text:
\section{Introduction and Motivation}
\label{secI}

Statistical mechanics can be derived from Jaynes' principle of maximum
entropy~\cite{Jaynes1957a, Jaynes1957b} given the constants of motion for any
generic many-body system. Remarkably, this principle may also be extended to
the level of a single eigenstate in a thermodynamically large, isolated
quantum system leading to the eigenstate thermalization hypothesis
~\cite{Deutsch1991, Srednicki1994, RigolDO2008}. The eigenstate thermalization
hypothesis implies that the expectation values of all operators which are
local in space equal those obtained from a maximum entropy statistical
description when {\it appropriate} conserved quantities are taken into account
for constructing the ensemble. The Hamiltonian is the sole such conserved
quantity in a generic chaotic system. Exceptions to the eigenstate
thermalization hypothesis are provided by disordered systems that show
localization, both for non-interacting models~\cite{Anderson1958}
and with interactions~\cite{BaskoAA2006} (i.e., many-body localized
systems~\cite{NandkishoreH2015}). Integrable models provide examples of
systems that are not localized and yet do not satisfy the eigenstate
thermalization hypothesis. However, Jaynes' principle may again
be applied here by considering the extensive number of conservation laws
due to integrability that then leads to a generalized
Gibbs ensemble (GGE)~\cite{RigolDYO2007, CassidyCR2011, CauxE2013}
instead of the standard Gibbs ensemble. These statistical descriptions are
expected to hold only for local properties (i.e., properties that are
determined by the degrees of freedom in a subsystem that is much smaller than
the rest of the system, as shown schematically in Fig.~\ref{fig0main}), and
not for the wave function of the entire system which is
after all in a pure state~\cite{PopescuSW2006}.

\begin{figure}
{\includegraphics[width=\hsize]{subsystem.pdf}}
\caption{The separation of a closed system into a subsystem with $l$ sites
(indicated in black) and the environment with $L-l$ sites (indicated in red).
Local properties only involve degrees of freedom on the sites of a subsystem
that satisfies $l \ll L$ with $L \gg 1$.} \label{fig0main} \end{figure}

Tremendous experimental progress in producing and manipulating well-isolated
quantum systems such as ultracold quantum
gases~\cite{Blochreview2005, Goldman2014, Langenetalreview2015, Eck17},
has also led to great interest in driven closed quantum systems. Removing
the time translational invariance of closed quantum systems via a
time-dependent Hamiltonian leads to the much richer possibility of stabilizing
non-equilibrium steady states with purely unitary dynamics.
Such non-equilibrium steady states may even have properties otherwise
forbidden in thermal equilibrium,
such as spontaneous time-translational symmetry breaking in many body localized
systems~\cite{ElseBN2016, KhemaniLMS2016} and dynamical topological
ordering~\cite{KitagawaBRD2010, NathanR2015}.
Many-body Floquet systems where the time-dependent Hamiltonian satisfies
\bea H(t)=H(t+nT), \label{periodicdrive1} \eea
with $T$ being the time period of the drive and $n$ being integer-valued,
is a well-studied case where such non-equilibrium steady states have been
shown to exist.
More precisely, the local properties of Floquet systems eventually synchronize
with the period of the drive that then allows the possibility of a ``periodic''
ensemble~\cite{LazaridesDM2014a, LazaridesDM2014b, Ponte15b, AlessioR2014}.
Propagating the wave function $|\psi \rangle$ of the system in a stroboscopic
manner
\bea |\psi(nT) \rangle = U(T)^n |\psi(0) \rangle, \label{periodicdrive2} \eea
where $U(T)$ is the time evolution operator for one time period, generates
a discrete quantum map indexed by $n$, and the steady state emerges as
$n \to \infty$. This steady state can again be described using
Jaynes' principle~\cite{LazaridesDM2014a} where only
quantities that are stroboscopically conserved need to be taken into account.
In the non-integrable case, it is expected that no such conserved quantity
exists~\cite{NandkishoreH2015} since the Hamiltonian is no longer conserved;
hence the system {\it should} locally mimic an infinite temperature ensemble
in the steady state by entropy maximization. This is indeed consistent
with the results of several recent studies~\cite{LazaridesDM2014b, Ponte15b,
AlessioR2014, Russomanno12, Bukov15, Ponte15a, Eckardt15}, but a complete
understanding is still lacking. The situation is markedly different for
integrable systems. For systems that are reducible to free fermions e.g.,
the prototypical one-dimensional (1D) spin-1/2 transverse field Ising model,
it has been shown~\cite{LazaridesDM2014a} that an extensive number of
quantities that are stroboscopically conserved continue to exist {\it even}
in the driven case; these prevent the system from approaching the infinite
temperature ensemble locally and instead lead to a {\it periodic} GGE
(p-GGE) description of the synchronized local properties at late times. The
fate of other types of integrable systems like Bethe-integrable ones (for
instance, the 1D spin-1/2 $XXZ$ model) under periodic drives is still an open
issue~\cite{BurkovH2012}.

\subsection{Synopsis}
\label{subsecIa}

In this work, we consider integrable models that are reducible to free
fermions, and our objective is to understand whether well-defined
non-equilibrium steady states exist for
continually driven protocols that are {\it not} periodic functions in time. We
restrict ourselves to drives that can be represented by discrete quantum maps
similar to Eq.~\eqref{periodicdrive2} for ease of analysis. For concreteness,
we work here with the 1D transverse field Ising model which is a standard
many-body integrable quantum system~\cite{Subirbook,SIBKCbook,Duttabook}. The
time-dependent Hamiltonian is given by
\bea H(t) ~=~ - ~\sum_{j=1}^L ~\left[ g(t) s_j^x+ s_j^z s_{j+1}^z \right],
\label{modeldef} \eea
where $s^{x,y,z}$ are the usual Pauli operators, $L$ denotes the number of
spins in the system under consideration, and we assume periodic boundary
conditions in space. $g(t)$ represents the time-dependent
transverse magnetic field through which the system is continually driven.

We start with a reference function $g_{\mathrm{ref}}(t)$ defined for
$0 \le t \le T$. Repeatedly using the corresponding time evolution operator
$U(T)$ for all values of $n>0$ in Eq.~\eqref{periodicdrive2} results in
periodic driving with period $T$. Consider now a different drive protocol
where $g(t)$ is constructed by patching together rescaled versions of
$g_{\mathrm{ref}}(t)$ in time as a function of $n$ such that
the period of $g_{\mathrm{ref}}(t)$ is stretched or reduced by $dT$
depending on $n$. The corresponding time evolution operator is
then denoted by $U(T+dT)$ or $U(T-dT)$. The new drive protocol $g(t)$ can now
be represented by a sequence in $n$ that takes a value of either $+1$
(representing $U(T+dT)$) or $-1$ (representing $U(T-dT)$) at each $n$.
There are an {\it infinite} number of ways to choose the sequence of $\pm 1$
as a function of $n$, each of which represents a different purely unitary
dynamics, such that the resulting driving protocol $g(t)$ cannot be
represented by {\it any} periodic function in time.
We note that the protocol described above involving two time periods
$T+dT$ and $T-dT$ is mathematically equivalent to a protocol in which
the time period $T$ is kept fixed but the Hamiltonian is scaled globally
by factors of $1+dT/T$ and $1-dT/T$ respectively. The important point about
the protocol is that it involves two possible time evolution operators which
do not commute with each other.

We will work out the nature of the resulting non-equilibrium steady states as
$n \to \infty$
in two cases here. First, we will consider a typical realization of a random
sequence of $\pm 1$ where the sign is chosen with probability 1/2 at each
$n$ with $dT \ll T$. This mimics a periodic drive protocol perturbed with
random noise due to the lack of perfect control over the time variation of
$g(t)$ in an experimental set-up. Secondly, we will consider a different
sequence of $\pm 1$ that is neither random nor periodic in $n$ but has
a fractal structure instead, and thus can be
thought of as an example of scale-invariant noise.

For a periodic drive protocol perturbed with random noise where $dT \ll T$,
we will show that the system, after initially approaching the p-GGE that
corresponds to $dT=0$, eventually heats up to an infinite temperature ensemble
as $n \to \infty$ in spite of the underlying integrability of the model. The
time spent by the system close to the p-GGE, which can now be thought of as
a quasisteady {\it prethermal} state, and the eventual manner of heating up to
an infinite temperature ensemble can both be understood from a simple random
walk argument, provided that $dT \ll T$. We will
further show that a {\it coarse-graining procedure} in momentum space
explains how the reduced density matrix of any subsystem with $l$ adjacent
spins approaches an infinite temperature ensemble as long as $l \ll L$
even though the dynamics is completely unitary.
It is well known that weakly interacting systems (i.e., systems close
to an integrable point) after a sudden perturbation show a prethermalization
regime that can be understood by a GGE based on perturbatively constructed
constants of motion~\cite{KollarWE2011}, before eventually thermalizing on
a much longer time scale~\cite{MoeckelK2008, Gring2012} with the energy being
the only constant of motion. In the perturbed Floquet integrable
systems with random noise that we consider, the integrability of the
model is never broken but the stroboscopically conserved quantities
at $dT=0$ no longer remain conserved when $dT \neq 0$.

However, it is important to note that the lack of conservation of these
quantities does {\it not} fix the form of the final non-equilibrium steady
state for an aperiodic
drive protocol. For a periodic drive protocol perturbed with scale-invariant
noise, we will show that if the local properties are observed not as a function
of $n$, but instead as a function of $2^n$ (thus, geometrically instead of
stroboscopically), a new steady state emerges when $n \to \infty$ which we call
``geometric'' GGE (g-GGE). Interestingly, this steady state can also be
understood using an effective p-GGE construction even though the drive in not
periodic in time because of a fixed point property of products of $2 \times 2$
$SU(2)$ matrices. An extensive number of conserved quantities appear when the
system is viewed after every $2^n$ drives, but only when $n$ is sufficiently
large, which shows that these conservations are {\it emergent}.

The rest of the paper is arranged in the following manner. In Sec.~\ref{secII},
we review some results relevant for our work and set the notations for the
rest of the paper. Sec.~\ref{subsecIIa} details the pseudospin representation
for the 1D transverse field Ising model that allows the wave function of the
system to be denoted in terms of points on Bloch spheres in momentum space.
Sec.~\ref{subsecIIb} details the aperiodic driving protocols that we study.
Sec.~\ref{subsecIIc} reviews the way to obtain the reduced density matrix of a
subsystem of $l$ adjacent spins using appropriate correlation matrices. In
Sec.~\ref{secIII} (including Sec.~\ref{subsecIIIa} and Sec.~\ref{subsecIIIb}),
we introduce certain coarse-grained quantities in momentum space which fully
determine the reduced density matrix of
a subsystem when $l \ll L$ and thus {\it all} local quantities, and we discuss
their behavior for time-periodic drive protocols. In Sec.~\ref{secIV}, we
present results for perturbed Floquet systems with random noise and use a
random walk argument to show how the system heats up to an infinite temperature ensemble in spite of its
integrability. In Sec.~\ref{secV}, we consider perturbed Floquet systems with
noise which is scale-invariant in time and show that for a certain type of
scale-invariant noise, the system reaches a steady state which is
{\it not} an infinite temperature ensemble, but it does so only
after an astronomically large number of drives. The final steady state is
explicitly constructed using a fixed point property of strings of $2\times 2$
$SU(2)$ matrices. Finally, we summarize our results and conclude with some
future directions in Sec.~\ref{secVI}.

