text:
\section{Introduction}
\label{sec:Introduction}

Of late, two different concepts in quantum many-body theory have elicited a surge of active research, partly stemming
from experimental advances in condensed matter and cold atomic systems – the concepts of quench dynamics ~\cite{Dziarmaga10,Polkovnikov11,Dutta10a,Kibble76,Kibble80,Zurek85,Zurek96,Dziarmaga05,Damski05,Damski06,Polkovnikov05,Polkovnikov08,
Calabrese05,Calabrese06,Cherng06,Mukherjee07,Divakaran08,Deng08,Divakaran09,Mukherjee10,Sengupta08,Mondal08,Sen08,Mondal09,DeGrandi08,Barankov08,
DeGrandi10,Patane08,Patane09,Bermudez09,Bermudez10,Perk09,Sen10,DeGottardi11,Pollmann10,Dutta10b,Hikichi10,Chandran12,Chandran13,Patel13,Mostame14,
Foster14,Dorner05,Rigol14,Canovi14,Mitra12,Iucci09,Flesch08,Khatami12,Gramsch12,He13}
 and topological order ~\cite{Prange87,Hasan10,Qi11,Bernevig2013,Alicea12,leijnse12}.
 Quenching, or ramping, concerns initializing a system in its equilibrium configuration at some point in parameter space followed by inducing non-equilibrium behavior via dynamic tuning of one of the parameters. When
the tuning occurs through a critical point separating two phases of matter, no matter how slow the tuning rate 1/$\tau$, the diverging time scale associated with the critical point and critical exponent $z$ always results in out-of-equilibrium dynamics in its vicinity. The quantum version of such Kibble-Zurek physics, initially studied as a thermal quench during the formation of the early universe ~\cite{Kibble76,Kibble80,Zurek85,Zurek96}, offers a mine of valuable information about the critical point in question.

 In the realm of topological systems, while quantum Hall systems have been hailed for their topological properties for over three decades ~\cite{Prange87}, the recent attention on other systems has also been spectacular ~\cite{Hasan10,Qi11,Bernevig2013,Alicea12,leijnse12}. On the theoretical front, among others, two paradigm low-dimensional models have been avidly studied for their topological properties – the Majorana wire proposed by Kitaev ~\cite{Kitaev01,Franz13}, which is effectively a lattice version of a spinless $p$-wave superconducting wire, and the two-dimensional Kitaev honeycomb model ~\cite{Kitaev06}.

Here we explore the synergy of these two concepts, namely quench dynamics ~\footnote{While the terms quench and ramp
are sometimes used to distinguish between instantaneous change of parameter versus a time-dependent change at some
given rate, respectively, here we use quench in a more general sense to encompass all such dynamic tuning. Our actual studies
 are restricted to the ramp case.} and topological order, a rich study that is still in its infancy. Such a marriage is exciting from at least two perspectives – can quench dynamics act as a probe for topological order? Can the existence of topological order lead to a different realm in non-equilibrium dynamics? Studies of such synergy ~\cite{Bermudez09,Rajak14,Bermudez10,Chandran12,Chandran13} have just begun to explore diverse and exciting phenomena with regards to dynamic evolution of topological features. In previous work by two of the authors of
this article and co-workers, Ref.~\cite{Kells14a}, the term `topological blocking' was coined with regards to the role played by a highlighting feature
of topological order - ground state degeneracies - in quench dynamics. It is known that a system can have several topological sectors which are associated with these degeneracies and are distinguished from each other by an invariant based on a discrete global symmetry~\cite{Kells08,Kells09}. In quenching between a topological and non-topological phase, if the ground states in the two phases belong to different topological sectors and the Hamiltonian commutes with the global symmetry at all times, the system
 never reaches the ground state of the final phase. As a result, the usual expectation that if the quench is sufficiently slow
 (i.e., almost adiabatic), the system always remains in the instantaneous ground state is violated. This topological blocking effect was demonstrated in Ref.~\cite{Kells14a} for the Majorana wire and the Kitaev honeycomb model constrained to
periodic boundary conditions.

The goal here is to explore quench dynamics in a finite-sized Majorana wire having open boundary conditions, a system that has come into the limelight for topological features that we expect to affect dynamics in a profound way. These features concern the presence of isolated zero energy Majorana fermion bound states at the wire ends within the topological phase; their possible experimental detection in the context of spin-orbit coupled wires ~\cite{Oreg10,Lutchyn10,Mourik12,Deng12,Das12,Finck13} has garnered much attention in terms of fundamental physics as well as
 implications for topological quantum computation~\cite{Nayak08}. In the thermodynamic limit, these Majorana end modes together form a Dirac fermion state that can either be occupied or empty. Thus, degenerate topological sectors are identified by fermion parity. In the finite-sized system, tunnel coupling between these end modes splits the degeneracy in a manner that can be tuned by changing
the parameters of the system. Here we explore the quench dynamics of tuning through a succession of parity flips of the ground state. The investigation involving open boundaries requires the formulation of new dynamic quantum many-body techniques, which we develop here. By investigating measures commonly studied in quench dynamics, we demonstrate that topological order drastically affects non-equilibrium behavior, the most dramatic signature stemming from quench-dependent switching of topological sectors.

Our presentation is as follows. In Sec.~\ref{sec:blocking}, we present a brief description of the salient features of topological blocking and the highlights of this work, including our results regarding parity switching and blocking in
a finite-sized Majorana wire. In Sec.~\ref{sec:pwave}, we begin our detailed exposition by reviewing the $p$-wave superconducting wire given by Kitaev's lattice Hamiltonian. We outline a derivation of its bulk spectrum and description of its topological phase diagram based on the presence or absence of Majorana end modes.
 In Sec.~\ref{sec:finitesize}, based on a transfer matrix formalism, we analyze the fate of the Majorana end modes for a finite-sized lattice. We obtain an approximate form for the degeneracy splitting and exact solutions for contours in the phase diagram where the splitting vanishes. We then invoke Kitaev's argument based on Pfaffian methods to determine the ground state parity of the system and confirm that parity switches occur at these degeneracy points. In Sec.~\ref{sec:quenching}, we begin our discussion of the quench dynamics associated with varying a parameter of the underlying Hamiltonian linearly in time.
We summarize the known results in the case of periodic boundary conditions, in particular, the Kibble-Zurek scaling of the post-quench excitations and the topological blocking phenomenon. In Sec.~\ref{sec:realspace}, we describe the real space time-dependent formalism that we use to study this problem numerically. We focus on two measures, namely the overlap between the time evolved state and the instantaneous ground state, which we refer to as adiabatic fidelity as in previous work~\cite{Polkovnikov11}, and the residual energy, which is the difference between the expectation value of the Hamiltonian in the time evolved state and the instantaneous ground state energy. In Sec.~\ref{sec:results}, we extensively discuss our numerical results in detail, pinpointing the effect of topological blocking associated with multiple parity switches. In Sec.~\ref{sec:discussion}, we present an overview of our study and connect it with related phenomena, such as the fractional Josephson effect in junctions of Majorana wires, as well as to experiments.

\section{Topological/Parity blocking in quenching dynamics}
\label{sec:blocking}

\begin{figure}[]
\centering
\includegraphics[width=0.5\textwidth]{cartoon.pdf}
\caption{(Color online) A finite size superconducting wire in the topological phase characterized
by Majorana fermionic end modes. While their wave functions may or may not oscillate
(red curves), they all decay into the bulk (envelope) over a characteristic length scale
that depends on system parameters. The overlap between the decaying oscillatory
wave functions of these two end modes gives rise to a tunnel splitting of
the otherwise doubly degenerate zero energy states.}\label{fig:cartoon}
\end{figure}

Here, we describe the essence of the topological features that we target with
regards to quench dynamics and present the highlights of this work before
embarking on a detailed exposition.

The stage is set by the concept of topological blocking, which, as mentioned in
the previous section, was studied in Ref.~\cite{Kells14a}. The study
involved quench dynamics in topological systems elicited by changing a parameter
of the Hamiltonian to tune from one quantum phase to another. In going
between a topological phase and a trivial phase, as in the Majorana
wire, or between two topological phases, as in the Kitaev honeycomb model, the focus was
on mismatch of degeneracies. It was shown that if the system was initialized in the ground state
in a phase with higher degeneracy and tuned to one with a lower degeneracy,
two to one in the former
case, and four to three in the latter, the phenomenon of topological blocking would occur.
Certain topological sectors characterized by topological quantum numbers, for instance,
fermion parity, would inhabit the ground state in the initial phase but would have no partner
in the ground state of the final phase. As a result, in tuning through a
quantum critical point separating the two phases, these sectors would evolve so as to have null overlap
with the final ground state no matter how slow the tuning rate, in stark contrast with Kibble-Zurek physics, where only a rate-dependent fraction of the time-evolved state overlaps with the excitation spectrum above the (gapped) final ground state. Moreover, it was shown that even if one took overlap
with the instantaneous sectoral ground state, different topological sectors would show quantitatively
different dynamic behavior, particularly in wave function overlap.

Here, we explore this notion of topological blocking with regards to a different but related aspect - the switching of topological sectors due to quench dynamics within a topological phase. In a Majorana wire with open boundaries, the topological degeneracy is associated with the presence of Majorana zero modes at the edges. The degeneracy is split due finite-size coupling of these edge modes, which induces fermionic parity sectors within the topological region of the phase diagram. Here we build on the notion of topological/parity blocking arising from tuning through these parity sectors.

\begin{figure*}[htp]
\centering
\subfloat[][Adiabatic fidelity for $N=34$.]{\includegraphics[width=0.5\textwidth]{OverlapOpen34.pdf}}
\subfloat[][Parity for $N=34$.]{\includegraphics[width=0.5\textwidth]{ParityOpen34.pdf}}

\caption{(Color online) Numerical results for the (a) adiabatic fidelity $\mathcal{O}(t)$ and (b) parity of
the instantaneous ground state for an even number of sites. The times at which
the parity switches its sign are exactly the points where parity blocking
occurs, resulting in the adiabatic fidelity plummeting down to zero.
Depending on the parameters chosen, the parity after crossing the quantum
critical point changes from the initial ground state parity thereby leading
to parity blocking for the entire topologically trivial region.}
\label{fig:OverlapOpen34}
\end{figure*}

 In Fig.~\ref{fig:OverlapOpen34}, we present some of our key results. Initializing the system in the ground state corresponding to a specific on-site chemical potential, and thus some fixed parity, we sweep the chemical potential to undergo several parity switches. As a measure of how closely the time evolved state tracks the instantaneous ground state, we evaluate the wave function overlap (adiabatic fidelity) associated with these two states. As seen in  Fig.~\ref{fig:OverlapOpen34}, the adiabatic fidelity plummets down to zero in certain chemical potential intervals that exactly correspond to the parity switched regions. The initial ground state, while being able to track some of the dynamic evolution, is thus forced to remain within its parity sector, an attribute of the topological phase.
This multiple parity blocked dynamics is a dramatic, topologically induced deviation from the continuous evolution expected in quench dynamics.

