigation of quantum quenches can be done via
Conformal Field Theories [13, 14, 15, 16] and tensor-network algorithms [17,
18, 19, 20], however, the investigations are hampered by the entanglement
growth. The analysis of the out-of-equilibrium dynamics of strongly-correlated
fermionic systems is still a challenging problem.

In this context, the non-equilibrium dynamics after a quench through a quantum
critical point is especially interesting. It has been shown that systems whose
low-energy spectrum is described by a conformal field theory feature a large
degree of universality even for late times after crossing the critical point
[21, 15]. Furthermore, for some systems that cross the criticality, the
crossover between adiabatic and non-adiabatic stages of time evolution has
been successfully described by a quantum extension of the Kibble-Zurek
mechanism, which is based on equilibrium critical exponents [22, 23]. However,
the complete theoretical description of the dynamics of strongly correlated
systems going through a quantum phase transition (QPT) constitutes an open
problem in many situations.

The extended Hubbard chain is a prototypical model for QPTs in strongly
correlated systems. Its equilibrium version, for repulsive interactions and at
half-filling, features a spin density wave (SDW), a charge density wave (CDW),
as well as a bond-order wave (BOW) insulating phases [24]. As the system is
metallic in the absence of interactions, it goes through metal-insulator QPTs
when the interactions are turned on. The study of CDW and SDW phases attracts
increasing attention due to experimental evidences of their interplay with the
superconducting phases of cuprates [25] and iron-pnictides [26]. Moreover,
recent ultrafast optical experiments have addressed the nonequilibrium CDW
phases presented in different families of correlated transition-metal
dichalcogenides. For instance, the physics related to the nonthermal melting
of CDW ordering in TiSe2 [27] and VTe2 [28] has triggered great interest and
is still under debate, since a purely electron-phonon mechanism can not
account for the CDW melt-state. The extended Hubbard model with an attractive
nearest-neighbor interaction is also of great interest. Recently, it was used
to describe the photoemission spectrum of the one-dimensional cuprate
Ba2-xSrxCuO3+δ [29]. According to Wang and co-workers [30], an attractive
interaction appears in this case due to a long-range electron-phonon coupling.

In the non-equilibrium scenario, it was observed that a transition between
correlated and uncorrelated states can be driven by a quench in an external
field added to the Hubbard model [31]. Further, the effects of many-body
interactions on the statistics of energy fluctuations were investigated in the
inhomogeneous version of the model, in which the system was submitted to an
out-of-equilibrium transient current along the chain [32]. In the case of the
extended Hubbard model, the non-equilibrium states generated by radiation
pulses were recently described by a generalized Gibbs ensemble [33]; the non-
equilibrium phase diagram of the system includes a $\eta$-paring
superconducting phase.

In this work, we investigate the formation of SDW and CDW ordering in the
extended Hubbard model submitted to interacting quenches. The initial state is
chosen to be a delocalized one, the ground state (GS) of the non-interacting
Hamiltonian, and we let it evolve under an interacting Hamiltonian. The onsite
and/or nearest-neighbor repulsive interactions increase linearly in time, such
that the final Hamiltonian is within the correlated SDW or CDW phase. At
equilibrium, in the thermodynamic limit, the system is metallic at the non-
interacting point and is an insulator for any finite interaction (within the
CDW phase, charge and spin excitations are gapped, while in the SDW phase spin
excitations are gapless [24]). Therefore, we start the evolution from a model
critical point and investigate the formation of ordered phases during the
quench.

We observe different regimes by varying the duration of the quench - we go
from sudden quenches to adiabatic ones. We start by turning on either the
onsite or the nearest-neighbor interaction, aiming to reach the SDW and the
CDW ordering, respectively. For the latter, the intermediate regime, that
precedes the adiabatic one, is characterized by an increase of the
entanglement entropy with respect to the initial state, indicating that the
evolved state includes excited disordered ones. As a consequence, we observe
that the adiabatic regimes are reached by distinct values of the ramping time
scales, longer for the CDW case than for the SDW one. Interestingly, the
inclusion of small onsite (nearest-neighbor) interactions - meaning that both
interactions are now turned on - hardly changes the dynamics during the quench
towards the CDW (SDW) phase.