\section{Preliminaries}
\label{secII}

\subsection{Pseudospin representation of dynamics for the 1D transverse
field Ising model}
\label{subsecIIa}

To solve the 1D transverse field Ising model (Eq.~\eqref{modeldef}), we
introduce the standard Jordan-Wigner transformation of spin-1/2's to spinless
fermions~\cite{Lieb61}
\bea s_n^x &=& 1-2c^\dagger_n c_n, \nonumber \\
s_n^z &=& -(c_n+c_n^\dagger)\prod_{m<n}(1-2c^\dagger_m c_m). \label{jw1} \eea
Writing $H$ in Eq.~\eqref{modeldef} in terms of these fermions, we obtain
\bea H ~=~ P^+ H^+ P^+ ~+~ P^-H^-P^-, \label{jw2} \eea
where $P^{\pm}$ are projectors on to subspaces with even ($+$)
and odd ($-$) number of fermions, and
\bea H^{\pm} &=& -\sum_{n=1}^L ~[g(t) ~-~ 2g(t) c^\dagger_n c_n ~+~
c^\dagger_n c_{n+1} ~+~ c_{n+1}^\dagger c_n \nonumber \\
&& ~~~~~~~~~+ ~c_{n+1}c_n ~+~ c_n^\dagger c_{n+1}^\dagger], \label{jw3} \eea
with antiperiodic (periodic) boundary conditions for the fermions for
$H^+$ ($H^-$). We now focus on the case of even $L$ with antiperiodic
boundary conditions for the fermions (which corresponds to
periodic boundary conditions for the spins). To go to $k$ space, we define
\bea c_k &=& \frac{e^{i\pi/4}}{\sqrt{L}} ~\sum_{n=1}^L ~e^{-ikn} ~c_n, \nonumber \\
c_n &=& \frac{e^{-i\pi/4}}{\sqrt{L}} ~\sum_k ~e^{ikn} ~c_k, \label{jw4} \eea
where $k = 2\pi m/L$ with $m = -(L-1)/2, \cdots, -1/2$, $1/2, \cdots, (L-1)/2$
for even $L$. Rewriting $H^+$ in terms of $c_k$ and $c_k^\dagger$, we get
\bea H^+ &=& \sum_k ~[2(g(t)-\cos k) ~c^\dagger_k c_k ~+~\sin k~ c_{-k}c_k \nonumber \\
&& ~~~~~~+ ~\sin k ~c^\dagger_k c^\dagger_{-k}-g(t)]. \label{fermionH} \eea
This Hamiltonian connects the vacuum $|0 \rangle$ of the $c$ fermions with the
two-particle state $|k,-k \rangle =c_k^\dagger c^\dagger_{-k}|0 \rangle$, and
$|k \rangle
= c_k^\dagger |0 \rangle$ with $|-k \rangle = c_{-k}^\dagger |0 \rangle$. It is
more convenient to write Eq.~\eqref{fermionH} as $H = \sum_{k>0} H_k$ where
\bea H_k &=& 2(g(t)-\cos k) [c^\dagger_k c_k - c_{-k}c^\dagger_{-k}] \nonumber \\
&+& 2 \sin k [c_{-k}c_k + c^\dagger_k c^\dagger_{-k}]. \label{pseudospin1} \eea

We now introduce a ``pseudospin representation'' $\sigma_k$ (which is different
from the original $s$ matrices in Eq.~\eqref{modeldef})
where $|\uparrow \rangle_k = |k,-k \rangle = c_k^\dagger c^\dagger_{-k}|0
\rangle$ and $|\downarrow \rangle_k = |0 \rangle$~\cite{KolodrubetzCH2012}.
Writing $H_k$ in Eq.~\eqref{pseudospin1} in this basis, we see that
\bea H_k ~=~ 2(g(t)-\cos k) ~\sigma^z_k ~+~ 2 \sin k ~\sigma^x_k.
\label{pseudospin2} \eea
Then the pseudospin state $|\psi_k \rangle$ for each $k$ mode evolves
independently as
\bea i \frac{\mathrm{d}}{\mathrm{dt}} |\psi_k \rangle ~=~ H_k(t)
|\psi_k \rangle, \label{pseudospin3} \eea
where
\bea |\psi(t) \rangle &=& \otimes_{k>0}|\psi_k(t) \rangle, \nonumber \\
|\psi_k (t)\rangle &=& u_k(t)|\uparrow \rangle_k + v_k(t)|\downarrow \rangle_k.
\label{pseudospin4} \eea
Thus, specifying $(u_k(t),v_k(t))^T$ (where the superscript $T$ denotes the
transpose of the row $(u_k(t),v_k(t))$) for each allowed $k>0$ for a finite
$L$ specifies the complete wave function $|\psi(t) \rangle$ through
Eq.~\eqref{pseudospin4}. The wave function of the system can be equivalently
characterized by a two-dimensional state space at each allowed positive $k$,
and $(u_k(t),v_k(t))^T$ can then be represented as a point that evolves with
time according to Eq.~\eqref{pseudospin3} on the corresponding Bloch sphere.

In the rest of the paper, we will take $(u_k,v_k)^T=(0,1)^T$ to be the starting
state at each $k$ from which the unitary dynamics proceeds. This corresponds
to the system being initially prepared in the pure state where all the spins
have $\sigma^x=+1$ (which corresponds to the ground state for $g \to \infty$).

\subsection{Details of the driving protocol}
\label{subsecIIb}

While our basic conclusions are protocol independent and do not depend
on the exact nature of $g_{\mathrm{ref}}(t)$, we take it to
have a square pulse variation in time for mathematical convenience, namely,
\bea g_{\mathrm{ref}} &=& g_i \mbox{~~~for~~~} 0 \le t<T/2 \nonumber \\
&=& g_f \mbox{~~~for~~~} T/2 \le t \le T. \label{squarepulse} \eea
The corresponding unitary time evolution operator $U_k(T)$
for the pseudospin mode with momentum $k$ equals
\bea U_k(T) = \exp \left(-iH_k(g_f)\frac{T}{2}\right) \exp \left(-iH_k(g_i)
\frac{T}{2} \right), \label{defU} \eea
where $H_k$ has the form given in Eq.~\eqref{pseudospin2}. For the perturbed
Floquet dynamics that we consider in this work, we need to define
$U_k(T \pm dT)$; these follow in a straightforward manner from Eq.~\eqref{defU}
by replacing $T \to T \pm dT$.

Rewriting Eq.~\eqref{periodicdrive2} for the perturbed Floquet dynamics for
the mode with momentum $k$, we get
\begin{widetext}
\bea |\psi_k(n) \rangle ~=~ U_k(T+ \tau_n dT) U_k(T + \tau_{n-1}dT) \cdots
U(T + \tau_1 dT) |\psi_k(0) \rangle ~=~ {\cal T} ~\prod_{i=1}^n ~U_k(T+\tau_i
dT) |\psi_k(0) \rangle, \label{defsequence} \eea
\end{widetext}
where the sequence ${\tau_i}=\tau_1, \tau_2, \tau_3, \cdots$ (with each
$\tau_i$ being equal to either $+1$ or $-1$) is the same for all the allowed
$k$ modes at a finite $L$. (In the last expression in Eq.~\eqref{defsequence},
${\cal T}$ denotes time-ordering).

For Floquet systems perturbed with random noise, the sequence
${\tau_i}$ can be taken to be any typical realization of a random
process where each element is chosen to be $\pm 1$ randomly and
independently with probability 1/2; we will discuss the results
for such a realization later in the paper. For Floquet systems perturbed with
scale-invariant noise, we take the well-known Thue-Morse sequence
~\cite{Thue1906, Morse1921a, Morse1921b} which is an infinite sequence of
$\tau_i = \pm 1$ that is obtained by starting with $\tau_1=-1$ and
successively appending the negative of the sequence obtained thus far.
The first few steps of this recursive procedure yield
\begin{widetext}
\bea
\label{TMS}
m=0:&& \mbox{~~} \tau_1 ~=~ -1 \\ \nonumber
m=1:&& \mbox{~~} \tau_1,\tau_2 ~=~ -1,+1 \\ \nonumber
m=2:&& \mbox{~~} \tau_1, \cdots, \tau_4 ~=~ -1,+1,+1,-1 \\ \nonumber
m=3:&& \mbox{~~} \tau_1, \cdots, \tau_8 ~=~ -1,+1,+1,-1,+1,-1,-1,
+1 \\ \nonumber
m=4:&& \mbox{~~} \tau_1, \cdots, \tau_{16} ~=~ -1,+1,+1,-1,+1,-1,-1,+1,+1,-1,
-1,+1,-1,+1,+1,-1 \\ \nonumber
\vdots
\eea
\end{widetext}
At each recursion level $m$, we thus obtain the first $2^m$ elements of this
infinite sequence. The self-similar structure of the Thue-Morse sequence can be
verified by removing every second term from it, which then results in the same
infinite sequence. This sequence is thus an example of a {\it quasiperiodic}
sequence in time which is neither periodic nor random. Note that the average
time period of this perturbed Floquet system equals $T$ (the time period
of the unperturbed Floquet system), just like in the random noise case.

\subsection{Reduced density matrices and the distance measure}
\label{subsecIIc}

Given the wave function $|\psi(n) \rangle$ of the entire system, which can be
described by the corresponding density matrix $\rho = |\psi(n) \rangle \langle
\psi(n)|$, all local properties within a length scale $l$ can be understood by
considering the reduced density matrix of a subsystem of $l$ adjacent
spins (Fig.~\ref{fig0main}); we denote this by $\rho_l$ which is given by
\bea \rho_l ~=~ \mathrm{Tr}_{L-l} (\rho),
\label{RDM} \eea
where all the other $L-l$ degrees of freedom have been integrated out. Even
though $\rho$ is a pure density matrix at any $n$, the reduced density matrix
$\rho_l$ is mixed since the pure state $| \psi(n) \rangle$ typically gets
more entangled as $n$ increases.

The reduced density matrix of $l$ adjacent spins when $|\psi(n) \rangle$ has
the form given in Eq.~\eqref{pseudospin4} is determined in terms of the
correlation functions of the $c$ fermions at these $l$
sites~\cite{VidalLRK2003}. For free fermions, since all higher-point
correlation functions are expressible using two-point correlations from
Wick's theorem, the reduced density matrix can be determined from the
knowledge of two $l \times l$ matrices~\cite{Peschel2003}, $\mathbf{C}$ and
$\mathbf{F}$, whose elements are constructed from $u_k(n)$ and $v_k(n)$ as
follows:
\bea C_{ij} &=& \langle c^\dagger_i c_j \rangle_n = \frac{2}{L} \sum_{k>0}
|u_k(n)|^2 \cos[k (i- j)], \label{matrices1} \nonumber \\
F_{ij} &=& \langle c^\dagger_i c^\dagger_j \rangle_n = \frac{2}{L}
\sum_{k >0} u^*_k(n) v_k(n) \sin[k(i-j)], \label{corrmat1} \eea
where $i,j$ refer to sites in the subsystem. Using these expressions,
we construct the $2l \times 2l$ matrix ${\mathcal C}_n(l)$ given by
\bea {\mathcal C}_n(l) &=& \left( \begin{array}{cc} \mathbf{I-C} &
\mathbf{F} \\ \mathbf{F}^{\ast} & \mathbf{C } \end{array} \right).
\label{corrmat2} \eea
The reduced density matrix $\rho_l$ is then completely fixed by the
eigenvalues and eigenvectors of $\mathcal{C}_n(l)$. For
example, the entanglement entropy $S_{\mathrm{ent}}(l)$ equals:
\bea S_{\mathrm{ent}}(l)&=& -~ \mathrm{Tr} [\rho_l \log(\rho_l)] \nonumber \\
&=& -~ \sum_{k=1}^{2l} ~p_k \log(p_k) \label{Sent} \eea
where $p_k$ denotes the $k$-th eigenvalue of $\mathcal{C}_n(l)$.

If a non-equilibrium steady state exists as
$n \to \infty$, then the convergence of the
local properties to their final steady state values as $n$ increases can be
characterized by defining a distance measure $\mathcal{D}_n(l)$ as
\bea \mathcal{D}_n(l) ~=~ \frac{1}{2l}\mathrm{Tr}\sqrt{[\mathcal{C}_\infty(l)-
\mathcal{C}_n(l)]^\dagger [\mathcal{C}_\infty(l)-\mathcal{C}_n(l)]}.
\label{distancemeasure} \eea
We note that $0 \leq \mathcal{D}_n(l) \leq 1$ and is identically zero only
if $\mathcal{C}_n(l) = \mathcal{C}_\infty(l)$. For a periodically driven
protocol with $dT=0$, $\mathcal{C}_\infty(l)$ can be calculated using the
appropriate p-GGE; the distance measure decays to zero as $\mathcal{D}_n(l)
\sim n^{-\alpha (T)}$ for large $n$ (and $l \ll L$),
with $\alpha(T)=1/2$ or $3/2$ depending on the time period $T$
of the drive~\cite{SenNS2016}. We note that $\mathcal{D}_n(l)$ does not
decay to zero as $n \to \infty$ if $l/L$ is finite; this explicitly
shows that only local quantities can be described by the p-GGE.

