text:
\section{Introduction}
Topological superconductivity is an interesting state of matter, partly because it is associated with quasiparticle
excitations which are Majorana fermions (MFs).
In particle physics, MFs are (fermionic) particles which are their own anti-particles~\cite{Wilczek09}.
It is still unclear if there are elementary particles which are MFs, but they are likely to exist as quasiparticle
excitations in certain condensed matter systems, where a MF is a quasiparticle which is its "own hole".
The condensed matter version of MFs have attracted massive theoretical interest, mainly because of their
special exchange statistics: They are non-abelian anyons~\cite{Stern10rev}, meaning that particle exchanges are non-trivial
operations which in general do not commute. This is unlike other known particle types where an exchange operation merely
has the effect of multiplying the wavefunction with +1 (for bosons) or -1 (for fermions) or a general phase factor
$\phi$ (for "ordinary" (abelian) anyons). Furthermore, a MF is in a sense half of a normal fermion, meaning that
a fermionc state is obtained as a superposition of two MFs.

It should be noted, however, that any fermion can be written as a combination of two MFs, which basically corresponds to splitting
the fermion into a real and an imaginary part, each of which is a MF. Normally this is a purely mathematical operation without physical
consequences, since the two MFs are spatially localized close to each other, overlap significantly, and cannot be addressed individually.
When we talk about MFs here, we mean that a fermionic state can be written as a superposition of two MFs which are \emph{spatially separated}
(or prevented from overlapping in some other manner). Such a highly delocalized fermionic state is protected from most
types of decoherence, since it cannot be changed by local perturbations affecting only one of its Majorana constituents.
The state can, however, be manipulated by physical exchange of MFs because of their non-abelian
statistics, which has lead to the idea of
low-decoherence \emph{topological} quantum computation~\cite{Nayak08rev}.

Being its own hole means that a MF must be an equal superposition of an electron and a hole state. It is natural to search for
such excitations in superconducting systems, where the wavefunctions of Boguliubov quasiparticles have both an electron and a hole
component. The most common type of superconducting pairing is of so-called $s$-wave symmetry, where Cooper pairs are formed
of electrons with opposite spin projections (forming a singlet).
In second quantization language, the annihilation operator of a Boguliubov quasiparticle in an $s$-wave superconductor has the form
$b = u c^\dagger_\uparrow + v c_\downarrow$,
where $c_\sigma$ annihilates a normal fermion with spin projection $\sigma = \uparrow,\downarrow$ (we neglect for simplicity to write
out additional quantum numbers which are irrelevant here).
Having \emph{equal} electron and hole components mean that MFs are instead associated with an annihilation operator
of the type $\gamma = u c^\dagger_\sigma + u^* c_\sigma$, which
is hermitian and therefore equal to the creation operator, $\gamma = \gamma^\dagger$.
Note that, in contrast to the $s$-wave Boguliubov quasiparticle operator, the fermion operators making
up the MF have equal spin projections.
Such quasiparticles do not occur in most types of superconductors and
were instead first predicted to occur in the $\nu = 5/2$ fractional quantum Hall
state~\cite{Moore91}. However, as we will discuss in more detail below, isolated MFs occur in general in
vortices and on edges of effectively spinless superconducting systems with
triplet pairing symmetry~\cite{Kopnin91, Read00, Sengupta01, Ivanov01, Zhang08, Kraus09, Wimmer10, Sato09, Sato10}
($p$-wave pairing symmetry in one dimension (1D) and $p_x \pm ip_y$ pairing symmetry in two dimensions (2D)).
Triplet pairing has been predicted for the ground state of the superconductor
Sr$_2$RuO$_4$~\cite{DasSarma06}, but is very sensitive to disorder and has never been observed experimentally.
However, the existence of (spatially separated) MFs is a topological invariant~\cite{Hasan10} (hence the name
\emph{topological} superconductors). As a result, they will exist in all
systems with the same topological properties as a $p$-wave or $p_x \pm i p_y$-wave superconductor
(we will come back to the precise meaning of this later).

A few years ago, the search for MFs took a big step forward when Fu and Kane~\cite{Fu08} showed that $p_x \pm ip _y$-wave-like
pairing may also occur for the surface states of a strong topological insulator when brought into tunneling
contact with an ordinary $s$-wave superconductor (giving rise to proximity-induced superconductivity~\cite{deGennes64, Doh05, vanDam06}
in the topological insulator).
The necessary underlying physical ingredient is the strong spin-orbit coupling of the topological insulator, giving rise to
split bands with momentum-dependent spin directions. In addition, by coupling the topological insulator also to a magnetic
insulator, the resulting induced Zeeman splitting lifts the Kramer's degeneracy and allows an effectively spinless regime to be reached.
However, as was realized shortly after the pioneering work of Kane and Fu, the strong spin-orbit coupling in
certain two-dimensional semiconductor quantum wells should also do the job~\cite{Sau10, Alicea10, Akhmerov11}.
Also here superconductivity can be induced through the proximity effect and magnetism either induced by a magnetic
insulator~\cite{Sau10} or provided by an external magnetic field~\cite{Alicea10}.
Shortly thereafter, two works~\cite{Oreg10, Lutchyn10} suggested a further simplification by instead using 1D
semiconducting wires.
There have also been proposals to create MFs e.g., in vortices in doped topological insulators~\cite{Hosur11},
on the interface between a ferromagnet and a superconductor deposited on a
two-dimensional topological insulator~\cite{Nilsson08, Fu09, Linder10}, in cold atomic gases~\cite{Gurarie05, Tewari07},
in carbon nanotubes~\cite{Sau11a, Klinovaja12, Egger12}, and using chains of quantum dots~\cite{Sau11b}, just to name a few.

During the last couple of years, a number of experimental groups have taken up the challenge to create MFs.
Very recently this quest may have seen success~\cite{Mourik12}, and several other groups have made observations which can be
interpreted as signatures of Majoranas~\cite{Williams12, Rokhinson12, Deng12, Das12}.
There is no doubt that these early findings will be scrutinized in future experiments and time will tell which
are genuine observations of Majorana physics.
In any case, once MFs have been conclusively produced and detected in the lab, the truly exciting experimental work begins, to
test their theoretically predicted properties and to design setups for ever more advanced manipulation of the quantum information
they can encode.

In this short review article we give an introduction to the topic of topological superconductivity and MFs, which is aimed at both
experimental and theoretical physicists without prior knowledge of the field. We explain the generic necessary
ingredients of topological superconductivity, as well as the basic properties of MFs, and how they can be realized in standard semiconductors
proximity-coupled to $s$-wave superconductors. Two excellent review articles discussing the subject have appeared
recently~\cite{Alicea12rev, Beenakker11rev}. Therefore, we do not attempt to give an exhaustive review of the subject, or to provide detailed derivations
of all results, but instead focus on providing physical insight into what we feel are the most important basic concepts.
We also try to convey the excitement over MFs by briefly discussing some of their exotic properties, such as the non-abelian
statistics and the resulting potential for topological quantum computation.
The text should be accessible for all readers with a basic knowledge of quantum mechanics and the second quantization formalism.

\section{Majorana fermions in $p$-wave superconductors}
We start our discussion by introducing a simple Hamiltonian, describing a spinless $p$-wave superconductor, which has eigenstates
which are spatially isolated MFs.
It is most intuitive to start from a 1D tight-binding chain with $p$-wave superconducting pairing,
as first introduced by Kitaev~\cite{Kitaev01}, described by the Hamiltonian
\begin{eqnarray}\label{eq:kitaevH}
	\mathcal{H}_\mathrm{chain} = -\mu \sum_{i=1}^N n_i - \sum_{i=1}^{N-1} \left( t c_i^\dagger c_{i+1} + \Delta c_i c_{i+1} + h.c. \right),
\end{eqnarray}
where $h.c.$ means hermitian conjugate, $\mu$ is the chemical potential, $c_i$ is the electron annihilation operator for site $i$,
and $n_i = c_i^\dagger c_i$ is the associated number operator. The superconducting gap, $\Delta$, and hopping, $t$,
are assumed to be the same for all sites.
We can then choose the superconducting phase $\phi$ to be zero, such that $\Delta = |\Delta|$.
Note that time-reversal symmetry is broken in Eq.~(\ref{eq:kitaevH}) since we only consider one value for the spin projection,
i.e., effectively spinless electrons (we suppress the spin label). Furthermore, the superconducting pairing is non-standard since it
couples electrons with the same spin (in contrast to standard $s$-wave pairing, which only couples electrons with
opposite spin projection). Note also that electrons on neighboring sites are paired: The sites cannot be doubly
occupied by the spinless electrons because of the Pauli exclusion principle.

We now want to rewrite Eq.~(\ref{eq:kitaevH}) in terms of Majorana operators (we will see shortly why this is useful).
It was mentioned above that MFs are basically obtained by splitting a fermion into its real and imaginary parts.
Therefore we write
\begin{eqnarray}
\label{eq:kitaevMFannihilation}
	c_i = \frac{1}{2} \left( \gamma_{i,1} + i \gamma_{i,2} \right), \\
\label{eq:kitaevMFcreation}
	c_i^\dagger = \frac{1}{2} \left( \gamma_{i,1} - i \gamma_{i,2} \right),
\end{eqnarray}
where $\gamma_{i,j}$ are Majorana operators living on site $i$.
That they are indeed Majorana operators is seen by inverting
Eqs.~(\ref{eq:kitaevMFannihilation})--(\ref{eq:kitaevMFcreation}), giving
\begin{eqnarray}
\label{eq:kitaevMFinv1}
	\gamma_{i,1} = c_i^\dagger + c_i, \\
\label{eq:kitaevMFinv2}
	\gamma_{i,2} = i \left( c_i^\dagger - c_i \right),
\end{eqnarray}
which are clearly hermitian and therefore Majorana operators.
Figure~\ref{fig:kitaev} shows a sketch of Kitaev's chain and the upper panel indicates how the fermion operators on each
site are split into Majorana operators.
\begin{figure}[h!]
  \includegraphics[height=0.32\linewidth]{chain.eps}
\begin{centering}
	\caption{\label{fig:kitaev}
	Sketch of Kitaev's 1D $p$-wave superconducting tight binding chain.
	Upper panel: The fermion operators on each site $i$ of the chain can be split into two Majorana operators,
		   $\gamma_{i,1}$ and $\gamma_{i,2}$.
	Lower panel: In the limit $\mu = 0$, $t = \Delta$, the Hamiltian is diagonal in fermion operators which are
		   obtained by combing instead Majorana operators on neighboring sites, $\gamma_{i+1,1}$ and $\gamma_{i,2}$.
		   This leaves two unpaired Majorana operators, $\gamma_{1,2}$ and $\gamma_{N,1}$, which can be combined to form
		   one zero energy, highly non-local fermion operator, $\tilde{c}_M$.}
\end{centering}
\end{figure}

The Majorana physics is most easily understood when $\mu =0$, $t = \Delta$, in which case inserting
Eqs.~(\ref{eq:kitaevMFannihilation})--(\ref{eq:kitaevMFcreation}) into the Hamiltonian~(\ref{eq:kitaevH}) results in
\begin{eqnarray}\label{eq:kitaevHMF}
	\mathcal{H}_\mathrm{chain} = -i t \sum_{i=1}^{N-1} \gamma_{i,2} \gamma_{i+1,1}.
\end{eqnarray}
In fact, Eq.~(\ref{eq:kitaevHMF}) is nothing but an alternative way of writing the diagonalized Hamiltonian.
To see this we go back to a fermionic representation by noting that, analogous to
Eqs.~(\ref{eq:kitaevMFannihilation})--(\ref{eq:kitaevMFcreation}), where a fermion
on site $i$ was split into two Majorana operators living on site $i$, we can construct new fermion operators, $\tilde{c}_i$,
by combining Majorana operators on neighboring sites
\begin{eqnarray}
	\label{eq:kitaevnewfermion}
	\tilde{c}_i = ( \gamma_{i+1,1} + i \gamma_{i,2} ) / 2.
\end{eqnarray}
This pairing is demonstrated in the lower panel of Fig.~\ref{fig:kitaev}.
In terms of these new fermions we find $-i \gamma_{i,2} \gamma_{i+1,1} = 2 \tilde{c}_i^\dagger \tilde{c}_i = 2 \tilde{n_i}$
and therefore
\begin{eqnarray}\label{eq:kitaevHdiag}
	\mathcal{H}_\mathrm{chain} = 2 t \sum_{i=1}^{N-1} \tilde{c}_i^\dagger \tilde{c}_i.
\end{eqnarray}
Thus, $\tilde{c}_i$ are the annihilation operators corresponding to the eigenstates and the energy cost of
creating a $\tilde{c}_{i}$ fermion is $2t$.
The Majorana operators are merely a formal way of writing the Hamiltonian and the physical
excitations are fermionic states at finite energy, obtained by a superposition of nearest neighbor MFs.

