# Many-body dynamical phase transition in quasi-periodic potential

Ranjan Modak1 and Debraj Rakshit1 1 Department of Physics,Institute of
Science, BHU, Varanasi-221005, India

###### Abstract

Much has been learned regarding dynamical quantum phase transition (DQPT) due
to sudden quenches across quantum critical points in traditional quantum
systems. However, not much has been explored when a system undergoes a
localization-delocalization transition. Here, we study one dimensional
fermionic systems in presence of a quasi-periodic potential, which induces
delocalization-localization transition even in 1D. We show signatures of DQPT
in the many-body dynamics, when quenching is performed between phases
belonging to different universality classes. We investigate how the non-
analyticity in the dynamical free energy gets affected with filling fractions
in the bare system and, further, study the fate of DQPT under interaction.
Strikingly, whenever quenching is performed from the low-entangled localized
phase to the high-entangled delocalized phase, our studies suggest an intimate
relationship between DQPT and the _rate_ of the entanglement growth – Faster
growths of entanglement entropy ensures quicker manifestation of the non-
analiticties in the many-body dynamical free energy.

##  I Introduction

Isolated out-of-equilibrium systems posses many challenges, and concurrently,
offers new possibilities: Whereas it gives rise to complex physical phenomena
beyond the reach of equilibrium statistical mechanics, it also opens door for
discerning equilibrium statistical properties in an unique manner Eisert15 .
Dynamical quantum phase transition (DQPT) Heyl18  constitutes a prime example
that has illuminated a new light on the traditional understanding of quantum
phase transition, which is developed on the pillars of equilibrium statistical
mechanics.

DQPT is built upon by drawing parallel ideas borrowed from equilibrium
physics, where Loschmidt amplitude, $\mathcal{L}(t)$, does a similar kind of
business that the partition function does in the equilibrium physics of
quantum phase transition. $\mathcal{L}(t)$ quantifies the overlap between an
initial state and the time-evolved state, and is defined as
$\mathcal{L}(t)=\langle\psi_{i}|\exp(-i\hat{H}t)|\psi_{i}\rangle$, where
$|\psi_{i}\rangle$ is an initial state, and $\hat{H}$ is the driving
Hamiltonian. The signature of a dynamical phase transition is imprinted in
form of the non-analyticities at certain time instances, $t=t^{*}$, in the
dynamical free energy, $f(t)=-2\lim_{L\to\infty}\ln|\mathcal{L}(t)|/L$.

DQPT has been investigated elaborately in various contexts of quantum phase
transitions in quantum model systems Heyl13 ; Sedlmayr19 ; Karrasch13 ;
Andraschko14 ; Halimeh18 ; Mishra18 ; Shpielberg18 ; Srivastav19 ; Zunkovic18
; Hagymasi19 ; Huang19 ; Soriente19 ; Cao19 ; Gurarie19 ; Abdi19 ; Haldar20 ,
and also for topological transitions Vajna15 ; Schmitt15 ; Jafari17 ;
Sedlmayr18 ; Jafari18 ; Zache19 . Along with sudden quench, periodically
driven systems have been studied as well Sharma14 ; Yang19 . Although DQPT is
mostly studied with the pure ground state as an initial state, conceptual
generalization has been established in the cases of degenerate states, mixed
state, or even the open systems Abeling16 ; Mera17 ; Bhattacharya17 ; Heyl17 ;
Lang18a ; Lang18b ; Kyaw18 . Importantly, recent experimental advances with
quantum simulators Lewenstein07 ; Lewenstein12 ; Georgescu14 , that can mimic
real-time dynamics of isolated quantum many-body systems, justifies such
theoretical investigations. DQPT has already been experimentally verified in
laboratory _via_ ultracold atom Smale19 , and ion traps Jurcevic17 .