  In what follows, we detail several aspects leading up to this quench behavior, including the formulation of the Majorana wire model, parity switching due to coupling of Majorana wavefunctions and a real space formalism to compute the many-body dynamics.

\section{$p$-wave superconducting wire}
\label{sec:pwave}
The system that forms our subject of study is a lattice version of the spinless fermionic
 one-dimensional $p$-wave superconducting wire with spinless fermions, also referred
to as the Kitaev chain~\cite{Kitaev01}.
This system can be mapped exactly to a spin-1/2 $XY$ model in a transverse
field (i.e., a magnetic field applied along the $z$ direction) via
 the Jordan-Wigner transformation ~\cite{Lieb61}. The
parameters of the system are the nearest-neighbor hopping amplitude $w$,
the superconducting pairing amplitude between nearest neighbors $\Delta$, and
a chemical potential $\mu$. This model is a paradigm system for demonstrating
numerous interesting topological properties, including the existence of
Majorana modes at the ends of an open chain in the topological phase.

The Hamiltonian of such a system with $N$ sites and open boundary conditions
is given by
\begin{eqnarray} H &=& - \sum\limits_{n=1}^{N-1} (-w f^{\dagger}_{n+1}f_n +
\Delta f^{\dagger}_{n+1}f^{\dagger}_n + H.c.) -\mu \sum_{n=1}^{N}(f^{\dagger}_n f_n-1/2), \label{ham1} \end{eqnarray}
where $f_m$ are Dirac fermion operators obeying the commutation
relations $\{f_m,f_n\}=0$ and $\{f_m,f_n^{\dagger}\}=\delta_{mn}$.
We now introduce $2N$ Majorana fermion operators as $a_{2n-1}=
f_n+f_n^{\dagger}$ and $a_{2n}=i(f_n^{\dagger}-f_n)$. These satisfy the
relations $a_m^{\dagger}=a_m$ and $\{a_l,a_m\}= 2 \delta_{lm}$.
In terms of these operators, the Hamiltonian takes the form

\begin{eqnarray} \nonumber
H&=&-\frac{i}{2} \sum\limits_{n=1}^{N-1}\bigg[(w-\Delta) a_{2n-1} a_{2n+2}
- (w+\Delta) a_{2n} a_{2n+1} \bigg] -\frac{i\mu }{2} \sum\limits_{n=1}^{N} a_{2n-1} a_{2n}. \label{ham2} \end{eqnarray}

\begin{figure}[hp]
\centering
\includegraphics[width=0.5\textwidth]{phasediagram.pdf}
\caption{The phase diagram of the one-dimensional Kitaev Hamiltonian for the Majorana wire. Phases I and
II are topologically non-trivial and have Majorana end modes, whereas phases
III and IV are topologically trivial. The thick lines $\mu=\pm 2w$ and
$\Delta=0$ are the quantum critical lines where the bulk gap vanishes.}
 \label{fig:phasediagram}
\end{figure}

The Hamiltonian in Eq. (\ref{ham1}) can be diagonalized, up to a constant, to
give the canonical form,
\begin{equation} H=4\sum\limits_{j=1}^N \lambda_j b^{\dagger}_j b_j, \end{equation}
where the $\lambda_j$ are non-negative real numbers. This can be done
through a transformation of the form
\begin{equation} \bar{b}=B\bar{a}, \label{bBa} \end{equation}
where $\bar{a}=(a_1,a_2, \cdots, a_{2N})^T$ and $\bar{b}=(b_1, \cdots,
b_N,b^{\dagger}_1, \cdots, b^{\dagger}_N)^T$ are column vectors with $2N$
components.
The $(2N)$-dimensional matrix $B$ comprises of the eigenvectors of
$H$ and their Hermitian conjugates, belongs to the unitary group
$U(2N)$, and has its determinant satisfying the property $det(B)= \pm 1$.
 The energy eigenvalues of the Hamiltonian are $0$ and $4 \lambda_j$.

The phase diagram of this model is shown in Fig.~\ref{fig:phasediagram}. The
phases I and II are topologically non-trivial and, in the thermodynamic limit,
 have zero energy Majorana
modes bound to the ends of the wire, whereas such modes are absent in the
topologically trivial phases III and IV. These Majorana modes have finite
support at the ends and decay rapidly into the bulk with a decay length
proportional to the reciprocal of the bulk gap. One can understand the
existence of the Majorana end modes by considering the extreme limit of
$w=\Delta$ and $\mu=0$. The Hamiltonian reduces to $H=iw\sum_n a_{2n}
a_{2n+1}$. The Majorana operators $a_1$ and $a_{2N}$ are not paired with
any other operators in the system and therefore do not appear in the
Hamiltonian. These isolated modes correspond to the zero energy eigenvectors
localized at the ends. The existence of these modes is robust even away
from this extreme limit and they only disappear with the closing of the
bulk gap.

The ground state of the system in the topological phase is thus doubly degenerate
and has two zero energy eigenvalues corresponding to the Majorana modes.
These Majorana modes can be combined to form a complex Dirac fermion state,
which can be either empty or occupied. Hence, each of the degenerate ground
states has a specific fermion parity and the system can be characterized by
a related $Z_2$-valued topological invariant. This ground state parity will
play an important role in the subsequent sections.

There are three phase boundaries, indicated by dark lines in
Fig.~\ref{fig:phasediagram}, where
the bulk gap vanishes.
These are the quantum critical lines across which there is a topological
phase transition. In the thermodynamic limit (infinite wire) or for a
closed chain,
one can transform the Hamiltonian into Fourier space, and the single particle
energy spectrum takes the form
\begin{equation} E_k = \pm \sqrt{(2w \cos k+\mu)^2+ 4 \Delta^2 \sin^2k}. \end{equation}
This spectrum has a finite superconducting gap in all the phases; the gap
vanishes as one crosses one of the critical lines and reopens upon entering
another phase. In the spin language, the topological phases correspond to
the ferromagnetic phases of the transverse field $XY$ model (where either
the $x$ or the $y$ component of the spins has long range order), and the
trivial phases are in the paramagnetic phase.

 These characteristic features of the system, namely, the topological
invariant, the spectrum of the bulk and end modes, and the wave functions of
the Majorana end modes have been discussed extensively in previous work
(see, for example, Ref.~\cite{Kitaev01,DeGottardi11,Alicea12}). So far,
all the above
mentioned characteristics of the model assume the size of the system to be
much larger than the decay length of the Majorana end modes. In the next
section we consider the
case when the Majorana modes at the two ends have a finite overlap,
giving rise to consequences such as parity blocking in quench dynamics.

\section{Finite size effects in the Majorana wire}
\label{sec:finitesize}

\subsection{Tunneling between Majorana end modes}

In a Majorana wire of finite length, the two Majorana end modes are no longer
completely decoupled since there is some overlap between their wave functions
(Fig.~\ref{fig:cartoon}); the overlap shifts their energies slightly away
from zero. In the extreme
limit of the topological phase with $\Delta = w$, the end modes are exactly
localized at the ends with no overlap between them;
hence the effective Hamiltonian governing these modes is
\begin{equation} H_f =iJ a_1 a_{2N}, \end{equation}
with $J=0$, But in general, the effective Hamiltonian has an
expression in terms of the (almost) zero energy eigenvectors
localized at the ends of a finite length wire~\cite{Kitaev01}
\begin{equation} H_f=iJ b'b'', \end{equation}
where $b'=\sum\limits_j(\alpha'_+ x^j_+ +\alpha'_- x^j_-)a_{2j-1}$,
$b''=\sum\limits_j(\alpha''_+ x^{-j}_+ +\alpha''_- x^{-j}_-)a_{2j}$. Here $J$
is in general a function of $\Delta/w$, $\mu/w$ and $N$. Due to this coupling
the two zero modes split in energy into a particle-hole symmetric pair of
eigenenergies $E =\pm J$, and the ground state is no longer
degenerate. The Majorana end modes can be combined into
non-local Dirac fermions as $\tilde{c}=(b'+ib'')/2$ and
$\tilde{c}^{\dagger}=(b'-ib'')/2$. The Hamiltonian can now be expressed as
\begin{equation} H_f =J(2\tilde{c}^{\dagger}\tilde{c}-1). \end{equation}
The occupation number $\tilde{c}^{\dagger}\tilde{c}$ can either be zero or 1.
Thus we see that the energies $\pm E$ come with corresponding eigenstates
with opposite fermion parities. The sign of $J$ decides which of these
states is the ground state. The parity of the states in the bulk being fixed,
the overall ground state parity is then decided by the lower one of the two
split energy levels (which lie inside the bulk gap).
The coupling $J$ is a function of the chemical
potential $\mu$ and oscillates, switching its sign at specific values of $\mu$.
Therefore the split energy levels cross zero at certain points in time
if $\mu$ is varied linearly in time.
This leads to oscillations in the overall parity of the ground state.
Even though these split energy levels are exponentially smaller (for large
system size $N$) than the energies in the bulk, the parity oscillations play
a key role in the time evolution of the ground state in the quenching dynamics.

