# Entanglement dynamics of a hard-core quantum gas during a Joule expansion

Filiberto Ares1, Stefano Scopa1 and Sascha Wald2 1 SISSA and INFN, via Bonomea
265, 34136 Trieste, Italy  2 Statistical Physics Group, Centre for Fluid and
Complex Systems, Coventry University, Coventry, England

(August 30, 2024)

###### Abstract

We study the entanglement dynamics of a one-dimensional hard-core quantum gas
initially confined in a box of size $L$ with saturated density $\rho=1$. The
gas is suddenly released into a region of size $2L$ by moving one of the box
edges. We show that the analytic prediction for the entanglement entropy
obtained from quantum fluctuating hydrodynamics holds quantitatively true even
after several reflections of the gas against the box edges. We further
investigate the long time limit $t/L\gg 1$ where a Floquet picture of the non-
equilibrium dynamics emerges and hydrodynamics eventually breaks down.

###### Contents

  1. 1 Introduction
  2. 2 The model and the quench protocol
  3. 3 Hydrodynamics in the phase space
    1. 3.1 Analogy with the multiple domain wall setting
  4. 4 Entanglement dynamics
    1. 4.1 Quantum fluctuations and entanglement spreading
    2. 4.2 Joule expansion vs multiple domain wall setting
  5. 5 Long time limit: reflections, relaxation and recurrence
    1. 5.1 Long time limit of the density
    2. 5.2 Long time limit of the entanglement
  6. 6 Summary and conclusion
  7. A Calculation of the Fermi contour
  8. B Exact lattice calculations

##  1 Introduction

The formulation of Generalized Hydrodynamics (GHD) (see e.g. Ref. [1] and Ref.
[2, 3, 4, 5] for recent reviews) – that is, an Euler hydrodynamic theory for
one-dimensional integrable models – is beyond doubt a recent milestone in the
theory of out-of-equilibrium properties of many-body quantum systems. Its
success is self-evident since GHD allows to easily obtain analytical results
for the conserved charges and currents of interacting integrable models after
a quantum quench, opening, in this way, a new possibility to move the research
beyond the Tonks-Girardeau limit. It was originally formulated in Ref. [6, 7]
as an emergent large-scale theory of the front propagation dynamics obtained
by joining together two Generalized Gibbs ensembles, extending previous
results for the dynamics of such bi-partite states, see e.g. [8, 9, 10, 11,
12, 13, 14]. From this dawn, GHD has been extensively put to test against
numerical simulations e.g. [15, 16, 17, 18, 19, 20, 21, 22, 23], cold atom
experiments [24, 25] and by relaxing some hypothesis that could have
undermined its validity (see e.g. Ref. [26, 27, 28, 29, 30]), but nonetheless
it always returned a predictive power beyond expectations. On the other hand,
it is also known that GHD is a semi-classical theory and so it can only act as
background for some quantum fluctuations in the analysis of intrinsically
quantum features, such as entanglement. However, a reliable description of the
system is obtained by focusing on its low-energy regime and by building a
conformal field theory on top of the GHD for the description of the missing
large-scale quantum fluctuations. This method is generically referred to as
quantum fluctuating hydrodynamics (or, more simply, as quantum GHD) see Ref.
[31, 32, 33, 34, 35, 36].

As a consequence, some simple platforms for the study of quantum transport
with non-interacting spin chains (or, equivalently, with impenetrable quantum
particles) gained a renewed interest, especially those in which a scaling
behavior of the propagating fronts was observed long ago, see e.g. Ref. [37,
38, 39, 8, 9, 10, 11, 40, 12, 13, 41]. In fact, if such a scaling theory turns
out to be essentially equivalent to a non-interacting limit of GHD, a careful
usage of quantum fluctuating hydrodynamics is instead needed to make
quantitative predictions on the entanglement dynamics beyond numerical and
heuristic considerations [42, 43, 44, 45].

In this context, we shall investigate the entanglement dynamics after a
geometric quench where the boundary region of a trapped one-dimensional gas is
suddenly changed [46, 47, 48, 49]. In particular, we wish to understand
whether the presence of boundaries for the expanding gas will eventually spoil
the validity of the hydrodynamic approach and, if so, when this is expected to
happen. An ideal testbed for these ideas is the so-called Joule expansion [47,
48, 50, 51, 52, 53, 54], where the system undergoes a sudden expansion of the
containing tank as detailed in the following section. For this problem, we
characterize the entanglement dynamics across the reflections in the Euler
scaling limit and we present a Floquet picture for its long time limit. We
also show that this quench protocol gives the same set of conserved quantities
of an infinite system prepared in a periodic fashion and let to relax freely.
But this analogy does not extend to the entanglement properties. We further
discuss the failure of hydrodynamics at times much larger than the system size
and the emergence of revivals of the initial configuration.

Oultine. In Sec. 2, we introduce the model and the quench protocol considered
in this work. Afterwards, in Sec. 3, we construct a large-scale description of
the model in terms of the local occupation function of fermionic modes and we
study its time evolution in phase space. With this method, we reconstruct the
behavior of the conserved quantities during the quench dynamics. In Sec. 4, we
proceed with the re-quantization of the theory and with the calculation of the
entanglement entropy, following the discussion of the recent Ref. [33, 34,
35]. Finally, in Sec. 5, we discuss the long time limit and the breakdown of
hydrodynamics. In Sec. 6, we report our conclusions while details on the
calculation of the Fermi contour and on the numerical implementation are left
to A and B respectively.

##  2 The model and the quench protocol

We consider a one-dimensional gas of hard-core quantum particles with nearest
neighbor hopping and coupled to a potential $V$. This system is described by
the Hamiltonian

$\displaystyle\hat{H}=-\frac{1}{2}\sum_{j\in\mathbb{Z}}\left(\hat{c}^{\dagger}_{j}\hat{c}_{j+1}+\hat{c}^{\dagger}_{j+1}\hat{c}_{j}\right)+V(j,t)\hat{c}^{\dagger}_{j}\hat{c}_{j}.$
(1)

Here, $\hat{c}^{\dagger}_{j}$ and $\hat{c}_{j}$ denote the creation and
annihilation operator of a spinless fermion at lattice site $j$, satisfying
the standard canonical anticommutation relations
$\\{\hat{c}_{i},\hat{c}^{\dagger}_{j}\\}=\delta_{ij}$. We further consider the
situation in which the model is initially prepared in the ground state of the
Hamiltonian (1) with a box potential $V={\cal V}_{[a,b]}$ of infinitely high
edges

$V(j,t<0)={\cal
V}_{[-L,0]}(j)\equiv\lim_{\Lambda\to\infty}\begin{cases}-1,\quad\text{if
$j\in[-L,0]$};\\\\[4.0pt] \Lambda,\quad\text{otherwise}\end{cases}$ (2)

Figure 1: Illustration of the quantum Joule expansion quench protocol
considered in this work: We initially load a hard-core quantum gas in the
region $-L\leqslant j\leqslant 0$ with density $\rho=1$. At time $t=0^{+}$ we
suddenly move the right edge of the box confinement from position $j=0$ to
position $j=L$ and we study the unitary dynamics in the doubled box generated
by the hopping fermionic Hamiltonian in Eq. (1).

which is the product state

$\ket{\Psi(0)}=\left(\bigotimes_{j<-L}\ket{0}_{j}\right)\otimes\left(\bigotimes_{j\in[-L,0]}\ket{1}_{j}\right)\otimes\left(\bigotimes_{j\in>0}\ket{0}_{j}\right),$
(3)

where $\ket{n=0,1}_{j}$ are the eigenstates of the number operator
$\hat{c}^{\dagger}_{j}\hat{c}_{j}$ with eigenvalues $0,1$ respectively.

At time $t=0^{+}$ we suddenly double the size of the initial box confinement,
i.e. $V(j,t>0)={\cal V}_{[-L,L]}$, and we investigate the unitary dynamics
$\ket{\Psi(t)}=\exp(-\mathbf{i}t\hat{H})\ket{\Psi(0)}$ generated by the
Hamiltonian (1) in the expanded region of size $2L$, see Fig. 1 for an
illustration. We shall refer to this quench protocol as quantum Joule
expansion (QJE), in analogy to the well-known classical process in statistical
mechanics. Intuitively, the non-equilibrium dynamics during the QJE can be
divided in different regimes. First, for times $t\leqslant L$, we observe the
free expansion of a hard-core quantum gas towards the right vacuum. This free
expansion has been thoroughly investigated in the literature, see e.g. Refs.
[8, 10, 11, 12, 13] and Refs. [42, 43, 44, 55, 56, 45, 57, 58, 59, 60, 33] for
studies on the entanglement. As we shall see further below, at time $t=L$ the
first front of excitations arrives at the infinite potential well and is
reflected. Thus, for $t>L$ we observe a regime where the dynamics of the
quantum gas is affected by multiple reflections against the left and right
walls located at the positions $j=\pm L$. While the non-equilibrium profiles
of charges and currents after such reflections has been understood in
literature using semi-classical hydrodynamic pictures (see Sec. 3 below and,
e.g., Ref. [13]), we investigate the behavior of intrisically quantum
properties such as entanglement. To this end, in Sec. 4, we shall employ the
recently developed framework of quantum fluctuating hydrodynamics for the
exact asymptotic calculation of the Rényi entropies $S_{n}(j,t)$ of a
bipartition $A\cup B$, $A=[-L,j]$ and $B=[j+1,L]$, defined as

$S_{n}(j,t)=\frac{1}{1-n}\log\tr\hat{\rho}_{A}(t)^{n},\quad n\in\mathbb{N},$
(4)

where $\hat{\rho}_{A}(t)\equiv\tr_{B}\ket{\Psi(t)}\bra{\Psi(t)}$ is the
reduced density matrix of subsystem $A$ at a certain time $t$. From Eq. (4),
the von Neumann entanglement entropy
$S_{1}(j,t)=-\tr\hat{\rho}_{A}(t)\log\hat{\rho}_{A}(t)$ is obtained by taking
the analytic continuation of $S_{n}(j,t)$ for $n\to 1$.