So far, conventional quantum phase transitions governed by Landau theory of
spontaneous symmetry breaking have enjoyed the bulk amount of attention. This
work, instead, considers localization-delocalization transition, which cannot
be described by simple Landau theory. Localization-delocalization transition
has been a subject of intense research since the seminal work of Anderson
(1958), which revealed that despite the quantum tunneling processes a quantum
particle may get localized in presence of a disorder Anderson58 ; Abrahams79 .
Many-body localization (MBL) further considers the effect of interaction in
addition to disorder, and off-late, has become a hot research topic Abani17 ;
Alet18 ; Abanin19 ; Nandkishore6 . Localization-delocalization transition
separates the localized phase from the thermal one, and a question that
naturally arises is whether the equilibrium physics of localization-
delocalization transition leaves a trace on the real-time dynamics, and if the
progressing time would play a role similar to the equilibrium control
parameter, i.e. the strength of randomness.

Several of the early works with Anderson localization and MBL were carried on
disordered quantum many-body systems, such as spin chains in presence of
random field or fermionic lattice systems with random onsite potential.
Another closely related setting considers quasi-periodic many-body systems
Aubry80 ; Modugno09 , also known as the the Aubry-André (AA) model, where
instead of pure randomness the incommensurate on-site potential drives a
system in the localized phase. In this model, the localization-delocalization
transition occurs for a finite incommensurate potential amplitude, say
$\Delta=\Delta_{c}$ Machida86 ; Geisel91 ; Roscilde08 ; Roux08 ; Deng08 ;
Albert10 ; Cai10 ; He12 ; Gullo15 . This is different from the usual Anderson
localization in one-dimension, which requires only an infinitesimal disorder
strength to localize all states. It also has been shown that by introducing
the interaction in such systems, the ergodic-MBL transition takes place at a
critical amplitude $\Delta>\Delta_{c}$ Iyer13 . Quasi-periodic on-site
potentials can be engineered in ultracold simulators in a controlled manner
Schreiber15 ; Luschen17 ; Luschen18 .

There has been a very recent effort in understanding the fate of DQPT in the
context of the single-particle localization-delocalization transition in case
of Aubrey-andre model Yang17 , where an analytical expressions of the
transition times ($t^{*}$) can be derived for a limiting situation of end-to-
end quench between two distinct phases. In order to build an in-depth
understandings of the _many-body aspects_ of the DQPT in localization-
delocalization transition, we consider fermions and probe the AA model at
finite filling fractions. Owing to the added complexity associated with many-
body generalization, we perform time extension of density matrix
renormalization group (tDMRG) calculations _via_ matrix product state (MPS)
formalism white-2004 ; daley-2004 ; uli-2011 . We investigate the appearances
of the non analytic points and identify their trends for varied filling
fractions, interaction strength and initial states. Similar to the single-
particle calculation, we also find that the nonanalytic points in dynamical
free energy bearing the signatures of the DQPT appear only if an sudden quench
is performed across the transition point, i.e. the system is suddenly driven
from the localized (delocalized) to the delocalized (localized) phase.
Moreover, we investigate entanglement entropy, which often offers useful
insights on quantum many-body phenomenology Eisert10 ; Amic10 . In particular,
it has been recently suggested that the DQPTs may breed enhanced entropy
production around the transition times Jurcevic17 ; Hagymasi19 . We find this
to happen whenever the system is driven from a low-entanglement (localized)
regime to high-entanglement (delocalized) regime. In-fact, our results provide
solid evidence, which suggest that occurrences of DQPTs are intimately related
to the _rate_ of information spreading, i.e. the entanglement growth rate.
However, this may not be uni-vocally true, particularly, decisive statements
cannot be passed for a sudden quench occurring other way around as highly-
entangled initial states are driven by low-entangled Hamiltonians, hardly any
entanglement production occurs. It’s now possible to measure entanglement
entropy in cold-atom experiments Islam15 ; Kaufman16 , and hence our results
can be experimentally verified.

Here is how the rest of the paper is organized. In Sec. II, we introduce the
model. Section III discusses DQPT in fractionally filled non-interacting
systems. In-particular, we concentrate on the situation when quenching is
performed between phases belonging to different universality classes, i.e. the
system is suddenly quenched from the (delocalized) localized to the
(localized) delocalized phase. Moreover, we also study entanglement production
associated with DQPT. In Sec. IV, we investigate the effects of interactions.
Finally, we draw conclusions in Sec. V.