So far there appears to be nothing special about the diagonal Hamiltonian in Eqs.~(\ref{eq:kitaevHMF}) and~(\ref{eq:kitaevHdiag}).
However, the Majorana operators $\gamma_{N,2}$ and $\gamma_{1,1}$, which are localized at the two ends of the wire,
are completely missing from Eq.~(\ref{eq:kitaevHMF})!
These two Majorana operators can equivalently be described by a single fermionic state with operator
\begin{eqnarray}\label{eq:kitaevfermion}
	\tilde{c}_M = ( \gamma_{N,2} + i \gamma_{1,1} ) / 2.
\end{eqnarray}
This is a highly non-local state since $\gamma_{N,2}$ and $\gamma_{1,1}$ are localized on opposite ends of the chain.
Furthermore, since this fermion operator is absent from the Hamiltonian, occupying the corresponding state requires zero energy.
Thus, in contrast to "normal" superconductors, where the ground state is non-degenerate and consists of a superposition
of even-particle-number states (condensate of Cooper pairs), the Hamiltonian~(\ref{eq:kitaevH}) allows for an odd number
of quasiparticles at zero energy cost. The ground state is therefore two-fold degenerate, corresponding to having in total
an even or odd number of electrons in the superconductor. This even or oddness, also called parity, corresponds to the
eigenvalue of the number operator of the zero-energy fermion, $n_M = \tilde{c}_M^\dagger \tilde{c}_M = 0 (1)$
for even (odd) parity.

The above argument was made for the very special case $\Delta = t$ and $\mu = 0$, but one can show that the Majorana end
states remain as long as the chemical potential lies within the gap~\cite{Kitaev01}, $|\mu| < 2t$.
In the general case, however, the MFs are not completely localized only at the two edge sites of the wire, but decay exponentially away
from the edges. The MFs remain at zero energy only if the wire is long enough that they do not overlap.

The Hamiltonians for the continuum version of a $p$-wave superconductor in 1D and 2D are
\begin{eqnarray}\label{eq:pgap1d}
	\mathcal{H}^\mathrm{pw}_{\mathrm{1D}} &=& \int d x \; \left[ \Psi^\dagger (x) \left( \frac{p_x^2}{2 m} - \mu \right) \Psi (x) +
				    \Psi (x) |\Delta| e^{i\phi} p_x \Psi(x) + h.c. \right], \\
	\label{eq:pgap2d}
	\mathcal{H}^\mathrm{pw}_{\mathrm{2D}} &=& \int d^2r \; \left[ \Psi^\dagger (\mathbf{r}) \left( \frac{\mathbf{p}^2}{2 m} - \mu \right)
				     \Psi (\mathbf{r}) + \Psi (\mathbf{r}) |\Delta| e^{i\phi}
				     \left( p_x \pm ip_y \right) \Psi(\mathbf{r}) + h.c. \right], \nonumber \\
\end{eqnarray}
where $\Psi^\dagger (\mathbf{r})$ is the real-space creation operator, $\mathbf{p}$ is the momentum, $m$ is the effective electron mass,
and we have re-introduced the superconducting phase $\phi$.
As we saw above, in the 1D case MFs appear at the edges of the wire. They will also appear at transition points between topological and
non-topological regions.
For example, if the chemical potential or the hopping amplitude varies along the wire and $|\mu| > 2t$ in some segment, two additional
MFs will appear at the transition points where the gap closes~\cite{Oreg10}.

Similarly, in a 2D $p_x \pm ip_y$-wave superconductor, MFs appear in vortices in the
superconducting pairing potential~\cite{Read00, Ivanov01, Kraus09}.
Alternatively, MFs can appear if the gap is closed by variations in the chemical potential or electrostatic potential~\cite{Wimmer10}.
It is worth noting that isolated MFs can appear even in a
spinful $p_x \pm ip_y$-wave superconductor, in so-called half-quantum vortices~\cite{Ivanov01}, where there is a vortex for only one direction of
the triplet (note that this also means a breaking of time-reversal symmetry).

If there are in total an odd number of gap closings, an additional MF will appear somewhere in the system
to guarantee that there is always an even number (exactly where this additional MF appears depends on the details,
see Ref.~\cite{Alicea12rev} for a detailed discussion).

\section{Properties of Majorana fermions}
Before explaining how MFs can be realized in more realistic systems, we now discuss in more detail some of their
generic properties, which are independent of the specific system in which they appear.

\subsection{Representation in terms of fermionic operators}
Let us assume that we have a system with $2N$ spatially well-separated MFs, $\gamma_1, \ldots \gamma_{2N}$.
The number of MFs is necessarily even since one MF contains half the degrees of freedom of a normal fermion.
Similar to the case of Kitaev's chain, the Majorana operators are obtained by
splitting a normal fermion $f_i$ in its real and imaginary parts (cf., Eq.~(\ref{eq:kitaevfermion}))
\begin{eqnarray}\label{eq:fermion}
	f_i &=& (\gamma_{2i-1} + i \gamma_{2i})/2.
\end{eqnarray}
The inverse relation is then
\begin{eqnarray}
\label{eq:inversefermion1}
	\gamma_{2i-1} &=& f^\dagger + f, \\
\label{eq:inversefermion2}
	\gamma_{2i} &=& i(f^\dagger - f).
\end{eqnarray}
Obviously, the Majorana operators are hermitian, $\gamma_j = \gamma_j^\dagger$. Using the fermionic anti-commutation relations for the
$f_i$-fermions, it is easily verified that the Majorana operators satisfy the anti-commutation relation
\begin{eqnarray} \label{eq:anticommutation}
	\{\gamma_i, \gamma_j\} = 2 \delta_{ij},
\end{eqnarray}
which is somewhat reminiscent of normal fermions. There are, however, important differences.
From Eq.~(\ref{eq:anticommutation}), we see that $\gamma_i^2 = 1$. Thus, acting twice with a Majorana operator, we get back the
same state we started with. There is therefore no Pauli principle for MFs (cf., $c^2 = (c^\dagger)^2 = 0$ for normal
fermion operators $c$).
In fact, we cannot even speak of the occupancy of a Majorana mode. We can try to count the occupancy by constructing a
"Majorana number operator", $n_i^\mathrm{MF} = \gamma_i^\dagger \gamma_i$.
However, using hermiticity, $\gamma_i^\dagger = \gamma_i$, together with $\gamma_i^2 = 1$, we find $n_{i}^\mathrm{MF} \equiv 1$.
Similarly, $\gamma_i \gamma_i^\dagger \equiv 1$. Thus, the Majorana mode is in a sense always empty and always filled and counting
does not make any sense.

It would still be very useful to use some sort of number states. These are instead provided through the $f_i$ fermions as
we already saw above in the example with Kitaev's chain.
Since these are normal fermions, there are number states $|n_1, \ldots, n_N\rangle$ which are eigenstates
of the fermionic number operators, $n_i = f_i^\dagger f_i$, with eigenvalue $n_i = 0, 1$ (by Pauli exclusion principle).
Note that the labelling of the $\gamma$'s, and thereby the pairing into fermionic states, is arbitrary and merely represents a
choice of basis for the number states. However, if two MFs come close enough to overlap, it is natural to choose to combine them into
a fermion. To describe an overlap, $t$, between $\gamma_{2i-1}$ and $\gamma_{2i}$, the only term one can introduce into the Hamiltonian is
\begin{eqnarray}\label{eq:overlap}
	\frac{i}{2} t \gamma_{2i-1} \gamma_{2i} &=& t \left(n_i - \frac{1}{2}\right),
\end{eqnarray}
which corresponds to a finite energy cost for occupying the corresponding fermionic state ($t>0$).
If the MFs do not overlap, the groundstate is $2^N$-fold degenerate, corresponding to each $n_i$ being equal to zero or one.
The sum of all occupation numbers, $\sum_{i=1}^{N} n_i$, being even or odd now reflects whether the total number of electrons
in the superconductor is even or odd (even or odd parity). To change the parity, electrons have to be physically added
to or removed from the superconductor.

The experimentally measurable quantities are the occupation numbers $n_i$. Such a measurement
can be done by bringing two MFs close together and measuring the energy of the corresponding state~\cite{Kitaev03}, which
reveals the occupation through Eq.~(\ref{eq:overlap}). One could also use interferometry~\cite{Bishara09} or coupling to
conventional superconducting qubits~\cite{Hassler10}.
Note that it does not make sense to talk about the "state of a MF" since a single MF contains only "half a degree of freedom".
The only physical observables are the fermionic occupation numbers.

\subsection{Non-abelian statistics}
It is a crucial ingredient for non-abelian statistics to have a degenerate groundstate, which is separated from all excited states
by a gap (ideally the induced superconducting gap, but it could also be a smaller gap to for example finite-energy vortex- or edge
states). Then adiabatic operations, such as the slow exchange of quasiparticle positions, can in principle bring the system from one
groundstate to another. It is, of course, not obvious that such a transformation indeed takes place, which will depend on the details
of the system. In the case of MFs in a $p_x \pm ip_y$ superconductor, Ivanov~\cite{Ivanov01} provided a simple and elegant proof of
the non-abelian statistics which we sketch here.
(The Supplementary Information of Ref.~\cite{Alicea10b} provides a proof in the
case of 1D wires, where MFs can be moved using closely spaced electronic "keyboard" gates and particle exchange is made possible
by connecting the 1D wires in T-junctions.)