\section{Coarse-graining in momentum space}
\label{secIII}

Even though the entire wave function $|\psi(n) \rangle$ needs the
specification of $(u_k(n),v_k(n))^T$ at $k=2\pi m/L$ with
$m=1/2,3/2,\cdots, (L-1)/2$, $\mathcal{C}_n(l)$ and consequently the reduced
density matrix for any $l \ll L$ depends {\it only} on certain
{\it coarse-grained variables} defined as follows~\cite{LaiY2015, NandySDD2016}
\bea \left(|u_k(n)|^2\right)_c &=& \frac{1}{N_c}\sum_{k \in k_{\mathrm{cell}}}
|u_k(n)|^2, \nonumber \\
\left(u^*_k(n)v_k(n)\right)_c &=& \frac{1}{N_c}\sum_{k \in k_{\mathrm{cell}}}
u^*_k(n) v_k(n). \label{coarsegrain1} \eea
These variables are defined using $N_c (\gg 1)$ consecutive $k$ modes that lie
within a cell ($k_{\mathrm{cell}}$) which has an average momentum $k_c$ in
momentum space and a size $\delta k$ where $1/L \ll \delta k \ll 1/l$. With this
condition on $\delta k$, the sinusoidal factors in Eq.~\eqref{corrmat1} can be
replaced to a very good approximation by
\bea \cos[k(i-j)] &\simeq& \cos[k_c(i-j)], \mbox{~~} \forall \mbox{~} 0
\leq |i-j| \leq l, \nonumber \\
\sin[k(i-j)] &\simeq& \sin[k_c(i-j)], \mbox{~~} \forall \mbox{~} 0 \leq |i-j|
\leq l, \eea
for all the $k$ modes within the cell.
The sum over momentum in Eq.~\eqref{corrmat1} can thus be carried out in two
steps, first summing over the consecutive momenta in a single cell, and then
summing over the different momentum cells. This gives
\bea C_{ij} &\simeq& \frac{1}{\mathcal{N}_{\mathrm{cell}}} ~\sum_{k_c}
~(|u_k(n)|^2)_c \cos [k_c(i-j)], \nonumber \\
F_{ij} &\simeq& \frac{1}{\mathcal{N}_{\mathrm{cell}}} ~\sum_{k_c}
~(u^*_k(n)v_k(n))_c \sin [k_c(i-j)], \label{coarsegrain2} \eea
where $\mathcal{N}_{\mathrm{cell}} \ll L/2$ represents
the number of momentum cells after the coarse-graining procedure.

Note that even though quantities like $|u_k(n)|^2$ and $u^*_n(k)v_n(k)$
continue to display Rabi oscillations and thus have no well-defined $n \to
\infty$ limit, the coarse-grained variables $(|u_k(n)|^2)_c$ and $(u^*_k(n)
v_k(n))_c$ must reach final steady state values for a non-equilibrium steady
state to exist.

If the system is locally described by an infinite temperature ensemble, then
$\rho_l =I$, where $I$ is
the identity matrix, for any $l \ll L$. This immediately implies that
$(|u_k(n)|^2)_c \to 1/2$ (also $(|v_k(n)|^2)_c \to 1/2$) and $(u^*_n(k)v_n
(k))_c \to 0$ for all the coarse-grained momentum cells as $n \to \infty$,
since $C_{ij}=(1/2) \delta_{i,j}$ and $F_{ij}=0$ for $\rho_l=I$. (A similar
behavior has been found for repeated quenching of the transverse field in the
1D spin-1/2 $XY$ model~\cite{Mukherjee08}). We calculate the $n$ dependence of
such coarse-grained quantities numerically by taking a sufficiently large
system size $L$ (typically $L=131072$, but $L=524288$ in some cases),
so that the allowed momentum modes are sufficiently dense in the
interval $k \in [0,\pi]$, and then dividing the Brillouin zone into
$\mathcal{N}_{\mathrm{cell}} =32$ cells, each of which has $N_c \gg 1$
consecutive momenta. For the periodic drive case where $dT=0$, we see that
although $(|u_k(n)|^2)_c$ indeed converges to a finite value
(Fig.~\ref{fig1main}) and thus has a well-defined non-equilibrium steady state,
this constant depends on the value of the average momentum of the
coarse-grained cell ($k_c$) and is different from 1/2, showing that the local
description is {\it different} from that of an infinite temperature ensemble.

\begin{figure}
{\includegraphics[width=\hsize]{uk2coarsedT0.pdf}}
\caption{Behavior of the coarse-grained quantities $(|u_k(n)|^2)_c$ as
a function of $n$ for a perfectly periodic drive ($dT=0$). The parameters
used are $L=131072$, $g_i=4$, $g_f=2$, and $T=0.2$. Each coarse-grained cell
contains $N_c = 2048$ consecutive momenta, and $k_c$ denotes the average
momentum of such a cell. Results for $k_c=31\pi/64$ (lower line in red) and
$k_c=15\pi/64$ (upper line in blue) are shown.} \label{fig1main} \end{figure}

\subsection{Probability distribution on the Bloch sphere}
\label{subsecIIIa}

At each $k$, the pseudospin state $|\psi_k (n)\rangle$ obtained after $n$
drives can be parameterized in terms of three angles $(\theta_k(n),\phi_k(n),
\beta_k(n))$,
\beq \left( \begin{array}{c}
u_k(n) \\
v_k(n) \end{array} \right) ~=~ e^{i\beta_k(n)} ~\left( \begin{array}{c}
\cos (\theta_k(n)/2) e^{-i\phi_k(n) /2} \\
\sin (\theta_k(n)/2) e^{i\phi_k(n) /2} \end{array} \right).
\label{blochsphere1} \eeq
The overall phase $\beta_k(n)$ is not important and we will ignore it
henceforth. We will concentrate on the angles $\theta_k(n)$ and $\phi_k(n)$
which define the motion of a point on the unit Bloch sphere for each momentum
$k$. From Eq.~\eqref{defU}, it is clear that $U_k(T)$ is an element of the
group $SU(2)$ (namely, $2 \times 2$ unitary matrices
with determinant equal to 1), and it can be written as
\beq U_k(T) ~=~ \exp (-i \gamma_k {\hat e}_k \cdot {\vec \sigma}), \label{uk2} \eeq
where $0 \le \gamma_k \le \pi$ and ${\hat e}_k = (e_{1k}, e_{2k}, e_{3k})$ is a unit
vector. For a drive protocol that is perfectly periodic, $(\theta_k(n),\phi_k(n))$
traces out a circle on the Bloch sphere formed by the intersection of this
unit sphere with the plane that contains the point $(\theta_k(0),\phi_k(0))$ and
whose normal vector equals ${\hat e}_k$. It is then clear that
the quantities $\mathcal{I}_k$ defined as
\bea \mathcal{I}_k ~=~ \langle \psi_k(n)| {\hat e}_k \cdot {\vec \sigma} | \psi_k(n) \rangle
\label{conservedIk} \eea
are all independent of $n$ for a periodic drive protocol and thus define
the extensive number of stroboscopically conserved quantities.

The behaviors of the coarse-grained quantities in Eq.~\eqref{coarsegrain1}
depend on the simultaneous positions of $(\theta_k(n), \phi_k(n))$ on the unit
sphere of all the momentum modes that lie within a coarse-grained cell.
Since their number $N_c \gg 1$ when $L \gg 1$, it is useful to instead
look at the probability distribution of these $N_c$ points on the unit sphere.
In particular, we will study the probability distribution of
$|u_k(n)|^2-|v_k(n)|^2 = \cos \theta_k$ for each such coarse-grained momentum
cell, which we denote as $P_n(\cos \theta_k)$, as a function of $n$.
We can also define another probability distribution from the motion
of a single $k$ mode for a large number of drive cycles and denote
it by $\overline{P}(\cos \theta_k)$. Remarkably, we find that
\bea P_n(\cos \theta_{k_c}) \simeq \overline{P}(\cos \theta_{k_c})
\label{twoprob} \eea
for large $n$, where $k_c$ denotes the average momentum of a coarse-grained
cell that has $N_c$ momenta on the left hand side of Eq.~\eqref{twoprob}, and
the right hand side is obtained from the motion of a single momentum mode at
$k_c$.

For an infinite temperature ensemble, we expect that the $N_c$ points will
cover the unit sphere uniformly
at large $n$ even though all these points started from the south pole at
$n=0$ due to the choice of $|\psi(0) \rangle$. Thus, $P_n(\cos \theta_{k_c})$
at large $n$ should be independent of the value of the average momentum of the
cell, $k_c$, and thus equal 1/2, if the local description is that of an
infinite temperature ensemble.
We will derive $\overline{P}(\cos \theta_{k_c})$ for the perfectly periodic case
in the next subsection and see that numerically $P_n(\cos \theta_{k_c})$
approaches $\overline{P}(\cos \theta_{k_c})$ for large $n$. It turns out that
$\overline{P}(\cos \theta_{k_c})$ depends on $k_c$; this again shows that the
description is not locally that of an infinite temperature ensemble for
periodic driving.

\subsection{Probability distribution for periodic driving: $dT=0$}
\label{subsecIIIb}

We will first derive the form of $\overline{P} (\cos \theta_k)$ when there is no
randomness,
i.e., when $dT=0$. In this case, the time evolution operator after one time
period $T$ for the momentum mode $k$ (where $0 \le k \le \pi$) is given by
\bea U_k(T) &=& \exp [-i(T/2)\{ 2(g_f - \cos k) \sigma^z + 2 (\sin k) \sigma^x \}]
\nonumber \\
&& \times \exp [-i(T/2)\{ 2(g_i - \cos k) \sigma^z + 2 (\sin k) \sigma^x \}]. \nonumber \\
&& \label{uk} \eea
Since $U_k(T)$ is an element of the group $SU(2)$, it can be written in the
form of Eq.~\eqref{uk2}, where $\gamma_k$ and the unit vector ${\hat e}_k = (e_{1k},
e_{2k},e_{3k})$ can be derived by using the expression given in Eq.~\eqref{uk}.
Then the two-component spinor obtained after $n$ drives is given by
\bea \left( \begin{array}{c}
u_k(n) \\
v_k(n)\end{array} \right) &=& (U_k(T))^n \left( \begin{array}{c}
0 \\
1 \end{array} \right) \nonumber \\
&=& \left( \begin{array}{c}
-i ~\sin (n \gamma_k) ~(e_{1k} - i e_{2k}) \\
\cos (n \gamma_k) ~+~ i ~\sin (n \gamma_k) ~e_{3k} \end{array} \right). \eea
Hence
\bea \cos \theta_{nk} &=& |u_k(n)|^2 - |v_k(n)|^2 \nonumber \\
&=& - e_{3k}^2 - \cos (2n \gamma_k) ~(1- e_{3k}^2), \label{costank} \eea
where we have used the identity $e_{1k}^2 + e_{2k}^2 + e_{3k}^2 = 1$. We now
assume that $\gamma_k/\pi$ is an irrational number; this is justified since
rational numbers lying in the range $[0,1]$ form a set of measure zero.
Hence the set of numbers $2 n \gamma_k ~{\rm mod}~ \pi$ covers the region
$[0,\pi]$ uniformly as $n \to \infty$. We now use the fact that if
$2 n \gamma_k$ is uniformly distributed from 0 to $\pi$, the probability
distribution of the variable $v=\cos (2n \gamma_k)$ is given by
\beq P(v) ~=~ \frac{1}{\pi ~\sqrt{1 - v^2}} \label{pv} \eeq
for $-1 \le v \le 1$. [This is because $P(v) = (1/\pi) \int_0^\pi ~d\theta ~
\delta (v - \cos \theta) = 1/(\pi \sqrt{1-v^2})$.] Using Eq.~\eqref{pv} in
Eq.~\eqref{costank}, we find that the probability distribution of
$\cos \theta_k$ is given by
\beq \overline{P}(\cos \theta_k) ~=~ \frac{1}{\pi ~\sqrt{(1 + \cos \theta_k)~
(1 - 2 e_{3k}^2 - \cos \theta_k)}} \label{pcostk} \eeq
for $-1 \le \cos \theta_k \le 1 - 2 e_{3k}^2$, while $\overline{P} (\cos \theta_k) =
0$ for $1 - 2 e_{3k}^2 < \cos \theta_k \le 1$. We observe that this distribution
has square root divergences at two points given by $\cos \theta_k = -1$ and
$1 - 2 e_{3k}^2$, and the range of $\cos \theta_k$ is given by $2(1-e_{3k}^2)$.
Further, the distribution is correctly normalized so that $\int_{-1}^1
d (\cos \theta_k) \overline{P} (\cos \theta_k) = 1$.

