\section{Introduction}

Since its discovery, the Einstein's general relativity(GR) is the core to understand the connection between spacetime and geometry. The area of a surface, a geometric quantity, plays an quite important role to promote our understanding of the fundamental laws of physics. Even though its geometric meaning is lucid, its physical interpretation is intriguing. In short, there are two remarkable results on the area $\mathcal{A}$ of a surface in GR. The first one is the Bekenstein-Harking formula of a black hole entropy \cite{Bekenstein, Harking}
 \be
 S_{BH}=\frac{\mathcal{A}}{4G_N}\label{BH}
 \ee
    It relates the area of a Killing horizon to the thermal entropy of the system, and plays the key role in the black hole thermodynamics. How to understand the area law of the black hole entropy is one of most important questions in quantum gravity. It inspired people to propose  the holographic principle in quantum gravity.  The other one is the Ryu-Takayanagi(RT) formula for the holographic entanglement entropy \cite{Ryu0603}
 \be
 S_{EE}(I)=\mathop{\mbox{ext}}_{m\sim I}\frac{\mathcal{A}(m)}{4G_N}.\label{RT}
 \ee
It gives a simple prescription relating the entanglement entropy of a submanifold $I$ in a conformal field theory(CFT) to the area of an extremal bulk surface $m$ which is homologous to the boundary region $I$.

 The surfaces appear in (\ref{BH}) and (\ref{RT}) are either a Killing horizon or an extremal surface, so they are quite special in some sense. In the most general case, we may choose a time slice $\Sigma_{0}$ of a spacetime and a region $V\subseteq \Sigma_0$. Now the region $V$ and its complement $\bar{V}$ compose the time slice $\Sigma_{0}$. Their common boundary is denoted as $\partial V$. On one hand, we can define  a natural geometric quantity $\mathcal{A}(\partial V)$,  the area of $\partial V$. On the other hand, we cannot always relate it to a physical quantity, except in the above two cases\footnote{Some interesting efforts can be found in \cite{Bianchi, Balasubramanian}.}. This is an interesting phenomenon, as it indicates that our understanding of $\mathcal{A}(\partial V)$ is incomplete. The lesson from (\ref{BH}) and (\ref{RT}) tells us that there may be some generalized gravitational entropy\cite{Lewkowycz:2013nqa,Balasubramanian:2013rqa}
 \be
 S_{gr}=\frac{\mathcal{A}(\partial V)}{4G_N}\label{gr}
 \ee
 for arbitrary $\partial V$\footnote{To make the picture clear, we always illustrate our examples in the Einstein-Hilbert theory, but  we can easily generalize the arguments to other theories.}. Though the meaning of the formula (\ref{gr}) is not as clear as (\ref{BH}) and (\ref{RT}), there is interesting progress recently. In \cite{Balasubramanian1310}, the authors considered the gravitational entropy (\ref{gr}) of an arbitrary bulk curves in $AdS_3$, and related it to the so called ``Differential Entropy''
 \be
 E=\lim_{n\to \infty}\sum_{k=1}^n[S_{EE}(I_k)-S_{EE}(I_k\cap I_{k+1})]\label{RE}
 \ee
 in the boundary conformal field theory. Here the intervals $I_k(k=1,2,\cdots,n)$ are constructed as follows. We denote the $AdS_3$ coordinates as $t,x,z$, among which $t,x $ are also the boundary $CFT$ coordinates, $z$ represents  the extra dimension. A closed bulk curve $z=z(x)$ is assumed to be smooth. We divide the curve according to $n$ points $(x_k,z(x_k))(k=1,2,\cdots,n)$\footnote{Here we omit the coordinate $t$ as it is the same for the points in the bulk curve.}. When $k>n$, we impose the periodic condition: $k\sim k+n$. For each point labeled by $k$, we search for a boundary interval $I_k$ such that
 \begin{enumerate}
 \item The corresponding bulk extremal curve goes through the point $(x_k,z(x_k))$.
 \item The corresponding bulk extremal curve is tangent to the original curve $z=z(x)$ at $(x_k,z(x_k))$.
 \end{enumerate}
After finding the intervals $I_k$, the authors  calculated the differential entropy (\ref{RE}) and found a remarkable equality
\be
S_{gr}=E.\label{equality}
\ee
In \cite{Balasubramanian1310}, it was argued that the differential entropy is a measure of uncertainty about the state of a system left by an infinite family of local, finite-time observables. It seems that the differential entropy is closely related to the causal holographic information proposed in \cite{Hubeny1204} and studied in \cite{Kelly:2013aja}

In \cite{Myers}, the concept of Differential Entropy (\ref{RE}) has been modified and generalized to higher dimensions when the bulk curve has planar symmetry\footnote{Correspondingly, the boundary ``interval'' now becomes ``strip''.}. The reason for the modification  in higher dimension is that using the causal holographic information associated with the boundary strip leads to divergent results, though it works fine in AdS$_3$. The investigation in \cite{Myers} relied more on geometric construction. The basic idea is that bulk surface could be taken as the outer envelope of the bulk regions associated with the boundary intervals. It has been checked that the relation (\ref{equality}) holds  in various situations, including other backgrounds which is asymptotically $AdS$ and the Lovelock gravity.  Though the studies in \cite{Myers} strongly suggest that there is a new holographic equivalence between the gravitational entropy of bulk surface and the Differential entropy in the boundary, the discussion were made case by case. It would be interesting to see  why the equivalence (\ref{equality}) holds in these cases and when (\ref{equality}) could break down.

In this note, we present a brief proof of the equivalence (\ref{equality}) based on strong sub-additivity of the entanglement entropy and some general properties of quantum field theory.  The key ingredients in our discussion are two concepts, which are called as the Upper Differential Entropy (UDE) $E_{u}$
\be
E_u=\lim_{n\to\infty}\sum_{k=1}^n[S_{EE}(I_k)-S_{EE}(I_k\cap I_{k+1})]\label{URE}
\ee
 and the Lower Differential Entropy(LDE) $E_{l}$ respectively
 \be
 E_l=\lim_{n\to\infty}\sum_{k=1}^n[S_{EE}(I_k\cup I_{k+1})-S_{EE}(I_k)]\label{LRE}.
 \ee
Note that $E_u$ is just the Differential Entropy defined previously. Then we show that the three quantity $E_u, S_{gr},E_l$ satisfy the inequality
\be
E_u\ge S_{gr}\ge E_l\label{inequality}.
\ee
On the other hand, in quantum field theory, we can prove under quite reasonable  assumption that  UDE and LDE are actually equal
\be
E_u=E_l.\label{U=L}
\ee
If the holographic entanglement entropy is the same as the one in boundary CFT, as has been proved in the case of AdS$_3$/CFT$_2$ correspondence\cite{Faulkner:2013yia,Hartman:2013mia}, or at most differs a global factor in higher dimensional cases, the relation (\ref{U=L}) leads to the equivalence (\ref{equality}). This provides another point of view on the holographic equivalence (\ref{equality}).

The structure of the paper is as follows. In section 2 we show the inequality (\ref{inequality}) based on the strong subadditivity in the bulk. Then we  check the equality of (\ref{U=L}) in some simple cases in section 3. This makes us to be familiar to the concepts given in section 2.  After that, we prove the relation  (\ref{U=L}) from the quantum field theory side in section 4. Some discussion and conclusion will be presented in the last section.

\section{Gravitational entropy between bounds}
In quantum information theory, there is a fundamental inequality \cite{Lieb73prl,Lieb73jmp} which is called strong subadditivity. One of its form is\footnote{Here and in the following, for simplicity we will use $S$ instead of $S_{EE}$ to denote the entanglement entropy.}
\be
S(A)+S(B)\ge S(A\cap B)+S(A\cup B)\label{ssb}
\ee
We  illustrate the inequality in Figure \ref{ssb1}. Since we are interested in the entanglement entropy in quantum field theory, the label $A$ and $B$ are chosen to be some spacelike region in the figure.  The original proof of strong subadditivity (\ref{ssb})  relies on some nontrivial properties of the entanglement entropy\cite{Lieb73jmp}. However, due to the RT formula, there indeed be a straightforward demonstration of this inequality\cite{Headrick0704}. The key point is shown in Figure \ref{ssb2}. It is easy to see
\be
S(A)+S(B)=(a+b)+(c+d)=(a+c)+(b+d)\ge e+f=S(A\cup B)+S(A\cap B)\label{proofssb}
\ee
The $\ge$ appear since $a+c$($b+d$) and $e$($f$) are homologous to  the same boundary $AB'$($BA'$) while $e$($f$) is the minimal surface according to RT formula. The holographic demonstration (\ref{proofssb}) is the simplest proof of the strong subadditivity we ever find. Here we use a similar argument to introduce the notions of UDE and LDE\footnote{The RT formula proposed in \cite{Ryu0603}mainly focus on static spacetime. The corresponding proof of strong subadditivity in \cite{Headrick0704} is only valid for static spacetime. A covariant formula is proposed in \cite{Hubeny:0705}, and the related work on strong subadditivity in this covariant framework can be found in \cite{Wall}. Our discussion in this paper focus on static spacetime.}.
\begin{figure}
\centering
\includegraphics[height=9cm,width=12cm]{circle6d.eps}
\caption{An illustration of the strong subadditivity. The two circle are the regions $A$ and $B$, there can be some overlap between $A$ and $B$ and it is denoted as $A\cap B$.  }\label{ssb1}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=3cm,width=12cm]{circle7d.eps}
\caption{Holographic proof of the strong subadditivity. The straight line $ABA'B'$ corresponds to the boundary space. The line $AA'$ is the region $A$ and $BB'$ is the region $B$. $AB'$ is the region $A\cup B$ and $BA'$ is the region $A\cap B$. According to the RT formula, the green curve,  blue curve, dashed red curve and dashed black curve represent the extremal surface which are homologous to $AA', BB',AB'$ and $BA'$ correspondingly. Their area are labeled as $a+b,c+d,e$ and $f$ correspondingly.  }\label{ssb2}
\end{figure}

\begin{figure}
\centering
\includegraphics[]{circle2.eps}
\caption{An illustration of the appearance of UDE and LDE. We use five interval $I_i=M_iN_i(i=1,2,\cdots,5)$ to present the idea. We still use the symbol $M_iN_i$ to denote the corresponding bulk extremal surface without confusion. Each pair of consecutive bulk surface $M_iN_i, M_{i+1}N_{i+1}$ will meet at a codimension 3 surface which we denote as a point $P_i$ in the figure. Then there is an emergent bulk surface $P_1P_2\cdots P_5$ as shown in the figure. We have divide the extremal bulk surface as figure \ref{ssb2}. For example, the area of the bulk surface $M_1N_1$ has been divided to three terms a,A,d.   }\label{skeleton}
\end{figure}

As shown in Fig. \ref{skeleton}, we choose a series of intervals $I_i=M_iN_i$ and denote the entanglement entropy as $S_i$ correspondingly. According to the RT formula, the entropy is given by the area of a bulk minimal surface which has the same boundary as the interval. Hence we have\footnote{Here the coefficient $1/4G_N$ has been omitted to simplify the notation.}
\bea
&&S_1=a+A+d\nn\\
&&S_2=c+B+f\nn\\
&&S_3=e+C+h\\
&&S_4=g+D+j\nn\\
&&S_5=i+E+b\nn
\eea
In the boundary, we use $I_{1\cup2}$ to represent $I_1\cup I_2=M_1N_2$ and $I_{1\cap2}$ to represent $I_1\cap I_2=M_2N_1$. Then we find the inequality from the geometry
\bea
S_{1\cup2}\le a+A+B+f,\hs{3ex} S_{1\cap2}\le c+d\nn\\
S_{2\cup3}\le c+B+C+h,\hs{3ex} S_{2\cap3}\le e+f\nn\\
S_{3\cup4}\le e+C+D+j,\hs{3ex} S_{3\cap4}\le g+h\\
S_{4\cup5}\le g+D+E+b,\hs{3ex} S_{4\cap5}\le i+j\nn\\
S_{5\cup1}\le i+E+A+d,\hs{3ex} S_{5\cap1}\le a+b\nn
\eea
The bulk surface $P_1P_2P_3P_4P_5$ has an area $\tilde{S}$ as
\be
\tilde{S}=A+B+C+D+E
\ee
Then one easily find
\be
S_u\ge \tilde{S}\ge S_l
\ee
where $S_u$ and $S_l$ are defined to be
\be
S_u=\sum_{r=1}^{r=5} (S_r-S_{r\cap r+1}),\ S_l=\sum_{r=1}^{r=5}(S_{r\cup r+1}-S_r)
\ee
They are respectively called the Upper and Lower Differential Entropy of the surface $P_1P_2P_3P_4P_5$.
A general definition of the $n$-th Upper and the $n$-th Lower Differential Entropy are
\be
S_u^{(n)}=\sum_{r=1}^{r=n} (S_r-S_{r\cap r+1}),\ S_l^{(n)}=\sum_{r=1}^{r=n}(S_{r\cup r+1}-S_r)
\ee
The subscript $n$ means that there are $n$ intervals and $S_{n+1}=S_1$. In the limit $n\to \infty$, the bulk surface becomes smooth and the inequality should hold still,
\be
E_u\ge \tilde{S}_{\infty}\ge E_l.
\ee
We have used $E_u$ and $E_l$ to denote the limiting quantity of the $n$-th Upper and Lower Differential Entropy and they are the quantity UDE and LDE we have shown in the introduction
\be
E_u=\lim_{n\to\infty} S_u^{(n)},\ E_l=\lim_{n\to\infty} S_l^{(n)},
\ee
and $\tilde{S}_{\infty}$ is just the gravitational entropy $S_{gr}$
\be
\tilde{S}_{\infty}=S_{gr}
\ee
introduced  in the Introduction. Hence, we have proven the inequality (\ref{inequality}).