##  3 Hydrodynamics in the phase space

Although the non-interacting nature of the quench problem under analysis might
call for an exact lattice description, we rather consider the non-equilibrium
dynamics of the QJE in the Euler scaling regime $j,t\to\infty$ at fixed
$j/t=\text{cst}$. In this regime, a hydrodynamic description of the initial
state in terms of the local Fermi occupation number $n(x,k)$ yields
asymptotically exact results for the conserved charges and currents of the
model. In this formulation, the initial state (3) corresponds to a macrostate
that can be expressed by the local Fermi occupation function [11, 10, 61, 33]

$n_{0}(x,k)=\begin{dcases}1,\quad\text{if $x\in[-L,0]$ and $-\pi\leqslant
k\leqslant\pi$,}\\\\[4.0pt] 0,\quad\text{otherwise},\end{dcases}$ (5)

where we replaced the lattice position $j\in\mathbb{Z}$ with a continuum
spatial variable $x\in\mathbb{R}$. Hence, the system is decomposed into non-
interacting fermionic modes that entirely fill the region $-L\leqslant
x\leqslant 0$, and leave the rest of the system empty. As the particles
populating the initial macrostate in Eq. (5) are non-interacting, one can
determine the time evolution of $n_{0}(x,k)$ by simply following the free
evolution $x_{t}=x+v(k)t$ of each particle with momentum $k$ from its initial
position $x$. This yields

$n_{t}(x,k)=n_{0}(x-v(k)t,k),$ (6)

with velocity $v(k)=\sin k$ for the free Fermi gas [62, 63].

Equivalently, due to the zero-entropy condition $n_{0}=\\{0,1\\}$ of Eq. (5)
which is preserved by the time evolution (cf Eq. (6)), one can conveniently
encode the information on the phase space dynamics in a local (split-)Fermi
sea $\Gamma_{t}(x)=\bigcup_{j=1}^{Q}[k_{F,2j-1},k_{F,2j}]$, with local Fermi
points $k_{F,j}(x,t)$ given as solution of the zero-entropy hydrodynamic
equation [18]

$\left(\partial_{t}+\sin k_{F,j}\partial_{x}\right)k_{F,j}=0,\qquad
j=1,\dots,2Q.$ (7)

In terms of $\Gamma_{t}(x)$, one can express the fermionic occupation function
(6) as

$n_{t}(x,k)=\begin{cases}1,\quad\text{if $k\in\Gamma_{t}(x)$};\\\\[4.0pt]
0,\quad\text{otherwise}\end{cases}$ (8)

and determine the profile of a conserved quantity $q$ as

$q(x,t)=\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}n_{t}(x,k)\
h_{q}(k)=\int_{\Gamma_{t}(x)}\frac{\mathrm{d}k}{2\pi}\ h_{q}(k),$ (9)

where $h_{q}(k)$ is the single-particle eigenvalue associated to $q$. For
concreteness, one finds that $h_{e}=-\cos k$ for the energy density $e$,
$h_{J_{\rho}}(k)=\sin k$ for the particle current $J_{\rho}$, $h_{\rho}=1$ for
the particle density $\rho$ and similarly for other quantities.

We first consider the regime $t\leqslant L$, characterized by the free
expansion of the quantum gas without reflections of the Fermi points. Since
the propagating modes on the initial Fermi contour are only those belonging to
the position $x=0$, it is easy to see that the entanglement spreads during the
quench dynamics inside a region $|x|\leqslant t$, the so-called light cone,
which is determined by the fastest particles $k=\pm\pi/2$ with velocity $v=\pm
1$. Hence, at position $0\leqslant x\leqslant t$, it is easy to show that Eq.
(7) is characterized by the two solutions

$\Gamma_{t}(x)\equiv\left[\arcsin\frac{x}{t};\pi-\arcsin\frac{x}{t}\right]$
(10)

while outside the light-cone

$\Gamma_{t}(x)=\emptyset,\quad\text{for}\ t>x\geqslant 0.$ (11)

An analogous treatment applies for positions $-L\leqslant x\leqslant 0$ by
exploiting the particle-hole symmetry of the problem. This leads to the useful
identity

$\Gamma_{t}(-x)=[-\pi,\pi]\setminus\Gamma_{t}(x),\quad\forall\ x\in[-L,L]\,.$
(12)

Using Eq. (10)-(11) in Eq. (9) with $h_{\rho}\equiv 1$, one obtains the
asymptotic density profile of the expanding gas before reflections as [8, 10,
64, 60, 34]

$\rho(x,t\leqslant
L)=\begin{cases}\bar{\rho}(x,t)\equiv\arccos(x/t)/\pi;\quad-t\leqslant
x\leqslant t;\\\\[4.0pt] 1,\quad-L\leqslant x<-t;\\\\[4.0pt]
0,\quad\text{otherwise.}\end{cases}$ (13)

At times $t>L$ , the system will undergo multiple reflections at the left and
right walls $x=\pm L$. Due to the absence of interactions and to the
particularly simple choice of potential, each particle with momentum $k$ is
elastically reflected at the wall and it simply inverts its trajectory,
$k\to-k$. This leads to the phase space picture depicted in Fig. 2.

Figure 2: Local occupation number $n_{t}(x,k)$ of the free Fermi gas in the
plane $x$-$k$ (gray region) and the associated Fermi contour $\Gamma_{t}$
(thick blue line) at time $t=0$ (left panel) and after the first reflection
$t>L$ (right panel). The green regions mark the reflected modes $k\to-k$ in
the hydrodynamic picture.

With this picture at hand, one can analytically determine the structure of the
Fermi contour $\Gamma_{t}(x)$ for given position $x$ and time $t$. Focusing on
$x>0$, one can write

$\Gamma_{t}(x)=\bigcup_{j=1}^{N_{+}+N_{-}}[k_{F,2j-1},k_{F,2j}]$ (14)

with

$N_{-}=\left\lfloor{\frac{t+x}{2L}}\right\rfloor,\quad
N_{+}=\left\lceil{\frac{t-x}{2L}}\right\rceil$ (15)

where $\lfloor x\rfloor$ (resp. $\lceil x\rceil$) is the standard floor (resp.
ceil) rounding of $x$ to integer value, and Fermi points defined as

$k_{F,\nu}(x,t)=\begin{cases}\arcsin\frac{x+2L(\nu-1)}{t},\quad\nu=1,\dots,N_{+};\\\\[4.0pt]
\pi-\arcsin\frac{x+2L(\nu-
N_{+}-1)}{t},\quad\nu=N_{+}+1,\dots,2N_{+};\\\\[4.0pt]
\arcsin\frac{x+2L(2N_{+}-\nu)}{t},\quad\nu=2N_{+}+1,\dots,2N_{+}+N_{-};\\\\[4.0pt]
-\pi-\arcsin\frac{x-2L(2Q-\nu+1)}{t},\qquad\nu=2N_{+}+N_{-}+1,\dots,2Q,\\\\[4.0pt]
\end{cases}$ (16)

sorted in increasing order to form the set $\\{k_{F,j}(x,t)\\}_{j=1}^{2Q}$,
with total number of Fermi points $2Q=2(N_{+}+N_{-})$.

For $x<0$, the Fermi contour can be obtained from Eq. (14) using (12). With
our notations, it is easy to recover the result of Eq. (10) since for
$t\leqslant L$ one finds that $N_{-}=0$ and $N_{+}=1$. Eqs. (14)-(16) are
obtained with simple but lengthy algebra that we report in A.

From the knowledge of $\Gamma_{t}(x)$, we evaluate the particle density using
Eq. (9) as

$\rho(x,t)=\sum_{j=1}^{Q}\frac{k_{F,2j-1}-k_{F,2j}}{2\pi}.$ (17)

In Fig. 3, we test our result against exact numerical calculations for the
lattice model finding an excellent agreement. We briefly discuss the
implementation used for the numerical calculations in B.

Figure 3: Particle density profile during a quantum Joule expansion at
different times (panels) as a function of the rescaled position $x/L$. The
asymptotic prediction (solid line) in Eq. (17) is compared with exact
numerical calculations (symbols) performed on a lattice with $300$ sites. The
agreement in each panel is extremely good.

###  3.1 Analogy with the multiple domain wall setting

It is instructive to notice that the phase space picture in Fig. 2 obtained
for the QJE at times $t>L$ can be equivalently interpreted as the result of an
initial configuration

$n^{(\text{copy})}(x,k)\equiv\tilde{n}(x\ {\rm mod}\
4L,k),\;\tilde{n}(|x|\leqslant 2L,k)=\begin{cases}1,\qquad\text{if
$|k|\leqslant\pi$ and $x\in[-2L,0]$};\\\\[4.0pt]
0,\;\text{otherwise}\end{cases}$ (18)

where the initial domain wall configuration of Sec. 3 is doubled and copied
over an infinite one-dimensional lattice with $4L$ periodicity in absence of
any confinement for $t\geqslant 0^{+}$, see Fig. 4 for an illustration. We
will often refer to this configuration as multiple domain wall setting (MDW).

Figure 4: Analogy of the phase space picture between the QJE and the MDW
configuration. By focusing on the region $|x|\leqslant L$ (yellow area), one
can equivalently interpret the superposition of the light cones (depicted with
solid lines) as result of reflections against walls at $x=\pm L$ (QJE) or as
the propagation of particles coming from other junctions located at
$x_{p}=2pL$, $p\in\mathbb{Z}$ in absence of confinement (MDW).

From this analogy, the density profile in Eq. (17) can be also written as the
sum of the density profiles coming from the junctions at $x_{p}=2pL$,
$p\in\mathbb{Z}$ and propagating up to position $|x|\leqslant L$ at time $t$
in absence of reflections (cf Eq. (13)) [47, 48]

$\rho(x,t)=\bar{\rho}(x,t)-\sum_{p=1}^{N_{-}}(-1)^{p}\bar{\rho}(x_{p}-x,t)+\sum_{p=1}^{N_{+}}(-1)^{p}\bar{\rho}(x_{p}+x,t),$
(19)

where we recall $\bar{\rho}(x,t)=\arccos(x/t)/\pi$ and
$N_{\mp}=\lfloor\frac{t\pm|x|}{2L}\rfloor$ is the farther particles source
that contribute to the charges profile at given $x$ and $t$, see A.