##  II Model

We study a system of fermions in an one-dimensional lattice of size $L$, which
is described by the following Hamiltonian:

$\displaystyle\hat{H}(\Delta)$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{L-1}(\hat{c}^{{\dagger}}_{i}\hat{c}_{i+1}+\text{H.c.})+\Delta\sum_{i=1}^{L}\cos(2\pi\alpha
i)\hat{n}_{i}$ (1) $\displaystyle+$ $\displaystyle
V\sum_{i}\hat{n}_{i}\hat{n}_{i+1},$

where $\hat{c}^{{\dagger}}_{i}$ ( $\hat{c}_{i}$) is the fermionic creation
(annihilation) operator at site $i$,
$\hat{n}_{i}=\hat{c}^{{\dagger}}_{i}\hat{c}_{i}$ is the number operator, and
$\alpha$ is an irrational number. Without loss of any generality, we choose
$\alpha=\frac{\sqrt{5}-1}{2}$ for all the calculations presented in this work.
In the absence of interaction i.e. $V=0$, the Hamiltonian $\hat{H}$ is known
as Aubry-André (AA) model. It supports a delocalization-localization
transition as one tunes $\Delta$. In the thermodynamic limit, $\Delta=2$
corresponds to the transition point Aubry80 .

For the most calculations in this paper, the system initially prepared in a
ground state $|\psi_{i}\rangle$ of the Hamiltonian $\hat{H}$ for
$\Delta=\Delta_{i}$. We study subsequent unitary dynamics of the initial state
following a sudden quench: $\hat{H}(\Delta_{i})\to\hat{H}(\Delta_{f})$. The
time evolved state is given by
$|\psi(t)\rangle=e^{-i\hat{H}(\Delta_{f})t}|\psi_{i}\rangle$. In order to
detect DQPT, we focus on dynamical free energy, which is described by,

$f(t)=-\lim_{L\to\infty}\frac{2}{L}\ln|\langle\psi(t)|\psi_{i}\rangle|.$ (2)

It has been recently reported that DQPT occurs in single particle Aubry-André
model, once the quenching is performed from the localized phase to the
delocalized phase or vice-versa Aubry80 . Here, our interest is to understand
the fate of the DQPT when finite number of particles are loaded in such
systems. The filling fraction is identified as $\nu=N/L$, where $N$ is the
total number of fermions. Given that we use tDMRG method for all our
calculations, we restrict our-selves to the open boundary condition.

##  III Fractionally filled bare systems

Given that our main aim is to understand the _many-body aspects_ in DQPT for
localization-delocalization transition, we first focus on the Hamiltonian
$\hat{H}$ in the absence of interaction i.e. $V=0$. Subsequently, we study two
distinct cases - quenching the Hamiltonian from (A) the delocalized to
localized phase and (B) the localized to the delocalized phase.

Figure 1: _Delocalized to localized quench – Varied filling fractions._
Variation of the dynamical free energy $f(t)$ as a function of time $t$ for
different values of the filling fractions, $\nu$, for (a) $L=40$ and (b)
$L=60$. In the inset of (a) we bench mark our tDMRG results with exact
diagonalization calculations for $L=16$ at half filling. Inset of (b) shows
the variation of the $1^{st}$ non-analytic point $t^{*}$ with filling fraction
$\nu$, where the triangle, circle, and diamond symbols correspond to $L=40$,
60, and 80 respectively. We choose $\Delta_{i}=0.2$ and $\Delta_{f}=100$.