\section{Examples}

In the following examples, we  find that when the number of the intervals tends to infinity, the inequality actually becomes equality.
\be
E_u=S_{gr}=E_l.
\ee

The first example is the $AdS_3$. We choose the Poincar\'e coordinate and set the AdS radius to be unit
\be
ds^2=\frac{-dt^2+dx^2+dz^2}{z^2}.
\ee
 The boundary space is at infinity where $z=0$, with its length being $l\to\infty$.
When the bulk surface $S_{\infty}$ is a circle $z=z_0$, we use $n$ polygons to  approximate it. Hence, for every vertex $x_k=k l/n$, we can find an interval $I_k$ whose extremal surface in the bulk is tangent to the bulk surface at $(x_k,z_0)$. This is shown in Figure \ref{ads3}.

Since the extremal surface is a half circle $(x-x_k)^2+z^2=z_0^2(z>0)$, we find the length of the interval is
\be
\Delta x_k=2z_0.
\ee
For the interval $I_{k+1}$, its center is at $x_{k+1}=(k+1)l/n$ and the length is the same. Then the length of the intersection $I_k\cap I_{k+1}$ is
\be
\Delta x_{I_k\cap I_{k+1}}=2z_0-l/n.
\ee
The length of the union $I_k\cup I_{k+1}$ is
\be
\Delta x_{I_k\cup I_{k+1}}=2z_0+l/n.
\ee
Hence, we find the upper bound and the lower bound are respectively
\bea
S_u^{(n)}=n\frac{c}{3}\ln \frac{2z_0}{2z_0-l/n}\\
S_l^{(n)}=n\frac{c}{3}\ln \frac{2z_0+l/n}{2z_0}
\eea
where $c=\frac{3}{2G_N}$. In the limit $n\to\infty$, the upper and the lower bound approach to each other and equal to the area of the bulk surface $z=z_0$
\bea
E_u=E_l=S_{gr}=\frac{l}{4G_N z_0}.
\eea
\begin{figure}
\centering
\includegraphics[]{linecircled.eps}
\caption{$E_u=E_l$ when the bulk surface is $z=z_0$ }\label{ads3}
\end{figure}

In general, the bulk surface can be $z=z(x)$. In this case, we replace $k$'s with the coordinates $\cdots,x,x+dx\cdots$. Then for an interval $I_x$, we need to find its center point $x_c(x)$ and its length $\Delta x(x)=2r(x)$. This is shown in the Figure \ref{ads3arb}.
The natural candidate of the extremal curve should be tangent to the curve $z(x)$, hence we find
\be
x_c=x+z z'(x),\ r(x)=z\sqrt{1+z'^2}.
\ee
For the interval $I_{x+dx}$, we just replace $x$ to $x+dx$. Then
\bea
&&S(I_x)-\frac{1}{2}(S(I_x\cap I_{x+dx})+S(I_x\cap I_{x-dx}))\nn\\&\simeq& \frac{1}{2}(S(I_x\cup I_{x+dx})+S(I_x\cup I_{x-dx}))-S(I_x)\nn\\&\simeq&\frac{1}{4G_N}\frac{1+z'^2+zz''}{z\sqrt{1+z'^2}}dx+\mathcal{O}(dx^2)
\eea
and
\be
E_u=E_l=\frac{1}{4G_N}\int \frac{1+z'^2+zz''}{z\sqrt{1+z'^2}}dx=S_{\infty}(z=z(x))+\frac{1}{4G_N}\arcsin (z')|_0^l
\ee
We assume the bulk curve is smooth and periodic such that the boundary terms in the right hands side vanish.
\begin{figure}
\centering
\includegraphics{linecirclepara2.eps}
\caption{$E_u=E_l$ when the bulk surface is $z=z(x)$}\label{ads3arb}
\end{figure}

 In the two examples given above, we find that the inequality from the strong subadditivity is actually an equality when the number of intervals tends to infinity. The fact that UDE and LDE are both equal to a bulk surface area holds for more general cases. One can check this point in all the examples given in \cite{Myers}.

From strong subadditivity, we only know that UDE is no smaller  than LDE. The fact that  the equality is actually saturated deserves interpretation. We will study this issue from the point of view of quantum field theory in the next section.

\section{$UDE=LDE$ in $QFT$}

In this section, we show that UDE and LDE are equal for a general local $QFT$\footnote{The $QFT$ we consider are continuous $QFT$. For lattice $QFT$, the equality found in this section is not valid.}. This relies on some simple assumptions which should hold for general $QFT$s. The entanglement entropy we consider corresponds to an interval in 2D QFT or an strip in $d>2$ QFT\footnote{We will use uniformly the word "interval"  even when it is a strip for higher dimensional quantum field theory.}. The intervals we place are in $x$ direction. The other spatial directions which are orthogonal to $x$ direction are written collectively as $y_i,i=2,3,\cdots,d-1$. All the spatial directions extend infinitely. However, we set an IR cutoff by restricting the length of each direction to be $l,l_2,l_3,\cdots,l_{d-1}$ respectively.  To construct the quantities UDE and LDE, we need an infinite set of intervals $I_x$. We use the point $x$ to label each interval as $I_x$, within it the center being $x_c(x)$ and the length being $\Delta x=2r(x)$. For consecutive intervals $I_x$ and $I_{x+dx}$, the center and the length of the interval should varies smoothly. When $x>l$, we require $I_x\sim I_{x+l}$.  In addition, we  assume that the $QFT$ satisfy two other requirements
\begin{enumerate}
\item $QFT$ is translational invariant.
\item As the intervals varies smoothly, we exclude the case that the entanglement entropy varies non-smoothly. In other words, the entanglement entropy (and all its derivatives\footnote{Actually, we only need it to be $C^2$-differentiable for our demonstration.}) is differentiable with respect to the interval.
\end{enumerate}
Since the $QFT$ is translational invariant, for an interval which is characterized by the boundary point $u$ and $v$, the entanglement entropy depends only  on the length of the interval $\Delta x=|u-v|$, namely $S_{EE}=S_{EE}(\Delta x)$. There are four kinds of intervals: $I_x$ and its neighbor $I_{x+dx}$, their intersection $I_x\cap I_{x+dx}$ and their union $I_x\cup I_{x+dx}$, which have the distances respectively
\bea
\Delta x(I_x)&=&2r(x),\\
\Delta x(I_{x+dx})&=&2r(x)+2r'dx+r''dx^2+\mathcal{O}(dx^3),\\
\Delta x(I_x\cap I_{x+dx})&=&2r(x)+(-x_c'+r')dx+\frac{1}{2}(r''-x_c'')dx^2+\mathcal{O}(dx^3),\\
\Delta x(I_x\cup I_{x+dx})&=&2r(x)+(x_c'+r')dx+\frac{1}{2}(r''+x_c'')dx^2+\mathcal{O}(dx^3).
\eea
 The prime in the superscript denotes the derivative with respect to $x$. Since $I_x\cap I_{x+dx}\subseteq I_x,\ I_x\cup I_{x+dx}\supseteq I_x$, we have
 \be
 r'^2-x_c'^2=(r'+x_c')(r'-x_c')\le0.
 \ee
As the entanglement entropy changes smoothly, we can safely do Taylor expansion for the entanglement entropy. We find that the mutual information
\bea
S(I_x,I_{x+dx})&=&S_x+S_{x+dx}-S_{x\cup x+dx}-S_{x\cap x+dx}\nn\\
&=&\frac{\p^2S(\Delta x)}{\p (\Delta x)^2}(r'^2-x_c'^2)dx^2+\mathcal{O}(dx^3)
\eea
vanishes as it is proportional to $(dx)^2$. Hence the upper and the lower bound should be equal. They are
\be
E_u=E_l=\int_0^l dx \frac{\partial S_{EE}}{\partial \Delta x}x_c(x)'.\label{URE}
\ee
Note that for the expansion should be consistent with the strong subadditivity $S(I_x,I_{x+dx})\ge 0$, we must have $\frac{\p^2S(\Delta x)}{\p (\Delta x)^2}\le 0$ as $r'^2-x_c'^2\le 0$.

 In the above discussion, we did not require the $QFT$ to be conformal invariant.  That means even though the theory is not a CFT, we can still find the equality $E_u=E_l$. For a theory which has a holographic dual, and the holographic entanglement entropy is given by the RT formula\footnote{Of course, we should demand the $QFT$ satisfy the previous two requirements. In the following, we always assume the two requirements to be satisfied when we consider a quantum field theory.},  since UDE and LDE are equal, there should be a surface whose area is the same as the two bounds when we take into account of the holographic inequality (\ref{inequality}) above. According to the RT formula, there is an emergent extremal surface corresponding to an entanglement entropy. Now, after one calculates the quantity $UDE$ (or LDE), there is a corresponding emergent bulk surface, which  need not to be extremal. Similar to the holographic entanglement entropy, the entropy of the surface could be understood as a generalized gravitational entropy as well.

 One may wonder why we need so many intervals  to reconstruct a bulk surface while for the extremal surface only one interval is needed. For an interval region $I$, we can always associate it a bulk region $r$, whose boundary in the bulk is $\partial r$, as we show in  Figure \ref{extr}.
    \begin{figure}
\centering
\includegraphics{circle4pd.eps}
\caption{The correspondence between boundary interval $I$ and bulk region $r$. The original RT formula provides an one to one correspondence between $I$ and $\partial r$. It is also an one to one correspondence between $I$ and $r$.}\label{extr}
\end{figure}
After we specify the definite boundary $QFT$ and its bulk dual theory, the bulk region $r$ and the boundary region $I$ are in one to one correspondence\footnote{ There are similar statement in \cite{Headrick1312} and some earlier discussions on this issue  in \cite{Czech}.}. However, for a region $\Sigma$ bounded by a circle(and boundary), as we show in Figure \ref{circle case}, there is no single interval $I$ dual to it. However, the region $\Sigma$ can be constructed as the union of $r_k$ as shown in the figure,
   \be
    \mathop\bigcup_{k}^{\infty} r_k=\Sigma.\label{bulk}
   \ee\
   Each $r_k$ corresponds to an interval $I_k$ such that the region $\Sigma$ corresponds to an infinite number of intervals $\{\cdots,I_k,\cdots\}$. Then the gravitational entropy of the dashed circle, which is the bulk part of the boundary of $\Sigma$, must map to a boundary observable which is related to $\{\cdots,I_k,\cdots\}$. This observable in QFT has been shown to be the $UDE(LDE)$. For more general bulk region,  one can have
 \be
  \Sigma\subseteq\mathop\bigcup_{k}^{\infty} r_k.\label{bulk}
\ee
The most natural candidate of $I_k$ is constructed as we have discussed in the Introduction.

 \begin{figure}
\centering
\includegraphics{circle3pd.eps}
\caption{ Bulk region $\Sigma$ from the boundary. $\Sigma$ is the region between the dashed circle and the boundary. }\label{circle case}
\end{figure}