##  4 Entanglement dynamics

In this section, we wish to investigate the entanglement spreading during the
quantum Joule expansion. We notice that for the non-homogeneous quench problem
under analysis, standard lattice and field theory techniques are typically not
valid and, although being highly desirable, a quantum hydrodynamic description
obtained directly from the lattice model in Eq. (1) is still out-of-reach.
However, a fruitful possibility is established by the recently developed
framework of quantum fluctuating hydrodynamics [31, 32, 33, 34, 36, 35] where
relevant quantum processes in the low-energy regime are modeled as a Luttinger
liquid on top of the phase space hydrodynamics that we previously determined.
More precisely, it is expected that relevant quantum processes at low energy
are in the form of particle-hole excitations around the Fermi points and can
be conveniently described introducing a density fluctuating field

$\delta\hat{\rho}=\frac{1}{2\pi}\partial_{x}\hat{\phi},$ (20)

which is typically further expressed as the derivative of a bosonic field
$\hat{\phi}$. Following the standard bosonization (see e.g. Ref. [65, 66,
67]), we expand the lattice fermionic operator in terms of vertex operators of
the bosonic field obtaining

$\hat{c}_{x}^{\dagger}(t)\propto\textbf{:}\,\exp\left(\frac{\mathbf{i}}{2}(\hat{\phi}_{+}(x,t)-\hat{\phi}_{-}(x,t))\right)\,\textbf{:}+\text{subleading
terms}$ (21)

up to a semi-classical phase that is unimportant for our scopes (see e.g. Ref.
[64, 60, 31] for more details) and at leading order in the scaling dimension
of the field theory operators. Here, we considered a chiral decomposition of
the bosonic field $\hat{\phi}=\hat{\phi}_{+}+\hat{\phi}_{-}$ where
$\hat{\phi}_{-}$ (resp. $\hat{\phi}_{+}$) is the left- (resp. right) moving
component of the fluctuating field along the Fermi contour. Crucially, the
effective Hamiltonian that governs the dynamics of the quantum fluctuations is
that of a non-homogeneous Luttinger liquid [60, 68, 69, 31, 70, 71, 33, 34,
36, 35]

$\hat{H}_{LL}[\Gamma]=\frac{1}{2\pi}\int_{\Gamma}\frac{\mathrm{d}\theta}{2\pi}\
{\cal J}(\theta)\sin
k(\theta)\left(\partial_{\theta}\hat{\phi}_{a(\theta)}(\theta)\right)^{2},\qquad(a(\theta)\equiv\pm={\rm
sign}(k(\theta)))$ (22)

here reported in terms of a coordinate $\theta$ that parametrizes the Fermi
contour $\Gamma$ as a simple curve in the $x$-$k$ plane, ${\cal J}(\theta)$ is
the Jacobian factor associated to the change of coordinates $x\mapsto\theta$,
see also Ref. [33] for more details.

We now apply this framework in the analysis of entanglement dynamics for the
specific quench setting discussed in Sec. 2. In particular, due to the non-
interacting nature of the problem considered, quantum fluctuations in the
initial state are obtained as the ground state of the Luttinger liquid
Hamiltonian (22) with Fermi contour

$\Gamma_{0}=\\{(x,k)\ :\ x=0;\;-\pi\leqslant k\leqslant\pi\\},$ (23)

which allows for a very intuitive parametrization as $\theta\equiv k$.
Furthermore, due to the non-interacting nature of the underlying quench
protocol, the effect of the time evolution is simply to transport these
quantum fluctuations along the curve $\Gamma_{t}$ that gets deformed in time
according to the semi-classical hydrodynamics of Sec. 3.

###  4.1 Quantum fluctuations and entanglement spreading

We now focus on the calculation of the Rényi entropy of a bi-partition $A\cup
B$ with $A=[-L,x]$ and $B=(x,L]$ at fixed time $t$, as defined in Eq. (4). In
Sec. 3, we showed that a cut in real space at position $x$ and time $t$
identifies a set of Fermi points $\\{k_{F,j}(x,t)\\}_{j=1}^{2Q}$ given by Eq.
(16). Due to momentum conservation, such Fermi points are transported backward
in time up to the initial Fermi contour in Eq. (23) as [31, 33, 34, 35]

$\\{k_{F,j}(x,t)\\}_{j=1}^{2Q}\;\overset{\text{backward
evo.}}{\longrightarrow}\;\\{\theta_{j}\\}_{j=1}^{2Q}=\\{(-1)^{q_{j}}\
k_{F,j}(x,t)\\}_{j=1}^{2Q},$ (24)

and the set $\\{\theta_{j}\\}_{j=1}^{2Q}$ can be used to identify the boundary
points separating $A$ and $B$ along $\Gamma_{t}$, as discussed in the previous
section. In Eq. (24), $q_{j}$ is the number of reflections of the $j^{\rm th}$
particle at the walls given by

$q_{j}=\begin{cases}0,\quad\text{if $|t\sin k_{F,j}|<L$};\\\\[4.0pt]
q\in\mathbb{N},\quad\text{if $(2q-1)L<|t\sin k_{F,j}|<(2q+1)L$}.\end{cases}$
(25)

We illustrate the backward time evolution procedure in Fig. 5.

Figure 5: Backward evolution of the Fermi points
$\\{k_{F,j}(x,t)\\}_{j=1}^{4}$ at given entangling point $x$ and time $t>0$.
Notice that 3 and 4 take an extra minus sign ($q_{j=3,4}=1$) due to the
reflection occurred.

Once determined the set of coordinates $\\{\theta_{j}\\}_{j=1}^{2Q}$ along
$\Gamma_{t}$, we are ready to compute the Rényi entropy. In particular, as
well-known from Refs. [72, 73, 74], the Rényi entropy (4) can be split as the
sum of two contributions

$\begin{split}&S_{n}(x,t)=\frac{1}{1-n}\log\left[\epsilon(x,t)^{\Delta_{n}}\braket{\hat{\cal
T}_{n}(x,t)}\right]\\\\[4.0pt]
&\quad=\tilde{S}_{n}(x,t)-\frac{n+1}{12n}\log\epsilon(x,t).\end{split}$ (26)

The first term $\tilde{S}_{n}(x,t)$ is universal in the sense that it can be
calculated within the effective field theory description of Eq. (22) as the
expectation value of a twist field $\hat{\cal T}_{n}$ inserted at the
entangling point $x$. The twist field is a primary with scaling dimension
$\Delta_{n}=\frac{1}{12}(n-1/n)$, see Refs. [73, 74] for a comprehensive
discussion. Therefore, by decomposing $\hat{\cal T}_{n}$ into its chiral parts
$\hat{\tau}^{\pm}_{n}$, with scaling dimension $d^{\pm}_{n}=\Delta_{n}/2$, and
by performing the transformation of Eq. (24), $\tilde{S}_{n}(x,t)$ can be
determined from the $2Q$-point correlation function of $\hat{\tau}^{\pm}_{n}$
inserted at the points $\theta_{j}$

$\tilde{S}_{n}(x,t)=\frac{1}{1-n}\log\left[\prod_{j=1}^{2Q}\
\left|\frac{\mathrm{d}\theta_{j}}{\mathrm{d}x}\right|^{\nicefrac{{\Delta_{n}}}{{2}}}\left\langle\prod_{j=1}^{Q}\hat{\tau}_{n}^{+}(\theta_{2j-1})\hat{\tau}^{-}_{n}(\theta_{2j})\right\rangle\right].$
(27)

The Jacobian factors can be explicitly written as (cf. Eq.(16))

$\left|\frac{\mathrm{d}\theta_{j}}{\mathrm{d}x}\right|\equiv\left|\frac{\mathrm{d}k_{F,j}(x,t)}{\mathrm{d}x}\right|=\left[t\sqrt{1-R_{j}^{2}/t^{2}}\right]^{-1},$
(28)

and $R_{j}$ associated to the Fermi points in Eq. (16)

$R_{\nu}=\begin{cases}|x|+2L(\nu-1),\quad\nu=1,\dots,N_{+};\\\\[4.0pt]
|x|+2L(\nu-N_{+}-1),\quad\nu=N_{+}+1,\dots,2N_{+};\\\\[4.0pt]
|x|+2L(2N_{+}-\nu),\quad\nu=2N_{+}+1,\dots,2N_{+}+N_{-};\\\\[4.0pt]
|x|-2L(2Q-\nu+1),\quad\nu=2N_{+}+N_{-}+1,\dots,2Q.\\\\[4.0pt] \end{cases}$
(29)

The $2Q$-point correlation function is given by [72, 75]

$\left\langle\prod_{j=1}^{Q}\hat{\tau}^{+}_{n}(u_{j})\hat{\tau}^{-}_{n}(v_{j})\right\rangle=\frac{\prod_{1\leqslant
i<j\leqslant
Q}g(u_{j},u_{i})g(v_{j},v_{i})}{\prod_{i,j=1}^{Q}g(v_{j},u_{i})},\quad
g(u,v)=\left|2\sin\frac{u-v}{2}\right|^{\Delta_{n}}\,.$ (30)

The second contribution comes instead from the short-distance regularization
$\epsilon(x,t)$ of the field theory result and it is therefore non-universal.
For the non-interacting spin chain, its value can be analytically derived with
the Fisher-Hartwig conjecture (see e.g. Ref. [76, 77, 78]) and reads [34, 35]

$\log\epsilon(x,t)=\sum_{1\leqslant i<j\leqslant
2Q}(-1)^{i+j}\log\left|\sin\frac{k_{F,i}-k_{F,j}}{2}\right|+\frac{Q}{2}(\Upsilon_{n}+\log(2)/3)$
(31)

with $\Upsilon_{n}$ a real number whose explicit expression can be found in
Ref. [76], for instance $\Upsilon_{1}\simeq 0.49502$.

Figure 6: Evolution of the entanglement entropy profiles during a QJE. Panels
show different times $t$ increasing from the left to the right and from the
top to the bottom, as written in each legend. The numerical data (symbols)
obtained with exact lattice calculations on a lattice of $300$ sites is found
in very good agreement with the hydrodynamic prediction in Eqs. (26)-(31)
(solid line) even after several reflections.

In Fig. 6 we compare the hydrodynamic prediction for the entanglement entropy
obtained combining Eqs. (26)-(31) with exact lattice calculations, finding an
extremely good agreement. Notice that the non-linear structure of the Fermi
points (16) does not allow for an easy manipulation of our results in order to
get a simpler and closed expression of $S_{n}$. To give a hint of the
complexity of this algebra, we report the expression for the entanglement
entropy at half-system ($x=0$) and at given time $2qL\leqslant t<(2q+2)L$
after $q$ reflections have occurred,

$\begin{split}S_{1}(0,t)&=\frac{2q+1}{6}\log(2t)+\frac{2q+1}{2}\left(\Upsilon+\frac{\log
2}{3}\right)-\frac{q}{3}\log\left(\frac{t}{2L}\right)-\frac{2}{3}\log\Gamma(q+1)\\\\[4.0pt]
&\quad+\frac{1}{2}\log\left(\frac{t+2Lq}{t}\frac{\Gamma(t/(2L)+q)}{\Gamma(t/(2L)-q)}\right)-\frac{4}{3}\sum_{j=1}^{q}(-1)^{j}\log\tan\left(\frac{\varphi_{j}}{2}\right)\\\\[4.0pt]
&\quad-\frac{4}{3}\sum_{1\leqslant j<k\leqslant
q}(-1)^{j+k}\left[\log\tan\frac{\varphi_{j}+\varphi_{k}}{2}+\log\tan\frac{\varphi_{k}-\varphi_{j}}{2}\right],\end{split}$
(32)

where $\Gamma(\cdot)$ is the Euler Gamma function and we defined the shorthand
$\varphi_{j}\equiv\arcsin(2jL/t)$. In Fig. 7, we show the growth of
entanglement at half-system obtained from Eq. (32) against numerical data. As
one can see, the half-system entanglement displays some jumps at times
$t=2\eta L$, $\eta\in\mathbb{N}$ (dashed vertical lines). These jumps are a
consequence of the merging at $x=0$ of the two light cones that reflected
against the walls in $x=\pm L$ at times $t=(2\eta-1)L$, compare with Fig. 6 to
visualize this process with the profile of entanglement. This pattern of
reflections and merging suggests a Floquet picture of the long time dynamics
in terms of the stroboscopic time $t\equiv\eta L$ that we discuss in Sec. 5.2.