###  III.1 Delocalized to localized case

In this section we primarily investigate the behavior of dynamical free energy
$f(t)$ as we quench the Hamiltonian (1) from the delocalized phase to the deep
into localized phase, i.e. $\Delta_{i}=0.2\to\Delta_{f}=100$. Figure 1 shows
the variation of $f(t)$ with time for different values of the filling
fractions. Two different system sizes are studied $L=40$ (Fig. 1(a)) and
$L=60$ (Fig. 1(b)). In the single-particle limit, an analytical expression for
$f(t)$ can be derived in the thermodynamic limit,
$f(t)=-\frac{2}{L}\ln|J_{0}(\Delta_{f}t)|$, where $J_{0}$ is is the zero-order
Bessel function, for end-to-end quench with $\Delta_{i}=0\to\Delta_{f}=\infty$
Yang17 . It implies that the non-analytic points in the dynamical free energy
appear at $t^{*}_{c}=x_{c}/\Delta_{f}$, where $x_{c}$ with $c=1$, 2, 3 .. are
zeroes of $J_{0}(x)$. Even for our choice of quench parameters (where
$\Delta_{i}=0.2$ and $\Delta_{f}=100$), it turns out that the non-analytic
points are $t^{*}_{c}\simeq x_{c}/\Delta_{f}$ for $N=1$. However, the non-
analytic points in $f(t)$ start shifting with increasing number of particles.
Given that in this work we primarily concentrate on the 1st non-analytic
point, it turns out that as we increase the filling fraction $\nu$, the 1st
non-analytic point starts moving towards smaller values of $t^{*}$. We find
that while for the single particle case, $\Delta_{f}t^{*}\simeq 2.41$, for
$\nu=1/2$, $\Delta_{f}t^{*}\simeq 1.95$. In the inset of Fig. 1(b), we show
the positioning of the $1^{st}$ non-analytic point as a function of filling
fraction for different system sizes, $L$ = 40, 60 and 80.

Next, we analyze finite size effects on the $1^{st}$ non-analytic point. For
this, first we consider different values of $\Delta_{i}$, while keeping the
driving Hamiltonian $(\Delta_{f}=100)$, filling fraction $(\nu=1/2)$, and the
system size $(L=40)$ fixed. Fig. 2 (a) demonstrates variations of $f(t)$ with
$\Delta_{f}t$ for different choices of the initial states, which are
determined by setting $\Delta_{i}=$ 0.1, 0.2, 0.5, 1.0, 2.5, and 5.0. It is
evident that the cases with $\Delta_{i}=0.1$ and $0.2$ are characterized by
presence of ”cusps” in time dynamics, providing clear signatures of the non-
analyticities. However, the non-analytic features are not very clear for other
values of $\Delta_{i}$. In contrast they appear like ”humps”.

In fig. 2 (b) and fig. 2 (c), we show the variation of these ”humps” as we
increase the system size $L$, for $\Delta_{i}=0.5$ and $\Delta_{i}=5.0$,
respectively. We find that while for $\Delta_{i}=0.5$ the ”hump” is getting
sharper as we increase $L$ (this feature has been observed for other values of
$\Delta_{i}$ as well as long as $\Delta_{i}\lesssim 2$). On the other hand for
$\Delta_{i}=5.0$, the peak flattens with increasing $L$. The trends clearly
indicate that in the thermodynamic limit the non-analytic behaviour survives
when the quench is performed from the delocalized phase to localized phase.
However, when we quench the Hamiltonian (1) without crossing the phase
boundary i.e. $\Delta_{i}>2$, the signature of non-analytic behaviour in
$f(t)$ tends to wash away. Similar conclusions also has been achieved for
$N=1$ as well in Ref. Yang17 .

Figure 2: _Delocalized to localized quench – Varied initial states._ (a) shows
the variation of the dynamical free energy as a function of time for a fixed
value of $\Delta_{f}$, but for different choices of $\Delta_{i}$. We choose
$\Delta_{f}=100$, $L=40$, and $\nu=1/2$. Analysis of the finite size effects
are presented in (b) for $\Delta_{i}=0.5$ and in (c) for $\Delta_{i}=5.0$,
which monitor the change in the sharpness of the $1^{st}$ peak of the
dynamical free energy with increasing system size. The chosen parameters are
again $\Delta_{f}=100$, and $\nu=1/2$.