Given the values of the parameters $g_i, ~g_f, ~T$ and $k$, one can calculate
${\hat e}_k$ and therefore $e_{3k}$ from Eqs.~\eqref{uk} and \eqref{uk2}.
Eq.~\eqref{pcostk} then gives the probability distribution of $\cos \theta_k$
which is generated by a large number of drives, as shown in
Fig.~\ref{fig2main}. We also calculate $P_n(\cos \theta_k)$ for the 1D transverse
field Ising model with
$L=131072$ where the positive momenta in the Brillouin zone are divided into
$\mathcal{N}_{\mathrm{cell}}=32$ cells, each thus containing $N_c=2048$
consecutive momenta; we display the result for a large $n$ ($=5 \times
10^5$) in Fig.~\ref{fig2main} which validates Eq.~\eqref{twoprob}.

\begin{figure}
{\includegraphics[width=\hsize]{probcoarsedT0.pdf}}
\caption{Plot of $P_n(\cos \theta_k)$ calculated at $n=5 \times 10^5$
for a system size of $L=131072$ spins, where the number of coarse-grained
cells equal $\mathcal{N}_{\mathrm{cell}}=32$ and each cell contains
$N_c=2048$ consecutive momenta. Results for $k_c=31\pi/64$ (red, solid)
and $k_c=15\pi/64$ (blue, solid) are shown, where $k_c$ denotes the average
momentum of the corresponding cell. $\overline{P}(\cos \theta_k)$ at
$k=31\pi/64$ (red, dashed) and $k=15\pi/64$ (blue, dashed) are also shown
using the analytical result in Eq.~\eqref{pcostk}. The drive parameters
are $g_i=4, g_f=2, T=0.2$ and $dT=0$.} \label{fig2main} \end{figure}

Eq.~\ref{twoprob} can be analytically understood as follows. For large
$n$, we have argued above that the right hand side of Eq.~\ref{twoprob} is
given by Eq.~\eqref{pcostk} with $k=k_c$. Now we look at the left hand side
of Eq.~\ref{twoprob}; here we have to consider a large but fixed value of
$n$ and average over a range of momenta $\delta k$ around $k=k_c$. We begin with
the expressions
\bea U_k(T) &=& \exp (-i \gamma_k {\hat e}_k \cdot {\vec \sigma}), \nonumber \\
U_{k + \delta k} (T) &=& \exp (-i \gamma_{k + \delta k} {\hat e}_{k + \delta k} \cdot {\vec \sigma}).
\label{ukt} \eea
For $\delta k \ll 1$, $\gamma_k$ will be close to $\gamma_{k + \delta k}$ and
${\hat e}_k$ will be close to ${\hat e}_{k + \delta k}$. However,
\bea (U_k(T))^n &=& \exp (-i n\gamma_k {\hat e}_k \cdot {\vec \sigma}), \nonumber \\
(U_{k + \delta k} (T))^n &=& \exp (-i n\gamma_{k + \delta k} {\hat e}_{k + \delta k} \cdot {\vec \sigma}),
\eea
and it is clear that if $n$ is very large, $n \gamma_k$ and $n \gamma_{k + \delta k}$
will not be close to each other. In other words, a small value of
$\gamma_{k + \delta k} - \gamma_k$ gets amplified as $n$ increases but a small value of
${\hat e}_{k + \delta k} - {\hat e}_k$ does not get amplified. For $\delta k \ll 1$, we can
therefore make the approximation of setting ${\hat e}_k = {\hat e}_{k + \delta k}$, but we
cannot set $\gamma_k = \gamma_{k + \delta k}$ if we are interested in the $n \to \infty$
limit. Hence,
\bea | \psi_k (n) \rangle &=& (U_k(T))^n ~\left( \begin{array}{c}
0 \\
1 \end{array} \right), \nonumber \\
| \psi_{k + \delta k} (n) \rangle &=& (U_{k + \delta k}(T))^n ~
\left( \begin{array}{c}
0 \\
1 \end{array} \right), \label{psikn} \eea
imply, using Eq.~\eqref{costank}, that
\bea \cos \theta_{nk} &=& - e_{3k}^2 - \cos (2n \gamma_k) ~(1- e_{3k}^2), \nonumber \\
cos \theta_{n,k+\delta k} &=& - e_{3k}^2 - \cos (2n \gamma_{k+\delta k}) ~(1- e_{3k}^2).
\eea
Now, we can write $\gamma_{k+\delta k} - \gamma_k = g_k \delta k$, where $g_k$ is a number
of order 1. Suppose that $n$ is large enough that $2n g_k \delta k \gg \pi$.
This implies that as $k$ goes over all the momenta in a cell lying in the
range $k_c - \delta k/2$ to $k_c + \delta k/2$, $2 n \gamma_k$ mod $\pi$ will cover
the range $[0,\pi]$ uniformly. Hence in this cell with an average
momentum $k_c$, the distribution of the variable $v = \cos (2n \gamma_k)$ will
again be given by Eq.~\eqref{pv}. Arguments similar to the ones leading from
Eq.~\eqref{costank} to Eq.~\eqref{pcostk} will then show that the left hand
side of Eq.~\eqref{twoprob} is also given by Eq.~\eqref{pcostk}.

We will now derive the value to which $(|u_k(n)|^2)_c$
converges as $n \rightarrow \infty$ (Fig.~\ref{fig1main}). It is
straightforward to see that $1/2-(|u_k(n)|^2)_c = -(1/2)(\cos \theta_k(n))_c$,
where the coarse-grained quantity $(\cos \theta_k(n))_c$ is given by
\bea (\cos \theta_k(n))_c &\equiv& \int_{-1}^1 d (\cos \theta_k) P_n (\cos \theta_k)
\cos (\theta_k). \eea
As $n \rightarrow \infty$, using Eq.~\ref{twoprob} and Eq.~\ref{pcostk}, we get
\bea (\cos \theta_k(n))_c &=& \int_{-1}^1 d (\cos \theta_k) \overline{P}
(\cos \theta_k) \cos (\theta_k) \nonumber \\
&=& - ~e_{3k}^2. \eea
This is in agreement with the results shown in Fig.~\ref{fig1main} for
$1/2 - (|u_k (n)|^2)_c$ which, in the limit $n \to \infty$, is equal to
$(1/2) e_{3k_c}^2$.

\section{Results for perturbed Floquet system with random noise}
\label{secIV}

We will now study the unitary dynamics when $dT \ll T$ and the $\tau_i$'s in
Eq.~\eqref{defsequence} are taken to be any typical realization of a random
sequence of $\pm 1$'s which are chosen with probability 1/2 each. We again
look at the behavior of the coarse-grained quantities to understand the nature
of the resulting non-equilibrium steady state. We will see that in spite of
the integrability of the model, the resulting non-equilibrium steady state
is locally described by an infinite temperature ensemble as $n \to \infty$,
unlike the case with $dT=0$.

\begin{figure}
{\includegraphics[width=\hsize]{chaotic.pdf}}
\caption{The overlap $|\langle \psi_k(n)| \psi_{k+\delta k}(n) \rangle|^2$
for two different kinds of unitary processes as a function of $n$. $dT=0$
(black lines) denotes a perfectly periodic drive protocol, while $dT=0.05$
(red lines) is a typical realization of a random process where $\tau_i$ are
chosen to be $\pm 1$ with equal probability (Eq.~\eqref{defsequence}). $T=0.2$
in both cases, and the starting wave function equals $(u_k(0),v_k(0))^T=
(0,1)^T$ for all momenta. $k=\pi/2$ is taken as the reference mode, and its
overlap calculated with the next ten consecutive momenta are shown where the
momentum spacing equals $\pi/65536$.} \label{fig3main} \end{figure}

The first quantity we look at is the overlap of a {\it reference} $k$ mode
$|\psi_k(n) \rangle$ with other $k+\delta k$ modes that lie very close to it,
i.e., where $\delta k \ll 1$, as a function of $n$ when the unitary dynamics is
of the form given in Eq.~\eqref{defsequence} with the $\tau_i$'s taken from
a typical realization of a random sequence of $\pm 1$'s.
Fig.~\ref{fig3main} shows a plot
of $|\langle \psi_k (n)|\psi_{k+\delta k} (n) \rangle|^2$ versus $n$. At $n=0$,
all the $|\psi_n(k)\rangle$ start from the same state $(0,1)^T$. We see that
while $|\langle \psi_k (n)|\psi_{k+\delta k} (n) \rangle|^2$ stays localized
between 1 and another value (which is derived in the next paragraph) for the
perfectly periodic drive with $dT=0$ (see Fig.~\ref{fig3main}), the $dT \neq 0$
case is extremely sensitive to initial conditions; we then find that
$|\langle \psi_k (n)|\psi_{k+\delta k} (n) \rangle|^2$ covers the entire range
$[0,1]$ at large $n$ when the overlap with different $k+\delta k$ modes is
considered where all $\delta k \ll 1$ (see Fig.~\ref{fig3main}). Note that if
$\delta k \ll 1$, then each individual $U_{k+\delta k}(T+\tau_n dT)$ is only
slightly different from $U_k(T+\tau_n dT)$ of the reference $k$ mode. This
behavior of the overlaps already suggests that the coarse-grained quantities
behave completely differently in the $dT \neq 0$ case at large $n$.

For the perfectly periodic drive with $dT=0$, we can understand the behavior
of $|\langle \psi_k (n)|\psi_{k+\delta k} (n) \rangle|^2$ versus $n$ as follows.
Following the arguments given in Eqs.~(\ref{ukt}-\ref{psikn}), we find that
\bea && |\langle \psi_k (n)|\psi_{k+\delta k} (n) \rangle|^2 \nonumber \\
&& = ~1 ~+~ \sin^2 [n(\gamma_k - \gamma_{k+ \delta k})] ~(e_{3k}^2 ~-~ 1).
\label{overlap} \eea
For $n \gg 1/|\gamma_k - \gamma_{k+ \delta k}|$, $\sin^2 [n(\gamma_k - \gamma_{k+ \delta k})]$
will oscillate between 0 and 1. Eq.~\eqref{overlap} then implies that
$|\langle \psi_k (n)|\psi_{k+\delta k} (n) \rangle|^2$ will oscillate between
1 and $e_{3k}^2$, and the range of values is given by $1-e_{3k}^2$. We note
that this is exactly half the range of values of $\cos \theta_k$ which is
given by $2(1-e_{3k}^2)$ according to Eq.~\eqref{pcostk}.

We will now study what happens to the probability distribution
$P_n(\cos \theta_k)$ for the unitary process with $dT \neq 0$. To
evaluate this quantity, we use a chain with $L=524288$ spins where each
coarse-grained cell in momentum space contains $N_c=8192$ consecutive momenta
(hence, $\mathcal{N}_{\mathrm{cell}}=32$) and the spinor at each $k$ equals
$(0,1)^T$ at $n=0$. The results for one such cell (with average momentum
$k_c=31\pi/64$) are shown in Fig.~\ref{fig4main}. For small $n$, we see that
$P_n(\cos \theta_k)$ approaches the distribution for the perfectly periodic
drive with the same $T$ but with $dT=0$ which has characteristic
square-root divergences as shown in Eq.~\eqref{pcostk}. However, as $n$
increases further, the probability distribution starts deviating strongly
from this form and approaches a constant distribution eventually
at large $n$ (see Fig.~\ref{fig4main}); this shows that the $N_c$
points that constitute the coarse-grained momentum cell are uniformly spread
over the unit sphere. The lower panel of Fig.~\ref{fig4main} shows the
positions of the $N_c$ points on the unit sphere, from which $P_n(\cos
\theta_k)$ is constructed, for $n=100$, $n=10000$ and $n=200000$.
We will now provide an understanding of this spreading based
on the idea of random walks.

\begin{figure}
{\includegraphics[width=\hsize]{probonBlochsphere.pdf}} \\
{\includegraphics[width=\hsize]{blochsphereinN.pdf}}
\caption{Plots of $P_n(\cos \theta_k)$ for various values of $n$.
The distributions are constructed using $N_c=8192$ points in a
coarse-grained momentum cell with average momentum $k_c=31\pi/64$;
the system size equals $L=524288$ and the number of coarse-grained momentum
cells equal $\mathcal{N}_{\mathrm{cell}}=32$. The drive parameters used are
$g_i=4, ~g_f=2, ~T=0.2$ and $dT =0.05$. For small $n (=100)$ (red, solid
line), $P_n(\cos \theta_k)$ approaches the $dT=0$ analytical result in
Eq.~\eqref{pcostk} (red, dashed line). At larger values of $n (=10000)$
(green, solid), the distribution strongly deviates from the $dT=0$ form, and
it approaches a constant function equal to 1/2 at still larger values of
$n(=200000)$ (blue, solid). The lower panel shows the positions of the
$N_c=8192$ points (in blue) on the unit sphere for different values of $n$.}
\label{fig4main} \end{figure}

We define two unitary matrices $U_k(T-dT)$ and $U_k(T+dT)$ given by
\begin{widetext}
\bea U_k(T \pm dT) = \exp [-(i/2)(T \pm dT)\{ (g_f - \cos k) \sigma^z + (\sin k)
\sigma^x \}] \times \exp [-(i/2)(T \pm dT) \{ (g_i - \cos k) \sigma^z + (\sin k)
\sigma^x \}], \nonumber \\
\label{ab0} \eea
\end{widetext}
These are the time evolution operators for the two possible time periods
$T-dT$ and $T+dT$ respectively. The time evolution operator $U_{nk}$ for
$n$ drives is obtained by multiplying $n$ matrices each of which is randomly
chosen to be either $U_k(T-dT)$ or $U_k(T+dT)$ with probability 1/2 each to
obtain a particular realization of this random process as shown in
Eq.~\eqref{defsequence}.