Figure 7: Half-system entanglement entropy as function of time. The
hydrodynamic prediction of Eq. (32) (solid line) is found in extremely good
agreement with the numerical data obtained for a lattice of 300 sites
(symbols). The dashed (resp. dot-dashed) vertical axes mark the times $t=2\eta
L$ (resp. $t=(2\eta-1)L$), $\eta\in\mathbb{N}$, corresponding to the merging
(resp. reflection) of the left and right light cones.

###  4.2 Joule expansion vs multiple domain wall setting

It is interesting to notice that the entanglement shows a qualitatively
different behavior in the quantum Joule expansion from that observed during
the quench of a multiple domain wall configuration, even though the two
settings are characterized by the same profile of conserved charges, see Sec.
3. This feature is easily explained by noticing that, although the two
settings share the same Fermi points $\\{k_{F,j}(x,t)\\}_{i=1}^{2Q}$ in Eq.
(14), the latter come from different points on the initial Fermi contour, and
so, they are characterized by very different quantum correlations in the two
cases. More precisely, while the boundary points for the QJE are given in Eq.
(24), the dynamics of modes in the MDW is characterized by the absence of
reflecting walls, therefore one can simply identify

$\\{\theta_{j}\\}_{j=1}^{2Q}\equiv\\{k_{F,j}(x,t)\\}_{j=1}^{2Q}.$ (33)

Quite nicely, we notice that the factor $(-1)^{q_{j}}$ appearing in Eq. (24)
is responsible for the correct implementation of the reflecting walls in the
QJE problem and, therefore, it alone distinguishes between the two quench
problems, ensuring (if present) that the entanglement entropy drops at
positions $x=\pm L$ because of the vanishing of the time-evolved wave
function. In Fig. 8 we show a comparison of the entanglement spreading in the
two problems.

Figure 8: Comparison of the entanglement dynamics in the MDW (top row) and in
the QJE (bottom row) quench protocols as function of $x/L\in[-1,1]$ at
different times increasing from the left to the right column. The two settings
show the same entanglement profiles at times $t\leqslant L$ i.e., before any
reflection, and significantly deviate at later times $t>L$. In both the
settings, the analytical prediction obtained with quantum fluctuating
hydrodynamics (solid line) is found in very good agreement with the numerical
data (symbols).

##  5 Long time limit: reflections, relaxation and recurrence

A natural question that might arise considering the Joule expansion protocol
is about the fate of the quantum gas at very large times $t/L\gg 1$, after it
undergoes several reflections against the walls. Naively, one could imagine
that the gas eventually relaxes towards a steady state where the system is in
a homogeneous configuration with half-filling. Although this classical picture
turns out to be somewhat correct for the conserved charges, understanding how
this relaxation takes place and the asymptotic behavior of the entanglement is
less intuitive. Moreover, due to the unitary character of the time evolution,
revival effects are expected to manifest at large but finite times. For sake
of clarity, it is useful to distinguish among three regimes occurring during
the non-equilibrium dynamics at different time scales. Precisely, we find

  * (i)

A reflection regime at $t/L\sim{\cal O}(1)$, i.e. at Euler scales, which is
captured by the hydrodynamics developed in the previous sections. Here, we
observe a Floquet picture of the large-time dynamics with periodicity $4L$.

  * (ii)

An equilibration regime for $1\ll t/L\ll L$, where the system already went
through several reflections and slowly relaxes towards a steady state
configuration. In this regime, we observe a breakdown of the hydrodynamic
theory.

  * (iii)

A recurrence regime for $t\sim{\cal O}(L^{2})$, over which the time-evolved
wave function goes arbitrarily close to the initial configuration, as expected
by standard ergodicity arguments.

(i) (ii) (iii)

Figure 9: Loschmidt echo ${\cal L}(t)$ in Eq. (34) as function of time for
different regimes: (i) reflection regime $t/L\sim{\cal O}(1)$; (ii)
equilibration regime $1\ll t/L\ll L$ and (iii) recurrence regime $t/L\sim{\cal
O}(L)$ from the left to the right panel. The curves are obtained numerically
for a lattice of $300$ sites.

In Fig. 9, we plot the Loschmidt echo, defined as

${\cal L}(t)=\left|\braket{\Psi(t)}{\Psi(0)}\right|^{2},$ (34)

at different time scales corresponding to the regimes (i)-(iii) discussed
above. From this figure, it is easy to see that the revival time scale is
$\sim{\cal O}(L^{2})$ and, therefore, a stationary behavior can only be
achieved in the scaling limit $t/L\to\infty$ such that $t/L^{2}\to 0$ [9, 48,
79, 80].

###  5.1 Long time limit of the density

We first focus on the particle density starting from the result in Eq. (19),
which is a particularly pleasant form for the study of the long time limit.
Indeed, one can asymptotically replace the series appearing in Eq. (19) using
the Euler-MacLaurin formula as

$I(\pm x,t)\equiv\sum_{p=1}^{N_{\mp}}(-1)^{p}\rho(x_{p}\mp
x,t)\approx\int_{1}^{N_{\mp}}\mathrm{d}p\
f(p)+\frac{f(N_{\mp})+f(1)}{2}+\text{subleading terms}$ (35)

where $f(p)=\cos(\pi p)\bar{\rho}(x_{p}\mp x,t)$. It is then easy to show that
$\lim_{t\to\infty}I(x,t)=1/4$ and, from Eqs. (13)-(35), to conclude that

$\lim_{t\to\infty}\rho(x,t)=\lim_{t\to\infty}\bar{\rho}(x,t)=1/2.$ (36)

Figure 10: Panel $(a)$: Particle density dynamics at fixed position $x$ as
function of time, obtained with exact numerical calculations on a lattice of
$500$ sites. The horizontal dashed line marks the value $\rho=1/2$ while
vertical axes are drawn at each period $4L$. Panel $(b)$-$(c)$: Same plot
where the numerical data (symbols) are compared with the hydrodynamic
prediction in Eq. (17) (solid line) for two different values of $x$. We can
see that by considering the subleading contributions in Eq. (35),
hydrodynamics captures the oscillations of density, going beyond the classical
result. Panel $(d)$ shows the breakdown of hydrodynamics at times $t/L\sim
0.3L$ for $x/L=-0.25$ and for the choice of parameters considered.

The result in Eq. (36) agrees with the expectation for the asymptotic density
in a classical Joule experiment. However, in Fig. 10$(a)$, we show that the
underlying unitary evolution manifests in persistent oscillations around the
classical result $\rho=1/2$ with period $4L$ and decreasing amplitude, see
also Ref. [47, 48]. These oscillations are well captured by the subleading
terms appearing in Eq. (35) that we previously neglected to recover the
classical result in (36), see Fig. 10 – panels $(b)$ and $(c)$. We numerically
observe the same qualitative behavior up to the equilibration regime $1\ll
t/L\ll L$, after which hydrodynamics deviates from the numerical data, see
Fig. 10 $(d)$.

(i) (ii) (iii1) (iii2)

Figure 11: Colormap plot of the particle density $\rho(x,t)$ in the $x$-$t$
plane at different time scales, increasing from panel (i) to panel (iii). The
data is obtained with exact lattice calculations on a system of $300$ sites.

We provide a thorough numerical analysis of the particle density profile in
Fig. 11, showing $\rho(x,t)$ as a colormap plot in the $x$-$t$ plane at
different time scales. In panel (i), corresponding to the reflection regime,
we clearly see the pattern of density oscillations with $4L$ periodicity. The
latter gradually smooth out on larger time scales corresponding to the
equilibration regime where $\rho\approx 1/2$ homogeneously along the chain, as
shown in panel (ii). Finally, new patterns in the density profile arise at
$t\sim{\cal O}(L^{2})$ and correspond to those times where ${\cal L}(t)\sim
1$, i.e. to recurrences, or to times where the wave function is orthogonal to
the initial state and ${\cal L}(t)\sim 0$, compare Fig. 11 (iii1) with Fig. 9.
Notice that the structure of the patterns in the recurrence regime is very
rich and some of the local minima and maxima of ${\cal L}(t)$ correspond to
the sub-patterns shown in Fig. 11 (iii2).

###  5.2 Long time limit of the entanglement

We finally discuss the long time behavior of the entanglement entropy. As
already commented in Sec. 4.1, from Fig. 6 and 7 one can easily see that the
entanglement dynamics during the QJE is made of two alternating stages: 1. the
spreading of the domain wall from $x=0$ to $x=\pm L$, where it reflects
against the walls; 2. the dynamics of the two reflected (half) domain walls
from $x=\pm L$ to $x=0$ and their subsequent merging. Focusing on the
asymptotic dynamics at half system $x=0$, we notice that these reflection
(merging) processes occur at odd (even) multiples of the rescaled time
$\eta=t/L$, hence suggesting a Floquet picture of the long-time dynamics in
terms of the stroboscopic time $\eta$. We show this analysis in Fig. 12, where
the exact lattice numerics matches the hydrodynamic prediction with excellent
agreement in the reflection regime $\eta\sim{\cal O}(1)$. Moreover, a careful
analysis of Eq. (32) reveals that the half-system entanglement grows for
$\eta\gg 1$ as

$S_{1}(0,\eta L)\overset{\eta\gg
1}{\sim}2\alpha(L)\eta-\frac{2}{3}\eta\log\eta-\frac{1}{6}\log\eta$ (37)

up to an additive constant, with

$\alpha(L)=\frac{1}{3}\log\left(\frac{4L}{\pi^{2}}\right)+\Upsilon+\frac{1}{3}.$
(38)

In Fig. 12, we observe that Eq. (37) captures extremely well the behavior of
entanglement even at relatively small values of $\eta$. In the absence of the
analytical result in Eq. (32), one might conjecture a power-law behavior
$S_{1}\sim C(t/L)^{a}$ for the asymptotic growth (see e.g. Ref. [44]), which
we find indeed numerically very close to the correct result in Eq. (37), see
Fig. 13.

Figure 12: Stroboscopic growth of the half-system entanglement $S_{1}(0,\eta
L)$ with $\eta\in\mathbb{N}$. The figure shows the exact numerical data
(symbols) for a lattice of $300$ sites, the hydrodynamic prediction of Eq.
(32) (solid line) and its asymptotic behavior given in Eq. (37) (dot-dashed
line), which are in extremely good agreement even after many periods.