Imagine that we have two vortices
in a two-dimensional topological superconductor, hosting MFs described by the operators $\gamma_1$ and $\gamma_2$ at the vortex cores,
see Fig.~\ref{fig:braid}. Each vortex is associated with a winding of $2\pi$ of the superconducting phase $\phi$.
We can choose $\phi$ to be single-valued everywhere, except for at branch cuts (red dashed lines in Fig.~\ref{fig:braid})
emanating from each vortex, such that $\phi$ changes by $2 \pi$ when crossing this
line (the direction of the branch cuts can be chosen arbitrarily).
Vortices could perhaps be moved using the tip of a scanning tunneling microscope, or by local magnetic gates.
If we now exchange vortices one and two in a clockwise manner
as indicated in Fig.~\ref{fig:braid}(a), vortex 1 crosses a branch cut and acquires a $2 \pi$ phase shift, while vortex 2 does not acquire a phase.
\begin{figure}[h!]
  \includegraphics[height=0.22\linewidth]{braid.eps}
\begin{centering}
	\caption{\label{fig:braid}
	Sketch of vortices in a 2D $p_x \pm ip_y$ superconductor, hosting MFs described by the operators $\gamma_1$ and $\gamma_2$.
	Inside the vortex core the superconducting gap vanishes, $\Delta \rightarrow 0$, and going around the vortex the phase of
	the superconducting condensate (Cooper pairs) increases by $2\pi$.
	Therefore we introduce branch cuts emanating from the vortex cores (red dashed lines), where the phase makes a
	$2\pi$ jump.
	(a) In a clockwise exchange, vortex one necessarily crosses the branch cut of vortex two.
	(b) When bringing vortex one around vortex two, both vortices cross the branch cut of the other vortex.}
\end{centering}
\end{figure}
The superconducting phase is the phase of the Cooper pairs in the condensate. The MF in vortex 1, which is made up
from single (rather than products of two) fermion operators, then acquires a phase of $\pi$ upon crossing the branch cut.
The result of this exchange operation is thus
\begin{eqnarray}
\label{eq:braid1}
	\gamma_1 &\rightarrow& - \gamma_2,\\
\label{eq:braid2}
	\gamma_2 &\rightarrow& + \gamma_1.
\end{eqnarray}
This transformation is described by $\gamma_i \rightarrow B_{12} \gamma_i B_{12}^\dagger$, where the so-called braid operator
is given by
\begin{eqnarray}\label{eq:exchange}
	B_{12} &=& \frac{1}{\sqrt{2}} \left( 1 + \gamma_1 \gamma_2 \right).
\end{eqnarray}
This choice of operator is made unique by requiring that, in a system with more than two vortices (and therefore MFs),
all the others are unaffected by the exchange of vortices one and two. Note that an anti-clockwise exchange instead results in
$\gamma_1 \rightarrow  \gamma_2$, $\gamma_2 \rightarrow - \gamma_1$, which is described by the operator
$\tilde{B}_{12} = ( 1 - \gamma_1 \gamma_2 ) / \sqrt{2}$. Of course, if we had chosen the branch cuts in a different direction the situation
could be reversed, but this does not matter as long as we make a choice and stick with that.

Next, we want to discuss the effect of bringing vortex one around vortex two and back to its original position. Topologically,
this is equivalent to two
successive exchanges. Thus, the associated operator is given by $B_{12}^2 = \gamma_1 \gamma_2$, leading to the transformation
\begin{eqnarray}
\label{eq:exchangesquare1}
	\gamma_1 \rightarrow \left( \gamma_1 \gamma_2 \right) \gamma_1 \left( \gamma_1 \gamma_2 \right)^\dagger = -\gamma_1, \\
\label{eq:exchangesquare2}
	\gamma_2 \rightarrow \left( \gamma_1 \gamma_2 \right) \gamma_2 \left( \gamma_1 \gamma_2 \right)^\dagger = -\gamma_2.
\end{eqnarray}
Bringing vortex one around vortex two thus has the effect of introducing a minus sign into each Majorana operator.
An alternative way of deriving Eqs.~(\ref{eq:exchangesquare1})--(\ref{eq:exchangesquare2}) is sketched in Fig.~\ref{fig:braid}(b).
When bringing vortex one around vortex two, it necessarily crosses the branch cut of vortex two, but in addition forces
vortex two to cross the branch cut of vortex one. Therefore, each Majorana operator acquires a phase shift of $\pi$.

We now go back to the case of vortex exchange. The effect of the braid operator acting on the number states is
\begin{eqnarray}
\label{eq:exchange0}
	B_{12} |0\rangle =  \frac{1}{\sqrt{2}} \left(1 + i \right) |0 \rangle, \\
\label{eq:exchange1}
	B_{12} |1\rangle =  \frac{1}{\sqrt{2}} \left(1 - i \right) |1 \rangle,
\end{eqnarray}
where $|1\rangle = f_1^\dagger |0\rangle$, with $f_1 = (\gamma_1 + i \gamma_2)/2$ as discussed above.
Thus, exchanging the two MFs has the rather mundane effect of multiplying the number states with a phase factor.

In fact, it is natural that the exchange operation cannot change the eigenvalue of the number operator in a system with
only two MFs, since this encodes whether
there are in total an even or odd number of particles in the superconductor, a quantity which is not changed by particle exchanges.
To find non-trivial effects of exchange operations we must consider a system with at least four MFs, described in terms of the
fermionic number states $|n_1 n_2\rangle$.
Let us now investigate the effect of exchanging neighboring MFs, described by braid operators $B_{i, i+1}$.
For simplicity we choose the branch cuts of all MFs to be in the same direction and number the MFs
based on their position orthogonal to this direction, such that when exchanging MFs $i$ and $i + 1$ in a clockwise manner,
vortex $i$ crosses \emph{only} the branch cut of vortex $i+1$, and no other vortices cross any branch cuts (crossing the same
branch cut twice in different directions is equivalent to not crossing any branch cuts at all).
Note that MFs which are not neighbors can always be exchanged through a sequence of neighbor exchanges.
Consider now the effect of braid operations on the number states
\begin{eqnarray}
\label{eq:exchange12}
	B_{12} |00 \rangle &=& \frac{1}{\sqrt{2}} \left( 1 + i \right) |00\rangle, \\
\label{eq:exchange23}
	B_{23} |00 \rangle &=& \frac{1}{\sqrt{2}} \left( |00\rangle + i |11\rangle \right), \\
\label{eq:exchange34}
	B_{34} |00 \rangle &=& \frac{1}{\sqrt{2}} \left( 1 + i \right) |00\rangle,
\end{eqnarray}
with analogous results for the other number states. Note especially that $B_{23}$, which involves MFs from different fermions, produces
a superposition state of different number states. However, the \emph{total} parity ($n_1 + n_2$ being even or odd) of each state in the
superposition must be the same.

With four MFs we can also demonstrate the non-abelian nature of braid operations (with two MFs there is only one possible exchange operation).
In general, two braid operations commute whenever they do not involve any of the same MFs, $[B_{12}, B_{34}] = 0$. This is easy to believe
on physical grounds, as there is no reason that the exchange of MFs three and four should care about whether MFs one and two have been
exchanged. However, whenever two exchanges involve some of the same MFs, the braid operators do not commute
\begin{eqnarray}\label{eq:noncommuting}
	\left[ B_{i-1,i}, B_{i, i+1} \right] &=& \gamma_{i-1} \gamma_{i+1}.
\end{eqnarray}
Equation~(\ref{eq:noncommuting}) expresses the non-abelian exchange statistics of MFs.

At this point the attentive reader might be slightly upset by a simple fact we have neglected. Namely the question of what
exactly qualifies as an exchange operation. If we define an exchange operation as bringing one vortex exactly to the old
position of another vortex, and vice versa, there is no problem. But clearly this is not possible in reality and certainly
goes against the idea of robust topological quantum information processing to be discussed below. (In networks of 1D wires it
is somewhat easier to find a satisfying definition of particle exchanges~\cite{Alicea10b}.)
Mathematically, the exchange process
happens when the branch cut is crossed, but since this is arbitrarily defined, it is not a good definition either. Physically,
the solution to this problem is that there is no measurable effect of the exchange process, unless it is followed by one of the
two MFs involved in the exchange being joined with a third MF to perform a measurement of the state of the fermion formed by
this pair.
This is demonstrated in Fig.~\ref{fig:exchange2}.
\begin{figure}[h!]
  \includegraphics[height=0.45\linewidth]{exchange2.eps}
	\caption{\label{fig:exchange2}
	Sketch demonstrating two equivalent sets of operations. In the upper panel, MFs 2 and 3 are first exchanged (black arrows),
	then the nearest neighbor MFs are brought together (magenta arrows) and the states of the corresponding fermions
	are measured.
	In the lower panel, there is no exchange, but instead we directly measure the fermions formed by pairing next-nearest
	neighboring MFs ($1 + 3$ and $2 + 4$).}
\end{figure}
In the upper panel, two neighboring MFs are first exchanged,
which is followed by measuring the fermionic states formed by pairing the nearest neighbor MFs.
In the lower panel, on the other hand, we do not exchange the MFs, but instead measure directly the fermionic states
formed by pairing next-nearest neighboring MFs.
Both these operations give the same result for the measurements of the fermionic states and are therefore equivalent.
Therefore, a "computation" can be defined either as a set of exchanges, or by defining the combinations of pairs that are being
measured at the end, which removes the ambiguity in defining an exchange process.

Another way of seeing this is to note that the result of a set of exchange operations depends on the topology of the "world lines"
of the MFs (or other particles). The world lines start when pairs of MFs are created from the vacuum, cross each other when MFs are exchanged, and
end when pairs of MFs are measured (fused). Only sets of world lines which are topologically distinct can lead to different results, which defines the
so-called braid group, of which the MF exchange statistics is one possible representation (fermions, bosons, and abelian anyons in general are the
more trivial one-dimensional representations). Non-abelian representations of the braid group exist only in 2D (networks of 1D wires are also
effectively 2D). Ref.~\cite{Nayak08rev} provides a more complete mathematical discussion of these issues.

\subsection{Majorana qubits and topological quantum computation}
We saw above that parity conservation prevented braiding operations from changing the state of a system with only two MFs.
For this reason, the two-level system spanned by the number operator $n_1$ is not suitable to use as a qubit.
To define a (topological) Majorana qubit, we should therefore use four MFs, meaning two normal fermions~\cite{Bravyi06}, and consider
the case of fixed parity (even or odd) of the total number of fermions.
Let us consider the even parity subspace and define a
qubit by $|\bar{0} \rangle \equiv |00\rangle$, $|\bar{1} \rangle \equiv |11\rangle$.

In the basis $\{\bar{0},\bar{1}\}$, the Pauli matrices can be represented in terms of products of Majorana operators
\begin{eqnarray}
\label{eq:pauliMFz}
	-i \gamma_1 \gamma_2 = \sigma_z, -i \gamma_3 \gamma_4 = \sigma_z \\
\label{eq:pauliMFx}
	-i \gamma_2 \gamma_3 = \sigma_x, \\
\label{eq:pauliMFy}
-i \gamma_1 \gamma_3 = \sigma_y, -i \gamma_2 \gamma_4 = \sigma_y
\end{eqnarray}
which is seen by calculating the corresponding matrix elements.
In the standard representation with $|\bar{0}\rangle$ and $|\bar{1}\rangle$ being respectively the north and south poles of the block
sphere, we can then identify the different braids with single-qubit rotations
\begin{eqnarray}
\label{eq:rotz}
	B_{12} = B_{34} = e^{-\frac{i\pi}{4}\sigma_z}, \\
\label{eq:rotx}
	B_{23} = e^{-\frac{i\pi}{4} \sigma_x},
\end{eqnarray}
Thus, by braiding operations we can only perform single-qubit rotations by an angle $\pi/2$.