\begin{figure*}[htp]
\centering
\subfloat[]{\includegraphics[width=0.4\textwidth]{Figure4aSplittinganalytic.pdf}\label{fig:Splittinganalytic}} \hspace{1cm}
\subfloat[]{\includegraphics[width=0.4\textwidth]{Figure4b.pdf}\label{fig:SplittingOpen35}}\\
\subfloat[]{\includegraphics[width=0.4\textwidth]{ParityOpen34Fig4c.pdf}\label{fig:ParityOpen35}}
\caption{(Color online) Oscillations of the split mid-gap energy levels for a linear time variation
of $\mu$. Figure (a) shows the overlap of analytically calculated Majorana wave functions as an indicator of the splitting, and (b) shows the variation of the mid-gap states obtained by numerical diagonalization of the Hamiltonian. Figure (c) shows
that the oscillations of the energy levels in (b) correspond exactly to the
parity oscillations of the ground state obtained numerically from Eq.\ref{detB} in the text.}
\end{figure*}

\begin{figure*}[htp]
\centering
\subfloat[][Energy splitting not crossing zero]{\includegraphics[width=0.4\textwidth]{Figure5aSplittingNoOsc.pdf}\label{fig:SplittingNoOsc}} \hspace{1cm}
\subfloat[][No parity switching]{\includegraphics[width=0.4\textwidth]{Figure5bParityNoOsc.pdf}\label{fig:ParityNoOsc}}
\caption{(Color online) Absence of oscillations in the mid-gap levels in the regime which lies outside the circle $\mu=2
\sqrt{w^2-\Delta^2}$. In this regime the parity of the ground state will not show oscillations.}
\end{figure*}

The energy splitting due to the coupling of the two Majorana end modes can be
derived by evaluating the overlap between their associated wave functions.
For a very long wire, the Majorana modes have zero energy; we calculate their
exact wave functions by using the Heisenberg
equations of motion $[H,a_n]=0$ and $[H,b_n]$, where $a_n$, $b_n$ are the
two Majorana operators at site $n$
which were denoted by $a_{2n-1}$ and $a_{2n}$ above. We obtain the difference
equations for these operators as:
\begin{eqnarray} (w+\Delta) b_{n-1} + \mu b_n + (w - \Delta)b_{n+1} &=& 0,
\nonumber \\
(w-\Delta) a_{n-1} + \mu a_n + (w + \Delta)a_{n+1} &=& 0, \label{abn1} \end{eqnarray}
for $2 \le n \le N-1$.

These difference equations can be solved exactly by using $z$-transform methods
and taking into account the form of the difference equations at the ends.
The wave functions $\alpha_n$ and $\beta_n$ for the $a_n$ and $b_n$ modes on
site $n$, respectively, are of the form
\begin{eqnarray} \alpha_n &=& \alpha_0 C^{n} \bigg[\cos(\omega n)+
\frac{1}{\tan\omega}\sin(\omega n) \bigg], \nonumber \\
\beta_n &=& \beta_0 C^{-n}[\cos(\omega n)+ \frac{1}{\tan\omega}\sin(\omega n)],
\label{alphabetan} \end{eqnarray}
where $C=\left( \frac{w+\Delta}{w-\Delta} \right)^{1/2}$, and
\begin{equation} \omega= \tan^{-1} \left( \frac{\sqrt{4w^2-4\Delta^2-\mu^2}}{
\mu} \right). \label{om} \end{equation}
Using these wave functions, we can now approximately calculate the energy
splitting by assuming that the wire has a finite length $N$
and computing the overlap
of the wave functions $\alpha_L$ at the left end and $\alpha_R$ at the
right end. This gives an expression of the form
\begin{eqnarray} \nonumber
\alpha_L \alpha_R &=&\frac{\alpha_{0L} \alpha_{0R} a^{N+1}}{4\sin^3\omega}
\bigg[2\sin(\omega N)\\ &&+ N\{\sin[(N+2)\omega] - \sin[(N+4)\omega]\}\bigg].
\label{aLR} \end{eqnarray}

Figure~\ref{fig:Splittinganalytic} shows the oscillations in the splitting as calculated above.
This is an approximate calculation because we have assumed the energies to
be zero in the Heisenberg equations of motion for the Majorana operators and
then calculated their overlap (which shifts the energies slightly away from
zero). But one can see that it qualitatively agrees
with the exact numerical calculation in Fig.~\ref{fig:SplittingOpen35}. Figure~\ref{fig:ParityOpen35} shows
the oscillations of the ground state parity as a function of $\mu$ which is
linearly varied with time. One sees an excellent correspondence with the
zero crossings of the energy splitting in Fig.~\ref{fig:SplittingOpen35}. Further, although
these oscillations appear due to the degeneracy splitting in the
topological phase, they do not exist in the entire topological region in the
phase diagram. To understand this, we stress the fact that the key ingredient
in getting these oscillations is the oscillatory component in the wave
functions of the Majorana modes given in Eqs. (\ref{alphabetan}). One can see
that the oscillatory functions $\sin(\omega n)$ and $\cos(\omega n)$ become
hyperbolic if $\omega$ given by Eq. (\ref{om})
becomes imaginary. The boundary at which this happens is given by the
circle $\mu^2=4(w^2-\Delta^2)$. Beyond this circle the Majorana wave functions
have only a decaying (but not oscillatory) component. This implies that
there would not be any parity oscillations in this region. This can be
clearly seen in Fig.~\ref{fig:SplittingNoOsc}.

We remark here that expressions for the energy splitting
of the Majorana end modes are known for a continuum
model~\cite{Pientka13,Thakurathi14}. These results are consistent with the
 splitting being both oscillatory
and decaying exponentially with increasing length.

{\bf Calculation of ground state fermion parity:}
To obtain a rigorous knowledge of the parity and its switching as a function
of the chemical potential, as has been used in other Majorana wire
contexts~\cite{Sau13},
we employ the measure introduced in Kitaev's well-known
work~\cite{Kitaev01}. Given a
Hamiltonian of the form in Eq. (\ref{ham1}), the transformation $B$, which
reduces the Hamiltonian to the canonical form, can be represented as a
conjugation by a parity preserving unitary operator if $B$ has the form
$B=e^D$ i.e., if $det(B)=1$. Otherwise $B$ changes the parity.
Therefore, the parity of the system is given by
\begin{equation} P(H)=sgn[det(B)]. \label{detB} \end{equation}
In Appendix B we illustrate this result with a simple problem of a
two-site effective Hamiltonian. This illustration is particularly useful in our
case since the effective Hamiltonian for only the coupled
Majorana modes is in fact a two-site problem given by $H_f=iJb' b''$.
Within the topological phase, the dynamics of only these end modes and their
associated Dirac fermions determine the overall parity as we saw above.

In terms of the Majorana operators, the parity of a $N$-site system is
given by
\begin{equation} P~=~ \prod_{j=1}^N ~(-ia_{2j-1}a_2j) ~=~ \prod_{j=1}^N
~(1 - 2 f_j^\dagger f_j). \label{parity} \end{equation}
We note that $P$ is both Hermitian and unitary, and it commutes with the
Hamiltonian in Eq. (\ref{ham2}); since $P^2 = I$, the eigenvalues of $P$ must
be $\pm 1$.
In the extreme case of $w=\Delta$ for the Majorana wire in Eq. (\ref{ham2}),
the terms of the form $a_{2j}a_{2j+1}$ with $j=1, \cdots, N-1$ are equal to 1.
Thus only the term $a_1a_{2N}$ remains, which is in fact the term in the
effective Hamiltonian. Now, within the topological phase, small deformations
of the parameters should not change this fact. Since the parity operator also
commutes with the Hamiltonian, we can see that in the topological phase
the end modes alone determine the parity of the ground state. The parity equivalence of ground state sectors has been studied in ~\cite{Kells14c}

\subsection{Exact expression for zero energy contours}

While we obtained an approximate result for the tunneling amplitude that
splits the degeneracy between Majorana end modes in the previous subsection,
a formalism involving transfer matrices~\cite{DeGottardi11,DeGottardi13}
enables to track the exact points in the parameter space at which this
energy changes
sign, restoring the zero energy degeneracy and resulting in a parity switch.
Previous work has presented similar derivations and results using the
equivalent method of chiral decomposition~\cite{Kao14}.

For a mode with energy exactly equal to zero, Eqs. (\ref{abn1}-\ref{alphabetan})
are applicable. We see that the $a$ and $b$ modes are decoupled; for
definiteness, let us consider a zero energy mode involving the $a_n$'s.
Eq. (\ref{abn1}) shows that for $2 \le n \le N-1$, the $a_n$'s are related to
each other by a transfer matrix $M_a$,
\begin{eqnarray} \left( \begin{array}{c}
a_{n+1} \\
a_n \end{array} \right) &=&~ M_a \left( \begin{array}{c}
a_n \\
a_{n-1} \end{array} \right), \nonumber \\
\mbox{where}~~ M_a &=& \left( \begin{array}{cc}
-\frac{\mu}{w + \Delta} & - \frac{w - \Delta}{w + \Delta} \\
1 & 0 \end{array} \right). \end{eqnarray}
If $\lambda_1$ and $\lambda_2$ are the two eigenvalues of $M_a$, the general
solution of Eq. (\ref{abn1}) is $a_n = c_1 \lambda_1^n + c_2 \lambda_2^n$.
Next we note that the boundary equations become satisfied
if we add fictitious sites with $n=0$ and $n=N+1$ at the two ends of the
system and demand that $a_0 = a_{N+1} = 0$. This is possible if and only if
\begin{equation} \left( \frac{\lambda_1}{\lambda_2} \right)^{N+1} ~=~ 1. \label{la12} \end{equation}

We can show that Eq. (\ref{la12}) holds and that hence there is a zero energy
mode if either \\
\noindent (i) the parameters lie insider the circular region $\mu^2 + 4
\Delta^2 < 4 w^2$, and
\begin{equation} \left( \frac{\mu + i \sqrt{4 w^2 - 4 \Delta^2 - \mu^2}}{\mu - i
\sqrt{4 w^2 - 4 \Delta^2 - \mu^2}} \right)^{N+1} ~=~ 1, \label{zeroen} \end{equation}
or \\
\noindent (ii) $\mu = 0$ (which implies that $\lambda_1 /\lambda_2 = -1$
regardless of the relative values of $w$ and $\Delta$) and $N$ is odd. \\
We get the same conditions if we look for a zero energy mode involving the
$b_n$'s.

In terms of $\omega$ defined in Eq. (\ref{om}), Eq. (\ref{zeroen}) is
equivalent to saying that $(N+1)\omega$ is an integer multiple of $\pi$,
namely, that $\sin [(N+1)\omega] = 0$. We see that this differs somewhat
from the approximate condition that $\alpha_L \alpha_R$ given in
Eq. (\ref{aLR}) should be equal to zero.