$(a)$ $(b)$

Figure 13: Panel $(a)$ — Stroboscopic half-system entanglement in a log-log
plot (symbols) and the fitting function $S_{1}(0,\eta L)\sim C\eta^{\alpha}$
(dashed line) with $\alpha\simeq 0.81$ and $C\simeq 1.25$ for our choice of
parameters. Panel $(b)$ — The fitting function is compared to the asymptotic
result in Eq. (37) of the stroboscopic entanglement for $\eta\gg 1$.

Moving to larger time scales $1\ll\eta\ll L$ we still observe a growth of the
entanglement associated with the relaxation dynamics in place but we also
witness to a significant deviation of $S_{1}(0,t)$ from the hydrodynamic
prediction, see Fig. 14$(a)$. Concluding, in Fig. 14$(b)$ we investigate the
regime $t\sim{\cal O}(L^{2})$ where we register oscillations of the
entanglement that correspond to the local minima and maxima of ${\cal L}(t)$
in Fig. 9 and, consequently, to the pattern in Fig. 11 (iii1) observed for the
particle density, see also Ref. [44].

$(a)$ $(b)$

Figure 14: Asymptotic behavior of the half-system entanglement beyond the
regime of hydrodynamics. Panel $(a)$ – Stroboscopic half-system entanglement
obtained with exact numerics on a lattice of $300$ sites (pale blue line)
significantly deviates from hydrodynamics (red line) at times $t/L\sim 100$
with our choice of parameters. Panel $(b)$ – Half-system entanglement entropy
in the recurrence regime.

##  6 Summary and conclusion

We investigated the non-equilibrium dynamics of a one-dimensional gas of
spinless fermionic particles with nearest-neighbor hopping Hamiltonian,
initially prepared in a box of size $L$ with saturated density $\rho=1$ and
subsequently let to expand in a doubled sized region by suddenly moving one of
the box’s edges. This simple quench protocol might be seen as a quantum
version of the Joule expansion, thus it was dubbed as quantum Joule expansion
in our analysis. For this problem, we employed tools stemming from phase-space
hydrodynamics to reconstruct the profile of conserved charges at Euler space-
time scales and subsequently, following the recent literature on quantum
fluctuating hydrodynamics [31, 32, 33, 34, 35, 36], we re-quantized the
hydrodynamic background to obtain an analytical prediction for the Rényi
entropies during the quench dynamics, which we tested against numerics for the
von Neumann entanglement entropy. Remarkably, we found that the prediction of
quantum hydrodynamics is able to reproduce the numerical data obtained for the
lattice model with great accuracy even after several reflections of the
quantum gas.
In the last part of this work, we studied the long time behavior of the
quantum gas, building on previous results appeared in literature, see e.g.
Ref. [9, 13, 47, 48]. We observed three time scales that characterize the long
time physics and we provided a careful analysis of each. First, we found a
reflection regime $t/L\sim{\cal O}(1)$ where the quench dynamics is still well
captured by our hydrodynamic theory. Here, we discussed the persistent
oscillations of the particle density around the classical value $\rho=1/2$ and
we calculated the half-system entanglement growth for $t\gg L$ to be the non-
trivial function $S_{1}\sim
2\alpha(L)t/L-\frac{2t}{3L}\log(t/L)-\frac{1}{6}\log(t/L)$, extracted from the
asymptotic expansion of our analytical result. At larger scales, we found an
equilibration regime $1\ll t/L\ll L$ where hydrodynamics breaks down and the
system relaxes to a stationary state and, finally, a recurrence regime for
$t\sim{\cal O}(L^{2})$ characterized by revivals of the many-body wave
function.
We envisage at least two follow-ups of this work. One future direction could
be the study of the entanglement spreading in systems with boundaries for
interacting integrable models (see e.g. [54]) or for more generic choice of
confinement. On the other hand, it would be interesting to better explore the
long time regime where hydrodynamics breaks down and the system eventually
relaxes to a non-equilibrium steady state, see e.g. Ref. [51, 52]. This path
would probably require to take into account higher order corrections to the
Euler hydrodynamics not considered in this work, see [62, 63].

Acknowledgments — FA and SS acknowledge support from ERC under Consolidator
grant number 771536 (NEMO). The authors are thankful to Dragi Karevski,
Pasquale Calabrese and Jérôme Dubail for useful discussions and insights on
the manuscript. FA acknowledges Nina Javerzat for helpful comments on Eq.
(37). SS acknowledges LPCT (Nancy) for the warm hospitality during the early
stage of this project.

##  Appendix A Calculation of the Fermi contour

After the sudden expansion of the box confinement, the propagating modes on
the initial Fermi contour $x=0$, $k\in[-\pi,\pi]$ in Eq. (5) move freely with
velocity $v(k)=\sin k$ up to the position of the two walls at $x=\pm L$, where
their trajectory is elastically reflected, $k\to-k$. As we have illustrated in
Fig. 2, each of these reflections further splits the Fermi contour. Focusing
on a position $x\in[-L,L]$ inside the box, we may employ the method of images
and consider the free propagation of a system with infinitely many virtual
copies of the Fermi contour rather than tracking the true reflection dynamics
in the presence of the walls, see Fig. 4. In this picture, while the initial
modes are allowed to leave the system, virtual modes will enter the system as
to mimic the reflection dynamics, see Fig. 15 for an illustration. The family
of images is parametrized by the equation of motion of the modes generated on
each of the junctions at $x_{p}=2Lp$ that arrived at position $x$ at a certain
time $t$

$\displaystyle x=x_{p}+t\sin(k)$ (39)

where $p$ is an integer number. We can find out the number of virtual sources
that are activated following the trajectory of the light cone $x=2pL\pm t$,
obtained from Eq. (39) with $k=\pm\pi/2$. Considering $x>0$, we find that
there are $N_{-}$ Fermi points in the interval $[-\pi,0]$ with

$(a)$ $(b)$ $(c)$

Figure 15:  Illustration of the time evolution of the Fermi contour –
Snapshots of the time evolution of the Fermi contour after the QJE. Panel
$(a)$ – $0<t<L$, no reflection has taken place yet and the quantum gas is
freely expanding. Panel $(b)$ – $L<t<2L$, the first reflection has taken place
and the Fermi sea is split. Panel $(c)$ – $3L<t<5L$, the fastest modes have
undergone two reflections. The blue line for $x=0.25L$ illustrates that the
solution (42) correctly reproduces the edges of the Fermi surface.

$\displaystyle N_{-}=\left\lfloor\frac{t+x}{2L}\right\rfloor$ (40)

and $N_{+}$ Fermi points in the interval $[0,\pi]$ with

$\displaystyle N_{+}=\left\lceil\frac{t-x}{2L}\right\rceil.$ (41)

Now that we know the valid range for $p$, we may solve for the Fermi points
using Eq. (39) but straightforwardly solving will yield the Fermi points
ordered by reflections and thus we need to rearrange them such that they are
ordered by magnitude. We readily find

$\displaystyle
k_{F,j}(x,t)=\begin{dcases}\arcsin\left(\frac{x+2Lj}{t}\right),\
&j=-N_{-},...,N_{+}-1\\\ \pi-\arcsin\left(\frac{x+2L(2N_{+}-1-j)}{t}\right),\
&j=N_{+},...,2N_{+}-1\\\ -\pi-\arcsin\left(\frac{x-2L(j+2N_{-}+1)}{t}\right),\
&j=-2N_{-},...,-N_{-}-1\end{dcases},$ (42)

which are related to Eq. (16) of Sec. 3 by a simple re-definition of the index
$j$.

##  Appendix B Exact lattice calculations

In this appendix, we briefly report the strategy of the exact numerical
lattice calculations that we performed to generate the data in the main text.
First, we write the free Fermi Hamiltonian in Eq. (1) as

$\hat{H}=\sum_{i,j=1}^{N-1}\hat{c}^{\dagger}_{j}\ \mathbb{H}^{(0)}_{ij}\
\hat{c}_{i},$ (43)

with $N$ the number of lattice sites and Hamiltonian matrix

$\mathbb{H}^{(0)}_{ij}=-\frac{1}{2}\left(\delta_{i,j+1}+\delta_{i+1,j}\right)+V(j,t<0)\delta_{i,j}.$
(44)

Next, we diagonalize the matrix in Eq. (44) as

$\mathbb{H}^{(0)}=\bm{w}^{\dagger}\ \text{diag}(\vec{E}^{(0)})\
\bm{w},\quad\bm{w}=\begin{pmatrix}\vec{w}_{1}\big{|}\vec{w}_{2}\big{|}\dots\big{|}\vec{w}_{N}\end{pmatrix};\quad\vec{E}^{(0)}=(E_{1}^{(0)},\dots,E^{(0)}_{N})$
(45)

with $\vec{w}_{n}$ (resp. $E^{(0)}_{n}$) the single-particle eigenvector
(resp. energy), and we construct the ground state of the model $\ket{\Psi(0)}$
by adding to the vacuum the negative-energy modes of the spectrum. The two-
point correlation matrix is then obtained as

$\mathbb{G}(0)=\left[\braket{\Psi(0)}{\hat{c}^{\dagger}_{i}\hat{c}_{j}}{\Psi(0)}\right]_{i,j=1}^{N}=\bm{w}\
\Pi\ \bm{w}^{\dagger}$ (46)

where $\Pi$ is a ground-state projector, defined as

$\Pi=\text{diag}(p_{1},\dots,p_{N}),\quad p_{j}=\Theta(-E_{j}^{(0)})$ (47)

and $\Theta$ is the Heaviside step function. At times $t>0$, we study the
quench dynamics generated by the (post-quench) Hamiltonian matrix

$\mathbb{H}_{i,j}=-\frac{1}{2}\left(\delta_{i,j+1}+\delta_{i+1,j}\right)+V(j,t>0)\delta_{i,j},$
(48)

which is diagonalized as

$\mathbb{H}=\bm{v}^{\dagger}\ \text{diag}(\vec{E})\ \bm{v}$ (49)

with energies $E_{n}$ and eigenstates $\vec{v}_{n}$. It is then easy to
exactly evolve the two-point function in Eq. (46) in the basis of post-quench
eigenstates, obtaining

$\mathbb{G}(t)=\bm{v}\ U(t)\ \bm{v}^{\dagger}\ \mathbb{G}(0)\ \bm{v}\
U^{\dagger}(t)\ \bm{v}^{\dagger},\qquad
U(t)=\text{diag}\left(\exp(-\mathbf{i}t\vec{E})\right).$ (50)