When considering a multi-qubit setup, the most obvious choice is to define each qubit in terms of four MFs~\cite{Bravyi06}.
However, it is not possible to construct a two-qubit gate based on braiding operations which is able to create entanglement
between two such qubits. In addition, this choice does not use all the degrees of freedom offered by the system.
Even with conservation of the parity of the total number of fermions, a system with $2N$ MFs has in principle enough degrees of freedom to
store $N-1$ qubits. Braiding-based gates acting on such "overlapping" qubit systems have been considered in Ref.~\cite{Georgiev06}.
No matter how the Majorana-based qubits are defined, braiding operations can only explore a tiny fraction of the total Hilbert space
and are insufficient for universal quantum computation, which can, however, be
achieved by including also non-protected operations~\cite{Kitaev03, Nayak08rev}, or by coupling Majorana qubits to other
qubit systems~\cite{Hassler10, Sau10b, Jiang11, Bonderson11, Leijnse11top}.

The advantage of Majorana-based qubits is that they keep the quantum information encoded in delocalized fermionic states.
Therefore, they are expected to be robust against most sources of decoherence which do not couple simultaneously to more
than one Majorana mode, i.e., decoherence requires perturbations of the form $\gamma_i \gamma_j$, which are suppressed
when MFs $i$ and $j$ are spatially separated.
An exception is processes which change the total parity of the superconductor, e.g., by electrons tunneling into a
Majorana mode~\cite{Leijnse11, Budich12, Rainis12}.
Such perturbations involve a single Majorana operator (they are $\propto\gamma_i$)
and are not suppressed by
keeping the MFs spatially separated. In fact, this phenomena, known as quasiparticle
poisoning~\cite{Mannik04, Aumentado04}, is a well-known problem in conventional (non-topological) superconducting qubits.

\section{Proximity-induced superconductivity in spin-orbit semiconductors}
Having seen how Majorana fermions appear in spinless $p$-wave (or $p_x \pm ip_y$-wave) superconductors,
we will now investigate how such exotic pairing can be engineered using more readily available ingredients.
We start, however, by considering the generic effects of proximity-induced superconductivity.

The system we have in mind is a $D$-dimensional semiconductor, where $D=1$ (wire) or $D=2$ (quantum well).
Neglecting electron--electron interactions (or including them in a mean-field manner), the system is described
by the Hamiltonian
\begin{eqnarray}\label{eq:H0}
	\mathcal{H}_0 &=& \sum_{\sigma = \uparrow,\downarrow}\int d^D r \; \Psi_\sigma^\dagger(\mathbf{r}) H_0(\mathbf{r}) \Psi_\sigma(\mathbf{r}),
\end{eqnarray}
where the first quantization single-particle Hamiltonian is given by
\begin{eqnarray}\label{eq:H0r}
	H_0(\mathbf{r}) &=& \frac{\mathbf{p}^2}{2 m} - \mu + V(\mathbf{r}) +
			    \alpha \left( \mathbf{E}(\mathbf{r}) \times \mathbf{p} \right) \cdot \bar{\sigma} +
			    \frac{1}{2} g \mu_B \mathbf{B}(\mathbf{r}) \cdot \bar{\sigma},
\end{eqnarray}
where $m$ is the effective electron mass, $\mathbf{B}$ is an applied magnetic field, $\mu_B$ is the Bohr magneton, $g$ is the Land\'{e} $g$ factor,
and $\bar{\sigma}$ is a vector of Pauli matrices.
The spin-orbit interaction with strength $\alpha$ has been written in the most general form in terms of the electric field
$\mathbf{E}$ felt by the valence electrons, and can involve both Rashba and Dresselhaus terms.

If a good interface is made between a semiconductor and a superconductor, electrons can tunnel between these two systems.
The effect is that the electrons in the semiconductor feels an effective "proximity-induced" superconducting pairing
field~\cite{deGennes64, Doh05, vanDam06}. The strength of this pairing field depends on the details of the semiconductor and
superconductor, as well as the interface. We do not attempt to make an accurate microscopic model, but instead include
the pairing effect in the semiconductor by the phenomenological Hamiltonian
\begin{eqnarray}\label{eq:HS}
	\mathcal{H}_S &=& \int d^D r \; d^D r' \; \Psi_\downarrow(\mathbf{r}) \Delta(\mathbf{r}, \mathbf{r'})
		\Psi_\uparrow(\mathbf{r'}) + h.c.,
\end{eqnarray}
where $\Delta(\mathbf{r}, \mathbf{r'})$ is the pairing potential.
The pairing symmetry is inherited from the superconductor, which we have here assumed to be $s$-wave,
inducing singlet pairing between spin-up and spin-down electrons.

When dealing with superconducting systems, it is standard practice to include both the electrons and the holes explicitly
by introducing so-called Nambu spinors
\begin{eqnarray}\label{eq:nambu}
	\bar{\Psi}(\mathbf{r}) = \left( \begin{array}{c}\Psi_\uparrow(\mathbf{r}) \\
				  			\Psi_\downarrow(\mathbf{r}) \\
							\Psi_\downarrow^\dagger(\mathbf{r}) \\
							-\Psi_\uparrow^\dagger(\mathbf{r}) \end{array} \right).
\end{eqnarray}
Matrices acting on the Nambu spinors must have dimension $4\times4$ and we introduce Pauli matrices $\tau_i$,
similar to $\sigma_i$, but acting in electron-hole space. Matrices such as $ \tau_j \otimes \sigma_i$ then have the
appropriate $4\times4$ structure.
The total Hamiltonian can be written as
\begin{eqnarray}\label{eq:H}
	\mathcal{H} &=& \mathcal{H}_0 + \mathcal{H}_S \nonumber \\
		    &=& \frac{1}{2} \int d^D r \; d^D r' \;
			\bar{\Psi}^\dagger(\mathbf{r}) \left[ \bar{H}_0(\mathbf{r}) \delta(\mathbf{r} - \mathbf{r'})
			 +  \bar{\Delta}(\mathbf{r}, \mathbf{r'}) \right] \bar{\Psi}(\mathbf{r}),
\end{eqnarray}
where
\begin{eqnarray}
\label{eq:H0bar}
	\bar{H}_0(\mathbf{r})  &=&  \left( \begin{array}{cc} H_0(\mathbf{r}) & \hat{0}_\sigma \\ \hat{0}_\sigma & - \sigma_y H_0^* (\mathbf{r}) \sigma_y \\
				  \end{array} \right), \\ \nonumber \\
\label{eq:Deltabar}
        \bar{\Delta}(\mathbf{r}, \mathbf{r'})  &=&  \left( \begin{array}{cc}
				  \hat{0}_\sigma & \Delta^*(\mathbf{r}, \mathbf{r'}) \hat{1}_\sigma \\
				  \Delta(\mathbf{r}, \mathbf{r'}) \hat{1}_\sigma & \hat{0}_\sigma \\ \end{array} \right).
\end{eqnarray}
Note that these are $4\times4$ matrices since $H_0(\mathbf{r})$ is a $2\times2$ matrix and $\hat{0}_\sigma$ and $\hat{1}_\sigma$
denotes respectively the zero and unit matrices in spin space.
The term $-\sigma_y H_0^* (\mathbf{r}) \sigma_y$ in Eq.~(\ref{eq:H0bar}) denotes the time-reversal of $H_0(\mathbf{r})$ and appears since
holes are time-reversed electrons. (Note, however, that we have not introduced any new physics with the matrices in
Eqs.~(\ref{eq:H0bar})--(\ref{eq:Deltabar}), they are simply defined to give
the correct total Hamiltonian, given by Eq.~(\ref{eq:H0}) + Eq.~(\ref{eq:HS}).)

The quasiparticle excitations of a superconducting system are given by solving the Bogoliubov-de Gennes equations for the eigenstates
$\bar{\psi}_i(\mathbf{r})$ (which are also four-component spinors)
\begin{eqnarray}\label{eq:BdG}
	\bar{H}_0 (\mathbf{r}) \bar{\psi}_i(\mathbf{r}) + \int d^D r' \; \bar{\Delta}(\mathbf{r}, \mathbf{r'}) \bar{\psi}_i(\mathbf{r'}) =
		E_i \bar{\psi}_i(\mathbf{r}),
\end{eqnarray}
and the diagonalized Hamiltonian becomes
\begin{eqnarray}
\label{eq:Hdiag}
	\mathcal{H} &=& \frac{1}{2} \sum_i  E_i \; \Psi_i^\dagger \Psi_i, \\
\label{eq:eigenop}
	\Psi_i &=& \int d^D r \; \bar{\psi}_i (\mathbf{r}) \cdot \bar{\Psi}(\mathbf{r}).
\end{eqnarray}

As was mentioned above, the Nambu spinors were only introduced to provide a convenient matrix representation of the Hamiltonian.
However, by explicitly including the hole states and thus doubling the dimension of the Hamiltonian, we have also artificially
doubled the number of eigenstates. Therefore, there must be some symmetry relation between the eigenstates such that the number
of independent solutions remains the same. This symmetry is electron-hole symmetry, which is expressed through the operator
\begin{eqnarray}\label{eq:Pop}
	P =  \tau_y \otimes \sigma_y K = \left( \begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0
					                         \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0\end{array} \right) K,
\end{eqnarray}
where $K$ is the operator for complex conjugation. It can then be verified that
\begin{eqnarray}\label{eq:ehsymm}
	P \bar{H}_0 (\mathbf{r}) P^\dagger &=& -\bar{H}_0 (\mathbf{r}), \\
	P \bar{\Delta} (\mathbf{r}, \mathbf{r'}) P^\dagger &=& -\bar{\Delta} (\mathbf{r}, \mathbf{r'}).
\end{eqnarray}
This means that if $\bar{\psi}_{i}(\mathbf{r})$ and $\Psi_i$ are solutions with positive energy $E_i$, then there
are also solutions $\bar{\psi}_{j}(\mathbf{r})$ and $\Psi_j$ with energy $E_j = -E_i$ which satisfy
\begin{eqnarray}
\label{eq:ehwavefuncsymm}
	\bar{\psi}_{j}(\mathbf{r}) = P \bar{\psi}_{i}(\mathbf{r}), \\
\label{eq:ehopsymm}
	\Psi_{j} = \Psi_{i}^\dagger.
\end{eqnarray}
In other words, creating a quasiparticle with energy $E$ or removing one with energy $-E$ are identical operations.