###  III.2 Localized to delocalized case

Now we focus on a quench from the localized phase to the delocalized phase.
Figure 3 shows the results for a quench from $\Delta_{i}=100$
$\to\Delta_{f}=0.2$. For the single-particle case i.e. $N=1$, it is straight
froward to analytically track the non-analytic points in $f(t)$ corresponding
to a quench from $\Delta_{i}=\infty\to\Delta_{f}=0$. The non-analytic points
in $f(t)$ corresponds to $t^{*}=x_{c}/2$, where $x_{c}$ are zeroes of
$J_{0}(x)$ Yang17 . Even for our choice of quench parameters i.e.
$\Delta_{i}=100$ $\to\Delta_{f}=0.2$, our single-particle numerical results
agrees well with the analytical results for the transition times, $t^{*}\simeq
x_{c}/2$. However, as we increase the filling fraction, $t^{*}$ starts
relocating. This is shown in the inset of fig. 3 (a). We further show, by
varying $\Delta_{i}$, while keeping $\Delta_{f}=0.2$ and filling fraction
$\nu=1/2$ fixed, that if we quench from the deep into the localized phase to
the delocalized phase, the non-analytic features become much sharper, at least
for our choice of system size (see Fig. 3 (b)). On the other hand, when
$\Delta_{i}\lesssim 2$, the signature of non-analyticity in $f(t)$ fades away.
This is once again an evidence of absence of the DQPT if the quench is
performed within the same phase.

Figure 3: _Localized to delocalized quench._ Variation of the dynamical free
energy $f(t)$ with time $t$ is shown in (a) for different filling fractions,
$\nu$ and (b) for different initial states. Inset of (a) shows the variation
1st non-analytic point $t^{*}$ with $\nu$, where $\Delta_{i}=100$ and
$\Delta_{f}=0.2$. (b) shows the Variation of $f(t)$ with $t$ for $\nu=1/2$,
$\Delta_{f}=0.2$ for different values of $\Delta_{i}$. $L=40$ for all the
cases.

###  III.3 Entanglement production

In this section, we investigate the connection between the DQPT and the
entanglement production. Given that here we quench from extremely deep
localized phase ($\Delta_{i}=100$) to the delocalized phase, the initial state
can be approximated quite accurately by a suitable product state. We have
checked this explicitly by choosing two states, 1) the exact ground state of
the pre-quench Hamiltonian $\hat{H}(\Delta_{i}=100)$, and 2) approximated
product state obtained by filling desired number (depends on the total number
of particles in the system) of lowest energy levels of the Hamiltonian
$\hat{H}_{0}=\sum_{i=1}^{L}\cos(2\pi\alpha i)\hat{n}_{i}$. We find that the
results of $f(t)$ vs $t$ for these two states are almost indistinguishable.
Hence, in Fig. 4 we don’t restrict ourselves to only the ground state of the
pre-quench Hamiltonian. We investigate the variation $f(t)$ with $t$ for
different initial product state (e.g. Neel state). Given that all these
product states have extremely high overlap with some excited eigenstates of
the pre-quench Hamiltonian $\hat{H}(\Delta_{i}=100)$, temporal variation of
$f(t)$ for such states are expected to be very similar to some highly excited
energy eigenstates of the Hamiltonian $\hat{H}(\Delta_{i}=100)$. Figure. 4(a)
shows that for many such states indeed one can find the clear signature of the
DQPT. However, the location of the non-analytic points $t^{*}$ varies quite a
lot as we change our initial states.

Next, we study the entanglement dynamics for the same set of initial states
once we let it evolve under the unitary evolution governed by a Hamiltonian
belonging in the delocalized phase $\hat{H}(\Delta_{f}=0.2)$. Given that the
dynamics is unitary, the entire system remains in a pure state. The reduced
density matrix $\rho_{A}$ of a finite subsystem A of length $L/2$ is defined
as $\rho_{A}=\mbox{Tr}\,_{B}|\psi(t)\rangle\langle\psi(t)|$, where the trace
is over the degrees of freedom of the complement B of A. Here we only restrict
ourselves to one of the most useful entanglement measures, the von Neumann
(entanglement) entropy $S=\mbox{Tr}\,[\rho_{A}\ln\rho_{A}]$.