From the knowledge of the two-point correlation matrix, other quantities can
be derived exploiting Wick’s theorem. In particular, the particle density is
read from the diagonal elements of the correlation matrix

$\rho(j,t)=\delta_{i,j}\mathbb{G}_{i,j}(t).$ (51)

and the entanglement entropy of a bi-partition $[1,\ell]\cup[\ell+1,N]$ is
obtained from the eigenvalues $\zeta_{i}$ ($i=1,\dots,\ell$) of the minor
$\mathbb{G}^{(\ell)}(t)=[\mathbb{G}_{i,j}(t)]_{i,j=1}^{\ell}$ as [81, 82, 83,
84, 85, 86]

$S_{1}(\ell,t)=-\sum_{i=1}^{\ell}\left(\zeta_{i}\log\zeta_{i}+(1-\zeta_{i})\log(1-\zeta_{i})\right).$
(52)

## References

## References

  * [1] A. Bastianello, B. Bertini, B. Doyon and R. Vasseur, Introduction to the Special Issue on Emergent Hydrodynamics in Integrable Many-Body Systems, J. Stat. Mech. (2022) 014001
  * [2] B. Doyon, Lecture notes on Generalised Hydrodynamics, Les Houches Lecture notes, SciPost Phys. Lect. Notes 18 (2020).
  * [3] B. Bertini, F. Heidrich-Meisner, C. Karrasch, T. Prosen, R. Steinigeweg and M. Znidaric, Finite-temperature transport in one-dimensional quantum lattice models, Rev. Mod. Phys. 93, 025003 (2021).
  * [4] V. Alba, B. Bertini, M. Fagotti, L. Piroli and P. Ruggiero, Generalized-Hydrodynamic approach to Inhomogeneous Quenches: Correlations, Entanglement and Quantum Effects, J. Stat. Mech. (2021) 114004
  * [5] I. Bouchoule and J. Dubail, Generalized Hydrodynamics in the 1D Bose gas: theory and experiments, J. Stat. Mech. (2022) 014003
  * [6] B. Bertini, M. Collura, J. De Nardis and M. Fagotti, Transport in out-of-equilibrium XXZ chains: Exact profiles of charges and currents, Phys. Rev. Lett. 117, 207201(2016).
  * [7] O. A. Castro-Alvaredo, B. Doyon and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6, 041065 (2016)
  * [8] T. Antal, Z. Rácz, A. Rákos and G. M. Schütz, Transport in the XX chain at zero temperature: Emergence of flat magnetization profiles, Phys. Rev. E 59, 4912 (1999)
  * [9] G. M. Schütz and S. Trimper, Relaxation and aging in quantum spin systems, EPL 47, 164 (1999)
  * [10] T. Antal, P. L. Krapivsky, and A. Rákos, Logarithmic current fluctuations in nonequilibrium quantum spin chains, Phys. Rev. E 78, 061115 (2008)
  * [11] D. Karevski, Scaling behaviour of the relaxation in quantum chains, Eur. Phys. J. B 27, 147 (2002)
  * [12] T. Platini and D. Karevski, Scaling and front dynamics in Ising quantum chains, Eur. Phys. J. B 48, 225 (2005)
  * [13] T. Platini and D. Karevski, Relaxation in the XX quantum chain, J. Phys. A 40, 1711 (2007)
  * [14] M. Collura and D. Karevski, Quantum quench from a thermal tensor state: Boundary effects and generalized Gibbs ensemble Phys. Rev. B 89, 214308 (2014)
  * [15] L. Piroli, J. De Nardis, M. Collura, B. Bertini and M. Fagotti, Transport in out-of-equilibrium XXZ chains: Nonballistic behavior and correlation functions, Phys. Rev. B 96, 115124 (2017)
  * [16] V. B. Bulchandani, R. Vasseur, C. Karrasch and J. E. Moore, Solvable Hydrodynamics of Quantum Integrable Systems, Phys. Rev. Lett. 119, 220604 (2017)
  * [17] A. De Luca, M. Collura and J. De Nardis, Nonequilibrium spin transport in integrable spin chains: Persistent currents and emergence of magnetic domains, Phys. Rev. B 96, 020403 (2017)
  * [18] B. Doyon, J. Dubail, R. Konik and T. Yoshimura, Large-Scale Description of Interacting One-Dimensional Bose Gases: Generalized Hydrodynamics Supersedes Conventional Hydrodynamics, Phys. Rev. Lett. 119, 195301 (2017)
  * [19] B. Doyon, T. Yoshimura, and J. S. Caux, Soliton Gases and Generalized Hydrodynamics, Phys. Rev. Lett. 120, 045301 (2018)
  * [20] V. B. Bulchandani, R. Vasseur, C. Karrasch, and J. E. Moore, Bethe-Boltzmann hydrodynamics and spin transport in the XXZ chain, Phys. Rev. B 97, 045407 (2018)
  * [21] A. Bastianello, V. Alba, and J. S. Caux Generalized Hydrodynamics with Space- Time Inhomogeneous Interactions, Phys. Rev. Lett. 123, 130602 (2019)
  * [22] I. Bouchoule and J. Dubail, Breakdown of Tan’s relation in lossy one-dimensional Bose gases, Phys. Rev. Lett. 126, 160603 (2021)
  * [23] F. S. Møller, G. Perfetto, B. Doyon and J. Schmiedmayer, Euler-scale dynamical correlations in integrable systems with fluid motion, SciPost Phys. Core 3, 16 (2020)
  * [24] M. Schemmer, I. Bouchoule, B. Doyon, and J. Dubail, Generalized Hydrodynamics on an Atom Chip, Phys. Rev. Lett.122, 090601 (2019)
  * [25] N. Malvania, Y. Zhang, Y. Le, J. Dubail, M. Rigol, and D. S. Weiss, Generalized hydrodynamics in strongly interacting 1D Bose gases, Science 373, 6559 (2021)
  * [26] A. Bastianello, J. De Nardis, and A. De Luca, Generalized hydrodynamics with dephasing noise, Phys. Rev. B 102, 161110 (2020)
  * [27] A. Bastianello, A. De Luca and R. Vasseur, Hydrodynamics of weak integrability breaking, J. Stat. Mech. (2021) 114003
  * [28] J. De Nardis, D. Bernard and B. Doyon, Hydrodynamic Diffusion in Integrable Systems, Phys. Rev. Lett. 121, 160603 (2018)
  * [29] J. De Nardis, D. Bernard and B. Doyon, Diffusion in generalized hydrodynamics and quasiparticle scattering, SciPost Phys. 6, 049 (2019)
  * [30] I. Bouchoule, B. Doyon and J. Dubail, The effect of atom losses on the distribution of rapidities in the one-dimensional Bose gas, SciPost Phys. 9, 044 (2020)
  * [31] P. Ruggiero, Y. Brun, and J. Dubail, Conformal field theory on top of a breathing one-dimensional gas of hard core bosons, SciPost Phys. 6, 051 (2019)
  * [32] P. Ruggiero, P. Calabrese, B. Doyon and J. Dubail, Quantum Generalized Hydrodynamics, Phys. Rev. Lett. 124, 140603 (2020)
  * [33] S. Scopa, A. Krajenbrink, P. Calabrese and J. Dubail, Exact entanglement growth of a one-dimensional hard core quantum gas during a free expansion, J. Phys. A 54, 404002 (2021)
  * [34] S. Scopa, P. Calabrese and J. Dubail, Exact hydrodynamic solution of a double domain wall melting in the spin-1/2 XXZ model, SciPost Phys. 12, 207 (2022)
  * [35] P. Ruggiero, P. Calabrese, B. Doyon and J. Dubail, Quantum generalized hydrodynamics of the Tonks-Girardeau gas: density fluctuations and entanglement entropy, J. Phys. A 55 024003 (2022)
  * [36] F. Rottoli, S. Scopa and P. Calabrese, Entanglement Hamiltonian during a domain wall melting in the free Fermi chain, J. Stat. Mech. (2022) 063103
  * [37] Y. Kagan, E. L. Surkov and G. V. Shlyapnikov, Evolution of a Bose-condensed gas under variations of the confining potential, Phys. Rev. A 54, R1753 (1996)
  * [38] A. Minguzzi and D.M. Gangardt, Exact coherent states of a harmonically confined Tonks-Girardeau gas, Phys. Rev. Lett. 94, 240404 (2005)
  * [39] P. G. L. Leach and H. R. Lewis, A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians, J. Math. Phys. 23, 2371 (1982)
  * [40] M. Rigol and A. Muramatsu, Emergence of Quasicondensates of Hard-Core Bosons at Finite Momentum, Phys. Rev. Lett. 93, 230404 (2004)
  * [41] L. Vidmar, J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, F. Heidrich-Meisner, I. Bloch and U. Schneider, Dynamical Quasicondensation of Hard-Core Bosons at Finite Momenta, Phys. Rev. Lett. 115, 175301 (2015)
  * [42] V. Eisler, D. Karevski, T. Platini and I. Peschel, Entanglement evolution after connecting finite to infinite quantum chains, J. Stat. Mech. (2008) P0102
  * [43] E. Vicari, Quantum dynamics and entanglement in one-dimensional Fermi gases released from a trap. Phys. Rev. A 85, 062324 (2012)
  * [44] V. Alba and F. Heidrich-Meisner, Entanglement spreading after a geometric quench in quantum spin chains, Phys. Rev. B 90, 075144 (2014)
  * [45] M. Gruber and V. Eisler, Magnetization and entanglement after a geometric quench in the XXZ chain, Phys. Rev. B 99, 174403 (2019)
  * [46] J.-S. Caux and R. M. Konik, Constructing the Generalized Gibbs Ensemble after a Quantum Quench, Phys. Rev. Lett. 109, 175301 (2012)
  * [47] M. Collura, S. Sotiriadis and P. Calabrese, Equilibration of a Tonks-Girardeau Gas Following a Trap Release,Phys. Rev. Lett. 110, 245301 (2013)
  * [48] M. Collura, S. Sotiriadis and P. Calabrese, Quench dynamics of a Tonks–Girardeau gas released from a harmonic trap, J. Stat. Mech. (2013) P09025
  * [49] R. Dubessy, J. Polo, H. Perrin, A. Minguzzi and M. Olshanii, Universal shock-wave propagation in one-dimensional Bose fluids, Phys. Rev. Res. 3, 013098 (2021)
  * [50] S. Camalet, Joule Expansion of a Pure Many-Body State, Phys. Rev. Lett. 100, 180401 (2008)
  * [51] J. D. Noh, E. Iyoda and T. Sagawa, Heating and cooling of quantum gas by eigenstate Joule expansion, Phys. Rev. E 100, 010106(R) (2019)
  * [52] J. Zhang, Y. Meurice and S.-W. Tsai, Quantum Joule expansion of one-dimensional systems, Phys. Rev. A 101, 033608 (2020)
  * [53] P. T. Grochowski, T. Karpiuk, M. Brewczyk,and K. Rzazewski, Fermionic quantum carpets: From canals and ridges to solitonlike structures, Phys. Rev. Research 2, 013119 (2020)
  * [54] M. Łebek, P. T. Grochowski and K. Rzazewski, Single- to many-body crossover of a quantum carpet, Phys. Rev. Research 3, 023009 (2021)
  * [55] V. Eisler, F. Maislinger and H. G. Evertz, Universal front propagation in the quantum Ising chain with domain-wall initial states, SciPost Phys. 1, 014 (2016)
  * [56] V. Eisler and F. Maislinger, Hydrodynamical phase transition for domain-wall melting in the XY chain, Phys. Rev. B 98,161117 (2018)
  * [57] V. Eisler and F. Maislinger, Front dynamics in the XY chain after local excitations, SciPost Phys. 8, 037 (2020)
  * [58] V. Eisler, Entanglement spreading after local and extended excitations in a free-fermion chain, J. Phys. A: Math. Theor. 54, 424002 (2021)
  * [59] M. Gruber and V. Eisler, Entanglement spreading after local fermionic excitations in the XXZ chain, SciPost Phys. 10, 005 (2021)
  * [60] J. Dubail, J.-M. Stéphan, J. Viti and P. Calabrese, Conformal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases, SciPost Phys. 2, 2 (2017)
  * [61] P. Wendenbaum, M. Collura, and D. Karevski, Hydrodynamic description of hard-core bosons on a Galileo ramp, Phys. Rev. A 87, 023624 (2013)
  * [62] M. Fagotti, Higher-order generalized hydrodynamics in one dimension: The noninteracting test, Phys. Rev. B 96, 220302 (2017)
  * [63] M. Fagotti, Locally quasi-stationary states in noninteracting spin chains, SciPost Phys. 8, 048 (2020)
  * [64] N. Allegra, J. Dubail, J.-M. Stéphan and J. Viti, Inhomogeneous field theory inside the arctic circle, J. Stat. Mech. (2016) 053108
  * [65] T. Giamarchi, Quantum Physics in One Dimension, Oxford Univ. Press (2007)
  * [66] M. A. Cazalilla, Bosonizing one-dimensional cold atomic gases, J. Phys. B 37, 1 (2004)
  * [67] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, One dimensional bosons: From condensed matter systems to ultracold gases, Rev. Mod. Phys. 83, 1405 (2011)
  * [68] Y. Brun and J. Dubail, One-particle density matrix of trapped one-dimensional impenetrable bosons from conformal invariance, SciPost Phys. 2, 012 (2017)
  * [69] Y. Brun and J. Dubail, The Inhomogeneous Gaussian Free Field, with application to ground state correlations of trapped 1d Bose gases, SciPost Phys. 4, 037 (2018)
  * [70] A. Bastianello, J. Dubail, and J.-M. Stéphan, Entanglement entropies of inhomogeneous Luttinger liquids, J. Phys. A 53, 23 (2020)
  * [71] S. Scopa, L. Piroli and P. Calabrese, One-particle density matrix of a trapped Lieb Liniger anyonic gas, J. Stat. Mech. (2020) 093103
  * [72] P. Calabrese and J. Cardy Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002
  * [73] J. L. Cardy, O. A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Stat. Phys. 130, 129 (2008)
  * [74] P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009)
  * [75] H. Casini, C. D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. (2005) P07007
  * [76] B. Q. Jin and V. E. Korepin, Quantum spin chain, Toeplitz determinants and the Fisher-Hartwig conjecture, J. Stat. Phys. 116, 79 (2004)
  * [77] P. Calabrese and F. H.L. Essler, Universal corrections to scaling for block entanglement in spin-1/2 XX chains, J. Stat. Mech. (2010) P08029
  * [78] F. Ares, J. G. Esteve, F. Falceto and A. R. de Queiroz, Entanglement entropy and Möbius transformations for critical fermionic chains, J. Stat. Mech. (2017) 063104
  * [79] E. Kaminishi, J. Sato and T. Deguchi, Recurrence Time in the Quantum Dynamics of the 1D Bose Gas, J. Phys. Soc. Jpn. 84, 064002 (2015)
  * [80] C. Rylands and N. Andrei, Quantum work of an optical lattice, Phys. Rev. B 100, 064308 (2019)
  * [81] I. Peschel, Entanglement in solvable many-particle models, Braz. J. Phys. 42, 267 (2012).
  * [82] I. Peschel, M. Kaulke and Ö. Legeza, Density-matrix spectra for integrable models, Ann. Phys. 8, 153 (1999)
  * [83] M. C. Chung and I. Peschel, Density-matrix spectra of solvable fermionic systems, Phys. Rev. B 64, 064412 (2001)
  * [84] I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36, L205 (2003)
  * [85] I. Peschel, On the reduced density matrix for a chain of free electrons, J. Stat. Mech. (2004) P06004
  * [86] I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42, 504003 (2009)