Next, we parameterize the two matrices as
\bea U_k(T-dT) &=& \exp (-i \alpha_k {\hat a}_k \cdot {\vec \sigma}), \nonumber \\
U_k(T+dT) &=& \exp (-i \beta_k {\hat b}_k \cdot {\vec \sigma}). \label{ab0k} \eea
For $dT=0$, these matrices reduce to $U_k(T)$ given in Eq.~\eqref{uk};
in that case, we have seen that the evolution operator $U_k(T)$
corresponds to a rotation about a unit vector ${\hat e}_k$ given in
Eq.~\eqref{uk2}. If $dT$ is non-zero but small, the unit vector corresponding
to each drive will have a small random component. To quantify this, we note
that if $dT$ is small, the unit vectors ${\hat a}_k$ and ${\hat b}_k$ given in
Eq.~\eqref{ab0k} differ by small angles $\Delta \phi_{1k} = \arccos
({\hat e}_k \cdot {\hat a}_k)$ and $\Delta \phi_{2k} = \arccos ({\hat e}_k \cdot {\hat b}_k)$
from the unit vector ${\hat e}_k$. These two small angles are of the same order
and we can denote their average as $\Delta \phi_k$. For $dT=0$ or $\sin k = 0$,
it is clear from Eqs.~\eqref{uk} and \eqref{ab0} that the unit vectors
${\hat e}_k, ~{\hat a}_k$ and ${\hat b}_k$ are identical, and therefore $\Delta \phi_k = 0$.
Hence, we see that
\bea \Delta \phi_k ~\propto~ dT \sin k \label{scalingphik} \eea
when either $dT$ or $\sin k \to 0$, and Eq.~\eqref{scalingphik} holds
independently of the exact form of the drive protocol.

If the drives have time
periods given by $T+dT$ and $T-dT$ randomly, we expect from the theory of
random walks that after $n$ drives, the unit vector corresponding to $U_{nk}$
will differ from ${\hat e}_k$ by an angle proportional to $\sqrt{n} \Delta \phi_k$,
assuming that $\Delta \phi_k$ is small (this will be true if $dT$ is
small). However, since the unit vectors lie on a compact space (given by the
points on the surface of the unit sphere), the deviation of the unit vector of
$U_{nk}$ from ${\hat e}_k$ cannot go to $\infty$ as $n \to \infty$; rather, the
unit vectors will cover the unit sphere uniformly giving rise to a uniform
probability distribution $\overline{P}(\cos \theta_k) = 1/2$. The behavior of
the probability distribution $P_n(\cos \theta_k)$ shown in Fig.~\ref{fig4main}
for large $n$ is thus consistent with Eq.~\eqref{twoprob}.

Furthermore, from the above argument, the spreading of the overlaps
$|\langle \psi_k(n)|\psi_{k+\delta k} (n) \rangle|^2$
as a function of $n$ (as shown in Fig.~\ref{fig3main})
when {\it many} neighboring momentum modes very close to the reference mode
$k$ are considered is controlled by a time scale $nT$, where
$n (\Delta \phi_k)^2 \simeq n (dT)^2 (\sin k)^2 \sim \mathcal{O}(1)$; hence
the time scale $nT$ varies with the momentum $k$. We explicitly verify
this by showing the behavior of the corresponding coarse-grained variables
$(|u_k(n)|^2)_c$ (defined in Eq.~\eqref{coarsegrain1})
in Fig.~\ref{fig5main}. The data is consistent with an exponential decay to
1/2; however, the rate of decay is clearly sensitive to the values of both
$dT$ and $k_c$ (the average momentum of the coarse-grained cell) as we
see in Fig.~\ref{fig5main} (a). However, when the data is plotted versus
$n(dT)^2(\sin k_c)^2$ (see Fig.~\ref{fig5main} (b)) instead of $n$, we clearly
see that the time scale of the exponential decay to 1/2 is controlled by
$\tau_{k_c,dT} = 1/\left[(dT)^2(\sin k_c)^2\right]$ as predicted by the
random walk argument. We have further checked that the coarse-grained
quantity $(u^*_k(n)v_k(n))_c$ (Eq.~\eqref{coarsegrain1}) relaxes to zero
for large $n$ for any value of $k_c$.

\begin{figure}
{\includegraphics[width=\hsize]{scalecollapsecombineduk2.pdf}}
\caption{(a) Behavior of the coarse-grained quantities $1/2-(|u_k(n)|^2)_c$
as a function of $n$ for different values of $dT$ and $k_c$ (average momentum
of the coarse-grained cell). The same random sequence $\tau_i$ has been used
in all the cases. The other parameters are $L=524288$,
$\mathcal{N}_{\mathrm{cell}}=32$ and $N_c=32$ with $g_i=4,g_f=2$ and $T=0.2$.
(b) Plot of $1/2-(|u_k(n)|^2)_c$ as a function of $n(dT)^2 (\sin k_c)^2$
shows that the decay of $(|u_k(n)|^2)_c$ to 1/2 is consistent with an
exponential decay with a time scale $\tau_{k_c,dT} = 1/\left[(dT)^2
(\sin k_c)^2\right]$. The fit to an exponential
function is displayed in (b) using open circles.
The color scheme used in both panels is:
$dT=0.05, k_c=15\pi/64$ (in black), $dT=0.05, k_c=31\pi/64$ (in red),
$dT=0.04, k_c=5\pi/64$ (in green), $dT=0.04, k_c=15\pi/64$ (in blue),
$dT=0.02, k_c=15\pi/64$ (in magenta), and $dT=0.02, k_c=31\pi/64$ (in orange).}
\label{fig5main} \end{figure}

The behaviors of both $P_n(\cos \theta_k)$ in Fig.~\ref{fig4main} and
$(|u_k(n)|^2)_c$ in Fig.~\ref{fig5main} as functions of $n$ show the
{\it irreversible} approach to an infinite temperature ensemble when local
quantities are probed.
Finally, we consider the implication of the dependence of the time scale
$\tau_{k,dT}=1/\left((dT)^2(\sin k_c)^2\right)$ (that controls the relaxation
of $(|u_k(n)|^2)_c$) on $dT$ and $k_c$ for the rate at which the trace distance
of the reduced density matrix of a subsystem of $l$ consecutive spins from an
infinite temperature ensemble (using the definition $\mathcal{D}_n(l)$ in
Eq.~\eqref{distancemeasure})
approaches zero after a large number of drives $n$. Since the trace distance
receives a contribution from all momenta $k$, it should approach zero as
\beq \mathcal{D}_{n,ITE}(l) ~\sim~ \int_0^\pi dk ~\exp (-n/\tau_{k,dT}),
\label{trace} \eeq
where $\mathcal{C}_{\infty}(l)$ in Eq.~\eqref{distancemeasure} is calculated
using an infinite temperature ensemble.

For large $n$, the integral in \eqref{trace} is dominated by the regions in
$k$ where $\tau_{k,dT}$ is large, namely, $k=0$ and $\pi$ where $\sin k \to
0$. Absorbing $n$ in $k$ (or $\pi -k$) in the exponential, we see that
the trace distance will scale with $n$ as $1/(\sqrt{n} dT)$ when $dT$ is
small. Thus, $\mathcal{D}_{n,ITE}(l) \sim \mathcal{F}_l(n(dT)^2)$, where
$\mathcal{F}_l(x) \sim 1/\sqrt{x}$ for $x \gg 1$. Secondly, $\mathcal{F}_l(x)
\sim \mathcal{O}(1)$ when $x \ll 1$ since in this limit the system must relax
to the p-GGE that emerges for $dT=0$. Thus, the prethermal regime
where the system initially approaches the p-GGE exists for time scales that
depend on $dT$ as $n \sim 1/(dT)^2$ for $dT \ll T$.
The numerical data for $dT \neq 0$ for the behavior of $\mathcal{D}_{n,ITE}
(l)$ as a function of $x =n(dT)^2$ is consistent with both these expectations,
i.e., $\mathcal{F}_l \sim 1/\sqrt{x}$ for $x \gg 1$ and $\mathcal{F}_l \sim
\mathcal{O}(1)$ for $x \ll 1$, as shown in Fig.~\ref{fig6main} for a fixed
$T=0.2$ and different values of $dT$. It is also instructive to look at
$\mathcal{D}_{n,GGE}$ for non-zero values of $dT$ (Fig.~\ref{fig6main},
inset) where $\mathcal{C}_\infty(l)$ is calculated
using the $dT=0$ p-GGE as a function of $n$. For time scales
$n \sim 1/(dT)^2$, $\mathcal{D}_{n,GGE}$ decreases as a function of $n$ which
shows that the system approaches the p-GGE in the prethermal regime. However,
at larger values of $n$, $\mathcal{D}_{n,GGE}$
starts increasing as a function of $n$ which explicitly shows that the
large-$n$ steady state is not described by a p-GGE. The initial relaxation to a
p-GGE in the prethermal regime scales either as $\mathcal{D}_{n,GGE} \sim
n^{-3/2}$ (the case shown in Fig.~\ref{fig6main}, inset) or as
$\mathcal{D}_{n,GGE} \sim n^{-1/2}$ depending on which dynamical phase the
periodic drive protocol ($dT=0$) belongs to~\cite{SenNS2016}.

For completeness, we point out that Ref.~\onlinecite{MarinoS2012} (see also
Ref.~\onlinecite{RooszJI2016}) also considered a noisy 1D transverse field
Ising model though not periodically driven on average, and showed that the
asymptotic steady state is an infinite temperature ensemble. However, they
considered the behavior of quantities averaged over different realizations of
noise which resembles an open quantum system with non-unitary dynamics. Here,
we show that if one restricts to understanding only {\it local} quantities
as functions of $n$ in a perturbed Floquet integrable system, even the unitary
dynamics of a single typical realization of a random sequence leads to an
infinite temperature ensemble.

\begin{figure}
{\includegraphics[width=\hsize]{tracenormplot.pdf}}
\caption{Plot of $\mathcal{D}_{n,ITE}$ as a function of $n(dT)^2$ for
different values of $dT$ ($dT=0.05$ in black, $dT=0.04$ in red, and $dT=0.02$
in green), where the other drive parameters are $g_i=4, g_f=2$
and $T=0.2$. The system size equals $L=524288$ spins and the subsystem
consists of $l=16$ consecutive spins. The same random sequence $\tau_i$ has
been used in all the cases. (Inset) Plot of $\mathcal{D}_{n,GGE}$ as a
function of $n$.} \label{fig6main} \end{figure}

\section{Results for perturbed Floquet system with scale-invariant noise}
\label{secV}

We will now study the unitary dynamics when $dT \neq 0$ but the $\tau_i$'s
in Eq.~\eqref{defsequence} are not a random sequence of $\pm 1$ but are instead
given by the scale-invariant Thue-Morse sequence in Eq.~\eqref{TMS} that is
neither periodic nor random. The average time-period of such a drive protocol
continues to equal $T$. To this end, we first show the coarse-grained quantity
$(|u_k(n)|^2)_c$ in Eq.~\eqref{coarsegrain1} as a function of $n$ where
the $\tau_i$'s are chosen to be $\pm 1$ according to the Thue-Morse,
with $T=0.2$ and $dT=0.05$ (Fig.~\ref{fig7main}). We see
that these quantities behave entirely differently from either the periodic
case ($dT=0$) in Fig.~\ref{fig1main} or the randomly perturbed case
in Fig.~\ref{fig5main}. From Fig.~\ref{fig7main}, it is not even clear whether
a well-defined non-equilibrium steady state exists as $n \to \infty$. We will
show below that in fact a non-equilibrium steady state does exist in the
large-$n$ limit if the local quantities are
observed not as a function of $n$, but instead as $2^n$ (hence, geometrically
instead of stroboscopically). New {\it emergent} conserved quantities, similar
to Eq.~\eqref{conservedIk} when $2^n$ is used instead of $n$, appear in this
unitary dynamics but only when $n$ becomes sufficiently large, which then allow
for the construction of a ``geometric'' GGE for the resulting steady state.

\begin{figure}
{\includegraphics[width=\hsize]{TMcoarse.pdf}}
\caption{Behavior of the coarse-grained quantities $(|u_k(n)|^2)_c$ as a
function of $n$ when the $\tau_i$'s are chosen according to the Thue-Morse
sequence.
The parameters used are $L=131072$, $g_i=4$, $g_f=2$, $T=0.2$ and $dT=0.05$.
Each coarse-grained cell contains $N_c = 2048$ consecutive momenta and $k_c$
denotes the average momentum of such a cell. Results for $k_c=31\pi/64$ (lower
red curve) and $k_c=15\pi/64$ (upper blue curve) are shown.}
\label{fig7main} \end{figure}