The solutions of Eq. (\ref{zeroen}) are given by
\begin{equation} 4 \Delta^2 + \mu^2 \sec^2 \left( \frac{\pi p}{N+1} \right) ~=~ 4 w^2,
\label{ellipse} \end{equation}
where $p$ is an integer equal to $1,2,\cdots,N/2$ if $N$ is even and $1,2,
\cdots,(N-1)/2$ if $N$ is odd. In terms of the variables $\mu/w$ and
$\Delta/w$, we observe that Eq. (\ref{ellipse}) defines a number of ellipses,
which are labeled by the integer $p$; these are shown in
Fig.~\ref{fig:oddNparitydiagram} (a) for $N$
even and Fig.~\ref{fig:oddNparitydiagram} (b) for $N$ odd. Note that all the ellipses pass through the two
points given by $\mu = 0$ and $\Delta = \pm w$.
Fig.~\ref{fig:oddNparitydiagram} (b) for $N$ odd also
contains a zero energy line lying at $\mu = 0$ for all values of $\Delta/w$.

\begin{figure*}[htp]
\centering
\subfloat[][Parity variations for even number of sites (N=10).]{\includegraphics[width=4in]{evenNparitydiagram.pdf}} \hfill
\subfloat[][Parity variations for odd number of sites (N=11).]{\includegraphics[width=4in]{oddNparitydiagram.pdf}}
\caption{(Color online) The phase diagram of the Kitaev Hamiltonian for the Majorana wire representing regions of odd and
even fermion parity ($\pm$) for finite length wires. The outer circle is the boundary at
which the oscillations in the Majorana wave function and the energy splitting
stops. Contours corresponding to exact zero energy states and parity switching
form ellipses. The parity corresponding to the specific parameters changes for
odd and even number of sites. As we can see there is a contrast in the behavior of parity for even and odd $N$ across the $\mu=0$ line. For odd $N$, the
parity actually changes across this line and thus starting from $\mu=0$ gives a special case of parity blocking. }\label{fig:oddNparitydiagram}
\end{figure*}

Eq. (\ref{ellipse}) can be understood in a simple way for the special case
$\Delta = 0$. Eq. (\ref{ellipse}) then reduces to
\begin{equation} \mu ~=~ \pm ~2 w ~\cos \left( \frac{\pi p}{N+1} \right). \label{Delta0}
\end{equation}
We can understand this as follows. For $\Delta = 0$, Eq. (\ref{ham1})
describes a non-superconducting tight-binding model whose single particle
spectrum is given, for an open chain with $N$ sites, by $E_q = 2 w \cos
[\pi q/ (N+1)] - \mu$, where $q = 1, 2, \cdots, N$. One of these energies
vanishes whenever $\mu$ satisfies the condition given in Eq. (\ref{Delta0}),
in particular, when $p$ is equal to the smaller of the two integers $q$ and
$N+1-q$. This is where the ranges of $p$ mentioned above, namely, $p=1,2,
\cdots,N/2$ for $N$ even and $1,2,\cdots,(N-1)/2$ for $N$ odd come from.
In addition, if $N$ is odd (but not if $N$ even), we have a zero energy
state at $\mu = 0$ corresponding to $q=(N+1)/2$.

Having found all the zero energy lines in the plane defined by $\mu/w$ and
$\Delta/w$, we observe that the parity of the fermion number of the ground
state flips sign whenever we cross one of these lines. As a check, this is
again easy to
see for the case $\Delta = 0$. The number of energy levels which are occupied
in the ground state changes by 1 and hence the fermion parity
changes sign whenever one of the single particle energies $E_q$ given above
crosses zero. For $\Delta = 0$ and very large negative values of $\mu$,
we can see that the ground state of Eq. (\ref{ham1}) contains no fermions;
hence, according to Eq. (\ref{parity}), the fermion parity is $+1$ for any
value of $N$. For very large positive values of $\mu$, the ground state of
Eq. (\ref{ham1}) is completely filled with $N$ fermions; hence the
fermion parity according to Eq. (\ref{parity}) is $(-1)^N$.
We remark that the oscillations in the parity, which are related to the Kitaev's
Pfaffian, map to spin-spin correlations in the transverse spin chain, and as with
much of the literature on Majorana wires, these oscillations have been discussed
in depth in the spin context~\cite{Barouch}.

\section{Quenching dynamics in the Majorana wire}
\label{sec:quenching}
Previous work involving the dynamics of quenching in the Majorana wire
described above has focused on tuning through quantum critical points
separating topological and trivial phases~\cite{DeGottardi11,Kells14a,Bermudez10,Rajak14}.
There have been recent works on the effect of quenching on Majorana modes, signatures of Majorana modes in quenching dynamics and Kibble-Zurek scaling ~\cite{Vasseur14,Perfetto13,MLee14,Karzig14,sacramento14}.
(As mentioned earlier, while the terms quench and ramp
are sometimes used to distinguish between instantaneous change of parameter
versus
a time-dependent change at some given rate, respectively, here we use quench
in a more general sense to encompass all such dynamic tuning. Our actual studies
 are restricted to the ramp case.)
While our analysis also explores non-equilibrium dynamics within a
particular topological phase, we use similar protocols for changing
parameters of the Hamiltonian to tune from one phase to another.

Specifically, we consider a linear variation with time of the chemical
potential of the system so as to go across the critical line $\mu = 2w$,
\begin{equation} \mu(t)= (2-\mu_i)t/\tau +\mu_i. \label{mut} \end{equation}
Here $\mu_i$ is the initial chemical potential at $t=0$ and $1/\tau$ is
the quench rate. Due to the finite rate of variation of $\mu$, the system
cannot remain exactly in its ground state and will exhibit
non-equilibrium behavior. Namely, excitations (or defects) will be produced
in the ground state; this lead to an excess energy of the system and may
also lead to a ground state which is in a different topological sector
than the initial ground state. These effects can be characterized by
the following quantities.
\begin{itemize}
\item Defect density: The number of defects produced in the ground state
configuration, which is given by the sum over all the excitations.
\item Adiabatic Fidelity $\mathcal{O}(t)$: This is the inner product of the
instantaneous ground state $\ket{\psi_{ins}(t)}$ of the time-dependent
Hamiltonian with the time evolved initial ground state $\ket{\Psi(t)}$,
\begin{equation} \mathcal{O}(t)=|\langle \Psi(t)\ket{\psi_{ins}(t)}|.
\label{eq:overlap} \end{equation}

\item Residual energy $E_{res}$: This is the energy in excess of the
instantaneous ground state of the system. We will define this as the
dimensionless quantity
\begin{equation} E_{res} = [\langle \Psi(t)| H(t)| \Psi(t)\rangle - E_G(t)]/|E_G(t)|,
\label{eq:eres} \end{equation}
 where $E_G$ is the energy of the ground state at time $t$.
\end{itemize}

{\bf Previous work:} Most of the analytical results for the above quantities
obtained in earlier work are in the limit of very large system size or
with periodic boundary conditions, where one can Fourier transform the
Hamiltonian to momentum space. This in fact reduces the calculation to a well
known problem of a transition between two states for each value of the
momentum $k$ in the Brillouin zone. This is the famous Landau-Zener-Majorana-Stueckelberg
problem~\cite{Landau32,Zener32,Majorana32,Stueckelberg32} which can,
under a few assumptions, be solved exactly to obtain the probability of
excitation from the ground state to the excited state. Using this probability,
we can obtain expressions for the defect density, adiabatic fidelity and residual energy.
In the limit of long time $t$, all these quantities
have a universal power law scaling as a function of the quench rate which
is related to the post-quench excitations. This is the well studied
Kibble-Zurek scaling~\cite{Dziarmaga10,Polkovnikov11,Dutta10a,Kibble76,
Kibble80,Zurek85,Zurek96,Dziarmaga05,Damski05,Damski06,Polkovnikov05,
Polkovnikov08,Calabrese05,Calabrese06,Cherng06,Mukherjee07,Divakaran08,Deng08,
Divakaran09,Mukherjee10,Sengupta08,Mondal08,Sen08,Mondal09,DeGrandi08,
Barankov08,DeGrandi10,Patane08,Patane09,Bermudez09,Bermudez10,Perk09,Sen10,
DeGottardi11,Pollmann10,Dutta10b,Hikichi10,Chandran12,Chandran13,Patel13,
Mostame14,Foster14}.
Given the quench rate $1/\tau$, the defect density and residual energy scale
as $1/\sqrt{\tau}$ and the adiabatic fidelity scales as $\exp (c/\sqrt{\tau})$
in this one-dimensional Majorana wire.

For the Majorana wire with open boundary conditions, there have been
investigations of the behavior of single particle states under a quench.
While it has been found that the single particle bulk states still obey the
Kibble-Zurek scaling for the defect density, the quench for an initial state
with a Majorana end mode has been found to be non-universal and dependent on
the topological
features of the system~\cite{Bermudez10}. The end states are not robust
with respect to the quench and they delocalize to merge with the bulk states.
This leads to a scaling of the defect density as $\tau^0$ (i.e., independent
of the quenching rate), which is very different from the Kibble-Zurek scaling.

Another investigation for the open chain looks into a quantity called the
Loschmidt echo, which is the survival probability of the Majorana end modes
under a quench~\cite{Rajak14,Vasseur14}. Upon quenching across the critical point, the
probability decays to extremely small values as the end modes merge with the
bulk states when the system is near the critical point where the gap between
the end and bulk states vanishes. But interestingly, when the system is
quenched to exactly the critical point, although the probability of survival
goes to zero initially, it revives itself completely at regular intervals of
time. This is attributed to the nearly equal spacing of the low-lying energy
levels in the bulk near the critical point; this spacing is of the order
of $1/N$ while the scaling of the gap at the critical point is also of order
$1/N$. This leads to oscillations in the survival probability with a time
period which is proportional to the system size $N$. We will see below the
remnant of this effect in the residual energy for an extremely slow quench.

In this work we will study the quenching dynamics in the above mentioned
quantities for an open Majorana wire. We will mainly focus on the many-body
states rather than the single particle states and the novel parity switching
mechanism discussed in the previous section, which comes about in the
topological phase due to the coupling between the Majorana end modes.