# Discovery of the Content and Engagement with the Content

Pushkal Agarwal1, Nishanth Sastry1, Edward Wood2
1King’s College London, 2House of Commons Research Library

## Introduction

For much of the 20th century, British politicians bemoaned the decline in
reporting of Parliament by the press. The UK Parliament’s _Modernisation
Committee_ noted ruefully in 2004 that “the House can no longer expect to
receive a certain amount of media coverage as of right. Parliamentary
proceedings must now compete with other potential news stories for
coverage”11126.5.04, para 108,
https://publications.parliament.uk/pa/cm200304/cmselect/cmmodern/368/36809.htm.
One obvious strategy to counter this trend is for Parliament to provide direct
coverage of debates and committee meetings, allowing people to get news
straight from the horse’s mouth. This window into the UK Parliament’s
activities is provided in three forms: _transcripts, video and social media_.
The first is _Hansard_ , which is a near verbatim transcript of the
Parliamentary debates and committee sessions. _Hansard_ , which took its name
from the printer of the early transcripts, is one of the oldest such series in
the World: regular reporting of the UK Parliament has been in operation since
the early 19th century. Access to _Hansard_ was increased dramatically when it
was published online from 1996. In the second half of the 20th century,
Parliament allowed broadcasters to transmit radio and eventually television
coverage of debates and meetings of select committees. More recently, in an
effort to further improve transparency and citizen engagement, the UK
Parliament started publishing videos of these debates and meetings itself, and
tweeting details of debates as they happened.

In this paper, we attempt to characterise how people engage with video data of
Parliamentary debates by using more than two years of Google Analytics data
around these videos222Data provided by the Video Analytics team of UK
Parliament. We analyse the patterns of engagement – how do they land on a
particular video? How do they hear about this video, i.e., what is the (HTTP)
referrer website that led to the user clicking on the video? Once a user lands
on a video, how do they engage with it? For how long is the video played? What
is the next destination? etc.

Answering these questions is an important first step towards understanding why
and how people use Parliamentary videos, and therefore, how the video delivery
platform should be adapted and personalised for the needs of the citizens of
the country. Taking inspiration from An et al. (2017), we employ Non-Negative
Matrix Factorization (NMF) (Lee and Seung, 1999) on the video views matrix to
identify different archetypes of users, and identify 5 archetypes. A deeper
examination of the archetypes we find reveals that they are primarily
distinguished by how they land on the video page: Search (i.e., through a
search engine), Referral (i.e., from other Parliamentary websites), Direct
(i.e., through a direct link, which is embedded on another website), Social
(i.e., through a social platform such as Facebook or Twitter) and Others.

Interestingly, these different archetypes appear to have different levels of
engagement with the Parliamentary videos: Those that land on a video from a
social website tend to watch the video for a shorter duration than those that
land from other sources such as referrals or direct links. When a larger
number of clusters is generated, further differences can be amongst users who
land from a social platform, with Twitter forming a separate cluster as
compared with Facebook and Reddit as more number of MPs are active on Twitter
(Agarwal et al., 2019). Also, the different archetypes have different viewing
interests which could be used for improving personalization for the end user.
As an example, during the period of our analysis, social media users were
mostly interested in debates relating to the ongoing separation of the United
Kingdom from the EU (also known as “Brexit”).

In the rest of this extended abstract, we first describe the dataset, and then
our initial results. Finally we sketch implications for personalisation.

## Dataset and Algorithm

UK parliament session video are broadcasted live at _parliamentlive.tv_ for
all to view. Subsequent to the live broadcast, an _archive_ of the video is
also saved for on-demand viewing at any later date. Citizens from UK and other
countries can view these live and archived sessions, check a calendar for the
schedule of sessions and so on. Each session also gets its own URL, and users
can share the URLs on OSMs (Online Social Media).

Our analysis is based on the _viewing logs_ and referral source of landing
users of these session videos. The details of _Google Analytics_ data which we
explored in the study are in Table 1

Duration |  23-02-2015 to 05-02-2017
---|---
Total Views |  5.96 Million
Total Viewers |  4.1 Million
Referral Groups |  1,056
Total Videos |  17,012
Total Video types |  6,251
Table 1: Dataset Details

One of the main reasons for running these video broadcasts and other online
parliamentary services is to promote user engagement. However, “the public”
does not seem to have an overwhelming interest in watching parliamentary
debates: the BBC Parliamentary channel had the least views in analysis of six
million users of iPlayer (Nencioni et al., 2016). So, one question to address
here is how can we make it more engaging?

Google Analytics’ dashboard of the aforementioned data claims that 42% users
landed from a _Direct Search_ _e.g.,_ Video link on homepage, 22% from other
parliament websites’ _Referral_ , 20% from _Social_ and 14.7% from organic
_Search_. For parliament websites’ referral, the average session duration is 3
minute 11 seconds while for social it is 1 minute 36 seconds. The highest
average duration for any class is 4 minute 11 second, for organic search. This
suggests that there are strong differences in consumption behavior. Also, a
thing to note here is that direct referral and search are somewhat offline
recommendation where users themselves browse for and start playing a video. In
these cases users discover the content with no online influence or
personalization. For users landing from social platform, their personalized
views from their social news feeds affect which videos are visible to them.

To understand more about these different patterns in viewing behavior we
grouped the raw data from Google Analytics and then extracted features of
users’ components. To achieve the first part we created viewership matrix $V$,
where $V_{ij}$ represents number of views coming from an $i^{th}$ referral
group (_e.g.,_ Google, Twitter, E-mails and so forth) for all $j^{th}$ video
type (_e.g.,_ all house of commons videos) for entire duration of the data.