The organization of the paper is the following: in Sec. II we present the
model and the quench protocol; details of the numerical calculations are also
mentioned. Our results are presented in Sec. III. In Sec. III.1 we discuss the
results obtained by turning on only one of the interactions, either the onsite
$U$ or the nearest-neighbor $V$ interaction. We observe the state time
evolution during the quench in different regimes: adiabatic, intermediate, and
impulse ones. Later, in Sec. III.2, we discuss the effects of turning on both
$U$ and $V$ simultaneously. Our conclusions are summarized in Sec. IV. The
dependence of our main results on the chain size are presented in Appendix A.

##  II Model and quench protocol

We investigate the time-dependent extended Hubbard model (EHM), given by the
Hamiltonian

$\displaystyle\hat{H}\left(t\right)$ $\displaystyle=$
$\displaystyle-J\sum_{j,\sigma}\left(a_{j,\sigma}^{\dagger}a_{j+1,\sigma}+H.c.\right)$
(1) $\displaystyle+$ $\displaystyle
U\left(t\right)\sum_{j}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow}+V\left(t\right)\sum_{j}\hat{n}_{j}\hat{n}_{j+1}$

which considers nearest-neighbor hopping of amplitude $J$ and time-dependent
onsite and nearest-neighbor interactions, given by $U$$\left(t\right)$ and
$V\left(t\right)$, respectively. In the equation above,
$a_{j,\sigma}^{\left(\dagger\right)}$ annihilates (creates) a fermion with
spin $\sigma=\uparrow,\downarrow$ on lattice site $j$,
$\hat{n}_{j,\sigma}=a_{j,\sigma}^{\dagger}a_{j,\sigma}$, and
$\hat{n}_{j}=\hat{n}_{j,\uparrow}+\hat{n}_{j,\downarrow}$.

Figure 1: Equilibrium values for the (a) CDW and (d) SDW order parameters, as
indicated by the color-codes, as a function of $U_{eq}$ and $V_{eq}$. The
white straight lines correspond to $U=2V$. The arrows indicate different
quench directions to be explored along the paper. The red (black) dot
identifies the final states corresponding to quench #1 (#6). The local
magnetization and charge density as a function of the lattice site $i$
corresponding to the final state of #1 quench are shown in panels (b) and (c),
respectively, whereas the behavior of the final state of #6 quench is depicted
in panels (e) and (f). Besides showing results concerning the final state for
quenches with different time scales ($\tau_{V}$ and $\tau_{U}$), we also
present the profiles corresponding to the equilibrium GS with $U_{eq}=U_{0}=0$
and $V_{eq}=V_{0}=0$ and with $U_{eq}=U_{f}$ and $V_{eq}=V_{f}$.

In this paper we study the time evolution of the system described by the
Hamiltonian of Eq. (1) during a finite-time quench in the interaction
strengths. We first address the case in which only one of the interactions,
either onsite or nearest-neighbor one, is turned on (Sec. III.1); later, we
turn on both of them simultaneously (Sec. III.2). We are particularly
interested in the formation of spin or charge ordered states and we thus start
from a delocalized state, that is, the GS of the Hamiltonian with no
interactions ($U_{0}=V_{0}=0$). Then over a finite time interval we evolve the
parameters $U(t)$ or/and $V(t)$ from their initial values, $U_{0}$ and
$V_{0}$, up to their final values, $U_{f}$ and $V_{f}$. To be precise, we
change the interactions linearly in time as follows:

$U\left(t\right)=U_{0}+sgn(U_{f}-U_{0})\frac{t}{\tau_{U}}$ (2)

or/and

$V\left(t\right)=V_{0}+sgn(V_{f}-V_{0})\frac{t}{\tau_{V}},$ (3)

where $t\in\left[0,t_{f}\right]$ and $\tau_{U}$ and $\tau_{V}$ are the ramping
time scales. We set $t_{f}$ such that at the end of quench the values of the
interactions are $U(t_{f})=U_{f}$ or/and $V(t_{f})=V_{f}$. If only one of the
interactions is turned on, for a final Hamiltonian within the SDW phase, we
have $t_{f}=|U_{F}-U_{0}|\tau_{U}$ and $V_{f}=V_{0}=0$; for a final
Hamiltonian in the CDW phase, we have $t_{f}=|V_{F}-V_{0}|\tau_{V}$ and
$U_{f}=U_{0}=0$. Finally, when both interactions are turned on simultaneously,
$\tau_{U}$ and $\tau_{V}$ are connected through
$\tau_{V}=\tau_{U}|U_{F}-U_{0}|/|V_{F}-V_{0}|$.

Throughout this paper energies are given in units of $J$, and, accordingly,
time is measured in units of $1/J$. The system is fixed at half-filling, the
total magnetization in the $z-$direction is conserved during the quench, and
we use open boundary conditions. Our results were obtained mainly for chains
of size $L=17$, but we have confirmed our main findings by increasing the
system length, as discussed in Appendix A. The non-equilibrium numerical
simulations of our global quench rapidly become computationally cumbersome
with the number of sites. In finite systems as those we simulate, the gap
remains non-zero for all the Hamiltonian parameters. This quantity is key in
the adiabatic theorem [34] \- it states that, in driven transitions to states
with non-zero gaps, it is always possible to reach the adiabatic limit if the
quench process is slow enough. Based on this theorem and on the finite $L$ of
our systems, we expect to be able to observe an adiabatic behavior, as we
indeed do (see the discussion of our results).

To obtain the system state at time $t$, we have performed time-dependent
density matrix renormalization group (DMRG) calculations [20, 18] with a first
order Suzuki-Trotter decomposition, meaning that the error in the time
evolution is of $\mathcal{O}\left(d\tau^{2}\right)$, where $d\tau$ is the time
step. We consider $d\tau$ between $10^{-3}$ and $10^{-2}$, depending on the
ramping time scale. Our implementation was built using the ITensor library
[35]. In addition, for a better comparison with the evolved state during the
quench, we have performed DMRG calculations to obtain the instantaneous
equilibrium GS corresponding to constant $U_{eq}=U(t)$ and $V_{eq}=V(t)$ at
each instant of time $t$. Our calculations were performed until the GS energy
convergence was of the order of $10^{-8}$.

In the following, we present our numerical results for the system dynamics
during the quenches towards different regions of the phase diagram. As we will
show, we find significant differences in the relationship between the ramping
time scale and the adiabatic behavior depending on which ordering (CDW or SDW
one) the final Hamiltonian corresponds to.

##  III Numerical results

We start by obtaining the equilibrium phase diagram of the EHM within our
implementation. To characterize both CDW and SDW phases, respectively, we use
the order parameters defined as

$m_{CDW}=\frac{1}{L}\sum_{j}\left(-1\right)^{j}\left(\left\langle\hat{n}_{j}\right\rangle-1\right)$
(4)

and

$m_{SDW}=\frac{1}{L}\sum_{j}\left(-1\right)^{j}\left\langle\hat{s}_{j}^{z}\right\rangle.$
(5)

As can be noticed in Figs. 1(a) and (d), for repulsive interactions, the EHM
features a CDW phase for $U<2V$ and a Mott insulator phase with SDW for
$U>2V$. The color-codes correspond to the calculated values of $|m_{CDW}|$
[panel(a)] and $|m_{SDW}|$ [panel (d)]. These results are well known and in
good agreement with early calculations [36, 24]. For small $U$ and $V$ around
the $U=2V$ line, an additional instability occurs due to the competition
between onsite and nearest-neighbor Coulomb interactions, giving rise to a BOW
[37], which is not addressed in our work.

###  III.1 Turning on either $U$ or $V$

Now we study the finite-time quench produced by turning on only one of the
interactions: the onsite interaction, $U$, for a final state in the SDW phase,
or the nearest-neighbor one, $V$, for a final state in the CDW phase. As
explained previously, in both cases, we start from a metallic state [GS of Eq.
(1) with fixed $U_{0}=V_{0}=0$] and analyze the system evolution as the
interactions increase linearly up to $U_{f}=6,V_{f}=0$ [#1 vertical line in
Fig. 1(a)] or $U_{f}=0,V_{f}=3$ [#6 horizontal line in panel (d) of the cited
figure].