Figure 4(b) shows the results for the entanglement growth for the same set of
initial states for which $f(t)$ has been plotted in the fig. 4 (a). We find an
extraordinary correlation, i.e. the appearance of the non-analytic point (at
least the 1st point) $t^{*}$ depends on the entanglement growth rate. At least
among the choices of our initial state it seems that the entanglement growth
rate for the Neel state i.e.
$|\psi(0)\rangle=\Pi_{i=1}^{L/2}\hat{c}^{{\dagger}}_{2i}|0\rangle=|1010..\rangle$
is maximum and where as, the entanglement growth rate of
$|\psi(0)\rangle=|11110000..\rangle$ is minimum. That is also reflected in the
$f(t)$ vs $t$ plot in fig. 4 i.e. the value of $t^{*}$ for the Neel state is
much smaller compare to other states. Intuitively we would expect that as
well. Given that the non-analytic points in $f(t)$ corresponds to the zero
fidelity $\mathcal{L}(t)$ in the thermodynamic limit, we would expect the
speed at which the memory of the initial state will be washed away should also
depends on the entanglement growth rate. If the propagation rate of quantum
correlation is higher, fidelity is expected to vanish in a much shorter time
compared to the case where the rate of propagation of quantum correlation is
smaller. This is precisely what has been manifested in Fig. 4.

Now given that the quench Hamiltonian $\hat{H}(\Delta_{f})$ is integrable in
the limit $\Delta_{f}\to 0$, the entanglement dynamics also can be described
extremely efficiently within quasi-particle picture calabrese-2005 . To
understand why this is the case, let us first briefly describe the
quasiparticle picture for the entanglement spreading which is applicable to
generic integrable models. According to this picture, the initial state acts
as a source of quasiparticle excitations which are produced in pairs and
uniformly in space. After being created, the quasiparticles move ballistically
through the system with opposite velocities. Only quasiparticles created at
the same point in space are entangled, and while they move far apart, they
carry forward entanglement and correlation in the system. A pair contributes
to the entanglement entropy at time $t$ only if one particle of the pair is in
A (of size $\ell$) and its partner is in B. Keeping track of the linear
trajectories of the particles, it is easy to conclude calabrese-2005 ;
alba-2016

$S(t)=\sum_{n}2t\int_{2|v_{n}|t\langle\ell}v_{n}(k)s_{n}(k)dk+\ell\int_{2|v_{n}|t\rangle\ell}s_{n}(k)dk.$
(3)

Here the sum is over the species of particles $n$ whose number depends on the
model, $k$ represents their quasimomentum (rapidity), $v_{n}(k)$ is their
velocity, and $s_{n}(k)$ their contribution to the entanglement entropy. The
quasiparticle prediction for the entanglement entropy holds true in the space-
time scaling limit, i.e. $t$,$\ell\to\infty$ with the ratio $t/\ell$ fixed.
When a maximum quasiparticle velocity $v_{M}$ exists (e.g., as a consequence
of the Lieb-Robinson bound), Eq. 3 predicts that for $t\leq\ell/(2v_{M})$, $S$
grows linearly in time. Conversely, for $t>\ell/(2v_{M})$, only the second
term survives and the entanglement is extensive in the subsystem size, i.e.,
$S\propto\ell$. The validity of Eq. 3 has been tested both analytically and
numerically in free-fermion and free-boson models calabrese-2005 ;
fagotti-2008 ; ep-08 ; nr-14 ; kormos-2014 ; leda-2014 ; collura-2014 ; bhy-17
; hbmr-17 ; buyskikh-2016 ; fnr-17 ; mm-20 ; knn-19 ; dat-19  and in many
interacting integrable models alba-2016 ; PVCP18_I ; PVCP18_II ; MBPC17 ;
modak.2019 ; dmcf-06 .

Hence, in Fig. 4(b) we also show the quasi-particle results using solid lines
for $\Delta_{f}=0$ and for different initial product state. Also since, our
tDMRG results are for open boundary condition, we replace $t\to t/2$ in Eq.
(3). It seems that entanglement growth rate obtained within this semi-
classical picture matches reasonably well with our numerical results even
though our simulation is for finite size system and $\Delta_{f}=0.2$. Note
that even though the correlations between the entanglement growth _rate_ and
the positioning of the non-analytic point $t^{*}$ is extremely apparent when
we quench from the localized phase to the delocalized phase, but entanglement
dynamics is featureless when quench is performed from the highly entangled
delocalized phase to the localized phase. However, signature of the DQPT
remains present in the temporal variation of $f(t)$ [see Fig. 1].

Figure 4: _Entanglement production and quasi-particle predictions._(a)
Dynamical free energy as a function of time for different initial product
states for $L=40$ and $\Delta_{f}=0.2$ at half filling. (b) shows the
entanglement entropy for the same set of initial states and parameters. Solid
lines correspond to quasi-particle predictions.