 The formula (\ref{URE}) provides a new interpretation of the quantity $\frac{\partial S_{EE}}{\partial \Delta x}$. To simplify discussion, we choose $x_c=x$ and $\Delta x=const$, which corresponds to  the situation that the bulk surface is a circle. We immediately see that\footnote{Actually, (\ref{prop}) is only valid up to a totally derivative. However, we choose the bulk curve to be a circle, we expect the totally derivative vanishes.}
    \be
    \frac{\partial S_{EE}}{\partial \Delta x}\propto Area(z^*)\label{prop}
    \ee
    or more precisely,
    \be
    \frac{\partial S_{EE}}{\partial \Delta x}=\frac{1}{4G_N} \frac{Area(z=z^*)}{l}=\frac{S_{gr}(z=z^*)}{l}\label{dEEformula}
    \ee
    where $l$ in the denominator  is just the length in the $x$ direction. If we view $\frac{1}{4G_N} Area(z=z^*)$ as a generalized gravitational entropy, then $\frac{\partial S_{EE}}{\partial \Delta x}$ is the gravitational entropy density.
    This is illustrated by Figure \ref{pspx}.
    \begin{figure}
    \begin{center}
\includegraphics{circle8d.eps}
\caption{An illustration of $\frac{\partial S_{EE}}{\partial \Delta x}$. $z=0$ is the boundary. $\Delta x$ is the length of the boundary interval which we use  green curve to represent it. The red curve is the bulk extremal surface corresponds to the interval. Its area is proportional to the entanglement entropy of the interval. The dashed blue circle $z=z*$ is determined by the interval in the boundary. The area of the bulk circle is proportional to the quantity $\frac{\partial S_{EE}}{\partial \Delta x}$. One should not be confused with the dimension matching problem, as is shown by (\ref{dEEformula}), $\frac{\partial S_{EE}}{\partial \Delta x}$ is actually the gravitational entropy density. }\label{pspx}
\end{center}
\end{figure}
 Roughly speaking, $\frac{\partial S_{EE}}{\partial \Delta x}$ has a geometrical meaning, as it is a measure of the area of the dual bulk circle. Several remarks are in order.
     \begin{enumerate}
     \item It is easy to see
     \be
     \frac{\partial S_{EE}}{\partial \Delta x}\ge0.\label{see>0}
     \ee
      This is obvious from the geometric meaning of $\frac{\partial S_{EE}}{\partial \Delta x}$, as the area is always non-negative. In the $QFT$ side, $\frac{\partial S_{EE}}{\partial \Delta x}$ is proportional to the central charge of the system, which counts the degrees of freedom of the system, so it should be non-negative.
      \item
      There are two possible ways to saturate the bound in (\ref{see>0}). The first one is to set the metric to be degenerate from the equation (\ref{dEEformula}). However, this is quite unusual so we will not consider this possibility. Another possible way
       to approach $0$ is when $\Delta x\to \infty$. In this limit, the dashed circle in Figure \ref{pspx} tends to a point if the bulk is global AdS. However, even though $\Delta x\to \infty$, the quantity $\frac{\partial S_{EE}}{\partial \Delta x}$ does not always tend to zero. When the background is an asymptotically $AdS$ black hole, for example, a $BTZ$ black hole,
       \be
       S_{gr}\to S_{BH},\ \Delta x\to \infty.\label{limit}
       \ee
       As $\Delta x\to \infty$, the extremal surface goes towards the horizon of the black hole. Combining (\ref{dEEformula}) with (\ref{limit}), one finds
    \be
    S_{EE}(\Delta x)\to s_{thermal}\Delta x,\ \Delta x\to \infty \label{EETh}
    \ee
    where $s_{thermal}$ is the thermal entropy density. This is consistent with the well known result.
    \item
     Let us give a precise realization of (\ref{dEEformula}). Choosing a general metric which is a holographic background,
    \be
    ds^2=-g_0(z)dt^2+g_1(z)dx^2+\sum_{i=2}^{d-1}g_i(z)dy_i^2+f(z)dz^2,
    \ee
    then we find
    \be
    \frac{\partial S_{EE}}{\partial \Delta x}=\frac{l_2\cdots l_{d-1}}{4G_N}G(z^*)
    \ee
    where $G(z)=\sqrt{g_1\cdots g_{d-1}}$. This is consistent with the result in \cite{Myers}.
    \item We can make use of the relation (\ref{dEEformula}) to reconstruct the black hole background. We take $AdS_3/CFT_2$ as the prototype. The one interval entanglement entropy in $CFT_2$ is  \cite{Holzhey,Calabrese}
        \be
        S(\Delta x)=\frac{c}{3}\ln \frac{\Delta x}{\delta}.\label{eeads3}
        \ee
        From holography, this comes from the minimal length of the curves which is homologous to the interval in the bulk of $AdS_3$ in Poincar\'e coordinate. After a conformal transformation, (\ref{eeads3}) becomes the finite temperature entanglement entropy
        \be
        S(\Delta x)=\frac{c}{3}\ln\frac{\beta}{\pi \delta}\sinh\frac{\pi \Delta x}{\beta}
        \ee
        As there is no rotation, we may assume the metric in the bulk to be\footnote{We do not have a good reason why the $g_{tt}$ should be the inverse(up to a minus sign) of $g_{rr}$ just from $CFT$ consideration yet. So we just assume it is like this.}
        \be
        ds^2=\frac{-dt^2}{f(r)}+f(r)dr^2+r^2 dx^2
        \ee
       The metric should be asymptotic $AdS_3$, so $f(r)\to \frac{1}{r^2}$ when $r\to\infty$. Then we can do power expansion of $f(r)$ around $r\to \infty$ as
      \be
       f(r)=\frac{1}{r^2}(1+\frac{\alpha_1}{r}+\frac{\alpha_2}{r^2}+\frac{\alpha_3}{r^3}+\frac{\alpha_4}{r^4}+\cdots).
           \ee
        From the RT formula, we find
         \be
          \Delta x=2\int_{r_0}^{\infty}\frac{dr r_0}{r^2\sqrt{r^2-r_0^2}}\sqrt{f(r)}\label{deltax1}
         \ee
where $r_0$ is the minimal value of the bulk coordinate where $r(x)'=0$\footnote{$r_0$ is similar to $z^*$ in Figure \ref{pspx}.}. On the other hand, from (\ref{dEEformula}),
\be
\frac{2\pi}{\beta}\frac{\cosh\frac{\pi\Delta x}{\beta}}{\sinh\frac{\pi\Delta x}{\beta}}=r_0,\label{deltax2}
\ee
we expand $\Delta x$ by the powers of $\frac{1}{r_0^n}$ in (\ref{deltax1},\ref{deltax2}), then the coefficient $\alpha_i$ are determined order by order
\be
\alpha_1=0,\ \alpha_2=\frac{4\pi^2}{\beta^2},\ \alpha_3=0,\ \alpha_4=\frac{16\pi^4}{\beta^4}\cdots
\ee
This determines $f(r)$ uniquely
\be
f(r)=\frac{1}{r^2-r_+^2},\ r_+=\frac{2\pi}{\beta}.
\ee
    \end{enumerate}

 The right hand side of (\ref{dEEformula}) is finite, hence for a general entanglement entropy of a strip
\be
S_{EE}=\frac{f_{d-2}}{\delta^{d-2}}+\cdots+\frac{f_{1}}{\delta}+p \ln \frac{g}{\delta}+ q, \label{EEstrip}
\ee
 the function $f_{d-2},\cdots,f_1,p$ should be independent of $\Delta x$. One can check this point for some simple $AdS$ cases\cite{Ryujhep}. In (\ref{EEstrip}) the last two terms are the universal terms, which are related to anomaly. The bulk surface which are only related to these universal terms deserves further study.

 If we define two new quantities
\bea
F_u=\lim_{n\to\infty}\sum_{k=1}^{n}\xi(I_{k})-\xi(I_k\cap I_{k+1})\\
F_l=\lim_{n\to\infty}\sum_{k=1}^{n}\xi(I_k\cup I_{k+1})-\xi(I_k)
\eea
where $\xi(I)$ is the causal information entropy, then we still find $F_u-F_l\to0$. However, as the causal information entropy does not satisfy the strong subadditivity in general\cite{Hubeny1204}, we cannot find a surface whose area is equal to them. This interprets why the causal information entropy is not a candidate in constructing the Differential Entropy\cite{Myers}.

\section{Discussion and Conclusion}

In this work, we have defined two bounds, Upper Differential Entropy $E_u$ and Lower Differential Entropy $E_l$,  in any quantum field theory. Both of them are constructed in terms of an infinite number of intervals(strips) which changes smoothly. When the quantum field theory is translational invariant and its entanglement entropy changes smoothly with respect to the length of the interval, the two bounds are actually equal. This has interesting implication when the quantum field theory has a gravitational dual. In the case that the RT formula is exact, we have shown that there is an emergent surface in the bulk whose gravitational entropy $S_{gr}$ falls exactly between the two bounds. In the infinite interval limit,  the three quantities must be equal. In this way, we have proven the holographic equivalence  found in
\cite{Balasubramanian1310} and extended in \cite{Myers}. The remarkable fact in the proof is that we just need some general properties of quantum field theory and the strong subadditivity of entanglement entropy. Another interesting corollary is encoded in the relation (\ref{dEEformula}), which intuitively interpret the quantity $\frac{\partial S_{EE}}{\partial \Delta x}$ to be the gravitational entropy density.

When the bulk theory is not the Einstein-Hilbert gravity, the area functional should change correspondingly. However, once the functional is extensive, the strong subadditivity is still satisfied\cite{Headrick0704}. In this case, one should replace $S_{gr}$ to a generalized area functional and then the identity (\ref{equality}) holds. However, there is indeed the case that the entropy  functional is not extensive. For example, the holographic higher spin entanglement entropy, which is proposed to be\cite{Boer, Ammon}
    \be
    S_{EE}(P,Q)=\frac{k_{cs}}{\sigma_{\frac{1}{2}}}\ln \lim_{\rho_0\to\infty} Tr_{\mathcal{R}}\mathcal{P}\exp(\int_Q^P\bar{A})\mathcal{P}\exp(\int_P^Q A)|_{\rho_Q=\rho_Q=\rho_0}
    \ee
     In the dual 2D CFT with W-symmetry, we can obtain the equality $E_u=E_l$ and their explicit forms from (\ref{URE}). However, it is not clear if  the functional is extensive for the bulk line which connects the point $P$ and $Q$. So whether the quantity $E_u$ or $E_l$  corresponds to some quantity in the bulk or not is a topic which deserves further exploration.

The proof of the equality $E_u=S_{gr}=E_l$ supports the idea that arbitrary bulk surface may connect to a gravitational entropy. However, this equality can only be proved when the boundary entanglement entropy is evaluated for the intervals or the strips. Correspondingly, the bulk surface $z=z(x)$  does not depend on other coordinates. For more general bulk surface, we have not found a suitable observable in the boundary theory. We believe some modification of the definition of $UDE$(or $LDE$) is inevitable as one cannot define two regions in succession in higher dimensions. For example, in three dimensional quantum field theory,  we need two coordinates $(x,y)$ to describe a general spacelike region. The interval $I_x$ in the definition of $UDE(LDE)$ need some generalization to $R_{x,y}$. There are at least two possible quantities $R_{x,y+dy}, R_{x+dx,y}$ analogue to $I_{x+dx}$. Consequently, we can construct out at least three entanglement entropies $S(R_{x,y}),S(R_{x+dx,y}),S(R_{x,y+dy})$. From the proof, $E_u=E_l$ is related to the fact that the mutual information $S(I_x,I_{x+dx})$ tends to zero in the second order of $dx$. So it seems that we need a more general definition of mutual information $S(R_{x,y},R_{x+dx,y},R_{x,y+dy})$. Since the mutual information is related to the strong subadditivity, maybe we need some generalization of strong subadditivity in order to include more objects.

 As noted in \cite{Myers}, when $x_c'\le0$, the definition of Differential Entropy should be modified. Hence, in this case, we should exchange $\cap$ and $\cup$ in $UDE$ and $LDE$ for the interval $I_x$'s which obey $x_c'\le0$. The equality between $UDE$ and $LDE$ still holds
\\\\

\noindent {\large{\bf Acknowledgments}} \\~\\
The work was in part supported by NSFC Grant No.~11275010, No.~11335012 and No.~11325522.


\section{Introduction}
The generalized hypergeometric function ${}_{A}\text{F}_{B}$  with A numerator and B denominator parameters is defined by
\begin{equation}
{}_{A}\text{F}_{B}\left[ \begin{array}{c} (a)_A \\ (b)_B \end{array} ; \begin{array}{cc} z \end{array}\right]=
{}_{A}\text{F}_{B}\left[ \begin{array}{c} a_1,a_2,\ldots,a_A \\ b_1,b_2,\ldots,b_B \end{array} ; \begin{array}{cc} z \end{array}\right]
=\sum_{k=0}^{\infty}\frac{\prod_{j=1}^{j=A}(a_{j})_{k}}{\prod_{j=1}^{j=B}(b_{j})_{k}}\frac{z^k}{k!}\label{eq1} \end{equation}
 where $a_i\in\bC$, $b_j\in\bC\setminus \bZ_0^-$, $1\leq i\leq A$, $1\leq j\leq B$ and

 $(\alpha)_{k}=\alpha(\alpha+1)\ldots(\alpha+k-1)=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}$ is the Pochhammer symbol.
    \\If any of the numerator parameters is a negative integer, say $a_1=-n,n\in\mathbb{N}$, the series terminates and reduces to a polynomial of degree $n$, called a {\it generalized hypergeometric polynomial}.
We are, in this note,  interested in  the asymptotic zero distribution of the polynomial coming from an ordinary hypergeometric polynomial
    \begin{equation}
   \label{hyp}
    p_n(z)={}_{2}\text{F}_{1}(-n,a_2(n);b_1(n);z)
   \end{equation}
   with complex parameters, dependent linearly on $n$, as $n\to \infty$.

 Based on experimental evidence  and results by previous authors, we made in \cite{AARB} a conjectures on the asymptotic behaviour of zeros of a certain class of such hypergeometric polynomials.   The  purpose of this note is to partially prove this conjecture.
   \\Our result extends results by earlier authors. We will describe them.