We begin by noting that the evolution operators for drives corresponding
to $T-dT$ and $T+dT$ at a particular $k$ are respectively given by the
matrices $U_k(T-dT)$ and $U_k(T+dT)$ given in
Eqs.~\eqref{ab0}; for simplicity of notation, we will not write the
subscript $k$ in this section henceforth. Furthermore, we will denote
$U(T-dT)$ by $A_0$ and $U(T+dT)$ by $B_0$ here. Then the operators
corresponding to the pair of drives $(\tau_i,\tau_{i+1})$ = $-1,+1$ and
$(\tau_i,\tau_{i+1})$=$+1,-1$ are given by $A_1 = B_0 A_0$ and $B_1 =
A_0 B_0$ respectively. We now focus on the case where $n=2^m$ where
$m=0,1,2,\cdots$. It is then easy to see that the unitary dynamics can be
entirely expressed in terms of the new matrices $A_1$ and $B_1$. We
illustrate this in the second line of Eq.~\eqref{coarsegrain_in_time} for
the case of $m=4$ (thus $n=16$).
\begin{widetext}
\bea \boxed{A_0 B_0} \boxed{B_0 A_0} \boxed{B_0 A_0} \boxed{A_0 B_0}
\boxed{B_0 A_0} \boxed{A_0 B_0} \boxed{A_0 B_0} \boxed{B_0 A_0} &|&\psi_k(n=0)
\rangle \nonumber \\
 \boxed{B_1 A_1} \boxed{A_1 B_1} \boxed{A_1 B_1} \boxed{B_1 A_1} &|&
\psi_k(n=0) \rangle \nonumber \\
 \boxed{A_2 B_2} \boxed{B_2 A_2} &|&\psi_k(n=0) \rangle \nonumber \\
 \boxed{B_3 A_3} &|&\psi_k(n=0) \rangle \nonumber \\
 A_4 &|&\psi_k(n=0) \rangle \label{coarsegrain_in_time} \eea
\end{widetext}
This motivates us to define matrices $A_m$ and $B_m$ recursively as
\bea A_{m+1} &=& B_m ~A_m, \nonumber \\
B_{m+1} &=& A_m ~B_m, \label{abn1} \eea
for all $m \ge 0$. We can then show that the evolution operator after exactly
$2^m$ drives is given by $A_m$. (Eq.~\eqref{coarsegrain_in_time} illustrates
this for $m=4$, i.e., $n=16$). Note that this provides a very efficient method
to calculate the dynamics after $n=2^m$ which only requires $\mathcal{O}(m)$
matrix multiplications at each $k$ instead of $\mathcal{O}(2^m)$. {\it
We will show that remarkably $A_m=B_m$ when $m$ becomes sufficiently large
and thus there is an emergent periodicity when the system
is viewed geometrically (i.e., after every $n=2^m$).}

We will now study how the matrices $A_m, ~B_m$ defined in Eqs.~\eqref{abn1}
behave in the limit $m \to \infty$. To this end, we parameterize these
matrices as
\bea A_m &=& \exp (-i \alpha_m {\hat a}_m \cdot {\vec \sigma}), \nonumber \\
B_m &=& \exp (-i \beta_m {\hat b}_m \cdot {\vec \sigma}). \label{abn2} \eea
We also define the angle $\phi_m$ between the unit vectors ${\hat a}_m$ and
${\hat b}_m$, namely,
\beq \phi_m ~=~ \arccos ({\hat a}_m \cdot {\hat b}_m), \label{phn1} \eeq
with $0 \le \phi_m \le \pi$.
Using Eqs.~(\ref{abn1}-\ref{phn1}), we find that
\bea \cos \alpha_{m+1} &=& \cos \beta_{m+1} \nonumber \\
&=& \cos \alpha_m \cos \beta_m ~-~ \sin \alpha_m \sin \beta_m \cos \phi_m, \nonumber \\
\sin \alpha_{m+1} ~{\hat a}_{m+1} &=& \sin \alpha_m \cos \beta_m ~{\hat a}_m ~+~ \sin \beta_m
\cos \alpha_m ~{\hat b}_m \nonumber \\
&& +~ \sin \alpha_m \sin \beta_m ~{\hat a}_m \times {\hat b}_m, \nonumber \\
\sin \beta_{m+1} ~{\hat b}_{m+1} &=& \sin \alpha_m \cos \beta_m ~{\hat a}_m ~+~ \sin \beta_m
\cos \alpha_m ~{\hat b}_m \nonumber \\
&& -~ \sin \alpha_m \sin \beta_m ~{\hat a}_m \times {\hat b}_m. \label{recur} \eea
{}From Eqs.~\eqref{recur}, we can prove the following results. First, we
find that $\alpha_m = \beta_m$ for all $m \ge 1$. We will therefore use only the
variable $\alpha_m$ henceforth. Next, we find the recursion relations
\bea \cos \alpha_{m+1} &=& \cos^2 \alpha_m ~-~ \sin^2 \alpha_m ~\cos \phi_m,
\label{aln} \\
\tan \left(\frac{\phi_{m+1}}{2}\right) &=& |\tan \alpha_m| ~\sin \left(
\frac{\phi_m}{2}\right). \label{phn2} \eea

Since $\alpha_m = \beta_m$ for all $m \ge 1$, Eqs.~\eqref{recur} imply that
\beq {\hat a}_{m+1} ~+~ {\hat b}_{m+1} ~=~ \frac{\sin (2 \alpha_m)}{\sin \alpha_{m+1}} ~
({\hat a}_m ~+~ {\hat b}_m). \label{ambm} \eeq
This means that the unit vector which points in the direction of ${\hat a}_m ~+~
{\hat b}_m$ is the same for $m$ and $m+1$, up to a sign; hence, up to a sign, the
direction of this unit vector does not change at all with $m$, all the way
from $m=1$ to $\infty$. These arguments will break down if either ${\hat a}_m$
or ${\hat b}_m$ is ill-defined; according to Eq.~\eqref{abn2}, this can only happen
if $\alpha_m$ or $\beta_m$ becomes exactly equal to 0 or $\pi$. However these cases
require very special choices of $A_0$ and $B_0$ which form a set of measure
zero; we will therefore ignore such special cases.

To summarize, we find that the dynamics given by Eq.~\eqref{abn1}
conserves two quantities at every iteration: $\alpha_m = \beta_m$, and the
direction of the unit vector ${\hat a}_m + {\hat b}_m$ is conserved for all
$m \ge 1$. The existence of these conserved quantities considerably
simplifies the analysis as we will see below.

For sufficiently large $m$, we find numerically that $\phi_m \to 0$
although $\phi_m$ does not approach zero
monotonically (see Fig.~\ref{fig8main}). We can understand
this as follows. If $\phi_m \to 0$, Eq.~\eqref{aln} implies that $\alpha_{m+1}
\to 2 \alpha_m ~{\rm mod}~ \pi$. Assuming that $\alpha_m/\pi$ is irrational, we see
that $\alpha_m$ will cover the region $[0,\pi]$ uniformly as $m \to \infty$.
Next, assuming $\phi_m, ~\phi_{m+1}$ to be small, we obtain
\beq \phi_{m+1} ~=~ |\tan \alpha_m| ~\phi_m \label{phn3} \eeq
from the lowest order expansion of the terms appearing in Eq.~\eqref{phn2}.
Since $\alpha_m$ covers $[0,\pi]$ uniformly, $|\tan (\alpha_m)|$ varies all the
way from zero to $\infty$; this explains why $\phi_m$ does not
monotonically approach zero. In fact,
\beq \frac{\phi_{m+N}}{\phi_m} ~=~ \exp [\sum_{j=0}^{N-1}
\log (|\tan \alpha_{m+j}|)], \eeq
and for large $N$, a uniform distribution of $\alpha_m$ implies that
$\sum_{j=0}^{N-1} \log (|\tan \alpha_{m+j}|) \to (N/\pi) \int_0^\pi d\alpha \log
(|\tan \alpha|)$ which is equal to zero. Thus we would expect that as
$m \to \infty$, $\phi_m$ should stay at a constant value if we stopped at
the first order expression in Eq.~\eqref{phn3}. We therefore have
to go to the next order in the expansion of Eq.~\eqref{phn2}. We find that
\beq \phi_{m+1} ~=~ |\tan \alpha_m| ~\phi_m ~\left(1 ~-~ \frac{\phi_m^2}{24}
\left( 1~+~2 \tan^2 \alpha_m \right) \right). \label{phn4} \eeq
The fact that the last factor on the right hand side is always less
than 1 explains why $\phi_m$ eventually goes to zero as $m \to \infty$;
this explains the numerical behavior shown in Fig.~\ref{fig8main}.

As $m \to \infty$, the fact that $\phi_m \to 0$ means that
${\hat a}_m - {\hat b}_m \to 0$. The discussion following Eq.~\eqref{ambm} then implies
that ${\hat a}_\infty = {\hat b}_\infty$ is equal, up to a sign, to the unit vector
which points in the direction of ${\hat a}_1 + {\hat b}_1$. Hence the value of
${\hat a}_\infty = {\hat b}_\infty$ can be found right in the beginning when we
calculate ${\hat a}_1$ and ${\hat b}_1$.

\begin{figure}
{\includegraphics[width=\hsize]{fixedpointapproachTM.pdf}}
\caption{Behavior of $\hat{a}_m \cdot \hat{b}_m$ as a function of $m$ for a
unitary process with $k=\pi/2$, $T=0.2$ and $dT=0.05$, where the $\tau_i$'s are
chosen according to the Thue-Morse
sequence. (Inset) Behaviors of the three components
of $\hat{a}_m$ ($a_{1m}$ (red), $a_{2m}$ (green), $a_{3m}$ (blue)) as functions
of $m$ for the same unitary process. The unit vector $\hat{a}_m$ points in the
direction of $\hat{a}_1+\hat{b}_1$ for large $m$ (from Eq.~\ref{ambm}) since
$\hat{a}_m=\hat{b}_m$ for $m \gg 1$.} \label{fig8main} \end{figure}

We now examine what happens if we view the system geometrically after $n=2^m$
drives where $m=0,1,2,\cdots$. In Fig.~\ref{fig8main} (inset), we show the
components of the unit vector ${\hat a}_m$ versus $m$ for $g_i=4, ~g_f=2, ~T=0.2, ~
dT =0.05$, and $k=\pi/2$. (In that figure, we have taken the third component
of ${\hat a}_m$, called $a_{3m}$, to be non-negative for all values of $m$.
According to Eq.~\eqref{abn2}, this can always be ensured,
by flipping $\alpha_m \to - \alpha_m$ and ${\hat a}_m \to - {\hat a}_m$ if necessary.)
We see that for $m$ up to about 1300, there are intermittent large
fluctuations in both ${\hat a}_m$ and ${\hat b}_m$. As Eq.~\eqref{phn3} shows, these
fluctuations occur when $|\tan \alpha_m|$ becomes large, namely, when $\alpha_m$
comes close to $\pi/2$. For very large values of $m$, ${\hat a}_m$ and ${\hat b}_m$
settle down to the same value (Fig.~\ref{fig8main}, main panel). We therefore
conclude that after an extraordinarily large number of drives given by $2^m$,
where $m \gtrsim 1300$ (we have checked this for other values of $k$ also),
${\hat a}_m$ and ${\hat b}_m$ become identical and independent of $m$.

Since the unitary
dynamics for $n=2^m$ can be represented by a single unitary matrix $A_m$ for
the Thue-Morse sequence, this immediately shows that $\mathcal{J}_k$ defined as
\bea \mathcal{J}_k (2^m)&=& \langle \psi_k(2^m)|\hat{a}_\infty \cdot \vec{\sigma}|
\psi_k(2^m) \rangle \nonumber \\
&=& \langle \psi_k(0)|A_m^\dagger (\hat{a}_\infty \cdot \vec{\sigma}) A_m |
\psi_k(0) \rangle \nonumber \\
&=& \langle \psi_k(0)|e^{i\alpha_m \hat{a}_m \cdot \vec{\sigma}} (\hat{a}_\infty
\cdot \vec{\sigma}) e^{-i\alpha_m \hat{a}_m \cdot \vec{\sigma}} |\psi_k(0)
\rangle \nonumber \\
\label{conservedJk} \eea
equals $\mathcal{J}_k \equiv \langle \psi_k(0)|\hat{a}_\infty \cdot \vec{\sigma} |
\psi_k(0) \rangle$
and becomes independent of $m$ for sufficiently large $m$ at each momentum
$k$. We should stress here, that unlike $\mathcal{I}_k$ defined in
Eq.~\eqref{conservedIk} for the $dT=0$ case which is conserved as a function of
$n$, the quantities $\mathcal{J}_k$ are conserved geometrically, i.e., when
$n=2^m$. Furthermore, one requires $m$ to be larger than a threshold value
($m \gtrsim 1300$ for the drive parameters shown in Fig.~\ref{fig8main}) for
these conserved quantities to emerge at all momenta $k$, and these conserved
quantities are not manifest in the unitary dynamics below this
threshold value of $m$. We have also verified numerically that $\hat{a}_m$
and $\hat{b}_m$ become identical and independent of $m$ at large $m$ not
only for $dT \ll T$ but for any value of $dT/T$.