As will become clear from our results discussed below, we can identify three
regimes in the quench evolution: impulse, intermediate, and adiabatic ones.
Since our main goal is the formation of the CDW or SDW ordering, we consider a
quench is adiabatic (rigorously quasi-adiabatic) if (1) the fidelity
calculated between evolved and equilibrium states [the precise definition of
this quantity is given in Eq. (6)] at the end of the quench evolution is
larger than $0.99$ and (2) the difference between quantities calculated from
the evolved state and the equilibrium GS is smaller than the uncertainties of
our numerical calculation. In this case, it is expected that the system hardly
evolves after the quench.

####  III.1.1 Spin and charge profiles

First, we look at the behavior of the local spin
$\langle\hat{s}^{z}_{i}\rangle$ and charge $\langle\hat{n}_{i}\rangle$ mean
values as a function of the chain site $i$, calculated from the states at
$t=t_{f}$. These quantities are displayed for different ramping time scales on
the middle and right panels of Fig. 1, respectively. For comparison, we also
show the spin and charge profiles corresponding to the GS of $H(t=0)$ and
$H(t=t_{f})$.

We observe that for a short quench time scale ($\tau_{U}=0.01$ or
$\tau_{V}=0.01$) the system goes through a sudden quench, a regime that we
identify as an impulse one. In this case the state is frozen at the starting
one: the observables at $t=t_{f}$ are equal to the ones at $t=t_{0}$, as can
be notice in Fig. 1(b),(c),(e), and (f). In contrast, if the quench happens
slowly ($\tau_{V}=12.0$ for the quench towards the CDW phase and
$\tau_{U}=2.5$ for the quench to SDW), an adiabatic process takes place. In
this case, we observe the formation of interchanging patterns in the charge
[panel (c)] and spin [panel (e)] mean values, which closely follow the profile
of the equilibrium GS with finite $V_{f}$ or finite $U_{f}$ and are
characteristic of the CDW and SDW phases, respectively. Between these two
extremes, we observe an intermediate regime, also displayed in the figure for
comparison.

For completeness, we also show in Fig. 1 the local magnetization for the
quench towards the CDW phase [panel (b)] and the charge density for the quench
towards the SDW phase [panel (f)]. For the latter, no charge fluctuations are
expected in the SDW phase and $\left\langle\hat{n}_{i}\right\rangle=1$ since
we consider a half-filled system. For the former, we find a residual, non-zero
magnetization, as it also occurs in the regime where the interactions are
turned off [first plot in panel (b)]. This is a result of the conservation of
total magnetization and the fact that we study an odd length chain at half-
filling - there is always a net magnetization equal to $1/2$ spreading along
the chain.

In Fig. 1 we have plotted observables as a function of the chain site
corresponding to the final state after the quench. Let us now look at
observables defined for the whole system (that involve a sum over the sites,
for example) and their evolution during the quench. In Figs. 2(a) and (b), we
show the calculated order parameters as a function of the interactions, $V(t)$
or $U(t)$, for distinct ramping time scales, $\tau_{V}$ or $\tau_{U}$. In
these figures, the green dashed lines represent the GS order parameters
corresponding to the instantaneous equilibrium Hamiltonian at time $t$, that
is, the Hamiltonian of Eq. (1) with constant $U_{eq}=U(t)$ or $V_{eq}=V(t)$.

We observe that both $m_{CDW}$ and $m_{SDW}$ increase as the respective
interaction increases. For our finite system, the former approaches the
maximum value ($m_{CDW}\approx 1$) for $V_{f}=3$. $m_{SDW}$, on the other
hand, saturates at $\approx 0.11$ for $U_{f}=6$. We recall that, in the limit
of large $U$ and $V=0$, the half-filled system is equivalent to a Heisenberg
chain [38, 39] with an exchange constant $\sim 4J{{}^{2}}/U$. Even in this
limit, we cannot observe a perfect Néel state because it is not an eigenstate
of antiferromagnetic Heisenberg model [40]. For this reason, in our problem,
$m_{SDW}$ never reaches the maximum possible value (of $0.5$) allowed by the
definition in Eq. (5).