  In 1999, K.Driver and P.Duren studied the zeros of the hypergeometric polynomial
  $F(-n, kn + 1; kn + 2; z)$  for integers $k,n>0.$ They used the  Euler integral representation together with a general theorem of Borwein and Chen and showed the following result.
  \begin{theorem}\label{Th1} Let $k ~\text{and}~  n \in \mathbb N.$  Then the zeros of the hypergeometric polynomial
  $$F(-n, kn + 1; kn + 2; z)$$
cluster on the loop of the lemniscate
 $$\{z:|z^k(z-1)|=\frac{k^k}{(k+1)^{k+1}};~Re(z)>\frac{k}{k+1}\}$$
 as $n \to \infty$.
 \begin{proof}
 See, \cite[Theorem 1]{KDPD1}

 \end{proof}
 \end{theorem}
 In 2001, K.Boggs and P.Duren gave the following extension of Theorem \ref{Th1}
\begin{theorem}
For arbitrary $k > 0$ and $\emph{l}>0$  the zeros of the hypergeometric polynomial $$F(-n, kn + l+1; kn +l+ 2; z)$$ cluster on the loop of lemniscate $$|z^k(z-1)|=\frac{k^k}{(k+1)^{k+1}}~ \text{with}~ Re(z)>\frac{k}{k+1}$$.
\end{theorem}
\begin{proof}
 See, \cite[Theorem 2]{KDPD1}

 \end{proof}
In this paper, we continue to study the asymptotic properties of zeros of hypergeometric function $F(-n, kn + 1; kn + 2; z)$ to allow $k$ to be any complex number with non negative real part. We have the following theorem.

\begin{theorem-non} {\it
 Let $\alpha = \eta + i \zeta $  where  $\eta > 0$ and $\zeta \neq 0$. The zeros of the hypergeometric polynomials
 $$p_n(z)={}_{2}\text{F}_{1}\left[ \begin{array}{c} -n, \alpha n + 1 \\ \alpha n+2 \end{array} ;
 \begin{array}{cc} z \end{array}\right],$$ asymptotically cluster on the level curve
  \begin{equation*}\label{lami}
{ |z^\alpha(1-z)|}=\vert(\frac{\alpha}{(\alpha+1)})^{\alpha}(\alpha+1)^{-1}\vert.
 \end{equation*}
}
 \end{theorem-non}

 In  \cite{AARB} it was proved, using indirect methods, that a convergent sequence of zeroes(described as a limit of corresponding measures has to cluster along a level curve of the function
 ${ |z^\alpha(1-z)|}$. This result identifies which level curve. As we will see, it is the unique level curve through the saddle point of the function. As seen from the results quoted, the zeroes will not cluster on the whole level curve. We prove in Lemma \ref{lemma:principalbranch} and Proposition \ref{lemma:zero free} that there is a certain region, containing the left half-plane that is zero-free. In  \cite{A}  we also obtain the density of the zeros. Some pictures of the accumulation of the zeroes may also be found there.

  The proof  uses, following the references above,   Euler integral representation of these hypergeometric polynomials and the saddle point method.
   For convenience we will give a short description below of the saddle point method and the Euler integral representation of Gauss hypergeometric functions.

   \section{ Integral representation of Hypergeometric function and the Saddle point method}
   \subsection{Integral Representation of Hypergeometric function}
The Gauss hypergeometric series ${}_{2}\text{F}_{1}(a,b;c;z)$ has the following integral representation due to Euler See \cite[Theorem 2.2.1]{A}. If $Re(c)>Re(b)>0,$ then
\begin{equation}
{}_{2}\text{F}_{1}\left[ \begin{array}{c} a,b \\ c \end{array} ; \begin{array}{cc} z \end{array}\right]
=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-tz)^{-a} dt,
 \label{eq2}
 \end{equation}
 in the $z$ plane cut along the real axis from 1 to $\infty.$  Here it is understood that $\arg t= \arg (1-t)=0$ and $(1-zt)^{-a}$ has its principal value.

 Choosing $a=-n$ and $c=b+1$ and $b=\alpha n+1$ for $n \in \mathbb N$ for an arbitrary complex number $\alpha,  Re(\alpha)>0$, we
get
$${}_{2}\text{F}_{1}\left[ \begin{array}{c} -n, \alpha n+1\\ \alpha n+2 \end{array} ; \begin{array}{cc} z \end{array}\right]
=(\alpha n+1)\int_0^1t^{\alpha n}(1-tz)^{n} dt.$$

 \subsection{The Saddle Point Method }
  We  use the saddle point method \cite{CBSO,NG} to get an asymptotic expansion for the Euler integral of the hypergeometric polynomial $F(-n, \alpha n+1; \alpha n+2; z)$, which will help us to understand the asymptotic distribution of the zeros.

  Let $\gamma$ be a contour in the complex plane and the function $\phi$ is
holomorphic in a neighborhood of this contour. We want to understand
the asymptotics, as $n \to \infty$, of  an integral:
$$I_n(z)=\int_{\gamma}e^{n\phi(z)}dz.$$
The idea of the saddle point method
is to deform the contour in such a way that the main contribution to the
integral comes from a neighborhood of a single point.
Consider the derivative $\phi'(z)$ of an analytic function $\phi(z)=u(x,y)+i v(x,y)$. It is an immediate consequence of the Cauchy-Riemann equations that the gradient of $u$ is given(as a complex vector) by $\overline \phi'(z)$. It is orthogonal to the gradient of $v$, and hence if $\phi'(z_0)\neq 0$, there is a unique curve through $z_0$ characterized by both the property that $Im \phi(z)=Im \phi(z_0)$ and that along this curve $Re \phi(z)$ grows fastest. This curve is called a curve of {\it steepest ascent}(or {\it descent,} if we reverse the direction).

A saddle point is a point $t_0$ where $\phi'(t_0)=0$, and if furthermore $\phi''(t_0)\neq 0,$ then it is called a simple saddle point.
Then in a small neighborhood of a simple saddle point $t_0$ the level curve $Im \phi(z)=Im \phi(t_0)$ consists
of two analytic curves that intersect orthogonally at the point $t_0$ and separate
the neighborhood of $t_0$ in four sectors. Seen from $t_0$ there are four curve segments starting at $t_0$, and along two of them $Re \phi(z)$ decreases from $Re \phi(z_0).$ Call these curves $\gamma_1,~\gamma_2$. We will use a simple form of the saddle point method, which can be found in \cite{CBSO, IA}

\begin{proposition}
\label{prop:saddlepoint}
Suppose there is only one simple saddle point $t_0$ and $\gamma_1$ contains the point $a$ and $\gamma_2$ contains the point $b$. Furthermore, we assume that the curves from $a$ to $t_0$ along $\gamma_1$ and from  $t_0$ to $b$ along $\gamma_2$ are finite. Then
$$I_n(z)=\sqrt{\frac{2\pi}{-\phi''(t_0)}}n^{-\frac{1}{2}}e^{n\phi(t_0)}[1+O(n^{-1}))]$$
\end{proposition}

\section{Proof  the Theorem}
We are (following \cite{LDJG}) going to prove the theorem by applying the saddle point method to evaluate the asymptotic expansion of the Euler integral of the polynomial

$${}_{2}\text{F}_{1}\left[ \begin{array}{c} -n, \alpha n+1\\ \alpha n+2 \end{array} ; \begin{array}{cc} z \end{array}\right]
=(\alpha n+1)\int_0^1t^{\alpha n}(1-zt)^{n} dt,$$  where $\alpha=\eta + i\zeta,$ $\eta>0$ and $\zeta\neq 0.$ We let
$$p_n=\int_0^1 [g(t)]^ndt,$$

where $g(t)=t^{\alpha }(1-tz)$ is a function of the complex variable $t$, and we have chosen the ordinary branch of the complex logarithm(which specializes to the real logarithm on $[0,1]$). The multivalued function $g(t)$ vanishes at $0$ and $1/z$, and has $t=0$ as the only branch point.

\subsection{The saddlepoint}
To find the saddle points consider $\phi(t)=\alpha\log t + \log (1-zt)$, for the principal branch determination of the logarithm. For each fixed $z\neq 0$, the function $\phi$ has two branch points, at $t=0$ and $t=\frac{1}{z}$. Since
\begin{equation}
\label{eq:dlog}
\phi'(t)=\frac{{\alpha-zt((\alpha+1))}}{t(1-zt)}
\end{equation}
we see that $\phi'(t)=0$ if and only if  $t=t_0:=\frac{\alpha}{(\alpha+1)z}$, this is the only saddle point of $\phi$.
Since $\phi''(t)=-(1+\alpha )^3z^2/\alpha $, it is a simple saddle point if $\alpha \neq -1$ and $z\neq 0$.(The first condition is true by assumption, and it is easy to see from the description of the hypergeometric polynomials that $z=0$ is not a zero of any $p_n$.) The fact that there is only one saddle point, makes it easy to exploit the local description for some global information. Note that the fact that the derivative $\phi'$ is defined in the whole $\mathbb C$ by (\ref{eq:dlog}) shows that paths of steepest ascent/descent are unambiguously defined, irrespective of choices of branches of logarithms. This follows since the gradient(considered as a complex number) at $t$ of $Re \phi$ by the Cauchy-Riemann equations is just $\overline{\phi'(t)}$.
 \begin{lemma}
 \label{lemma:steepest descent}  Let $t_0=\frac{\alpha}{(\alpha+1)z}$ as above be the saddle point of $g(t)$, where $\alpha \neq -1$ and $z\neq 0$.
 \begin{enumerate}
   \item[i)]  There are exactly two paths, of steepest descent,  starting at the saddle point $t_0$ along which $Re \phi(t)$ decreases,  one,  say $\gamma_1$ goes to $0$ and the other, say $\gamma_2$, to $\frac{1}{z}.$
    \item[ii)] There are exactly two paths $\delta_1$ and $\delta_2$, of steepest ascent,  starting at the saddle point $t_0$ along which $Re \phi(t)$ increases,  both going to infinity. Together they split the plane into two (closed) simply-connected regions $D_0$ and $D_{1/z}$, with the property that if $t\in D_0$, there is a path of steepest descent from $t$ to $0$, and correspondingly for $D_{1/z}$.
     \end{enumerate}
     \end{lemma}
     \begin{proof} i) By the fact that $\phi''(t_0)=\frac{z^2(\alpha+1)^3}{\alpha}=|\phi''(t_0)|e^{i\beta}\neq 0,$ we know the local behavior of $\phi(t)$ at $t=t_0$(it will behave like $\beta t^2,\ \beta\in \mathbb C$ around the origin.) We see in particular that there are indeed locally two paths with $\text{Im} \phi(t)=\text{Im}  \phi(z_0)$ along which  $\text{Re} \phi(t)$ decreases. By the fact that $t_0$ is the only saddle point, these paths have to end up in either of the two points $0,1/z$, where $\text{Re} \phi(t)=-\infty$(the paths cannot vanish to infinity since we have that $\alpha=\eta+\zeta i, $ with $\eta>0$, so that $\lim_{t\to\infty}\text{Re}\phi(t)=\infty$.) It remains to check that they do not end up in the same point. They cannot intersect, again since there is only one saddle point.  By calculating the imaginary part $\text{Im}  \phi(t) $ around each branch point we find that there is a most one path {\it starting} at the branch point with constant $\text{Im}  \phi(t) $. Around $t=1/z$ the function will behave as $\log w+\delta$, for som $\delta\in \mathbb C$, and so there will at most be one ray corresponding to constant imaginary value. Around $t=0$, the function will behave as $t^\alpha$, with imaginary part $\zeta log\vert t\vert+\eta Arg t$, and this will be constant along a spiral approaching  $t=0$, (since $\zeta\neq 0$ by assumption) with a unique spiral corresponding to a fixed imaginary value. Hence the two paths have to end up at different points.

     ii) The second part follows similarily since the two paths cannot intersect themselves or each other, and since $\lim_{t\to\infty}\text{Re}\phi(t)=\infty$.
\end{proof}

 There is given $z\in \mathbb C$ two cases that may occur: either $1\in D_0 $ or $1\in D_{1/z}$. Define $E=\{ z: 1\in D_{1/z}\}$. In the case that $\alpha\in \mathbb R$, $z\in E$ is described by the condition $Re z\geq \alpha/(\alpha+1)$(see \cite{LDJG}), but we do not  have a good description of it for general $\alpha$. However we have an indirect description of it in terms of the descent picture of $w^\alpha(1-w)$. As above this function has a unique simple saddle point at $\alpha/(\alpha+1)$, and the paths of steepest ascent split the complex plane into two regions $\tilde D_0$ and $\tilde D_1$, containing the two branch points of the logarithm $0$ and $1$ respectively.
 \begin{lemma} $E=\tilde D_1$.
 \end{lemma}
\begin{proof} We use the notation of the preceding lemma. Now $z^\alpha g(t)=z^\alpha t^{\alpha }(1-tz)=w^\alpha(1-w)$, with $w=zt$(using a suitable determination of $\log$). There is then a path of steepest descent from $t=1$ to $t=0$, with respect to $g(t)$, if and only if there is a path of steepest descent from $w=z$ to $w=0$, with respect to $w^\alpha(1-w)$. In more detail, putting $\psi(w)=\log w^\alpha(1-w) $, we see that $z\psi'(zt)=\phi'(t)$, so that the tangent vectors of steepest descent paths are related to each other by multiplication by $\bar z$(compare the note just before the previous lemma). So if $\gamma(s),\ s\in [0,1]$ is a path from $z=\gamma(1)$ to $\gamma(0)=0$ such that the tangent $\gamma'(s)=\overline{\psi'(\gamma(s))}$, then $\delta(s)=z^{-1}\gamma(s)$  is a path from $1$ to $0$, such that $\delta'(s)=z^{-1}\overline{\psi'(z^{-1}\gamma(s))}=\vert z\vert ^{-2} \overline{\phi'(\delta(s))}$. Clearly $\delta$ will then be a parametrization of the path of steepest descent for $g(t)$. Hence  $1\in D_0\iff z\in \tilde D_0,$ which proves the lemma.
\end{proof}
We can see that $E^c$ contains the left half plane, using that $\eta>0$, where $\alpha=\eta+i\zeta$.\begin{lemma}
\label{lemma:principalbranch}
Suppose that $z=x+iy$ with $x\leq 0$. Then $z\in E^c$.
\end{lemma}
 \begin{proof}It suffices to show that the segment $\gamma(s)=sz, s\in [0,1]$ is an ascending curve relative the function $\psi(w)=w^\alpha(1-w)$.
 For $\gamma$ to be ascending from $0$, it is enough to show that $Re(\overline{\psi'(\gamma(s))\gamma'(s)})>0$, which after some simplifications is equivalent to $E(s):= \eta -2\eta xs+(-x+\eta (x^2+y^2))s^2+(x^2+y^2)s^3\geq 0$. This is a degree three curve in the plane and clearly $\lim_{s\to\infty}E(s)=\infty$, and it will have at most two stationary points, and between them an inflection point.
Now $E''(s)=0$ has the solution $s=\frac{x-\eta(x^2+y^2)}{3(x^2+y^2)}\leq 0$, by assumption. Since furthermore $E'(0)=-2\eta x\geq 0$ and $E(0)=n>0$, $E(s)$ will be strictly increasing with $s\geq 0$.