The motion of the $N_c (\gg 1)$ consecutive momenta contained
in any of the coarse-grained momentum cells on the unit sphere as a function
of $n$ provides another way to geometrically see the emergent conserved
quantities of this unitary dynamics (with the $\tau_i$'s chosen according to
the Thue-Morse sequence). As shown in Fig.~\ref{figpanelTM}, when the system
is viewed geometrically, i.e., with $n=2^m$, the $N_c$ points that belong to a
momentum cell with average momentum $k_c$ initially perform a complicated
dynamics on the unit sphere but eventually settle down to a very simple motion
along a circle in the plane defined by $\hat{a}_\infty$ (at $k_c$) for large
enough values of $m$ beyond a particular threshold.

\begin{figure}
{\includegraphics[width=\hsize]{blochsphereTMslides.pdf}}
\caption{The positions of the $N_c$ points (in blue) contained in a
coarse-grained momentum cell on the unit sphere for various values of $n=2^m$,
where the system is viewed geometrically, and the $\tau_i$'s are chosen
according to the Thue-Morse sequence. The parameters used are $L=131072$, $g_i
=4$, $g_f=2$, $T=0.2$ and $dT=0.05$. Each coarse-grained momentum cell contains
$N_c=2048$ consecutive momenta and the results displayed are for a cell with
average momentum of $k_c=31\pi/64$.} \label{figpanelTM} \end{figure}

Finally, we show in Fig.~\ref{fig9main} the components of $\delta n_k={\hat a}_k(T,dT)
-{\hat a}_k(T,0)$ versus $k$ when both ${\hat a}_m$ and ${\hat b}_m$ have settled down to the
same value for a unitary process with the $\tau_i$'s chosen according to the
Thue-Morse sequence for a given $T$ and $dT$. Since ${\hat a}_k(T,0)$ represents
the unit vector that
corresponds to the unitary matrix $U_k(T)$ for the perfectly
periodic case ($dT=0$) given in Eqs.~\eqref{uk2} and \eqref{uk}, we must have
$\delta n_k \to 0$ as $dT \to 0$. In Fig.~\ref{fig9main} (a), we show the
variation of $\delta n_k$ versus $k$ for $dT=0.05, ~T=0.2$, and the corresponding
results for $dT=0.8, ~T=0.2$ (thus, $dT > T$) in Fig.~\ref{fig9main} (b). We
see that indeed ${\hat a}_\infty (={\hat b}_\infty)$ equals (up to a sign) the unit
vector in the direction of ${\hat a}_1+{\hat b}_1$. The nature of
the variation with $k$ is rather different for $dT \ll T$ (Fig.~\ref{fig9main}
(a)) and $dT \gg T$ (Fig.~\ref{fig9main} (b)). Also, we note that $\delta n_k$
equals $0$ for $k=0$ and $k=\pi$ as expected since $|a_3|=1$ and $a_1=a_2=0$
independently of the value of $dT$ at these two momenta.

\begin{figure}
{\includegraphics[width=\hsize]{nvectors_gGGE.pdf}}
\caption{$\delta n_k={\hat a}_k(T,dT)-{\hat a}_k(T,0)$ shown
as a function of $k$. Here ${\hat a}_k(T,dT)$ refers to the final value
${\hat a}_\infty={\hat b}_\infty$ to which ${\hat a}_m$ and ${\hat b}_m$ settle down at large $m$,
where the $\tau_i$'s in the unitary process are chosen according to the
Thue-Morse sequence with given values of $T$ and $dT$. The drive parameters
are (a) $g_i=4, g_f=2, T=0.2$ and $dT=0.05$, and (b) $g_i=4, g_f=2, T=0.2$ and
$dT=0.8$. The components of $\delta n_k$ are indicated as follows: $\delta n_{1k}$
(red), $\delta n_{2k}$ (green) and $\delta n_{3k}$ (blue).}
\label{fig9main} \end{figure}

Knowing the value of ${\hat a}_\infty={\hat b}_\infty$ as a function of $k$
(Fig.~\ref{fig9main}) allows us to construct the local description of the
final non-equilibrium steady state. The construction is completely analogous
to the perfectly
periodic driven case ($dT=0$) where the relevant integrals of motion are now
$\mathcal{J}_k$ (in Eq.~\eqref{conservedJk}) at each momentum instead
of $\mathcal{I}_k$ (in Eq.~\eqref{conservedIk}). The density matrix of the
g-GGE then equals
\bea \rho_{\mathrm{g-GGE}} ~=~ \frac{1}{Z} \exp(-\sum_k \lambda_k \mathcal{J}_k),
\label{gGGE1} \eea
where the Lagrange multipliers $\lambda_k$ are fixed by the condition
\bea \mathrm{Tr}[\rho_{\mathrm{g-GGE}}\mathcal{J}_k] ~=~ \langle \psi_k(n=0)|
\mathcal{J}_k |\psi_k(n=0) \rangle, \label{gGGE2} \eea
and the normalization constant $Z$ ensures that $\mathrm{Tr}[\rho_{\mathrm{
g-GGE}}]=1$. The final steady state value of any local operator can then be
evaluated by using this ensemble.

We show the results for one such local quantity
$\langle \psi (n)|s^x |\psi(n) \rangle$ in Fig.~\ref{fig10main}.
Fig.~\ref{fig10main} (a) shows the convergence of this quantity
to the final steady state value calculated using $\rho_{\mathrm{g-GGE}}$
for two different cases ($dT \ll T$ and $dT \gg T$) as a function of $n=2^m$;
the convergence to the final g-GGE is extremely slow as we have already
discussed in this section. Fig.~\ref{fig10main} (b) shows the
data for the case $dT \ll T$ as a function of $n$ from which
it is clear that even after a large number of drives $n \sim 10^5$, the
system does not seem to relax to the final non-equilibrium steady state in
any simple manner if $dT \neq 0$ (whereas the relaxation of
local quantities at $dT=0$ is much faster as is clear from the inset of
Fig.~\ref{fig10main} (a)), and the relaxation to the non-equilibrium steady
state takes place over a much longer time scale, as we see in
Fig.~\ref{fig10main} (a).
We note here that this extremely slow relaxation of local quantities to
their final steady state appears to be quite insensitive to the value of $dT$
as long as it is non-zero. Fig.~\ref{fig10main} (b) also shows the
perfect agreement of the data calculated at $n=2^m$ using the unitary matrix
$A_m$ at each $k$ to the results of the more direct calculation at each $n$
(using Eq.~\eqref{defsequence}) without using the recursive structure of the
Thue-Morse sequence.

\begin{figure}
{\includegraphics[width=\hsize]{gGGE.pdf}}
\caption{(a) Convergence of $\langle
\psi(n)|s^x|\psi(n) \rangle$ to the final steady state value calculated
using the g-GGE. The steady state values are shown as dashed lines and are for
drive parameters $g_i=4,g_f=2, T=0.2$ and $dT=0.05$ (in black) and $dT=0.8$
(in red), with a system size equal to $L=131072$. Note
that $m$ denotes exponentially growing time steps since $n=2^m$. The inset of
(a) shows the rapid convergence of the same local quantity as a function of
$n$ for a perfectly periodic protocol with $T=0.2$.
(b) The data for $dT=0.01$ (in green) and $dT=0.05$ (in black)
now shown as a function of $n$ and the dashed line represents the
value obtained from the g-GGE (the difference between the steady state values
at $dT=0.01$ and $dT=0.05$ is small and not visible on this scale).
The open circles represent the
calculation at $n=2^m$ using the unitary matrix $A_m$ at each $k$
for $dT=0.05$. The direct calculation using Eq.~\eqref{defsequence}
at each $n$ (black line) agrees perfectly with the results at $n=2^m$
(open circles) using $A_m$.} \label{fig10main} \end{figure}

\section{Conclusions and outlook}
\label{secVI}

In this work, we have considered a prototypical integrable model, the
one-dimensional transverse field Ising model, which is continually driven with
a time-dependent Hamiltonian that results in a unitary dynamics. Since the
model is translation invariant, the modes can be labeled by their momentum $k$.
For a subsystem whose size is much smaller than that of the total system
(shown schematically in Fig.~\ref{fig0main}), the reduced density matrix
of the subsystem is controlled by certain coarse-grained quantities in
momentum space as defined in Eq.~\eqref{coarsegrain1}. Even though the
corresponding quantities for a single momentum continually fluctuate as $n \to
\infty$ and do not have a well-defined large-$n$ limit, these coarse-grained
quantities have a well-defined $n \to \infty$ limit if a non-equilibrium
steady state exists. Furthermore, we define a probability distribution on the
unit sphere at each average momentum $k_c$ by employing a pseudospin
representation at each $k$ and using the large number of momenta lying in a
small cell in momentum space centered around a momentum $k_c$ during the
coarse-graining procedure. This probability distribution at each $k_c$
changes in an {\it irreversible manner} to the final $n \to \infty$
distribution as a function of $n$. At large $n$, this distribution generated
from the points in a cell (centered at $k_c$) at that $n$ has
the property stated in Eq.~\eqref{twoprob} that it equals the probability
distribution generated from the motion of the pseudospin at a single momentum
$k_c$ (thus, with no coarse-graining in momentum) for a large number of drives.

We first recapitulate the results known for the well-studied case in which
the system is driven perfectly periodically with a time period $T$.
Since the model is completely integrable with an
infinite number of conserved quantities $\mathcal{I}_k$ (defined in
Eq.~\eqref{conservedIk}), the system does not locally heat up to infinite
temperature, and instead this driving leads to the periodic generalized Gibbs
ensemble. We also calculate the behavior of the coarse-grained quantities in
momentum space and the probability distributions on the unit sphere as a
function of $k_c$ which immediately shows that the non-equilibrium steady
state is not an infinite temperature ensemble.

We then consider two new driving protocols both of which involve driving with
two possible periods $T+dT$ and $T-dT$, such that the unitary time evolution
operators corresponding to the two periods do not commute with each other;
these driving protocols can be thought of as perturbations of the perfectly
periodic driving if $dT \ll T$. In the first driving protocol, the successive
drive periods are randomly chosen to be $T+dT$ and $T-dT$ with probability
1/2 each. In this case, we find that after a large number of drives the system
eventually approaches an infinite temperature ensemble.
Remarkably, this ensemble also has the
property that the probability distribution of a coarse-grained quantity
averaged over many momenta in a cell is equal to the probability distribution
of the same quantity generated by many drives for a single momentum.
The second driving protocol that we consider is one in which the time
periods $T+dT$ and $T-dT$ are chosen according to the Thue-Morse sequence.
This sequence is neither periodic nor random, but it has the property of
being scale-invariant if the number of drives is doubled. In this case, we
find that the system approaches a non-equilibrium steady state with conserved
quantities $\mathcal{J}_k$ (defined in Eq.~\eqref{conservedJk}) which are
smooth functions of $k$ if we view it after $n = 2^m$
drives rather than after $n$ drives. We call this steady state a geometric
generalized Gibbs ensemble; to the best of our knowledge, this
is the first known example of such an ensemble. We note however that this
ensemble emerges only when $m$ is large (of the order of 1300 for one
particular set of parameters) which implies that the number of drives
$n$ required is astronomically large.

An interesting problem for future studies may be to understand how the local
properties relax in time to the final geometric generalized Gibbs ensemble
that we have discussed here. Such a study would lead to a
deeper understanding of the fixed point of a sequence of $SU(2)$ matrices
which is generated from two arbitrary initial matrices $A_0$ and $B_0$
(which do not commute with each other) by the recursion relations given in
Eq.~\eqref{abn1}. The numerical results that we have presented in
Sec.~\ref{secV} provide only a glimpse of this general problem which may
possibly involve a very rich mathematical structure.

More generally, it will be interesting to understand the conditions which
determine whether a continually driven integrable system locally heats up to
an infinite temperature ensemble or not~\cite{ongoing}. For the latter
case, it will be instructive to find other examples of {\it quasiperiodic}
drive protocols~\cite{Sutherland1986}, apart from the one we have pointed out
here, where well-defined non-equilibrium steady states
may emerge without the restriction of a time-periodic drive.