Figure 2: (a) CDW and (b) SDW order parameters and (c) and (d) the $EE$, all
displayed along the quenches, that is, as a function of $V(t)$ for the
quenches towards the CDW phase and as a function of $U(t)$ for the quenches
aiming the SDW phase.

By comparing the different curves in Fig. 2(a), we conclude that $m_{CDW}$
calculated from the evolved state corresponding to $\tau_{V}=12.0$ approaches
the equilibrium GS curve, revealing that $\tau_{V}\geq 12.0$ is necessary for
our system to reach the adiabatic regime (we will come back to this point when
discussing the fidelity in the next subsection). In contrast, for
$\tau_{V}=0.01$ the quench is sudden and the state does not evolve at all. The
quenches with $0.01<\tau_{V}<12.0$ correspond to an intermediate regime; the
order parameter increases but does not follow the GS value, indicating that
the evolved state includes excited ones. This is clear in the case of
$\tau_{V}=5.0$ (pink curve), for example, where we still observe some
deviations between the results obtained from the evolved state and those of
the equilibrium case. Interestingly, the evolution of $m_{CDW}$ at the
intermediate regime starts to exhibit some oscillatory behavior and is thus
non-monotonic. The overall time evolution of $m_{SDW}$ [Fig. 2(b)] is similar
to the one of $m_{CDW}$, however a smaller ramping time scale, $\tau_{U}=2.5$,
is already enough to ensure the adiabatic regime in this case. We can
partially explain these different time scales looking at the interactions
associated with the ordering of charges and spins along our quantum quenches.
The rearrangement of charges towards the CDW ordering is dominated by the
nearest-neighbor interaction $V$, while the rearrangement of spins towards SDW
is governed approximately by $4J^{2}/U$ (see Fig. 1). As a consequence, the
quenches towards the CDW ordering need to be longer than those towards SDW, in
such a way that the excitations produced by the charge rearrangements are
suppressed.

Figure 3: In (a) and (b) we present the fidelity $f(t)$ as a function of
$V(t)$ and $U(t)$, respectively. In (c) and (d) we plot the fidelity at the
end of the quench evolution $f(t_{f})$ as a function of $\tau_{V}$ and
$\tau_{U}$, respectively. The insets show a zoom of the data in panels (a) and
(b).

####  III.1.2 Fidelity between evolved and equilibrium states

To better characterize the observed regimes, we calculate the fidelity $f(t)$
between the evolved state at time $t$ and the GS corresponding to
$U_{eq}=U(t)$ or $V_{eq}=V(t)$. $f(t)$ is defined as:

$f(t)=\left|\left\langle\Psi(t)|\Psi_{GS}\right\rangle\right|^{2},$ (6)

where $\left|\Psi\left(t\right)\right\rangle$ is the evolved state and
$\left|\Psi_{GS}\right\rangle$ the instantaneous GS. We consider that when the
fidelity at the end of the quench evolution remains larger than a threshold
value close to unity (e.g. $0.99$, with a tolerance of $0.01$), the process
can be viewed as adiabatic. This assures us that the system will follow the GS
behavior in a free evolution after the quench since no additional excitations
were created by it. Otherwise, the evolved state includes excited ones and the
post quench evolution is nontrivial.

In Fig. 3(a) and (b), we display the calculated fidelity for the different
regimes observed for quenches towards the CDW and SDW phases, respectively. In
the impulse regime $(\tau_{V}=0.01$ and $\tau_{U}=0.01)$, we observe that the
fidelity vanishes in the CDW phase “earlier” [around $V(t)=2$] than when the
quench is towards the SDW phase, where $f(t)$ decays more slowly. It happens
because the equilibrium GS of the instantaneous Hamiltonian in the SDW portion
of the phase diagram keeps a certain overlap with the initial state, at which
$\left|\Psi\left(t\right)\right\rangle$ is frozen.