It is possible to engineer the parameters in the Hamiltonian~(\ref{eq:H}) such that it resembles a spinless $p$-wave
or $p_x \pm ip_y$ superconductor and has eigenstates which are MFs. We will discuss how to accomplish this in the following section,
but can already now derive a few of the generic properties Majorana solutions to Eq.~(\ref{eq:BdG}) must have,
\emph{if} they exist.
Let us assume that there is a solution to Eq.~(\ref{eq:BdG}) which corresponds to a Majorana operator
$\gamma_i \equiv \Psi_i = \Psi_i^\dagger$.
According to Eq.~(\ref{eq:ehopsymm}), this is only possible at $E=0$, so a MF
is a zero-energy solution to the Bogoliubov-de Gennes equations.
The corresponding real-space Majorana spinor
satisfies $\bar{\psi}_M(\mathbf{r}) = P \bar{\psi}_M(\mathbf{r})$, and its most general form is therefore
\begin{eqnarray}
\label{eq:realspaceMF}
	\bar{\psi}_M(\mathbf{r}) = \left( \begin{array}{c} f(\mathbf{r}) \\ g(\mathbf{r}) \\ g^*(\mathbf{r}) \\ -f^*(\mathbf{r}))
			     \end{array}\right).
\end{eqnarray}

Let us now assume that the Hamiltonian respects time-reversal symmetry, $H = T H T^\dagger$, where
the time-reversal operator is $T = i \hat{1}_\tau \otimes \sigma_y K$.
Then, if $\bar{\psi}_i(\mathbf{r})$ is an eigenstate, also $T \bar{\psi}_i(\mathbf{r})$ must be
an eigenstate with the same energy. This is nothing else than Kramer's theorem for a superconductor.
Specifically for MFs, the Kramer's partner to the Majorana spinor $\bar{\psi}_M(\mathbf{r})$ in
Eq.~(\ref{eq:realspaceMF}) is
\begin{eqnarray}
\label{eq:realspaceMFpartner}
	\bar{\psi}_{M'}(\mathbf{r}) = \left( \begin{array}{c} g^*(\mathbf{r}) \\ -f^*(\mathbf{r}) \\ -f(\mathbf{r}) \\ -g(\mathbf{r})
			     \end{array}\right).
\end{eqnarray}
Importantly, the probability densities associated with each Kramer's pair are identical
\begin{eqnarray}\label{eq:kramersspatial}
	|\bar{\psi}_M(\mathbf{r})|^2 = |\bar{\psi}_{M'}(\mathbf{r})|^2 = 2|g(\mathbf{r})|^2 + 2|f(\mathbf{r})|^2.
\end{eqnarray}
Thus, if we want to create spatially isolated MFs (not accompanied by a Kramer's partner with identical probability density),
we need a Hamiltonian which breaks time-reversal symmetry.

We note that it is possible for pairs of MFs to remain at zero energy in the presence of time-reversal symmetry even if they are
not spatially separated. For a full classification of different topological classes, see e.g.,~\cite{Ryu10, Fulga11}.

\section{Induced $p$-wave-like gap in semiconductors}
The strategy is now to engineer the parameters in the semiconductor Hamiltonians~(\ref{eq:H0})--(\ref{eq:HS}), such that
the total Hamiltonian~(\ref{eq:H}) is close enough to that of a spinless $p$-wave
superconductor (Eq.~(\ref{eq:pgap1d}))
to also host MFs (we focus here on the simplest case of a 1D nanowire, but the case of a 2D quantum well is rather similar).
The precise meaning of "close enough" is here that the two Hamiltonians can be continuously transformed into each other, without
ever closing the gap. In this case Eq.~(\ref{eq:H}) and Eq.~(\ref{eq:pgap1d}) describe topologically
equivalent systems. Since the presence of spatially separated MFs
is a topological property of the system~\cite{Hasan10}, they should then appear also as solutions to Eq.~(\ref{eq:H}).

A 1D semiconducting wire with strong spin-orbit coupling has been put forward as an experimentally attractive setting in which
to induce topological superconductivity.
We follow here closely the proposals originally put forward in Refs.~\cite{Oreg10, Lutchyn10}. The experimental geometry is
sketched in Fig.~\ref{fig:wire}.
\begin{figure}[h!]
  \includegraphics[height=0.42\linewidth]{wire.eps}
\begin{centering}
	\caption{\label{fig:wire}
	Sketch of setup for engineering topological superconductivity in a 1D nanowire.
	The nanowire (e.g., InAs or InSb) with strong spin-orbit coupling is proximity coupled to
	a bulk $s$-wave superconductor (e.g., Nb or Al). A set of gate electrodes are used to control the chemical potential inside the wire
	and bring it into the topological regime. MFs then form at the ends of the wire. The weight of the Majorana wavefunction decays
	exponentially inside the wire, indicated here in black (this is just an approximate form, the real wavefunction depends on the details
	and often exhibit oscillations). }
\end{centering}
\end{figure}
The nanowire is proximity-coupled to a $s$-wave superconductor and exposed to an external magnetic field (not shown). The chemical potential
of the wire is controlled by a set of gate electrodes. The wire is assumed to be long enough that we can ignore size quantization along the
wire direction and thin enough that the 1D subbands are well separated on the relevant energy scales. For simplicity, we also assume that
the chemical potential can be tuned to a regime where only a single 1D subband is occupied (MFs can also be found in multi-channel
wires~\cite{Potter10, Stanescu11, Fulga11} provided that the channel number is odd and the effective width of the wire is smaller
than the superconducting coherence length). The Hamiltonian is a special case of Eq.~(\ref{eq:H0r})
\begin{eqnarray}\label{eq:Hwire}
	H_{0}(x) = \frac{k_x^2}{2 m} - \mu + \tilde{\alpha} k_x \sigma_y + \frac{1}{2} \tilde{B} \sigma_z,
\end{eqnarray}
where we have taken $\hbar = 1$. $\tilde{\alpha} = \alpha E_\perp$, with $E_\perp$ the electric field perpendicular to the wire
direction, is the strength of the spin-orbit field originating
from Rashba spin-orbit coupling and $\tilde{B} = g \mu_B B$ is the Zeeman field. Note that the direction of the electric field
in a realistic setup is unknown due to the presence of the
superconductor and electric gates. However, since the spin-orbit field is given by a cross product of the electric field and the momentum,
it is orthogonal to the wire and we choose it to
be along the $y$-direction. Only the magnetic field component orthogonal to the spin-orbit field will help to induce topological
superconductivity and we choose the field to be along the $z$-axis.
We assume the proximity-induced pairing field to be homogeneous and to couple only electrons at the same position,
$\Delta(x,x') = \Delta \delta(x - x')$ in Eq.~(\ref{eq:HS}).

The eigenspectrum of Eq.~(\ref{eq:Hwire}) is shown in Fig.~\ref{fig:bands} as a function of the momentum along the wire.
\begin{figure}[h!]
  \includegraphics[height=0.65\linewidth]{bands.eps}
\begin{centering}
	\caption{\label{fig:bands}
	Bandstructure, $E(k_x)$, of a spin-orbit coupled nanowire with applied magnetic field, $\tilde{B}$.
	(a) $\tilde{B}=0$. The two parabolic bands are split depending on their spin polarization (red arrows)
	along the axis of the spin-orbit field.
	(b) Small $\tilde{B} > 0$. The field opens a gap at $k_x = 0$ and thereby a region without spin degeneracy.
	Each band holds only a single spin projection, but the direction of the spin polarization depends on the momentum.
	(c) Large $\tilde{B}$. The effectively spinless regime grows with magnetic field and also forces the spins within each band to
	increasingly align either parallel or anti-parallel to the field.
	(d) Same as (c), but with proximity-induced superconductivity, $\Delta > 0$. Here we plot all the solutions of the Bogoliubov-de
	Gennes equations (Eq.~\ref{eq:BdG}), giving twice the number of solutions. Note the symmetry between positive and negative
	$E$ due to the particle-hole symmetry.
	}
\end{centering}
\end{figure}
When $\tilde{B}=0$, see Fig.~\ref{fig:bands}(a), the spin-orbit coupling shifts the two parabolic bands depending on their spin polarization
along the axis of the spin-orbit field. However, at any given energy there is still spin degeneracy since time-reversal symmetry is
not broken.
Switching on a small $\tilde{B}$, see Fig.~\ref{fig:bands}(b), the crossing at zero momentum turns into an anti-crossing.
The lower band has a double-well shape and the gap
to the upper band is determined by $\tilde{B}$ (only the component of the magnetic field orthogonal to the spin-orbit field opens up a gap).
The Hamiltonian~(\ref{eq:Hwire}) is easily diagonalized, resulting in
\begin{eqnarray}\label{eq:bands}
	E_{\pm}(k_x) = \frac{k_x^2}{2m} - \mu \pm \sqrt{(\tilde{\alpha} k_x)^2 + \tilde{B}^2},
\end{eqnarray}
where $E_{+}(k_x)$ and $E_{-}(k_x)$ correspond to the upper and lower bands, respectively.
Inside the gap, there is only one effective spin direction (although this direction depends on
momentum). Therefore, if $\mu$ is placed inside the gap, spinless superconductivity can be induced by the proximity effect. Note that without the
$k_x$-dependence of the spin direction, it would not be possible to induce superconductivity by proximity with a $s$-wave superconductor,
since the pairing term in Eq.~(\ref{eq:HS}) only couples the components of the spins which are anti-parallel.
A larger $\tilde{B}$, as in Fig.~\ref{fig:bands}(c), increases the gap between the bands. This makes the effectively spinless regime larger and
therefore provides a larger window in which to place the chemical potential.
This is important if disorder causes the the chemical potential to vary over the length of the wire.
However, the larger field also enforces an increased
alignment of the spins within each band and therefore makes it harder to induce superconductivity.

Next we switch on also the proximity-induced superconducting pairing, $\Delta > 0$, see Fig.~\ref{fig:bands}(d).
Here we plot all the solutions of the Bogoliubov-de Gennes equations, Eq.~(\ref{eq:BdG}), which doubles the number of bands
(there are still only two independent solutions since particle-hole symmetry forces the positive and negative energy solutions to be
identical).
For small $\Delta$, the superconducting state is topological and associated with Majorana edge states, provided that
the chemical potential is placed within the spinless regime, $|\tilde{B}| > |\mu|$.
The gap at zero momentum decreases with increasing $\Delta$ and closes completely when $|\tilde{B}| = \sqrt{\Delta^2 + \mu^2}$.
For larger values of $\Delta$ the gap opens again, but now in a non-topological superconducting state.
The criteria for topological superconductivity is therefore
\begin{eqnarray}\label{eq:topcrit}
	|\tilde{B}| > \sqrt{\Delta^2 + \mu^2}.
\end{eqnarray}
The phase transition between the topological and non-topological superconducting states can \emph{only} take place at the
point where the gap closes. Therefore, to show that the criteria~(\ref{eq:topcrit}) indeed guarantees topological superconductivity,
it is sufficient to consider a certain limit of the topological phase and there provide a mapping to the spinless $p$-wave
superconducting wire in Eq.~(\ref{eq:pgap1d}) (and thereby to Kitaev's chain described by Eq.~(\ref{eq:kitaevH})).

Following Ref.~\cite{Alicea10b}, we consider the limit $|\tilde{B}| \gg E_{so}, |\Delta|$ and $\mu = 0$, where
$E_{so} = m \tilde{\alpha}^2 / 2$ is the spin-orbit energy.
With such a large magnetic field, the spins within each bands are nearly completely polarized and since the gap is large
we can consider a single-band model and ignore the higher band. Let $\Psi_{-}(x)$ be the annihilation operator
for electrons at position $x$ in the lower band of the wire. The bands are nearly spin polarized in the direction
of the magnetic field and therefore $\Psi_-(x) \approx \Psi_\downarrow (x)$. However, we need to take into account leading order
corrections due to spin-orbit couplings, since the induced superconductivity is $\propto \Psi_\downarrow \Psi_\uparrow (x)$.
The effective Hamiltonian for the lower band with induced superconcductivity then becomes
\begin{eqnarray}\label{eq:Heff}
	\mathcal{H}_\mathrm{eff} = \int dx \; \left[ \Psi^\dagger_-(x)\left( \frac{k_x^2}{2 m} - \frac{1}{2} \tilde{B} \right)\Psi_-(x)
				    + i \frac{\tilde{\alpha}}{\tilde{B}} \Delta \Psi_-(x) k_x \Psi_-(x) \right].
\end{eqnarray}
This is equivalent to the Hamiltonian~(\ref{eq:pgap1d}).
Note that the effective strength of the superconducting pairing term has been reduced by a factor $\tilde{\alpha} / \tilde{B}$,
corresponding to the polarization of the spin within the lower band, suppressing superconducting pairing.