 \end{proof}

\subsection{Zero-free region}

 We sketch a proof that the complement to $E$ describes a zero-free region.

 \begin{proposition}
 \label{lemma:zero free} If $z\notin E$, then $z$ is not the limit of zeroes of $p_n(z)$.
 \end{proposition}
 \begin{proof}
(sketch) Assume that $z_n\to z$, is a sequence of zeros $p_n(z)=0$, where $z\notin E$. Hence $z_n\notin E$ for large enough $n$, and so there will be paths $\gamma_n$ of steepest ascent from $0$ to $1$, and this means, using standard Laplace-type techniques, that the integral
 $$
0=(\vert  \int_0^1 [t^{\alpha }(1-tz_n)]^ndt\vert )^{1/n},
 $$
 will have limit $\vert 1-z\vert $ (compare the proof of Lemma \ref{lemma:general}) which will be non-zero. This is a contradiction.
 \end{proof}

\subsection{The first integral} Now assume that $z\in E$. Then there is a path of steepest ascent from $1/z$ to $1$, and a path from $0$ to $1/z$, first as a path of steepest ascent to the saddle point, and then as a path of steepest descent to $1/z$.
We deform the integral accordingly to
\begin{equation}\label{deform}
\int_0^1 [g(t)]^ndt=\int_0^{\frac{1}{z}} [g(t)]^ndt+\int_{\frac{1}{z}}^1 [g(t)]^ndt=I_1+I_2
\end{equation}

We will now detail the path from $0$ to $1/z$ in the integral. Note that the spiral $\gamma_1$ around $0$ will intersect the line $L$ between $0$ and $1/z$ in an infinite set of points clustering towards $0$. Choose an arbitrary $\epsilon>0$. Start in $0$, walk along $L$  a distance at most $\epsilon $, until one of the intersection points $w_\epsilon$with $\gamma_1$ is reached. Call this part $L_\epsilon$, and continue along the part $\gamma_\epsilon$ of $\gamma_1$ that remains to $z_0$ and then to $1/z$ along $\gamma_2$. Starting with the principal branch of the logarithm that is defined $]0,1]$, at $z_0=\alpha/(\alpha+1)1/z$ and at $1/z$(by lemma \ref{lemma:principalbranch}), we may analytically continue this one along $\gamma_1$ to $w_\epsilon$; we interpret $t^\alpha:=e^{\alpha \log t}$ in this sense. Now if $\epsilon$ is small enough, $\vert1-zt\vert <2$, for $t\in L_\epsilon $, and so there exists $M>0$ such that $\vert t^{\alpha n}(1-zt)^{n}\vert <M t^\eta$ , for $t\in L_\epsilon $($M$ will depend on which branch of the logarithm we use). As a consequence, using that $\alpha=\eta+\zeta i$, with $\eta>0$,
the integral
\begin{equation}
\label{eq:Lepsilon}
\vert \int_0^{w_\epsilon}t^{\alpha n}(1-zt)^{n} dt\vert \leq \frac{M\epsilon^{\eta +1}}{\eta+1}.
\end{equation}

The integral between $w_\epsilon$ and $1/z$ is the one we will estimate with the saddle point method. Call it $I_1$. Note that the paths $\gamma_\epsilon$ and $\gamma_2$ are finite.

Now we rewrite the integral as

$$ \int_0^{\frac{1}{z}} [g(t)]^ndt=\int_0^{\frac{1}{z}}e^{n\phi(t)}dt,$$

where $\phi(z)=\log (t^\alpha(1-zt)).$ Recall that the saddle point was $z_0=\frac{\alpha}{(\alpha+1)z}$. As a consequence of the description of the paths of steepest descent, and standard techniques we may now deduce the value of the integral from Lemma \ref{prop:saddlepoint}.

\begin{lemma}
\label{lemma:saddlepoint} Let $z\neq 0$, and use the notation above. For any $\epsilon>0$,
\begin{equation}
\label{eq:firstintegral}
\vert \int_0^{\frac{1}{z}} [g(t)]^ndt\vert = g(z_0)^n\sqrt{\frac{2\pi}{n|\phi''(t_0)|}}+O(1/n)+\epsilon.
\end{equation}
\end{lemma}

\subsection{The Second integral}
Now we turn to the second integral
$$I_2(z)=\int_{\frac{1}{z}}^1[g(t)]^ndt,$$
 where $g(t)=t^\alpha(1-zt),$ and we assume that $z\in E$, so that we have a path of steepest ascent from $1/z$ to $1$. In particular, the path will never wind around $0$, so we may use the principal value determination of $\log$ for the definition of $t^\alpha$. We heuristically see that then the value $(1-z)^n$ at $t=1$ should dominate the integral asymptotically, and we will also provide details of a proof of this. We start with the following change of variables idea that we have taken from \cite{KDPD1}.

Letting  $$g(t)=t^\alpha(1-zt)=s(1-z), 0\leq s\leq 1,$$ we have that $s=0\Leftrightarrow t=\frac{1}{z}$ and $ s=1\Leftrightarrow t=1$
and that this definition implicitly defines  a path, say $\Delta,$ by  $t=t(s),$ between $\frac{1}{z}$ and $1$.
\begin{lemma}
\label{lemma:changepath}
$\int_{\frac{1}{z}}^1[g(t)]^ndt=\int_\Delta [g(t)]^ndt$
\end{lemma}
\begin{proof}
Since $$|g(t)|=O(|t|^\eta)~\text{and}~\eta>0,$$
the integral on a circle around $0$ goes to $0$ as the radius of the circle decreases. Hence we may move the path of integration freely in the complex plane.
\end{proof}
We have that
$$t^{\alpha-1}(\alpha-(\alpha+1)zt)dt=(1-z)ds.$$
Hence
$$I_2=(1-z)^n\int_0^1\frac{t(1-zt)s^{n-1}}{\alpha-(\alpha+1)zt}ds=(1-z)^nK(z)$$
\begin{equation}
\label{eq:second integral}
(I_2)^\frac{1}{n}=(1-z)(K(z))^\frac{1}{n}
\end{equation}

 The calculation of the limit of $I_2^\frac{1}{n}$ is completed by showing
 \begin{equation}\label{lim1}
 \lim_{n \to \infty}|K(z)|^\frac{1}{n}=1.
 \end{equation}

  Set
  $$ f(s)=\frac{t(1-zt)}{\alpha-(\alpha+1)zt}$$
so that
  $$k(z)=\int_0^1 f(s)s^{n-1}ds.$$
Then \eqref{lim1} is a consquence of the following general lemma.
   \begin{lemma}
   \label{lemma:general}
   Assume that $f(z)$ is a non zero function analytic in a domain $D,$ containing the unit interval $[0,1]$ on the real axis. Then
   $$\lim_{n\to \infty}\Bigg(\int_0^1f(s)s^{n-1}ds\Bigg)^{\frac{1}{n}}=1$$
   \end{lemma}
   \begin{proof}
   Let $I(n)=\int_0^1f(s)s^{n-1}ds.$ Since there is a positive number $M$, $0<M$ such that $|f(s)|\leq M$ if $s\in[0,1],$ we have that $$|I(n)|^{\frac{1}{n}}\leq M^{\frac{1}{n}},$$
and so
$$\limsup_{n\to \infty}|I(n)|^{\frac{1}{n}}\leq 1.$$
We may assume that $Re f(z)$ is not identically zero on $[0,1]$, since not both $Re f(z)$ and $Im f(z)$  can be identically zero on $[0,1]$ and multiplication by $i$ exchanges $Re f(z)$ and $Im f(z)$  and does not change $|I(n)|.$ We may also assume that $f(z)$ is not constant, since otherwise the result is immediate.
Then $\{z:Re f(z)=0\}$ has a finite intersection with$[0,1]$. Let $\delta$ be the largest element in this intersection which is strictly less than 1, so that $h(z):=Re f(z)$ is of constant sign in $[\delta,1)$.

We may assume that $h(s)> 0$ in $[\delta,1],$ if necessary, by multiplying $f(z)$ with $-1$.
\\ Now, $h(s)$ is bounded on $[0,1]$ so there is $M>0$ such that $-M\leq h(s)\leq M.$ Furthermore, there is to each $C:$
$$0<C<\max \{h(s), s\in[\delta,1]\},$$
an interval $[\delta_2,\delta_1]\subset [\delta,1)$ such that $\delta_2\leq s\leq \delta_1\Longrightarrow  h(s)\geq C.$ We may take $\delta_1$ maximal with this properties (if $h(1)\ne 0,$ then $\delta_1=1$ if $C $ is small enough.)
\\ Now estimate
$$Re(I(n))\geq
\int_0^\delta h(s)s^{n-1}ds+\int_\delta^1h(s)s^{n-1}\geq \int_0^\delta-Ms^{n-1}ds+\int_{\delta_2}^{\delta_1}Cs^{n-1}ds$$
$$ =-M\frac{\delta^n}{n}+C\frac{\delta_1^n}{n}-C\frac{\delta_2^n}{n}
\\=\frac{\delta_1^n}{n}(-(\frac{\delta}{\delta_1})^nM-(\frac{\delta_2}{\delta_1})^nC+C).$$
Since $\frac{\delta}{\delta_1}<1$ and $\frac{\delta_2}{\delta_1}<1,$ there exists $n_0$ such that $n>n_0$
$$Re I(n)\geq \frac{\delta_1^n}{n}\frac{C}{2}\Longrightarrow \liminf|I(n)|^{\frac{1}{n}}\geq \delta_1$$
Now, $\delta_1$ depends on the choice of C and clearly $\lim_{c\to 0}\delta_1=1.$ Hence the lemma is proved.
  \end{proof}
  \subsection{Proof of the theorem}
  A zero $z_n$ of the polynomial $p_n(z)$ in (\ref{hyp}) satisfies, by  (\ref{deform}),   the equation $I_1=-I_2$.
  Supposing that we have a sequence of zeros $z_n\to z\in E$, we get that
  $$
  \lim_{n\to\infty}\vert I_1\vert^{\frac{1}{n}} = \lim_{n\to\infty}\vert I_2\vert^{\frac{1}{n}} =\vert 1-z\vert,
  $$
  by (\ref{lim1}) and (\ref{eq:second integral}).
 It follows from formula (\ref{eq2}), that  $z=0$ is not a zero of $p_n$.
  Hence Lemma \ref{lemma:steepest descent} and \ref{lemma:saddlepoint} apply, and we have that
 $$   \lim_{n\to\infty}\vert I_1\vert^{\frac{1}{n}} =\vert t_0^\alpha(1-zt_0)\vert.$$

 Hence $$
\vert(\frac{\alpha}{(\alpha+1)})^{\alpha}(\alpha+1)^{-1}\vert=\vert z^\alpha\vert \vert1-z\vert.
 $$
 (the form of the left hand side is chosen to avoid problems with changing the branch of the logarithm, note that $\frac{\alpha}{\alpha+1}$ is in the right half-plane, as well as $z$, by Lemma \ref{lemma:principalbranch}.)
This finishes the proof.