After computing users’ engagement with respective content as matrix $V$, we
use Non-Negative Matrix Factorization(NMF) to obtain $W$ and $H$ such that
$V\approx WH$. NMF is a state of the art technique to separate features from
groups based on a rank of the matrix supplied as $p$. (An et al., 2017) used
this to propose a model of Automatic Persona Generation (APG). Since Google
Analytics gives _five_ types of online user groups (Search, Referral, Direct,
Other, Social) we use $p=5$. Complete NMF is in equation 1, where Error matrix
$E$ balances the approximation on RHS.

$V_{gXc}=W_{g\times p}XH_{p\times c}+E_{g\times c}$ (1)

The application of NMF on our data and features of users’ group is in the next
Application section.

## Application

Using the two factors $W$ and $H$ of $V$ (from equation 1), here we check
discovery and engagement with the content. Matrix $W$ represents interaction
of referral source with that user groups (here five user groups as p=5). Top
15 referrals are plotted as a heatmap in Fig. 1. Higher the heat of a
component, the higher the interaction between the referral source and the
particular user group. This grouping based on referral source through light on
view pattern and conversion of users. The Figure clearly shows that different
user groups have different video preference and interests. Even Twitter (here
t.co) have different user referrals from corresponding social. On the one hand
online (social) users could be influenced with the personalization of their
online feeds while on the other hand offline (organic search) and recommended
(direct and referral) users are mostly those who hear about the event, see the
event going on parliamentary website or calenders. Given their prior knowledge
of the existence of the debate and their interest in finding the video, these
users are more committed and stay for a longer duration.

Another factor of matrix $V$ is $H$ which represents the engagement of user
groups with the video content. Table 2 shows the type of top video on which
the groups land. Interestingly, social is consuming more of Brexit-related
videos, which highlights the interest of the general public and especially the
social networkers in this topic.

Figure 1: Top fifteen referral sources (in rows) with heat of user groups (in columns). Component |  Top Video Type
---|---
Search |  business, innovation and skills committee and work and pensions committee
Referral |  house of commons
Direct |  westminster hall
Other |  house of lords
Social |  exiting the european union committee
Table 2: Preferred video for user groups from matrix $H$.

## Prospects for personalisation

These preliminary results present several opportunities in the context of
increasing user engagement. We identified that there are different kinds of
users who consume the Parliamentary videos in different ways. It may help
engagement with the Parliament if different videos were highlighted
differently to these different archetypes of users. For instance, it may be
useful to promote Brexit-like videos to users on social platforms. Other kinds
of debates may be more interesting to other kinds of users At the same time,
different modes of delivery may be employed for different user archetypes –
for users arriving from social media platforms, since their attention span is
limited, a “summary” or “highlights” version of a debate, similar to
highlights shown in sports matches, may be a useful addition.

## References

  * (1)
  * Agarwal et al. (2019) Pushkal Agarwal, Nishanth Sastry, and Edward Wood. 2019\.  Tweeting mps: Digital engagement between citizens and members of parliament in the uk. In Proceedings of the International AAAI Conference on Web and Social Media, Vol. 13. 26–37.
  * An et al. (2017) Jisun An, Haewoon Kwak, and Bernard J Jansen. 2017.  Personas for Content Creators via Decomposed Aggregate Audience Statistics. In Proceedings of the 2017 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2017. ACM, 632–635.
  * Lee and Seung (1999) Daniel D Lee and H Sebastian Seung. 1999.  Learning the parts of objects by non-negative matrix factorization.  Nature 401, 6755 (1999), 788.
  * Nencioni et al. (2016) Gianfranco Nencioni, Nishanth Sastry, Gareth Tyson, Vijay Badrinarayanan, Dmytro Karamshuk, Jigna Chandaria, Jon Crowcroft, Gianfranco Nencioni, Nishanth Sastry, Gareth Tyson, Vijay Badrinarayanan, Dmytro Karamshuk, Jigna Chandaria, and Jon Crowcroft. 2016.  SCORE: Exploiting Global Broadcasts to Create Offline Personal Channels for On-Demand Access.  IEEE/ACM Trans. Netw. 24, 4 (Aug. 2016), 2429–2442.  DOI:http://dx.doi.org/10.1109/TNET.2015.2456186


* Ayala  _et al._ (2017) V. Ayala, H. Román-Flores, and A. Da Silva, Chaos, Solitons & Fractals 94, 37 (2017).
  * Rahmani  _et al._ (2013) A. Rahmani, T. Kitagawa, E. Demler, and C. Chamon, Physical Review A 87, 043607 (2013).
  * Bao  _et al._ (2018) S. Bao, S. Kleer, R. Wang, and A. Rahmani, Physical Review A 97, 062343 (2018).
  * Karzig  _et al._ (2015) T. Karzig, A. Rahmani, F. von Oppen, and G. Refael, Physical Review B 91, 201404 (2015).
  * E.B. Davies (1976) E.B. Davies,  _Quantum Theory of Open Systems_ (Academic Press, London, 1976).
  * Alicki and Lendi (2007) R. Alicki and K. Lendi,  _Quantum Dynamical Semigroups and Applications_ (Springer Science & Business Media, 2007).
  * Rivas and Huelga (2012) A. Rivas and S. F. Huelga,  _Open Quantum Systems: An Introduction_, SpringerBriefs in Physics (Springer-Verlag, Berlin Heidelberg, 2012).
  * Lidar (2019) D. A. Lidar, arXiv preprint arXiv:1902.00967 (2019).
  * Fiacca  _et al._ (1998) A. Fiacca, N. S. Papageorgiou, and F. Papalini, Czechoslovak Mathematical Journal 48, 291 (1998).
  * Campos Venuti and Lidar (2018) L. Campos Venuti and D. A. Lidar, Phys. Rev. A 98, 022315 (2018).
  * Childs  _et al._ (2001) A. M. Childs, E. Farhi, and J. Preskill, Phys. Rev. A 65, 012322 (2001).
  * Alicki  _et al._ (2006) R. Alicki, D. A. Lidar, and P. Zanardi, Phys. Rev. A 73, 052311 (2006).
  * Albash  _et al._ (2012) T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, New J. Phys. 14, 123016 (2012).
  * Mozgunov and Lidar (2020) E. Mozgunov and D. Lidar, Quantum 4, 227 (2020).
  * Yamaguchi  _et al._ (2017) M. Yamaguchi, T. Yuge, and T. Ogawa, Physical Review E 95, 012136 (2017).
  * Dann  _et al._ (2018) R. Dann, A. Levy, and R. Kosloff, Physical Review A 98, 052129 (2018).
  * Nathan and Rudner (2020) F. Nathan and M. S. Rudner, Physical Review B 102, 115109 (2020).
  * Davidović (2020) D. Davidović, Quantum 4, 326 (2020).
  * Winczewski  _et al._ (2021) M. Winczewski, A. Mandarino, M. Horodecki, and R. Alicki, “Bypassing the intermediate times dilemma for open quantum system,”  (2021), arXiv:2106.05776 [quant-ph] .
  * Gorini  _et al._ (1976) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).
  * Lindblad (1976) G. Lindblad, Comm. Math. Phys. 48, 119 (1976).
  * Chruściński and Pascazio (2017) D. Chruściński and S. Pascazio,  _Open Systems & Information Dynamics_, Open Systems & Information Dynamics 24, 1740001 (2017).
  * Fleming and Rishel (1975) W. H. Fleming and R. W. Rishel,  _Deterministic and Stochastic Optimal Control_, Stochastic Modelling and Applied Probability (Springer-Verlag, New York, 1975).
  * Zanardi (1998) P. Zanardi, Physical Review A 57, 3276 (1998).
  * Lidar  _et al._ (1998) D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998).
  * Bacon  _et al._ (2000) D. Bacon, J. Kempe, D. A. Lidar, and K. B. Whaley, Phys. Rev. Lett. 85, 1758 (2000).
  * Kjaergaard  _et al._ (2020) M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Annual Review of Condensed Matter Physics 11, 369 (2020).
  * Harris  _et al._ (2010) R. Harris, M. W. Johnson, T. Lanting, A. J. Berkley, J. Johansson, P. Bunyk, E. Tolkacheva, E. Ladizinsky, N. Ladizinsky, T. Oh, F. Cioata, I. Perminov, P. Spear, C. Enderud, C. Rich, S. Uchaikin, M. C. Thom, E. M. Chapple, J. Wang, B. Wilson, M. H. S. Amin, N. Dickson, K. Karimi, B. Macready, C. J. S. Truncik, and G. Rose, Phys. Rev. B 82, 024511 (2010).
  * Yan  _et al._ (2016) F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, Nature Communications 7, 12964 EP (2016).
  * Smirnov and Amin (2018) A. Y. Smirnov and M. H. Amin, New Journal of Physics 20, 103037 (2018).
  * Khezri  _et al._ (2021) M. Khezri, J. A. Grover, J. I. Basham, S. M. Disseler, H. Chen, S. Novikov, K. M. Zick, and D. A. Lidar, npj Quantum Information 7, 36 (2021).
  * Harrigan  _et al._ (2021) M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. Dunsworth, D. Eppens, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, S. Habegger, A. Ho, S. Hong, T. Huang, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, C. Jones, D. Kafri, K. Kechedzhi, J. Kelly, S. Kim, P. V. Klimov, A. N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, M. Lindmark, M. Leib, O. Martin, J. M. Martinis, J. R. McClean, M. McEwen, A. Megrant, X. Mi, M. Mohseni, W. Mruczkiewicz, J. Mutus, O. Naaman, C. Neill, F. Neukart, M. Y. Niu, T. E. O’Brien, B. O’Gorman, E. Ostby, A. Petukhov, H. Putterman, C. Quintana, P. Roushan, N. C. Rubin, D. Sank, A. Skolik, V. Smelyanskiy, D. Strain, M. Streif, M. Szalay, A. Vainsencher, T. White, Z. J. Yao, P. Yeh, A. Zalcman, L. Zhou, H. Neven, D. Bacon, E. Lucero, E. Farhi, and R. Babbush, Nature Physics 17, 332 (2021).
  * Marshall (1955) W. Marshall, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 232, 48 (1955).
  * Lieb and Mattis (1962) E. Lieb and D. Mattis, Journal of Mathematical Physics 3, 749 (1962).
  * Agrachev and Sachkov (2004) A. A. Agrachev and Y. Sachkov,  _Control Theory from the Geometric Viewpoint_, Encyclopaedia of Mathematical Sciences, Control Theory and Optimization (Springer-Verlag, Berlin Heidelberg, 2004).
  * M.S. Byrd and N. Khaneja (2003) M.S. Byrd and N. Khaneja, Phys. Rev. A 68, 062322 (2003).
  * Borisov (2000) V. F. Borisov, J Math Sci 100, 2311 (2000).
  * Venuti (2015) L. C. Venuti, “The recurrence time in quantum mechanics,”  (2015), arXiv:1509.04352 [quant-ph] .