From the above discussion and from Eq.~(\ref{eq:topcrit}), it may seem like an arbitrarily small proximity-induced gap is
sufficient for topological superconductivity. However, the stability of the superconducting phase depends also on the size
of the gap at finite momentum (see Fig.~\ref{fig:bands}(d)), which is $\propto \Delta$, see Eq.~(\ref{eq:Heff}).
If the gap is too small, the topological
superconducting phase, although reached in the idealized model considered here, will in reality be destroyed e.g., by finite
temperature or disorder.
Clearly it is important to induce a significant Zeemann splitting in the wire. Since this has to be done without destroying
superconductivity, a semiconductor with a large $g$-factor is desirable. A large Zeeman splitting ensures a large gap at zero
momentum, which is important since the chemical potential must be placed within the gap.
Finally, it is important to have a large spin orbit coupling, since the gap at finite momentum is otherwise suppressed by the
magnetic field, see Eq.~(\ref{eq:Heff}). Furthermore, the sensitivity to disorder increases when time-reversal
symmetry is strongly violated for electrons close to the Fermi level~\cite{Potter11, Sau11c}, meaning when the spins at $\pm k_x^F$ are
almost aligned. To avoid this, and keep robostness against disorder, one should not allow the ratio $\tilde{B} / E_{so}$,
to become too large.
Suitable candidate materials for nanowires, with large $g$-factors and strong spin-orbit coupling, are for example InAs and InSb.

\section{Conclusions and outlook}
In this short review article we have tried to give a pedagogical introduction to the exciting field of topological
superconductivity and MFs in condensed matter systems.
We saw that MFs, being equal superpositions of electrons and holes, can appear as Bogoliubov quasiparticles in effectively
spinless superconductors. Their peculiar non-abelian (non-commutative) exchange statistics was discussed, and the basic concepts
of Majorana qubits and topological quantum computation was introduced. Finally, we discussed the possibility to realize topological
superconductivity by bringing semiconductors with strong spin-orbit coupling into proximity with standard $s$-wave superconductors
and exposing them to a magnetic field.

While this manuscript was being finalized, a report of the possible observation of MFs in 1D nanowires was published~\cite{Mourik12},
and around the same time several other groups reported observations which can be interpreted as signatures of
Majoranas~\cite{Williams12, Rokhinson12, Deng12} (for space reasons we have here not discussed the issue of Majorana detection,
see e.g.,~\cite{Beenakker11rev, Alicea12rev}).
Clearly the types of systems investigated in the hunt for Majoranas contain a lot of new and exciting physics, but which experimental
observation are indeed genuine Majorana sightings remains to be seen. We also mention that in a real system, with a finite size
and interactions which may not be well described within a mean-field picture, there is not necessarily a perfectly clear and
unique definition of a MF.

In any case, once MFs can be reproducibly realized and detected in the lab, the work has only started.
Further studies will have to investigate their properties in detail, and fabrication and measurement techniques will
have to be perfected. No doubt, there will also be a need for additional theoretical work to understand the experimental findings.
On a longer timescale, the goal is of course to be able to control and manipulate quantum information stored in
Majorana-based qubit systems. If there will be a useful technological application at the end of this long road is too
early to predict, but there is certainly a lot of interesting and new physics to explore.

\section*{References}

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


text:
Q: Editing the metadata of a PDF file using PDFBox I am able to edit the metadata for a PDF file using PDFBox3.x.  However, at the final step I am being forced to save the changes to a different file.
Is there a way to make changes to the original file itself?

Sample code:
  PDDocument doc = Loader.loadPDF(inFileObj);

  PDDocumentInformation pdd = doc.getDocumentInformation();

  pdd.setAuthor("Mr. Stack Overflow");
  // ...

  doc.save(outFileObj); // This works, however doc.save(inFileObj) makes the PDF document a blank document

redpajama_set_name: RedPajamaStackExchange


Alexandra Kotelnitskaya is a tango DJ, or as she likes to describe herself a DJ Tangess, from Ukraine. Alexandra has an extensive experience and knowledge of music. Since 2007, her DJ experience has expanded from Kiev Ukraine and she has been performing in a large number of tango festivals, marathons and encuentros in Europe and around the world. Alexandra is also one of the organisers of the Tango Remolino in Ukraine, one of the most famous tango festivals in Eastern Europe, where the DJs are the main and only stars. After her long experience as a DJ, she knows what tango to play at what moment, simply based on the smiles and reactions from the dancers. She uses all the well known tangos from the Golden Age to fill the dance floor with energy and passion. At the festival, Alexandra will be playing at the Sunday Gala Milonga.


I'm reading by JRR Tolkien's The Lord of the Rings. One of the most compelling themes is the Ring itself. It possesses its owner by generating a psychotic jealousy. Many who desire the ring are willing to do anything to have it. One particularly loathsome character, Gollum, calls the Ring "My Precious".
It makes me recall the rings in my life. I remember when I used to smoke cigarettes. I cut way back after a while, but I still smoked one before bedtime. Just one.
Meanwhile, I would preach the dangers of tobacco to my children. I was so desperate to have that smoke that I would hurry my children anxiously or even aggressively to bed. The Marlboro Man waited for me. But it's just one lousy smoke, what's the big deal? Beyond the fact that even just one cigarette per day significantly increases the risk of heart and lung disease, there was a deeper problem. It was my Precious.
The rings in our lives make us behave irrationally. We spit in the eye of reason. Rather than rely on God's grace, we hobble around on a crutch. The drinker or addict convinces him or herself that they can handle it. They convince themselves of this every day. And – speaking from personal experience – they might even convince themselves that God has cured them of their alcohol problem to justify their weekend-only beer drinking.
The 'ring' can morph into very wicked forms. One subtle but very real danger for the Christian is the desire to be the one that does/creates good rather than being an instrument of good or letting good happen on its own. The overwhelming desire to prove a point is many a Christian's Precious as well.
We are much better off tending the sheep.
A Gollum lives in each and every one of us. Some are more obvious than others. But he's there sniveling and plotting how to protect his Precious no matter what the cost. He hurts himself. He hurts others. Just so long as he can keep his Precious nothing else matters. Pride and shame drive us into oblivion.
We might reduce these rings, we might whittle them down so as to appear non-existent upon casual observation. But minimization is not the same as eradication. As long as they live within us we are at risk of transforming into a blood thirsty Gollum.
The only way to purge ourselves is honesty. It appears terrifying and exasperates us. What will I do without my Precious?
Examine your conscience. Admit your problem. Then grab it by the neck and yank it out by the root. God will never force you to be healed, but He will never turn you away either. Recognize the Ring, and let it be cast into the fiery pit. Let God's healing fire consume it and obliterate it.
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Amen to that! I though the author really did a great job paralleling the two points. You should check out his blog too. I think you will enjoy! Blessings on your day.


We offer a complete service for qualification of your clean room installation.
Qualifications are conducted in close cooperation with our customers. We work on the basis of your individual desires and requirements.
We are also happy to prepare flow visualisations of filter processes.


text:
5 Actors Who Deserve an Oscar But Don't Have One
Oscar season is upon us, and that means talking about films and actors. This post is certainly off-topic for my site, but I am quite a movie buff so I enjoy talking (writing) about films. Every year, a bunch of actors/actresses are nominated and ultimately win Oscars. There are several who, for whatever reason have not won this prestigious award. So many actors have been good enough to win but haven't due to either timing (another actor had a better role/performance) or academy ignorance (a few thespians are just not well liked by the academy – see below for more). Below are the five that I believe should have won by now. All are at least 50 years old and have been on the big screen several times. I had a difficult time narrowing it down to five so I'll put my "honorable mentions" at the end. Enjoy!
Of the folks on this list, Mr. D'Onofrio is probably my favorite actor. He seems to choose his roles very carefully and while he has not really been a "leading man" in most films, his roles are always memorable. His skill is tremendous. Many times, I've caught myself wondering who is playing a particular part only to find out it is him. He got his big break as Private Pyle in Full Metal Jacket. What followed were dozens of stand-out roles where he was arguably the best part of a film where he was not the star. Here are few of my favorites:
Full Metal Jacket – Private Pyle
Men in Black – Edgar/Bug
Feeling Minnesota – Sam Clayton
The Break Up – Dennis Grobowski
Daredevil (TV series) – Wilson Fisk
The Magnificent Seven (2016) – Jack Horne
Mr. D'onofrio is the perfect Wilson Fisk (yes, I'm a comic book nerd also). He is an imposing figure with a calm ruthlessness about him. Rarely does he show his temper, but when he does, it's explosive. The same is true for his other roles. Usually they are low-key, but then the volcano erupts. The eruptions are all the more special because of the calmness he portrays normally.
Honestly, I didn't know he was in The Magnificent Seven until I looked at the credits. His range is absolutely outstanding. Put his Wilson Fisk next to Jack Horne and they are polar opposites. Unfortunately, Mr. D'Onofrio is in that purgatory between "character actor" and "leading man." He'll probably never get an acting Oscar, but I guarantee he'll get a lifetime achievement award from the academy in a few years.
Here is an actor so revered, so loved that he had a feature film named after him. Being John Malkovich is just one of dozens of virtuoso performances for this brilliant actor. Many of his best performances are from his early work (The Killing Fields, Places in the Heart, and Dangerous Liaisons). My favorite performance of his is In The Line of Fire where he plays an assassin hell-bent on killing the president while Clint Eastwood's grizzled secret service agent tries to stop him.
His more recent work also deserves attention. Obviously Being John Malkovich is terrific, but also films like Rounders, Burn After Reading, and Red all have him at his best. He has leaned a little more toward comedy lately which shows even more of his talent. His character in Burn After Reading is both hilarious and tragic at the same time. I leave out Con Air because that movie sucks like a chest wound despite his excellent performance. NSFW language below. Just FYI…
I saw the above film when it was released and was awe-struck by Mr. Malkovich's performance. The range, just in that one phone call is amazing. He goes from zero to sixty just like that. Great stuff!
John Malcovich's on-screen partner in Dangerous Liaisons also deserves better attention from the academy. As with Mr. Malcovich, Glenn Close's most critically acclaimed performances are early in her career. She was nominated for her roles in The World According to Garp, The Big Chill, The Natural, Dangerous Liaisons, and Fatal Attraction. All of these films were released before 1990.
Her performance in Fatal Attraction stands out to me as her best. She sheds the stereotype given her by Hollywood as the good-natured but dowdy actress and turns in a performance that is just plain hot. She plays the role with a subtlety rarely seen in these types of films. Her character is certainly a complete nut-job, but she also shows frailty, dominance, and vulnerability all in one fantastic performance.
In just that one scene she goes from begging to demanding and finally desparate. When I look at the list of films she has been in, I see a lot of critically acclaimed films with great casts and tons of Academy Awards. Sadly, none for her yet.
Gary Oldman has been nominated for an Oscar one time. Once. And that was just a couple of years ago for Tinker, Tailor, Soldier, Spy. What the hell, man!?! Seriously, no actor (besides maybe Paul Newman or Al Pacino) has been screwed by the academy more than Mr. Oldman. Millenials probably know him as Commissioner Gordon in the Nolan Batman films. Others might remember him as a terrific villain in either The Professional or The Fifth Element. The performance I remember him best for is Beethoven in Immortal Beloved. A good film? No. But his performance is so authentic as the tortured Ludwig, that it almost rescues an otherwise terrible (and bafflingly historically inaccurate) film.
Of course, the music in the film is outstanding because it is, duh, Beethoven. He immerses himself completely in the role, and he not only looks the part, but plays it as accurately as I've seen. Among his other great films are Bram Stoker's Dracula, Sid and Nancy, JFK, and (another favorite of mine) The Contender. Again, how has he not won the big trophy yet?
Sigh. I will admit I am not a fan. I don't like his phony-baloney Scientology nonsense or his controlling attitude toward co-workers and spouses. Putting that aside, he has had more than one performance worthy of an Oscar. Of all the actors on this list, he is certainly the most popular. He has more blockbusters under his belt than almost anyone. For that reason, I think the academy has sadly overlooked his better performances.
The two films most associated with him and fine acting are Born on the Fourth of July and Magnolia. These are both excellent films, for which he should have won. These roles were ten years apart with Born in 1989 and Magnolia in 1999. His monologue over his dying father in Magnolia is arguably the finest performance in a film filled with them. Personally, I find his best performance to be one that is almost universally overlooked – Rain Man. Again, the subtlety in his character is amazing. He is a total dick at the beginning of the film, and gradually turns into a loving and caring brother at the end. Dustin Hoffman gets the recognition here (and the Oscar), but Cruise is really really good.
I almost hate to admit it, but I also like Tom Cruise in the Mission Impossible films and Edge of Tomorrow (a movie that I thought I would hate and wound up loving). Take away the schlock like Top Gun, Cocktail, and Days of Thunder, and remember films like Minority Report and A Few Good Men. That's what I choose to do.
Deserving Actors – Honorable Mentions
There are dozens of great actors who have yet to win the big prize. Here are my honorable mentions along with their best performances:
William H Macy – Fargo , Magnolia
John Cusack – Identity, Being John Malkovich, Grosse Point Blank
Samuel L Jackson – Pulp Fiction, Django Unchained
Ed Harris – Pollock, Apollo 13, A History of Violence
Harrison Ford – Witness, The Fugitive
Annette Bening – American Beauty, Being Julia
Robert Downey Jr. – Less Than Zero, The Judge
There are some pretty heavyweight actors on this list, all 50 years of age or older (Mr. Cusack is 50 – I looked it up). The 40-somethings will have their day, though. Yes that means you Amy Adams, Edward Norton, and Bradley Cooper. In another 10 years, I'll probably be writing about you too.
EDIT – Turns out RDJ is 51. Who knew?