\bibliographystyle{plain}


\section{Introduction}

In the search for a hardware for implementing quantum computation, Majorana fermions~\cite{majorana} have emerged as ideal candidates.
These quasi-particles appear in pairs and in case they are sufficiently separated they should be robust to quantum decoherence,
a major obstacle for achieving a quantum computer~\cite{quantum}.
Majorana fermions have been shown to exist in the ends of $p$-wave superconducting chains~\cite{kitaev}. A model proposed by Kitaev
has played an important role in clarifying the nature of these quasi-particles and how they can appear in condensed matter
systems~\cite{kitaev}. Majorana fermions are zero energy solutions of the Dirac equation with the special property that they
are their own anti-particles~\cite{shen,bernevig}. The excitations at the ends of a $p$-wave superconducting chain depend
on the quantum state of the system. This is determined by the ratio $|\mu|/2t$ between the chemical potential $\mu$ and
the electron hopping $t$. If $|\mu|/2t<1$ the chain is superconducting with non-trivial topological properties.
In this {\it weak pairing} phase it presents Majorana fermions at its ends. Otherwise, if $|\mu|/2t>1$, the chain is in
the  {\it strong coupling} superconducting phase with trivial topological properties and has no end states~\cite{alicea,cdc}.
This result can be obtained using either  Bogoliubov-de Gennes or Dirac equations with a space dependent order parameter~\cite{shen},
or more directly using a Majorana representation~\cite{kitaev}.

In this paper, we study a non-interacting two-band model of a chain of atoms with electronic orbitals of angular momentum
$l$ and $l+1$. Besides the hopping of electrons in the same orbital, we consider hybridization between different orbitals
in neighboring sites. Since these orbitals have angular momentum differing by an odd number, their wave-functions have
opposite parities. Then,  the hybridization between the orbitals in neighboring sites $i$ and $j$ is  antisymmetric,
i.e, $V_{ij}=-V_{ji}$ or in $k$-space $V(-k)=-V(k)$. For simplicity, we consider a $sp$-chain for which the importance
of the mixing of $sp$ bands for topological insulators has already been pointed out in different contexts, including that
of the spin quantum Hall effect~\cite{zhang} and cold atoms~\cite{liu}.  The interest in these chains goes back to Schokley's
work on surface states~\cite{shockley}. These states were investigated in detail~\cite{foo}, but until recently it lacked
the theoretical tools for their understanding, specially
of the zero energy surface states~\cite{kane}.

A cold atom version of the $sp$-chain, the $sp$-ladder, has recently been investigated in light of the new concepts of
topological insulators~\cite{liu}. Also a one dimensional Kondo lattice has been considered~\cite{coleman}.
Both works consider interactions among the quasi-particles besides the topological aspects of the problem.

In this work, we concentrate on the simplest, non-interacting condensed matter system, a chain with two atomic
orbitals per site with an antisymmetric hybridization between these orbitals.  We show that by tuning the parameters of this chain it is driven,
through a quantum phase transition~\cite{livro}, from a trivial  to a topological
insulator~\cite{kane}. The nature of the end states in  semi-infinite $sp$-chains, is studied using a scattering
approach that considers the end of the chain as a defect in a perfect crystal~\cite{foo}. The  scattering problem
due to the defect has an associated $T$-matrix whose poles give the energies of the end states.

In spite of the different symmetries of the $sp$ and $p$-wave chains, we find a mapping between these problems.
This mapping is purely formal as the two systems have different symmetries and physical properties.
The former is a non-interacting band insulator and the latter a $p$-wave superconductor. This mapping in the case of semi infinite chains leads to interesting results regarding the nature of the end states. We show that
both the semi-infinite $p$-wave  and  $sp$-chains  have zero energy end modes when these systems are in
non-trivial topological phases. For the $p$-wave chain, this zero energy mode is immediately recognized as
a Majorana fermion. For the $sp$-chain, the zero energy state share many properties with those of a Majorana mode.
For example, the penetration length of the zero energy end modes of both models diverge at the topological
transition separating the trivial from the non-trivial topological phases. Exact diagonalization results
confirm the correlation length exponent obtained analytically and are able to extend the results to parameter
values away from the specific symmetric point and also include the effects of disorder.

Our paper is organized as follows.  In Sec. II, we  show that the eigenspectrum of the $sp$-chain with odd parity hybridization is
identical to that of the $p$-wave superconducting chain for a specific choice of parameters.
In Sec. III, using a $T$-matrix scattering approach we calculate the decay length of the end mode
and compare with exact diagonalization results. While the analytical method is limited to a very
specific choice of the parameters, the numerical method can also be extended to arbitrary values of these parameters and gives some insights on the nature of the
end states in the topological phase. In Sec. IV, we study robustness of the end states against disorder
using exact diagonalization approach. We find that the end states survive for any values of the strength of
the symmetric disorder. In Sec. V, we show that there are two unpaired Majorana fermions in the each end of the
$sp$-chain. These Majorana fermions can combine to form a conventional fermion with hybrid $sp$ character.
We also show that if the ends of the $sp$-chain  are connected o a bulk metallic reservoir with $s$-states only, one $p$-type Majorana appears on each end of the chain.     Finally, we conclude in Sec. VI.

\section{The $sp$ and the $p$-wave superconducting chains}

Let us consider a one dimensional atomic system with two orbitals per site of angular momenta $l$ and $l+1$.
Due to their different parities the hybridization between these orbitals in neighboring sites is antisymmetric.
This system can be described by a two-band Hamiltonian with an antisymmetric hybridization. For simplicity,
we imagine an $sp$-chain with the Hamiltonian given by~\cite{foo},
\begin{eqnarray}
\label{sp}
\mathcal{H}_{sp}&=& \epsilon^0_s \sum_j c^{\dagger}_j c_j + \epsilon^0_p \sum_j p^{\dagger}_j p_j-
\sum_j t_s (c^{\dagger}_j c_{j+1} + c^{\dagger}_{j+1} c_{j})+ \sum_j t_p (p^{\dagger}_j p_{j+1} + p^{\dagger}_{j+1} p_{j}) \nonumber \\
&+&V_{sp} \sum_j (c^{\dagger}_j p_{j+1} -c^{\dagger}_{j+1} p_{j})) - V_{ps} \sum_j (p^{\dagger}_j c_{j+1} - p^{\dagger}_{j+1} c_{j})
\end{eqnarray}
where $\epsilon^0_{s,p}$ are the centers of the $s$ and $p$ bands respectively. The $t_{s,p}$ represent
the hopping of electrons to neighboring sites in the same orbital. $V_{sp}=V_{ps}$ and the antisymmetric
nature of the hybridization between the $s$ and $p$ states in neighboring sites is taken into account
explicitly in the Hamiltonian (see Fig.~\ref{fig1}). Due to the different parities of the orbital states,
this hybridization is odd-parity, such that, $V_{sp}(-x)=-V_{sp}(x)$ or in momentum space $V_{sp}(-k)=-V_{sp}(k)$.
Thus the mixing term breaks parity symmetry in spite of the fact that the chain is centro-symmetric. We take here the chemical potential of the $sp$-chain $\mu_{sp}=0$.

\begin{figure}[th]
\centering{\includegraphics[scale=0.3]{fig1.eps}}
\caption{(Color
online) The $sp$-chain with four sites showing the parity of the orbitals and their hybridization. }%
\label{fig1}%
\end{figure}

One of our aims in this paper is to compare the $sp$-chain with Kitaev's model, so for completeness,
we give below the Hamiltonian of the $p$-wave one-dimensional supercondutor~\cite{kitaev}.
This model has been extensively studied and is the archetypical example of a system exhibiting Majorana fermions.
It's Hamiltonian is given by~\cite{kitaev},
\begin{equation}
\label{kita}
\mathcal{H}=- \mu \sum_j c^{\dagger}_j c_{j} - t \sum_{j} ( c^{\dagger}_j c_{j+1}+ c^{\dagger}_{j+1} c_{j})  + \sum_{j} \Delta \left(    c_j c_{j+1} -      c_{j+1} c_{j} \right) +    \sum_{j}   \Delta^{*} \left(c^{\dagger}_{j+1} c^{\dagger}_{j} -c^{\dagger}_{j} c^{\dagger}_{j+1} \right)
\end{equation}
where $t$ is a nearest neighbor hopping, $\mu$ the chemical potential. The anomalous nearest neighbor
correlation function  $\Delta_{ij}=\Delta$ ($\Delta_{ji}=-\Delta$) is a pairing correlation that arises
due to an  effective  antisymmetric attractive interaction between fermions in neighboring sites.
The operators $c_i$ and $c_i^{\dagger}$ destroy and create fermions on site $i$ of the chain, respectively.

At first sight the two Hamiltonians above have nothing in common since they represent two completely
different problems. In particular the Kitaev superconducting model, Eq.~\ref{kita} has a broken U(1) symmetry,
which is not the case of the $sp$-chain.  The hybridized two-band system described by Hamiltonian Eq.~\ref{sp} is a non-interacting problem,
while that given by Eq.~\ref{kita} has a condensate ground state of electron pairs due to the attractive interaction.
The only apparent similarity of the two problems is the presence of a term which breaks the parity symmetry,
the odd-parity hybridization and the $p$-wave anomalous correlation function in Eqs.~\ref{sp} and~\ref{kita}, respectively.
As we show below this feature alone will give rise  to important similarities in the electronic states of these systems.

Both Hamiltonians above can be easily solved to yield the energies of the new quasi-particles. For chains with
periodic boundary conditions, this is accomplished using the translation invariance of the system and a simple
Fourier transformation. Here we introduce the Greenian operator~\cite{foo} defined by, $\mathbf{G}=(\mathbf{1}-\mathbf{\mathcal{H}})^{-1}$.

For the $sp$ model, this is given by,

\begin{eqnarray}
\mathbf{G}_{sp}(k, \omega)=\frac{1}{D_{sp}}
\left( \begin{array}{cc} \omega-\epsilon^0_p -2t_p \cos ka & 2iV_{sp} \sin ka \\ -2iV_{sp} \sin ka &   \omega -\epsilon^0_s + 2 t_s \cos ka  \\ \end{array} \right) \\  \nonumber
\end{eqnarray}
where
\begin{equation}
\label{reldisp}
D_{sp}(\omega)=(\omega -\epsilon^0_s + 2 t_s \cos ka)(\omega-\epsilon^0_p -2t_p \cos ka) - 4 V_{sp}^2 \sin^2 ka.
\end{equation}

For   the $p$-wave superconducting chain described by the Hamiltonian, Eq.~\ref{kita}, the Greenian can be written as:

\begin{eqnarray}
\mathbf{G}_{pw}(k, \omega)=\frac{1}{D_{pw}}
\left( \begin{array}{cc} \omega + \epsilon_k & 2 i \Delta_0 \sin ka \\ -2 i \Delta_0 \sin ka &   \omega - \epsilon_k   \\ \end{array} \right)  \\ \nonumber
\end{eqnarray}
where
\begin{equation}
D_{pw}(\omega)=(\omega -\epsilon_k)(\omega+\epsilon_k) -4 \Delta_0^2 \sin^2 ka
\end{equation}
with $\epsilon_k= 2 t \cos ka - \mu$ and  $\Delta_0=|\Delta|$.

The Greenians above yield a description of the excitations
of the infinite $sp$ and $p$-wave superconducting chains. The energies of the quasi-particles for each model
are obtained from the poles of the Greenians, i.e., from the equations $D_{sp}(\omega)=0$ and $D_{pw}(\omega)=0$.
We can use them to establish a mapping between  the two models. For a special choice of the parameters of the
$sp$-chain $t_s=t_p=t$ and $\epsilon^0_s=-\epsilon^0_p=\mu$, we find a complete formal correspondence between the two problems.
The role of the antisymmetric hybridization $V_{sp}$  in the $p$-wave model is played by the antisymmetric gap $\Delta$,
such that $V_{sp}=\Delta$.  With these identifications the eigenspectra of the two models are formally equivalent.

\section{Localization length of end mode:}

A fundamental feature of the $p$-wave superconducting model is that it exhibits Majorana fermions at the ends of
the chain~\cite{kitaev}.
In this section, we investigate the nature of the end states  by two methods: (A) a $T$-matrix scattering
approach, and (B) exact diagonalization approach.