##  IV Effects of interaction

For all our previous calculations, we set the interaction term in the
Hamiltonian $\hat{H}$ (see Eq. (1)) to zero, i.e. $V=0$. Here we investigate
the effect of interactions. While we know the Hamiltonian in (1) exhibits
localization-delocalization transition at $\Delta=2$ in the non-interacting
limit, the effect of interaction usually tends to delocalized the system.
However, it has been shown that even in the presence of interaction for the
sufficiently large value of the disorder strength (in this case the strength
of incommensurate potential) localization-delocalization transition takes
place, which is also known as ergodic to Many-body localization transition
Iyer13 .

Here, we perform a quench from the deep in to the many-body localized phase to
the delocalized ergodic phase phase corresponding to $\Delta_{i}=100$ $\to$
$\Delta_{f}=0.2$ and $V\neq 0$. We find that as we increase the interaction
strength $V$, the signature of the non-analyticity in $f(t)$ slowly fades way
as shown in fig. 5 (a). However interestingly we also observe that the value
of $t^{*}$ becomes smaller with the increase of $V$. These results can also be
complimented by the entanglement dynamics results in Fig. 5 (b), which show
that indeed the entanglement growth is comparatively faster if we increase the
interaction strength.

Figure 5: _Effects of interaction._ (a) Dynamical free energy, $f(t)$,
variation with time, $t$, for different interaction strengths, $V$, for
$\Delta_{i}=100$, and $\Delta_{f}=0.2$. (b) shows entanglement dynamics for
the same set of interaction strengths and system parameters.

##  V Summary and Discussions

The primary goal of this is work to understand the many-body aspects in the
context of DQPT, for which we turned our focus to a less explored quasi-
periodic system supporting a localization-delocalization transition. Our
_many-body_ results indicate that DQPT is a generic feature of sudden quenches
across the localization-delocalization transition point.

First we consider the non-interacting systems with finite particle densities,
and investigate how the DQPT gets modified as a function the filling fraction
when the system is quenched from the delocalized (localized) to the localized
(delocalized) phase. Further investigation is followed in order to understand
the effects of interaction. In the presence of interaction, single particle
localized phase modifies to athermal MBL phase. Here, we find a clear
signature of non-analyticity in the dynamical free energy, as well. Since,
unlike the usual quantum phase transition (which involves only the ground
states of the Hamiltonian), the highly excited eigenstates of the system are
also involved in MBL transition, we do not restrict ourselves to the only
ground state of the pre-quench Hamiltonian – We find that the signature of
DQPT persists even for the highly excited states.

While quenching from a low-entangled localized phase to high-entangled
delocalized phase, we find that DQPT is accompanied with entanglement
production. Crucially, following the trends of entanglement growth, we
conclude that the positioning of the non-analytic points $t^{*}$ intrinsically
depend on the _rate_ of entanglement growth.

However, the rate of entanglement growth does not solely determines the
positions of $t^{*}$. Clearly, the onsets of $t^{*}$ do not require the quasi-
particles to travel across the entire system and coming back to the initial
point in the space. The time required for quasi-particles to do so is given
by, $[t_{rev}\sim nL/(v_{M})]$ with $n$ as positive integers modak.2020 ,
which is much larger (also scales linearly with system size $L$) than the
transition times.

This is even more clear for the case when quenching is performed from the
delocalized phase (initial state is highly entangled) to the deep into
localized phase, which is only associated with a negligible entanglement
growth. However, even there the values of $t^{*}$ changes with the filling
fraction. Hence, further investigations are required to understand the
correlation between the entanglement growth and the DQPT. It will be
interesting to explore the roles of local entanglement or global genuine
multiparty entanglement. Even identifying the relations between quantum
scrambling and DQPT is also another potential future direction chen.2020 .

##  VI Acknowledgements

The authors acknowledges the support of IOE start-up grant. RM acknowledges
the support of DST-Inspire fellowship, by the Department of Science and
Technology, Government of India. Super computing Facilities at IIT (BHU) were
used to perform numerical computations.

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