For intermediate $\tau_{U}$ or $\tau_{V}$, the fidelity initially decreases
and then stabilizes (with small oscillations) at a finite value, indicating
that the evolved state is not orthogonal to the equilibrium GS. As the ramping
time scale increases even further, we observe $f\thickapprox{\rm 1}$,
characteristic of an adiabatic evolution. The behavior of the fidelity at the
end of the quench evolution, $f(t_{f})$, as a function of $\tau_{V}$ and
$\tau_{U}$ is plotted in panels (c) and (d) of Fig. 3. For the quench towards
SDW ordering [see panels (b) and (d)], the process with $\tau_{U}=2.0$ is
nearly adiabatic, while the one with $\tau_{U}=2.5$ is already in the
adiabatic regime [$f(t_{f})=0.992$]. For the quench towards the CDW ordering
[panels (a) and (c)], on the other hand, we obtain $f(t_{f})=0.903$ for
$\tau_{V}=5.0$, which is still in the intermediate regime. The adiabatic
evolution is reached only with a larger ramping time, $\tau_{V}=12.0$
[$f(t_{f})=0.996$], as already pointed out when we discussed the order
parameter in the previous subsection.

We note that, for $\tau_{V}=12.0$, at the beginning of the quench, the
fidelity is slightly smaller than our adiabatic criterium ($f>0.99$), with
minor excitations created close to the critical point, however, the system
equilibrates during the finite-time quench. Such a behavior is also obtained
for $\tau_{U}=2.5$ in the quench towards the SDW phase, as can be seen in the
insets of Fig. 3, which depict a zoom in the region of $f\approx 1$ of the
data in panels (a) and (b). For larger $\tau$ (see $\tau_{V}=20$ and
$\tau_{U}=5.0$), the fidelity remains above 0.99 during the whole quench. The
differences between the evolution driven by the onsite or the nearest-neighbor
interaction will be further explored below.

####  III.1.3 Entanglement entropy

To quantify how non-local correlations between parts of our system evolve
along the applied quenches, we evaluate the bipartite entanglement entropy,
defined as $EE=-\sum_{i}\lambda_{i}{\rm log}_{2}\lambda_{i}$, where the set
$\left\\{\lambda_{i}\right\\}$ is the so-called entanglement spectrum of
eigenvalues of the reduce density matrix $\hat{\rho}_{{\rm A}/{\rm B}}={\rm
Tr}_{{\rm B}/{\rm A}}\left|\Psi\right\rangle\left\langle\Psi\right|$. Here, we
consider that the subsystem A contains the $(L-1)/2$ leftmost sites and B the
$(L+1)/2$ rightmost one. We checked that the behavior of $EE$ observed by us
does not qualitatively change if we take other subsystem sizes.

The evolution of the $EE$ throughout the quenches we consider is shown in Fig.
2(c) and (d). For comparison, we also display the $EE$ corresponding to the
instantaneous equilibrium GS (green lines). For the smallest values of the
quench time ($\tau_{U}=0.01$ and $\tau_{V}=0.01$), that is, in the impulse
regime, the entropy does not evolve from the respective initial values, those
that correspond to $U_{0}=0$ or $V_{0}=0$. More interestingly, in quenches to
the CDW phase, we observe an enhancement of the $EE$ for $\tau_{V}=0.33$,
$\tau_{V}=2.0$, $\tau_{V}=3.5$, and $\tau_{V}=5.0$ [cyan, red, orange, and
magenta lines in Fig. 2(c)]. For $\tau_{V}=0.33$ the $EE$ increases
monotonically during the quench. For $\tau_{V}=2.0$, it increases up to
$\approx 1.8$ and roughly saturates - in this case the CDW is not well-formed,
as indicated by the small values of $|m_{CDW}|$ observed in panel (a). In the
case of $\tau_{V}=3.5$ and $\tau_{V}=5.0$, we observe a smaller enhancement of
the $EE$, which is followed by a sudden suppression towards the GS behavior.
These results indicate that during a quench in the intermediate time regime,
before approximating the CDW ordering, the system access more entanglement
excited states. On the other hand, we do not find any considerable enhancement
of the $EE$ along the formation of the SDW phase, as can be noticed in Fig.
2(d). These observations suggest that the formation of the CDW ordering along
a linear quench requires a larger rearrangement of our system than the
formation of the SDW one. An increase of $EE$ close to a critical point was
also observed in the dynamics of the Ising model with a time-dependent
transverse field [41].