\section{Real space formalism for studying quenching dynamics for
open boundary conditions}
\label{sec:realspace}

 In comparison to the translationally invariant systems with periodic boundary conditions studied in previous work, a challenge encountered in these finite-sized systems with open boundary conditions is that one cannot exploit the momentum basis, which in previous works highly simplified the quench dynamical problem. Here,in principle, we are faced with the full-fledged $2^N$ -dimensional Hilbert space associated with fermions on a $N$-site lattice associated with the Fock space formed by fermion occupancy on each site.  Here, we develop and present a dynamic many-body technique to reduce the problem to a numerically tractable form. Our technique hinges on two principles in calculating expectation values or overlaps between states in this time-dependent setting.  The first is to use the Heisenberg picture so that the crux of the information on the time evolution is given by the relation between fermionic creation/annihilation operators at different times. The second is to invoke an analog of Wick's theorem for Majorana operators. The computation then reduces to dealing with time-dependent $2N \times 2N$ matrices, allowing us to embark on an exhaustive analysis of adiabatic fidelity and residual energies and to pinpoint attributes of parity blocked dynamics.

  Let us start with a general time-dependent Hamiltonian which is
quadratic in Majorana operators $a_j$ ($j=1,2,\cdots,2N$),
\begin{equation} H=i\sum\limits_{i,j=1}^{2N} a_i M_{ij}(t) a_j. \end{equation}
Here $M(t)$ is a real antisymmetric matrix; its elements will be functions
of $w$, $\Delta$ and $\mu$ for the Majorana wire Hamiltonian in Eq.
(\ref{ham2}). This can be converted to the canonical form
\begin{equation} H=4\sum\limits_{j=1}^N \lambda_j(t) b^{\dagger}_j(t)b_j(t), \end{equation}
up to a constant, by a time-dependent transformation $B(t)$,
\begin{equation} \bar{b}(t)=B(t)\bar{a}. \label{matB} \end{equation}
Here $\bar{b}(t)$ is a $(2N)$-component vector $\bar{b}=(b_1,b_2,\cdots,
b_N,b^{\dagger}_1,\cdots,b^{\dagger}_N)^T$ and so is $\bar{a}=(a_1,a_2,
\cdots,a_{2N})^T$. The $(2N)$-dimensional matrix $B(t)$ comprises of the
eigenvectors of $H(t)$ and it belongs to the group $U(2N)$ with $det(B)=
\pm 1$. The eigenvalues of the Hamiltonian are $\pm \lambda_j$.

{\bf Adiabatic fidelity calculation:} As defined in a previous section the
adiabatic fidelity is given by $\mathcal{O}(t)= |\langle \psi_{ins}(t)\ket{\Psi(t)}|$. The corresponding annihilation operators of these many-body states
satisfy the relations
$b_j(t)\ket{\psi_{ins}(t)}=0$ and $\beta_j(t)\ket{\Psi(t)}=0$, where
$\ket{\Psi(t)}=S(t,0)\Psi(0)$ and $\bar{\beta}(t)=B(0)S(t,0)\bar{a}$. Here
$S(t,0)=\mathcal{T}exp(-4 \int_{0}^t M(t')dt')$ is the evolution operator,
with $\mathcal{T}$ denoting time ordering. The two sets of annihilation
operators are related by
\begin{equation} \bar{\beta}(t)=B(0)S(t,0)[B(t)]^{-1}\bar{b}(t)=G(t)\bar{b}(t). \label{matS}\end{equation}
The key idea underlying the calculation in real space is to express the
quantities of interest to us in terms of objects which can be calculated
numerically in a simple way. Given the form of the initial Hamiltonian $H(0)$
and the time-dependent $H(t)$, we can see that the quantities $B(0)$, $B(t)$,
$S(t,0)$ and
$G(t)$ can be easily computed. Given these and the annihilation operators for
the ground states, the calculation of the adiabatic fidelity reduces to a computation of
the determinant of an antisymmetric matrix $A$ given by
\begin{eqnarray*}
A_{jk} &=& \bra{\psi_{ins}(t)}\bar{\beta}_j(t) \bar{\beta}_k(t)
\ket{\psi_{ins}(t)} ~~{\rm for}~~ j<k, \\
&=& -\bra{\psi_{ins}(t)}\bar{\beta}_k(t) \bar{\beta}_j(t)\ket{\psi_{ins}(t)}
~~{\rm for}~~ j>k, \\
&=& 0 ~~{\rm for}~~ j=k. \end{eqnarray*}
We now directly state an important result, deferring the detailed derivation
to Appendix A. The adiabatic fidelity defined in Eq. (\ref{eq:overlap}) is given by
\begin{equation} \mathcal{O} (t) = |det(A)|^{1/4}. \label{detlap}\end{equation}
Given this relation and the Hamiltonian $H(t)$, we can numerically calculate
the adiabatic fidelity as a function of time for a system with open boundary conditions.
This can naturally be extended to periodic/antiperiodic boundary conditions as well.

{\bf Residual energy calculation:}
Another quantity of interest, the residual energy, defined in Eq. (\ref{eq:eres}),
measures the excess energy contained in the time-evolved quench dependent state
compared to the instantaneous ground state energy. This quantity can also
be calculated with the real space formalism developed in this section.
Following the same strategy as for the adiabatic fidelity, the final expression can
simply be expressed in terms of the matrix $G(t)$,
\begin{equation} E_{res}=[4 \sum\limits_{j,k}^N \lambda_j G_{N+j,k}^{-1}(t)
G_{j,k+N}^{-1}(t)]/|E_G(t)|. \label{resen} \end{equation}
The details of the derivation are given in Appendix C.

\section{Results}
\label{sec:results}
We now present the results that we obtain for an open Majorana wire with
$N$ sites with the Hamiltonian given in Eq. (\ref{ham2}), where $\mu$ varies
linearly in time as shown in Eq. (\ref{mut}). Given this specific form of
$H(t)$, we numerically calculate all the quantities $B(0)$, $B(t)$ in
Eq. (\ref{matB}),
$S(t,0)$, $G(t)$ in Eq. (\ref{matS}) and finally the adiabatic fidelity
$\mathcal{O}(t)$ in Eq. (\ref{detlap}) and the residual
energy $E_{Res}$ in Eq. (\ref{resen}). In what follows, we provide
an in-depth discussion of the novel phenomenon of topological and associated
parity blocking as elucidated in Sec.~\ref{sec:blocking}.

For periodic boundary conditions, the effect of the number of sites on
the fermion parity of the ground state and the consequent phenomenon called
topological blocking on the quenching dynamics of the ground state has
been discussed in detail in Ref.~\cite{Kells14a}. For an open chain,
we saw in Sec.~\ref{sec:finitesize} that the Majorana end modes play an
important role in
determining the parity. For a fixed $\Delta$ and $N$, the parity changes
sign as we sweep across the topological phase by varying $\mu$. Here we
will explicitly see this parity blocking effect in the evolution of the
ground state within the topological phase. The initial parity of the
system is set by the value of $\mu_i$ and the number of sites $N$. As
we will see below, the choice of $\mu_i$ can have drastic consequences,
especially for an odd number of sites.

\subsection{Adiabatic fidelity and Parity blocking}

\begin{figure*}[htp]
\centering
\subfloat[][Adiabatic fidelity for $N=17$ with $\mu_i \neq 0$.]{\includegraphics[width=0.4\linewidth]{OverlapOpen17.pdf}} \hspace{1cm}
\subfloat[][Parity for $N=17$ with $\mu_i \neq 0$.]{\includegraphics[width=0.4\linewidth]{ParityOpen17.pdf}}
\caption{(Color online) Numerical results for (a) adiabatic fidelity $\mathcal{O}(t)$ and (b)parity of the
instantaneous ground state for an odd number of sites. In this case the system
has the same parity as the initial ground state on crossing the quantum
critical point (Figure (b)) and therefore has a non-vanishing overlap with it.}
\label{fig:OverlapOpen13}
\end{figure*}

Figures ~\ref{fig:OverlapOpen34} and ~\ref{fig:OverlapOpen13} show the numerical results for the
adiabatic fidelity $\mathcal{O}(t)$ along with the parity of the instantaneous ground state
for an open chain with an even and odd number of sites, respectively. The
case of the initial value $\mu_i=0$ for an odd number of sites will be
discussed later.

We can see from Figs. ~\ref{fig:OverlapOpen34} and ~\ref{fig:OverlapOpen13} that for both even
 and odd number of sites, the system starts in a particular fermion parity sector, and as it
moves within the topological phase the instantaneous ground state switches
parity regularly. On crossing the critical point it can either have opposite
parity from the initial state or the same parity, depending on the initial
parity sector. On the other hand, as we are dealing with parity
conserved systems, the state which is time evolved from the
initial ground state continues to have the same fermion parity. Thus the overlap
of the time evolved state with the
instantaneous ground state plunges to zero at times when the instantaneous parity
becomes opposite to the initial parity. We call these parity oscillations, which
occur for an open Majorana wire, as the parity blocking effect. The initial
ground state is blocked from having any non-zero overlap with the instantaneous ground
state for certain values of $\mu$. Finally, on crossing the quantum critical
point it becomes zero at all times if the parity is flipped from the initial
ground state; this is also a manifestation of parity blocking.
Hence, in Fig. ~\ref{fig:OverlapOpen34}, the case of an even number
of sites, the parities of the initial and final ground states are the opposite and
the system shows parity blocking for the entire topologically trivial phase, while in
Fig. ~\ref{fig:OverlapOpen13}, the parities are matched and there is some residual
overlap in the trivial phase.

\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{OverlapNoOsc.pdf}
\caption{(Color online) Numerical results for quenching with $\mu_i> 2\sqrt{1-\Delta^2}$. i.e outside the domain of oscillations as shown in the phase diagram. Here the nature of Majorana wave functions at the edges are purely decaying and their coupling would not have any oscillations, which would result in the ground state parity not switching as one sweeps through the parameter space. }
\label{fig:OverlapNoOsc}
\end{figure}

\textbf{Domain with no parity blocking:}
As we showed in Sec. ~\ref{sec:finitesize}, the oscillations in the parity do
not occur throughout the parameter space corresponding to the topological
phase. Namely, there are no oscillations if $\mu^2 > 4 (w^2 -\Delta^2)$.
This implies that there ought to be no parity blocking in the adiabatic
fidelity. We indeed see this in the numerical results shown in
Fig.~\ref{fig:OverlapNoOsc}.