\section*{Acknowledgments}
A.S. is grateful to Kedar Damle and Pushan Majumdar for useful discussions.
The work of A.S. is partly supported through the Partner Group program
between the Indian Association for the Cultivation of Science (Kolkata)
and the Max Planck Institute for the Physics of Complex Systems (Dresden).
D.S. thanks Department of Science and Technology, India for Project
No. SR/S2/JCB-44/2010 for financial support.

redpajama_set_name: RedPajamaArXiv


text:
Organizational musings
Curved Phones? Innovations from near and far away
Today the long rumored announcement came: LG is launching a mobile phone with a curved screen, shortly after Samsung made the same announcement. The phones are not imitations of each other. The Samsung phone has curved sides, which changes the grip and visibility. The LG phone is curved lengthwise so that it can be held more snugly against the head. I admit some puzzlement at the need for these designs, but OK. Seen as innovations, it is pretty impressive that the screens can be curved. This is done as an OLED, or organic light-emitting diode, which is a new, difficult, and expensive technology.
Are these introductions right after each other just evidence that firms in the same industry will act as rivals and pursue the same technologies? Maybe, but the story has more details. LG and Samsung are Korean companies, are well aware of the screen technology each of them is developing, and most likely they get some information from the same sources. Such a high technology is not developed solely inside one firm, even a giant firm like Samsung; alliances are used to develop it. Indeed, Samsung is one of the key examples of a good user of alliances in the book that I, Tim Rowley, and Andrew Shipilov wrote and will publish soon. When firms are developing new products, a network of alliances gives a firm an advantage, but so does the proximity to other innovators. This is the kind of benefit that has made Silicon Valley so powerful, as firms there learn through alliances and through being near each other.
But now there is also research on how firms can be far away from other firms and still do well. The answer is again related to information flows through networks. Russel Funk has an article coming out in Academy of Management Journal that looks at innovations by nanotech firms. He finds that firms that are near other innovators have an advantage if they have a well-integrated network internally so they can integrate information they get from the outside. This is well known, and we write about it in our book. But he also finds a different form of advantage. Firms that are away from other innovators have an advantage if they have a loose network internally, so that innovators inside the firms can be more independent from each other. So, learning from the outside and developing in the inside calls for the exact opposite internal structure.
This new research is interesting, and gives some hope to firms that are away from other innovators. Such firms often have disadvantages in innovation races, and need every trick they can get. For Samsung, it is less interesting because they already have the advantages of proximity to others and a large network – and internally they are tightly integrated.
Funk, R. (2013). "Making the Most of Where You Are: Geography, Networks, and Innovation in Organizations." Academy of Management Journal forthcoming.
Greve, H. R., Rowley, T., and Shipilov,A. Forthcoming. Network Advantage: How to Unlock Value from Your Alliances and Partnerships. Jossey-Bass.
Sangani, B. 2013. LG curved G Flex smartphone 'to rival Samsung Galaxy Round'. The Telegraph, 28.10.2013.
Posted by Henrich R. Greve at 7:30 AM No comments:
"We are nice too": How Firms Deal with Problems
A famous firm has a business built on the principles of a good natural environment, community development and affordable housing, and engagement and philanthropy in society. It is the sponsor of cultural, scientific, and sports events. Its people (those who would have been called employees in many other firms) are its most valuable assets, and it treasures diversity and nourishes strong values among its people. All of this is on its web site.
I suppose you have probably already guessed that the firm is J.P. Morgan, the bank that just reached a deal to pay the US government $13 billion to settle government civil claims related to J.P. Morgan not being sufficiently truthful in its marketing of mortgages ahead of the 2008 financial crisis. The deal limits the financial repercussions of the government investigation, but the bank was unable to convince the US that it should also stop criminal investigations against its employees (sorry, people) that are apparently ongoing.
So what is going on with the difference between all these good things in the first paragraph and the problems in the second? Well, for many firms they might be unrelated, because it is easy to make neat web sites that trumpet various virtues; especially because the makers of those web sites are likely to be totally unaware of dirtier activities elsewhere in the firm. But actually, critiques and bragging could be related. The case for linking them is from research by Mary-Hunter McDonnell and Brayden King in Administrative Science Quarterly. They did not look at civil and criminal investigations, but rather announced boycotts of firms by social movements. Boycotts are less burdensome than fines and imprisonment, but are still enough of a problem that firms respond. In fact, a common response is to talk about all the good they do for society, and this is a stronger response among firms that have a past record of making such claims of helping society. So if you have heard a lot of talk from a firm about good deeds recently, one reason might be good deeds, the other reason might be the opposite.
The research shows clearly that some firms really like to talk about their good deeds, and those firms will talk more if others expose them to critiques. Will the firms also fix the problems that trigger critiques? Well, that is another question. There is also evidence that firms do this, but it takes solid press coverage for a firm to respond to a social critique. As the $13 billion payment suggests, investigations of criminal acts get a lot more attention. So, talk seems to be pretty reliable; responses to investigations quite solid; but the fixing of problems raised by social movements a little less so.
King, B.G. 2008. A political mediation model of corporate response to social movement activism. Administrative Science Quarterly 53(3) 395-421.
McDonnell, M.-H., B. King. 2013.Keeping up Appearances: Reputational Threat and Impression Management after Social Movement Boycotts. Administrative Science Quarterly 58(3)387-419.
Diapers and Alliance Advantage: Amazon connects to P & G
Procter & Gamble (P & G) makes and sells all sorts of essential goods that we don't necessarily talk about so much: diapers, paper towels, toilet paper, laundry detergent, cleaning liquids. Some of these goods are heavy and bulky, and there are probably some parents who wish they had not needed to carry all the diapers back home from the store. Well, now Amazon is there to help: they can deliver these goods, in fact as subscriptions that arrive regularly. Now this seems a bit odd because these products are cheap to buy and expensive to ship, at least compared to books and other Amazon goods, but Amazon do mean to profit from it.
Part of the ambition comes from how they do it. Now there are little Amazon centers in some P & G warehouses, and these pack and ship goods directly from the warehouse instead of having it sent to the Amazon warehouse. The cost savings are obvious. And, this is not just P & G; Amazon is said to be pursuing similar arrangement with other manufacturers too.
What will these alliances do to the market? The complementarity between Amazon and P & G in this arrangement will make them stronger than other firms trying to sell similar goods online; that is the easy part. It is likely that other firms will be either inspired or threatened enough to try the same strategic move. Interestingly enough, I found in research published in Strategic Management Journal that even strong competitive moves are not necessarily imitated fast. That research was about technologies that gave major advantages, but the results are likely to be similar for new alliance types. There are enough doubts about their value and too little information about how they are done, making other firms reluctant to move.
But to the extent that other firms will try to get the same benefits, the competition is likely to make things harder for the small firm and the late firm. P & G are happy to have Amazon in their warehouse, but how many additional alliance partners do they need? Amazon is happy to have P & G goods for sale, but how many brands of diapers and toilet paper do they need? The low need for replicating this arrangement with others will reduce competition, especially because people are more likely to use the web services that they already use for other goods. This will mean that many firms on both the producer and the web sales side will be left out in a game where the big firms win. And that's probably fine with the big firms P & G and Amazon.
Greve, H.R. 2009. Bigger and safer: The diffusion of competitive advantage. Strategic Management Journal30(1) 1-23.
Ng, Serena. 2013. Soap Opera: Amazon Moves in With P&G. Wall Street Journal, October 14, 2013.
Posted by Henrich R. Greve at 11:04 PM No comments:
Heart Medicine Cheating: Are the Top Firms the Worst Firms?
Today's news contains a note on well-known pharmaceutical company Novartis punishing some of its executives in Japan as a result of a scandal related to its heart medicine Diovan. Both the content of the scandal and the firm raises worrying questions. Novartis is being checked by Japan Health Ministry after two independent studies found that data in research on Diovan had been altered to produce inaccurate results, which in turn were used in Japanese advertising for the drug. Misleading advertising for heart medicine is a serious breach of trust. Novartis is a major pharmaceutical company with more than 100,000 employees that is marketing a broad range of medicines.
But the case is not unique, and Novartis is not the only well-known pharmaceutical firm in the news these days. China is moving against corruption in the medical industry, and firms suspected of bribing doctors to favor their medicines include GlaxoSmithKline, Novartis, AstraZeneca, Sanofi, Eli Lilly, and Bayer. Although the evidence is not clear yet, the accusation is familiar – authorities in the USA monitor pharmaceutical sales practices closely because of the potential for corruption.
So is there something special about large and famous firms that make them particularly drawn towards questionable practices? One would think so based on the familiarity of the names in the previous paragraph. But actually a lot of less-known firms have problems too. For example, Ranbaxy, an Indian maker of generic medicines, has repeatedly had unacceptable production quality in its factories, and was caught manipulating data for an application for drug approval. Ranbaxy also makes Diovan, but is likely to have to wait for approval in the USA given its history of problems.
Recently published research in Administrative Science Quarterly gives an interesting view on the problem of size, fame, and misbehavior. In the research, Scott D. Graffin and his coauthors examined the 2009 expense scandal in the UK Parliament, finding that the better-known members of parliament were just as likely to cheat on their expense reporting as the less-known members. So not more likely, nor less likely. But fame did have one effect: the best-known members were much more likely to be reported in the press, so for a newspaper reader they would appear to be much more likely to cheat.
Is this the case for firms too? It seems likely that the press would investigate and report the best-known firms first, and possibly journalists would even choose the best-known firms to name in a report if they had names of many cheaters available. In fact, the corruption investigation in China also involves a number of Chinese firms, but these are less known abroad and are usually not mentioned in articles. But does that mean we can be sure that the largest, most famous companies are just as good (just as bad?) as the others? Not really; unfortunately it is hard to be sure about something that is revealed as rarely as cheating and misbehavior.
Ranbaxy's Chronic Maladies. The Economist, September 21 2013.
Graffin, Scott D., Jonathan Bundy, Joseph F. Porac, James B. Wade, and Dennis P. Quinn. 2013. "Falls from Grace and the Hazards of High Status: The 2009 British MP Expense Scandal and Its Impact on Parliamentary Elites." Administrative Science Quarterly 58(3):313-45.
Inagaki, Kana. 2013. Novartis Sanctions Executives at Japan Unit amid Research Scandal. Wall Street Journal, October 4, 2013.
Henrich R. Greve
I am a Professor of Entrepreneurship at INSEAD. I have spent my career in the USA, Norway, Japan, and Singapore. My research is on how organizations change and how they influence society and are influenced by it.
Diapers and Alliance Advantage: Amazon connects to...
Heart Medicine Cheating: Are the Top Firms the Wor...

redpajama_set_name: RedPajamaCommonCrawl


At the start of the Summer Holidays all those weeks ago the girls and I brainstormed a list of things they would like to do during the holiday.
It wasn't just 'expensive days out' we added things like painting, and sticking, going to the library, going on a bike ride etc etc.
Today was the turn of 'Sticking'.
We're always saving bits and bobs to add to our sticking box, sweet wrappers, feathers, jewels, confetti along with having a pot of dry rice, lentils, pasta shapes and spaghetti.
Children really don't need much to be entertained.


Dr. Kowalski will present recent advances on contention resolution and packet queuing on a multiple access channel. His will include description of algorithms, both already known and newly developed, accompanied by theoretical analysis of queue sizes and latency. He will conclude with showing and elaborating on selected simulation results that support theoretical formulas.
Institute IMDEA Networks has announced that Dr. Dariusz Kowaslki has joined the team in September 2011 in the role of Visiting Researcher, bringing with him extensive experience in algorithms and data structures, fault-tolerant aspects of computer science, and distributed, parallel, network and mobile computing. His research expertise earned the recognition of his alma mater in 2001 and 2005, when he received the Award of the Rector of the Warsaw University, in Poland, for research excellence. At the Institute Dr. Kowalski will be focusing on areas such as reliability of wireless broadcast protocols with block acknowledgment, design and analysis of better epidemic communication protocols, energy aspects of dynamic distributed work scheduling protocols, queuing aspects of distributed multi-access communication processes, models and algorithms for micro- and nano-scale wireless communication and computing. As well as pursuing his own interests, Dr. Kowalski will be involved in the mentoring of young researchers.
Dr. Kowalski is currently a Reader in Computer Science at the University of Liverpool, United Kingdom, association which will mean new collaboration opportunities for Institute IMDEA Networks. He received his MSc. in Mathematics in 1996 and his Ph.D. in Computer Science in 2001, both from the University of Warsaw. Dr. Kowalski maintains an active profile within the international scientific community and has been a postdoctoral researcher at the University of Quebec, Canada, at the University of Connecticut, USA, and in the Max-Planck Institute fuer Informatik, Germany.
In the last few years Dr. Kowalski has developed and analyzed several algorithms for topics including communication in wireless and ad hoc networks (including multi-broadcast, leader election and routing algorithms); contention-resolution in a multiple-access channel; traffic-scheduling; classical distributed computing problems, such as consensus, work scheduling and shared-memory store&collect; dynamic distributed computing problems, including group communication, rumor and work scheduling; and other related aspects of distributed and network computing, including quantum model of computation, fault-tolerance, and combinatorial group testing.
Dr. Kowalski has authored nearly a hundred peer-reviewed publications, including thirty one journal articles. He is a reviewer for technical journals, for example IEEE Transactions on Parallel and Distributed Systems or the Journal of the ACM, as well as for conferences including some of the most referred international events in the field, such as the IEEE International Conference on Computer Communications (INFOCOM) and the IEEE Global Communications Conference (GLOBECOM). His service to the research profession takes the shape, amongst other activities, of participation on over thirty technical program & organizational committees for scientific events. Dr. Kowalski is a recipient of the prestigious NSF-NATO's Award.