In the intermediate regime, we also observe oscillations in the $EE$
throughout the quench evolution - look at the results for $\tau_{V}=3.5$ and
$\tau_{U}=1.0$ (orange lines). These oscillations decrease in amplitude as we
approach the adiabatic regime. In this case, the $EE$ follows closely the
results for the equilibrium GS state (green lines), as expected - both of them
decrease linearly when the system goes from the metallic to the (ordered)
insulating state. Oscillations in the $EE$ as well as in the expectation
values of operators that do not commute with the Hamiltonian were observed in
spin chains after sweeping the Hamiltonian through a critical point [42, 41].
These features can be viewed as consequences of the fact that the time evolved
state includes excited states of the equilibrium Hamiltonian.

####  III.1.4 Deviations with respect to equilibrium ground state

To summarize the main differences between the quenches towards the CDW and SDW
phases, we define the deviation of a given quantity $\Theta$ as

$\Delta\Theta=\frac{\sum\left|\Theta_{t}-\Theta_{GS}\right|}{\sum\left|\Theta_{GS}\right|},$
(7)

where $\Theta_{t}$ and $\Theta_{GS}$ refer to the values of $\Theta$ evaluated
with the evolved state and the equilibrium GS, respectively, and the sum is
over the instant of times between $t=0$ and $t=t_{f}$, that is, over the
quench time evolution. In Fig. 4(a) we present our results for the deviations
of $m_{CDW}$ and $m_{SDW}$ as a function of the respective quench ramping
times, $\tau_{V}$ for the quenches towards the CDW phase and $\tau_{U}$ for
quenches aiming the SDW phase. Here, we focus on the curves at which only one
of the interactions, either $U$ or $V$, are turned on (solid lines); the other
results will be discussed in the next section.

Figure 4: Deviations as defined in the text calculated for (a) the CDW and SDW
order parameters and (b) the $EE$ as a function of the ramping times,
$\tau_{V}$ for the quench towards the CDW phase (curves in red) and $\tau_{U}$
for the quench towards the SDW phase (curves in black).

On one hand, in both quench directions we observe that the order parameter
deviations decrease monotonically as the ramping time increases. However, they
vanish at a smaller $\tau$ when the quench is towards the SDW phase (black
curves) than when it is towards the CDW phase (red curves), emphasizing that
the adiabatic regime can be reached for smaller quench time scales in the
former. On the other hand, the $EE$ deviations shown in Fig. 4(b) features a
non-monotonic behavior for the quench towards the CDW ordering, with enhanced
deviations in the intermediate regime, compatible with the $EE$ behavior
observed in Fig. 2(c). In the case of the quench towards the SDW ordering,
practically no enhancement is observed in the EE, in agreement with Fig. 2(d).
A comparison between the red (CDW) and black (SDW) curves emphasizes the fact
that in the former excited states are accessed in the intermediate regime and
thus the adiabatic case is reached with a larger ramping time.

###  III.2 Turning on $U$ and $V$ simultaneously

We now investigate the effects of turning on both interactions, $U$ and $V$,
simultaneously. At equilibrium, the Hubbard model [Eq. (1) with
$U(t)=U_{eq}\neq 0$ and $V(t)=V_{eq}=0$] can be exactly solved by the Bethe
ansatz method [43]. Its extended version ($V_{eq}\neq 0$), on the other hand,
is non-integrable for general values of the model parameters [44]. It was
shown that integrable many-body quantum systems in one dimension could undergo
relaxation to an equilibrium state described by the Generalized Gibbs ensemble
[45] after a quench. The same does not apply to non-integrable models [46,
47]. Such a difference has motivated us to address the effects of small $V$
values on the quench towards the SDW phase and small $U$ on the quench towards
the CDW phase.