\textbf{Fermion parity degeneracy for an odd number of sites:}
For an open chain with an odd number of sites, and for states which belong to
the odd fermion sector, the $\mu_i=0$ point is special in that it has two
degenerate ground states. To see this, let us define an operator
\begin{equation} C ~=~ i^{N(N-1)/2} ~a_1 a_4 a_5 a_8 \cdots , \end{equation}
where the last term on the right hand side is given by $a_{2N-1}$ if $N$
is odd and $a_{2N}$ if $N$ is even. We note that $C$ is both Hermitian
and unitary, so that $C^2 = I$. Recalling that $f_n = (1/2) (a_{2n-1} + i
a_{2n})$ and $f_n^\dagger = (1/2) (a_{2n-1} - i a_{2n})$, we find that $C$
generates the particle-hole transformation
\begin{eqnarray} C f_n C &=& (-1)^{n+N-1} ~f_n^\dagger, \nonumber \\
C f_n^\dagger C &=& (-1)^{n+N-1} ~f_n. \label{charge} \end{eqnarray}
This is a symmetry of the Hamiltonian in Eq. (\ref{ham1}) if $\mu = 0$.
We now note that the parity and charge-conjugation operators $P$ and $C$
satisfy $P C = (-1)^N C P$. Thus $P$ and $C$ anticommute if the number of
sites $N$ is odd. Since $P$ and $C$ both commute with $H$ if $\mu=0$,
every energy of a system with an odd number of sites will have a two-fold
degeneracy with the two eigenstates having opposite fermion parities.
(This can be shown as follows. If $|\psi_+>$ is an eigenstate of $H$ and $P$
with eigenvalues $E$ and $+1$ respectively, the relations $PC = -CP$ and
$HC = CH$ imply that $|\psi_-> = C|\psi_+>$ is an eigenstate of $H$ and
$P$ with eigenvalues $E$ and $-1$ respectively).
\begin{figure*}
\subfloat[][Partial blocking in adiabatic fidelity for different $N$.]{\includegraphics[width=0.3\textwidth]{OverlapOpen35mu0.pdf}}\hfill
\subfloat[][No energy splitting at $t=0$.]{\includegraphics[width=0.3\textwidth]{SplittingOpen35mu0.pdf}}\hfill
\subfloat[][Usual parity switching of the instantaneous ground state.]{\includegraphics[width=0.3\textwidth]{ParityOpen35mu0.pdf}}
\caption{(Color online) Numerical results for quenching with $\mu_i=0$ for the odd
sector. This is the special case where the initial state is in a superposition of the odd and even parity states.Thus the time evolved states will not be completely 'parity blocked' but the amplitude of adiabatic fidelity will be reduced. As we go to smaller $N$ the splitting is exponentially enhanced and one can clearly see the effect of it in 'skewing' the superposition towards the state which
contributes to the ground state.}\label{fig:OverlapOpen35mu0}
\end{figure*}

Therefore, starting from $\mu_i=0$ means starting with parity states whose degeneracy is
not split. The time evolution of an arbitrarily chosen ground state would
therefore be that of a linear combination of both parity states. Even though the parity of the
instantaneous ground state would keep switching as one sweeps through the
topological phase, the overlap would be finite as the time evolved state
will be in a superposition of both parities. But the value of the adiabatic fidelity
will be smaller than in the parity blocked case as the amplitude is split
between the two superposed states. Figure ~\ref{fig:OverlapOpen35mu0} shows the results for this unique
case of quenching. We can clearly see in Fig. ~\ref{fig:OverlapOpen35mu0} that the splitting is zero
at time $t=0$. This change in the initial condition drastically affects the
evolution of the ground state and we do not see complete parity blocking in
this case.

\begin{figure*}
\subfloat[][]{\includegraphics[width=0.5\linewidth]{Overlapper.pdf}}\hfill
\subfloat[][]{\includegraphics[width=0.5\linewidth]{ApOverlap.pdf}}
\caption{(Color online) Numerical calculation of $\mathcal{O}(t)$ for a closed chain with $34$ sites. The periodic and antiperiodic closed chains represent even and odd fermion sectors respectively. One can see that there is blocking in the second case, whereas a small amount of overlap persists in the first case after crossing the critical point. Also the envelope of the adiabatic fidelities for closed chains is compared with that of the open chain for the same number of sites. Even though there is no 'parity blocking' within the topological phase in the case of closed chain due to absence of the edge modes, the overall the behavior remains qualitatively the same.}
\label{fig:Overlapper}
\end{figure*}

\textbf{Comparison with a closed chain:}
The case of parity blocking in closed chains with periodic and antiperiodic
boundary conditions has been studied in detail in a previous
work~\cite{Kells14a}. While the previous analysis involved momentum modes,
simplifying the problem to a set of two-level Landau-Zener systems, the
real space formalism is easily extended here to compute all the quantities
for a closed chain. The numerical results are shown in Fig. ~\ref{fig:Overlapper}. Even though
there is parity blocking in the case of an open chain, we may still expect
the envelope of the adiabatic fidelity to be comparable with that of the
periodic case. In Fig.~\ref{fig:Overlapper}, we see a good match between the
envelopes in the two cases.
For a closed chain, the final parity can flip from the initial state
depending on the boundary conditions. As can be seen, the parity does not
change for the periodic closed chain and therefore one
still has a finite overlap. The match between the envelopes in the open and
closed chain cases is very close. This suggests that overall the open chain
would also respect the Kibble-Zurek behavior for the defect production and
excitation density that was found in the closed chain case.
The crucial difference between open and closed chains is the parity blocking
and switching due to the coupling of the two Majorana end modes.

\subsection{Residual Energy}
The numerical results for the variation of the residual energy with time using
the full many-body formulation in Eq. (\ref{resen}) is shown in Fig.~\ref{fig:EresOpen34}
for both open and closed chains. The two cases have the same average behavior.
Both show a rapid increase in the energy of the system above the instantaneous many-body
ground state as they approach the critical point. This rapid rise is due to the system
falling out of equilibrium upon approaching the critical point and thus losing
adiabaticity. Far beyond the critical point, we find that the energy asymptotes to
a fixed average value. The scaling analysis of this quantity for the transverse field Ising chain has been studied numerically in ~\cite{Kolodrubetz12}.

\begin{figure*}[htp]
\centering
\subfloat[][Residual energy for an open chain with $N=36$ and $N=18$.]{\includegraphics[width=0.4\textwidth]{EresOpen34.pdf}}\hspace{1cm}
\subfloat[][Residual energy for a periodic closed chain with $N=34$]{\includegraphics[width=0.4\textwidth]{EresPer34.pdf}}
\caption{(Color online) Residual energy plots with the critical point occurring at $t=2$. One can notice in the case of open chain the oscillations before crossing the critical point, which arise due to the oscillation of mid-gap states.One can see that the steps arising due to the splitting scales inversely with the system size.}
\label{fig:EresOpen34}
\end{figure*}

\textbf{Effect of Majorana end modes:} The crucial difference between the open and
closed chains in Fig.~\ref{fig:EresOpen34} is the presence of the abrupt jumps at
small times in the case of the open chain. Reflecting the behavior of adiabatic fidelity, these
jumps correspond to switching back and forth between the ground state and excited
states due to parity blocking. Upon comparing the behavior of the residual energy
with the energy splitting and parity switching of Fig.~\ref{fig:SplittingOpen35},
we find that there is a complete match between the points at which the jumps
in the residual energy take place and the points where the parity switching
occurs.

\textbf{Signatures of Loschmidt echoes:}
In addition to parity blocking, in our many-body system, we find evidence for the
Majorana-mode related physics found in previous work on single particle dynamics in quenching~\cite{Rajak14}. The Loschmidt echo studied in this work calculates the probability for an initial Majorana mode
to become a single-particle bulk excitation as a function of time as one sweeps through the critical point. If one varies $\mu(t)$ extremely slowly so as to be close to the adiabatic limit, the gap
between the mid-gap end states and bulk states scales as $1/N$ on approaching the critical point since the dynamical critical exponent is equal to 1.
The level spacing of the low lying bulk states also scales as $1/N$.
Hence the Loschmidt echo turns out to be a periodic function with period
$N$ as one quenches to or across the critical point.

\begin{figure*}[htp]
\centering
\subfloat[][$N=18$]{\includegraphics[width=0.4\linewidth]{Eres18v1.pdf}} \hspace{1cm}
\subfloat[][$N=36$]{\includegraphics[width=0.4\linewidth]{Eres36v1.pdf}}
\caption{(Color online) The residual energy plots for a very small quench rate which is
nearly adiabatic. In this case one can see the periodic recurrence of
excitations at times after crossing the critical point. The period is doubled if we double the system size.}
\label{fig:Eres18v1}
\end{figure*}

In our full-fledged many-body treatment, these echoes appears as `chirps' of excitations whose occurrence has a period of $N$. These excitations contribute to the overall energy of the state. Note that this is true only if the quench rate is extremely slow. As shown in Fig.~\ref{fig:Eres18v1}, one can see these oscillations in the numerical results at a low quench rate like $1/\tau=0.1$.
The frequency of occurrence of these chirps in the excitations is indeed
halved when the system size is doubled. These chirps also appear in the
adiabatic fidelity.

To summarize, open and closed chains broadly show similar average behaviors
as expected for quench dynamics, However, both in the adiabatic fidelity and
residual energy, distinct non-analytic features arise in the form of jumps
only for an open chain, and these can be attributed directly to the presence
of end modes and their associated fermion parity.

\section{Discussion}
\label{sec:discussion}

In conclusion, our study of non-equilibrium behavior in finite-sized Majorana
wires demonstrates that the presence of topological order can dramatically
alter quench dynamics. Previous work involving Majorana wires having periodic
boundary conditions brought to light the notion of topological blocking in
tuning between different quantum phases ~\cite{Kells14a}. In contrast,
we have seen here that the coupling between Majorana end modes and the
associated ground state parity flips as a function of a tuning parameter
gives rise to a more drastic manifestation of topological blocking due to a
succession of switches between topological sectors within a single topological
phase. As a result, some common measures studied in the quench dynamics
literature, such as wave function overlaps between time-evolved states and
instantaneous ground states (the adiabatic fidelity), and residual energies,
show a series of non-analytic structures in the form of characteristic
jumps which are not observed in standard Kibble-Zurek physics.