redpajama_set_name: RedPajamaCommonCrawl


The Sk:n Clinic Canary Wharf has had an exciting new refurbishment, providing upgraded and improved facilities, including an additional treatment room equipped with a new Elite laser which allows for an even better quality of service and more availability during peak times.
Make an appointment at the Sk:n Clinic Canary Wharf and visit the refreshed, vibrant clinic based in the Canary Wharf shopping centre. Offering a range of skin treatments including anti-ageing injectables, acne, laser hair removal and dermatology treatments.
Sk:n Clinics are the country's leading skin clinic offering more highly medically trained staff than any other chain of clinics. They pride themselves on having the most regulated treatment approach in a meticulously clinical environment – nationwide. No one offers more specialised skin treatments, from mole excisions to anti-ageing injectables, tattoo removals to facial resurfacing, or uses more advanced technology to do so.


text:
The Indiana Health Information Exchange stays on the cutting edge
December 9, 2022 by publisher team
Indiana has a long history of health information exchange, shaped with the help of the Regenstrief Institute and the Indiana Health Information Exchange.
This history began in the 1970s, when the Regenstrief Institute created the first electronic health record system. In 1993, Regenstrief created the Indiana Network for Patient Care to enable data sharing between emergency departments of hospitals serving Indianapolis – perhaps the first "health information exchange" in the country.
This data-sharing clinical repository was formed from an alliance of the five largest hospital groups in Indianapolis and supported by the Regenstrief Institute.
A neutral steward of data
"As the value of the INPC became apparent and EHRs became more common, Indiana researchers, business owners and hospitals met and determined the INPC needed a neutral steward of the data with a business focus," explained John Kansky, president and CEO of the Indiana Health Information Exchange.
"In 2004, the IHIE was founded as a separate 501(c)(3) not-for-profit organization to securely manage and grow the INPC," he continued. "To this day, the INPC remains at the core of how to fulfill our mission – to improve health and healthcare through information exchange."
The INPC is the largest inter-organizational clinical data repository in the nation, according to Kansky. IHIE, as its data steward, is vendor- and customer-agnostic and does not own any of the data in the INPC.
"IHIE manages the data and ensures that it is protected on behalf of our customers," he said. "We have a customer-based governance committee to ensure data is used in the way in which customers approve. IHIE is responsible for the development and maintenance of products and services derived from the unique INPC data asset.
"While we are working intentionally to build and expand our relationship with state government, we believe many HIEs would benefit from expanding their service offerings to the market and building their base of revenue from fees paid in exchange for value."
John Kansky, Indiana Health Information Exchange
"As the statewide health information exchange and health data utility for Indiana, IHIE leverages the INPC's data, governance processes and fabric of trust among its participants to meet as many of the health data needs of the state as possible," he continued.
This means developing and delivering services to healthcare providers, but also to multiple state government agencies, federal government agencies, health insurance companies and other stakeholders in the health of citizens, he added.
More than 30 years of data
The INPC includes more than 30 years of data, which the IHIE manages within a dynamic yet production-oriented HIE environment. The HIE's domain expertise in acute and ambulatory EHR integration (including lab systems, health plan administrative claims systems, medication data, public health) positions the IHIE as one of the largest and most mature health information exchanges in the US, Kansky said.
"We have been able to amass more than 13 billion elements of clinical and claims data including public health data and payer information and patient data for things such as lab results, radiology reports, CCDs, transcriptions, cardiology reports and more," he explained.
"IHIE creates products and services to help customers leverage the data in ways that will be beneficial to them," he continued. "IHIE's services are organized into three product suites: OneCare, PopCare and GovCare."
OneCare, PopCare, GovCare
OneCare offers systems to improve the health of the individual patient. The products provide interoperability and seamless delivery with EHRs and technologies that organize information around patients when and where they seek care.
PopCare includes systems that offer insight into population health. The technologies are designed to support population health efforts that serve to strengthen the community. The data and products ensure health information follows the patient, regardless of treatment location, and help providers identify patients needing testing, screenings and other care interventions.
GovCare services provide a complete, coordinated effort to connect healthcare professionals to the state through communicable disease reporting, immunization registry programs, making clinical data available to Medicaid providers and supporting statewide data analysis.
"At IHIE's inception, we started with two products: CareWeb, a secure application that displays patient information from the INPC; and Docs4Docs, a clinical results delivery service that sends reports to providers around the country via a no-cost web inbox or EHR interface Kansky noted.
"Since then, we have expanded our offerings and now provide six products in our OneCare suite, four products in our PopCare suite and four products/services in our GovCare suite," he continued. "We are intentional about only developing those products and services that our customers need or specifically request."
IHIE makes information available to approximately 50,000 healthcare providers in Indiana and neighboring states.
In 2020, IHIE consolidated with the Michiana Health Information Network (MHIN). Unifying under IHIE created a single, statewide health information exchange serving the entire population of the state.
"Also in 2020, IHIE joined the fervent response to COVID-19," Kansky recalled. "We worked, and continue to work, closely with the Indiana Department of Health and Indiana Family and Social Services Administration to support COVID surveillance activities, processing and sharing testing results with clinicians and the state, and supporting dashboards for public consumption of COVID data.
"The work continued into 2021 as IHIE added vaccine data to the dashboards," he added. "We received a grant from HHS for an ONC STAR project to make vaccination data more readily available to local health and community leaders."
Through this initiative, IHIE provided county-level COVID-19 vaccination data on its website in seven different languages. The data reflects not only the number of vaccinated individuals in each geographic area, but also the social vulnerability index of those census tracts.
Generating revenue through products and services
"As an independent HIE that is not directly funded by the state, IHIE focuses on delivering value to its participants," Kansky said. "For the past 18 years, IHIE has been a self-sustaining, not-for-profit health information exchange. We generate revenue by offering products and services our participants value and will pay to use.
"While we are working intentionally to build and expand our relationship with the state government, we believe many HIEs would benefit from expanding their service offerings to the market and building their base of revenue from fees paid in exchange for value," he advised.
IHIE takes its data custodian role and data security very seriously.
"We are currently going through HITRUST recertification," Kansky noted. "We spend a lot of time and effort making sure that our staff is up to date on privacy and security policies and that we follow strong and secure processes and procedures.
"We work closely with legal and the INPC governance committee to make sure we are using and sharing data in ways that are both legally and organizationally acceptable," he continued.
Rapidly changing technology
Technology to collect and share data changes rapidly.
"IHIE is always seeking to understand and adapt to new and different data standards and technologies, such as APIs, including but not limited to FHIR," he concluded. "The ability to take, store and share data is the lifeblood of our organization, so we always need to be ready and able to do so in the manner and timing that makes sense for our customers."
Twitter: @SiwickiHealthIT
Email the writer: bsiwicki@himss.org
Healthcare IT News is a HIMSS Media publication.
'Building standards failing to deliver healthy ventilation', say experts
Workers who rely on tips, commissions also impacted, study finds — ScienceDaily


text:
Lok Satta demands speedy trial in corruption cases
Lok Satta Party national President Dr. Jayaprakash Narayan today demanded that various wings of the Union Government initiate action against the guilty if there is any truth in the allegations made by India against Corruption (IAC) against Mr. Robert Vadra.
Dr. JP said that action is warranted irrespective of evidence of quid pro quo and added that the issue need not be politicized.
Addressing a media conference, Dr. JP said that the IAC accused Mr. Robert Vadra of receiving an unsecured interest-free loan of Rs.65 crore from real estate firm DLF and that he acquired real estate from DLF at less than the market value. If the charges are true, they must be dealt with beyond politics.
Dr. JP asked what the independent directors in the public listed company were doing when the company advanced an interest-free loan to the detriment of its shareholders. The Registrar of Companies is duty bound to inquire into the irregularity. If the allegation is correct, the Income Tax Department should treat the receipt of money by Mr. Robert Vadra as unassessed income and impose tax and penalty. The IT Department is entitled to acquire the property for the Government if it is registered for less than the market value.
Dr. JP said that people were losing faith in the Government and the judiciary because the wheels of justice grind very slowly in India. He said that the trial in Satyam Ramalinga Raju's case is yet to begin and the investigation into cases against YSR Congress President Jaganmohan Reddy seems to last for years going by CBI's submissions in the Supreme Court.
In contrast, justice is rendered swiftly in countries like China and the U. S. He cited the instance of Bo, a rising star and Politburo member of the Chinese Communist Party and his wife being punished in a case of corruption and murder of a British businessman, Rod Blagojevich, the Illinois Governor, being jailed for seeking donation to fund his election in return for nomination to a Senate seat vacated by President Barrack Obama and an NRI couple being jailed and deprived of all their property for indulging in medical fraud. In India, it took more than ten years to bring the guilty in Rajiv Gandhi assassination to book whereas it took only one year in the Mahatma Gandhi assassination.
Dr. JP said people would regain faith in justice only if the guilty are meted out deterrent punishment swiftly. He demanded that special courts be constituted on a permanent basis to try corruption cases. Investigative agencies like the CBI and the ACB should be freed from seeking prior Government clearances for taking up investigation into cases against public servants. A law should be passed to facilitate attachment of properties in corruption cases. Prosecutors should be independent and immunized from political interference.
Dr. JP welcomed TDP President N. Chandrababu Naidu's letter to the Prime Minister seeking autonomy for the CBI. In this connection, Dr. JP recalled that he had submitted a private Bill for amending the Prevention of Corruption Act but it is yet to see the light of the day because neither the ruling Congress nor the Opposition TDP is interested in it.
In reply to a question on the Government of India permitting FDI in retail, Dr. JP said that while he welcomed it as a means to reduce the country's current account deficit, he did not subscribe to the Government flaunting it as a piece of radical reform and the Opposition rubbishing it as a catastrophe. The country is crying for reforms in fields like education, healthcare, agriculture, police, judiciary, elections and corruption. But the Governments would not undertake them.
In reply to a question, he said that the Lok Satta would welcome formation of Telangana as part of a comprehensive and amicable solution. He appealed to all parties and organizations not to fan hatred among brethren since they cannot change geography even after the State is carved out into two or three pieces.
He reiterated the Lok Satta Party is prepared to work together with the party proposed to be grounded by Mr. Avind Kerjiwal since they shared common goals. Informal consultations are going on right now, Dr. JP added.
Lok Satta leaders Katari Srinivasa Rao and E. Chennayya took part in the conference.