In Fig. 4 we present the deviations of the different quantities when
considering the quenches paths numbered 2, 3, 4, and 5 in panels (a) and (d)
of Fig. 1. We compare them with the results for the vertical and horizontal
quenches (paths 1 and 6) described in the previous section. We observe that
the deviations of the order parameter and of the $EE$ decrease as $U_{f}$
increases for fixed $V_{f}=3$ (quenches towards the CDW phase) or $V_{f}$
increases for fixed $U_{f}=6$ (quenches towards the SDW phase). More
importantly, the three distinct regimes - sudden quench, intermediate, and
adiabatic - are still clearly observed as a function of $\tau$ for the
quenches with non-zero $U_{f}$ and $V_{f}$. In fact, we find very similar
results to the ones discussed in the previous section, when only one
interaction is turned on, indicating that the behavior of the evolved state
during the finite-time quench does not depend on the integrability of our
model.

##  IV Conclusions

In this work we have performed time-dependent DMRG calculations to study the
formation of CDW and SDW phases within the extended Hubbard model. To this
aim, we have considered that our system is subjected to an interaction quench,
that is, it is prepared in an initial non-interacting state and the
interactions then change over a finite interval of time until their final
values are reached.

For the quenches we have analyzed, three different quench regimes - impulse,
intermediate, and adiabatic - are observed depending on the ramping quench
time $\tau$. For small $\tau$, we have an impulse regime, in which the system
remains frozen in the initial state. In the intermediate regime, for the
quench towards the CDW phase, we observe an increase of the entanglement
entropy with respect to the initial value, not seen for the quench towards the
SDW phase. This suggests that, during the time evolution towards the
electronic CDW phase, more entanglement excited states of the equilibrium
Hamiltonian are accessed, which does not happen during the formation of the
SDW phase. As a consequence, we observe that the third regime, the adiabatic
one, is reached with smaller ramping time scale if a SDW ordered state is
formed as compared to the formation of a CDW state - the latter has to happen
slowly to prevent entangled excited states from being accessed during the
quench.

Finally, our findings show that the breaking of the system integrability,
produced by turning on the nearest-neighbor interaction $V$, does not induce
significant changes in the non-equilibrium behavior during our quench. We
believe, however, that the non-integrability can affect the free evolution
after the quench, especially for the states generated in the intermediate
regime, since the system is excited at $t=t_{f}$. This effect can be
investigated in future work.

###### Acknowledgements.

The authors acknowledge financial support from the Brazilian agencies CAPES,
CNPq (in particular CNPq INCT-IQ and grant 422300/2021-7), FAPDF, and FAPEMIG.

##  Appendix A Dependence of the results on the system size

To analyze how our results depend on the number of chain sites, we have
selected the quench with $U_{f}=1,V_{f}=3$ [path #2 in Fig. 1(a)] as
representative of the formation of the CDW ordering and that with
$U_{f}=6,V_{f}=1$ [path #5 in Fig. 1(d)] for the case of the SDW ordering. In
Fig. 5, we show the deviations of the order parameters and of the EE as a
function of the quench time for different chain length $L$. As can be noticed
in the figure, the increase in the number of sites does not qualitatively
affect our findings. Moreover, before achieving the adiabatic behavior, we
observe that chains with more sites present larger deviations of both
quantities analyzed. According to the adiabatic theorem, small energy gaps
between the ground and first excited states lead to more excitations when the
system crosses a critical point, which in our case occurs when the
interactions are turned on (the system initial state coincides with the
critical point). For systems with a finite number of sites, the gap decreases
as the chain size increases [42], which leads to more excitations at the
beginning of the quench evolution and thus to an increase in the deviations of
quantities along the quench, as observed in our results.

Figure 5: Deviations of (a) the order parameters and (b) the $EE$ as a
function of the quench ramping time $\tau_{V}$ or $\tau_{U}$, depending on
whether the quench is towards the CDW (red curves, for which
$U_{f}=1,V_{f}=3$) or the SDW (black curves, for which $U_{f}=6,V_{f}=1$)
phase. Results for different chain length $L$ are shown, including the one,
$L=17$, considered in the other plots.

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