\subsection {T-matrix scattering approach:}
In order to study the edge excitations in the $sp$-chain, we consider a semi-infinite chain. The idea is to
consider the {\it end} of the chain as a defect and treat this problem using a scattering approach following
Ref.~\cite{foo}. The real space Greens function $G_S$ for the edge of the semi-infinite chain is obtained
from a Dyson equation,
$$
G_S=G_0+G_0VG_S
$$
where $G_0$ is the Greens function for the perfect crystal (infinite chain) in the site representation.
The potential $V$ is such that it creates a defect by {\it cleaving} the crystal say, between the sites $-1$
and $0$ of the infinite chain. It is given by,
 $$
V(0,-1)= \left( \begin{array}{cc} t_s & -V_{sp} \\ V_{ps} & -t_p \\ \end{array} \right)  \\ \nonumber
 $$
Using that the elements of $G_s$ between the two halves of the chain are zero one gets the end Greens function as
$$
G_S(m,m)=G_0(0)+G_0(m) T G_0(m)
$$
where $m$ is a site index. The poles of the  matrix $T(\omega)$ give the energies $E_S$ of the end states. These are obtained from the equation,
\begin{equation}
1-t_s G_{ss}(E_S)+2 V_{sp} G_{sp}(E_S) + t_p G_{pp}(E_S)=0
\end{equation}
where $G_{\alpha \beta}(E_S)=\sum_k G_{\alpha \beta}(k,E_S)e^{ika}$ with $a$ the lattice spacing ($\alpha$ and $\beta$=$s$,$p$).
For the  particle-hole {\it  symmetric} $sp$-chain, with $t_s=t_p=t$ and $\epsilon^0_s=-\epsilon^0_p=\epsilon$,
the  above equation can be solved   analytically for $t=\epsilon$ and numerically for general $\epsilon \neq t$,  yielding
\begin{equation}
\label{esp}
E_S= 0.
\end{equation}
for $\epsilon/2t < 1$, independent of the value of $V_{sp}$.
\begin{figure}[htb]
\centering
\subfigure[]{
\includegraphics[width=0.4\linewidth]{fig2a.eps}
\label{fig:subfig2a}
}\\
\subfigure[]{
\includegraphics[width=0.4\linewidth]{fig2b.eps}
\label{fig:subfig2b}
}\\
\subfigure[]{
\includegraphics[width=0.4\linewidth]{fig2c.eps}
\label{fig:subfig2c}
}
\caption{(Color
online) Dispersion relations of the 1d $sp$-model (roots of Eq.~\ref{reldisp}) for $V_{sp}=t$ and different
values of the ratio $\epsilon/2t$. a) For $\epsilon/2t >1$, the system is a standard  band insulator with a gap
between the valence and conduction bands. In this case the original bands in the absence of hybridization
(dashed lines) do not overlap.  b) For $\epsilon/2t=1$ there is a topological transition between the trivial
and non-trivial insulating states. The transition occurs with the closure of the gap between the bands.
Near the center of the band, the excitations are Dirac-like fermions with a linear dispersion given by,
$\omega =2 V_{sp} k$. c) For $\epsilon/2t<1$, the system is a topological insulator
with a zero energy mode on the end of the semi-infinite chain.}
\end{figure}
The decay of these end states,  assuming it is exponential inside the crystal, can be obtained using Schr\"odinger's equation~\cite{foo}.
The characteristic penetration length $\xi$ for $V_{sp}=t$ is given by,
\begin{equation}
\label{xi}
\xi^{-1} \propto|\ln \frac{\epsilon}{2t}|.
\end{equation}
 The condition $\epsilon/2t<1$ constraints the argument of the logarithmic in Eq.~\ref{xi} being less
than one and implies an overlap of the bands in the absence of hybridization (see Figs.~\ref{fig:subfig2a}).

The end states in the case of a generic $sp$-chain are the equivalent of the Shockley states in surface physics~\cite{shockley,beenakker}.
However, in the particle-hole symmetric case, i.e., for $t_s= t_p=t$ and $\epsilon^0_s=-\epsilon^0_p=\epsilon$ with $\epsilon/2t<1$, such that $E_S=0$, the zero energy modes correspond to the edge excitations of a topological phase of the $sp$-chain.  As we show in Sec. IV, for $\epsilon/2t=1$ the symmetric $sp$-chain has a quantum  phase transition from a topological insulator to a trivial insulating phase. The penetration length of the end mode  defined in Eq.~\ref{xi} diverges at this transition as,
\begin{equation}
\xi \propto  \frac{1}{(1-\frac{\epsilon}{2t})^{\nu}}
\label{xisp}
\end{equation}
with a critical exponent $\nu=1$.
For $\epsilon/2t<1$ (Fig.~\ref{fig:subfig2c}), where the original bands in the absence of hybridization overlap,
the symmetric $sp$-chain is in a topologically non-trivial phase that corresponds to a topological insulator (see in Sec. IV).
This phase has zero energy end modes, which disappear for $\epsilon/2t>1$, i.e., in the topologically trivial insulating phase (Fig.~\ref{fig:subfig2a}).
Now, using the mapping between the $sp$-chain and Kitaev's model  the results of the scattering approach
for the former can be immediately transposed for the $p$-wave superconducting chain.
In the case of the $p$-wave model, the mapping implies $t_p=t_s=t$ and $\epsilon^0_s=-\epsilon^0_p=\mu$.
This leads for the energy of the end mode,
\begin{equation}
\label{epw}
E_S=0,
\end{equation}
where the condition for the existence of this zero energy mode is $\mu/2t<1$. Comparing this result
with  Kitaev's  solution of the $p$-wave model model, we notice that the $E_S=0$ mode can be recognized
as the Majorana fermion at the end of the chain~\cite{kitaev}. Furthermore, the condition $|\mu|<2t$ refers to the region
of the phase diagram corresponding to the topologically non-trivial weak pairing phase where zero energy modes
are expected to exist.  Then, in the present approach,  the Majorana mode appears as an  end state arising
from a scattering problem.
As is known for Kitaev's model~\cite{kitaev}, and is obtained here for the $sp$-chain, the conditions for these systems to be in their topological non-trivial phases  with zero energy end modes do not
involve $\Delta_0$ or $V_{sp}$, respectively, although these quantities must be finite.
Then,  in the case of Kitaev's model, the scattering approach that considers the end of a semi-infinite chain
as a scattering center leads to well known results for this model~\cite{kitaev}.

As for the $sp$-chain, the exponential decay of the end state inside the superconductor has a characteristic penetration length $\xi$ given by,
 $\xi^{-1}=|\ln ( |\mu| /2t)|$ ($|\mu|<2t$). This  diverges at the weak-to-strong-pairing transition when $\mu=\mu_c=2t$ as,
\begin{equation}
\label{ximu}
\xi \propto \frac{1}{(\mu_c-\mu)^{\nu}}.
\end{equation}
with~\cite{exponent} $\nu=1$.

\subsection {Exact diagonalization approach:}

To verify the predictions of the  $T$-matrix approach  and to illustrate our results, we perform exact numerical diagonalization
of the Hamiltonian in Eq.~\ref{sp} using the transformation

\begin{align}
\label{tr1}
c_{j}&=\sum_{n}(u_{n}(j)\alpha_n+v_n(j)\beta_n)\nonumber\\
p_{j}&=\sum_{n}(-v_{n}(j)\alpha_n+u_n(j)\beta_n) \,,
\end{align}
where $\alpha$ and $\beta$ are the quasiparticle operators.
$u_n(j)$ and $v_n(j)$, which satisfy $u^2_n(j)+v^2_n(j)=1$ for
each site $j$, can be obtained from
\begin{align}
\label{bdg}
 \left( \begin{array}{cc}
h_1 & V_1  \\
V_2 & h_2  \end{array} \right)
\left(\begin{array}{c}
u_n(j)   \\
v_n(j)   \end{array}\right)=E_n
\left( \begin{array}{c}
u_n(j)   \\
v_n(j)   \end{array} \right)\,,
\end{align}
where $h_{1}u_n(j)=\epsilon^0_{s}u_n(j)-t_{s}(u_n(j+1)+u_n(j-1))$, $h_{2}v_n(j)=\epsilon^0_{p}v_n(j)+t_{p}(v_n(j+1)+v_n(j-1))$,
$V_{1}v_n(j)=-V_{sp}v_n(j-1)+V_{sp}v_n(j+1))$ and $V_{2}u_n(j)=V_{ps}u_n(j-1)-V_{ps}u_n(j+1))$.
We numerically determine the eigenvalues $E_n$ and eigenvectors $(u_n(j),v_n(j))$ of Eq.~\ref{bdg} for the chain length  $L=256$.

\begin{figure}[th]
\centering{\includegraphics[scale=0.3]{fig3.eps}}
\caption{(Color
online) Eigenvalues versus  indices  for  different system parameters. (a)
 $t_s=t_p=V_{sp}=V_{ps}=1$  and $\epsilon _s^0=-\epsilon_p^0=0.5$, (b)
all parameters  as in (a) except $V_{sp}=V_{ps}=0.1$, (c) all parameters  as in (a) except  $t_p=0.4$,
and (d) all parameters  as in (a) except $\epsilon_p^0=0$. Number of modes is 512 and a limited number of them around zero is shown. }%
\label{fig3}%
\end{figure}

\begin{figure}[htb]
\centering
\subfigure[]{
\includegraphics[width=0.4\linewidth]{fig4a.eps}
\label{fig:subfig4a}
}
\subfigure[]{
\includegraphics[width=0.4\linewidth]{fig4b.eps}
\label{fig:subfig4b}
}
\caption{(Color
online) (a) The site dependence of the weights of  the zero modes in the
topological phase. The chain with  the length of $L=256$ is considered with the  system parameters: $t_s=t_p=V_{sp}=V_{ps}=1$ and for several values of $\epsilon=\epsilon _s^0=-\epsilon_p^0$. The fit by the solid straight lines imply that the quantity in the ordinate is proportional to $\exp(-l_i/\xi)$ where $l_i=i a$ is the position of the $i$th atom and $a$ is the lattice parameter.
(b) figure shows the values of $\xi$ obtained from these curves versus distance $\delta$ from quantum criticality. The solid line is a fit to a power law, resulting in the correlation length exponent $\nu= 1.01$).}%
\end{figure}

In Fig.~\ref{fig3}, we consider different choices for the parameters and
calculate the eigenvalues $E_n$ and eigenvectors $(u_n(j),v_n(j))$ of Eq.~\ref{bdg}
in the topological phase $(|\epsilon|<2t)$. As the result,
we obtain two zero eigenvalues as expected (see Figures~\ref{fig3}(a,b)).
Notice that if the system is not in the symmetric case the modes move away from zero energy as shown in
Figs~\ref{fig3}(c,d).

Next, we  consider $t_s=t_p=V_{sp}=V_{ps}=1$ and $\epsilon=\epsilon _s^0=-\epsilon_p^0$ and calculate the weight  of zero modes defined by $n_0(i)=\sum_{k=1}^2(u^2_k(i)+v^2_k(i))$
as a function of site number $i$ for different values of $\epsilon$ in the topological phase $(|\epsilon|<2t)$. Here, $u_{1,2}(i)$ and $v_{1,2}(i)$ are eigenvectors
corresponding to two zero energies. The resulting data is described
by Eq.~\ref{xisp} very well (see Figure~\ref{fig:subfig4a}).  The penetration length $\xi$ can be determined
from fits to Eq.~\ref{xisp} (Fig.~\ref{fig:subfig4a}). Figure.~\ref{fig:subfig4b} shows how the penetration length $\xi$ changes
with distance from criticality $\delta= 2t-\epsilon$. The data can be fitted to the power-law
$\xi\sim\ \delta^{-\nu}$. By fitting we extract the exponent $\nu=1.01$ which is in agreement
with the prediction Eq.~\ref{xisp}.

\begin{figure}[th]
\centering{\includegraphics[scale=0.3]{fig5.eps}}
\caption{(Color
online) The site dependence of weights of  the zero modes  in the
topological phase. The zero mode weights   undergo damped oscillations. The period of oscillations increases as  approaching criticality.
 The chain with  the length of $L=256$ is considered with parameters: $t_s=t_p=1$, $V_{sp}=V_{ps}=0.1$ and
several values of $\epsilon=\epsilon _s^0=-\epsilon_p^0$.}%
\label{fig5}%
\end{figure}

We now solve  Eq.~\ref{bdg} for  $t_s=t_p=1$ and $V_{sp}=V_{ps}=0.1$ for several values of $\epsilon=\epsilon _s^0=-\epsilon_p^0$.
In this case, the zero modes undergo damped oscillations (Fig.~\ref{fig5}) in contrast with the case considered previously where zero modes decay without
oscillations (Fig.~\ref{fig:subfig4a}).   For $\epsilon=0$,   weights of  the zero modes are zero on even lattice sites
and decay exponentially (Fig.~\ref{fig5}(a)) in agreement with the result given in
Ref.~\cite{coleman}. This is in contrast with cases when $\epsilon \neq 0$ where zero modes decays in different manner. The zero modes on each even lattice sites
are not zero anymore. The period of the oscillations increases as approaching the critical point (Fig.~\ref{fig5} (b,c,d)).