Our work has shown that there is a much richer texture in the phase diagram
of the Kitaev chain or the Majorana wire than has been presented earlier.
The circle which separates the regions of oscillating and purely decaying
wave functions of the Majorana end modes exists in the thermodynamic limit.
In addition, we have shown that a coupling between the end modes, due to
a finite length of the chain, leads to further divisions within the circle
in the form of ellipses, each division corresponding to a particular fermion
parity of the ground state. Although the energy splitting between the
Majorana modes goes to zero as we increase the chain length, the number of
fermion parity switches increases linearly with the length. This has a
dramatic consequence for the adiabatic fidelity under a quench, namely,
parity blocking occurs more frequently as we increase the system size.
(This is very different from conventional finite size effects which
typically vanish in the thermodynamic limit). We therefore see that
parity blocking is not merely a finite size effect, but is a relevant
manifestation of the physics of Majorana modes and their topological
nature in any real system.

This study shows that quench dynamics serves well as a probe of topological
order. While blocking features need not be unique to topological systems in
that quantum invariants in other systems can possibly have similar effects,
they are necessary conditions under appropriate circumstances (for instance,
open boundary condition in the case studied here). Moreover, unlike in most
other systems, such as ferromagnets having local order, we expect this blocking
phenomenon to be robust against local perturbations. In the case of a
finite-sized Majorana wire, the succession of parity switches associated with
topological sectors is a crucial aspect of topological order; while studies of
the static behavior have been extensive (see, for instance,
~\cite{Burnell14,Zazunov13,Bonderson11,Hassler11}), here we have performed
the nearly unexplored study of their effect on quench dynamics.

In fact, the issue of parity forms the basis of several discussions and proposals
for Majorana wires, particularly in light of the potential experimental discovery
of isolated Majorana end modes and their implications for topological braiding
and quantum computing. Several schemes involve changing the on-site chemical
potential at specific locations as a means of manipulating and dynamically moving
the isolated end modes. A popular study regarding the end modes is the fractional
Josephson effect
(see, for instance, ~\cite{Kitaev01,Fu09,Law11,Kwon04,Beenakker13, Crepin14}),
which involves parity switches between two finite-sized Majorana wires
connected to each other at their ends and their effect on Josephson physics
(in principle, other zero energy end bound states could mediate such an effect).
Our study here is highly relevant to these lines of investigation and it provides
a dynamic quantum many-body formulation that goes beyond quasi-static approximations.

While our study primarily aims to understand the effects of parity switching on
issues typically studied in the literature on quench dynamics, an experimental
setup probing the predictions would be remarkable. While the arena of cold atomic
gases is more ideally suited for measurements of residual energy and adiabatic fidelity, realizing topological order in these systems is still in its initial phases \cite{Mazza12}.
In the setting involving spin-orbit coupled wires, where the isolated end modes have
potentially been observed, in principle, ground state parity switches can be
observed by coupling the wire to another system. For instance, a possible read-out
could involve tunnel-coupling to a quantum dot or STM tip (see, for example,
~\cite{Flensberg12,Lee14}).
Further studies would involve pinpointing ways of measuring the behavior of adiabatic fidelity predicted here
in such a setup.

Finally, the study of topological/parity blocking in quench dynamics presented here
and the associated quantum many-body formulation offer wide scope for further
exploration. Several aspects of this initial study require more detailed investigation,
for instance, more involved studies of system size, and further connections with
Kibble-Zurek scaling and single-particle physics, including anomalous scaling due to
boundary effects and appearance of Loschmidt echoes. Oscillations have been found in the derivative of the Renyi entropy with respect to the chemical potential in \cite{Ming15} and it may be interesting to see if there are such effects related to the oscillations in the ground state parity. A host of open issues related to topological blocking in Majorana wires include constraints on thermalization imposed by topological order, effects of external potentials, such as quasiperiodic
potentials and disorder, and higher dimensional analogs, such as the Kitaev
honeycomb model.

\verb"Acknowledgements"
We thank Bryan Clarke, Graham Kells and Jeffrey Teo for illuminating
conversations. This work is supported by the National Science Foundation under the grants DMR 0644022-CAR (SH) and DMR-0906521 (VS) , the U.S. Department of Energy, Division of Materials Sciences under Award No. DE-FG02-07ER46453 (SV), and by the Department of Science and Technology, India: SR/S2/JCB-44/2010. For
their hospitality during the course of this work, D. S. gratefully
acknowledges the Department of Physics, University of Illinois at
Urbana-Champaign and S.V. the Aspen Center for Physics.

\section*{Bibliography}
\bibliographystyle{iopart-num}

redpajama_set_name: RedPajamaArXiv


Samsung is about to throw in another tablet to the mix, the Galaxy Note 8.0.
The device is neatly positioned to start to offer Samsung users a range of devices that vary merely in their size and unlike the Apple world these in the main offer the same or similar functionality without the defined walls created by Apple. In other words you have the Galaxy phone with pen, the Note II, 7 & 8 with pen all with the same SIM size and all capable of being tethered or being a phone. Apple like to separate their media light device iPod, from their phone, Iphone from the Tablet, iPad.
The point is that all mobile devices are converging in technology and are all underpinned by a common platform but separating in how we use them. Do we want different devices to do different things or do we want all devices to inherently offer the same but vary in their form and footprint? We may want a basic phone tethered to a sophisticated larger form tablet / server. We may want this reversed so we have the smaller footprint on the go with the option to tether the tablet when required. The world is not about having devices that perform X or Y but a mobile platform that offers choices and interoperability.
It is not hard to envisage a mobile server which is 'connected' to slave mobile devices, each with its own specialised usage and with a common set of functionality. This year we shall see the 'glasses' and the wrist 'watch' and we already have the earpiece.
Ask yourselves what the difference really is between ASUS, Acer, HP, Dell, Samsung, laptops? Then what is the difference between these and a Mac?
Maybe it is down to Android versus IOS and for some that may be Déjà vu.
Hello, I love reading through your blog, I wanted to leave a little comment to support you and wish you a good continuation.
So What Does LG know that the rest of us don't?
So What Does Nook Want To be When It Grows Up?
What determines the device and format of music?
Should We Condemn DRM to Room 101?
Is The BBC Losing Its Way?


How to create a Wall Boundary between Solid Zones in FLUENT?
Walls between Fluid - Solid zones are created automatically when imported into FLUENT even if not set in the geometry software, but interfaces between two solids are treated as interior zones by default.
The boundary between two solid zones is treated as interior surface by default. In the boundary conditions panel, boundary type can be changed from interior type to wall.


text:
SwimSwam Pulse: 85% Would Support ISL Athletes In Boycott Scenario
While word of a possible ISL boycott made headlines a few weeks ago, things appear to be on track to have the playoffs commence on November 11. Archive photo via Giorgio Scala / Deepbluemedia / Insidefoto
by James Sutherland 4
November 05th, 2021 Lifestyle, News, Opinion
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SwimSwam Pulse is a recurring feature tracking and analyzing the results of our periodic A3 Performance Polls. You can cast your vote in our newest poll on the SwimSwam homepage, about halfway down the page on the right side, or you can find the poll embedded at the bottom of this post.
Our most recent poll asked SwimSwam readers if they would support ISL athletes if they opted to boycott from competing until they were fully paid for the 2020 season:
Question: Would you support ISL athletes holding out from competing in the playoffs if they haven't been fully paid for last season?
Yes – 85.4%
No – 14.6%
Word of a possible athlete boycott from the International Swimming League (ISL) first seriously arose on October 23 when The Sydney Morning Herald reported that clubs and athletes were considering holding out of competing if they weren't paid in full prior to the playoffs beginning on November 11.
At the time, London Roar general manager Rob Woodhouse was quoted as saying: "It's not good enough. I'm really hoping all athletes are paid in full from last season before the playoffs commence on November 11."
The poll asked readers if they would support athletes in holding out from competing if Season 2 payments had not been received. An overwhelming majority, just over 85 percent, said they would.
Since the initial report came out, some of the league's veteran swimmers have spoken to SwimSwam regarding the situation.
Aqua Centurion captain Fabio Scozzoli and Swimmers' Alliance member Cody Miller both said that a boycott was never a viable option for the swimmers, noting that it would not be in the best interest of the sport to put the league's existence in jeopardy.
"As someone that's a part of this league, we're all just kind of rolling with the punches and understanding that there are some growing pains and they're trying to work things out," Miller said on the SwimSwam Podcast.
"And hopefully, it gets better. From what I have heard, I think everybody — at least the athletes — have been paid out almost fully. So they're remedying some of those late payments."
Perhaps a better poll question would've been: Is boycotting the right move for swimmers in this situation?
Of course, it's no surprise that most fans would support athletes in holding out—they deserve to be paid and shouldn't be expected to continue to compete at the highest level without receiving what they're owed.
The problem lies in the fact that, as Miller noted Tom Shields said in one of their threads, boycotting/striking is essentially a nuke and would have severe ramifications on the league's existence.
More than 14 percent of voters said no, they wouldn't support a boycott. This is likely a group that basically believes it's just not the right move for the swimmers to make at this juncture.
The ISL playoffs are scheduled to begin on November 11 in Eindhoven, with six matches set to determine the four clubs advancing to the ISL Final. The final, originally expected for January 2022, has been bumped up to December 3-4 in Eindhoven.
Below, vote in our new A3 Performance Poll, which asks: Assuming the Cali Condors, Energy Standard and London Roar are near locks, which of the remaining ISL clubs has the best chance of joining the aforementioned three in the ISL Final?
Outside of Energy Standard, Cali and London, which club do you think has the best chance of qualifying for the ISL Final?
LA Current
Toronto Titans
Aqua Centurions
DC Trident
ABOUT A3 PERFORMANCE
A3 Performance is an independently-owned, performance swimwear company built on a passion for swimming, athletes, and athletic performance. We encourage swimmers to swim better and faster at all ages and levels, from beginners to Olympians. Driven by a genuine leader and devoted staff that are passionate about swimming and service, A3 Performance strives to inspire and enrich the sport of swimming with innovative and impactful products that motivate swimmers to be their very best – an A3 Performer.
The A3 Performance Poll is courtesy of A3 Performance, a SwimSwam partner.
« Thomas Ceccon Nuota Il Nuovo Record Italiano 50 Farfalla In Semifinale Europea
Silvia Scalia, Ai Piedi Del Podio Nuota Il Record Italiano Nei 50 Dorso »
Cody Miller
Fabio Scozzoli
Tom Shields
So what I hear people saying is that they don't want to see the ISL succeed and have never had the experience of a business startup.
Reply to Cate
Where are you hearing people say that? What I'm hearing is that decent startup businesses pay their staff and suppliers, I don't think it's too much to ask, it's a disgrace not to.
Bye ISL
Troyy
85% support the killing off of ISL.
Last edited 1 year ago by Troyy
About James Sutherland
James swam five years at Laurentian University in Sudbury, Ontario, specializing in the 200 free, back and IM. He finished up his collegiate swimming career in 2018, graduating with a bachelor's degree in economics. In 2019 he completed his graduate degree in sports journalism. Prior to going to Laurentian, James swam …
More from James Sutherland
arena Swim of the Week: Lindsay Looney Hits Back-To-Back 1:53s In 200 Fly
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redpajama_set_name: RedPajamaCommonCrawl