April 15, 2013, 3:33 pm -- /EPR NETWORK/ -- Laundry 365 were set up to supply all sorts of commercial laundry equipment by a group of dedicated service people with years of experience in the trade with the added advantage that the service element was supplied 365 days a year. The company has been very successful in its approach and have realised that they have years of experience supplying not only the commercial laundry business but also the service industry as a whole, where that service industry supplies services to people in bulk. To this end Laundry 365 now supply commercial warewashers, which can clean bulk loads of crockery and glasses efficiently and with a high quality finish.


Static vs Dynamic Website — What's the Difference in 2019?
Lately, there has been a lot of talk across the dev community related to traditional web development and how things are shifting in a certain direction. Static websites are back, big time!
Thanks to modern web architecture like JAMstack, static websites are becoming a better solution for web developers.
For quite a while, I've been trying to spread the word about JAMstack, simply because I have been relying on its architecture and the JAMstack in general, due to its many advantages. Gradually focusing on JAMstack improved my workflow and dev process but more importantly, it helped me to leave the problems of standard web-server centric architectures behind.
Affordability: My budget was tight and focused on various aspects of the project and even so, I had a strong desire to improve how I code; in short, any unnecessary server and bandwidth costs were a pain and nuisance. By making the transition to JAMstack, my infrastructure expenses were reduced significantly.
Simplicity: With JAMstack things are simple. It's simple for devs to set up, and there are very few moving parts, as such, there is less room for an error and screw-ups.
Security and reliability: Specific APIs are now used and the mistakes that I make are very rare indeed. The absence of server-side and database operations means additional security and immunity to SQL injection.
One of the additional benefits of JAMstack is that provides an extremely practical and effective alternative to LAMP and MEAN stacks. With LAMP stack being useful back during the early 2000s, it was comprised of open-source components, which many devs used to construct websites (PHP language, Linux OS, the Apache HTTP Server, and the MySQL RDBMS). Come 2013, the MEAN stack arrived and it included MongoDB, Express.js, Angular.js, and node.js.
To cut to the chase, LAMP was reliable, but it was reliable back in the day, and MEAN stack was an improvement. Both conventional systems can now be swiftly replaced by the speedy and more modern system such as the JAM stack.
It is a long-running practice to rely on dynamic websites in the current Web development scene, primarily due to the fact that you get plenty of customization, simple updating and so on. You can really easily make complex pages. However, all of this comes at a high price.
The truth of the matter is that a majority of dynamiс websites are liable to drain system resources because it denotes the incorporation of web-server and database.
Also, a conventional database-driven CMS (think WordPress, Drupal etc.) with third-party plugins is vulnerable to malicious attacks from many endpoints. Keeping all these moving parts is mandatory if you aim at keeping your website safe.
I've come to realize over the years that static websites have significant advantages denote ready web-pages (HTML and JS) that are downloaded and kept on the server. In this day and age, lack of speed and huge load times is a disadvantage you cannot afford. With mobile-first indexing and page speed as one of the top ranking factors, it is easy to understand why for websites or online business of any kind it's like shooting yourself in the foot.
Static websites will grant you greater performance overall and a drastic reduction of load times — not to mention, better resistance to hacker attacks. As a result, using static websites can lead you to better functionality and easier maintenance.
They do not have long load times and any devs who are working on it won't have a hard time to set them up. Also, it is easy to include diverse forms, payments or authentication if needed. You can do this by utilizing ubiquitous APIs.
JAMstack certainly streamlined my development process a great deal last year, but things don't stop there. For those of you that are not into knowhow, after all said and done, why should you resort to JAMstack?
For one thing, it's a pretty solid strategy to have when it comes to SEO. Google and other powerful search engines prefer static websites, seeing as they load very fast. What's more, Google is constantly motivating site owners to incorporate Accelerated Mobile Pages (AMP), and that particular open-source effort is generally static in nature (equivalent to the dynamic pages on the desktop). On the development side, PWAs (Progressive Web Apps) are gaining momentum. Afterall with mobile web taking over from desktop publishers are opting out to bring native-like experience to users on all platforms and all devices.
I really hope I've given you some insight about how this particular system is valuable if you want to improve your workflow as a web developer. I will most certainly continue my journey with JAMstack, so stay put.


text:
Q: Loop increasing value by 1 each time I am trying to start with a value of 1 in A1 and then increase the value by 1 and move to A2, ie A1 = 1, A2 = 2 etc
right now i only have code increasing the value in A1. How can i increase the value and drop down a cell each time?
Sub test()
Range("A1").Value = Range("A1").Value + 1
End Sub

A: Try this
Sub Demo()
    Dim rng As Range
    Set rng = Range("A1")
    If IsEmpty(rng) Then
        rng = 0
    Else
        If Not IsEmpty(rng.Offset(1, 0)) Then
            Set rng = rng.End(xlDown)
        End If
        rng.Offset(1, 0) = rng.Value + 1
    End If
End Sub

A: Sub test()
  start = Val(Range("A1").Value)
  For row = 1 To 1000
    Range("A" & row).Value = start + row
  Next
End Sub

A: Range("a1:a100").formula = "=row()"

or
Range("a1").Value2 = 1
Range("a1").AutoFill Range("a1:a20"), xlFillSeries

or
for counter = 0 to 20
    range("a1").offset(counter).value2 = counter +1
    range("b1").offset(counter+1).Value2 = range("b1").offset(counter).Value2 + 1
next counter


text:
Online TV is the Topic of Social Media Conversation
Research from VAB has found that video programming and content has become the primary social media conversation fuel during the time of social distancing and lockdowns in the US.
The AloneTogether: Culture in the Time of Covid-19 examined the topics people talked about on Twitter during the home isolation period and found that over a six-week period, three-quarters of the top trending topics on Twitter at night were related to people watching video content.
The study analysed the top ten trending Twitter topics each night over the first six weeks of country-wide lockdown orders from 16 March 26 April 2020. It monitored two points in time each night to capture what people are talking about online throughout the evening at 21:30 and 23:00. Trending topics based on video content included two types which have been aggregated within this analysis: direct, which were specific 'official' hashtags of entertainment shows, sports content or news programming; related, pertaining to topics associated with specific video programming including athletes, show characters, celebrity personalities, and specific video platform-related news references.
VAB also found that TV has become the center piece of the household, with four-fifths of people saying they couldn't imagine not having a television during the pandemic. As many as 83% said that they couldn't imagine not having a television set right now and 70% said that they binge-watched more TV shows or films during the lockdown period. Since the start of Covid-19, live TV and social media were found to be experiencing the biggest increase in consumption. Overall, 84% of respondents say they have more time to watch/listen/read media since the Covid-19 outbreak.
Looking at social media, VAB found that at least one video-related topic trended each day of the six-week analysis, covering a variety of genres including scripted dramas, reality TV, live news specials and sports-related updates, among others. Ad-supported TV accounted for 72% of total video-related trending topics and filling the void created by the absence of live sports on TV, entertainment content made up over three-fourths (78%) of ad-supported TV programmes trending in the top 10 on Twitter.
The AloneTogether: Culture in the Time of Covid-19AloneTogether: Culture in the Time of Covid-19 study also revealed that ad-supported TV programming specials, including live music performances, late night fundraisers and televised town halls, also were top trending Twitter topics on the nights that they ran. Prime examples cited included Fox's iHeart Living Room Concert music event on 29 March; CBS' Homefest: James Cordon's Late, Late Show special on 30 March; The CNN Town Hall hosted by Anderson Cooper and Dr. Sanjay Gupta providing a forum to discuss issues related to the pandemic on 2 April.
Joseph O'Halloran| 17 June 2020
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William Cabell Rives (1793-1868)
Descendants of William Cabell Rives
Shown here in front of an idealized "Castle Hill," the Albemarle County home of William C. Rives, are his three sons: Francis Robert Rives, William Cabell Rives, Jr., and Alfred Landon Rives. Oil on Canvas. George Cooke at the University of Virginia.
William C. Rives (1793-1868) married Judith Page Walker (1802-1882) in 1819. The couple had five children:
Francis Robert Rives (1822-1891) m. (1848) Matilda Barclay (?-1888)
George Lockhart Rives (b. 1849)
Ella Louisa Rives (b. 1851)
Francis Robert Rives (b. 1853)
Maud Antonia Rives (b. 1855)
Constance Evelyn Rives (b. 1855)
Reginald William Rives (b. 1861)
William Cabell Rives, Jr. (1825-1889) m. (1849) Grace Winthrop Sears (?)
William Cabell Rives (b. 1850)
Alice Rives (b. 1852)
Arthur Landon Rives (b. 1853)
Col. Alfred Landon Rives (1830-1903) m. (1859) Sarah Catherine MacMurdo (?-1909)
Amelie Louise Rives (1863-1945)
Gertrude Rives (b. c. 1866)
Sarah Landon Rives (b. 1874)
Amelie Louise Rives (1832-1873) m. (1854) Henry Sigourney (?-1873)
Henry Sigourney (b. 1855)
Three younger Sigourney children perished with their parents at sea in 1873.
Ella Rives (1834-1892)
Additional Source Consulted:
James Childs Rives, Reliques of the Rives (Ryves) (1929)
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Beyond Making the Grade: Student and Life success at UVA (in 1854 and 2022) December 7, 2022
Special Collections Catalogs and Catalogers December 5, 2022
Nick Biddle and the Commemorative Power of the Carte de Visite November 30, 2022
150 Years Ago This Month: A Civil War Blog
Friday, 1865 December 22, Charlottesville, Albemarle County, Virginia: "As Brother James Fife may be Pastor of the Charlottesville African Church, it is Competent for him, when in his judgement advisable, to invite any Preacher of the Gospel, without regard to Colour" December 2, 2015
Monday, 1865 November 20, Huntley, Fairfax County, Virginia: "My whole heart was in our struggle . . . God had a purpose in scourging us which I cannot now see, but it will doubtless be revealed in the future" November 2, 2015
Thursday, 1865 October 12, Decatur, Morgan County, Alabama: "He will see that his command conduct themselves in an orderly manner . . ." October 1, 2015
Longtime supporters of the University and the Library, Albert ('46) and Shirley Small made a substantial gift toward the construction of the special collections library that bears their name and houses Mr. Small's remarkable collection of autograph documents and rare, early printings of the Declaration of Independence
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