\section {Robustness of the edge modes against disorder}

In this section,  we explore the robustness of the edge modes against a diagonal disorder. In the absence of disorder zero modes are localized on
the ends of the chain (Fig. \ref{fig6}(a)).

\begin{figure}[th]
\centering{\includegraphics[scale=0.3]{fig6.eps}}
\caption{(Color
online) The figures show, going from (a) to (d),  how the weight of  the zero mode
shifts to the second site when the strength of disorder on the first site
increases.  The chain with  the length of $L=256$ is considered with  parameters:  $t_s=t_p=V_{sp}=V_{ps}=1$, $\epsilon _s^0=-\epsilon_p^0=0$
and several values of disorder strength $V_{{\rm{dis}}}(1)$ presented only on the first site
 $(\epsilon _s^0 \rightarrow \epsilon _s^0 +V_{{\rm{dis}}}(1)$ and  $\epsilon_p^0  \rightarrow  \epsilon _p^0 -V_{{\rm{dis}}}(1)))$.}%
\label{fig6}%
\end{figure}

Let us now consider the effect of single symmetric disorder $V_{{\rm{dis}}}(1)$, which is present only
on the first site. The symmetric disorder means that disorder acts on the centers of the $s$ and $p$ bands in the
following way:  $\epsilon _s^0 \rightarrow \epsilon _s^0 +V_{{\rm{dis}}}(1)$
and  $\epsilon_p^0  \rightarrow  \epsilon _p^0 -V_{{\rm{dis}}}(1)$. As the result, we find that both zero modes always
survive for any value of the disorder strength (Fig.~\ref{fig6}). The zero mode weight
is shared between the first and second sites for small disorder potential (Fig.~\ref{fig6}(b,c)), and is totally shifted to the second
site for large disorder strength (Fig.~\ref{fig6}(d)).
The zero mode localized on the last site is not affected  when disorder is
present on the first site (Fig.~\ref{fig6}).  When   non-symmetric disorder (e.g. $\epsilon _s^0 \rightarrow \epsilon _s^0 +V_{{\rm{dis}}}(1)$
and  $\epsilon_p^0  \rightarrow  \epsilon _p^0$) is present on the first site, only one zero mode localized on the last site
survives (Fig.~\ref{fig7}).

\begin{figure}[th]
\centering{\includegraphics[scale=0.3]{fig7.eps}}
\caption{(Color
online) Eigenvalues versus  indices. Only one zero mode survives when disorder is
presented non-symmetrically.  The following parameters are taken:
$t_s=t_p=V_{sp}=V_{ps}=1$  and $\epsilon _s^0=-\epsilon_p^0=0.5$ in the presence of disorder
 $V_{{\rm{dis}}}(1)=1$ only on the first site
$(\epsilon _s^0 \rightarrow \epsilon _s^0 +V_{{\rm{dis}}}(1)$ and  $\epsilon_p^0  \rightarrow  \epsilon _p^0$).
 Number of modes is 512 and a limited number of them around zero is shown. }%
\label{fig7}%
\end{figure}

\section{ End modes: Majoranas or conventional fermions?}

In the discussion below, we investigate the topological character of the elementary excitations.
The topological nature of the insulating phases of the $sp$-chain for the symmetric case $t_s= t_p=t$ and $\epsilon^0_s=-\epsilon^0_p=\epsilon$ can be established by calculating the winding number of these phases~\cite{shen,alicea}.
We start writing the fermion operators of the $sp$-chain Hamiltonian in terms of Majorana fermions~\cite{kitaev,shen}.
These are given by the following definitions,
\begin{eqnarray}
\label{majo}
c_i=\frac{\alpha_{Bi} + i \alpha_{Ai}}{\sqrt{2}} \nonumber \\
p_i=\frac{\beta_{Bi} + i \beta_{Ai}}{\sqrt{2} }
\end{eqnarray}
and similar equations for their complex conjugates, noticing that the Majorana particles are their own antiparticles, i.e., $\alpha_{Ai}^{\dagger}=  \alpha_{Ai}$, $\alpha_{Bi}^{\dagger}=  \alpha_{Bi}$, $\beta_{Ai}^{\dagger}=  \beta_{Ai}$ and $\beta_{Bi}^{\dagger}=  \beta_{Bi}$. In the Majorana basis the Hamiltonian of the $sp$-chain can be written as (for $t_s= t_p=t$ and $\epsilon^0_s=-\epsilon^0_p=\epsilon$),
\begin{eqnarray}
H_{sp}&=&i \epsilon \sum_{i=1}^N (\alpha_{Bi} \alpha_{Ai} - \beta_{Bi} \beta{Ai} )-it \sum_{i=1}^{N-1} (\alpha_{Bi} \alpha_{Ai+i} - \alpha_{Ai} \alpha_{Bi+1} )
+it \sum_{i=1}^{N-1} (\beta_{Bi} \beta_{Ai+i} - \beta_{Ai} \beta_{Bi+1} ) \nonumber \\
&+&iV \sum_{i=1}^{N-1} (\alpha_{Bi} \beta_{Ai+1}-\beta_{Bi}\alpha_{Ai+1}-\alpha_{Ai}\beta_{Bi+1}+\beta_{Ai}\alpha_{Bi+1}).
\end{eqnarray}
Next we apply a Fourier transformation. For this purpose we introduce,
\begin{eqnarray}
\alpha_{Bi}=\sum_k \alpha_k^B e^{ikx_i}
\end{eqnarray}
and similarly for the other Majoranas. Using the properties of the Majorana operators we see that $\alpha_{-k}^B=\alpha_{k}^{B \dagger}$. On the other hand,  the $k=0$ mode is a Majorana since,
$\alpha_{0}^B=\alpha_{0}^{B \dagger}$.
We finally get,
\begin{eqnarray}
H_{sp}= i \left[ \sum_k E_k \alpha^B_k\alpha^A_{-k} - \sum_k E_k \beta^B_k \beta^A_{-k}-2i \sum_k V_{sp}(k)\beta^A_k \alpha^B_{-k}+2i \sum_k V_{sp}(k)\beta_k^B \alpha^A_{-k}\right]
\end{eqnarray}
where $E_k=\epsilon -2t \cos ka$ and $V_{sp}(k)=V_{sp} \sin ka$. This Hamiltonian can be written in matrix form in the following way, $H_{sp}=\Psi H \Psi^{\dagger}$, where

\[H(k)=
\left(
\begin{array}{ccccc}
  0   &  -E_k & 0 & -2i V_{sp}(k) \\
 E_k & 0 & 2i V_{sp}(k) & 0 \\
 0 & -2iV_{sp}(k) & 0 & E_k \\
 2i V_{sp}(k) & 0 & - E_k & 0
\end{array}
\right)
\]
and $\Psi= (\alpha^A_k, \alpha^B_k, \beta^A_k, \beta^B_k)$. This Hamiltonian  belongs to the class AIII of chiral Hamiltonians~\cite{aiii}. The winding number $\mathcal{M}$ is defined as the product of the signs of the Pfaffians of the matrix $H(k)$ at $k=0$ and $k=\pi/a$. We have, $Pf[H(k=0)]=\epsilon-2t$ and $Pf[H(k=\pi/a)]=\epsilon+2t$, such that $\mathcal{M}=$sgn$(\epsilon-2t)$sgn$(\epsilon+2t)$. The trivial non-topological phase is characterized by $\mathcal{M}=+1$ while the topological phase is characterized~\cite{shen,alicea} by $\mathcal{M}=-1$. Then, in the case $t_s= t_p=t$ and $\epsilon^0_s=-\epsilon^0_p=\epsilon$ and for $\epsilon<2t$ the $sp$-chain is a topological insulator and for $\epsilon=2t$ there is a quantum topological transition from this state to a trivial insulating one with no special (topological)  properties, as shown in~Fig.\ref{fig:subfig2b}. Notice that, if we take $t_s= t_p=t$ and $\epsilon^0_s=-\epsilon^0_p=\mu$, with $t$ and $\mu$ referring to parameters of Kitaev's $p$-wave model, the results above translate directly into known results for this model~\cite{alicea}.  As can be seen in the derivation above, the criterion for the system to be in the topological phase does not involve directly $V_{sp}$.

In order to obtain the end modes of the chain, we introduce two new operators~\cite{coleman}
\begin{eqnarray}
l_i=c_i+p_i  \nonumber \\
r_i=c_i-p_i.
\end{eqnarray}
Using Eqs.~\ref{majo} these can be rewritten in terms of Majorana operators as,
\begin{eqnarray}
l_i=\gamma^{+}_{Bi}+i \gamma^{+}_{Ai}  \nonumber \\
r_i=\gamma^{-}_{Bi}+i \gamma^{-}_{Ai}
\end{eqnarray}
where we introduced new composite or hybrid $sp$ Majorana operators given by,
\begin{eqnarray}
\gamma^{\pm}_{A/Bi}= \alpha_{A/Bi} \pm \beta_{A/Bi}.
\end{eqnarray}
In terms of these hybrid $sp$ Majorana operators, the symmetric ($t_s=t_p=t$,  $\epsilon_0^s = -\epsilon_0^p=\epsilon$)  $sp$-chain Hamiltonian can be written as,
\begin{eqnarray}
\mathcal{H}_{sp}&=& i\frac{\epsilon}{2}\sum_{i=1}^{N}(\gamma^{+}_{Bi} \gamma^{-}_{Ai}-
\gamma^{+}_{Ai} \gamma^{-}_{Bi}) +
2i(V-t)\sum_{i=1}^{N-1}(\gamma^{-}_{Bi}\gamma^{+}_{Ai+1}-\gamma^{-}_{Ai}\gamma^{+}_{Bi+1})\nonumber \\
&&-2i(V+t)\sum_{i=1}^{N-1}(\gamma^{+}_{Bi}\gamma^{-}_{Ai+1}-\gamma^{+}_{Ai}\gamma^{-}_{Bi+1}).
\end{eqnarray}
Let us consider the point in the phase diagram ($\epsilon=0$, $V=t$) inside the topological insulating phase. The Hamiltonian is given by,
\begin{eqnarray}
\mathcal{H}_{sp}= -4it\sum_{i=1}^{N-1}(\gamma^{+}_{Bi}\gamma^{-}_{Ai+1}-\gamma^{+}_{Ai}\gamma^{-}_{Bi+1}).
\end{eqnarray}
Notice that the Majorana operators $\gamma^{-}_{B1}$ and $\gamma^{-}_{A1}$ do not enter the Hamiltonian
so that there are two unpaired hybrid $sp$ Majorana fermions in the end of the chain. The same occurs for
$\gamma^{+}_{BN}$ and $\gamma^{+}_{AN}$ on the opposite end of the chain. However, two Majoranas  in the
same extremity can combine to form a single conventional fermionic quasi-particle with hybrid $sp$ character. On the
left end, we have $l_1= \gamma^{-}_{B1}+i \gamma^{-}_{A1}=c_1-p_1$.
On the right end, we have, $r_N=\gamma^{+}_{BN}+ i \gamma^{+}_{AN}=c_N+p_N$, a total of two electronic
states one in each end of the chain. These zero energy modes are the protected modes of the topological
insulator phase of the symmetric $sp$-chain as shown in previous sections.

Now imagine the ends of this finite $sp$-chain are connected to a bulk metallic reservoir with $s$-states only. Referring to the $s$-electrons of the reservoir  as $b$, the Hamiltonian which couples both systems can be written as~\cite{hibri,vernek},
 \begin{equation}
 H_{cb}= v(l^{\dagger}b+b^{\dagger}l) +v(r^{\dagger}b+b^{\dagger}r),
\end{equation}
where $v$ is the tunneling element connecting the electrons of the reservoir  and those, one at each end, of the chain.
Substituting the expressions above for  {\it left} $l$ and {\it right} $r$ electrons, $l=c-p$ and $r=c+p$ in this equation, we obtain,
 \begin{equation}
 H_{cb}= 2v(c^{\dagger}b+b^{\dagger}c).
 \end{equation}
Then one electron $s$, half from each end of the chain tunnels to the reservoir leaving two $p$-type Majoranas, one at each end of the chain.

\section{Conclusions}

We have shown that the infinite, spinless, symmetric $sp$-chain maps formally in a model for
a $p$-wave superconducting chain. In the region of parameter space where this mapping holds,
the $sp$-chain presents a quantum phase transition (QPT)  from a topological insulator to a
topologically trivial insulating phase. This QPT occurs when  the ratio  between the center
of the bands and the hopping integrals, $\epsilon/2t=1$.  At the QFT, near the $\Gamma$
point ($k=0$), the spectrum is  linear with Dirac-like excitations. In the topological
insulating phase, in the end of $sp$-chain there are two unpaired hybrid $sp$ Majorana fermions
which can combine to form a single conventional fermion. This fermion mode shares many properties with the
Majorana modes at the ends of a $p$-wave superconducting chain such as the behavior of the localization length,
the non-trivial topological index and robustness to disorder.
We also show that in the topological insulator phase, if the ends of the $sp$-chain  are connected to a metallic reservoir with $s$-states only,  $p$-type Majoranas appear on each end of the chain.

Generically, for values of the parameters of the $sp$-chain
where the mapping with Kitaev's does not hold, the $sp$-chain may still have end modes not
necessarily of zero energy. In this case these modes can be identified with Schokley
states~\cite{shockley},  as studied in surface physics~\cite{shockley,beenakker,coleman}.

The  important feature of both $sp$-chain and $p$-wave models that give rise to a common
behavior is an antisymmetric term in their Hamiltonian that breaks inversion symmetry in
an otherwise centro-symmetric lattice.

We have shown that the scattering approach that treats the end of a semi-infinite chain
as a defect in an infinite chain turns out to be useful  for studying end modes.
It can be extended to higher dimensions~\cite{foo} and represents an additional tool
to study these modes besides the usual BdG~\cite{read} or the Jackiw-Rebbi methods~\cite{jak}
used in conjunction with Dirac equation~\cite{shen}.

 {\acknowledgments   We wish to thank the Brazilian agencies, CAPES, FAPERJ, CNPq and FAPEMIG for financial support. We would like to thank Jason Alicea and Alexander Altland for useful discussions. HC acknowledges the hospitality of CBPF where this work was done.
We (DN and NT) would like to acknowledge funding from Grant No. NSF  DMR-1309461.}