It seems like big businesses are preparing to solidify a place in the US online gambling market which is interesting because it doesn't seem like there is any impending breakthroughs when it comes to legalizing and regulating gambling in the USA. Many are wondering if the big business buzz could be coming from impending legislation that perhaps us laymen citizens aren't in the loop about. Perhaps the preparations that big names in casino-related business like Trump Entertainment, bwin.party and Bally Technologies are making are tell-tale signs of big things to come!
The name "Donald Trump" is synonymous with big business and upscale entertainment. Trump Entertainment and Avenue Capital Group (who own 22% of Trump Entertainment) have recently established a joint venture between the two businesses and have filed statutory documents with the Securities and Exchange Commission outlining that their joint venture will be specifically for online gambling. Ivanka Trump, the Trump Organizations Executive VP of Development and Acquisition (she also happens to be Donald's daughter), has been quoted as saying that they are planning to be "the largest online gaming company if the USA industry is legalized".
The Trump dynasty aren't the only ones after this coveted title. It's been reported that the joint CEO of bwin.party, which currently enjoys the reputation of being the largest listed global online gambling operator), has been rounding up US investment for a similar agenda. It seems that bwin.party might have a better stronghold given their experience in the global online gambling space. However, Bally Technologies, another US based company specializing in brick and mortar gambling products has also expressed interest in going after the US market once it's legalized, specifically in the online social and mobile gaming arena.
Is Malta Threatened by Denmark's New Online Gambling Regulations?
The Biggest Online Gaming Awards are Around the Corner!


Follow this link to explore the many career opportunities offered by the construction industry and how to get there.
Click here for the latest video from CAREERS IN CONSTRUCTION about the growing demand across Canada and around the world for skilled tradespeople.
Visit the Calgary Construction Association's careers in construction website, designed for the youth, to learn more about construction trades, apprenticeship, and potential earnings.
Skills Canada also has useful information about pursuing a career in the skilled trades and technology, in the construction industry and beyond.
Follow this link for the NWT Jobs in Demand Handbook. This handbook highlights job opportunities in the NWT over the next 15 years, as well as other labour market information to help you choose the right career.
Follow this link for In Demand Career Opportunities in Nunavut, as well as other resources, to help you make an informed decision.


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text:
Visitor Centre and Girls School
August 23, 2019 By Admin 0 Comment
The Visitor Centre was originally built as a Girls School by Seigneur Ernest Le Pelley in 1841, completing the project which had been envisaged by his brother Pierre Le Pelley in consultation with his good friend Rev Cachemaille, which was to use to the best advantage for Sark's community the surplus of money being generated by the new silver mining venture. The School mistress was to be appointed by the…
La Coupee
Is an Isthmus (narrow strip of land with sea either side) made up of sedimentary rock. The rock is made up of sediments of various types of rock but is crumbly and the sea is eroding both sides of the this walkway. Until 1900 there were no railings extending the whole length and on windy days the school children would crawl over the crumbling path on their hands and…
Ramsar Site and Gouliot Headland
The Gouliot Headland is a RAMSAR site (it gained this status on 9th April 2007), recognised worldwide as a Wetland of International importance under the International Ramsar Convention. This recognises and protects the thriving wealth of species of this special Marine environment. The Gouliot Caves, a large sea cave system on the Sark side of the Gouliot passage. Gouliot caves lie to the right of the small headland which is…
Window in the Rock
250FT DROP – WARNING In the 1850s, Rev Collings was Seigneur of Sark. He inherited the title from his Mother Marie in 1853. His mother brought the Fief in 1852 from Pierre Le Pelley (The 2nd one) who was forced to sell to repay the debts of his father. Debts which accrued through the folly to mine Silver. He encouraged this further, seeing the benefit of having additional revenue stream….
La Seigneurie
Today's main house dates from 1675 and has been home to two of Sark's three Seigneurial families: the Le Pelleys (from 1730) and the Collings (from 1852), ancestors of our current Seigneur Because the Seigneurie, residence of the Seigneur, in 1730 when Susanne Le Pelley became the Dame of Sark. Rather then move to the Manoir she chose to stay in her own residence, La Perronnerie. She gave Le Manoir…
Creux Harbour
visit the old harbour, Creux. The original entrance to the bay was blasted through the rock in 1588 and is still in the same position today. The main entrance that we now use, along with the protective harbour wall was not built until 1866. This harbour allowed easy access to Jersey. The Creux harbour is mainly used by local fishermen now, the service boat will use the harbour if the…
Helier de Carteret found the ruins of this property when he landed on Sark in 1563. He rebuilt this property and carved his coat of arms into the stone just below the roof in the top left hand corner of the main house. The cottages to the right were built by Helier as servants quarters. Le Manoir remained the residence of the Seigneur on the island until 1730 and The…
L'Eperquerie
L'Eperquerie Common is the open healthland on the top of the northern headland. The monks (present on Sark between the 6th and 14th centuries) used the openness of the Common to dry their fish and conger eels on wooden stakes. The old Norman French word for the stake was 'perques', giving the headland its name. The Eperquerie Landing was the main port until Creux harbour was opened up in 1588….
October 21, 2016 By Admin 0 Comment
Sark Windmill 1571 Until the 20th century, Sark's economy was based on grain; reliable mill was essential. Domestic handmills (like the pot quern at La Seigneurie) were small and inefficient, and the monks' water mill above Port du Moulin could not serve forty settler families. Within six years of obtaining Queen Elizabeth's grant of Sark (1565), Helier De Carteret built a windmill. The DE Carteret arms carved on the north…


text:
Zenith Jet Increases its Business Aviation Delivery Forecast
Zenith Jet has issued a mid-year update for its annual business aviation forecast. The Montreal-based business aviation services company forecasts delivery of 736 business jets this year, a number that, if accurate, would represent an increase over the 703 handed over last year and the first year-over-year improvement in deliveries since 2008. Zenith views this as the start of a five-year recovery run that will see deliveries grow at a compound average rate of 17 percent each year, with a peak of 1,392 deliveries anticipated in 2016. As a result, the company revised the number of private jet deliveries it had predicted for the next 10 years, from 10,120 earlier in the year to 10,377, representing approximately $220 billion in revenue. Those aircraft deliveries will spawn an additional $37 billion in revenue for engine manufacturers and $8 billion for avionics makers. Over the life of the forecast, the category accounting for most aircraft deliveries will be the ultra-long-range segment at nearly 1,500 aircraft, testimony to the international market's preferences, while light jet deliveries are estimated at 1,363, providing optimism for a rebound in this category as well. In terms of geographic delivery distribution, according to the forecast, North America's dominance will continue to erode to a ratio of 43 percent versus 57 percent international deliveries predicted over the next decade.
All of Us at JetOptions Wish you a Healthy, Prosperous and…
Happy New Year 2022 to you and your loved ones from…
Business Aviation Demand, XB-1 Supersonic Demonstrator, FAA…
Happy New Year 2023 from JetOptions Private Jets!
Tags: business aviation forecast, zenith jet


text:
AQB – codice aeroportuale IATA per l'aeroporto di Santa Cruz del Quiché
 AQB – sigla inglese per "quantizzazione adattativa all'indietro" (Adaptive Quantization with Backward estimation), opposta a AQF, per "quantizzazione adattativa in avanti" (Adaptive Quantization with Forward estimation)

redpajama_set_name: RedPajamaWikipedia


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Components of a business plan appendix format
Comparison between conventional bank and islamic
Therapeutic nurse patient relationship
Home homework help witches 03 06 review critical thinking music of
03 06 review critical thinking music of
Looking Around In the beginning Yes started out as a conversation between Anderson and self-taught bassist Chris Squire at a bar where Anderson was essentially the janitor.
The lyrics of the song, described as "racy" sees Rowland asking her partner to move his hands over her body.
Problems playing this file? He just does it. A real concern for us was to get that smash for her The music video for "Motivation" was directed by Sarah Chatfieldat the end of March With Sarah she understood; she got the concept.
It was great the way everything came together. The clip then shows Lil wayne in his own scene rapping his verse to the song. He can then be seen leaning against a wall and standing on a bed with the lights flashing while Rowland is simultaneously being entertained by a male dancer as she sits upon a chair.
After more dancing from Rowland and her dancers, the video then ends with Rowland singing her final lines, surrounded by the sexually charged crowd.
Kellywhose seductive bedroom-themed remix of the song was applauded by Rowland. The unofficial remix, features a new introduction from Da Brat, and was released online to Rap-Up. Kelly joined the list of singers to create their own remix for "Motivation". In his version, R. Kelly adds additional verses and references his own songs such as "Pregnant".
Using her official Twitter account, Rowland said, "R. Hollywood News reported that Rowland was among the most anticipated performers of the evening. Additionally Markman said Rowland "received one of the loudest ovations of the night.With that – let's dive into the full in-depth review below.
Unboxing: To get started, let's look at what's in the box. Which of course starts by looking at the box itself. Cuckoo is a Tamil romantic drama film written and directed by Raju Murugan and produced by Fox Star timberdesignmag.com film stars Dinesh and Malavika Nair as a visually challenged pair, while Aadukalam Murugadoss plays a pivotal role.
The film was released on 21 . Ryan Brockington, from the New York Post, was amongst the first to review "Motivation." In his review, he said "The single is a big leap from the dance club synth pop that Kelly has been releasing the past year and it seems like she might be trying to shimmy her way back into the hip-hop world by teaming up with the Midas touch vocals of Lil Wayne.".
timberdesignmag.com is a curated music destination for those looking for something outside of the mainstream.
News, reviews, forums, articles, and commentary. Founded by Jason Tate. At one point in "Paul, Apostle of Christ," a Roman prefect (Olivier Martinez, sporting a Caesar haircut and a nearly unintelligible accent) refers to rumors that his captive, the early.
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Book Review: Twelve Rules For Life | Slate Star Codex